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Models for estimating solar irradiance at the earth’s surface from satellite data : an initial assessment Raphael, Clifford 1982

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MODELS FOR ESTIMATING SOLAR IRRADIANCE AT THE EARTH'S SURFACE FROM SATELLITE DATA: AN INITIAL ASSESSMENT by CLIFFORD RAPHAEL B.A. (Hons.), McMaster University, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © C l i f f o r d Raphael, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Departme nt of &MMfM The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date QI N i i ABSTRACT The performance of three models which use s a t e l l i t e data to estimate s o l a r irradiance at the Earth's surface i s assessed using measured rad i a t i o n data from a mid-latitude l o c a t i o n . In addition the mesoscale s p a t i a l v a r i a b i l i t y in the global solar irradiance resolvable by the three models i s also evaluated. The data are drawn from a twelve s t a t i o n pyranometric network and represent a variety of sky conditions at d i f f e r e n t times of the year. Assessment of the models depends upon the accurate Earth lo c a t i o n of the s a t e l l i t e imagery and the merging of s a t e l l i t e and pyranometric data. The resolution of the s a t e l l i t e data for the study area is determined to be 0.82 km in the East-West d i r e c t i o n and 1.67 km in the North-South d i r e c t i o n . Based on this resolution the s a t e l l i t e imagery can be Earth located to within 1.25 km in an East-West d i r e c t i o n and to within 1.71 km in a North-South d i r e c t i o n . Merging of the s a t e l l i t e and pyranometric data results in the use of as many as three images to represent the radiation c h a r a c t e r i s t i c s of a given hour. I n i t i a l applications of the three models reveal that the o r i g i n a l regression c o e f f i c i e n t s for both the Tarpley (1979) and Hay and Hanson (1978) models are inappropriate for the study area because of the bias introduced. Subsequent r e v i s i o n of these c o e f f i c i e n t s leads to s i g n i f i c a n t improvements under most conditions. Seasonal assessments of the three models demonstrate that on an hourly basis the overall performance of the Gautier et a l . (1980) model under partly cloudy and overcast conditions i s superior to that of the other two models. However, compared to the clear sky case a l l i i i three models give poor results under partly cloudy and overcast conditions. An increase in the averaging period leads to marked decreases in the RMS errors observed for the three models under a l l conditions with the greatest improvement occurring for the Hay and Hanson (1978) model. Changes in temporal and s p a t i a l averaging configurations reveal that temporal averaging could have an important influence on the radiation estimates under partly cloudy and overcast conditions. Spatial averaging in the context of the Gautier et a l . (1980) model does not support the use of an 8 x 8 pixel array to improve the temporal representation of the s a t e l l i t e data. In terms of the mesoscale spatial v a r i a b i l i t y in the global so l a r i r r a d i a n c e , the best resolution i s provided by the Gautier et a l . (1980) model. For hourly values an average RMS error of ±17.1% l i m i t s the s p a t i a l resolution to approximately 15 km; for d a i l y values an average RMS error of 8.2% l i m i t s the s p a t i a l r esolution to approximately 12 km. Suggestions for improvements in the three models include; 1) a more accurate and e x p l i c i t treatment of cloud absorption; 2) the consideration of the effects of changing S u n - s a t e l l i t e azimuth angle under overcast conditions and 3) in the context of both the Gautier et al . (1980) and Tarpley (1979) models there i s the need for the i n c l u s i o n of the e f f e c t s of aerosols under c l e a r skies and the accurate and objective s p e c i f i c a t i o n of a cloud threshold. iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv L i s t of Tables v i i i L i s t of Figures xv Acknowledgements xxv Chapter One: Introduction 1 1.1 Objectives 1 1.2 Background 3 Chapter Two: Data Sets and Data Processing 10 2.1 Radiation Data Set 10 2.2 S a t e l l i t e Data Set 13 2.3 Problem Areas 16 2.3.1 Spectral Ranges and C a l i b r a t i o n 16 2.3.2 Earth Location of the S a t e l l i t e Imagery 17 2.3.3 Temporal Sampling and Viewing Angles of 17 Sensors 2.4 Precipitable Water 18 2.5 Earth Location of S a t e l l i t e Imagery 27 2.6 Validation of the Earth Location Routine 35 2.7 Pixel Size Determination 38 2.8 L i n e a r i t y 45 2.9 Merging of the S a t e l l i t e and Solar Radiation 48 Data Sets 2.10 V e r i f i c a t i o n S a t i s f i e s 61 V Page Chapter Three: The Models 52 3.1 The Tarp ley Model - Overview 52 3.1.1 P h y s i c a l / E m p i r i c a l Bas i s 55 3.2 The Gau t ie r Model - Overview 57 3.2.1 P h y s i c a l / E m p i r i c a l Bas is 59 3.3 The Hay - Hanson Model - Overview 62 3.3.1 P h y s i c a l / E m p i r i c a l Bas i s 63 Chapter Four: A p p l i c a t i o n o f the Tarp ley Model 65 4.1 Implementation 65 4.1.1 Minimum Br igh tness Paramete r i za t i on 66 4 .1 .2 Data S t r a t i f i c a t i o n and Image-Related Inputs 70 4 . 1 . 3 Atmospheric Transmit tance 73 4 . 1 . 4 Regress ion Equat ions f o r P r e d i c t i n g I n s o l a t i o n 77 4.2 Resu l ts 77 4 .3 Model M o d i f i c a t i o n 97 4.3.1 Generat ion o f New Regress ion C o e f f i c i e n t s 97 4 .3 .2 Changes in Temporal Averaging 111 4 . 3 . 3 Changes in S p a t i a l Averaging 115 4.4 Seasonal Assessments 118 4.5 D a i l y S t a t i s t i c s 134 4.6 Imp l i ca t i ons o f Model Accuracy f o r S p a t i a l Sampling 135 Requirements 4.7 Summary and Conc lus ions 138 v i Page Chapter Five: Application of the Gautier Model 140 5.1 Implementation 140 5.1.1 Clear Sky Model Development 141 5.1.1.1 Surface Albedo 141 5.1.1.2 Scattering C o e f f i c i e n t s 142 5.1.1.3 Water Vapour Absorption 143 5.1.1.4 Cloud Threshold 145 5.1.2 Cloudy Atmosphere Model Development 147 5.1.2.1 Cloud Albedo 147 5.1.2.2 Cloud Absorption 148 5.1.2.3 Water Vapour Absorption 149 5.1.3 Radiation Calculations 150 5.2 Results 151 5.3 Model Modification 162 5.3.1 Changes in Temporal Averaging 162 5.3.2 Changes in Spatial Averaging 166 5.4 Seasonal Assessments 168 5.5 Daily S t a t i s t i c s 187 5.6 Implications of Model Accuracy for Spatial Sampling 190 Requi rements 5.7 Summary and Conclusions 191 Chapter Six: Application of the Hay and Hanson Model 194 6.1 Implementation 194 6.1.1 C a l i b r a t i o n of the S a t e l l i t e Data I 9 5 6.1.2 Radiation Calculations I 9 5 v i i Paae 6.2 Results 196 6.3 Model Modification 2 0 6 6.3.1 Generation of New Regression Coefficients 206 6.3.2 Changes in Spatial Averaging 217 6.4 Seasonal Assessments 220 6.5 Daily S t a t i s t i c s 238 6.6 Implications of Model Accuracy for Spatial 241 Sampling Requirements 6.7 Summary and Conclusions 242 Chapter Seven: Conclusions 245 References 254 Appendix A . l : L i s t of Symbols 260 Appendix A.2: Equations for Calculating S a t e l l i t e Zenith 265 and Azimuth Angles and Water Vapour Absorption for the Cloudy Atmosphere Model Appendix A.3: Hourly S t a t i s t i c s for the Individual Stations 268 From the Tarpley (1979) Model (Original and Revised) Appendix A.4: Hourly S t a t i s t i c s for the Individual Stations 279 From the Gautier et a l . (1980) Model Appendix A.5: Hourly S t a t i s t i c s for the Individual Stations 287 From the Hay and Hanson (1978) Model (Original and Revised) Appendix A.6: Computer Routines for the Implementation of 294 the Three Models vi i i LIST OF TABLES Page Table 1.1 Errors (E) Associated With the Spatial 4 Interpolation of Hourly and Daily Values of Global Solar Radiation Table 2.1 Distance Separation Matrix for Mesoscale 11 Network Table 2.2 Days Used in the Mesoscale Study 14 Table 2.3 Estimated and Computed Values of P r e c i p i t a b l e 20 Water Table 2.4 Variations in Estimated P r e c i p i t a b l e Water 21 With Dew Point Temperature Table 2.5 Landmarks Used in Earth Location Routine 29 Table 2.6 V e r i f i c a t i o n S t a t i s t i c s f o r the Cubic 37 Interpolation Routine Table 2.7 Data Set for S a t e l l i t e Resolution 40 Determination Table 2.8 Results from the Assessment of L i n e a r i t y 47 Table 4.1 Minimum Brightness Regression C o e f f i c i e n t s 71 and Other S t a t i s t i c s f o r the 12 Stations Table 4.2 Measurement Stations in Mesoscale Network 75 Table 4.3 Original Regression C o e f f i c i e n t s f o r the 78 Tarpley (1979) Model Table 4.4 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from 80 the Application of the Original Tarpley (1979) Model to Six (6) Days. ix Page Table 4.5 Data Sets for Regression 99 Table 4.6 Hourly Average V e r i f i c a t i o n S t a t i s t i c s 101 from the Application of the Revised Tarpley (1979) Model Table 4.7 Hourly Average V e r i f i c a t i o n S t a t i s t i c s 113 from the Revised Tarpley (1979) Model Based on a Flux Averaging Approach and a Spatial Averaging Approach Table 4.8 Daily Average V e r i f i c a t i o n S t a t i s t i c s from 136 the Application of the Revised Tarpley (1979) Model Table 5.1 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from 152 the Application of the Gautier et a l . (1980) Model Table 5.2 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from 164 the Application of the Gautier et a l . (1980) Model Based on a Pixel Averaging Approach and a Spatial Averaging Approach Table 5.3 Daily Average V e r i f i c a t i o n S t a t i s t i c s from the 188 Application of the Gautier et a l . (1980) Model Table 6.1 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from 197 the Application of the Original Hay - Hanson (1978) Model X Page Table 6.2 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from 208 the Application of the Revised Hay - Hanson (1978) Model Table 6.3 Hourly Average V e r i f i c a t i o n S t a t i s t i c s for the 218 Revised Hay - Hanson (1978) Model Based on a Spatial Averaging Approach Table 6.4 Daily Average V e r i f i c a t i o n S t a t i s t i c s from the 239 Application of the Revised Hay - Hanson (1978) Model Table 7.1 Summary S t a t i s t i c s for the Three Models 247 Table A.3.1 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 270 Individual Stations from the Application of the Original Tarpley (1979) Model to (3) Days in Summer Table A.3.2 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 271 Individual Stations from the Application of the Original Tarpley (1979) Model to (3) Days in Spring Table A.3.3 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 272 Individual Stations from the Application of the Revised Tarpley (1979) Model to (3) Days in Summer xi Page Table A.3.4 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 273 Individual Stations from the Revised Tarpley (1979) Model Based on a Flux Averaging Approach Table A.3.5 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 274 Individual Stations from the Revised Tarpley (1979) Model Based on a Spatial Averaging Approach Table A.3.6 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 275 Individual Stations from the Application of the Revised Tarpley (1979) Model to (3) Clear Days Table A.3.7 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 276 Individual Stations from the Application of the Revised Tarpley (1979) Model to (2) Partly Cloudy Days Table A.3.8 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 277 Individual Stations from the Application of the Revised Tarpley (1979) Model to (3) Overcast Days Table A.3.9 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 278 Individual Stations from the Application of the Revised Tarpley (1979) Model With a Modified threshold. x i i Page Table A.4.1 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 280 Individual Stations from the Application of the Gautier et a l . (1980) Model to (3) Days i n Summer Table A.4.2 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 281 Individual Stations from the Application of the Gautier et a l . (1980) Model Using a Pixel Averaging Approach Table A.4.3 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 282 Individual Stations from the Application of the Gautier et a l . (1980) Model Based on a Spatial Averaging Approach Table A.4.4 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 283 Individual Stations from the Application of the Gautier et a l . (1980) Model to (3) Clear Days Table A.4.5 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 284 Individual Stations from the Application of the Gautier et a l . (1980) Model to (3) Overcast Days Table A.4.6 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 285 Individual Stations from the Application of the Gautier et a l . (1980) Model to (3) Partly Cloudy Days x i i i Pa£e Table A.4.7 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 286 Individual Stations from the Application of the Gautier et a l . (1980) Model Using a Modified Threshold Table A.5.1 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 288 Individual Stations from the Application of the Original Hay - Hanson (1978) Model to (3) Days i n Summer Table A.5.2 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 289 Individual Stations from the Application of a Revised Hay - Hanson (1978) Model to (3) Days i n Summer Table A.5.3 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 290 Individual Stations from the Application of a Revised Hay - Hanson (1978) Model Based on a Spatial Averaging Approach Table A.5.4 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 291 Individual Stations from the Application of a Revised Hay - Hanson (1978) Model to (3) Clear Days Table A.5.5 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 292 Individual Stations from the Application of a Revised Hay - Hanson (1978) Model to (2) Partly Cloudy Days x i v Page Table A.5.6 Hourly V e r i f i c a t i o n S t a t i s t i c s for the 293 Individual Stations from the Application of a Revised Hay - Hanson Model to (3) Overcast Days XV LIST OF FIGURES Page Figure 1.1 Map Showing Location of Stations i n the Mesoscale 2 Network Figure 2.1 Percentage Difference Between Measured and 23 Calculated Radiation f o r Two Di f f e r e n t Inputs of P r e c i p i t a b l e Water at Three Di f f e r e n t Times of the Year Figure 2.2 Map of the Area Covered by the Largest S a t e l l i t e 34 Image Showing Landmarks T y p i c a l l y Used i n the Assessment of L i n e a r i t y Figure 2.3 Example of S a t e l l i t e Image Weighting Scheme 50 for Merging S a t e l l i t e Data to Hourly Integrated Radiation Data Figure 4.1a Observed Radiation and Calculated Radiation 81 from the Original Tarpley Model at P i t t Meadows for a Clear Day in Summer Figure 4.1b Observed Radiation and Calculated Radiation from 82 the Original Tarpley Model at Vancouver A i r p o r t f o r a Clear Day in Summer Figure 4.2a Observed Radiation and Calculated Radiation from 83 the Original Tarpley Model at Vancouver A i r p o r t f o r a Par t l y Cloudy Day in Summer Figure 4.2b Observed Radiation and Calculated Radiation 84 from the Original Tarpley Model at Abbotsford Library f o r a Par t l y Cloudy Day in Summer xvi Figure 4.3a Observed Radiation and Calculated Radiation from the Original Tarpley Model at Vancouver A i r p o r t f o r an Overcast Day in Summer Figure 4.3b Observed Radiation and Calculated Radiation from the Original Tarpley Model at BC Hydro fo r an Overcast Day in Summer Figure 4.4a Observed Radiation and Calculated Radiation from the Original Tarpley Model at P i t t Meadows for a Clear Day in Spring Figure 4.4b Observed Radiation and Calculated Radiation from the Original Tarpley Model at Vancouver A i r p o r t f o r a Clear Day in Spring Figure 4.5a Observed Radiation and Calculated Radiation from the Original Tarpley Model at Vancouver A i r p o r t f o r a Par t l y Cloudy Day in Spring Figure 4.5b Observed Radiation and Calculated Radiation from the Original Tarpley Model at Abbotsford Library f o r a Par t l y Cloudy Day in Spring Figure 4.6a Observed Radiation and Calculated Radiation from the Original Tarpley Model at Vancouver A i r p o r t f o r an Overcast Day in Spring Figure 4.6b Observed Radiation and Calculated Radiation from the Ori g i n a l Tarpley Model at BC Hydro f o r an Overcast Day in Spring XVI 1 Figure 4.7a Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Vancouver Air p o r t f o r a Clear Day in Summer Figure 4.7b Observed Radiation and Calculated Radiation from a Revised Tarpley Model at P i t t Meadows for a Clear Day in Summer Figure 4.8a Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Vancouver Ai r p o r t f o r a Partly Cloudy Day in Summer Figure 4.8b Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Abbotsford Library f o r a Partly Cloudy Day in Summer Figure 4.9a Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Vancouver Ai r p o r t f o r an Overcast Day in Summer Figure 4.9b Observed Radiation and Calculated Radiation from a Revised Tarpley Model at BC Hydro f o r an Overcast Day i n Summer Figure 4.10a Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Vancouver A i r p o r t f o r a Clear Day in Spring Figure 4.10b Observed Radiation and Calculated Radiation from a Revised Tarpley Model at P i t t Meadows fo r a Clear Day in Spring xvi i i Page Figure 4.11a Observed Radiation-and Calculated Radiation 121 from a Revised Tarpley Model at Vancouver A i r p o r t f o r a Clear Day i n F a l l Figure 4.11b Observed Radiation and Calculated Radiation 122 from a Revised Tarpley Model at P i t t Meadows for a Clear Day in F a l l Figure 4.12a Observed Radiation and Calculated Radiation 123 from a Revised Tarpley Model at Vancouver Ai r p o r t f o r a Clear Day in Winter Figure 4.12b Observed Radiation and Calculated Radiation 124 from a Revised Tarpley Model at P i t t Meadows for a Clear Day in Winter Figure 4.13a Observed Radiation and Calculated Radiation 127 from a Revised Tarpley Model at Abbotsford Library f o r a Partly Cloudy Day in Spring Figure 4.13b Observed Radiation and Calculated Radiation 128 from a Revised Tarpley Model at Vancouver Ai r p o r t f o r a Partly Cloudy Day in Spring Figure 4.14a Observed Radiation and Calculated Radiation 129 from a Revised Tarpley Model at Vancouver A i r p o r t f o r an Overcast Day in Spring Figure 4.14b Observed Radiation and Calculated Radiation 130 from a Revised Tarpley Model at BC Hydro f o r an Overcast Day i n Spring x i x Figure 4.15a Observed Radiation and Calculated Radiation from a Revised Tarpley Model at Vancouver Airport for an Overcast Day in Fall Figure 4.15b Observed Radiation and Calculated Radiation from a Revised Tarpley Model at BC Hydro f o r an Overcast Day in Fall Figure 5.1a Observed Radiation and Calculated Radiation from the Gautier Model at Grouse Mountain for a Clear Day in Summer Figure 5.1b Observed Radiation and Calculated Radiation from the Gautier Model at Vancouver Airport for a Clear Day in Summer Figure 5.1c Observed Radiation and Calculated Radiation from the Gautier Model at P i t t Meadows for a Clear Day in Summer Figure 5.2a Observed Radiation and Calculated Radiation from the Gautier Model at Vancouver A i r p o r t for a Partly Cloudy Day in Summer Figure 5.2b Observed Radiation and Calculated Radiation from the Gautier Model at Abbotsford Library for a Partly Cloudy Day i n Summer Figure 5.3a Observed Radiation and Calculated Radiation from the Gautier Model at Abbotsford Library for an Overcast Day in Summer XX Page Figure 5.3b Observed Radiation and Calculated Radiation 159 from the Gautier Model at Vancouver Ai r p o r t for an Overcast Day in Summer Figure 5.4a Observed Radiation and Calculated Radiation 170 from the Gautier Model at Vancouver Ai r p o r t for a Clear Day in Spring Figure 5.4b Observed Radiation and Calculated Radiation 171 from the Gautier Model at P i t t Meadows for a Clear Day in Spring Figure 5.5a Observed Radiation and Calculated Radiation 172 from the Gautier Model at P i t t Meadows for a Clear Day in Fall Figure 5.5b Observed Radiation and Calculated Radiation 173 from the Gautier Model at Vancouver Airport for a Clear Day in Fall Figure 5.6a Observed Radiation and Calculated Radiation 174 from the Gautier Model at P i t t Meadows for a Clear Day in Winter Figure 5.6b Observed Radiation and Calculated Radiation 175 from the Gautier Model at Vancouver Ai r p o r t for a Clear Day in Winter Figure 5.7a Observed Radiation and Calculated Radiation 177 from the Gautier Model at Vancouver Ai r p o r t for a Partly Cloudy Day in Spring xx i Page Figure 5.7b Observed Radiation and Calculated Radiation 178 from the Gautier Model at Abbotsford Library for a Partly Cloudy Day in Spring Figure 5.8a Observed Radiation and Calculated Radiation 179 from the Gautier Model at Vancouver Airport for a Partly Cloudy Day in Fall Figure 5.8b Observed Radiation and Calculated Radiation 180 from the Gautier Model at Abbotsford Library for a Partly Cloudy Day in F a l l Figure 5.9a Observed Radiation and Calculated Radiation 182 from the Gautier Model at Vancouver Ai r p o r t for an Overcast Day in Spring Figure 5.9b Observed Radiation and Calculated Radiation 183 from the Gautier Model at Abbotsford Library for an Overcast Day in Spring Figure 5.10a Observed Radiation and Calculated Radiation 184 from the Gautier Model at Vancouver Airport for an Overcast Day in Fall Figure 5.10b Observed Radiation and Calculated Radiation 185 from the Gautier Model at Abbotsford Library for an Overcast Day in Fall Figure 6.1a Observed Radiation and Calculated Radiation 198 from the Original Hay - Hanson Model at Vancouver A i r p o r t for a Clear Day in Summer xx i i Page Figure 6.1b Observed Radiation and Calculated Radiation 199 from the Original Hay - Hanson Model at P i t t Meadows for a Clear Day i n Summer Figure 6.2a Observed Radiation and Calculated Radiation 200 from the Original Hay - Hanson Model at Vancouver Airport for a Partly Cloudy Day in Summer Figure 6.2b Observed Radiation and Calculated Radiation 201 from the Original Hay - Hanson Model at Abbotsford Library for a Partly Cloudy Day in Summer Figure 6.3a Observed Radiation and Calculated Radiation 202 from the Original Hay - Hanson Model at Abbotsford Library for an Overcast Day in Summer Figure 6.3b Observed Radiation and Calculated Radiation 203 from the Original Hay - Hanson Model at Vancouver Ai r p o r t for an Overcast Day in Summer Figure 6.4a Observed Radiation and Calculated Radiation 209 from a Revised Hay - Hanson Model at Vancouver Ai r p o r t for a Clear Day in Summer Figure 6.4b Observed Radiation and Calculated Radiation 210 from a Revised Hay - Hanson Model at P i t t Meadows for a Clear Day in Summer x x i n Figure 6.5a Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Vancouver Airport for a Partly Cloudy Day in Summer Figure 6.5b Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Abbotsford Library for a Partly Cloudy Day in Summer Figure 6.6a Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Vancouver Airport for an Overcast Day in Summer Figure 6.6b Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Abbotsford Library for an Overcast Day in Summer Figure 6.7a Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at P i t t Meadows for a Clear Day in Spring Figure 6.7b Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Vancouver Air p o r t for a Clear Day in Spring Figure 6.8a Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at P i t t Meadows for a Clear Day in Fall Figure 6.8b Observed Radiation and Calculated Radiation from a Revised Hay - Hanson Model at Vancouver Ai r p o r t for a Clear Day in Fall xx iv Page Figure 6.9a Observed Radiation and Calculated Radiation 226 from a Revised Hay - Hanson Model at Vancouver' Airport for a Clear Day in Winter Figure 6.9b Observed Radiation and Calculated Radiation 227 from a Revised Hay - Hanson Model at P i t t Meadows for a Clear Day in Winter Figure 6.10a Observed Radiation and Calculated Radiation 230 from a Revised Hay - Hanson Model at Vancouver Airport for a Partly Cloudy Day in Spring Figure 6.10b Observed Radiation and Calculated Radiation 231 from a Revised Hay - Hanson Model at Abbotsford Library for a Partly Cloudy Day in Spring Figure 6.11a Observed Radiation and Calculated Radiation 232 from a Revised Hay - Hanson Model at Vancouver Air p o r t for an Overcast Day in Spring Figure 6.11b Observed Radiation and Calculated Radiation 233 from a Revised Hay - Hanson Model at Abbotsford Library for an Overcast Day in Spring Figure 6.12a Observed Radiation and Calculated Radiation 234 from a Revised Hay - Hanson Model at Vancouver Air p o r t for an Overcast Day in Fall Figure 6.12b Observed Radiation and Calculated Radiation 235 from a Revised Hay - Hanson Model at Abbotsford Library for an Overcast Day in F a l l XXV ACKNOWLEDGEMENTS Financial support f o r t h i s study was provided by two UBC Graduate Fellowships and by the Canadian Atmospheric Environment Service through a contract (DSS NO. 01 SE. KM 147-2-1206) with Dr. J.E. Hay. I would l i k e to thank my supervisor, Dr. J.E. Hay for his invaluable support and d i r e c t i o n . The i n s i g h t f u l comments and prompt reading of the f i n a l manuscript by Dr. T.R. Oke must be recognized. I would further l i k e to acknowledge Mr. M. Roseberry for his assistance with computations and programming. I extend thanks to the UBC I n t e r d i s c i p l i n a r y Program i n Remote Sensing f o r the l i b e r a l use of t h e i r f a c i l i t i e s (and e s p e c i a l l y the Comtal image processor) f o r analysis of the s a t e l l i t e imagery. I am indebted to Mr. N. Wanless f o r his cooperation on some aspects of the research and for many helpful discussions. I would also l i k e to acknowledge my colleagues i n the Geography Department and numerous other friends f o r t h e i r advice and encouragement. F i n a l l y , I am grateful to God without whose strength i t would not have been possible. 1 CHAPTER ONE 1. Introduction 1.1 Objectives This i n v e s t i g a t i o n has two primary objectives: 1) To assess the performance of three models which use s a t e l l i t e data to calculate solar radiation incident upon the earth's surface; and 2) To consider the a b i l i t y of these models to determine the mesoscale sp a t i a l v a r i a b i l i t y of incoming solar radiation over Greater Vancouver and a section of the Lower Fraser Valley (see Figure 1.1). The models chosen f o r assessment are those of Hay and Hanson (1978), Tarpley (1979), and Gautier et a l . (1980). These models were chosen because they represent three d i s t i n c t approaches to the problem of estimating incident solar radiation at the surface using s a t e l l i t e data. Assessment w i l l include the v e r i f i c a t i o n of these models under regimes of c l e a r , p a r t l y cloudy and overcast skies and an examination of the seasonal behaviour of the models. The s p a t i a l v a r i a b i l i t y of the s o l a r irradiance over the study area has been well documented in the work of Hay and Tooms (1979), Hay (1981) and Hay (1982). The nature of the network established to monitor the s p a t i a l v a r i a b i l i t y i n the s o l a r irradiance i s described in Hay and Tooms (1979) and Hay (1982) (see Section 2.1 f o r further discussion). The work of Hay (1981), based on hourly and d a i l y c o r r e l a t i o n s between observations at the U.B.C. Climate Station and the other stations i n the network, reveals strong anisotropy i n the Figure 1.1 Map of the Study Area Showing Locations of Measurement Stations 3 r a d i a t i o n f i e l d with stations in the North of the study area (Figure 1.1) having s u b s t a n t i a l l y reduced c o r r e l a t i o n s compared with equidistant stations in the South of the study area. Hay (1982), has shown that the differences in the s p a t i a l d i s t r i b u t i o n of the observed solar irradiance are best developed under average conditions ( i . e . conditions of p a r t i a l cloud cover). Implications of the mesoscale s p a t i a l v a r i a b i l i t y in terms of s p a t i a l sampling requirements have been assessed via the technique of optimum i n t e r p o l a t i o n described by Alaka (1970). The r e s u l t i n g Root Mean Square Errors of i n t e r p o l a t i o n for hourly and d a i l y data are given in Table 1.1 taken from Hay (1981). On the basis of these results Hay (1981), concludes that on average a ±10% error in the estimated hourly radiation at a l l locations in the network can be achieved for a sampling resolution of approximately twelve (12) km (smaller errors w i l l be observed f o r the d a i l y estimates and i n t e r p o l a t i o n distance w i l l vary with season). In the context of the Hay (1981) study j u s t described, and in view of the high resolution associated with the s a t e l l i t e data (from 2 - 8 km) the challenge i s whether the models to be assesed can indeed characterize the mesoscale s p a t i a l v a r i a b i l i t y and, i f so, with what accuracy? 1.2 Background The need f o r r a d i a t i o n data for solar energy and other applications i s becoming inc r e a s i n g l y apparent. As a consequence, in recent times much work has been channeled into improving the methods 4 Table 1.1(From Hay 1981)  Errors(E)associated with the sp a t i a l interpolation of hourly and  daily values of global solar radiation at Vancouver,Canada.  Analyses based on distance co r r e l a t i o n data for the mesoscale  network(excluding Grouse Mountain) and the technique of A l a k a ( 1 9 7 0 T : HOURLY DISTANCE(km) E(KJm-2hr-l) % of MEAN 0 2 5 . 8 2 . 9 5 7 8 . 8 8 . 9 1 0 1 2 0 . 6 1 3 . 6 2 0 1 6 8 . 7 1 9 . 0 . 3 0 2 2 2 . 4 2 5 . 1 4 0 2 2 6 . 3 2 5 . 5 5 0 2 3 4 . 8 2 6 . 5 DAILY DISTANCE(km) E(MJm -2day-1) % of MEAN 0 0 . 3 3 2 2 . 7 5 0 . 6 3 7 5 . 7 1 0 0 . 8 7 8 7 . 2 2 0 1 . 4 2 7 1 1 . 7 3 0 1 . 5 7 4 1 3 . 0 4 0 1 . 5 9 1 1 3 . 1 5 0 1 . 7 8 6 1 4 . 7 5 for determining the s o l a r irradiance at the earth's surface. Data a v a i l a b i l i t y i s l i m i t e d by the s p a r s i t y of both the e x i s t i n g measurement network, and the data required by numerical models. The p o s s i b i l i t y of u t i l i z i n g s a t e l l i t e measurements to provide the necessary r a d i a t i o n data appears to be a viable a l t e r n a t i v e to, and an appropriate extension of the e x i s t i n g networks. Early studies u t i l i z i n g s a t e l l i t e data involved the use .of a va r i e t y of systems of varying resolutions. They included the ITOS (Improved Ti r o s Operational S a t e l l i t e ) and NOAA (National Oceanic and Atmospheric Administration) series of polar o r b i t i n g meteorological s a t e l l i t e s . Two major f o c i can be i d e n t i f i e d f o r these early studies: 1) The determination of the earth's r a d i a t i o n budget and albedo climatology and 2) i n i t i a l i n s o l a t i o n estimates. The work of Curran et a l . (1978) Gruber (1977), Flanders and Smith (1975), and Vonder Haar (1972) represents the f i r s t major focus. Using both v i s i b l e (0.5 - 0.7 ym) and i n f r a r e d (10.5 - 12.0 ym) data they attempted to measure and describe; 1) The annual, seasonal and l a t i t u d i n a l v a r i a t i o n s in albedo, (Curran et a l . , 1978); 2) The natural v a r i a t i o n of the zonal radiation budget of the earth highlighting areas that contributed s i g n i f i c a n t v a r i a t i o n s , (Vonder Haar, 1972); and 3) The various components of the earth's radiation budget including the mean monthly zonal and meridional r a d i a t i o n p r o f i l e s , long-wave flu x and the net r a d i a t i o n on the scale of continents and oceans (Flanders and Smith, 1975; Gruber, 1977). The second major focus i s represented by the work of Hanson (1971), Vonder Haar and E l l i s (1975), E l l i s and Vonder Haar (1978), and Vonder Haar (1973). Hanson (1971), showed that on a monthly time 6 scale s a t e l l i t e parameterization of i n s o l a t i o n was two to three times better than other conventional techniques. His work was based on the p r i n c i p l e of conservation of energy i n an atmospheric column. Given a value of the incident e x t r a t e r r e s t r i a l solar irradiance, a quantitative estimate of atmospheric absorption and a s a t e l l i t e radiance measurement, absorbed i n s o l a t i o n at the surface could be calculated as a residual from the following expression. : • lg '0 AR XA I = energy absorbed at the surface I R = energy r e f l e c t e d back to space 1^ = energy absorbed in the atmosphere FQ = energy incident at the top of the atmosphere With a knowledge of the surface albedo the incident radiation at the surface i s given by: I = I (1 - a) (1.2.2) -> y a = surface albedo E l l i s and Vonder Haar u t i l i z e d a s i m i l a r approach to the one outlined above to a r r i v e at estimates of incident radiation at the surface for the United States. Vonder Haar and E l l i s (1975) attempted to show that a solar energy 'microclimate' could be developed for s p a t i a l scales smaller than that of the conventional pyranometric network. * A f u l l l i s t of symbols in given in Appendix A.l 7 This required a knowledge of cloudiness (derived from the s a t e l l i t e data) on the smaller scale along with the knowledge of surface s o l a r energy beneath an average cloud and under clear skies. A major drawback to these early i n s o l a t i o n studies was the li m i t e d temporal coverage provided by the pol a r - o r b i t i n g meteorological s a t e l l i t e s (one daylight image per day). This meant that the question of diurnal variations in cloud cover could not be addressed within the framework of these early studies. This l i m i t a t i o n is of importance since clouds are by far the major attenuators of incoming solar r a d i a t i o n . Recent research on the u t i l i z a t i o n of s a t e l l i t e data to estimate incident s o l a r radiation at the surface has attempted to address this question of cloud cover v a r i a b i l i t y through the use of data from the Geostationary Operational Environmental S a t e l l i t e s (GOES), and the Synchronous Meteorological S a t e l l i t e s (SMS). The data from these s a t e l l i t e s have high temporal (30 minutes) and spatial (approximately two km at 45° l a t i t u d e ) resolutions, (Gautier et a l . , 1980), and are considered to be ideal for addressing the question of cloud cover v a r i a b i l i t y . Models developed using GOES data are the f i r s t real attempts to rigorously address the question of the estimation of incident s o l a r r a d i a t i o n at the surface from s a t e l l i t e data. They can be considered as being representative of the "state of the a r t " . Only a few such models e x i s t . They include those of Tarpley (1979), Hay and Hanson (1978), Hiser and Senn (1980), Brakke and Kanemasu (1981), and Gautier et a l . (1980). A s t a t i s t i c a l approach was adopted by a l l the researchers with the exception of Gautier et a l . (1980). They adopted a semi-empirical 8 ("physically based") approach to the determination of i n s o l a t i o n at the surface using the s a t e l l i t e data. Tarpley (1979), used hourly geostationary s a t e l l i t e data together with a s t a t i s t i c a l model r e l a t i n g s a t e l l i t e brightness to i n s o l a t i o n to derive d a i l y i n s o l a t i o n estimates that were within 10% of the pyranometer measurements. Brakke and Kanemasu (1981), followed a s i m i l a r approach to that of Tarpley (1979), d i f f e r i n g only in the f i n a l form of the regression equation. Their estimates were within 12% of the measured data. Hiser and Senn (1980) outlined a s t a t i s t i c a l method to map available solar energy on the mesoscale from s a t e l l i t e -derived cloud cover/sunshine d i s t r i b u t i o n . Hay and Hanson (1978), developed a simple s t a t i s t i c a l model r e l a t i n g normalized s a t e l l i t e r e f l e c t i v i t y to normalized atmospheric transmissivity. With this model hourly radiation could be estimated to within ±22% decreasing to 8% on a d a i l y basis. The models described above have been developed and tested under a variety of radiation conditions in d i s t i n c t l y d i f f e r e n t environments: the Tropical A t l a n t i c , Hay and Hanson (1978), the U.S. Great Pl a i n s , Tarpley (1979), Brakke and Kanemasu (1981), and Southern Central Canada, Gautier et a l . (1980). (The method outlined by Hiser and Senn (1980), has been assessed for d i f f e r e n t environments in the contiguous United States). However, the models have yet to be tested and compared using a s i n g l e homogenous data set. According to Hay (1981), s a t e l l i t e s are most appropriate for assessing the d i s t r i b u t i o n of solar radiation on the meso scale. Oke (1978), defines the mesoscale to include phenomena occurring at •9 s p a t i a l scales from 10> - 2 x 10 5 m. In the context of this study the term mesoscale i s used to include phenomena occurring on time scales of 10 3 - 10 6 s and s p a t i a l scales of 10 3 - 10 5 m. This d e f i n i t i o n represents substantial overlap with the lo c a l scale (10 2 - 5 x 10 4 m) proposed by Oke (1978), but does not d i f f e r s i g n i f i c a n t l y from the d e f i n i t i o n of the mesoscale given by Steyn et a l . (1981) (see t h e i r Figure 1) and i s also included in the mesoscale Gamma (2 x 10 3 - 2 x ]0k m) and mesoscale Beta (2 x l O 4 - 2 x 10 5 m) categories proposed by Orlanski (1975). The claim made by Hay (1981) i s based on the following reasons: 1) The inadequacy of the e x i s t i n g network (both instrumented and model-based) to define anything more than the macroscale structure of the solar r a d i a t i o n f i e l d ; 2) The costliness of increasing the density of the pyranometric network to provide adequate mesoscale coverage; 3) The high resolution of the s a t e l l i t e data and 4) The vast volume of s a t e l l i t e data that must be processed favours mesoscale applications (data processing techniques need to be proven on small data sets before they are directed toward the massive volume of data required to define the s o l a r radiation climatology of larger areas). This thesis w i l l report on an attempt to characterize the mesoscale v a r i a b i l i t y of s o l a r radiation using s a t e l l i t e data. It w i l l provide some i n i t i a l v a l i d a t i o n or r e f u t a t i o n of the claim that s a t e l l i t e s are the most appropriate means with which to assess mesoscale . v a r i a t i o n s . Another aspect of the thesis w i l l be an intercomparison of three of the models described previously (see Section 1.1). 10 CHAPTER TWO 2. Data Sets and Data Processing 2.1 Radiation Data Set The radiation data for this study come from a twelve station pyranometric network spanning a 45 km by 70 km (3150 sq km) section of Greater Vancouver and the adjacent lower Fraser Valley. The locations of these stations are shown on Figure 1.1. The separation distance between stations varies from a minimum of four (4) km to a maximum of 74 km (Table 2.1). The network i s designed to maximise the mesoscale sp a t i a l variations in sol a r radiation and has been f u l l y operational since June 1979 (Hay and Tooms, 1979). According to Hay (1982), mesoscale s p a t i a l variations in sola r radiation are lar g e l y influenced by the spa t i a l v a r i a b i l i t y in cloud cover. Factors influencing cloud v a r i a b i l i t y on the mesoscale include orographic effects and topographic controls (e.g. land/sea contrasts). These factors are well represented in the study area. In add i t i o n , mesoscale s p a t i a l variations i n sol a r radiation due to urban/ rural differences are being investigated. At each l o c a t i o n , the sol a r irradiance on a horizontal plane at the earth's surface i s measured by a Kipp and Zonen pyranometer and recorded by a Campbell S c i e n t i f i c CR21 data logger. The data are sampled with a 10 second frequency and integrated to provide hourly t o t a l s of the sol a r i r r a d i a n c e . Hourly averaged values of the ambient a i r temperature are also a v a i l a b l e at each s i t e and are used in a Table 2.1 Distance Separation Matrix (km) f o r Lower Mainland Mesoscale R a d i a t i o n Network. STATION/NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 NAME 1 GROUSE x MOUNTAIN 2 NORTH 5.6 X MOUNT 3 BC HYDRO 14.5 4.2 x 4 VANCOUVER 20.9 15.3 11.0 x INT 1 L AIRPORT 5 TSAWWASSEN 38.5 32.9 28.7 17.7 x FERRY TERMINAL 6 PITT MEADOWS 32.3 28.9 30.9 33.4 38.2 x AIRPORT 7 HABITAT 62.2 59.4 61.6 63.0 61.7 30.8 x APARTMENTS Table 2.1 (continued) STATION/NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 NAME 8 ABBOTSFORD LIBRARY 6 8.0 64. 5 65. 8 65. 6 61. 4 35. 8 9 . 8 X 9 ABBOTSFORD AIRPORT 63.9 60. 0 60. 9 59. 8 54. 7 31. 7 13. 2 7.0 X 10 LANGLEY 43.8 39. 6 40. 1 39. 2 37. 3 13. 1 23. 1 26.3 20. 8 X 11 LANGARA 17.2 11. 7 6. 3 6. 2 24. 1 29. 0 59. 3 62.7 56.5 36.6 x 12 U.B.C. CLIMATE STATION 18.1 15. 2 9 . 6 10. 7 29. 5 39. 7 70. 3 73.8 68.5 47.6 11.2 X 13 temperature correction of the radiation data (Hay, 1982). Measurements of normal incidence s o l a r radiation are collected at one of the s i t e s (University of B r i t i s h Columbia Climate Station). Data processing and quality control procedures are described in Hay (1982). The accuracy associated with the radiation data set i s in accord with the ±5% error s p e c i f i e d by Latimer (1972). Hay and Wardle (1982), have shown that this RMS error is appropriate for hourly values but reductions occur for d a i l y and longer time periods. For d a i l y time periods a RMS error of ±3% is appropriate. The Mean Bias Error is on average within ±2.5%. A subset of these radiation data, consisting of twenty-three (23) days, is chosen for analysis (Table 2.2). These days represent radiation conditions of c l e a r , partly cloudy and overcast skies. The bases for the choice were; an examination of both the normal incidence data available at the UBC Climate Station and the hourly data available for each location and a desire to have a data set incorporating variable sky cover conditions in a l l seasons. From the normal incidence data there i s evidence of cloud contamination on the days c l a s s i f i e d as c l e a r . Following the d e f i n i t i o n given in the Manual of Standard Procedures for Public Weather Services (Man Pub, 1977), " c l e a r " i s used here to mean a sky condition with 2/10 cloud cover during a given radiation hour. 2.2 S a t e l l i t e Data Set The s a t e l l i t e data used in t h i s study are from a Geostationary Operational Environmental S a t e l l i t e (GOES). The s a t e l l i t e (GEOS-2) i s situated at 135 degrees west and zero (0) degrees l a t i t u d e at an 14 T a b l e 2.2 Days u s e d i n t h e M e s o s c a l e S t u d y w i t h v a l u e s o f P r e c i p i t a b l e W a t e r ( u ) from S m i t h ' s (1966) F o r m u l a t i o n . ' JULIAN DAY DATE SKY CONDITION U(mm) 196/79 1 5/7 c l e a r 26.7 200/79 19/7 1! 31.8 257/79 14/9 ti 25.8 263/79 20/9 o v e r c a s t 27.3 276/79 3/1 0 c l e a r 19.4 304/79 31/10 n 15.3 361/79 27/12 14.0 363/79 29/12 o v e r c a s t 13.4 025/80 25/1 c l e a r 03.7 121/80 30/4 17.1 105/80 1 4/4 o v e r c a s t 17.4 126/80 05/5 p / c l o u d y 26.7 141/80 20/5 o v e r c a s t 20. 5 160/80 08/6 p / c l o u d y 24. 1 171/80 19/6 c l e a r 24.2 183/80 01/7 it 26.2 185/80 03/7 o v e r c a s t 23.9 196/80 1 4/7 p / c l o u d y 27.0 197/80 1 5/7 o v e r c a s t 27.4 231/80 18/8 p / c l o u d y 25.4 239/80 26/8 Tl 25.5 245/80 01/9 o v e r c a s t 23.6 261/80 17/9 p / c l o u d y 25.5 15 a l t i t u d e of approximately 36,000 km above the earth ' s surface (Johnson et a l . , 1976). The primary instrument on board the s a t e l l i t e is a V i s i b l e and Jjifrared Spin Scan Radiometer (VISSR). The sensor measures radiance emerging from the Earth Atmosphere System in the v i s i b l e (0.55 - 0.75 um) and in f r a r e d (10.5 - 12.6 ym) regions of the electromagnetic spectrum. For the purposes of the present study only the v i s i b l e data are of i n t e r e s t . The v i s i b l e data are in the form of count values (measures of r e l a t i v e brightness) on an 8-bit scale ranging from 0 - 255 counts. These data are acquired at half-hourly i n t e r v a l s . An image consisting of a f u l l earth disc but covering about one quarter of the earth ' s surface is provided. (Corbel! et a l . 1981). Assessments of the resolution of the v i s i b l e data show some discrepancies: Hambrick and P h i l l i p s (1980), quote the highest resolution at the s u b - s a t e l l i t e point to be approximately 0.8 km; Johnson et a l . (1976), quote a value of one (1) km. Resolution deteriorates as the area viewed moves away from the s u b - s a t e l l i t e point (Corbell et a l . 1981). According to Gautier et a l . (1980), the resolution i s approximately 2 km at 45° (degrees) North. A s i m i l a r resolution was o r i g i n a l l y assumed for the l a t i t u d e s of the present study area (between 49°N and 50°N) (see Section 2.7 for s p e c i f i c determinations of the spa t i a l r e s o l u t i o n ) . The s a t e l l i t e data are archived at a few locations one of them being the Space Science and Engineering Centre of the University of Wisconsin at Madison, Wisconsin. The data are archived in d i g i t a l form on magnetic tapes and can be used both as image and d i g i t a l data. To obtain data for a given area, the number of l i n e s and elements in 16 an image ( t y p i c a l l y 1000 by 1008, r e s p e c t i v e l y ) , and the l a t i t u d e and longitude coordinates of the image centre must be s p e c i f i e d . S a t e l l i t e data corresponding to the days l i s t e d in Table 2.2 were obtained from the University of Wisconsin. The area covered by these data is much larger than our study area. Consequently, procedures have been developed to s e l e c t sub-images of this larger area. These sub-images are usually centered on Vancouver (49° 16'N, 123° 15'W) and contain a l l the stations in the radiation network. 2.3 Problem Areas The models mentioned in Chapter One and further developed in Chapter Three, rely for t h e i r effectiveness on the combination of the data sets described in Sections 2.1 and 2.2. These data sets are d i s t i n c t l y d i f f e r e n t leading to a number of problem areas. These are outlined in subsequent sections. 2.3.1 Spectral Ranges and C a l i b r a t i o n The f i r s t major problem area i s that of c a l i b r a t i o n . The s a t e l l i t e data are measured over a narrow spectral band (0.55 - 0.75 ym). These data are related to pyranometer measurements taken over the entire short-wave spectrum (0 - 3.5 ym). The i m p l i c i t assumption here i s that the narrow band measurement can adequately represent the entire short-wave spectrum. This is c l e a r l y not the case because of wavelength dependencies in the absorption and scattering processes. Additional problems a r i s e from the fa c t that there i s no d i r e c t way to accurately monitor the c a l i b r a t i o n of the VISSR sensor. The problem of 17 c a l i b r a t i o n , which i m p l i c i t l y includes the question of spectral s e n s i t i v i t y , is addressed in d i f f e r e n t ways by the three models. These are discussed in Chapters Four, Five and Six. 2.3.2 Earth Location of the S a t e l l i t e Imagery Another problem area, and one which i s of great importance in the context of mesoscale studies such as the present, is the earth l o c a t i o n of the s a t e l l i t e imagery. This requires the accurate aligning of points on the s a t e l l i t e image to the same points on the earth's surface. This is to ensure consistency of position of a given feature from one image to the next ( i . e . the same points should over l i e when sequential images are compared). The accuracy usually quoted for earth location (navigation) routines is on the order of one or two pixels (Gautier et a l . , 1980). The issue of earth l o c a t i o n of the s a t e l l i t e imagery i s discussed further in Section 2.5. 2.3.3 Temporal Sampling and Viewing Angles of Sensors The s a t e l l i t e measurement is an instantaneous measurement over a small s o l i d viewing angle. On the other hand the pyranometer measurement i s integrated over an hour and a s o l i d angle of 2TT. The comparison of these two measurements represents another problem. An assumption i s made that half-hourly instantaneous measurements can be used to represent the rad i a t i o n c h a r a c t e r i s t i c s of an hour. Gautier et a l . (1980), attempt to address this time and angle discrepancy by averaging the radiation estimates over an 8 x 8 pixel box (averaging over the 8 x 8 array also attempts to account for the variations in s e n s i t i v i t y of the eight v i s i b l e sensors). This i s supposed to 18 correspond to a space/time t r a n s l a t i o n of 4.5 m s" 1 (16 km for 1 hour). The v a l i d i t y of th i s space/time t r a n s l a t i o n i s yet to be established. In spite of t h i s , we have adopted a s i m i l a r s p a t i a l averaging procedure but using a 5 x 5 pixel box. The choice of t h i s array size i s based on a ± 1 - 2 pixel accuracy obtainable from our earth location routine (see Section 2.6). In addition, we consider the s e n s i t i v i t y of our radiation c a l c u l a t i o n s to changes in the size of our sampling array (see Section 5.3.2). A measure of integration i s introduced into the radia t i o n c a l c u l a t i o n s through the use of as many as three images to represent the radiation conditions f o r an hour. This approach i s used i n both the Gautier et a l . (1980) and the present studies (see Section 2.9 f o r further discussion). 2.4 Precipi table Water Pr e c i p i t a b l e water i s one of the two surface-based measurements required by the models of Tarpley (1979), and Gautier et a l . (1980). Gautier et a l . (1980) use Smith's (1966), empirical formulation of the dewpoint temperaturG to ca l c u l a t e values f o r p r e c i p i t a b l e water, whereas Tarpley (1979), uses measured p r e c i p i t a b l e water data from the National Meteorological Centre data f i l e . Atwater and Ball (1976), show that f o r eleven (11) locations i n the U.S.A. the method of determining p r e c i p i t a b l e water (either from the surface dewpoint temperature or radiosonde data) has l i t t l e e f f e c t on the calculated s o l a r r a d i a t i o n . Since both approaches are represented in the models under study, we f e l t i t necessary to check the v a l i d i t y of the Atwater and Ball (1976), f i n d i n g . The model by Gautier et a l . (1980) 19 i s the obvious choice f o r such t e s t i n g because of i t s physically-based framework, and the f a c t that p r e c i p i t a b l e water i s a d i r e c t input. Four c l e a r days, r e f l e c t i n g conditions at d i f f e r e n t times of the year are chosen f o r the analysis (see Table 2.3). There are no upper a i r stations within the study area. Data f o r computing p r e c i p i t a b l e water are obtained from two locations; Port Hardy on the North East coast of Vancouver Island, and Quillayute on the Olympic Peninsula Washington U.S.A.. These data are archived at the Atmospheric Environment Service Weather O f f i c e , Vancouver, B r i t i s h Columbia as twice d a i l y readings (0400 and 1600 LST) of dewpoint and dry bulb temperatures at 14 lev e l s in the atmosphere (from the surface to 300 mb). These data are converted to p r e c i p i t a b l e water values using formulae described in the Atmospheric Environment Service's Monthly B u l l e t i n of Canadian  Upper A i r Data. We estimate that errors of up to four (4) mm are possible i n the p r e c i p i t a b l e water values. P r e c i p i t a b l e water values f o r the study area are determined as weighted averages of the mean d a i l y values at both locations. Weights are assigned on the basis of distance with Port Hardy (at a distance of approximately 360 km from the study area) assigned a weighting of 0.5 and Quillayute (at a distance of approximately 171 km) assigned a weighting of 1.0. Values calculated f o r the four days are l i s t e d in Table 2.3. Dewpoint temperatures are available at three of the stations in the network, U.B.C. Climate Station, Vancouver International A i r p o r t and Abbotsford A i r p o r t . Variations in the dewpoint temperatures between the three locations lead to a maximum difference of four (4) mm when Smith's (1966) formulation i s used to calculate the p r e c i p i t a b l e water (see Table 2.4). Table 2.3 Precipitable Water (u) Estimated Using Smith's (1966) Formulation and Computed from Radiosonde Data for (4) Clear Days u FROM SMITH'S u FROM DATE SMITH'S T, d (1966) U.B.C. (1966) FORMULATION RADIOSONDE (CM) COEFFICIENT (CM) (CM) DIFFERENCE % DIFFERENCE 276/79 2.71(FALL) 47.3°F 3RD OCT. (8.5°C) 121/80 2.95(SPRING) 45.7°F 30TH APRIL (7.6°C) 171/80 2.77(SUMMER) 53.4°F 19TH JUNE (11.9 °C) 183/80 2.77(SUMMER) 55.4°F 1ST JULY (13.0°C) 1.94 1. 71 2.42 2.62 1. 27 1. 29 1.11 1. 32 .67 .42 1. 31 1. 30 52. 8 32.6 118 98.5 Table 2.4 V a r i a t i o n s i n Estimated P r e c i p i t a b l e Water (Smith 1966) Using Dewpoint Temperatures (T ) from Three Locations i n the Mesoscale Network. DATE UBC CLIMATE STATION VANCOUVER AIRPORT ABBOTSFORD AIRPORT DEW POINT PRECIPITABLE DEW POINT PRECIPITABLE DEW POINT PRECIPITABLE TEMP. WATER (CM) TEMP. WATER (CM) TEMP. WATER (CM) °C °F °C °F °C °F 276/79 8.5 47.3 1.94 3RD OCT. 121/80 7.6 45.7 1.71 30TH APRIL 11 51.8 2.31 9 48.2 1.88 10 50 2. 15 8 46.4 1.76 171/80 11.9 53.4 2.42 19TH JUNE 183/80 13 55.4 2.62 1ST JULY 11 51.8 2. 28 12 53.6 13 55.4 2.62 15 59 2.44 3.02 22 In (11) = [ 0.1133 - In (x)] + 0.0393 T d (2.4.1) U = p r e c i p i t a b l e water in cm T"d = surface dewpoint temperature (°F) x = l a t i t u d e and seasonally adjusted c o e f f i c i e n t (dimensionless) Ideally, p r e c i p i t a b l e water ca l c u l a t i o n s (for t h i s analysis) should be based on dewpoint temperatures from each of our twelve (12) locations. However, on the basis of the small variations in p r e c i p i t a b l e water values shown in Table 2.4, we decided to use the values from one location (U.B.C. Climate Station) to represent the en t i r e network. The dewpoint temperature used i s the average of two observations taken at 0900 and 1600. Values of p r e c i p i t a b l e water must be adjusted for the Grouse Mountain location which i s at an elevation of 1128 m. Lowry (1980), has shown that decreases in p r e c i p i t a b l e water of up to 39% are associated with a l t i t u d e s s i m i l a r to that of Grouse Mountain. On the basis of his f i n d i n g , and from an examination of the p r e c i p i t a b l e water p r o f i l e s at Port Hardy and Quillayate (to assess the decrease in p r e c i p i t a b l e water for pressure l e v e l s s i m i l a r to the height of Grouse Mountain), the p r e c i p i t a b l e water value at Grouse Mountain i s estimated to be 80% of the surface value (a decrease of 20%). Estimates of the incoming solar radiation f o r the four days were made using the Gautier model with both the e m p i r i c a l l y calculated and radiosonde based values of p r e c i p i t a b l e water. These were compared with measured r a d i a t i o n values. The % differences between modelled and measured values are plotted in Figures 2.1a - 2.Id. These show that, with few exceptions, the model cons i s t e n t l y over estimates the 23 Day 183/80 W= 13.2 mm, (radiosonde) i }W = 26.2 mm \ / calculated (Smith, 1966) -i 1 1 1 i i_ 2. la(Summer) Day 171/80 SW = 24.2 mm calculated (Smith, 1966) 2.lb(Summer) CO 3 O DC U I- O < > z Q o CO DC 5 tr O DC 1- UJ < 1 ABBAI X Q z DC 5 X m ABBAI NRT BC HY LA VAr UJ LL • u PIT SSIIAI ABI ABBAI UJ -J o z < STATIONS la - 2.Id: Showing the % Difference Between Measured and Calculated Radiation f o r Two Di f f e r e n t Inputs of Pr e c i p i t a b l e Water at Three D i f f e r e n t Times of the Year 24 STATIONS 25 measured r a d i a t i o n . Over estimation i s as much as 13% using the radiosonde-based values (see Figure 2.1b). Also obvious from the graphs i s that; 1) use of p r e c i p i t a b l e water values determined from the surface dew-point temperature y i e l d s estimates which are c l o s e r to the observed solar r a d i a t i o n . (The differences between the observed solar r a d i a t i o n and these estimates are generally comparable to the ±5% instrument error) and 2) values of p r e c i p i t a b l e water derived from the radiosonde data are largely inappropriate f o r use in t h i s study due to the large errors introduced. Atwater and Ball (1976), found differences of 1 - 2% between the r a d i a t i o n calculated using estimated and computed p r e c i p i t a b l e water. The differences in t h i s present study are often as great as 5%. The discrepancy between t h e i r study and the present one may be due to the f a c t that Atwater and Ball (1976), had on-site observations of both surface dewpoint temperature and radiosonde data. In addition, t h e i r comparisons were made over a longer time period. Table 2.3 shows the absolute and percentage differences between the two values of p r e c i p i t a b l e water for the four days analysed. Differences range from 32.6% (4.2 mm) to 118% (13.1 mm). However, the change in the estimated ra d i a t i o n as a r e s u l t of the largest change in p r e c i p i t a b l e water i s within 5%. It appears from t h i s that the model i s r e l a t i v e l y i n s e n s i t i v e to large changes in p r e c i p i t a b l e water, a f a c t which supports the use of a single dewpoint temperature to represent the e n t i r e network given the v a r i a t i o n s documented in Table 2.4. Another feature of note in the analysis i s shown on Figures 2.1a and 2.1b. Stations 1 - 7 are the coastal and higher elevation stations. Stations 8 - 1 2 are the i n t e r i o r locations. Model over estimation 26 i s greater at these i n t e r i o r locations. Table 2.4 shows that the p r e c i p i t a b l e water values estimated f o r Abbotsford A i r p o r t are higher than those at the U.B.C. Climate Station for the four days. On the basis of t h i s observation i t might be argued that the lower p r e c i p i t a b l e water values used in the c a l c u l a t i o n s are the source of t h i s over estimation. However, there are several factors that m i l i t a t e against such an explanation; 1) the i n s e n s i t i v i t y of the model to p r e c i p i t a b l e water indicates that changes in the calculated radiation a r i s i n g from the use of p r e c i p i t a b l e water values at Abbotsford A i r p o r t w i l l be minimal (calculations f or day 183/80 (re s u l t s not shown) support t h i s conclusion) and 2) the normal gradient in p r e c i p i t a b l e water from the ocean to the land would c a l l f o r lower values of t h i s parameter at the i n t e r i o r locations ( i . e . the higher values at Abbotsford A i r p o r t could be e n t i r e l y f o r t u i t o u s ) . A more plausible explanation would be to a t t r i b u t e the over estimations at the i n t e r i o r locations to the e f f e c t s of aerosols. Hay (1982), has noted the occurence of lower measured radiation values at these locations under c l e a r summer conditions (the conditions represented by the data in Figures 2.1a and 2.1b). A postulated cause for t h i s minima i s the bui l d up of aerosols (pollutants) at these i n t e r i o r locations. The Gautier et a l . (1980), model does not include the e f f e c t s of aerosols. Therefore, a reduction in measured values due to the influence of aerosols w i l l not be r e f l e c t e d in a corresponding reduction in the calculated values, leading to the overestimations Based on some of the r e s u l t s from the preceding analyses, p r e c i p i t a b l e water values were calculated f o r the days in the study using Smith's (1966) empirical formulation along with dewpoint temperatures 27 from the U.B.C. Climate Station. These values are l i s t e d in Table 2.2. 2.5 Earth Location of S a t e l l i t e Imagery Before the s a t e l l i t e data can be u t i l i z e d , i t i s imperative that the earth location of each image be known. That i s , which pixel on the image corresponds to a measurement s i t e or other landmark at the earth's surface. This issue i s of importance because any errors in the earth location of the s a t e l l i t e imagery w i l l a f f e c t the rad i a t i o n c a l c u l a t i o n s based on these data. This i s e s p e c i a l l y s i g n i f i c a n t under p a r t l y cloudy skies where the ra d i a t i o n c h a r a c t e r i s t i c s may be changing ra p i d l y over small distances. I n i t i a l earth location of the s a t e l l i t e imagery was performed using a model described by Hambrick and P h i l l i p s (1980). The model i s based on the use of both s t e l l a r and t e r r e s t r i a l navigation points and a knowledge of the departure of the s a t e l l i t e from a true geosynchronous o r b i t . The accuracy of t h i s model (subsequently referred to as " F i r s t Order Navigation", FON) i s claimed to be within one pixel RMS accuracy. However, i n i t i a l comparisons of the location of landmark determined v i a FON and vi s u a l navigation using the COMTAL image processor (UBC Computing Centre) frequently revealed much larger errors. Visual navigation involves comparison of the predicted position of a landmark based on the F i r s t Order Navigation, to the actual position of the same landmark as seen on the COMTAL image processor. Errors were as large as one hundred and f i f t y two pixels in the l i n e (N-S) d i r e c t i o n for day 171/80 and seventy-three pixels in the element (E-W) d i r e c t i o n f o r day 361/79, with errors of twenty p i x e l s and above, not uncommon. 28 The errors determined were unacceptable for our studies. Therefore, "Second Order Navigation" was developed to correct f o r the earth location errors s t i l l present following the F i r s t Order Navigation. The objective was to achieve earth location of landmarks to within one or two p i x e l s . The Second Order Navigation involved r e l a t i n g the l a t i t u d e and longitude coordinates of a landmark (see Table 2.5 for landmarks used in earth location) to the pixel coordinate as v i s u a l l y determined on the image processor, and subsequent to the F i r s t Order Navigation. This procedure i s repeated f o r at l e a s t three images spread throughout the study day. That i s , one in the early morning, near noon and late afternoon, but preferably for one image per hour during the day and for at least one landmark per image. For each image the difference between the pixel coordinates of the landmark (or an average of the difference i f multiple landmarks are used), produced by the F i r s t Order Navigation and the Second Order Navigation i s calculated. This difference i s the t r a n s l a t i o n that must be applied to give the 'true' location of landmarks on that image. The remaining images f o r a given day are subsequently earth-located by using a cubic i n t e r p o l a t i o n function generated from the navigated images to estimate the correction to be applied to the image which has received only F i r s t Order Navigation. This i n t e r p o l a t i o n procedure i s based on time and incorporates the three images that have undergone Second Order Navigation and are closest to the unnavigated image. The t r a n s l a t i o n s associated with these three images are used by the cubic i n t e r p o l a t i o n function to estimate the t r a n s l a t i o n f or the unnavigated image. Translations f o r images that occur e a r l i e r than the f i r s t Table 2.5 Landmarks Used i n Earth L o c a t i o n Routine LANDMARK NAME LATITUDE (N) 48° 41' 40" 48° 56' 12" 48° 18' 54" 48° 18' 56" 49° 29' 26" 48° 22' 50" 49° 30' 00" 50° 19' 54" 49° 35' 48" LONGITUDE (W) 123° 28' 54" 122° 48' 36" 123° 38' 39" 123° 34* 18" 124° 08' 03" 124° 43' 48" 119° 34' 24" 119° I V 24" 116° 49' 12" LOCATION MOSES POINT BIRCH POINT BEECHY HEAD CHURCH POINT TEXADA ISLAND (SOUTHTERN TIP) CAPE FLATTERY LAKE OKANAGAN (SOUTHERN END) LAKE OKANAGAN (NORTHERN END WESTERN ARM) KOOTENAY LAKE (CAPE HORN) SAANICH PENINSULA VANCOUVER ISLAND WASHINGTON U.S.A. VANCOUVER ISLAND VANCOUVER ISLAND VANCOUVER ISLAND OLYMPIC PENINSULA (WASHINGTON) BRITISH COLUMBIA (OKANAGAN VALLEY) BRITISH COLUMBIA (OKANAGAN VALLEY) S.E. BRITISH COLUMBIA Table 2.5 (continued) LANDMARK NAME LATITUDE (N) LONGITUDE (W) CAPE BLANCO 42° 50 ' 27" 124° 30 ' 33" COOS BAY (SOUTH) 43° 20 1 01" 124° 19 • 05" COOS BAY (NORTH) 43° 21' 09" 124° 19 • 18" TILLAMOOK BAY (HEAD OF BAY) 45° 29 ' 34" 123° 53' 05" WALDO LAKE (SOUTH) 43° 41' 14" 122° 06 ' 31" LAKE PEND OREILLE 48° 09 ' 24" 116° 14' 06" LAKE PEND OREILLE (SOUTH WEST) 47° 58' 01" 116° 32' 04" HOOD CANAL (PUGET SOUND) 47° 20 ' 12" 123° 07' 06" GRAYS HARBOUR 46° 56' 36" 124° 09 ' 12" COLUMBIA RIVER 46° 14' 12" 124° 00 ' 00" MOUTH (SOUTH SPIT) LOCATION OREGON COAST OREGON COAST OREGON COAST OREGON COAST co o CENTRAL OREGON NORTHERN'IDAHO NORTHERN IDAHO N.W. WASHINGTON U.S.A. N.W. WASHINGTON U.S.A. WASHINGTON/OREGON U.S.A. LANDMARK NAME Table 2.5 (continued) LATITUDE (N) LONGITUDE (W) KOOTENAY LAKE 50° 10' 18" 116° 55* 48" (NORTH) KOOTENAY LAKE 49° 11' 34" 116° 39' 29" (SOUTH) SLOCAN LAKE 49° 46' 13" 117° 28' 12" (SOUTH) WILLISTON LAKE 55° 56' 30" 122° 08' 24" (EASTERN ARM) STUART LAKE 54° 23' 30" 124° 16' 54" PRIEST LAKE 48° 28' 59" 116° 51' 52" (SOUTH) COEUR D'ALENE LAKE 47° 26' 58" 116° 54' 55" (WESTERN TIP) FERN RIDGE LAKE 44° 07' 25" 123° 18' 18" (NORTH-WEST TIP) CAPE ARAGO 43° 17' 54" 124° 23' 18" LOCATION S.E. BRITISH COLUMBIA S.E. BRITISH COLUMBIA S.E. BRITISH COLUMBIA NORTHERN BRITISH COLUMBIA NORTHERN BRITISH COLUMBIA NORTHERN IDAHO WESTERN IDAHO WESTERN OREGON OREGON COAST Table 2.5 (continued) LANDMARK NAME LATITUDE (N) LONGITUDE (W) LOCATION BANKS LAKE (NORTHERN END) 47° 55 • 57" 119° 05' 4 8" CENTRAL WASHINGTON BANKS LAKE (SOUTH) 47° 37' 04" 119 ° 17' 13" CENTRAL WASHINGTON PRIEST RAPIDS LAKE 46° 38' 08" 119° 85' 0 3" CENTRAL WASHINGTON WANAPUM LAKE (SOUTH) 46° 52' 05" 119° 57' 09" CENTRAL WASHINGTON POTHOLES RESERVOIR (SOUTH END) 46° 59 • 19" 119 ° 17' 18" CENTRAL WASHINGTON 33 navigated image, or l a t e r than the l a s t navigated image, are determined by extrapolation. As an example (of the in t e r p o l a t i o n procedure); given a sui t e of four (4) earth located (navigated) images (16:45, 18:15, 19:15 and 20:15). The estimated translations (E-W and N-S) for an image at 18:45 w i l l be determined using the images at 18:15, 19:15 and 20:15. The accuracy of t h i s i n t e r p o l a t i o n function i s discussed in Section 2.6. The Second Order Navigation procedure r e l i e s t o t a l l y on visual i d e n t i f i c a t i o n of landmarks on the earth's surface and therefore depends on comparatively c l e a r skies. The present area of i n t e r e s t incorporates part of the Lower Fraser Valley where suitable land-water boundaries along the southern B.C. coastline provide c l e a r l y v i s i b l e landmarks. However, clouds are prevalent in t h i s area and frequently obscure the coastal landmarks. The s a t e l l i t e images used extend into the western edge of Alberta and some of the larger images extend i n t o Northern B r i t i s h Columbia and Southern Oregon. Therefore, i f the Vancouver area i s cloudy a l t e r n a t i v e areas maybe s u f f i c i e n t l y c l e a r to permit the i d e n t i f i c a t i o n of landmarks. This procedure was car r i e d out by f i r s t examining hardcopy images at the Atmospheric Environment Service Weather O f f i c e , Vancouver, to locate possible cloud free areas and then inspecting these areas on the image processor. I n t e r i o r lakes (e.g. Kootenay Lake B.C. and P r i e s t Lake Idaho U.S.A.) and the Oregon coast (e.g. Cape Arago) are examples of landmarks that have been used to navigate the images (see Figure 2.2). 34 Figure 2.2 Map of the Area Covered by the Largest S a t e l l i t e Images (1000 Lines x 1008 Elements) Showing Landmarks T y p i c a l l y Used in the Assessment of L i n e a r i t y . 35 2.6 V a l i d a t i o n of the Earth Location Routine The i n t e g r i t y of the radiation c a l c u l a t i o n s to be performed at some subsequent stage in t h i s study r e l i e s heavily on the v a l i d i t y of our earth location routine. The earth location routine i s l a r g e l y dependent on the 'visual' processing of a number of images coupled with the use of a cubic i n t e r p o l a t i o n procedure to account f o r those images that have not been v i s u a l l y earth located (see Section 2.5). We assessed the accuracy of t h i s i n t e r p o l a t i o n procedure using a sample of 34 images selected from seven c l e a r days; J u l i a n days 196/79, 257/79, 276/79, 304/79, 361/79, 121/80 and 183/80. The images were selected s o l e l y on the basis that they had not been previously earth-located using our '2nd order navigation'. During t e s t i n g , landmarks on these images were processed as outlined in Section 2.5, and the observed t r a n s l a t i o n s calculated. Estimated tr a n s l a t i o n s were also determined f o r these images using the cubic i n t e r p o l a t i o n procedure. Two s t a t i s t i c s , the Mean Bias Error and the Root Mean Square, were used as the indicators of the goodness of f i t of the i n t e r p o l a t i o n procedure. They are based on the residuals of the estimated minus the observed t r a n s l a t i o n s . MBE = N. = 1 (F! - F.) / N (2.6.1) RMSE = (FI - F.)2 / N ] 1 / 2 (2.6.2) F! = the i t h value of the estimated t r a n s l a t i o n s F. = " " " " " observed N = sample siz e = 34 36 Calculations were performed f o r both l i n e (N-S) and element (E-W) tr a n s l a t i o n s . From the r e s u l t s l i s t e d in Table 2.6 we can note the following: the RMSE calculated f o r the l i n e i s 1.026 pixels whilst the MBE i s -0.17 p i x e l s . The maximum difference between the estimated and observed t r a n s l a t i o n s i s j u s t under two pixels (1.65), while the minimum difference i s .03 p i x e l s . This means that, on average, the cubic i n t e r p o l a t i o n procedure can estimate the l i n e coordinate of a given landmark to an accuracy of ±1.026 p i x e l s . In terms of absolute distances (using the values of pixel s i z e i n Section 2.7) we can estimate the position of the landmark in the North-South dimension to within 1.71 km. The negative MBE means that the tendency would be to estimate a position North of the true location. The RMSE calculated f o r the element (E-W) i s 1.53 pixels whilst the MBE i s .302 p i x e l s . The maximum difference between estimated and observed t r a n s l a t i o n s i s j u s t under (4) p i x e l s , while the minimum difference i s -0.03 p i x e l s . The values f o r the element coordinate (E-W) are larger in most instances than those f o r the l i n e . The RMSE of 1.53 pixels means that on average we can estimate the element coordinate of a given landmark to an accuracy of ±1.53 p i x e l s . In terms of absolute distances (based on a pixel s i z e in the E-W of .82 km; Section 2.7) we can estimate the position of a landmark to within 1.25 km. The p o s i t i v e MBE means that the tendency would be to estimate to the East of the actual l o c a t i o n . We conducted a 'worst case' analysis (based on one landmark per image) of the cubic i n t e r p o l a t i o n procedure using a sample of 24 images. Again the r e s u l t s are l i s t e d in Table 2.6 (numbers in parentheses). Our r e s u l t s show that except in the case of the RMSE f o r the l i n e 37 Table 2.6 V e r i f i c a t i o n S t a t i s t i c s for the Cubic Interpolation Routine used  in the Earth Location of the S a t e l l i t e Imagery.N=39.Values are  quoted in pix e l s . LINE(N-S) ELEMENT(E~W) RMSE 1.026 1.53 M0.736) (1.53) MBE -0.170 0.302 (0.198) (0.290) MAXIMUM 1.65 3 770 DIFFERENCE . (1.75) (3.'60) MINIMUM 0.03 -0.03 DIFFERENCE (0.03) ( - C L 0 2 ) *Data for a "worst case" analysis based on only one landmark per image with N=24(see text). 38 coordinate, only minor changes occur. The accuracy of the cubic i n t e r p o l a t i o n procedure for earth location of s a t e l l i t e imagery demonstrated by the r e s u l t s in Table 2.6 i s s i m i l a r to that quoted by Gautier et a l . (1980) f o r t h e i r navigation procedure and i s considered to be more than adequate for a mesoscale study such as the present. However, t h i s accuracy could be reduced by the e f f e c t s of n o n - l i n e a r i t y across the image (see Section 2.8). A more d i r e c t technique f o r t e s t i n g the earth location procedure involved using forty-three images and numerous landmarks e.g. Birch Point (Washington), Mosses Point (Vancouver Island), Cape Arago (Oregon), see Figure 2.2, giving a t o t a l of f i f t y - t h r e e observations. Only those images which were not used to derive the Second Order Navigation were incorporated i n t h i s t e s t . This second technique gave r e s u l t s of 0.34 pixels (0.28 km) and 0.13 p i x e l s (0.22 km) f o r the Mean Bias Error in the element and l i n e d i r e c t i o n r e s p e c t i v e l y , and 1.20 pixels (1.06 km) and 0.63 pixels (1.05 km) f o r the Root Mean Square Error in the element and l i n e d i r e c t i o n r e s p e c t i v e l y . These re s u l t s confirm those of the previous technique, though, t h i s second technique i s favourable due to i t being a more e f f i c i e n t procedure. The accuracy of the earth location i n t e r p o l a t i o n method i s more than adequate f o r our research requirements. 2.7 Pixel Size Determination We noted e a r l i e r (Section 2.2) that the s p a t i a l resolution assumed f o r the l a t i t u d e s of the study area i s approximately 2 km (or 4 sq km). The mesoscale nature of the study area requires an accurate determination of the s p a t i a l resolution of the s a t e l l i t e data 39 (pixel size) f or the study area. The data base for t h i s determination consisted of two c l e a r images on J u l i a n day 276/79 at 10:45 and 11:45 LST and two topographic map sheets ( V i c t o r i a sheet 92B, scale 1:250,000 or 1 cm - 2.5 km by U.S. A.C.E.) and Vancouver sheet 92 G scale 1:250,000. Pairs of landmarks were chosen from the images and the pixel separation distance between pairs was noted. Point pairs were chosen so that determinations could be made both in the East-West and North-South d i r e c t i o n s . To ensure that points were in a s t r a i g h t l i n e on the image, the element coordinate had to be the same f o r the North-South pairs and the l i n e coordinate had to be the same for the East-West pairs. Landmarks were also located on the topographic maps and distances between the point pairs were determined. The si z e of an individual pixel was calculated by d i v i d i n g the map separation distance between point pairs by the number of pixels between the corresponding points on the image. The r e s u l t s of these c a l c u l a t i o n s (along with the relevant data on the various point pairs) are l i s t e d i n Table 2.7. The average pixel s i z e i n the East-West d i r e c t i o n i s approximately 0.82 km, whilst in the North-South d i r e c t i o n i t i s approximately 1.67 km. According to these c a l c u l a t i o n s the smallest area resolved by the s a t e l l i t e sensor for the l a t i t u d e s of the study area i s approximately 1.37 sq km; considerably less than the 4 sq km o r i g i n a l l y assumed. The shape of a pixel i s not a 2 km by 2 km square but a rectangle elongated in the North-South and compressed in the East-West d i r e c t i o n s . According to Corbell et a l . (1981), the res o l u t i o n of the s a t e l l i t e deteriorates as the area viewed moves away from the s a t e l l i t e Table 2.7 Data Set f o r S a t e l l i t e R e s o l u t i o n ( P i x e l Size) Determination Landmarks Taken from V i c t o r i a Sheet 92 B and Vancouver Sheet 92 G Sca l e : 1:250,000. : ~ IMAGE DATE/TIME LANDMARKS COORDINATES SEPARATION DISTANCE RESOLUTION ELEMENTS LINE PIXEL MAP (km) (km) 23 39 1.7 (E-W) (N-, 79/276/21: 45 BEECHY HEAD 114 186 MUDBAY 114 163 79/276/18: 45 DIAMOND PT. 188 206 (MILLER PENIN.) COLVILLE 188 182 PT. (LOPEZ IS. ) 79/276/18: 45 MOSES PT. 124 164 BAY TO EAST 124 189 OF SMYTH HEAD 79/276/21: 45 CHASE RIVER 070 130 (mth) SECHELT 070 107 PENINSULA 24 38.8 1.61 25 42.3 1.69 23 38.1 1.66 Table 2.7 (continued) IMAGE DATE/TIME LANDMARKS COORDINATES ELEMENTS LINE (E-W) (N-S) SEPARATION DISTANCE RESOLUTION PIXEL MAP (km) (km) 79/276/21:45 79/276/21:45 79/276/21:45 79/276/21:45 COQUITLAM LAKE (N) BLACKIE SPIT (MUDBAY) WHITECLIFF PT. WATTS POINT 164 164 121 121 183 S.E. END PITT LAKE CENTRAL BEND 183 PITT LAKE WINTER COVE 142 MAYNE IS. S.E. TIP 142 VANCOUVER IS. 110 138 116 098 119 112 148 180 28 18 32 45.5 31 15 44. 88 1.63 1. 72 2. 14 1.40 Table 2.7 (continued) IMAGE DATE/TIME LANDMARKS COORDINATES SEPARATION DISTANCE RESOLUTION ELEMENTS LINE PIXEL MAP (km) (E-W) (N-S) (km) 79/276/21:45 BIRCH POINT 171 145 S.E. TIP 33 56.5 1. 71 SAN JUAN IS. 171 178 79/276/21:45 SHAW IS. 171 178 (S.E. TIP) S.E. TIP 171 171 7 10 .9 1.55 SAN JUAN IS. 79/276/21:45 NORTHERN TIP 105 145 SALT SPRING 68 IS. 56.5 . 83 BIRCH POINT 173 145 79/276/21:45 MOSES POINT 119 162 EASTERN HEADLAND 86 71. 2 . 83 BELLINGHAM 205 162 BAY Table 2.7 ( c o n t i n u e d ) IMAGE DATE/TIME LANDMARKS COORDINATES ELEMENTS LINE (E-W) (N-S) SEPARATION DISTANCE PIXEL MAP (km) RESOLUTION (km) 79/276/21:45 79/276/21:45 79/276/21:45 79/276/21:45 S.E. TIP 142 VANCOUVER IS. LANGLEY PT. 19 9 WHIDBEY IS. DEPARTURE 06 2 BAY POINT GREY 126 POINT GREY 126 S. END STEVE 211 LAKE PT. COWAN 114 (BOWEN IS.) PT. ATKINSON 123 (W. VAN.) 180 180 124 124 124 128 118 118 57 64 85 46.3 50 73 7.1 .81 781 . 86 . 792 Table 2.7 (continued) IMAGE DATE/TIME LANDMARKS COORDINATES SEPARATION DISTANCE RESOLUTION ELEMENTS LINE PIXEL MAP (km) (E-W) (N-S) (km) 79/276/21:45 INDIAN ARM 157 112 (NORTH) PITT LAKE 183 112 26 25.5 .981 (NORTH BEND) 79/276/21:45 STAVE LAKE (BEND NEAR 211 122 S. END) 17 15. 8 .926 ALOUETTE 194 122 LAKE (S. END) 79/276/21:45 SOUTHERN TIP 145 182 VANCOUVER IS. LANGLEY PT. 202 182 57 46. 3 .81 WHIDBEY IS. Average P i x e l S i z e : E-W .82 km N-S 1.6 7 km 45 sub-point (resolution at sub-point i s approximately 0.8 km). The sub-point of the s a t e l l i t e used i n t h i s study i s at 135° West and 0.0° North. The distance of the study area away from the s a t e l l i t e sub-point i s greater in the North-South dimension than i t i s in the East-West, t h i s means that degradation of the resolution should be greater in the North-South dimension; t h i s i s borne out by the present r e s u l t s . 2.8 L i n e a r i t y The earth-location of imagery on p a r t l y cloudy and overcast days (occasions when landmarks in the local area are not v i s i b l e ) e n t a i l s the use of landmarks from other areas (see Table 2.5) that are cloud free (Section 2.5). We make the assumption that the tr a n s l a t i o n s associated with these landmarks can be applied to the study area without any loss in accuracy. That i s , that the tr a n s l a t i o n s behave in a l i n e a r manner across a given image. A l t e r n a t i v e l y , any no n - l i n e a r i t y introduced w i l l be of l i t t l e importance. The issue of l i n e a r i t y i s an important one because i t i s one of the sources of error a f f e c t i n g the accurate earth-location of the s a t e l l i t e imagery (see Section 2.5). To be useful in the assessment of l i n e a r i t y , images must s a t i s f y two basic requirements; 1) they must be large enough to include landmarks outside the study area; 2) they must be p r a c t i c a l l y cloud-free to enable the i d e n t i f i c a t i o n of landmarks over a large area. These requirements constrain our assessment to clear days f o r which our largest images (1000 l i n e s x 1008 elements) were a v a i l a b l e . We assessed the v a l i d i t y of our assumption of l i n e a r i t y using imagery from several c l e a r days (183/80, 121/80 and 276/79). Images 46 from these days were earth-located using landmarks from the study area. The t r a n s l a t i o n s determined from these landmarks were applied to landmarks from another area of i n t e r e s t (Figure 2.2 shows some of the landmarks t y p i c a l l y used). The t r a n s l a t i o n determined f o r the new landmark outside the study area provides a measure of the departure from l i n e a r i t y . L i n e a r i t y was assessed in two dimensions, the North-South dimension and the East-West dimension. Results from our assessments are l i s t e d in Table 2.8. Analyses f o r the East-West dimension are based on a sample of (13) images. They indicate that, when using landmarks some distance from the study area, on average the element position of a landmark in the study area can be estimated to within ±5 pixels (4.1 km). The l i n e position of that same landmark can be estimated to within ±4.6 pixels (8 km). Analyses f o r the North-South dimension are based on a sample of (22) images. They indicate that the element position of a landmark in the study area can be estimated to within ±1.57 pixels (1.28 km), while the l i n e position of that same landmark can be determined to within ±1.23 pixels (2 km). The preceding analyses have been based on small data sets. Their s i z e has been l a r g e l y influenced by the inhomogenous occurrence of cloud on most of the ava i l a b l e imagery. However, we believe that some tentative conclusions can be drawn on the basis of our r e s u l t s : 1) The e f f e c t s of no n - l i n e a r i t y across the image in the North-South dimension i s n e g l i g i b l e . The errors determined here are of s i m i l a r magnitude to those associated with our cubic i n t e r p o l a t i o n procedure (see Section 2.6). In the worst case (when i n t e r p o l a t i o n and l i n e a r i t y errors are additive) the true position of a landmark w i l l be o f f by 47 T a b l e 2.8 R e s u l t s from t h e A s s e s s m e n t s o f L i n e a r i t y a c r o s s t h e S a t e l l i t e  Image b a s e d on d a t a from t h r e e c l e a r d a y s . V a l u e s a r e i n p i x e l s LINE(E-W) ELEMENT(E-W) LINE(N-S) ELEMENT(N~S) RMSE 4.62 5.0 1.23 1.57 N=13 N=22 MBE -3.8 -4.8 -0.08 -0.07 48 approximately three (3) pixels (2.5 km) in the element (E-W) d i r e c t i o n and 2.3 pixels (3.8 km) in the l i n e (N-S) d i r e c t i o n ; 2) The e f f e c t s of n o n - l i n e a r i t y across the image in the East-West d i r e c t i o n i s of greater s i g n i f i c a n c e . This i s r e f l e c t e d in the large MBE. The average error determined f o r the l i n e (N-S) d i r e c t i o n i s greater than the separation distance calculated f o r some of the stations i n the network (see Table 2.1). In the worst case (when i n t e r p o l a t i o n and l i n e a r i t y errors are additive) the true position of a landmark w i l l be o f f by up to 10 km. The errors in the calculated radiation under p a r t l y cloudy and overcast conditions due to errors i n earth-location of the s a t e l l i t e imagery could be greater when the landmark used to earth-locate the imagery l i e s to the East of the study area rather than to the North or South. 2.9 Merging of the S a t e l l i t e and Solar Radiation Data Sets The two data sets used i n t h i s study are based on d i f f e r e n t time scales. Therefore to r e l a t e the measured solar radiation data to that estimated from s a t e l l i t e data i t i s necessary to merge the two data sets. The measured r a d i a t i o n data i s an hourly integrated value and the s a t e l l i t e data i s e s s e n t i a l l y an instantaneous measurement taken every half hour. In making the data sets compatible, i t was decided f o r two reasons to f i t the s a t e l l i t e data to the radiation data. F i r s t l y , the majority of solar r a d i a t i o n studies use hourly or d a i l y q u a n t i t i e s , while instantaneous estimates are rare. Secondly, the s a t e l l i t e data are not s t r i c t l y instantaneous due to the scan time of the sensor (they are instantaneous f o r a given l o c a t i o n ) . However, 49 t h i s f a c t i s never e x p l i c i t l y taken into account. Thus we developed a simple technique which uses the most appropriate s a t e l l i t e images to represent conditions through the hour over which the solar radiation measurement i s integrated. Since the s a t e l l i t e images are taken every ha l f hour, with a complete data set a maximum of three images w i l l be used. The s a t e l l i t e data for each image i s then weighted with respect to time and to the amount of e x t r a t e r r e s t r i a l r a d i a t i o n . The time weighting i s best described by reference to Figure 2.3. Images two, three and four would be chosen to represent conditions for the hour between 11:00 and 12:00. Of the s i x t y minute period, image two would receive a weighting of 24/60, image three, 30/60 and image four 6/60. Use of the e x t r a t e r r e s t r i a l weighting places greater importance on the images closer to s o l a r noon. The r a d i a t i o n weighting is based on a c a l c u l a t i o n of the e x t r a t e r r e s t r i a l r a d i a t i o n at the time of each image. This l a t t e r weighting scheme i s of secondary importance and modifies the weight based on time. If an e n t i r e s a t e l l i t e image i s unavailable or there are missing data over the study area, images adjacent to the missing image time are assigned appropriately increased weights. For example i f image two (see Figure 2.3) was missing then image three would be used to represent conditions back to 11:09 while image one would be used to represent conditions between 11:00 and 11:09. However, there are l i m i t a t i o n s to t h i s correction procedure. If an image has to represent an i n t e r v a l greater than one hour the hourly period in which t h i s image i s being used to represent r a d i a t i o n conditions, i s c l a s s i f i e d as a missing data period. 50 Satellite image 10=39 1 11=09 11=39 112=09 12 = 39 time I K 1 »+*-! ^ »+« 1 — | >+« 1 x Image number' ' ' Time I I I 10=00 11=00 12=00 13=00 F igure 2 . 3 : Example o f S a t e l l i t e Image Weight ing Scheme f o r Merging S a t e l l i t e Data to Hour ly In tegra ted Rad ia t i on Data (Arrows Ind i ca te Time I n te r va l Each Image Represents) 51 2.10 V e r i f i c a t i o n S t a t i s t i c s Evaluations of the models outlined i n Section 1.1 and further developed in Chapter Three, w i l l be based on v e r i f i c a t i o n s t a t i s t i c s generated from a comparison of modelled and measured values. The s t a t i s t i c s to be used include the Mean Bias Error, which describes the long term performance; the Root Mean Square Error, which describes the short term performance and the c o r r e l a t i o n c o e f f i c i e n t (r) which i s a measure of the degree of association between the modelled and measured values. A s i g n i f i c a n t portion of our assessments w i l l be based on the average s t a t i s t i c s calculated f o r a combination of the 12 network stations. However, when the average s t a t i s t i c s do not t r u l y represent the behaviour of the models at s p e c i f i c stations within the network i t w i l l be necessary to consider the s t a t i s t i c s f o r the individual s t a t i o n . This i s e s p e c i a l l y important when considering the Mean Bias Error since a small average value may in f a c t r e s u l t from the c a n c e l l a t i of p o s i t i v e and negative values occurring at d i f f e r e n t stations. Since the average RMS error i s not influenced by such cancellations i t w i l l often be more representative of the average behaviour across the network Following the work of Hay and Wardle (1982), Mean Bias Errors of within ±2.5% (see Section 2.1) are considered to represent e f f e c t i v e equality between modelled and measured values. 52 CHAPTER THREE 3. The Models 0 3.1 The Tarpley Model - Overview The s t a t i s t i c a l model developed by Tarpley (1979), represents part of the outcome of an experiment conducted in the summer of 1977 over the U.S. Great Plains. The study area spanned ten (10) degrees of longitude (95°W - 105°W) and 20 degrees l a t i t u d e (29°N - 49°N), an areal coverage of approximately 1,560,000 sq km (Tarpley et a l . 1978). The main objectives of the experiment were: 1) To determine how accurately i n s o l a t i o n could be inferred from s a t e l l i t e and supplementary data ( i . e . p r e c i p i t a b l e water and surface pressure) and 2) To assess the problem of deriving i n s o l a t i o n in a "timely manner" using operationally available data. Data from three sources were u t i l i z e d in the development of the model: hourly GOES data measured in the 0.55 - 0.75 ym region of the spectrum at a resolution of approximately eight (8) km; measured radi a t i o n data from a 22 s t a t i o n pyranometer network, and surface data ( p r e c i p i t a b l e water and surface pressure) obtained from the National Meteorological Center (NMC) data f i l e . The modelling procedure required the development of a c l e a r ( i . e . minimum) brightness parameterization, the determination of cloud amounts and other image related quantities (mean target brightness and mean cloud brightness), the determination of atmospheric transmittance and the formulation of the f i n a l regression equations. 5 3 The c l e a r brightness parameterization i s represented by the equation: B = a + b cos e + c sin e cos $ + d sin e cos 2 <|> ( 3 . 1 . 1 ) B = predicted minimum brightness 6 = local solar zenith angle <}> = the azimuth angle between sun and s a t e l l i t e a, b, c, d = regression c o e f i c i e n t s (the predicted minimum brightness r e l a t i o n s h i p i s developed in Section 4 . 1 . 1 ) . The second term on the rig h t of equation 3 . 1 . 1 accounts for the changing incident flu x . The other two terms are introduced to account for changes i n target brightness due to shadowing at the surface and anisotropic s c a t t e r i n g . Cloud amount i s determined via a two-threshold method proposed by Shenk and Salomonson ( 1 9 7 2 ) . Three categories are distinguished by this method; c l e a r , 5 0 % cloud covered (partly cloudy) and 1 0 0 % cloud covered (cloudy). The cloud f r a c t i o n (n) is computed via the expression: 0 . 5 N , + N - N 0 + 2N-n = - * -± - ( 3 . 1 . 2 ) N 1 + N 2 + N 3 2 N n = cloud f r a c t i o n N = total number of pixels in target area N-j, N ^ J N^. = number of pixels in c l e a r , p a r t l y cloudy and cloudy categories respectively The c l e a r / p a r t l y cloud threshold (T^) i s the predicted c l e a r brightness (Equation 3 . 1 . 1 ) plus three counts (on a 6 - b i t scale) and any pixel 54 value < T.| i s considered to be c l e a r . The pa r t l y cloudy/cloudy threshold (T 2) i s the predicted c l e a r brightness plus f i v e counts. Any pixel value > T-j but < T 2 i s considered to be pa r t l y cloudy. Pixels > T^ are classed as cloudy. In a l l cases the counts are measures of r e l a t i v e brightness on a 6-bit scale ranging from 0 - 6 3 counts. Atmospheric transmittance (ty) i s computed by the following equation: iLr = ty • ty • ty (3.1.3) $~ = transmission due to water vapour s c a t t e r i n g , us ua r water vapour absorption and Rayleigh scattering respectively. * u s = 1 - 0.00225 um (3.1.4) (Davies et a l . , 1975) • 0.00933 m2 (Davies et a l . , 1975) * = 0.972 - 0.0826 m + m2 (3.1.5) * u a = 1 - 0.077 (um) 0' 3 (3.1.6) ua (McDonald, I960) u = p r e c i p i t a b l e water in (cm) m = o p t i c a l a i r mass = e " ^ z / 8 2 4 3 V c o s e (3.1.7) z = st a t i o n elevation (m) The broad band transmissions (equations 3.1.4 - 3.1.6) are assumed representative of the transmissions i n the v i s i b l e wavelengths. Three d i s t i n c t regression equations are developed corresponding to the three cloud categories (see Section 3.1.1 f o r the physical/ empirical bases of these equations) as follows: 55 I s = a ] + b 1 cos e + c ] $ + + e ] ( ^ ) 2 (3.1.8) n < 0.4 c l e a r I s = a 2 + b 2 cos e + c 2n ( ^ - ) 2 (3,1.9) 0.4 < n < 1.0 pa r t l y cloudy I = a 3 + b 3 cos e + c 3 ( ^ - ) 2 (3.1.10) n = 1.0 cloudy I = hourly surface irradiance s I = mean target brightness B = predicted c l e a r brightness Ic l d = mean cloud brightness B Q = normalized c l e a r brightness [= B (e = 45°, <j> = 105°)] a n ' ^n' c n ' ^ n' e n = r e 9 r e s s i o n c o e f f i c i e n t s n = 1, 3 These equations are used to calculate hourly values of surface irradiance. For the independent data analysed by Tarpley estimates are within 10% of the measured radiation f o r d a i l y i n s o l a t i o n and within 5% on clear days. 3.1.1 Physical/Empirical Basis Any attempts to estimate the incoming radiation at the surface must consider the r a d i a t i v e t r a n s f e r problem; that i s , the processes of absorption and scattering by the various atmospheric constituents. In the context of models which use s a t e l l i t e data as a major input, two streams of interactions are important: 1) radiation as i t passes through the atmosphere to the surface; 2) radiation as i t i s scattered by the atmosphere and surface to the s a t e l l i t e sensor. 56 Although ultimately s t a t i s t i c a l in nature, Tarpley's (1979) model has a sound physical base. The choice of the regression variables displays an awareness of the importance of the atmospheric processes and t h e i r influence on the r a d i a t i o n . The'model follows the arguments on the conservation of energy in an atmospheric column proposed by Hanson (1971) where by energy incident at the top of the atmosphere (FQ) i s divided into three components: F 0 = I R + I A + I g f 3 ' 1 - 1 - 1 ' Ip, 1^, I = energy r e f l e c t e d to space, absorbed in the atmosphere and absorbed by the surface, respectively. In the narrow band formulation of 3.1.1.1, I R i s measured at the s a t e l l i t e . Manipulation of the Hanson (1971), equation y i e l d s a form that provides the basis f o r the f i n a l regression equations: I g = I s (1 - o) (3.1.1.2) and i = _ L _ ( F . I - I ) (3.1.1.3) 1 - a I = irradiance at the surface s a = surface albedo An important feature of empirical modelling i s the need to i d e n t i f y s uitable surrogates f o r the physical processes of a given system. This requirement i s met in the Tarpley (1979) model. The regression variables in equations (3.1.8 - 3.1.10) can be related to the physical processes described by equation (3.1.1.1). The object i s to emp i r i c a l l y r e l a t e the narrow band measurement of the s a t e l l i t e to the broad band 57 formulations of the short wave radiation balance. The cosine of the zenith angle i s proportional to IQ, while atmospheric transmittance, cloud amount, mean target brightness and mean cloud brightness are the surrogates that account for scattering and absorption described by the I R and 1^ terms of equation 3.1.1. 3.2 The Gautier Model - Overview The model developed by Gautier et a l . (1980), represents a more 'physical' approach to the question of in s o l a t i o n estimation from s a t e l l i t e data. Within the model's framework an attempt is made to e x p l i c i t l y handle the physical processes operative in the atmosphere. This e n t a i l s the s p e c i f i c a t i o n of absorption, r e f l e c t i o n and scattering quantities. The model u t i l i z e s the highest s p a t i a l resolution of the GOES data (approximately 2 km at 45° l a t i t u d e ) along with images from every hour i n an attempt to address the issue of cloud v a r i a b i l i t y . There are two components to the Gautier et a l . (1980), model; a 'clear' a i r formulation and a cloudy atmosphere formulation. The 'clear' a i r formulation requires a knowledge of the flux measured at the s a t e l l i t e . This flux (SW+) i s described by the following equation SW+ = F Q 3 + F Q (1 - 3) [1 - a ( U l ) ] [1 - a ( u 2 ) ] (1 - 3-, ) • a (3.2.1) a = albedo of the surface FQ = instantaneous shortwave flux at the top of the atmosphere [= I Q COS e (W m" 2)] 3 , 3 - 1 = r e f l e c t i o n c o e f f i c i e n t s for beam and d i f f u s e r a d i a t i o n r e spectively. 58 a(u-j), a(u 2) = absorption c o e f f i c i e n t s for slant water vapour paths (sun and s a t e l l i t e r e s p e c t i v e l y ) . Values are assigned for each term in the equation except the surface albedo (a). From 3.2.1: sw+ - F n 3 a = (3.2.2) F Q (1 - 6) [1 - a ( U l ) ] [1 - a ( u 2 ) ] (1 - 3-,) The irradiance at the surface (I ) i s then defined by I s = F Q (1 - 3) [1 - a ( U l ) ] (1 + B 1 a) (3.2.3) a = surface albedo (the remaining terms have been defined e a r l i e r ) . The cloudy atmosphere model retains the clear a i r formulation with the added ef f e c t s of clouds. The clouds are assumed to occur in a d i s c r e t e layer. Scattering and absorption c o e f f i c i e n t s are assigned to above, below and within cloud layers. Cloud absorption is s p e c i f i e d as a l i n e a r function of cloud brightness, ranging from 0.0 for no clouds to 0.2 for the deepest clouds (see Section 5.1.2.2 for the c a l c u l a t i o n of cloud absorption). Cloud amount i s assumed to be a continuous variable rather than the d i s c r e t e form as in the case of Tarpley (1979). The cloudy atmosphere i s represented by a s i m i l a r s u i t e of equations as the c l e a r a i r model. SW+ = F Q 3 + F Q (1 - B) [1 - a ( U ] ) t ] (1 - B ]) • Ac [1 - a ( u 2 ) t ] + F Q (1 - 3) [1 - a ( U l ) t ] • (1 - A c ) 2 [1 - a ( U l ) b ] a (1 -• [1 - a ( u 2 ) t ] (1 - abs) 2 [1 - a ( u 2 ) b ] (3.2.4) 59 I s = F 0 (1 - B) Ll - a ( U l ) t ] (1 - Ac) (1 - abs) [1 - a ^ ) b] (3.2.5) I = irradiance at the surface under cloudy conditions s c Ac = cloud albedo abs = cloud absorption a ( ) , a{u^)^ - absorption c o e f f i c i e n t s above cloud level For the sun's path and s a t e l l i t e path respectively a(u^ ), a ^ ) ^ = b absorption below cloud level for sun and s a t e l l i t e path respectively (other terms in the equations are defined e a r l i e r ) . A c r u c i a l aspect of the cloudy atmosphere model i s the determination of a threshold separating c l e a r from cloudy observations. This threshold requires a knowledge of the minimum brightness of the surface determined under clear sky conditions. The methodology used to define this threshold is outlined in Section 5.1.1.4. In Gautier's study (Gautier et a l . , 1980) estimates obtained from the model were compared to pyranometer measurements. For cle a r days they were within 5% of the mean measured i n s o l a t i o n decreasing to 14% and 15% for cloudy and completely over cast days, respectively. For a l l days combined (clear, cloudy and overcast) estimates were within 9% of the mean measured i n s o l a t i o n . The model was shown to reproduce quite c l o s e l y the hourly variations in incident r a d i a t i o n . (Gautier et a l . , 1980). 3.2.1 Physical/Empirical Basis In a manner s i m i l a r to Tarpley (1979), Hanson's (1971), 60 arguments on the conservation of energy in an atmospheric column provide the physical basis for the model developed by Gautier et a l . (1980). If the r a d i a t i v e transfer i s viewed in terms of the conservation of energy arguments proposed by Hanson (1971) then a l i n k , however tenuous, can be established between the present model and the ra d i a t i v e transfer equation. As proposed by Kondrat'yev (1969), the ra d i a t i v e transfer equation attempts to account for sc a t t e r i n g and absorption over a l l possible angles (zenith and azimuth), wavelengths and layers in the atmosphere. Therefore to determine the irradiance at the surface the equation must f i r s t be solved and then integrated over a l l angles, wavelengths and. layers in the atmosphere. The immensity of this requirement has been the impetus for various approximations to the fundamental r a d i a t i v e transfer equation. The most common of these allow for the decomposition of the radiation into d i r e c t beam and di f f u s e components and the use of broad band approximations to describe the scattering and absorption behaviour of the various atmospheric components (Davies et a l . , 1975; Davies and Hay, 1980; Atwater and Brown, 1974). The model of Gautier et a l . (1980), represents a further departure from the r a d i a t i v e transfer equation. Equation 3.2.3 describes global irradiance at the surface under clear skies. The three terms in the equation account for transmission a f t e r scattering (1 - 3), transmission a f t e r absorption [1 - a ( u ^ ) ] , and a d i f f u s e radiation component due to multiple r e f l e c t i o n ( a B ^ ) . A l l scattering in the atmosphere i s assumed to be due to dry a i r molecules (Rayleigh scattering) 61 and a l l absorption i s assumed to be due to water vapour. For a c l e a r atmosphere the water vapour assumption i s quite reasonable, but scattering under clear skies should include the effects of aerosols (see Section 2.4). Moreover, the d i f f u s e radiation components are either i m p l i c i t l y included or completely ignored except in the case of the multiple r e f l e c t i o n term. At best the physically-based model i s an over s i m p l i f i e d representation of the r a d i a t i v e transfer equation. Although the framework i s that of a physical model ( i . e . requiring a s p e c i f i c a t i o n of scattering and absorption quantities) the elements within that framework are s o l e l y dependent on empirical determinations. The scattering c o e f f i c i e n t s (g and B^) are obtained from the work of Coulson (1959) , who relates albedo to sun elevation for a Rayleigh scattering atmosphere. The water vapour absorption term is ^determined via a n a l y t i c a l expressions derived by Paltridge (1973) based on the e a r l i e r work of Yamamoto (1962) which describes water vapour absorption as a function of atmospheric p r e c i p i t a b l e water path length. Precipitable water (the only surface data used) i s determined using Smith's (1966) empirical formulation based on the surface dewpoint temperature. The i n c l u s i o n of the effects of clouds in the model's formulation increases the degree of empiricism. Quantification of the effects of clouds follows the work of Monteith (1962). Cloud i s assumed to occur in one d i s c r e t e layer and only the bulk properties of absorption, transmission and scattering are of i n t e r e s t . These are determined empirically i n the Gautier et a l . (1980) model (see Chapter Five). The transformation of the narrow band measurement of 62 the s a t e l l i t e to the broad band irradiance required in the model is accomplished via the c a l i b r a t i o n procedure described in Section 5.1.1.1. 3.3 The Hay - Hanson Model - Overview The model developed by Hay and Hanson (1978), i s by far the simplest of the three models to be considered. The model formed part of a study undertaken to map the d i s t r i b u t i o n of shortwave radiation incident at the sea surface for the GATE (1974). (GLOBAL ATMOSPHERIC RESEARCH PROGRAM ATLANTIC TROPICAL EXPERIMENT), B scale study area. The objective of the study was to derive a r e l a t i o n s h i p between hourly s a t e l l i t e observations from SMSI in the v i s i b l e region of the spectrum (0.55 - 0.75 p m ) , and solar radiation measured at the surface. A simple l i n e a r r e l ationship was developed between the v i s i b l e radiance measured by the s a t e l l i t e sensor and atmospheric short-wave transmi ttance. T = a - bSR (3.3.1) r v T^ = atmospheric transmittance = I / IQ cos 9 SR = normalized s a t e l l i t e measured brightness a, b = regression c o e f f i c i e n t s (intercept and slope respectively) To determine the irradiance at the surface ( l s ) equation 3.3.1 can be rewritten: I / I Q COS 9 = a - bSR (3.3.2) which y i e l d s I = I Q COS 6 [a - bSR] (3.3.3) 63 IQ = solar constant ( 1 3 5 3 W m'11) e = local solar zenith angle V e r i f i c a t i o n of the Hay and Hanson ( 1 9 7 8 ) model showed that on an hourly basis the incident radiation could be estimated to within ± 2 2 % of the measured radiation with a marked improvement to ± 8 % for dai l y time periods. 3 .3 .1 Physical/Empirical Basis The model by Hay and Hanson ( 1 9 7 8 ) , i l l u s t r a t e s the 'black box' approach to estimating incident radiation at the surface. According to Haggett and Chorley ( 1 9 6 7 ) , models of this i l k are s t r i c t l y concerned with the input and output variables of the system with l i t t l e emphasis on the internal status variables. The present model maintains no e x p l i c i t l i n k s with the rad i a t i v e transfer equation. However, the choice of the variables (atmospheric transmittance, s a t e l l i t e measured radiance) and the r e l a t i o n s h i p derived between them indicate i m p l i c i t consideration of r a d i a t i v e transfer. The model therefore has a sound empirical basis. Both atmospheric transmittance and s a t e l l i t e measured radiance are i n d i c a t i v e of the 'state' of the atmosphere (also the state of the surface in the case of the radiance measurement). Atmospheric transmittance changes as the various constituents of the atmosphere change. An increase in the concentrations of aerosols and dust p a r t i c l e s , water vapour and increasing cloud cover leads to increased scattering and absorption in the atmosphere. This results in a decrease in the atmospheric 64 transmittance, and in the case of s c a t t e r i n g , to an increase in the s a t e l l i t e measured radiance. Increased absorption w i l l reduce the radiation measured at the surface and in some instances (e.g. when absorption is large) w i l l reduce the radiance measured at the s a t e l l i t e . An inverse r e l a t i o n s h i p between atmospheric transmittance and s a t e l l i t e measured radiance should capture the variations in scattering and absorption which influence the irradiance measured at the surface but would be less r e l i a b l e when absorption i s s i g n i f i c a n t . The rel a t i o n s h i p breaks down under conditions of high surface albedo (e.g. a snow-covered surface). The high albedo w i l l increase the brightness measured by the s a t e l l i t e (except under heavy overcast) and increase the radiation measured at the surface (through the e f f e c t of multiple r e f l e c t i o n ) . This would lead to s i g n i f i c a n t under estimation of the radiation at the surface p a r t i c u l a r l y under clear skies (a s i m i l a r s i t u a t i o n would occur for both the Tarpley, 1979 and Gautier et a l . ; 1980 models). The transformation of the narrow band measurement of the s a t e l l i t e to f i t the broad band irradiances represented by the atmospheric transmittance, is accomplished via the c a l i b r a t i o n procedure described in Section 6.1.1. 65 CHAPTER FOUR 4. Application of the Tarpley Model 4.1 Implementation An important requirement of this study is that the models be applied as developed by the o r i g i n a l researchers thereby involving no major modifications. This ensures that any differences in results cannot be att r i b u t e d to changes i n the model i t s e l f but must r e f l e c t some other conditions (e.g. a d i f f e r e n t environment). We are interested here in the general a p p l i c a b i l i t y of a given model. Before applying the Tarpley (1979), model to the study area various factors have to be considered; 1) difference in sp a t i a l scale (the model was developed over a 1.5 x 10^ sq km area, while the present study area spans 3150 sq km). 2) differences in brightness scale (a 6-b i t scale is used in the Tarpley (1979), study with a count range from 0 - 6 3 while an e i g h t - b i t scale i s used in the present study with a count range from 0 - 255). Where relevant, values must be mu l t i p l i e d by a factor of (4) to make the two scales compatible. 3) the present study has a more comprehensive radiation data base which permits a more accurate assessment of the radi a t i o n c h a r a c t e r i s t i c s of any day used in this study (see Section 2.1 for further discussion of the radiation data base). The procedures to implement the model are grouped as follows: 1) Determination of a Minimum Brightness Parameterization; 2) Data s t r a t i f i c a t i o n and generation of image related inputs; 3) Atmospheric 66 Transmittance; 4) Regression Equations to estimate i n s o l a t i o n using Tarpley's (1979) c o e f f i c i e n t s . Variables in the regression equations are hourly measured r a d i a t i o n (the dependent variable) the cosine of the loca l solar zenith angle, atmospheric transmittance, mean cloud amount, mean cloud brightness, mean target brightness, predicted minimum brightness and normalized minimum brightness (the independent v a r i a b l e s ) . 4.1.1 Minimum Brightness Parameterization The minimum brightness of the surface can be determined using equation (3.1.1). B i s the predicted minimum brightness of the surface and depends upon functions of the solar zenith angle and the s u n - s a t e l l i t e azimuth. The brightness of a given surface i s known to vary with changing incident f l u x wnich i s proportional to the cosine of the zenith angle. The signal received by the s a t e l l i t e sensor i s influenced by changing incident f l u x , the scattering properties of the surface ( i s o t r o p i c / a n i s o t r o p i c ) and the e f f e c t s of shadowing. Since most natural surfaces behave as anisotropic scatterers at the wavelengths of i n t e r e s t here (Eaton and Dirmhirn, 1979), the azimuth angle between the sun and s a t e l l i t e i s therefore an important determinant of the signal received by the s a t e l l i t e sensor. Equation (3.1.1) attempts to parameterize the influence of these factors (changing incident f l u x , shadowing and anisotropic scattering) on the minimum brightness of the surface. The important variables i n Equation (3.1.1) are the cosine of the zenith angle and the s u n - s a t e l l i t e azimuth. Determination of 67 these depends upon the l a t i t u d e and longitude of a given l o c a t i o n . We noted e a r l i e r (Section 3.1) that, since Tarpley's study area spanned 20° of la t i t u d e and 10° of longitude, these variables would show much v a r i a t i o n . The present study spans 23 minutes of l a t i t u d e (49°N -49° 23'N) and 58 minutes of longitude (122° 17' 18"W - 123° 15'W). Because of the smaller s p a t i a l scale the l a t i t u d e and longitude coordinates are assumed to be constant across the study area. Solar zenith angle and s a t e l l i t e azimuth calculations are carried out for the P i t t Meadows loca t i o n and used to represent conditions at the other loc a t i o n s . At 49 degrees 13 minutes North and 122 degrees 42 minutes West, P i t t Meadows represents approximately the mid-point of the study area. (See Figure 1.1 and Table 4.2 for the locations of stations and t h e i r l a t i t u d e and longitude coordinates). The cosine of the sol a r zenith angle (e) is determined from Se l l e r s (1965): cos e = s i n l a t • s i n 6 cos l a t • cos 6 cos h (4.1.1) l a t = st a t i o n l a t i t u d e d = de c l i n a t i o n of the sun in degrees h = hour angle of the sun h = 15 |12 - LAT (4.1.2) LAT = local apparent time LAT = LST + ET + 4(L g - L) LST = local standard time (4.1.3) ET = equation of time in minutes s t a t i o n longitude 68 \ L = longitude of standard meridian <5 = 0.006918 - 0.399912 cos g + 0.070257 s i n 5 - 0.006758 cos 2? + 0.000907 s i n 2? - 0.002697 cos 3£ + 0.001480 s i n 3? (4.1.4) Spencer (1971) £ = 27id/365 (4.1.5) d = the day number which runs from 0 (Jan. 1) to 364 (Dec. 31). The s u n - s a t e l l i t e azimuth depends upon both the s a t e l l i t e ' s azimuth from South and the sun's azimuth from South. The s a t e l l i t e azimuth from South is assumed to be a constant, and i s determined from a knowledge of the longitude of the s a t e l l i t e (135° West), the station's longitude (122° 42'W) and the approximate distance of the s a t e l l i t e from the earth's centre (approximately 42,167 km). Using the relationships given by Equation A.2.1 ( i n Appendix 2) the azimuth i s calculated to be 16.1 degrees. The sun's azimuth from South (az) is given by ( S e l l e r s , 1965): cos (az) = s i n l a t • cos 6 - sin 6 ( 4 J > 6 ) cos l a t • sin e The azimuth i s considered to be negative when the sun i s to the East. The s u n - s a t e l l i t e azimuth (<}>) is then given by: <t> = |16.1 - az| (4.1.7) 4> = s u n - s a t e l l i t e azimuth (Since the cosine of the s u n - s a t e l l i t e azimuth i s the quantity of in t e r e s t only the absolute value of the azimuth i s considered to be important) 69 To check the v a l i d i t y of the assumptions of a constant s a t e l l i t e azimuth and a constant s o l a r zenith angle (at a given time) across the network, values of the two parameters were determined for the end points of the study area. The s a t e l l i t e azimuth is calculated for the U.B.C. Climate Station and Abbotsford Library. The respective values are 15.3 degrees and 16.6 degrees differences of 0.80 and 0.50 degrees from the central value. This results in an i n s i g n i f i c a n t change of .003 in the cosine of the s a t e l l i t e azimuth. The solar zenith angles for s o l a r noon on J u l i a n day 196 1979 (see Table 2.2 for a l i s t of days used in the study) were calculated f or Grouse Mountain, the Tsawwassen Ferry Terminal and P i t t Meadows. Assuming a value of 1353 (W m ) for the solar constant we found that the cosine of the zenith angle changes by .003 at most and the incoming flux by four (4) W m or 0.3% of the value at P i t t Meadows. This shows that our assumptions of constant s a t e l l i t e azimuth angle and constant s o l a r zenith angle across the study area would not a f f e c t the r e s u l t s obtained. We developed our minimum brightness parameterization from data for three c l e a r days in 1979. Julian days 196, 257 and 304. The days are chosen at d i f f e r e n t times of the year to incorporate a wide range of solar zenith angles and in order to sample i f possible, changing atmospheric and surface conditions (e.g. pollutants, water vapour content, vegetation cover) from one season to the next. Such a choice should r e s u l t in a more representative minimum brightness parameterization. The independent variable in the regression i s the average brightness of a 5 x 5 pixel array centered on a given station (choice of the array s i z e i s based on the accuracy of our earth location 70 routine Section 2.7). A l l available images between loca l sunrise and sunset f o r the three days are used. The data (average brightness along with the variables in Equation 3.1.1) are analysed using a Triangular Regression Package (TRP) a v a i l a b l e at the University of B r i t i s h Columbia Computer Science Centre. Individual" regression equations are developed f o r each of twelve stations. The c o e f f i c i e n t s of determination, Standard Errors of the estimate and the number of observations are l i s t e d in Table 4.1. The regression c o e f f i c i e n t s and the normalized predicted minimum brightness [Bg = B( e = 4 5 ° <j> = 105°)] f o r each station are also presented in that table. The normalized predicted minimum brightness (on an 8-bit scale) e x h i b i t a wide range of v a r i a t i o n , from 63 counts at the Tsawwassen Ferry Terminal to 78 counts at Abbotsford Library. This range of v a r i a t i o n highlights the v a r i a b i l i t y in minimum brightness across the network due to both atmospheric and surface factors (sea/land, urban/rural contrasts). 4.1.2 Data S t r a t i f i c a t i o n and Image-Related Inputs As noted e a r l i e r (Section 3.1), data f o r analysis in the Tarpley (1979) model are separated into three categories based on a two-threshold approximation proposed by Shenk and Salomonson (1972). Based on an 8-bit brightness scale, T^ (the c l e a r / p a r t l y cloudy threshold) i s B plus 12 counts, while 1^ (the p a r t l y cloudy/cloudy threshold) i s B plus 20 counts. The i n i t i a l data set for the generation of the image-related inputs consists of three days; J u l i a n days 183/80, 197/80 and 239/80. These days are chosen because they represent c l e a r , cloudy and p a r t l y Table 4.1 Minimum Brightness Regression Coefficients and Other S t a t i s t i c s  for the 12 Stations in the Mesoscale Network Based on a 5 x 5  Pix e l Array. Units: Counts; Scale: 8-bit. STATION N R2 SEE a b c d B Q NAME GROUSE 67 0.91 4.39 37.37 40.29 7.48 15.12 65.20 MOUNTAIN NORTH 67 0.93 4.25 39.09 43.35 8.60 12.87 68.78 MOUNT BC HYDRO 67 0.91 4. 82 41. 23 42. 44 9. 85 10. 82 69 .95 VANCOUVER 68 0.95 3.82 40.16 52.74 9.37 8.99 76.16 INT'L AIRPORT TSAWWASSEN 68 0.81 4.03 42.32 29.59 2.77 2.57 62.86 FERRY TERMINAL PITT MEADOWS 68 0.94 3.85 40.13 50.87 7.90 9.07 75.08 AIRPORT HABITAT 68 0.93 4.51 39.55 52.29 6.00 12.64 76.02 APARTMENTS Table 4.1 (continued) STATION N R2 SEE a b c d B o NAME ABBOTSFORD LIBRARY 68 0.93 4. 17 41.44 49 . 78 5.32 12. 34 76.23 ABBOTSFORD AIRPORT 68 0.94 3.93 41. 04 52.34 5.52 10.91 77.55 LANGLEY 68 0.95 3.84 39. 26 51.30 8.14 10. 16 74.52 LANGARA 68 0.94 3.93 40.32 50. 45 8.65 10. 16 74. 89 U.B.C. CLIMATE STATION 68 0. 89 4.52 39. 80 38. 18 7. 79 7. 16 65.71 73 cloudy conditions respectively. Computer routines were developed to process the useable images av a i l a b l e f o r these days. The c l a s s i f i c a t i o n of images into the various categories proceeds as follows: For a given image, each pixel of a 5 x 5 array centered on a given s t a t i o n i s compared to the two thresholds. If the pixel value i s < T^ i t i s recorded as a c l e a r p i x e l , i f the pixel value i s > T-| but < T^ i t i s recorded as a partly cloudy p i x e l , i f the pixel value i s > .1^ i t i s recorded as a cloudy p i x e l . Cloud amount, which determines whether the e n t i r e image i s considered c l e a r , p a r t l y cloudy or cloudy i s determined from Equation (3.1.2). When n < 0.4 the image is c l e a r , when 0.4 < n < 1.0 the image i s p a r t l y cloudy and for n = 1.0 the image i s cloudy. From the above procedure the cloud amount emerges as an input variable in the f i n a l regression equation. The mean target brightness, another image related input, i s simply the mean brightness of the 5 x 5 pixel array centered on a given station . The predicted brightness i s determined f o r each image based on the equations developed in Section 4.1.1. The mean cloud brightness i s the f i n a l image-related input derived (cloud brightness i s used as an indica t o r of cloud thickness). Mean cloud brightness i s computed by averaging the brightness values of a l l pixels in the 5 x 5 array that are brighter than 1^ ( t n e Partly cloudy/cloudy threshold). . 4.1.3 Atmospheric Transmittance Clear a i r transmittance i s the only non-image-derived independent variable used in the model. Following Tarpley (1979), i t i s determined using a series of Equations (3.1.4 - 3.1.7) outlined i n Section 3.1. P r e c i p i t a b l e water (u) and o p t i c a l a i r mass (m) are the 74 two important quantities to be determined in the c a l c u l a t i o n of the atmospheric transmittance. Both p r e c i p i t a b l e water (u) and opti c a l a i r mass (m), are known to decrease with height (see Section 2.4 f o r further discussion on p r e c i p i t a b l e water).- The e f f e c t on the atmospheric transmittance (ty) of a decrease in op t i c a l a i r mass (m) with height, i s assessed as follows: For Grouse Mountain, which i s 1128 m above sea-level (Table 4.2), and given Equation 3.1.7 with a value of .7015 fo r cos e (corresponds to a zenith angle of 45.5° f o r a s a t e l l i t e observation at 09:45 on day 196/79), m i s determined to be 1.24 at Grouse Mountain, a difference of .19 from a sea-level value of 1.43. Under the above conditions and assuming a value of u = 1.3 cm atmospheric transmittance (ty) increases by 2.0% above the sea-level value. By way of comparison, o p t i c a l a i r mass (m) changes a maximum of .003 between the other eleven (11) stations and t h i s has l i t t l e or no e f f e c t on the transmittance (ty). We conclude from the above that a height correction in op t i c a l a i r mass (m) i s only necessary f o r Grouse Mountain. A l l other stations are assumed to be at sea l e v e l . At large zenith angles the r e l a t i o n s h i p in (m) breaks down due to r e f r a c t i o n and other considerations. To improve t h i s s i t u a t i o n the equation f o r (m) used by Tarpley (1979) ( i . e . Equation 3.1.7) i s modified following McDonald (1960) and Kasten (1966) and becomes; m = e " [ z / 8 2 4 3 ] / [cos e + 0.15 / [93.885 - e ] 1 ' 2 5 3 ] (4.1.3.1) This modified form of the opti c a l a i r mass (m) i s applied to a l l zenith angles. Table 4.2 Measurement S t a t i o n s In Mesoscale Network STATION NAME LATITUDE LONGITUDE ELEVATION (m) ABBREVIATION 1 GROUSE MOUNTAIN 2 NORTH MOUNT 3 BC HYDRO 4 VANCOUVER INT'L AIRPORT 49° 23' 00" N 49° 19' 13" N 49° 16' 54" N 49° 11' 00" N 5 TSAWWASSEN 49° 00' 00" N FERRY TERMINAL 6 PITT MEADOWS 49° 13' 00" N AIRPORT 7 HABITAT APARTMENTS 8 ABBOTSFORD LIBRARY 49° 08' 11" N 49° 02' 55" N 123° 05' 00" W 123° 04' 12" W 123° 07' 24" W 123° 10' 00" W 123° 08' 00" W 122° 42' 00" W 122° 17' 55" W 122° 17' 30" W 1, 128 114 122 05 03 05 125 60 GRSMT NRTHMT BC HYDRO VANAIR FERRY PITMED MISSHAB ABBLIB Table 4.2 (continued) STATION NAME LATITUDE LONGITUDE ELEVATION ABBREVIATION (m) 9 ABBOTSFORD AIRPORT 49° 01* 00" N 122° 22 ' 00" w 61 ABBAIR 10 LANGLEY 49° 06 • 44" N 122° 38 ' 36" w 11 LANGLEY 11 LANGARA 49° 13' 24" N 123° 06 ' 12" w 69 LANG A 12 U.B.C. CLIMATE STATION 49° 16' 00" N 123° 15' 00" w 93 CLISTN 77 P r e c i p i t a b l e water (u) i s determined via Smith's (1966) empirical formulation based on the dewpoint temperature (Section 2.4). 4.1.4 Regression Equations f o r Estimating Insolation The f i n a l regression equations used to estimate i n s o l a t i o n are presented as Equations (3.1.8 - 3.1.10). I n i t i a l l y , Tarpley's o r i g i n a l c o e f f i c i e n t s (modified to the appropriate units) are used in the application of the model to data from the present study. Table 4.3 contains a l i s t i n g of these c o e f f i c i e n t s . Modelling of i n s o l a t i o n at the surface depends upon the combination of the s a t e l l i t e and solar r a d i a t i o n data sets. These data sets are merged in the manner outlined in Section 2.9. We deviate from Tarpley (1979), at th i s point in that we use as many as three images to represent conditions f o r a given ra d i a t i o n hour whereas Tarpley's o r i g i n a l study i s limited to the use of one image per hour. We expect t h i s change to improve the q u a l i t y of our r e s u l t s since we have more information on the radi a t i o n conditions for a given hour. To compute the ra d i a t i o n f o r any hour the various independent variables are determined. These are combined to y i e l d weighted averages f o r each variables and are subsequently used i n the in s o l a t i o n c a l c u l a t i o n (the weights are applied as outlined in Section 2.9). 4.2 Results The o r i g i n a l Tarpley (1979) model was developed using data from the summer of 1977. To minimize any bias that could be introduced due to seasonal e f f e c t s , we f i r s t applied the model to three days chosen from the summer of 1980. J u l i a n days 183/80, 197/80 and 239/80 78 T a b l e 4.3 O r i g i n a l R e g r e s s i o n C o e f f i c i e n t s f o r t h e T a r p l e y ( 1 9 7 9 ) model  b a s e d on a 6 - b i t b r i g h t n e s s s c a l e . U n i t s : K J m 2 h r ' 1 . COEFFICIENTS CLEAR n<0.4 P/CLOUDY CLOUDY n=1.0 0.4<n<0.9 a -809.54 -400.79 -274.73 * ( - 195.67 ) (-199.30) (-49.80) b 3646.91 3959.34 3672.04 (3722.93) (4047.97) (2187.16) c 1155.10 -319.13 -314.10 (85.98) (-329.30) (-168.80) d -438.90 (151.10) e -266.78 (-90.86) NUMBER OF 5736(638) 2127(183) 822(671) OBSERVATIONS CORRELATION .94(.92) ,77(.89) .70(.64) (R) COEFFICIENT COEFFICIENT .88(.84) .59(.79) .49(.41) (R*R) OF DETERMINATION * Numbers i n p a r e n t h e s e s a r e f o r a r e v i s e d v e r s i o n of t h e T a r p l e y (1979) model b a s e d on a 5x5 a r r a y and an 8 - b i t b r i g h t n e s s s c a l e ( s e e t e x t ) . 79 represent c l e a r , overcast and p a r t l y cloudy skies, respectively. To assess the influence of seasonal e f f e c t s on performance, the model i s also applied to three days chosen from the spring of 1980. J u l i a n days 121/80, 105/80 and 160/80 represent c l e a r , overcast and p a r t l y cloudy skies, respectively. Hourly estimates for the six days are compared to the measured data at the twelve locations in the network. The average s t a t i s t i c s f o r the twelve stations f o r each day are given in Table 4.4 (complete s t a t i s t i c s f o r the individual stations f o r a l l days analysed are given in Appendix A.3). Figures 4.1a - 4.6b g r a p h i c a l l y depict the model and measured rad i a t i o n behaviour at selected locations f or the days analysed. From Table 4.4 we note the following: on average for the clear days (183/80 and 121/80) the model systematically underestimates the measured r a d i a t i o n , the bias being greater for the springtime s i t u a t i o n . Systematic underestimation also describes the average behaviour for the p a r t l y cloudy days (239/80 and 160/80). The small average MBE (-2.79%) on day 160/80 indicates that the behaviour f o r the twelve stations i s that of e f f e c t i v e equality between the estimated and measured radi a t i o n . On average the model overestimates the measured rad i a t i o n on day 105/80 (overcast conditions) and i s i n perfect long term agreement on day 197/80. Figures 4.1a - b and 4.4a show that the underestimation observed f o r the c l e a r sky days occurs mainly at smaller zenith angles ( i . e . under high r a d i a t i o n inputs) and the consistency suggests that the source i s inappropriate regression c o e f f i c i e n t s . An examination of Table A.3.1 (see Appendix A.3) reveals that the average MBE (-4.75%) 80 T a b l e 4.4 H o u r l y A v e r a g e V e r i f i c a t i o n S t a t i s t i c s from t h e A p p l i c a t i o n of  t h e o r i g i n a l T a r p l e y ( 1 9 7 9 ) . model t o a sample of s i x (6) days  r e p r e s e n t i n g v a r i a b l e sky c o v e r c o n d i t i o n s i n s p r i n g and summer  U n i t s KJm-^hr' 1. A r r a y size : 5 x 5 . : DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL SUMMER 183/80 192 -89.34 180.56 -4.75 9.61 .993 1879.7 1790.4 ( c l e a r ) 239/80 132 -132.85 333.81-17.07 42.78 .842 780.3 647.4 ( p / c l o u d y ) 197/80 191 4.30 236.38 1.05 57.67 .708 409.9 414.2 ( o v e r c a s t ) SPRING 121/80 155 -166.26 204.88 -8.33 10.26 .994 1995.9 1829.7 ( c l e a r ) 160/80 168 -18.48 371.70 -2.79 56.07 .868 663.0 644.5 ( p / c l o u d y ) 105/80 142 79.46 264.03 20.07 66.68 .767 395.6 475.4 ( o v e r c a s t ) Figure 4.1a: Observed radiation and calculated radiation from the ori g i n a l Tarpley (1979) model at P i t t Meadows for a clear day (183/80) in summer. Figure 4.1b: Observed radiation and calculated radiation from the or i g i n a l Tarpley (1979) model at Vancouver Ai r p o r t for a clear day (183/80) in summer. o - — VRNR.TR 239 ^ OBSERVED o CRLCULRTED 0 3 G 9 12 .15 18 21 24 LOCRL RPPRRENT TIME Figure 4.2a: Observed radiation and calculated radiation from the o r i g i n a l Tarpley (1979) model at Vancouver A i r p o r t for a partly cloudy day (239/80) in summer. LOCAL RPPRRFNT TIME Figure 4.2b: Observed radiation and calculated radiation from the or i g i n a l Tarpley (1979) model at Abbotsford Library for a partly cloudy day (239/80) in summer. O VRNRJR 197 -* OBSERVED o CRLCULRTED oo cn O 3 G 9 12 15 18 21 24 LOCRL RPPRRENT TIME Figure 4.3a: Observed radiation and calculated r a d i a t i o n from the or i g i n a l Tarpley (1979) model at Vancouver Ai r p o r t for an overcast day (197/80) in summer. cr. ZD o eg L U CJ cn i—i Q cr cr. cr ro OBSERVED o BCHYDRO 197 - _ 0 CRLCULRTED 6 9 12 15 18 LOCRL RPPRRENT TIME 0 0 (Tl 21 24 Figure 4.3b: Observed radiation and calculated radiation from the or i g i n a l Tarpley (1979) model at BC Hydro for an overcast day (197/80) in summer. Figure 4.4a: Observed radiator) and calculated r a d i a t i o n from the or i g i n a l Tarpley (1979) model at P i t t Meadows for a cl e a r day (121/80) in spring. LOCAL APPARENT TIME Figure 4.4b: Observed radiation and calculated r a d i a t i o n from the o r i g i n a l Tarpley (1979) model at Vancouver Ai r p o r t for a clear day (121/80) in spring. OBSERVED CALCULATED LOCAL APPARENT TIME Figure 4.5a: Observed radiation and calculated r a d i a t i o n from the o r i g i n a l Tarpley (1979) model at Vancouver Airport for a partly cloudy day (160/80) in spring. C \ J RBBL.IB * OBSERVED o CRLCULRTED o 12 15 18 21 24 LOCRL RPPRRFNT TIME Figure 4.5b: Observed radiation and calculated r a d i a t i o n from the o r i g i n a l Tarpley (1979) model at Abbotsford Library for a partly cloudy day (160/80) in spring. C \ J o 3 6 9 12 .15 18 21 LOCAL APPARENT TIME * OBSERVED o- 0 CALCULATED VANAIR 24 Figure 4.6a: Observed radiation and calculated radiation from the o r i g i n a l Tarpley (1979) model at Vancouver Airport for an overcast day (105/80) in spring. ro C M _ o o-BCHYDRO 105 _* OBSERVED o CRLCULRTED ro 6 9 12 15 18 21 24 LOCRL RPPRRENT TIME Figure 4.6b: Observed radiation and calculated radiation from the o r i g i n a l Tarpley (1979) model at BC Hydro for an overcast day (105/80) in spring. 93 on day 183/80 does not t r u l y r e f l e c t conditions occurring across the network. Large differences are observed to occur at Grouse Mountain, Tsawwassen Ferry and UBC Climate Station while small values of the MBE (in some cases i n d i c a t i n g e f f e c t i v e e q u a lity between modelled and measured radiation) occur at the i n t e r i o r locations. The smaller bias observed at these i n t e r i o r stations could be attributed to the f a c t that the measured ra d i a t i o n i s generally lower at these locations due to the e f f e c t s of aerosols (Hay, 1982). Since the model by Tarpley (1979) does not e x p l i c i t l y account f o r the e f f e c t s of aerosols there w i l l be no corresponding reduction in the model values at these locations thereby leading to a closer correspondence between modelled and measured values. Larger values of the MBE ( r e l a t i v e ) occur at Grouse Mountain (-10%), Tsawwassen Ferry (-7%) and UBC Climate Station (-6%). The increased bias may be influenced by the contrasting conditions occurring at these locations (complex topography at Grouse Mountain and sea/land contrasts at Tsawwassen Ferry and UBC Climate Station). The influence of these contrasting conditions w i l l be introduced into the c l e a r sky c a l c u l a t i o n (Equation .3.1.8) via the predicted minimum brightness r a t i o T 2 (gjg-) » where BB i s the predicted minimum brightness calculated in Section 4.1.1. Table 4.1 shows that the greatest amount of v a r i a b i l i t y in the predicted minimum brightness r e l a t i o n s h i p occur at these three locations. This may be s u f f i c i e n t to explain the larger biases observed. On the c l e a r spring day (121/80) the model has a larger MBE with t h i s poorer performance occurring more co n s i s t e n t l y over the e n t i r e study area. This i s l i k e l y a consequence of the use of summer regression c o e f f i c i e n t s f or a spring analysis. 94 The short term (hourly) performance of the model (as assessed by the RMSE) i s more consistent over the study area and thus the average c l e a r sky values of 9.'6% f o r day 183/80 and 10.3% f o r 121/80 maybe considered to represent the model performance under such conditions. These comparatively low values r e f l e c t the s i m p l i c i t y of the radi a t i o n regime under such sky conditions. The s i m i l a r i t y of c o r r e l a t i o n c o e f f i c i e n t s f o r these two days provides evidence to support the claim that the higher RMSE for spring i s due to poorer model performance in the long term rather than in the short term. Systematic underestimation i s also observed for the p a r t l y cloudy days (239/80 and 160/80) although the. small MBE on day 160 indicates a value within measurement error. However, a comparison of the re s u l t s given in Table 4.4 with those given in Table A.3.1 and A.3.2 reveals that the average errors f o r the par t l y cloudy and overcast conditions quoted i n Table 4.4 do not t r u l y represent conditions occurring across the network. The MBE on day 239/80 i s adversely affected by the large negative values at Grouse Mountain, Tsawwassen Ferry and UBC Climate Station. S i m i l a r l y the average RMSE (43%) i s larger than those observed at most of the stations since i t i s again influenced by the larger values at the three stations mentioned above. The small MBE (-2.79%) on day 160/80 i s as a r e s u l t of can c e l l a t i o n of the Bias Errors ( p o s i t i v e and negative) occurring at d i f f e r e n t l o c a t i o n s . Average s t a t i s t i c s f o r the overcast days (197/80 and 105/80) suggest excellent long-term model performance in summer but a problem with over estimation i n spring and large short term errors i n both seasons. However, the e s s e n t i a l l y perfect agreement observed for day 197/80 i s d e f i n i t e l y the r e s u l t of the can c e l l a t i o n of the Bias Errors 9 5 occurring at the various s i t e s (note the large negative errors at Grouse Mountain, Tsawwassen Ferry and UBC Climate Station). The average MBE (20%) on day 105/80 i s much smaller than than those commonly observed at the various stations f o r t h i s day. In i n t e r p r e t i n g the r e s u l t s obtained f o r the p a r t l y cloudy and overcast conditions we must account f o r the underestimation under p a r t l y cloudy conditions, the anomalies occurring at stations such as Grouse Mountain, Tsawwassen Ferry and UBC Climate Station on a l l four days, and the overestimation of the model under overcast conditions. The underestimation of the model under partly cloudy conditions ( e s p e c i a l l y evident on day 239/80) indicates that more rad i a t i o n i s being received at the surface than the model estimates. This may be the r e s u l t of an inadequate threshold separating p a r t l y cloudy from overcast conditions. With a threshold that i s too low, many observations that should be c l a s s i f i e d as p a r t l y cloudy w i l l be c l a s s i f i e d as overcast and cal c u l a t i o n s w i l l be performed using Equation 3.1.10 (for cloudy conditions) instead of 3.1.9 (for p a r t l y cloudy conditions) leading to lower estimates of the r a d i a t i o n . An increase i n the p a r t l y cloudy/overcast threshold leads to a decrease in underestimation (see Table A.3.9). However, the changes are minor and other factors must account f o r the major discrepancies. The anomalous values occurring at locations such as Grouse Mountain, Tsawwassen Ferry and UBC Climate Station on a l l four days are probably due to the formulation of the model f o r p a r t l y cloudy and cloudy (overcast) conditions. Given Equations 3.1.9 and 3.1.10 the calculated r a d i a t i o n at the surface i s influenced by the cosine of the solar zenith angle and the r a t i o ( ^ ^ ) 2 where B„ i s the normalized 96 predicted minimum brightness. For a given value of the mean cloud brightness the r a t i o w i l l be larger with smaller values of B N . Since a U a negative c o e f f i c i e n t is attached to this ratio,, the larger the r a t i o the lower the estimated value of the r a d i a t i o n . Table 4.1 shows that the smallest BQ values occur at these locations. It i s possible therefore that the influence of the smaller BQ values leads to larger underestimations of the measured radiation (-under partly cloudy and cloudy conditions) at these locations. This explanation i s supported by the behaviour of the MBEs at both Northmount and BC Hydro, where the pattern i s generally one of smaller underestimations (than at the three stations) or small overestimations. BQ values at these two locations are s l i g h t l y larger than those at Grouse Mountain, Tsawwassen Ferry and UBC Climate Station (Table 4.1). The e f f e c t i v e equality of estimated-and measured radiation at Grouse Mountain on day 105/80 suggests that underestimations and overestimations i n the hourly values are o f f s e t t i n g each other r e s u l t i n g in the small MBE observed. An examination of the hourly values supports this conclusion. In addition the (r) value of (.63) (see Table A.3.2), indicates large random fluctuations in the differences between observed and calculated r a d i a t i o n . Model overestimations under overcast conditions indicates that less r a d i a t i o n is being received at the surface than is estimated by the model. We a t t r i b u t e t h i s to the inadequate handling of cloud absorption within the model's framework. Tarpley (1979) has pointed out this l i m i t a t i o n based on the results of his study. While absorption processes may be i m p l i c i t l y included in the regression model, the neglect of an e x p l i c i t absorption parameterization (which could decreases 97 fluxes by as much as 25% for some clouds; Liou, 1976) can p a r t i a l l y explain the overestimations observed under overcast conditions. The behaviour of the model on day 160/80 does not conform to the pattern expected for a pa r t l y cloudy day. The overestimations observed (Table A.3.2) are more inkeeping with overcast conditions. Figures 4.5a and 4.5b show heavily overcast conditions with substantial model overestimation, between 0600 - 1200 on day 160/80 followed by substantial clearing between 1300 - 1500 which in turn i s followed by partly cloudy conditions between 1600 - 1900. The s i g n i f i c a n t overestimation observed during the morning period on day 160/80 i s the major feature influencing the po s i t i v e Mean Bias Errors on this day (note that s i g n i f i c a n t overestimations also occur on the morning of day 105/80). The smaller MBEs on day 160/80 are due to the fact that i t was not a completely overcast day. The i n i t i a l assessments of the Tarpley (1979) model have shown that the c o e f f i c i e n t s developed for the U.S. Great Plains are not e n t i r e l y appropriate for the present study area. This i s r e f l e c t e d i n the biases observed f o r a l l conditions assessed. Model bias increases from the clear to the partly cloudy and overcast conditions and from summer to spring. We w i l l attempt to solve the problem of inappropriate c o e f f i c i e n t s by generating regression c o e f f i c i e n t s that are representative of the average conditions occurring i n the study area. 4.3 Model Modification 4.3.1. Generation of New Regression C o e f f i c i e n t s To generate regression c o e f f i c i e n t s for the study area we 98 divided our data set into two; a developmental data set and an independent data set for testing the model. The developmental data set consisted of twelve (12) days representing c l e a r , partly cloudy and overcast conditions. The two data sets are l i s t e d i n Table 4.5. The days in the developmental data set were chosen to incorporate a variety of conditions occurring at d i f f e r e n t times of the year. By se l e c t i n g our days in this manner we are attempting to provide the model with some degree of v a l i d i t y over the entire year. The s a t e l l i t e data were s t r a t i f i e d into c l e a r , p a r t l y cloudy and overcast cases in the manner outlined in Section 4.1.2. Values of the various independent variables (the cosine of the zenith angle, mean cloud brightness, mean target brightness, predicted brightness and atmospheric transmittance) were assessed for the twelve (12) days as outlined i n Sections 4.1.1 - 4.1.3. These were combined with hourly values of the measured radiation (the dependent variable) at the twelve stations in the network. The regression c o e f f i c i e n t s were generated using a Triangular Regression Package (TRP) a v a i l a b l e at the University of B r i t i s h Columbia Computer Centre. The new regression c o e f f i c i e n t s for the three equations are l i s t e d in Table 4.3 along with n (the number of cases analysed) and R and R2 the multiple c o r r e l a t i o n c o e f f i c i e n t and the c o e f f i c i e n t of determination, respectively. For the clear sky equation, 98% of the v a r i a t i o n i n the measured rad i a t i o n i s explained by the independent var i a b l e s . The R value of .994 i s larger than that obtained by Tarpley (1979) for his c l e a r sky regression equation. The R values for the partly cloudy and overcast equations decrease to 0.89 and 0.64 99 T a b l e 4.5  Data S e t s F o r R e g r e s s i o n DEVELOPMENTAL CLEAR DAYS P/CLOUDY DAYS OVERCAST DAYS * l 9 6 / 7 9 * 196/80 126/80 *257/79 •231/80 *141/80 *304/79 * 2 6 l / 8 0 *245/80 361/79 _ * 185/80 *171/80 VERIFICATION CLEAR DAYS P/CLOUDY DAY OVERCAST DAYS *121/80 * 160/80 *263/79 *276/79 *239/80 * 105/80 *025/80 * 197/80 * 183/80 *363/79 200/79 * I n d i c a t e s d a ys used i n t h e e v a l u a t i o n of t h e G a u t i e r e t a l . ( 1 9 8 0 ) model. 1 0 0 respectively (the values obtained by Tarpley 1979 were .77 and .70). The smallest number of observations analysed was for the partly cloudy condition (n = 183). The r e l a t i v e l y small si z e of this sample may r e s t r i c t the p r e d i c t i v e power of the r e l a t i o n s h i p because of the greater v a r i a b i l i t y encountered under these conditions. The revised version of Tarpley's (1979) model was applied to a s u i t e of nine days. The average s t a t i s t i c s for the twelve s t a t i o n for each day are l i s t e d in Table 4.6 (complete s t a t i s t i c s for the individual stations are given in Appendix A.3). Our present discussions w i l l consider the results from the summer time application ( i . e . days 183/80, 197/80 and 239/80) (results for the spring, f a l l and winter days w i l l be discussed in-Section 4.4). Figures 4.7a - 4.9b show the model's behaviour at selected stations in the network for days 183/80, 197/80 and 239/80. From a comparison of the results for the three days in Tables 4.4 and 4.6 we note the following: on average the bias in the model for the clear sky condition has been completely removed. In addition, there has been a concomittant decrease in the Root Mean Square Error. The RMSE of 5.3% observed for the c l e a r sky condition i s s i m i l a r to the ±5% measurement error quoted for hourly radiation values (Latimer, 1972). The pattern of underestimation for the partly cloudy day i s s t i l l evident but there has been a decrease in the Mean Bias Error and an increase in the Root Mean Square Error. For the overcast day model overestimation i s s t i l l apparent with an increase in the average MBE but a decrease in the Root Mean Square Error. 101 Table 4.6 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the revised Tarpley(1979) model to a sample of nine (9) days  representing variable sky cover conditions at di f f e r e n t times of  the year. Units KJm'2hr~ . Array size:5x5T DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL SUMMER 183/80 192 18.19 100.12 0.97 5.33 .996 1879.7 1897.9 (clear) 239/80 132 -81.68 369.67 10.47 47.38 .789 780.3 698.6 (p/cloudy) 197/80 191 21.42 177.94 5.23 43.41 .778 409.9 431.3 (overcast) SPRING 121/80 155 -40.80 111.95 -2.04 5.61 .995 1995.9 1955.1 (clear) 160/80 168 88.88 393.89 13.41 59.41 .865 663.0 ' 751.9 (p/cloudy) 105/80 142 101.57 264.69 25.65 66.85 .757 395.6 497.5 (overcast) FALL 276/79 120 -39.58 108.81 -2.63 7.23 .991 1505.7 1466.1 (clear) 263/79 105 29.92 202.68 7.77 52.64 .491 385.0 415.0 (overcast) WINTER 025/80 86 -129.31 164.90 -13.00 16.55 .968 996.2 866.8 (clear) .(9-day)l29l 5.50 236.20 0.54 23.00 .967 1027.1 1032.6 (sample) OBSERVED CRLCULRTED LOCRL RPPRRENT TIMF Observed radiation and calculated radiation from a revised Tarpley (1979) model at Vancouver A i r p o r t for a c l e a r day (183/80) in summer. Figure 4.7b: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at P i t t Meadows for a clear day (183/80) in summer. CO C\J _ CD OBSERVED - c CRLCULRTED LOCRL RPPRRENT TIME Figure 4.8a: Observed radiation and calculated radiation from a revised Tarpley (1979) model at Vancouver A i r p o r t for a partly cloudy day (239/80) in summer. QZ ZD o nz c\i i C J cr 1—1 Q CT oz oz CM CD _ L OBSERVED o CflLCULflTED o 24 l o c a l apparent time Figure 4 .8b : Observed r a d i a t i o n and c a l c u l a t e d r a d i a t i o n from a rev i sed Tarp ley (1979) model a t Abbots ford L i b r a r y f o r a p a r t l y c loudy day (239/80) i n summer. o-VRNRIR 197 OBSERVED o CRLCULRTED o cn 6 9 12 15 ' 18 LOCRL RPPRRENT TIME 21 24 Figure 4.9a: Observed radiation and calculated radiation from a revised Tarpley (1979) model at Vancouver Ai r p o r t for an overcast day (197/80) in summer. co _ j o-BCHYDRO .197 ^ OBSERVED c CRLCULRTED o —I CD 12 15 18 21 24 LQCRL RPPRRENT TIME Figure 4.9b: Observed radiation and calculated radiation from a revised Tarpley (1979) model at BC Hydro f or an over-cast day (197/80) i n summer. 1 0 8 The removal of the bias in the model f o r the c l e a r sky condition i s due to the f a c t that the c o e f f i c i e n t s are more representative of the average conditions occurring in the study area. However, the average behaviour of the model obscures the anomalies occurring at some of the stations. Table A.3.3 (see Appendix A.3) shows that s i g n i f i c a n t model underestimation i s occurring at Grouse Mountain while s i g n i f i c a n t overestimation i s occurring at a l l the i n t e r i o r locations. The anomaly at Grouse^Mountain was noted in the a p p l i c a t i o n of the o r i g i n a l Tarpley (1979) model in conjunction with anomalies at the Tsawwassen Ferry and UBC Climate Station (Section 4.2). Although the bias at Grouse Mountain has been s i g n i f i c a n t l y reduced i t i s evident that,the problem of minimum brightness (see Section 4.2) has not been adequately accounted f o r in the r e v i s i o n of the regression c o e f f i c i e n t s . On the other hand revised c o e f f i c i e n t s have removed the biases at the Tsawwassen Ferry and UBC Climate Station. This suggests that problems involved in the use of the s a t e l l i t e data in complex t e r r a i n such as the Grouse Mountain area cannot be accommodated within the r e l a t i v e l y simple s t a t i s t i c a l framework of the Tarpley model. The overestimations observed f o r the i n t e r i o r locations are consistent with the smaller underestimations at these locations noted in the a p p l i c a t i o n of the o r i g i n a l model (Section 4.2). Again we appeal to the influence of aerosols to explain the present findings. A pattern of overestimation at these stations due to a reduction in the measured values because of aerosols, would be a more consistent explanation given the f a c t that the model does not e x p l i c i t l y include the e f f e c t s of aerosols. 109 The reduced average MBE (Table 4.6) f o r the pa r t l y cloudy day (239/80) r e f l e c t s the general trend i n t h i s s t a t i s t i c at many of the stations though there are again the e f f e c t s of cancel l a t i o n of pos i t i v e and negative errors (e.g. at Grouse Mountain and Vancouver Air p o r t ; see Table A.3.3). The increase in the overall RMSE i s consistent with the increases observed at many of the stations. The average Mean Bias Error for the overcast condition day 197/80 i s not t r u l y representative of conditions at most of the stations. The Mean Bias Errors are generally larger and the"smaller average again r e s u l t s from the ca n c e l l a t i o n of negative and po s i t i v e errors (see Table A.3.3). The improved average RMSE r e f l e c t the improvement in RMSEs at most of the stations. The average error (43.4%) i s influenced by the larger values occurring at stations such as Langara. From the o r i g i n a l a p p l i c a t i o n of the model we recognize four possible sources of bias under pa r t l y cloudy and overcast conditions: 1) the o r i g i n a l regression c o e f f i c i e n t s ; 2) an inadequate threshold between p a r t l y cloudy and overcast conditions; 3) bias r e s u l t i n g from small BQ values in the r a t i o (-^|p^-)2 and 4) bias r e s u l t i n g from the inadequate handling of cloud absorption (note we also recognize the p o s s i b i l i t y of combinations of these f a c t o r s ) . With the application of new regression c o e f f i c i e n t s we would expect a reduction i n the under-estimations that occurred with the o r i g i n a l c o e f f i c i e n t s . Unfortunately t h i s w i l l also r e s u l t in increased overestimations f o r stations already experiencing a p o s i t i v e bias error. Except at P i t t Meadows and Abbotsford Library the changes observed f o r the pa r t l y cloudy day are consistent with t h i s statement (Table A.3.1 and A.3.3). For example, 110 at Grouse Mountain the revised coefficients lead to a decrease in underestimation while at Vancouver Airport they lead to an increase in the overestimation of the model. At Pitt Meadows the change in coefficients leads to an increase of the already substantial underestimation. It is clear that one or more of the other sources of bias are occurring at this location. The increase in the RMSE at many of the stations may be related to the size of the sample used to generate the new coefficients. We used a total of 183 cases (Tarpley, 1979 used 2127) to describe our partly cloudy condition. This limits the range of conditions that that can be accommadated by our coefficients. That is a larger sample is required to truly describe the diversity of conditions inherent in the partly cloudy situation. For the overcast day (197/80) there are two distinct features of the model's response; decreased overestimation at the interior stations and decreased underestimation at the coastal and mountain stations (including urban and suburban sites). The only station that departs from this pattern is Vancouver Airport where decreased overestimation is observed to occur. (Although the MBE is positive at Langara and increases with the application of the new coefficients the direction of change is consistent with one of decreased underestimation). In an attempt to reconcile these two features we offer the following explanation; at coastal 'and mountain stations the ratio (-^r^-)2 is the important B0 source of bias. Since BQ values are smaller at these locations this will result'in large underestimation (Grouse Mountain) or small I l l overestimations (BC Hydro). 'Tuning' of the regression c o e f f i c i e n t s reduces the bias introduced by t h i s r a t i o leading to a decrease in the underestimations (Grouse Mountain) and an increase i n overestimations (BC Hydro). Since BQ values are much higher f o r the i n t e r i o r locations the r a t i o (-^§^-)2 i s not as s i g n i f i c a n t . Reduction in the Mean Bias Error may be due simply to the use of revised regression c o e f f i c i e n t s and t h e i r a b i l i t y to incorporate in a more e x p l i c i t l y adequate manner such processes as cloud absorption. The-behaviour of the model f o r Vancouver A i r p o r t i s s i m i l a r to that observed f o r the i n t e r i o r locations (conditions at Vancouver A i r p o r t such as higher BQ value are s i m i l a r to the i n t e r i o r l o c a t i o n s ) . Therefore we tender the same explanation. The increased overestimation at Langara i s an anomaly and i s l i k e l y a response to the complex i n t e r a c t i o n of the sources of bias i d e n t i f i e d e a r l i e r . Revised regression c o e f f i c i e n t s f o r the Tarpley (1979) model has led to s i g n i f i c a n t improvements in both long term and short term model performance under a v a r i e t y of conditions. This i s p a r t i c u l a r l y apparent f o r the c l e a r sky case (183/80), although s i g n i f i c a n t bias i s s t i l l observed at some locations. Performance of the model under par t l y cloudy and overcast conditions i s confounded by sources of bias that are inherent in the model's formulation and by the small sample used to generate the regression c o e f f i c i e n t s (as i s the case f o r the pa r t l y cloudy regression equation). 4.3.2 Changes in Temporal Averaging To t h i s stage c a l c u l a t i o n s involving the Tarpley (1979) model 112 are based on an averaging procedure we refer to as 'pixel averaging'. That i s , c a l c u l a t i o n s are performed using weighted hourly average values of the parameters in our i n s o l a t i o n equations (see Section 4.1.4). In t h i s section we examine the s e n s i t i v i t y of the model to a change in the temporal averaging procedure. We adopt an a l t e r n a t i v e procedure which we r e f e r to as 'flux averaging' whereby the instantaneous incident shortwave f l u x i s determined for the individual images. The instantaneous fluxes are then combined as a weighted average to y i e l d the calculated radiation f o r the'hour. A s i m i l a r procedure i s followed in our c a l c u l a t i o n s f o r the Gautier et a l . (1980) model (see Section 5.1.3). The advantage of t h i s temporal averaging procedure i s that more information i s a v a i l a b l e on the radiation c h a r a c t e r i s t i c s from the individual images than would be the case i f an average i s used to represent the hour. The f l u x averaging procedure was applied to the three days used in the e a r l i e r assessment (183/80, 197/80 and 239/80). The version of Tarpley's (1979) model with revised regression c o e f f i c i e n t s provided the basis f or our comparisons. Average s t a t i s t i c s generated for the three days are l i s t e d i n Table 4.7a. Complete s t a t i s t i c s f o r the ind i v i d u a l stations are l i s t e d in Table A.3.4 (see Appendix A.3). A comparison of the r e s u l t s i n Table 4.6 and Table 4.7a reveals the following: Changes in the model's behaviour for the c l e a r sky from one configuration to the next are minimal (only a s l i g h t increase in the average RMS error i s observed). The r e s u l t s in Table A.3.4 show that the pattern of overestimation at the i n t e r i o r stations and underestimation at Grouse Mountain i s maintained. The model's i n s e n s i t i v i t y under c l e a r skies i s not e n t i r e l y unexpected considering the conservative temporal 113 Table 4.7 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the revised Tarpley (1979) model based on a Flux Averaging Approach (4.7a) and a Spatial Averaging Approach (4.7b). Units K ,1m ' 2 h r " a1 ~ — " — — • KJm- 'hr"*1 . DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL (a) FLUX AVERAGING 183/80 192 (clear) 18.96 1 02.84 1.01 5.47 .996 1879. 7 1898.7 239/80 132 (p/cloudy) -161.87 341.22 -20.75 43.70 .824 780. 3 618.4 197/80 191 (overcast) -3.39 157. 10 -0.83 38.30 .809 409. 9 406. 5 (b) SPATIAL AVERAGING 183/80 192 (clear) 17.96 100.66 0.96 5.35 .996 1879. 7 1897.9 239/80 132 (p/cloudy) -100.50 377.65 - 12.88 48.40 .781 780. 3 679.8 197/80 191 (overcast) 21 .43 181.78 5.23 44.35 .762 409. 9 431 .3 114 v a r i a b i l i t y of the radiation within a given hour. This means that the half-hourly s a t e l l i t e data w i l l be highly correlated and hence averaging techniques w i l l have l i t t l e influence. For the pa r t l y cloudy and overcast conditions flux averaging leads to decreases in the average RMS e r r o r s , an increase in the average Mean Bias Error f o r pa r t l y cloudy conditions and a decrease in the MBE for the overcast conditions. In t h i s l a t t e r case the e f f e c t i v e agreement between estimated and measured values i s v a l i d f o r the study area as a whole but i s not an accurate r e f l e c t i o n of the s i t u a t i o n at the individual locations. Flux averaging leads to reduced estimates of the surface irradiance under partly cloudy and overcast conditions. This w i l l r e s u l t in an increase in the MBEs which are already negative but w i l l reduce i n i t i a l l y p o s i t i v e MBEs. Because of the consistent changes brought about by fl u x averaging i t would be necessary to revise the regression c o e f f i c i e n t s before the f i n a l conclusions can be made about the ultimate nature of the errors. I n t u i t i v e l y the improvement in the short term performance of the model i s what we would expect from the ap p l i c a t i o n of the fl u x averaging approach to p a r t l y cloudy and overcast conditions ( i . e . i n contrast to pixel averaging). Flux averaging means an increase in the a v a i l a b l e information on the cloud conditions and the associated v a r i a b i l i t y in those conditions within the= hour. An improvement in the handling of the cloud v a r i a b i l i t y shown by an improvement in model prediction should be the end r e s u l t . This i s demonstrated by the re s u l t s presented. The anomalous increases i n the RMS error (e.g. 115 Grouse Mountain on day 197/80) are i n v a r i a b l y associated with increases in the c o r r e l a t i o n c o e f f i c i e n t (r) which would indi c a t e that increases in the RMSE are due to increases in the MBE (Table A.3.3 and A.3.4). Thus our r e s u l t s support the e a r l i e r contention that the use of 1/2 hourly s a t e l l i t e data i s desirable f o r addressing the issue of cloud v a r i a b i l i t y (Gautier et a l . , 1980). 4.3.3 Changes in Spatial Averaging In the preceding section we examined the s e n s i t i v i t y of a revised version of Tarpley's (1979) model to changes in temporal averaging. In t h i s section, we consider the s e n s i t i v i t y of the model to changes in s p a t i a l averaging. Our previous evaluations were based on a 5 x 5 pixel array centered on the station (Tarpley 1979 used a 7 x 6 pixel array where each pixel was 8 km). We w i l l evaluate the model for a 3 x 3 pixel array centered on the s t a t i o n . The 5 x 5 pixel array represents the outer l i m i t s of the accuracy of our earth location routine ( i . e . a RMS error of ± two pixels see Section 2.6) whilst the 3 x 3 pixel array represents the highest accuracy associated with our, earth location routine ( i . e . a RMS error of + one p i x e l ) . Under consideration in t h i s analysis i s the s e n s i t i v i t y of the model's performance to errors in our earth location routine. Model evaluations were performed f o r days 183/80, 197/80 and 239/80 using the 3 x 3 pixel configuration. Average s t a t i s t i c s f o r the twelve stations f o r the three days are l i s t e d i n Table 4.7b (complete s t a t i s t i c s f o r the i n d i v i d u a l stations are given i n Table A.3.5). A comparison of the r e s u l t s in Table 4.7b with those in Table 4.6 reveals 116 that the change in sp a t i a l configuration has l i t t l e e f f e c t on the average errors observed. The largest change occurs f o r the p a r t l y cloudy condition"with a 2% increase in the Mean Bias Error. For the cl e a r sky case (183/80) the res u l t s in Tables A.3.3 and A.3.5 indicate that the model's behaviour at the individual stations remains v i r t u a l l y unchanged. The cl e a r sky r e s u l t i s the expected behaviour given the conservative s p a t i a l c h a r a c t e r i s t i c s of the radiation under these conditions. Even though the regression c o e f f i c i e n t s are based on a 5 x 5 array the high c o r r e l a t i o n between"adjacent pixels under c l e a r skies means that changes in the s p a t i a l configuration w i l l have l i t t l e e f f e c t on the calculated r a d i a t i o n . The average errors quoted in Table 4.7b indicate l i t t l e change in the behaviour of the model f or pa r t l y cloudy and cloudy conditions. However, a comparison of the res u l t s in Table A.3!3 and A.3.5 reveals important changes occurring at many of the stations. The basic pattern of model underestimation i s maintained f o r the p a r t l y cloudy day. Spatial averaging leads to an increase in both the Mean Bias Error and Root Mean Square Error at many stations. The basic pattern of model overestimation i s maintained for,the overcast day. However, s p a t i a l averaging leads to a decrease in both the Mean Bias Error and Root Mean Square Error at -> most of the stations. On day 197/80 the increases in the RMS errors observed appear to be due la r g e l y to an increase in the random component of t h i s error r e f l e c t e d in a decrease i n the c o r r e l a t i o n c o e f f i c i e n t (r) (e.g. at the Tsawwassen Ferry and Abbotsford A i r p o r t . The RMSE at the Tsawwassen Ferry increases from 59% to 63% and (r) decreases from .925 to .702; at Abbotsford A i r p o r t the RMSE increases from 26% to 61% (r) 117 decreases from .938 to .729). On day 239/80 increases in the RMS errors are due both to increases in the Mean Bias Error and increases i n the random component of the RMSE. An example of the former i s the model's behaviour at Tsawwassen Ferry where the MBE increases from -30% to -35% and the RMSE increase from 53% to 55% with l i t t l e change (an increase) in the value of ( r ) . The performance of the model at Abbotsford A i r p o r t and BC Hydro i l l u s t r a t e the e f f e c t s of an increase in the random component of the RMS error. The r e s u l t s f o r the p a r t l y cloudy and overcast conditions are d i f f i c u l t to reconcile. Since we are dealing with the average value of a given array we expect our errors to be reduced f o r the larger array (because the influence of extreme values w i l l be greater the smaller the array s i z e ) . The model's behaviour at most of the stations on the p a r t l y cloudy day supports t h i s conclusion (while the behaviour on the overcast day contradicts t h i s conclusion). The increases in the Mean Bias Error on day 239/80 may be influenced by the regression c o e f f i c i e n t s which are developed f o r a 5 x 5 pixel array. However, the behaviour of the model may not be influenced by the change in array size as much as by the v a r i a b i l i t y in the r a d i a t i o n conditions represented in a given array. That i s , minor changes in the model's performance from one averaging configuration to the next may r e f l e c t f a i r l y homogenous conditions. Large changes in the model's performance between averaging configurations suggest heterogenous radiation conditions ( i . e . rapid changes are occurring over small distances). For the days analysed we conclude that an error in our earth location routine w i l l be of importance under p a r t l y cloudy and overcast 118 conditions e s p e c i a l l y when the radiation c h a r a c t e r i s t i c s are changing r a p i d l y over small distances. In addition, our r e s u l t s tend not to support the use of s p a t i a l averaging to improve the temporal representation of the s a t e l l i t e data as proposed by Gautier et a l . (1980). 4.4 Seasonal Assessments Complete v e r i f i c a t i o n of any model requires that the model be tested under a l l conditions f o r which i t i s to be applied. In our r e v i s i o n of Tarpley's (1979) model (i.e.-generation of new regression c o e f f i c i e n t s ) we endeavoured to include the e f f e c t s of changing season by using a suite of days covering various times of the year. With t h i s s i t u a t i o n in mind we attempted to v e r i f y the revised version of Tarpley's (1979) model using an independent set of days (Table 4.5) representing variable sky cover conditions in the d i f f e r e n t seasons. In performing a seasonal assessment such as the present one, we recognize the problem posed by d i f f e r i n g sample size imposed by varying daylengths. However, 'we believe that t h i s w i l l not s u b s t a n t i a l l y a f f e c t the v a l i d i t y of any conclusions that we present. A suite of four c l e a r days, two p a r t l y cloudy days and three overcast days i s used in the analysis. Average s t a t i s t i c s f o r the 12 stations f o r the nine days analysed are l i s t e d in Table 4.6 (complete s t a t i s t i c s f o r the 12 stations are given in Appendix A.3). Figures 4.7a - b and 4.10a - 4.12b show the behaviour of the model at selected stations. For the c l e a r days, the r e s u l t s in Table 4.6 show that on average the model i s without bias f o r the spring and summer applications (days 121/80 and 183/80) but s i g n i f i c a n t underestimation i s observed for the winter CO _ CM _ * CD- — VRNRIR OBSERVED CRLCULRTED 3 6 9 12 15 18 21 24 LOCRL RPPRRENT TIME Figure 4.10a: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Vancouver A i r p o r t for a clear day (121/80) in spring. cr o JZ C M ! L U (_j C E Q C E cr CNJ _ 9 12 15. 18 21 ^ OBSERVED o CflLCULflTED ro o 24 LOCAL APPARENT TIME Figure 4.10b: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at P i t t Meadows for a clear day (121/80) i n s p r i n g . cr o zc C\J I C J cr o cr cr. cr CO I CM _ C D _L 0 6 9 12 15 o- — •- -VRNR.TR 276 _ ^ OBSERVED _ 0 CRLCULRTED 18 21 24 LOCAL ,RPPRRFNT TIME Figure 4.11a: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Vancouver A i r p o r t for a c l e a r day (276/79) in f a l l . CO C\J X-0 3 6 9 12 15- . 18 LOCAL RPPRRENT TIMF 21 OBSERVED o CRLCULRTED ro ro 24 Figure 4.11b: Observed radiation and calculated radiation from a revised Tarpley (1979) model at Pitt Meadows for a clear day (276/79) in f a l l . cr ZD O •Z CM LU C J cr czi cr cr cr CO CM _ 0 o- — — -VRNfllR 25 S 9 12 1 5 . 1 8 LOCAL APPARFNT TIMF 21 OBSERVED c CALCULATED ro 24 Figure 4.12a: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Vancouver A i r p o r t for a clear day (025/80) in winter. CO _ CNJ I CD 3 o- — — -PITMED 25 ^ OBSERVED o CRLCULRTED 18 21 24 LOCRL RPPRRENT TIME Figure 4.12b: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at P i t t Meadows for a clear day (025/80) in winter. 125 a p p l i c a t i o n (day 025/80). The model only marginally underestimates the measured f o r the f a l l a p p l i c a t i o n (day 276/79). Both absolute and r e l a t i v e average RMS errors are greater f o r the f a l l and winter applications. The magnitude of the c o r r e l a t i o n c o e f f i c i e n t (r) suggests that a s i g n i f i c a n t portion of the increase in the RMS error observed ( e s p e c i a l l y f o r day 025/80) may be attributed to the increase i n the Mean Bias Error. A comparison of the results in Table A.3.3 and A.3.6 with those in Table 4.6 reveals that except in the case of day 025/80 the average s t a t i s t i c s obscure s i g n i f i c a n t aspects of the model's performance at various stations at d i f f e r e n t times of the year. The overestimations noted at the i n t e r i o r stations on day 183/80 (discussed in Section 4.3.1) is not evident at any other time of the year. Hay (1982) suggests that substantial aerosol attenuation i s only a summer time phenomenon. S i g n i f i c a n t underestimation i s observed at a l l but the inland locations on day 121/80. S i g n i f i c a n t underestimation i s also observed at the coastal and mountain stations and at Abbotsford Ai r p o r t and Langara on day 276/79. Part of the underestimation at Grouse Mountain has been attributed to problems in the predicted minimum brightness r e l a t i o n s h i p (Section 4.2). Another possible source of the bias observed at these stations i s an inadequate threshold between p a r t l y cloudy and c l e a r conditions. The occurrence of high c i r r u s clouds which would increase the brightness observed by the s a t e l l i t e but only marginally reduce the incoming radiation could be the i n f l u e n t i a l f a c t o r (note that the measured radiation at these stations i s higher than at the other l o c a t i o n s ) . An examination of the normal incidence data at the UBC Climat Station indicates the presence of clouds ( c i r r u s ) on these two days. 126 The large bias observed on day 025/80 may be due to the temporal l i m i t a t i o n s of our regression c o e f f i c i e n t s . A consideration of our developmental data set (Table 4.5) shows that the data f o r c l e a r days i s heavily weighted in favour of the summer conditions. This could explain the d e t e r i o r a t i o n in the model's performance observed for the winter s i t u a t i o n . Figures 4.8a - 4.9b and Figures 4.13a - 4.15b show the behaviour of the model at selected locations f o r the p a r t l y cloudy and overcast days. From Table 4.6 we note the following; on average the model overestimates the radiation f o r the three overcast days. Both the Mean Bias Error and the RMS error are largest on day 105/80 (overcast day in spring). For the p a r t l y cloudy days the model underestimates the measured on day 239/80 and overestimates the measured on day 160/80. The RMS error (59%) i s larger f o r the spring a p p l i c a t i o n . Table A.3.8 shows that the Mean Bias Errors at many of the stations are larger than the average errors quoted in Table 4.6 for the overcast days. Model underestimation at some of the stations suggests the reason f o r the smaller average errors. Explanations of the pattern of overestimation and underestimation observed on overcast days have been dealt with in Section 4.2 and 4.3.1. However, two features are of note: the large overestimation at Grouse Mountain on day 105/80 (where underestimation usually occurs), and the very small (r) values observed at Northmount and BC Hydro on day 263/79. The Grouse Mountain anomaly i s p a r t i a l l y explained by the f a c t that the revised regression c o e f f i c i e n t s have reduced the influence of the r a t i o ( ^ p ^ ) 2 on the estimated radiation B0 (see Section 4.2). The small values of (r) at Northmount and BC Hydro oz ZD o CM I CD cr i — i CZ) cr QZ cr on _ C M CD 0 ^ OBSERVED o CRLCULRTED r o —i 12 15 18 21 24 LOCRL RPPRRFNT TIME Figure 4.13a: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Abbotsford Library for a partly cloudy day (160/80) in spring. OBSERVED CRLCULRTED 0 3 6 9 12 1.5 18 21 24 LOCRL'RPPRRENT TIME Figure 4.13b: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Vancouver Ai r p o r t for a partly cloudy day (160/80) in spring. /—> HQUR-_ C\J i —) ro _ x o x OBSERVED c CRLCULRTED VRNR.TR IRRRDIRNCE o // \ 105 e ® IRRRDIRNCE 3 1 3 i i i i 6 9 12 .15 .18 21 l 24 LOCRL RPPRRENT TIME Figure 4.14a: Observed radiation and calculated radiation from a revised Tarpley (1979) model at Vancouver A i r p o r t for an overcast day (197/80) in spring. ro c\J _ J o BCHYDRO 105 ^ OBSERVED c CRLCULRTED CO o LOCRL RPPRRENT TIME Figure 4.14b: Observed radiation and calculated radiation from a revised Tarpley (1979) model at BC Hydro for an overcast day (105/80) in spring. I cr ZD O X r\j UJ CJ cn i—i Q CE cr. cn o _ L 6 o VflNRIR 263 ^ _OBSERVED c CRLCULRTED 18 21 24 LOCRL RPPRRENT TIME Figure 4.15a: Observed radiation and calculated r a d i a t i o n from a revised Tarpley (1979) model at Vancouver A i r p o r t for an overcast day (263/79) in f a l l . ZD o C\J LU CJ CE I— I Q CT c n c n X-o- — — — BCHYDRQ 263 -x o OBSERVED CALCULATED CO ro 12 15 18 21 24 LOCAL APPARENT TIME Figure 4.15b: Observed radiation and calculated radiation from a revised Tarpley (1979) model at BC Hydro for an . overcast day (263/79) in f a l l . 133 indicate very l i t t l e c o r r e l a t i o n between estimated and measured radiation values. The low c o r r e l a t i o n at BC Hydro i s l a r g e l y influenced by the anomalous value i n the estimated radiation at 0800 (see Figure 4.15b). The individual s t a t i o n results in Table A.3.7 f o r the p a r t l y cloudy days show that the Mean Bias Errors at the various locations are not t r u l y represented by the averages quoted in Table 4.6. A wide v a r i a t i o n in the MBEs are noted f o r both days. The behaviour of the model on day 239/80 has already been discussed in Section 4.3.1. We noted e a r l i e r that the behaviour of the model on day 160/80 i s much clo s e r to that observed f o r the overcast condition (Section 4.2). This i s further confirmed by the results in Table A.3.7 f o r the Tarpley (1979) model with revised c o e f f i c i e n t s . The pattern of overestimation at most stations i s evident on day 160/80. The Mean Bias Errors on day 160/80 are generally smaller than on the other overcast days. This we a t t r i b u t e to the f a c t that the heavy overcast conditions evident during the morning period did not p e r s i s t throughout the e n t i r e day. The differences in model performance under p a r t l y cloudy and overcast conditions f o r the three d i f f e r e n t times of year considered, may be due to a number of f a c t o r s : 1) D i f f e r e n t radiation c h a r a c t e r i s t i c s . For example, the spring days are characterized by heavily overcast conditions with s i g n i f i c a n t model overestimation (at most locations) in the morning period followed by a close correspondence between estimated and measured values (see Figure 4.14a f o r day 105/80). This feature i s not present on the summer days and not as well developed on day 263/79 (see Figure 4.15a). A tentative explanation of t h i s behaviour i s that the azimuth angle between sun and s a t e l l i t e i s important under overcast 1 3 4 conditions. In Section 4 . 1 . 1 we argued that the azimuth angle between sun and s a t e l l i t e influences the signal received by the s a t e l l i t e under c l e a r skies,and the use of a constant B Q value i s therefore inadequate ( B Q i s used in the Tarpley ( 1 9 7 9 ) model because there i s no angular model to describe scattering under cloudy conditions). Under some cloud regimes the cloud configuration i s such that under a morning sun s c a t t e r i w i l l be dominated by backscatter away from the s a t e l l i t e leading to a darker scene being viewed by the sensor. In the present formulation t h i s w i l l lead to overestimation. The same scene, viewed with sun and s a t e l l i t e in the same quadrant w i l l appear brighter leading to e i t h e r underestimation or closer correspondence between estimated and measured values. This i s what i s observed f o r day 1 0 5 / 8 0 and to some extent day 2 6 3 / 7 9 . This suggests that a change in the model formulation i s necessary to incorporate scattering under cloudy conditions. 2 ) A second fa c t o r influencing the differences in model performance f o r the d i f f e r e n t times i s related to the a p p l i c a b i l i t y of the pa r t l y cloudy equation outside the summer context. This i s questionable considering the f a c t that the developmental data l a r g e l y r e f l e c t summer time conditions. 4 . 5 Daily S t a t i s t i c s As part of our v e r i f i c a t i o n , in t h i s section we examine the e f f e c t s of increasing the averaging period on the performance of the Tarpley ( 1 9 7 9 ) model with revised c o e f f i c i e n t s . The model was evaluated f o r four periods; c l e a r p a r t l y cloudy and overcast days and f o r the t o t a l sample of nine ( 9 ) days. Values at the twelve stations were combined to y i e l d an average d a i l y value f o r each period. These s t a t i s t i c s 135 are l i s t e d in Table 4.8. From a comparison of the res u l t s in Table 4.8 with the hourly r e s u l t s given i n Table 4.6 we note the following; there i s an ove r a l l decrease i n the Mean Bias Error and the Root Mean Square Error f o r the c l e a r , p a r t l y cloudy and overcast days. The model i s without bias f o r clear and p a r t l y cloudy conditions but shows s i g n i f i c a n t bias f o r the overcast days. The model i s without bias f o r the t o t a l sample of 9 days. For th i s same sample, the RMSE of 12.0% i s much smaller than those observed f o r the p a r t l y cloudy and overcast days r e f l e c t i n g the influence of the much smaller error observed f o r the cl e a r day period. The overall decrease in the RMS errors f o r the d a i l y analyses i s what i s commonly observed f o r this s t a t i s t i c as the averaging period increases. The RMS error of 4% for the clear day period i s s i m i l a r to that obtained by Tarpley (1979) f o r the cl e a r days used in his analysis. The RMS error computed f or the c l e a r days i s quite s i m i l a r to the hourly RMS error calculated f o r the summer s i t u a t i o n (Table 4.6). The larger RMS errors observed f o r the p a r t l y cloudy and overcast days a t t e s t to the reduced a b i l i t y of the model to estimate the rad i a t i o n under those conditions. The RMS error of 12.0% f o r the to t a l sample i s larger than that quoted by Tarpley (1979) f o r his to t a l sample. This difference could be due to the size and composition of the two samples used ( i . e . the r a t i o of c l e a r , p a r t l y cloudy and overcast days). 4.6 Implications of Model Accuracy f o r Spatial Sampling Requirements In Section 1.1 we outlined the character of the mesoscale v a r i a b i l i t y documented f o r the study. Table 1.1, taken from Hay, 1981, 136 Table 4-8 Daily Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the revised Tarpley (1979) model to periods of clear, p a r t l y and  overcast days and to an a l l day sample of nine (9) days. U n i t s -MJm^day 1. Array size:5x5. PERIOD N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL CLEAR 48 -0.389 0.794 -2.00 4.07 .998 19.5 19.1 DAYS P/CLOUDY 24 0.173 2.130 1.94 23.90 .642 8.9 9.1 DAYS OVERCAST 36 0.602 1.690 12.38 34.86 .642 4.9 5 5 DAYS 9-DAY 108 0.066 1.500 0.54 12.20 .987 12.3 12.3 SAMPLE 137 contains the Root Mean Square Errors of Interpolation associated with various sampling distances f o r hourly and d a i l y data based on the technique of Alaka (1970) and 2.5 years of measured data at a l l stations excluding Grouse Mountain. To determine the sampling distances associated with the accuracy of the model, we evaluated the model f o r the network as a unit ( i . e . we are interested in the average accuracy using the nine (9) days of data). To be consistent with the Hay (1981) study, values f o r Grouse Mountain are excluded. The RMS error f o r hourly values from t h i s assessment i s 23% of the mean measured global solar r a d i a t i o n . The RMS e r r o r decreases to 12.0% f o r the d a i l y values. From a consideration of Table 1.1 we note that an error of 23% for hourly values i s associated with a sampling distance of approximately 30 km. This suggest, that on average to maintain hourly estimates of the r a d i a t i o n to within ±23% we are limited to a s p a t i a l resolution of approximately 30 km. At shorter distances the actual variations in solar r a d i a t i o n could not be distinguished from the variations that occur due to errors in the c a l c u l a t i o n procedures. An i n t e r p o l a t i o n technique based on measured data would be the more appropriate method at these shorter distances. For d a i l y values an error of 12.0% i s associated with a sampling distance of approximately 25 km. On average at sampling distances less than 25 km an i n t e r p o l a t i o n technique should be used i f the er r o r desired i s less than ±12.0% f o r d a i l y values. The mesoscale s p a t i a l v a r i a b i l i t y in the solar irradiance f o r the study area can be resolved by the s a t e l l i t e model at distances of 30 km (to within ±23%) and 25 km (to within ±12.0%) f o r hourly and d a i l y values, respectively. 138 4.7 Summary and Conclusions Implementation of the Tarpley (.1979) model f o r the study area has shown that the o r i g i n a l c o e f f i c i e n t s developed f o r the U.S. Great Plains are inappropriate f o r the study area because of the s i g n i f i c a n t bias introduced. This bias increases when the model i s applied outside the summertime context. It i s also evident from the o r i g i n a l a p p l i c a t i o n that l i m i t a t i o n s in the formulation of the model (inadequate cloud threshold etc.) lead to increased bias under p a r t l y cloudy and overcast conditions. Subsequent r e v i s i o n of the regression c o e f f i c i e n t s to represent conditions i n the study area r e s u l t s in a complete removal of the model's bias under clear skies except in cases where the bias i s inherent in the model's formulation (e.g. at the i n t e r i o r stations under summer conditions 183/80) or where the temporal extension of the revised c o e f f i c i e n t s i s inappropriate (e.g. on day 25/80 f o r winter conditions). Hourly RMS errors f o r the c l e a r sky summer s i t u a t i o n are much smaller than f o r Tarpley (1979). This could be due to the larger number of images used in t h i s study to represent an hour. The model's performance f o r overcast days i s consistent f o r the three seasons of the year examined. The r e s u l t s from the spring and f a l l applications indicate that the changing azimuth angle between sun and s a t e l l i t e may be important under overcast conditions. We a t t r i b u t e the d e t e r i o r a t i o n in the model's short term performance under p a r t l y cloudy conditions to the small sample si z e used to generate our c o e f f i c i e n t s . Changes in the temporal and s p a t i a l averaging conditions reveal that the model i s i n s e n s i t i v e to such changes under c l e a r sky conditions but s e n s i t i v e to changes under p a r t l y cloudy and overcast conditions. 139 Our r e s u l t s would indicate that the temporal dimension i s more important since i t influences the degree of information available from the data whereas s p a t i a l averaging r e a l l y depends not on the changes in the array size but on the r a d i a t i o n c h a r a c t e r i s t i c s included in a given array. Assessments of the model f o r d a i l y time periods and f o r the t o t a l sample of nine (9) days show the expected decreases in RMS errors associated with an increase in the averaging period. The RMS error of ±4% ( f o r c l e a r days) i s s i m i l a r to that quoted by Tarpley (1979) while the ±12% e r r o r f o r the t o t a l sample i s s l i g h t l y larger than in Tarpley's 1979 study. In terms of the mesoscale s p a t i a l v a r i a b i l i t y i n the solar irradiance resolvable by t h i s p a r t i c u l a r model, the hourly average RMS error of ±23% for the nine (9) day sample r e s t r i c t s the model to a s p a t i a l resolution of 30 km. On a d a i l y basis the average RMS error of ±12.0% l i m i t s the s p a t i a l resolution to 25 km. Recommendations f o r improvements in the model's performance include: 1) The use of a larger sample to generate regression c o e f f i c i e n t s ( e s p e c i a l l y f o r the p a r t l y cloudy condition); 2) Improvement in the cloud thresholds ( i . e . a more rigorous, objective d e f i n i t i o n ) ; 3) Consideration of the e f f e c t s of changing azimuth angle between sun and s a t e l l i t e under overcast conditions; and 4) The i n c l u s i o n of the e f f e c t s of aerosols f o r c l e a r skies, and absorption under overcast conditions. 140 CHAPTER FIVE 5. Application of the Gautier Model 5.1 Implementation The implementation of the Gautier et a l . (1980) model requires many of the same assumptions made in connection with the Tarpley (1979), model. We assume that the zenith angle and the s a t e l l i t e azimuth angle are the same across the network and that Grouse Mountain (1128 m) is the only location where height must be taken into consideration. The question of brightness scale i s not relevant because the same 8-bit scale i s used in both the o r i g i n a l study and the present one. The physically-based model has two major facets; a.clear sky model and a cloudy atmosphere model. The clear sky model is described by a series of equations outlined in Section 3.2, namely Equations 3.2.1 -3.2.3. The cloudy atmosphere model i s e s s e n t i a l l y the c l e a r a i r formulation with the i n c l u s i o n of the effects of a cloud layer. The relevant equations for the cloudy atmosphere formulation are Equations 3.2.4 - 3.2.5. The important inputs for the c l e a r a i r formulation include water vapour absorption, the scattering c o e f f i c i e n t s , surface albedo, a cloud threshold and the e x t r a t e r r e s t r i a l i r r a d i a n c e . The e x t r a t e r r e s t r i a l irradiance is simply the product of the solar constant (assigned a value of 1353" W m ) and the cosine of the zenith angle (Equation 4.1.1). The cloudy atmosphere model has the additional requirements of cloud albedo and cloud absorption. 141 5.1.1 Clear Sky Model Development 5.1.1.1 Surface Albedo The signal received at the s a t e l l i t e sensor under clear sky conditions i s the sum of two components; energy scattered out of the radi a t i o n stream on i t s i n i t i a l passage through the atmosphere and shortwave radiation r e f l e c t e d by the surface back to the s a t e l l i t e sensor (Gautier et a l . , 1980) (some of this l a t t e r radiation w i l l be scattered out by the atmosphere i . e . not a l l w i l l reach the s a t e l l i t e sensor). The surface albedo describes the amount of radiation r e f l e c t e d by a given surface while Equation 3.2.1 defines the signal received at the s a t e l l i t e sensor in terms of an energy flux. Since the s a t e l l i t e data are expressed in counts (measures of r e l a t i v e brightness), some form of c a l i b r a t i o n i s necessary to convert the signal into energy u n i t s . We follow the work of Smith and Vonder Haar (1980) who provide a curve r e l a t i n g the s a t e l l i t e brightness counts to normalized reflectance (albedo). This curve i s defined by means of the following equation: RNORM = 0.00154 + 0.000166 • SC +.0.0000137 • SC 2 (5.1.1.1) RN0RM - normalized reflectance SC = measured s a t e l l i t e brightness in counts Values of normalized reflectance (albedo) obtained using this equation are converted into energy fluxes by multiplying by the appropriate value of the e x t r a t e r r e s t r i a l irradiance ( i . e . the value relevant to a p a r t i c u l a r s a t e l l i t e observation). I f the other terms in Equation 3.2.1 (scattering c o e f f i c i e n t s and water vapour absorption) are known (see Sections 5.1.1.2 and 5.1.1.3) then the surface albedo can be determined using Equation 3.2.2. 142 The determination of the surface albedo requires a c l e a r ' a i r ' image; that i s an image that has been determined to be cloud free. In the original" formulation of the model outlined by Gautier et a l . (1980), the clear ' a i r ' image is a composite image obtained by taking minimum brightness values for several images at the same time of day. The albedo values calculated from this composite i s then applied to a l l images used in a given day. The obvious l i m i t a t i o n of this albedo c a l c u l a t i o n , as pointed out by Gautier et a l . (1980), is that i t does not take into consideration variations in albedo with changing zenith angle. We deviate from the o r i g i n a l model formulation by using Tarpley's (1979) minimum brightness parameterization (Equation 3.1.1) to generate minimum brightness values from which the surface albedo can be determined. This equation gives us the c a p a b i l i t y to account for changing surface t albedo with changing zenith angles and therefore represents an improvement on Gautier et a l . (1980). The accuracy of the surface albedo calculated in this way w i l l depend upon the representativeness of our minimum brightness parameterization ( i . e . how cl o s e l y i t represents the apparent changing c h a r a c t e r i s t i c s of the surface). 5.1.1.2 Scattering C o e f f i c i e n t s The scattering c o e f f i c i e n t s for direct-beam and diffuse r a d i a t i o n are based on the empirical determinations of Coulson (1959). His work involved the approximation of the earth's planetary albedo due to the scattering of a Rayleigh atmosphere. The disc of the earth as seen from the sun i s divided into concentric rings and the albedo calculated f o r each r i n g . The sun's elevation for each ring i s 143 considered constant. The albedo of a given ring can then be considered to be representative of the atmospheric scattering at the p a r t i c u l a r zenith angle for that r i n g . The scattering c o e f f i c i e n t for diffused radiation is assumed to be constant with zenith angle. The value assigned to this c o e f f i c i e n t is 0.076. The c o e f f i c i e n t for d i r e c t beam radiation varies with zenith angle. Gautier et a l . (1980) present values for zenith angles ranging from 0.0° to 66.4°. We deviate s l i g h t l y from the o r i g i n a l researchers' use of the Coulson (1959) data by using the following equation to describe the r e l a t i o n s h i p between zenith angle and d i r e c t beam s c a t t e r i n g ; 6 = 0.0467563 + 0.0014173 e - 0.00005258-e2 + 0.000000651 e 3 (5.1.1.2.1) 3 = scattering c o e f f i c i e n t for beam radiation (dimensionless) e = s o l a r zenith angle in degrees This equation extends the c a l c u l a t i o n of the scattering c o e f f i c i e n t beyond the l i m i t of 66.4° used by Gautier et a l . (1980) and in fact i s v a l i d to a zenith angle of approximately 85° (degrees) the l i m i t imposed by Coulson's (1959) o r i g i n a l data. The equation y i e l d s s l i g h t l y higher values for the s c a t t e r i n g c o e f f i c i e n t , but we do not expect t h i s to s i g n i f i c a n t l y a f f e c t our r e s u l t s . 5.1.1.3 Water Vapour Absorption Calculation of the water vapour absorption u t i l i z e s a n a l y t i c a l expressions derived by Paltridge (1973) for Yamamoto's (1962) theoretical water vapour absorption as a function of p r e c i p i t a b l e water path. Under 144 cLear skies water vapour plays a major role in the depletion of incoming so l a r r a d i a t i o n . The o p t i c a l path for water vapour i s given by: u' = u • m (5.1.1.3.1) u = p r e c i p i t a b l e water @ zenith angle = 0° (cm) u' = slant path of p r e c i p i t a b l e water (cm) m = o p t i c a l a i r mass Values of-.u are determined via Smith's (1965) empirical formulation of the surface dewpoint temperature (Section 2.4). The slant path of p r e c i p i t a b l e water must be evaluated for two radiation streams: the primary stream of radiation in the sun-earth path and a secondary stream in the e a r t h - s a t e l l i t e path. In both instances the o p t i c a l a i r mass (m) is important. For the primary stream the optical a i r mass i s simply 1/cos e given by Equation 4.1.1. For the secondary stream the zenith angle (viewing angle) of the s a t e l l i t e must be evaluated. Using the relationships in Equation A.2.2 (see Appendix A.2), the s a t e l l i t e zenith angle is calculated to be approximately 58° (degrees). Gautier et a l . (1980) use the r e l a t i o n s h i p 1/cos e to describe the o p t i c a l a i r mass. However, th i s formulation i s limited because i t does not take into consideration the reduction in the opt i c a l a i r mass with height (Section 4.1.3) and in addition the r e l a t i o n s h i p breaks down at large values of the zenith angle. We overcome t h i s l i m i t a t i o n by using a combination of McDonald's (1960) expression for opt i c a l a i r mass which includes a height correction and Kasten's (1966) expression for o p t i c a l a i r mass which includes a correction for large zenith angles. 145 The r e s u l t i n g expression is given by Equation 4.1.3.1. The water vapour absorption as a function of slant path p r e c i p i t a b l e water (u 1) is given by the following equations from Paltridge (1973): a(u') = .099 u " 3 4 for u' > 0.5 cm (5.1.1.3.2) a(u') = .14 u " 4 4 for u 1 < 0.5 cm (5.1.1.3.3) a(u') = slant path water vapour absorption u' = slant path p r e c i p i t a b l e water (cm) These expressions are used to calculate water vapour absorption for the clear sky model. 5.1.1.4 Cloud Threshold Determination of the cloud threshold is a key feature of the modelling procedure. Clouds s i g n i f i c a n t l y increase the brightness recorded by the s a t e l l i t e sensor over the clear sky value and reduce the r a d i a t i o n received at the surface below the clear sky value. The cloud threshold acts as a switching c r i t e r i o n between the clear sky model and the cloudy atmosphere model. That i s , when brightness values are less than or equal to the threshold the clear sky model i s used, but when values are greater than the threshold the cloudy atmosphere model i s applied. The cloud threshold i s evaluated from a c l e a r ' a i r ' brightness value (equivalent to the minimum brightness under clear skies) to which a confidence margin i s added. According to Gautier et a l . (1980) this confidence margin is supposed to account for small changes in surface albedo, variations in atmospheric water vapour and the presence of aerosols. 146 A l t e r n a t i v e l y , the confidence margin i s added to ensure that brightness values are not erroneously designated as being cloud contaminated. Gautier et a l . ( 1 9 8 0 ) present no objective c r i t e r i o n for assigning the confidence margin. * The cloud threshold for th i s study is evaluated from the minimum brightness of the surface to which we also add a confidence margin. As noted in 5 . 1 . 1 . 1 the minimum brightness value is calculated using Tarpley's ( 1 9 7 9 ) minimum brightness parameterization (Equation 3 . 1 . 1 ) . The confidence margin i s evaluated as three times the average standard error of the estimate of the minimum brightness parameterizations developed for our 1 2 stations (Table 4 . 2 ) . This y i e l d s a value of 1 2 counts (on an 8 - b i t scale) for our confidence margin. The standard error of the estimate i s used here because i t is a measure of the scatter in the minimum brightness r e l a t i o n s h i p . This scatter can be attributed to changing atmospheric water vapour, aerosols and small changes in surface albedo. The cloud threshold i s evaluated from the following equation; THR = I Q COS 9 (B + A' • DENOM) ( 5 . 1 . 1 . 3 . 4 ) ' 6 = sol a r zenith angle I Q = sol a r constant ( 1 3 5 3 W m ) B = scattering c o e f f i c i e n t for. d i r e c t beam radiation (dimensionless) A' = a + . 0 0 5 6 6 ( 5 . 1 . 1 . 3 . 5 ) (the value . 0 0 5 6 6 i s the normalized confidence margin of 1 2 counts). a = surface albedo DENOM = ( 1 - B) [ 1 - a ( u ] ) ] [ 1 - a ( u g ) ] ( 1 - B ]) ( 5 . 1 . 1 . 3 . 6 ) 147 3-| = s c a t t e r i n g c o e f f i c i e n t for diffuse radiation (dimensionless) a(u^), a(u 2) = absorption c o e f f i c i e n t s for s l a n t water vapour paths (sun and s a t e l l i t e r espectively) The appropriateness of the cloud threshold i s further discussed in Section 5.2. 5.1.2 Cloudy Atmosphere Model Development 5.1.2.1 Cloud Albedo We noted e a r l i e r that the cloudy atmosphere model i s e s s e n t i a l l y the clear a i r formulation with the added effects of a cloud layer. The signal received by the s a t e l l i t e sensor under cloudy conditions consists of: 1) energy scattered from the atmosphere to the s a t e l l i t e , 2) energy re f l e c t e d from the cloud to the s a t e l l i t e and 3) energy that reaches the ground and i s r e f l e c t e d back through the cloud to the s a t e l l i t e (Gautier et a l . , 1980). Except in the case of very thin clouds the cloud albedo ( r e f l e c t i o n ) usually dominates the radiation c h a r a c t e r i s t i c s under cloudy conditions. Equation 3.2.4 describes the energy flux at the s a t e l l i t e under cloudy conditions. The cloud albedo can be evaluated in a manner s i m i l a r to the c a l c u l a t i o n of the surface albedo (Section 5.1.1.1) i f a l l the other variables in the equation are known. These variables are s p e c i f i e d in Sections 5.1.1, 5.1.2.2 and 5.1.2.3. The solution takes the form of a quadratic, the p o s i t i v e root being the value assigned to the cloud albedo. To avoid anomalous values of cloud albedo due to increased 148 d i r e c t i o n a l r e f l e c t i o n at large zenith angles, the cloud albedo is cons'trained to an upper l i m i t of .85 (85%) ( F r i t z 1954). 5.1.2.2 Cloud Absorption Another variable introduced by the cloud layer i s the cloud absorption. The model is tuned to best describe the behaviour of low .and middle str a t i f o r m clouds. Absorption in the Gautier et a l . (1980) model i s calculated on the basis of cloud brightness, since cloud brightness i s assumed to be an in d i c a t o r of cloud type and thickness. Deducing the thickness of.a cloud s o l e l y on the basis of i t s brightness does not take into consideration the fact that clouds of d i f f e r e n t thickness may have s i m i l a r brightness when seen by the s a t e l l i t e sensor. This w i l l lead to errors in c a l c u l a t i n g cloud absorption. Nevertheless the cloud absorption i s assumed in the model to be l i n e a r l y dependent on cloud brightness. Absorption i s allowed to vary from 0.0 for no clouds to 0.2 (20%) for the deepest (brightest) clouds. We evaluated cloud absorption i n the following way: the contribution of the cloud to the brightness of a p a r t i c u l a r pixel i s assumed to be the pixel brightness minus the cloud threshold (Section 5.1.1.4). Following Gautier et a l . (1980), we assume that for a p a r t i c u l a r observation the maximum possible cloud brightness i s the e x t r a t e r r e s t r i a l r adiation minus the cloud threshold. Since absorption is assumed to be l i n e a r l y dependent on brightness we calculate the increment in brightness that w i l l bring about a change in absorption. The change in absorption from 0.0 to 0.2 i s divided into twenty l i n e a r 149 steps. Therefore, the increment in brightness that w i l l bring about a change in absorption is calculated to be 1/20 of the difference between the e x t r a t e r r e s t r i a l radiation and the cloud threshold. The cloud absorption for a p a r t i c u l a r observation then becomes: abs = [(REFL - THR) / INCR] • 0.01 (5.1.1.2.1) REFL = pixel brightness THR = cloud threshold calculated via equation 5.1.1.3.4 INCR = increment in brightness that w i l l bring about a change in absorption (see text) 5.1.2.3 Water Vapour Absorption Calculation of the effects of water vapour in the cloudy atmosphere model e n t a i l s the s p e c i f i c a t i o n of absorption above and below cloud l e v e l . Again, absorption calculations must be made for the two slant water vapour paths; the sun's path and the s a t e l l i t e path. Following Paltridge (1973), the bulk of the water vapour is assumed to l i e below the cloud base. The p a r t i t i o n i n g for the Gautier et a l . (1980) model is given as 70% of atmospheric water vapour below cloud leve l and 30% above cloud l e v e l . Estimation of the absorption c o e f f i c i e n t s in the cloudy atmosphere model takes into account the cumulative character of the absorption. The equations used to determine the absorption are given in Appendix A.2. We deviate s l i g h t l y from the Gautier et a l . (1980) method of determining the absorption c o e f f i c i e n t s . Our method simply involves the p a r t i t i o n i n g of the water vapour value calculated using Smith's (1966) empirical formulation (Equation 2.4. ), into the fractions 150 described e a r l i e r (that is 70% below cloud level and 30% above cloud l e v e l ) . Calculation of the absorption c o e f f i c i e n t s above and below the cloud layer is s i m i l a r to the clear sky model evaluation (Section 5.1.1.3). Because of the i n s e n s i t i v i t y of the model to p r e c i p i t a b l e water (Section 2.4) any errors introduced as a r e s u l t of our calculations should have l i t t l e e f f e c t on the f i n a l output of the model. 5.1.3 .Radiation Calculations The radiation calculations for the clear sky and cloudy atmosphere models are performed on a 5 x 5 pixel array centered on a given s t a t i o n . The choice of the 5 x 5 array i s based on the accuracy obtainable using our earth location routine (Section 2.6). We d i f f e r from Gautier et a l . (1980) who use an 8 x 8 pixel array for t h e i r r a d i a t i o n c a l c u l a t i o n s (see Section 5.3.1 for further discussion). The radi a t i o n i s estimated for each pixel i n the 5 x 5 array and the average flux is calculated. The estimation of the radiation on a pixel by pixel basis is an attempt to capture some of the v a r i a b i l i t y that exists in the radiation f i e l d e s p e c i a l l y under partly cloudy conditions. Following Gautier et a l . (1980) we use as many as three images to calculate the radiation for a given hour. The images are weighted and merged in the manner outlined i n Section 2.9. To obtain the calculated r a d i a t i o n for any hour we determine the fluxes for the relevant images and apply the appropriate weights. The re s u l t i n g calculated r a d i a t i o n for a given hour and location i s compared with the corresponding measured value. 151 5.2 Results To f a c i l i t a t e comparison, i n i t i a l assessments of the Gautier et a l . (1980) model were conducted for the same days used in the assessment of the Tarpley (1979) model (origi n a l and revised); namely Julian days 183/80, 197/80 and 239/80. Average s t a t i s t i c s calculated for the three days are l i s t e d in Table 5.1 (complete s t a t i s t i c s f o r the individual stations are given in Appendix A.4). Figures 5.1a - 5.3b show the behaviour of the model at selected locations in the network. From Table 5.1 we note the following; on average the model systematically overestimates the measured under c l e a r skies, underestimates the measured under partly cloudy conditions and i s e s s e n t i a l l y in agreement under overcast conditions. Average RMS errors are smallest for the clear sky (±5%), increasing to 33% for partly cloudy conditions. On the basis of the errors observed here the average performance of the model for the three days i s better than that observed for the o r i g i n a l Tarpley (1979) model (on day 197/80 average long term performance is i d e n t i c a l for both models). The overestimation observed for the clear sky i s in contrast to the underestimation observed for the o r i g i n a l Tarpley (1979) model (Table 4.4). The average performance of the Gautier et a l . (1980) model is also better than that observed for the revised Tarpley (1979) model except under clear sky conditions (Table 4.6). Although the average s t a t i s t i c s quoted in Table 5.1 give a good i n d i c a t i o n of the model's performance on the three days they obscure s i g n i f i c a n t aspects of the model's behaviour at various locations (see Table A.4.1). For the clear sky case there are two features of note: 152 Table 5.1 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from the Application of , the Gautier et a l . (1980) model to a sample of ten (10) days  representing variable sky cover conditions at diff e r e n t times of  the year.Units KJnr'hr' 1 ..Numbers in parentheses are for a 1 9-day  sample.Array size;5x5. DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL SUMMER 183/80 192 61.74 116.34 3.28 6.19 .996 1879.7 1941.5 (clear) 239/80 132 -86.34 257.18 -11.10 33.00 .889 780.3 693.9 (p/cloudy) 197/80 191 -5.60 114.10 -1.37 27.82 .892 409.9 404.3 (overcast) SPRING 121/80 155 71.22 134.95 3.57 6.76 .992 1995.9 2067.1 (clear) 160/80 168 29.30 269.60 4.42 40.66 .936 663.0 692.3 (p/cloudy) 105/80 142 161.32 229.44 40.74 58.00 .899 395.6 557.3 (overcast) FALL 276/79 120 21.88 119.78 1.45 7.96 .987 1505.7 1527.6 (clear) 261/80 90 -161.60 412.70 -19.40 49.40 .819 834.8 673.1 (p/cloudy) 263/79 105 -12.53 140.86 -3.26 36.58 .779 385.0 372.5 (overcast) WINTER 025/80 86 -9.81 103.83 -0.99 10.42 .962 996.2 986.3 (clear) (9-day)l29l 30.00 178.30 2.90 17.40 .982 1027.1 1057.1 (sample) (2638)(30.00)(230.10) (218)(21.4)(.972)(1073.1)(1103.1) Figure 5.1a: Observed radiation and calculated radiation from the Gautier et a l . (1980) model at Grouse Mountain f o r a c l e a r day in summer (183/80). Figure 5.1b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a c l e a r day in summer (183/80). LOCRL APPARENT TIME Figure 5.1c: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at P i t t Meadows f o r a cl e a r day in summer (183/80). ro r\j 3 * OBSERVED c CRLCULRTED t n 6 9 12 1-5 18 LOCRL RPPRRENT TIME 21 24 Figure 5.2a: Observed radiation and calculated radiation from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a pa r t l y cloudy day in summer (239/80). OBSERVED CRLCULRTED 0 3 6 9 . 12 15 . 18 2 1 ' 24 LOCRL RPPRRFNT TIME Figure 5.2b: Observed radiation and calculated radiation from the Gautier et a l . (1980) model at Abbotsford Library f o r a pa r t l y cloudy day in summer (239/80). CM _ CD X~ o- — — -RBBLIB 197 OBSERVED CRLCULRTED tn Oo 24 LOCRL APPARENT TIME Figure 5.3a: Observed radiation and calculated radiation from the Gautier et a l . (1980) model at Abbotsford Library f o r an overcast day in summer (197/80). C O O N ) 3 9 * CD — -VRNRIR 197 12 15 18 21 CD OBSERVED CRLCULRTED tn 24 LOCRL RPPRRENT TIME Figure 5.3b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r an overcast day in summer (197/80). 160 1) the increase in both the Mean Bias Error and RMS error at the i n t e r i o r locations and 2) the perfect long term agreement between estimated and measured values at Grouse Mountain, UBC Climate Station and Tsawwassen Ferry Terminal. The increased MBEs and RMSEs observed at the i n t e r i o r locations are in contrast to the finding for the o r i g i n a l Tarpley (1979) model (Section 4.2) where values of both s t a t i s t i c s were lower at the i n t e r i o r l o c a t i o n s . However, the observation i s consistent with the behaviour of the revised Tarpley (1979) model (Section 4.3.1). Again we appeal to the influence of aerosols as postulated by Hay (1982). Since the model's formulation doesn't account for the effects of aerosols, a reduction in the measured values due to t h e i r influence w i l l not be accompanied by a corresponding reduction in the calculated values. Since the model has a general tendency to overestimate the r a d i a t i o n , neglecting aerosol e f f e c t s w i l l lead to further overestimation and hence an increase in the error s t a t i s t i c s . If we a t t r i b u t e the bias in the model mainly to the exclusion of aerosols then the close agreement between estimated and measured values observed at Grouse Mountain, UBC Climate Station and Tsawwassen Ferry i s consistent with the fact that these areas are least l i k e l y to be influenced by the build-up of aerosols. The model's longterm behaviour under partly cloudy conditions is dominated by underestimation while under overcast conditions there is a mixture of both overestimation and underestimation (see Table A.4.1). The average RMS errors on both days (239/80 and 197/80) are representative of conditions across the network. Since the model overestimates the rad i a t i o n for the clear sky the underestimation observed for the pa r t l y 161 cloudy day (at a l l stations except Grouse Mountain) suggests that the problem l i e s with the cloud attenuation calculations ( i . e . an inadequate cloud threshold) (note that stations such as Abbotsford Library and Abbotsford A i r p o r t in close proximity and with s i m i l a r d a i l y totals have markedly d i f f e r e n t errors Table A.4.1). If the cloud threshold (Section 5.1.1.4) is too low, then pixels that are clear w i l l be classed as cloudy p i x e l s . Radiation calculations w i l l be performed using the .cloudy Atmosphere rather than the clear sky model. This w i l l r e s u l t in a lower calculated r a d i a t i o n value leading to the underestimation. Figures 5.2a and 5.2b show that partly cloudy conditions e x i s t at Vancouver Airport-and Abbortsford Library. However, the hourly data for these two stations reveal that'many of the'5 x 5 pixel arrays are'cl'assi f i e d as being completely cloudy even when the measured radi a t i o n input i s quite s u b s t a n t i a l . This supports our i n t e r p r e t a t i o n . The variable pattern of overestimation and underestimation on the overcast day could be explained as follows: where overestimation occurs the bias i s l a r g e l y due to the inappropriate handling of cloud absorption and r e f l e c t s the l i m i t a t i o n s of the use of cloud brightness to denote cloud absorption; where underestimation occurs conditions are closer to those observed for the partly cloudy day and the e a r l i e r explanation would be appropriate. Table A.4.1 shows that values of the measured rad i a t i o n are much higher at these locations than at the locations where the model overestimates the measured r a d i a t i o n . The absolute values of the Mean Bias Error and RMS error on the overcast day are smaller than both the c l e a r sky case and the partly cloudy case. The r e l a t i v e RMS errors are greater than in the clear sky 162 case because of the reduced radiation input under overcast conditions. The smaller absolute RMSEs indicate that the model's performance i s better for the overcast day (197/80) than for the other days. The s i m i l a r i t y in absolute RMS errors for the clear and overcast days attests to the consistency in model performance between the cle a r sky and the overcast sky condition. The quality of this performance deteriorates for the partly cloudy day (239/80). The reduction in the Mean Bais Error for the overcast day can be p a r t i a l l y explained i f we assume the following: 1) part of the bias in the model's formulation under clear skies i s due to the exclusion of aerosols and 2) Part of the bias in the model's formulation for cloud s i t u a t i o n s is due to the cloud threshold and that the threshold is more important under partly cloudy conditions. The effects of aerosols w i l l be outweighed by the effects of the clouds in the overcast s i t u a t i o n . In addition, the l i k e l i h o o d of m i s c l a s s i f y i n g pixels because of an inappropriate threshold would be reduced under overcast conditions because of the larger brightness values (much greater than the threshold) usually associated with those conditions. The combination of these two factors should lead to a reduction in both the Mean Bias Error and the RMS e r r o r . 5.3 Model Modification 5.3.1 Changes in Temporal Averaging In our i n i t i a l assessments of the Gautier et a l . (1980) model (Section 5.2), the calculated radiation was determined through a procedure we refer to as 'flux averaging'. That i s , the incident 163 shortwave flux is determined for each image relevant to a given hour, and the various fluxes are subsequently weighted and combined to y i e l d the calculated radiation for the entire hour. In this section we examine the effects on the model's performance, r e s u l t i n g from changes in the temporal averaging scheme. We use a procedure referred to as 'pixel averaging' by which the radiation calculations are performed on a weighted average image formed from the combination of the two or more images available for a given hour. Values of the parameters used in equation 3.2.3 (the c l e a r sky model) and Equation 3.2.6 (the cloudy atmosphere model) are also weighted averages while in this case the e x t r a t e r r e s t r i a l radiation i s calculated f o r the mid-point of the hour. We applied this revised temporal averaging scheme to the same three summer days (183/80, 197/80 and 239/80). Average s t a t i s t i c s for the three days are l i s t e d in Table 5.2a (complete s t a t i s t i c s for the individual stations are given in Table A.4.2 see Appendix A.4) and show that both the Root Mean Square and Mean Bias Errors increase r e l a t i v e to the i n i t i a l averaging scheme (Table 5.1). However, the MBE for day 197/80 s t i l l indicates equality between estimated and measured values. The decreases in the c o r r e l a t i o n c o e f f i c i e n t (r) for the c l e a r sky and overcast days indicate an increase in the random fluctuations between measured and estimated values. The changes r e s u l t i n g from a re v i s i o n of the temporal averaging configuration are further defined from a comparison of Tables A.4.1 and A.4.2. Under c l e a r skies (day 183/80) increases (up to 3%) occur in both the Mean Bias and Root Mean Square errors at a l l s t a t i o n s . These increases are inconsistent with expectations and with the re s u l t s using 164 Table 5.2 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the Gautier et al.(l980) model based on a Pixel Averaging  Approach (4.7a) and a Spatial Averaging Approach ( 4 . 7 b 7 T u n i t s KJnr 2 h r • 1 . ~ DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL (a) PIXEL AVERAGING 183/80 192 99.53 183.78 5.29- 9.78 .990 1879.7 1979.2 (clear) 239/80 132 -100.27 258.52 -12.85 33.10 .893 780.3 679.9 (p/cloudy) 197/80 191 -8.70 119.62 -2.12 29.20 .878 409.9 401.2 (overcast) (b) SPATIAL AVERAGING 8x8 ARRAY 183/80 192 65.16 112.44 3.47 5.98 .997 1879.7 1944.9 (clear) 239/80 132 -90.87 248.71 -11.65 31.88 .900 780.3 689.4 (p/cloudy) 197/80 167 -2.65 137.30 -0.57 29.69 .779 409.9 459.6 (overcast) *(-14.93) (119.46)(-3.23)(25.8)(.842) (447.4) * Numbers in parentheses are based on a 5x5 pixel array. o 165 the revised Tarpley.(1979) model (Section 4.3.2). Given the high c o r r e l a t i o n between the half-hourly s a t e l l i t e data under clear sky conditions, changes i n temporal averaging should have l i t t l e e f f e c t on the calculated r a d i a t i o n therefore, these r e s u l t s may be unrepresentative. By contrast only small changes are observed for the par t l y cloudy and overcast days. Pixel averaging leads to lower values for the calculated irradiance thereby decreasing i n i t i a l overestimations but further increasing underestimations. The only exceptions to th i s pattern are at Abbotsford Library and Abbotsford A i r p o r t on day 197/80 where decreased underestimation i s observed. Small increases in the RMS errors i s the dominant pattern on both days. On day 197/80 increases in the RMS errors are associated with decreases i n -the c o r r e l a t i o n c o e f f i c i e n t (r) (e.g. at Northmount RMSE increases from 27% to 29% and (r) decreases from .917 to .895). For the par t l y cloudy day (239/80) increases i n the RMSE i s related both to a decrease in the c o r r e l a t i o n c o e f f i c i e n t (r) and an increase i n the Mean Bias Error (e.g. at Tsawwassen Ferry the RMSE increases by 2%, the MBE increases by 1% and (r) decreases from .909 to .888). The decrease i n the RMSE at Grouse Mountain i s associated with a decrease i n the Mean Bias Error. The r e s u l t s for the par t l y cloudy and overcast days indicate that the possible source of bias observed could be the loss of information that i s an integral part of the pixel averaging procedure. The d i r e c t i o n of the changes observed t e n t a t i v e l y support the Gautier et a l . (1980) contention that the high temporal resolution used in t h e i r model takes into account the cloud v a r i a b i l i t y . The results are in harmony with those obtained for the revised Tarpley (1979) 166 model but the extent of the changes observed i s far less than what might be expected given the loss of cloud information incurred in the averaging process. 5.3.2 Changes in Spatial Averaging Evaluation of the model by Gautier et a l . (1980) has been based on a 5 x 5 pixel array centered on each s t a t i o n . The choice of t h i s array s i z e i s based on the accuracy of our earth l o c a t i o n routine (see Section 2.6). In t h i s section we investigate the e f f e c t s of a change in array si z e on the performance of the model. S p e c i f i c a l l y , we w i l l evaluate the model for an 8 x 8 pixel array. This array s i z e was used by the o r i g i n a l researchers to partly account for the time discrepancies r e s u l t i n g from the d i f f e r e n t types of measurements represented by the pyranometric and s a t e l l i t e data and for the multi-channel nature of the VISSR sensor. Our comparisons of the results from the 5 x 5 and 8 x 8 s p a t i a l configurations may help to provide some insights into the usefulness of this i m p l i c i t space/time t r a n s l a t i o n involved in the changes in s p a t i a l averaging. Model evaluations were again conducted for 183/80, 197/80 and 239/80. Average s t a t i s t i c s f o r the three days are l i s t e d i n Table 5.2b (complete s t a t i s t i c s for the individual stations are given in Tables A.4.3 and A.4.7). Spatial averaging leads to small increases in the average Mean Bias Errors and small decreases in the average RMS errors for the partly cloudy and c l e a r days r e l a t i v e to the i n i t i a l averaging configuration (Table 5.1). The average RMS error increases for the 8 x 8 array on day 197/80. A comparison of Table A.4.3 and A.4.1 shows 167 that for the c l e a r sky case only minor changes occur in the Mean Bias Errors and Root Mean Square Errors at a l l the stations except for the UBC Climate Station and the Tsawwassen Ferry. The changes in the MBEs at these two stations are in opposite directions and suggest that the 8 x 8 array samples a more representative area at the Tsawwassen Ferry than at UBC Climate Station. Part of the difference may be due to the smaller array s i z e (5 x 5) used in the i n i t i a l determinations of the minimum brightness (and hence of the surface albedo). Minimum brightness values would tend to be more conservative over larger areas at locations such as the Tsawwassen Ferry than they would be at the UBC Climate Station. The o v e r a l l pattern of minor changes is what i s to be expected for the clear sky case given the s p a t i a l l y conservative nature of the ra d i a t i o n f i e l d under these conditions. Our results are consistent with those obtained for the revised Tarpley (1979) model under c l e a r sky conditions.(Section 4.3.2). The effects of a change in the s p a t i a l averaging vary across the network for the partly cloudy day (239/80) with no consistent pattern d i s c e r n i b l e . Table A.4.3 shows that the changes that do occur are quite small. Decreases in the RMS errors and Mean Bias Errors are noted at many of the st a t i o n s . Under overcast conditions the change in s p a t i a l averaging leads to more unambiguous changes in the model's performance (Tables A.4.3 and A.4.7). RMS errors increase at a l l stations with a corresponding decrease in the c o r r e l a t i o n c o e f f i c i e n t (r) (e.g. at Langara the RMSE increases from 16% to 24% and (r) decreases from (.981) to (.904)). This suggests that the increase in the RMS error may be due to greater v a r i a b i l i t y in conditions over the larger array. 168 Consistent changes i n Mean Bias Errors are observed since increasing the array si z e always leads to lower values for the calculated irradiance. A s i m i l a r pattern was observed as a r e s u l t of changes in the temporal averaging (Section 5.3.1). The b e n i f i t s to be accrued from the use of an 8 x 8 pixel array for the appl i c a t i o n of the Gautier et a l . (1980) model has not been demonstrated. Instead our results have revealed some ambiguity in model response; that i s both improvements and deteriorations have resulted from a change in the spa t i a l averaging configuration. Our results favour the use of a 5 x 5 pixel array for model evaluations. This i s well supported by the model's performance for the overcast day (197/80). The marginal improvements observed for some locations under clear and partly cloudy conditions are well within the 5% error associated with our measured data and do not imply the need to downgrade the sp a t i a l resolution of the data. 5.4 Seasonal Assessments The o r i g i n a l model as developed by Gautier et a l . (1980), was v e r i f i e d using data from the spring of 1978 and the summer of 1979. We attempt to v e r i f y the model for s i m i l a r conditions ( i . e . spring and summer) with an extension to include conditions occurring in the f a l l and winter. Analyses were performed on the same data used in an e a r l i e r Section (4.4) for analyses of the revised Tarpley (1979) model. This enables us to compare the re s u l t s obtained from both models. Hourly average s t a t i s t i c s for the ten (10) days assessed are l i s t e d in Table 5.1 (since day 261/80 i s used in the developmental data set for the revised Hay and Hanson 1978 and Tarpley 1979 models, 169 s t a t i s t i c s for this day are not used in the intercomparisons) (complete s t a t i s t i c s for a l l days are given in Appendix A.4). From Table 5.1 we note the following; on average and under clear skies the model systematically overestimates the radiation in summer, spring and f a l l (the MBE of 1.45% for the f a l l indicates essential agreement between estimated and measured values). In winter the average MBE indicates agreement between estimated and measured values. The average RMS errors for the clear days are s i m i l a r for the spring, summer and f a l l but the r e l a t i v e value increases s i g n i f i c a n t l y for the winter time s i t u a t i o n . Figures 5.1a - 5.1c and 5.4a -'5.6b show the model's performance at selected stations in the network, while Table A.4.1 and A.4.4 show the model's behaviour at the individual stations for the four c l e a r days (183/80, 121/80, 276/79 and 025/80). These generally support the re s u l t s given i n Table 5.1 but some additional features are of note. The consistent pattern of larger errors at the f i v e i n t e r i o r stations noted for the summer time s i t u a t i o n i s not evident i n spring and f a l l although at some stations (e.g. P i t t Meadows) the large overestimation is maintained. In winter the Mean Bias Errors are in some cases s u b s t a n t i a l l y larger than at other times of the year. The occurrence of large p o s i t i v e and negative values account for the small average error quoted in Table 5.1. The only consistent s p a t i a l pattern is for overestimation in the urban and suburban areas. , The underestimati observed for the winter s i t u a t i o n must be due at le a s t in part to an inadequate threshold. An examination of the hourly data for Abbotsford A i r p o r t (the location where the greatest underestimation is observed) revealed that many of the pixels in the 5 x 5 array for a p a r t i c u l a r Figure 5.4a: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a c l e a r day in spring (121/80). L O C R L R P P R R F N T T I M F Figure 5.4b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at P i t t Meadows f o r a c l e a r day in spring (121/80). C M _ 3 C 9 12 OBSERVED _ c CRLCULRTED 15 18 21 24 L O C R L R P P R R E N T T I M E Figure 5.5a: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at P i t t Meadows f o r a c l e a r day in f a l l (276/79). L O C A L A P P A R E N T T I M E Figure 5.5b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a c l e a r day in the f a l l (276/79). cr o •z C\J cr i — i Q cr cr cr ro _ O x ) 0 3 x. o — -PITMED 2 5 OBSERVED CRLCULRTED 18 21 - • j 24 L O C R L R P P R R E N T T I M E Figure 5.6a: Observed r a d i a t i o n and c a l c u l a t e d r a d i a t i o n from the Gau t i e r e t a l . (1980) model a t P i t t Meadows f o r a c l e a r day i n w in te r (025/80) . CO _ C\J CD X-VANAIR 2 5 ^ OBSERVED o CALCULATED 6 9 12 15 18 L O C A L A P P A R E N T T I M E 2 24 Figure 5.6b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a clear day in winter (025/80). 176 hour are c l a s s i f i e d as being cloudy which means that c a l c u l a t i o n s w i l l be performed using the cloudy atmosphere model thereby generating the underestimation. Absolute RMS error values are seasonally consistent except in the winter time when values are lower. However, the reduced radiation input i n winter time would explain the larger r e l a t i v e errors observed. The RMSE differences between stations tend to r e f l e c t differences in the MBE. This suggests that the random component in the error i s s p a t i a l l y conservative. The cons i s t e n t l y high (r) values at the individual stations support t h i s conclusion. From Table 5.1 the average behaviour of the model f o r the pa r t l y cloudy days i s characterized by underestimation i n summer and f a l l with overestimation occurring in the spring. The average RMS error i s largest f o r the f a l l (49.4%) and smallest f o r the summer (33.0). Figures 5.2a - 5.2b and 5.7a - 5.8b show the model's performance at selected stations in the network, while Table A.4.6 shows the model's behaviour at the individual stations f o r the three partly cloudy days. The r e s u l t s i n Table A.4.6 reveal that the average RMS erro r on day 261/80 i s not t r u l y representative. At most stations the RMS error i s within 30% of the mean measured rad i a t i o n . The large values at Grouse Mountain, Northmount and BC Hydro give r i s e to the large average RMS error observed. The average MBE does not r e f l e c t the unbiased behaviour of the model at Vancouver A i r p o r t , Tsawwassen Ferry, Abbotsford Library and UBC Climate Station on day 261/80. The res u l t s in Table A.4.6 also reveal that the pattern of underestimation observed f o r the summertime at a l l stations excluding Grouse Mountain i s not retained in spring Figure 5.7a: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a p a r t l y cloudy day in spring (160/80). c r ZD o LxJ CJ cr . i-—i Q c r c r c r CO C M CD 6 9 .12 15 18 21 -x CD OBSERVED CRLCULRTED oo 24 . O C R L R P P R R E N T T I M E Figure 5.7b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Abbotsford Library f o r a partly cloudy day in spring (160/80). cr o LU L J CE Q CE cr cr ro C\j CD 0 3 G x-CD VRNRIR 2 6 1 -X o OBSERVED CRLCULRTED 12 15 18 21 24 LOCAL RPPRRFN T T.IMF Figure 5.8a: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r a pa r t l y cloudy day in f a l l (261/80). I or ZD O ZC CM LU C_J CE t ! Q CE c r C O CM _ 3 12 15 O-^ OBSERVED o CRLCULRTED RBBLIB 261 18 21 0 0 o 24 LOCRL RPPRRFNT T IME Figure 5.8b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Abbotsford Library f o r a partly cloudy day in f a l l (261/80). 181 and f a l l . The s i g n i f i c a n t biases in spring are overestimates while in f a l l s i g n i f i c a n t biases are underestimates. The differences in behaviour observed f o r the three p a r t l y cloudy days suggest that part of the problem l i e s with the cloud threshold. We argued e a r l i e r (Section 4.3.2) that the behaviour of the model on day 160/80 i s strongly influenced by the heavily overcast conditions occurring during the morning period (Figures 5.7a and 5.7b) and hence more c l o s e l y resembles the behaviour expected on an overcast day. The pattern of overestimation observed f o r the Gautier et a l . (1980) model on t h i s day tends to support t h i s argument. The average errors for the overcast days (Table 5.1) indicate s i g n i f i c a n t model overestimation in spring, with some underestimation in f a l l and l i t t l e or no underestimation f o r the summer (day 197/80). The average RMS error i s largest f o r the spring s i t u a t i o n and smallest fo r the summertime. Figures 5.3a - 5.3b and 5.9a - 5.10b shows the model's performance f o r the three days (263/79, 105/80 and 197/80) at selected locations in the network. Table A.4.5 l i s t the i n d i v i d u a l station s t a t i s t i c s f o r the three days. These show that in spring substantial overestimation occurs at a l l locations, the most noteworthy being Grouse Mountain (MBE 170%, RMSE 200%). The pattern of behaviour i s s i m i l a r f or the summer and spring; that i s s i m i l a r areas experience the large under and overestimates that do occur. The largest overestimates are observed at the urban and suburban stations (excluding Langara on day 263/79). We a t t r i b u t e part of the overestimation observed on the overcast days to the way absorption by clouds have been handled in the model. It appears that the use of cloud brightness to indicate absorption I cr ZD o C M I L U L D cr CD CT cr C O _ C M CD- — •- -VRNRIR * OBSERVED CRLCULRTED 0 0 I X ) L O C R L R P P R R E N T T I M E Figure 5.9a: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r an overcast day in spring (105/80). cr. o ZC-CM LU CJ CU. a cr az cr. C O C\J CD 0 3 cr . 0 . x-O-RBBLIB 105 12 1 5 ' 18 21 -X OBSERVED CRLCULRTED 0 0 CO 24 L O C R L R P P R R E N T T I M E Figure 5.9b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Abbotsford Library f o r an overcast day in spring (105/80). cr ID nz CXI L u cr I ! Q cr cr. cr. co I ONJ C D 3 x-o- — — -V R N R I R 2 6 3 18 21 -x OBSERVED CRLCULRTED 24 L O C R L R P P R R E N T T I M E Figure 5.10a: Observed radiation and calculated radiation from the Gautier et a l . (1980) model at Vancouver A i r p o r t f o r an overcast day in f a l l (263/79). * * O B S E R V E D c 0 C f l L C U L R T E D R B B L I B • 2 6 3 0 3 6 9 12 15 18 21 24 L O C R L R P P R R E N T T I M F Figure 5.10b: Observed radiation and calculated r a d i a t i o n from the Gautier et a l . (1980) model at Abbotsford Library f o r an overcast day in f a l l (263/79). 186 does not account f o r a l l the absorption occurring under cloud conditions. The measured t o t a l d a i l y r a d i a t i o n i s generally higher at locations where underestimation i s observed. This suggests that conditions at these locations are closer to the pa r t l y cloudy s i t u a t i o n hence the reason for the underestimation. The substantial overestimations observed f o r day 105/80 i s s i m i l a r to that noted in the assessments of the revised Tarpley (1979) model (Section 4.3.4). The consideration of the e f f e c t s of changing s u n - s a t e l l i t e angle under overcast conditions (Section 4.7) may also be important in the context of t h i s model. We a t t r i b u t e the anomalous values at Grouse Mountain (MBE 170%, RMSE 200%) to markedly d i f f e r e n t cloud c h a r a c t e r i s t i c s occurring at t h i s location than at the other locations (note that the measured radiation i s less than half the value of the measured radiation at most of the other s t a t i o n s ) . Seasonal analyses f o r the Gautier et a l . (1980) model show that under c l e a r skies, a revised Tarpley (1979) model y i e l d s smaller average MBE and RMS errors under summer and spring conditions. For a c l e a r day in the f a l l the average RMS errors are quite s i m i l a r while a small average MBE i s observed f o r the Gautier et a l . (1980) model. The average performance of the Gautier et a l . (1980) model i s better fo r the c l e a r day in winter. The pattern of overestimation observed for the Gautier et a l . (1980) model in summer, spring and f a l l i s in contrast to the underestimation observed f o r the revised Tarpley (1979) model under s i m i l a r conditions. Average RMS errors f o r the p a r t l y cloudy days are much smaller f o r the Gautier et a l . (1980) model than for the revised Tarpley (1979) model while average Mean Bias Errors 187 are s i m i l a r f o r day 239/80 but larger f o r the revised Tarpley (1979) model on day 160/80. For the overcast days average RMS errors and Mean Bias Errors are smaller f o r the Gautier et a l . (1980) than for the revised Tarpley (1979) model. Our r e s u l t s indicate that f o r c l e a r skies the revised Tarpley (1979) model gives s l i g h t l y better estimates than the Gautier et a l . (1980) model (at l e a s t in summer and spring) while the Gautier et a l . (T980) model provides f a r better estimates than the revised Tarpley (1979) model under p a r t l y cloudy and overcast conditions. 5.5 Daily S t a t i s t i c s As part of our v e r i f i c a t i o n and to f a c i l i t a t e comparison with the e a r l i e r r e s u l t s of Gautier et a l . (1980) we examine the e f f e c t s on the model 1s performance of an increase i n the averaging period. S p e c i f i c a l l y , we evaluate the model's performance for periods of c l e a r , p a r t l y cloudy and overcast days and f o r two samples of nine and nineteen days incorporating a va r i e t y of sky cover conditions. Values at the twelve stations were combined to y i e l d an average value f o r each period. Daily average s t a t i s t i c s f o r the f i v e periods are l i s t e d in Table 5.3. These show small overestimation f o r the c l e a r sky days, small underestimation f o r the p a r t l y cloudy days and substantial overestimation f o r the overcast days. The large bias in the model f o r the overcast days i s d e f i n i t e l y influenced by day 105/80 see Table 5.1. Substantial reduction in both the RMS errors and Mean Bias Errors f o r the d a i l y periods are observed f o r a l l conditions r e l a t i v e to the hourly s t a t i s t i c s quoted in Table 5.1. The RMS e r r o r f o r c l e a r days i s within ±5%. This i s s i m i l a r to both 188 Table 5.3 Daily Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the Gautier et a l . -(1980) model to periods of cl e a r , p a r t l y  cloudy and overcast days and to an a l l day sample of nine (9) days. Units MJm'^day'1. Array size;5x5." PERIOD N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL CLEAR 48 0.514 0.839 2.63 4.30 .999 19.5 20.0 DAYS P/CLOUDY 24 -0.270 1.070 -3.02 11.95 .751 8.9 8.7 DAYS OVERCAST 36 0.570 1.280 11.73 26.39 .743 4.9 5.4 DAYS 9-DAY 108 0.358 1.060 2.90 8.60 .995 12.3 12.6 SAMPLE *(208) (0.410) (1.400) (3.31)(11.23)(.990)(12.4) (12.8) *Numbers in parentheses are for a 19-day sample. 189 the error f o r c l e a r days observed f o r the revised Tarpley (1979) model and to that reported i n the o r i g i n a l Gautier et a l . (1980) study. The overestimation observed f o r the Gautier et a l . (1980) model i s in contrast to the underestimation observed f o r the revised Tarpley (1979) model under s i m i l a r conditions. The large value of (r) (.999) suggests that a large portion of the RMS error i s due to the bias i n the model. The average RMS error f o r p a r t l y cloudy days i s within ±12% and represents a s i g n i f i c a n t improvement over the error observed f o r the revised Tarpley (1979) model f o r the"same conditions. A ±12% error f o r p a r t l y cloudy days i s smaller than the error quoted by Gautier et a l . (1980) f o r t h e i r p a r t l y cloudy condition. The s l i g h t underestimation f o r the d a i l y values i s in contrast to the overestimation observed f o r the revised Tarpley (1979) model. The average' Mean Bias Error f o r the overcast days are s i m i l a r f o r both the Gautier et a l . (1980) model and the revised Tarpley (1979) model. However, the average RMS error quoted in the o r i g i n a l Gautier et a l . (1980) study i s s u b s t a n t i a l l y smaller. Gautier et a l . (1980) reported a RMS error of ±15% f o r t h e i r overcast days. This i s much smaller than the RMS error (±26% observed in t h i s study. The larger errors observed f o r the p a r t l y cloudy and overcast days r e f l e c t the det e r i o r a t i o n in the model's performance under these conditions. A further decrease in the RMS error i s observed f o r the nine (9) day averaging period. The RMS error i s now within ±9% of the mean measured d a i l y r a d i a t i o n . This error i s consistent with that reported by Gautier et a l . (1980) and i s much smaller than the RMS error f o r the revised Tarpley (1979) model f o r the same sample of nine days. The 190 size of the RMS erro r i s d e f i n i t e l y influenced by the smaller values occurring under c l e a r sky conditions. The Mean Bias Error f o r the nine (9) day period s t i l l indicates small overestimation by the model. Since the model by Gautier et a l . (1980) i s physically-based, the days l i s t e d i n Table 4.6 are a l l a v a i l a b l e f o r model evaluations; hence the 19-day sample. Both the average MBE and RMSE increase f o r the 19-day sample. The larger errors r e f l e c t the i n c l u s i o n of a greater number of p a r t l y cloudy and overcast days to c l e a r days. Although the RMS e r r o r and MBE increase f o r the 19-day sample they are s t i l l smaller than the hourly values quoted i n Table 5.1. The decrease i n the error s t a t i s t i c s reported f o r the d a i l y average i s t y p i c a l of an increased averaging period. This r e s u l t s from a c a n c e l l a t i o n of random errors over the longer time period. Daily averaging has resulted i n s i m i l a r performances f o r both the revised Tarpley (1979) model and the Gautier et a l . (1980) model under c l e a r sky conditions and superior performances f o r the Gautier et a l . (1980) model under p a r t l y cloudy and overcast conditions. The r e s u l t s from the 19-day sample confirm that the averaging period as well as the composition of the sample are important determinants of the model's performance. 5.6 Implications of Model Accuracy f o r Spatial Sampling Requirements Using the technique of Alaka (1970), Hay (1981) has quantified the mesoscale s p a t i a l v a r i a b i l i t y i n the study area in terms of the Root Mean Square Errors of i n t e r p o l a t i o n and associated separation distances fo r hourly and d a i l y values (Table 1.1). The analysis i s based on 2.5 years of measured data f o r a l l the stations i n the network excluding Grouse Mountain. To ascertain the sampling distances congruent with the 191 accuracy of the model, we evaluated the performance f o r the network as a unit. To be consistent with the Hay (1981) study values f o r Grouse Mountain were not included. The RMS error f o r hourly values f o r the 9-day sample i s ±17.1% of the mean measured global r a d i a t i o n , the average RMS error decreases to 8.2% f o r the d a i l y values. For the 19-day sample the average hourly RMS error i s ±20% of the mean measured global r a d i a t i o n decreasing to ±11.3% f o r d a i l y values. From Table 1.1 we note that a RMS error of ±17.1% f o r hourly values l i m i t s the spa t i a l resolution of the s a t e l l i t e model to a djstance of approximately 15 km while a RMS error of ±8.2% f o r d a i l y values l i m i t s the s p a t i a l resolution to a distance of approximately 12 km. At shorter distances'1 the actual fluctuations i n the r a d i a t i o n , and errors in the model calc u l a t i o n s w i l l be indistinguishable. For the 19-day sample a ±20% RMS error f o r hourly values i s associated with a s p a t i a l r e s o l u t i o n of approximately 25 km while a ±11.3% RMS error f o r d a i l y values i s associated with a spat i a l resolution of approximately 15 km. The RMS errors calculated f o r the Gautier et a l . (1980) model are smaller than those associated with the revised Tarpley (1979) model a t t e s t i n g to the f i n e r s p a t i a l resolution which may be achieved by the Gautier et a l . (1980) model. We hypothesize that the decrease in model resolution f o r the larger sample r e f l e c t s the influence of the larger number of overcast and pa r t l y cloudy days to cl e a r days included and further suggests the need to specify the sample composition when quoting model accuracy. 5.7 Summary and Conclusions * The physically-based framework of the Gautier et a l . (1980) model ensures that only a change in the relevant parameters (scattering 192 and absorption) i s necessary before i t can be applied to a p a r t i c u l a r area. Application of the model to summertime conditions shows that the model overestimates the measured under c l e a r and overcast conditions but underestimates f o r the p a r t l y cloudy case. Model bias i s att r i b u t e d to factors such as the exclusion of aerosols, and to problems in cloud attenuation c a l c u l a t i o n s (including an inadequate threshold and problems of cloud absorption). The pattern of overestimation under clear skies i s maintained f o r the spring and f a l l but s i g n i f i c a n t underestimation is observed f o r the winter time s i t u a t i o n . For the p a r t l y cloudy and overcast days analysed the problems of an inadequate threshold and inappropriate cloud absorption c a l c u l a t i o n s are apparent at other times of the year. The s i g n i f i c a n t overestimation observed f o r day 105/80 (spring) suggests that f o r some cloud regimes a consideration of the ef f e c t s of changing s u n - s a t e l l i t e azimuth angle i s warranted. Comparisons with the res u l t s from a revised Tarpley (1979) model reveal that the present model i s superior in i t s performance except under c l e a r skies i n summer and spring. Evaluations of the model's s e n s i t i v i t y to changes in the temporal and s p a t i a l averaging configurations indicate that temporal averaging (pixel averaging) leads to unexpected increases under c l e a r skies with smaller changes consistent with expectations under overcast and p a r t l y cloudy conditions. However, the extent of the changes observed only lend tentative support to the claim that a high temporal reso l u t i o n takes into account cloud v a r i a b i l i t y . In a s i m i l a r manner, changes i n the s p a t i a l configuration do not support the use of an 8 x 8 pixel array to improve the temporal representation of the s a t e l l i t e data. 193 Assessments of the model f o r d a i l y and longer time periods r e s u l t in substantial reductions in the RMS and Mean Bias Error s t a t i s t i c s observed, consistent with expectations. The superior performance of the Gautier et a l . (1980) model over the revised Tarpley (1979) model i s maintained f o r the p a r t l y cloudy and overcast conditions. In a d d i t i o n , the performance f o r c l e a r days i s s i m i l a r f o r both models. In terms of the mesoscale s p a t i a l v a r i a b i l i t y in the solar irradiance resolvable by the model; model accuracy would di c t a t e a s p a t i a l r esolution of 15 km f o r hourly values and 12 km f o r d a i l y values. The implementation of the Gautier et a l . (1980) model f o r the study area has shown that the major focus f o r model improvements must be the p a r t l y cloudy and overcast s i t u a t i o n s . The problems that heed addressing include 1) an improved cloud threshold; 2) more accurate cloud absorption c a l c u l a t i o n s and 3) a consideration of the importance of s u n - s a t e l l i t e azimuth on the signal received by the s a t e l l i t e under cloudy conditions. In addition, for the c l e a r sky model the e f f e c t s of aerosols need to be included. 194 CHAPTER SIX 6 . Application of the Hay and Hanson Model 6 . 1 Imp!ementation In contrast to the Tarpley ( 1 9 7 9 ) model, the model by Hay and Hanson ( 1 9 7 8 ) requires l i t t l e in the way of adaptation before i t can be applied to the study area. The model i s described by the Equation 3 . 3 . 3 , where the irradiance at the surface ( I g ) i s written: I S = I Q cos 6 (a - bSR) ( 6 . 1 . 1 ) I Q = solar constant ( 1 3 5 3 W m ) SR = normalized s a t e l l i t e measured reflectance a,b = regression c o e f f i c i e n t s (intercept and slope respectively) e = local s o l a r zenith angle We make the same assumption that the solar zenith and s a t e l l i t e azimuth angles are constant across the network. No height correction i s necessary because this i s i m p l i c i t l y included in the model's formulation via the atmosphere transmittance c a l c u l a t i o n . The brightness scales ( 8 - b i t ) are the same for both the o r i g i n a l formulation and the present model ap p l i c a t i o n . The model does not require any supplementary surface based observational data. The only input required for app l i c a t i o n to the study area are values of normalized s a t e l l i t e reflectance. These are evaluated in Section 6 . 1 . 1 . 195 6.1.1 C a l i b r a t i o n of the S a t e l l i t e Data The c a l i b r a t i o n of the s a t e l l i t e data involves the conversion of the brightness counts- into a normalized reflectance or albedo. The c a l i b r a t i o n procedure used here was provided by E. Smith, Colorado State University (personal communication). Using t h i s procedure a computer-based "look-up table" is generated r e l a t i n g the brightness counts to relevant normalized reflectance and irradiance values. The normalized reflectance for any brightness value (from 0-255 counts) i s obtained simply by locating this brightness value in the look-up table and r e t r i e v i n g the appropriate normalized reflectance. The computer routines necessary to generate the look-up table are given in Appendix A.6. Ideally, the contents of the look-up table should be replaced every time the model is applied to a new data set ( i . e . a new day of data). This would mean that the model is s e l f - c a l i b r a t i n g . In r e a l i t y however, there i s only one set of c a l i b r a t i o n data. Thus i t i s assumed that the c a l i b r a t i o n i s constant with time. 6.1.2 Radiation Calculations To evaluate the rad i a t i o n for a given hour we need values of the e x t r a t e r r e s t r i a l global irradiance, the regression c o e f f i c i e n t s and normalized reflectance. In the i n i t i a l applications of the model we use the o r i g i n a l c o e f f i c i e n t s generated by Hay and Hanson (1978). The (a) c o e f f i c i e n t has a value of 0.79, and the (b) c o e f f i c i e n t a value of -0.71. The e x t r a t e r r e s t r i a l global irradiance is simply the product of the solar constant (1353 W m ) and the cosine of the zenith angle at the time of the s a t e l l i t e image. The normalized reflectance is evaluated 196 from a 5 x 5 pixel array centered on the s t a t i o n . The value used i s the average value of the 5 x 5 array. The calculated r a d i a t i o n for an hour i s based on the flux averaging approach. Equation 6.1.1 is used to estimate the "instantaneous" value of the fluxes at the times of the s a t e l l i t e data. These are then weighted an averaged to give the radiation for an hour. The r e s u l t i n g calculated radiation for a given hour and location i s compared with the corresponding measured value. 6.2 Results The i n i t i a l assessments of the Hay and Hanson (1978) model were performed using the days (Julian days 183/80, 197/80 and 239/80) u t i l i z e d in e a r l i e r evaluations of the Tarpley (1979) model (Section 4.2) and the Gautier et al . (1980) model (Section 5.2). This allows for the intercomparison of the results obtained and also minimizes the effects that may be introduced as a r e s u l t of seasonal changes (the Hay and Hanson (1978) model was o r i g i n a l l y developed for summertime conditions). Hourly ra d i a t i o n values were generated for the three days and compared to.the measured radiation at the twelve stations in the network. Hourly average s t a t i s t i c s from these comparisons are l i s t e d in Table 6.1. Figures 6.1a - 6.3b show the behaviour of the model at selected locations in the network. The results in Table 6.1 indicate that on average the model s l i g h t l y overestimates the radiation underclear sky conditions with substantial increases i n overestimation for both p a r t l y cloudy and overcast days. The average RMS errors behave in a s i m i l a r fashion, showing marked increases for the p a r t l y cloudy and overcast conditions 197 Table 6.1 Hourly Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the o r i g i n a l Hay-Hanson (1978) model to three(3) days  representing variable sky cover conditions in the summer (see tex t ) . Array size:5x5. Units KJm-'hr'1. DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL 183/80 192 56.33 171.70 3.00 9.14 .996 1879.7 1936.0 (clear) 239/80 132 419.80 489.90 53.80 62.80 .879 780.3 1200.0 (p/cloudy) 197/80 191 570.00 626.10 139.1*0 152.80 .853 409.9 979.9 (overcast) 0 O B S E R V E D C A L C U L A T E D L O C A L A P P A R E N T T I M E Figure 6.1a: Observed radiation and calculated r a d i a t i o n from the ori g i n a l Hay - Hanson (1978) model at Vancouver A i r p o r t f o r a clear day (183/80) in summer. Figure 6.1b: Observed radiation and calculated r a d i a t i o n from the or i g i n a l Hay - Hanson (1978) model at P i t t Meadows f o r a clear day (183/80) in summer. I cr ZD CD C\J ro _ J CM _J az CD CE cr cr 0 ^ A w / > 2 3 9 X o_ _ _ _ VflNRJR — o OBSERVED CRLCULRTED ro o o '12 15 18 21 24 L O C R L R P P R R E N T T I M E Figure 6.2a: Observed radiation and calculated r a d i a t i o n from the or i g i n a l Hay - Hanson (1978) model at Vancouver A i r p o r t f o r a parly cloudy day (239/80) in summer. I cr ZD o zr. CM co _ J C M _ LU C J CE i—i Q CE cr c r o X- ^ O B S E R V E D c C R L C U L R T E D ro o 12 1'5 18 21 24 . O C R L R P P R R E N T T I M E Figure 6.2b: Observed radiation and calculated r a d i a t i o n from the or i g i n a l Hay - Hanson (1978) model at Abbotsford Library for a pa r t l y cloudy day (239/80) i n summer. ro C\J X- ^ O B S E R V E D o- i - c C R L C U L R T E D R B B L I B 1 9 7 12 15 18 21 24 L O C R L R P P R R E N T T I M E Figure 6.3a: Observed radiation and calculated r a d i a t i o n from the ori g i n a l Hay - Hanson (1978) model at Abbotsford Library for an overcast day (197/80) i n summer. CT Z D o C\J ro CM _J LU CJ 2: CE I 1 Q CE Cr: cr C D -X- ^ O B S E R V E D O - — — — — CD V A N R I R 1 9 7 C A L C U L A T E D ro o co 5 18 21 24 L O C A L A P P A R E N T T I M E Figure 6.3b: Observed radiation and calculated r a d i a t i o n from the or i g i n a l Hay - Hanson (1978) model at Vancouver A i r p o r t f o r an overcast day (197/80) in summer. 204 over the clear sky condition. The average Mean Bias Error for the clear sky case i s s i m i l a r to those observed for the Gautier et a l . (1980) model and for the Tarpley (1979) model (the pattern of overestimation is in contrast to the underestimation observed for the Tarpley (1979) model). The average RMS error for the clear sky case i s s i m i l a r in magnitude to that observed for the Tarpley (1979) model but larger than the RMS error for the Gautier et a l . (1980) model. In contrast to both Tarpley's (1979) model and the Gautier et a l . (1980) model, Table A.5.1 shows that there are no anomalies in the behaviour of the model at the i n t e r i o r l ocations. This i s probably due to the fac t that the model's formulation i m p l i c i t l y includes the eff e c t s of a l l the atmospheric constituents on the incoming r a d i a t i o n . Figures 6.1a and 6.1b show that the model tends to underestimate the measured radiation in the middle of the day and overestimate the measured rad i a t i o n in the early and l a t e r parts of the day. In some instances i t appears that these cancel out to give small Mean Bias Errors while in other instances i t appears that they do not. These over and underestimates w i l l give larger RMSEs than would be expected given the high values of (r) and the low Mean Bias Errors (Table A.5.1). From Table A.5.1 we also note that the average MBE does not r e f l e c t the e f f e c t i v e equality between modelled and measured values at Grouse Mountain, Langara and Vancouver Ai r p o r t for the clear day. The increase in the Mean Bias and Root Mean Square Errors for the pa r t l y cloudy and overcast days i s much larger than that observed f o r both the Gautier et a l . (1980) model and Tarpley's (1 979) model. In sp i t e of the substantial overestimations, Figure 6.2a - 6.3b 205 show that the variations in the measured radiation are quite c l o s e l y mirrored by the variations in the estimated r a d i a t i o n . The r e l a t i v e importance of the systematic and random errors for both days can be shown by the (r) values in Table A.5.1. The large (r) values at BC Hydro on both days indicate that the large RMS errors are mainly influenced by the large Mean Bias Errors, and not random f l u c t u a t i o n s . The (r) values of .72 at Tsawwassen Ferry on day 197/80 and .79 at Langley on day 239/80, indicate that although the Mean Bias Errors are quite large, random fluctuations contribute s i g n i f i c a n t l y to the RMS errors observed. The systematic bias observed in the Hay and Hanson (1978) model points to the i n a b i l i t y of the regression c o e f f i c i e n t s developed for the t r o p i c a l A t l a n t i c to describe the conditions in a mid-latitude location such as the study area. It i s not s u r p r i s i n g that the best agreement i s obtained under clear s k i e s , when variations from one location to the next tend to be quite conservative. The real l i m i t a t i o n s of the o r i g i n a l regression c o e f f i c i e n t s become apparent under the partly cloudy and overcast conditions. Cloud regimes are generally d i f f e r e n t for the mid-latitude and t r o p i c a l environments with the former being dominated by s t r a t i f o r m clouds and the l a t t e r by cumuliform clouds (usually scattered and deep). The behaviour of the model indicates that for s i m i l a r brightness more radiation i s available at the surface under t r o p i c a l conditions than under mid-latitude conditions. Some of the bias in the model can be removed i f the c o e f f i c i e n t s are 'tuned' to f i t the conditions in the study area (Section 6.3.1). 206 6.3 Model Modification 6.3.1 Generation of New Regression Coefficients The developmental data set used to produce the revised regression c o e f f i c i e n t s for Tarpley's (1979) model i s the same one used here to produce the new c o e f f i c i e n t s f o r the Hay and Hanson (1978) model. This allows us to make d i r e c t comparisons of the results obtained from the revised version of both models. The independent variable in the Hay and Hanson formulation (normalized s a t e l l i t e measured reflectance) i s assessed as per Section,6.1.2. The dependent v a r i a b l e , normalized atmospheric transmittance is the hourly measured global radiation divided by the e x t r a t e r r e s t r i a l global irradiance calculated for the mid-point of the hour. Values of both variables are determined for the twelve days in Table 4.5 at a l l stations in the network. The regression c o e f f i c i e n t s are generated using a Triangular Regression Package available at the University of B r i t i s h Columbia Computing Centre. To avoid anomalous values of both the atmospheric transmittance and normalized s a t e l l i t e reflectance due to i n s t a b i l i t i e s in both the dependent and independent variables which occur at large zenith angles when ra d i a t i o n inputs are exceedingly low, values at zenith angles greater than 80° were discarded. A total of 1373 cases were used in the f i n a l regression analysis. The new values for the c o e f f i c i e n t s are (a) = 0.788 and b = -1.078. The (a) c o e f f i c i e n t i s i d e n t i c a l to the o r i g i n a l value given by Hay and Hanson (1978) but the absolute value of the (b) c o e f f i c i e n t which represents the slope of the regression l i n e i s much larger than the Hay and Hanson (1978) value. The standard error of y i s 0.11 the same as for the or i g i n a l formulation. The c o e f f i c i e n t of determination ( r 2 ) show that 86% of the v a r i a t i o n in 207 the atmospheric transmittance is explained by the v a r i a t i o n in the normalized s a t e l l i t e reflectance. The c o r r e l a t i o n c o e f f i c i e n t (r) has a value of 0.93, while Hay and Hanson (1978) quote a value for (r) of 0.85 for t h e i r regression equation. o The revised version of the Hay and Hanson (1978) model was applied to the same suite of independent days used in the e a r l i e r a analyses (183/80, 239/80 and 197/80). The average s t a t i s t i c s generated for the three days are l i s t e d in Table 6.2 (Table A.5.2 contains the complete s t a t i s t i c s for the individual stations) while Figures 6.4a - 6.6b show the model's performance at selected locations in the network. From the re s u l t s i n Table 6.2 the improvement in the average long-term performance of the model over the i n i t i a l a p p l i c a t i o n i s re a d i l y apparent. The model now underestimates the measured radiation on the partly cloudy day in contrast to the substantial overestimation indicated in Table 6.2 and i s without bias for the clear and overcast days. The average RMS error i s larger for the clear day but markedly reduced for the partly cloudy and overcast days over the i n i t i a l a p p l i c a t i o n (Table 6.1). RMS errors observed for the revised Hay and Hanson (1978) model are larger than those observed for the Gautier et a l . (1980) on a l l three days and larger than those observed for the revised Tarpley (1979) model on clear and overcast days. The average Mean Bias Errors tend to be quite s i m i l a r f or the three models. Figures 6.4a and 6.4b show that the pattern of underestimation during the middle of the day and overestimation during the early and l a t e r parts of the day observed for the i n i t i a l a p p lication (on day 183/80) i s maintained. However, there i s a noticeable increase in 208 T a b l e 6.2 H o u r l y A v e r a g e V e r i f i c a t i o n S t a t i s t i c s from t h e A p p l i c a t i o n of  the r e v i s e d Hay-Hanson (1978) model t o a sample o f n i n e (9) days  r e p r e s e n t i n g v a r i a b l e sky c o v e r c o n d i t i o n s a t d i f f e r e n t t i m e s of  the y e a r . U n i t s KJm'^hr"^ A r r a y size;5x~57 DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL SUMMER 183/80 192 -32.70 214.40 -1.74 11.40 .995 1879.7 1846.9 ( c l e a r ) 239/80 132 -104.70 302.30 -13.47 38.70 .848 780.3 675.6 ( p / c l o u d y ) 197/80 191 8.97 204.70 2.20 50.00 .586 409.9 418.9 ( o v e r c a s t ) SPRING 121/80 155 -118.50 256.30 -5.90 12.80 .992 1995.9 1877.5 ( c l e a r ) 160/80 168 -68.40 276.50 -10.30 41.70 .932 663.0 594.6 ( p / c l o u d y ) 105/80 142 149.70 237.10 37.80 59.90 .872 395.6 545.6 ( o v e r c a s t ) FALL 276/79 120 -71.60 215.60 -4.80 14.30 .989 1505.7 1434.1 ( c l e a r ) 263/79 105 46.00 211.40 12.00 54.90 .419 385.0 431.0 ( o v e r c a s t ) WINTER 025/80 86 -128.21 197.40 -12.90 19.80 .942 996.2 867.9 ( c l e a r ) ( 9 - d a y ) l 2 9 l 32.40 238.70 -3.20 23.20 .972 1027.1 994.7 (sample) Figure 6.4a: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver A i r p o r t for a clear day (183/80) in summer. Figure 6.4b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at P i t t Meadows for a clear day (183/80) in summer. HQUR-<d C\J i ro _ * * O B S E R V E D 2 : © - — - — — 0 C R L C U L R T E D —) 2 : V R N R J R 111 A / \ 2 3 9 C J CE 1—1 \ O CE (—\ cn *> ! 1 1 1 1 1 1 1 cn 0 3 6 9 12 L O C R L R P P R R E N T 15 18 21 T I M E 24 Figure 6.5a: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver Airport for a partly cloudy day (239/80) in summer. HOUR-\r _ C M i n —) r o _ x-o- — — — x O B S E R V E D _ Q C A L C U L A T E D O x ) _ A B B L I B IflNCE - - 7 \ 2 3 9 IRRRD o IRRRD D 1 3 l 1 6 9 L O C A L i 12 A P P A R E N T 1 1 15 18 21 T I M E I 24 Figure 6.5b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Abbotsford Library for a partly cloudy day (239/80) in summer. HOUR-<d- _ CM i ro _ x — * O B S E R V E D —•> CD — CD C R L C U L R T E D C\J _ V R N R I R IRRRDIflNCE CD 1 9 7 *f * ^ ^^^ ^ o ^ o - o * - ^ L J 2 IRRRDIflNCE 0 1 3 1 1 1 1 1 6 9 12 15 18 21 L O C R L R P P R R E N T T I M E 1 24 Figure 6.6a: Observed radiation and calculated radiation from the revised Hay - Hanson (1978) model at Vancouver Airport for an overcast day (197/80) in summer. .—„ HOUR-M" C M i ro _ x — * O B S E R V E D 2 1 o — — o C R L C U L R T E D ) C\J _ R B B L I B IRRRDIRNCE o 197 IRRRDIRNCE 0 1 3 1 1 1 1 6 9 12 1 LOCRL RPPRRENT 1 1 5 18 21 T I M E 1 24 Figure 6.6b: Observed radiation and calculated radiation from the revised Hay - Hanson (1978) model at Abbotsford Library for an overcast day (197/80) in summer. 215 the underes t imat ion and decrease in the o v e r e s t i m a t i o n . This i s exp la ined from a c o n s i d e r a t i o n o f the (b) va lues i n the o r i g i n a l and r e v i s e d ve rs i ons o f the Hay and Hanson (1978) model . The sma l l e r (b) value f o r the r e v i s e d model means that the overes t imat ions w i l l decrease but there w i l l be an i nc rease i n underes t ima t ion . Th is w i l l , r e s u l t i n sma l l e r MBEs (as overes t imates and underest imates cance l ) but l a r g e r RMS e r r o r s . This i s the pa t te rn observed fo r the c l e a r day. The r e s u l t s i n Table A . 5 . 2 support t h i s reason ing . These show tha t s i g n i f i c a n t model underes t imat ion i s on ly occu r r i ng at those s t a t i o n s (Grouse Mounta in , Langara and Vancouver A i r p o r t ) tha t were wi thout b ias i n the e a r l i e r a p p l i c a t i o n (Table A . 5 . 1 ) . The r e s u l t s i n Table A . 5 . 2 show that the average MBE e r ro r s quoted in Table 6.2 f o r the p a r t l y c loudy and overcas t days do not t r u l y represent the model 's behaviour at many of the s t a t i o n s . The unbiased performance o f the model on day 197/80 i s on ly r e p r e s e n t a t i v e o f the behaviour o f the model at Vancouver A i r p o r t (1.61%) and Abbots ford A i r p o r t (-1.71%). The smal l average Mean B ias E r ro r r e s u l t s from the c a n c e l l a t i o n o f negat ive and p o s i t i v e va lues o c c u r r i n g at d i f f e r e n t s t a t i o n s ( e . g . Langara and Lang ley ) . The on ly c o n s i s t e n t pa t te rn i s tha t o f ove res t ima t i on i n the urban and suburban a r e a s . The average MBE f o r day 239/80 ( p a r t l y c loudy) i s g rea te r than the Mean B ias E r ro r a t seven o f the twelve s t a t i o n s and i s i n f l u e n c e d by the l a r g e r e r r o r s o c c u r r i n g at l o c a t i o n s such as P i t t Meadows (MBE - 27%). The Mean B ias E r ro rs tend to be l a r g e r a t the i n t e r i o r l o c a t i o n s . F igures 6 . 5 a , 6.6a and 6.6b show tha t the c l e a r day pa t te rn o f ove res t ima t i on i n the e a r l y and l a t e par ts o f the day w i th underes t imat ion 216 during the middle of the day i s repeated at some locations on the partly cloudy and overcast days. For the clear day (183/80), the overall e f f e c t of the smaller (b) c o e f f i c i e n t was to reduce the overestimation of the model (or increase the underestimation). The behaviour of the model on the partly cloudy day (underestimation) and the overcast day (under and overestimations) suggests a s i m i l a r e f f e c t . This i s supported by the results in Table A.5.2 where on day 197/80 the overestimates tend to occur at the locations which experienced the largest overestimates in the i n i t i a l a p p lication and on day 239/80 the large underestimates tend to occur at the locations which experienced the smallest overestimates i n the i n i t i a l a p p l i c a t i o n (Table A.5.1). Variations in the model's behaviour across the network are complicated by factors such as cloud absorption (which the model does not e x p l i c i t l y handle) and downward r e f l e c t i o n from the sides of clouds which would help to explain some of the underestimation under partly cloudy conditions (multiple r e f l e c t i o n from the sides of clouds would increase the measured radiation but cannot be incorporated in the simple l i n e a r formulation of the Hay and Hanson (1978) model). The absolute value of the RMS and Mean Bias Errors are s i m i l a r for c l e a r and overcast days a t t e s t i n g to the consistency in model performance between these two regimes. The performance of the model deteriorates for the p a r t l y cloudy condition. The pattern of underestimation for the p a r t l y cloudy day i s s i m i l a r to that observed for the other models (Gautier et a l . (1980) and Tarpley (1979)) while the pattern of over and underestimates observed for day 197/80 i s s i m i l a r to that observed for the Gautier et a l . (1980) model. 217 The revised c o e f f i c i e n t s used in the Hay and Hanson (1978) model have led to s i g n i f i c a n t reduction in the bias observed in the model. This improvement i s p a r t i c u l a r l y noticeable for the partly cloudy and overcast days. The reduction in model bias indicates that the c o e f f i c i e n t s are more suited to describe the average conditions occurring in the study area. 6.3.2 Changes in Spatial Averaging Values of normalized s a t e l l i t e measured reflectance used i n e a r l i e r assessments have been determined based on a 5 x 5 pixel array centered on the s t a t i o n . Since our calculated radiation i s d i r e c t l y dependent on the value of the normalized reflectance we examine the s e n s i t i v i t y of the revised version of the Hay and Hanson (1978) model to changes in normalized s a t e l l i t e reflectance r e s u l t i n g from a change in our spatial averaging configuration. S p e c i f i c a l l y , we consider the •A effects of a reduced array si z e ( 3 x 3 pixel array) on the q u a l i t y of performance of the Hay and Hanson (1978) model. Assessments using t h i s reduced array si z e were performed for days 183/80, 239/80 and 197/80. The average s t a t i s t i c s derived from these assessments a r e - l i s t e d in Table 6.3 (complete s t a t i s t i c s for the individual stations are given in Table A.5.3). The data in Table 6.3 show that only minimal changes have occurred in the average behaviour of the model r e l a t i v e to the o r i g i n a l a p p l i c a t i o n (Table 6.2). On average the model now s l i g h t l y overestimates the measured radiation on the overcast day with a small increase in the average RMS error observed. Similar small increases in the average RMS errors are 218 T a b l e 6.3 H o u r l y A v e r a g e V e r i f i c a t i o n S t a t i s t i c s from t h e A p p l i c a t i o n of  t h e r e v i s e d Hay-Hanson (1978) model b a s e d on a S p a t i a l A v e r a g i n g  A p p r o a c h ( s e e t e x t ) . A r r a y s i z e : 3 x 3 . U n i t s K J m ^ h r " 1 DAY N MBE RMSE MBE% RMSE% r MEAN MEAN OBS CAL 183/80 192 -35.20 217.00 -1.90 11150 .994 1879.7 1844.5 ( c l e a r ) 239/80 132 -101.60 306.70 -13.00 39.30 .842 780.3 678.7 ( p / c l o u d y ) 197/80 191 12.10 206.40 3.00 50.40 .585 409.9 422.0 ( o v e r c a s t ) 219 observed for the partly cloudy and clear days along with a small decrease in the average MBE observed on the partly cloudy day. The results i n Table A.5.3 confirm the i n s e n s i t i v i t y of the model to s p a t i a l averaging under c l e a r skies ( i . e . there are only minor changes at the individual s t a t i o n s ) . This i s the expected behaviour considering the spatially_ conservative nature of the radiation under clear skies. A s i m i l a r r e s u l t was obtained for the revised Tarpley (1979) model (Section 4.3.3). Larger variations in the model's behaviour are observed for the partly cloudy and overcast days. Although no s p a t i a l pattern is . apparent, the general tendency i s for increases in both the RMS error and Mean Bias Error at many of the st a t i o n s . Some of the changes observed may be. due to the fact that the regression c o e f f i c i e n t s are based on a 5 x 5 array while the present assessments are based on a 3 x 3 array. ; A t some stations (e.g. Grouse Mountain and Mission Habitat Apartments) the cancellation of the over and underestimates ( s i m i l a r to what, was observed for the assessments based on a 5 x 5 array) leads to agreement between modelled and measured values. The preceding analyses have shown that a change in sp a t i a l averaging has l i t t l e e f f e c t on the performance of the revised Hay and Hanson (1978) model for clear sky conditions but results in somewhat larger changes under partly cloudy and overcast conditions. The changes fo r the p a r t l y cloudy and overcast days can hardly be considered to be s i g n i f i c a n t since they are within 4% of the mean measured rad i a t i o n at most loca t i o n s . The substantial changes observed for the revised Tarpley (1979) model (Section 4.3.3) are larg e l y absent from the present 220 analyses leading us to conclude that the Hay and Hanson (1978) model is less s e n s i t i v e to changes in spatial averaging under partly cloudy and overcast conditions. 6.4 Seasonal Assessments The o r i g i n a l model by Hay and Hanson (1978) has only been v e r i f i e d for what were e s s e n t i a l l y summertime conditions. In our present study we extend this v e r i f i c a t i o n to include a wider variety of conditions. V e r i f i c a t i o n i s performed on the revised version of the Hay and Hanson (1978) model. The same sui t e of days used i n an e a r l i e r analysis (Section 4.4) is used here. The li m i t a t i o n s imposed on the extent of our testing of the Tarpley (1979) model by the need for independence in our test data is also applicable in this context. Average v e r i f i c a t i o n s t a t i s t i c s for the nine (9) days analysed are given in Table 6.2 (complete s t a t i s t i c s for the individual stations are given in Appendix A.5). The results in Table 6.2 show that on average for clear skies the model underestimates the measured radiation in spring, f a l l and winter with agreement between modelled and measured values i n summer. The average RMS error increases to a maximum for the winter conditions. For the p a r t l y cloudy days the average behaviour of the model i s to underestimate the r a d i a t i o n . Under overcast conditions the average behaviour of the model i s to overestimate the radiation in spring and f a l l with agreement between modelled and measured values in the summer. Under clear skies the average RMS errors are largest for the revised Hay and Hanson (1978) model except in the wintertime when the 221 RMS e r r o r fo r the r e v i s e d Tarp ley (1979) i s the l a r g e s t . The abso lu te va lues o f the average Mean B ias Er ro rs f o r c l e a r s k i e s are l a r g e s t f o r the Hay and Hanson (1978) model except i n the summer and w i n t e r . The pa t te rn o f underes t imat ion on day 160/80 i s i n con t ras t to the ove res t ima t i on observed i n both the Gau t ie r e t a l . (1980) and the r e v i s e d Tarp ley (1979) models. The average e r r o r s are l a r g e r than those fo r the Gau t ie r et a l . (1980) model but sma l l e r than those fo r the rev i sed Tarp ley (1979) model . The average long- term performance o f the three models are qu i te s i m i l a r f o r day 239/80 but the shor t term performance o f the r e v i s e d Hay and Hanson (1978) model i s on ly be t t e r than tha t o f the rev i sed Tarp ley (1979) model. Under overcas t cond i t i ons the average RMS and Mean Bias Er ro rs are l a r g e s t fo r the r ev i sed Hay and Hanson (1978) model except- i n sp r i ng when the RMS e r r o r f o r the r e v i s e d Tarp ley (1979) model i s l a r g e r and i n summer when the average MBE fo r the rev i sed Tarp ley (1979) model i s l a r g e r . The i n d i v i d u a l s t a t i o n r e s u l t s i n Tables A . 5 . 2 and A . 5 . 4 conf i rm the pa t te rn of underes t imat ion f o r the c l e a r days r e f l e c t e d i n the average s t a t i s t i c s but a l s o Veveal that the MBEs a t some l o c a t i o n s are not represented by the average e r r o r . Two important departures from the average behaviour are o f no te : 1) s i t u a t i o n s where the i n d i v i d u a l s t a t i o n MBE i n d i c a t e s agreement between model led and measured va lues and 2) s i t u a t i o n s where the i n d i v i d u a l s t a t i o n MBE i s much g rea te r than the average e r r o r . F igures 6.7a - 6.9b show tha t the pat tern , o f over and underest imates observed fo r the summer a p p l i c a t i o n (F igures 6.4a and 6.4b) i s mainta ined i n the sp r i ng and f a l l but i s l a r g e l y absent from the L O C A L R P P R R F N T T I M E Figure 6.7a: Observed radiation and calculated radiation from the revised Hay - Hanson' (1978) model at P i t t Meadows for a clear day (121/80) in spring. O B S E R V E D C R L C U L R T E D 6 9 12 15 18 21 L O C R L R P P R R E N T T I M E Figure 6.7b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver A i r p o r t -for a clear day (121/80) in spring. ro _ C M _ CD _ 3 X- . * O B S E R V E D o C R L C U L R T E D ro ro 12 15 .18 21 24 L O C R L R P P R R E N T T I M E Figure 6.8a: Observed radiation and calculated radiation from the revised Hay - Hanson (1978) model at P i t t Meadows for a clear day (276/79) in f a l l . ro CM _ X- ^ O B S E R V E D c C R L C U L R T E D ro cn o 3 G 9 12 15, 18 21 L O C R L R P P R R E N T T I M E 24 Figure 6.8b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver A i r p o r t for a c l e a r day (276/79) in f a l l . C O C\J o _ L ca X-o vnNniR 2 5 G 9 12 15 18 L O C R L R P P R R F N T T I M F 21 -* OBSERVED o CRLCULRTED 24 Figure 6.9a: Observed radiation and calculated radiation from the revised Hay - Hanson (1978) model at Vancouver A i r p o r t for a clear day (025/80) in winter. cn ZD o CXI UJ CJ ^' CE 1 1 CD CE cn cn. co _ CD X-O P . I T M E D 2 5 — * O B S E R V E D _o C R L C U L R T E D S 9 12 • 15 18 21 2 4 L O C R L R P P R R E N T T I M E Figure 6 .9b : Observed r a d i a t i o n and c a l c u l a t e d r a d i a t i o n from the rev i sed Hay - Hanson (1978) model at P i t t Meadows f o r a c l e a r day (025/80) i n w in te r . 228 wintertime a p p l i c a t i o n . The r e l a t i v e magnitude of the Mean Bias Errors and RMS errors and the large value of (r) in spring and f a l l i s s i m i l a r to that observed for the summertime suggesting a s i m i l a r causal factor. The overestimates during the early and l a t e r parts of the day indicate that the estimated transmittance i s higher than the actual transmittance. This suggests that the problem could be the i n s e n s i t i v i t y of the model to increasing o p t i c a l depths which occur toward sunrise and sunset. With increasing o p t i c a l depth, both r e f l e c t i o n and absorption increase. The increase in absorption w i l l r e s u l t in a lower normalized reflectance and a higher estimated transmittance which would explain the overestimation. Since o p t i c a l depths are smaller around the middle of the day actual and estimated transmittances w i l l be quite s i m i l a r . The underestimation observed during the middle of the day i s therefore due to the adjustment of the regression c o e f f i c i e n t s to compensate for the overestimations during the early and l a t e r parts of the day. The s i m i l a r i t y in the magnitude of the Mean Bias Errors and RMS errors on day 025/80 suggests another source of bias. A probable source is the lack of temporal representativeness of the regression c o e f f i c i e n t s . The developmental data set (Table 4.5) reveals that the c o e f f i c i e n t s are heavily weighted toward summertime conditions. The data in Tables A.5.5 and A.5.6 show that the average errors for the partly cloudy and overcast days quoted in Table 6.2 obscure some features of the model's performance at individual s t a t i o n s . As in the clear sky case, the average errors are influenced by the large values occurring at some s i t e s and in most instances do not represent the e f f e c t i v e agreement between modelled and measured values 229 occurring at other loc a t i o n s . In addition ( e s p e c i a l l y under overcast conditions) the average Mean Bias Errors are affected by the cancel l a t i o n of negative and p o s i t i v e values recorded at d i f f e r e n t s i t e s . From Table A.5.5 a s i m i l a r pattern of underestimation i s observed for both p a r t l y cloudy days except at BC Hydro where overestimation occurs on day 160/80. Larger underestimation at the i n t e r i o r locations are observed on both days. The c o r r e l a t i o n c o e f f i c i e n t (r) indicates that there is greater spatial v a r i a t i o n in the random error on day 239/80 than on day 160/80. The high value of (r) at some stations (e.g. P i t t Meadows r = .977) on day 160/80 suggests that a large portion of the RMS error observed is due to the bias in the model. The variations in the re l a t i o n s h i p between the Mean Bias Error, RMS error and the co r r e l a t i o n c o e f f i c i e n t (r) (on day 160/80) suggest two basic patterns; at Langara and UBC Climate Station the unbiased performance of the model coupled with a Targe RMS error and a large (r) indicate that shorterm under and overestimates are occurring, s i m i l a r to the model's behaviour on day 239/80. At stations such as P i t t Meadows and Mission Habitat Apartments the large MBEs indicate that the model is c onsistently underestimating the measured ra d i a t i o n . An examination of the hourly data f o r day 160/80 confirms the occurrence of these patterns. Figures 6.10a and 6.10b show a close correspondence between modelled and measured values f or the morning period of day 160/80. This is in contrast to the pattern of overestimation observed for the revised . Tarpley (1979) and the Gautier et a l . (1980) models. The summer pattern of over and underestimations at many stations for the overcast day (197/80) i s not repeated i n the spring and f a l l . cr ZD CD ZE. C M a: I— I (ZD cr c r c r M- __, CO CM _ J o 0 . 6 © - — — -VRNRJR 160 21 OBSERVED o CflLCULflTED ro co o 24 LOCRL A P P A R E N T T IME Figure 6.10a: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver Airport for a partly cloudy day (160/80) in spring. L O C R L R P P R R E N T T I M E Figure 6.10b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Abbotsford Library for a partly cloudy day (160/80) in spring. M- _ CD _ CM _ 3 x- * O B S E R V E D c C R L C U L R T E D L O C R L R P P R R E N T T I M E Figure 6.11a: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Vancouver Ai r p o r t for an overcast day (105/80) in spring. x- -X o A B B - L I B 105 CD O B S E R V E D C A L C U L A T E D JD -© 21 24 L O C A L A P P A R E N T T I M E Figure 6.11b: Observed radiation and calculated radiation from the revised Hay - Hanson (1978) model at Abbotsford Library for an overcast day (105/80) in spring. I cr. ZD CD zc CM ~D LU CJ ~Z CE i — i Q CE. cr. cr. M-c n (XI CD 0 x-o- — VRNRIR 2 6 3 OBSERVED CRLCULRTED ro co 18 21 2 4 LOCRL RPPRRENT TIME Figure 6.12a: Observed radiation and calculated r a d i a t i o n from the revised Hay- Hanson (1978) model at Vancouver A i r p o r t for an overcast day (263/79) in f a l l . HOUR-\f _ C\J i 2 : —) m _ x 0- — — x O B S E R V E D C A L C U L A T E D C\l _ A B B L I B fl.NCE *—1 2 6 3 Q CE CT c r CD 3 1 3 I I G 9 L O C A L 1 12 A P P A R E N T I l 1 15 18 2.1 T I M E 1 2 4 Figure 6.12b: Observed radiation and calculated r a d i a t i o n from the revised Hay - Hanson (1978) model at Abbotsford Library for an overcast day (263/79) in f a l l . 236 There i s s i g n i f i c a n t overestimation at a l l stations on day 105/80. Thi pattern i s s i m i l a r to that observed for both the Gautier et a l . (1980) and the revised Tarpley (1979) models. In the context of the Hay and Hanson (1978) model we also suggest that i t i s important to consider the e f f e c t s of changing s u n - s a t e l l i t e azimuth angle under overcast conditions (see Section 4.4 and 5.4). The pattern of overestimation observed on day 263/79 is more s i m i l a r to the conditions on day 105/80 than day 197/80. On day 263/79 the largest overestimations concide with the lowest inputs of the measured radiation except at P i t t Meadows (the lower measured radiation at P i t t Meadows i s due in part to the three hours of missing data). The low values of the measured radiation at the other s i t e s indicate heavily overcast conditions and possibly s i g n i f i c a n t cloud absorption. Since the model by Hay and Hanson (1978) does not e x p l i c i t l y account for cloud absorption (though increased absorption w i l l be re f l e c t e d in lower brightness values) a reduction in the measured ra d i a t i o n w i l l not necessarily be accompanied by a s i m i l a r reduction in the estimated r a d i a t i o n leading to the large overestimations. The large values of both the Mean Bias and Root Mean Square Errors at Grouse Mountain on day 105/80 are s i m i l a r to those observed for the Gautier et a l . (1980) model (Section 5.4). The small value of the measured radiation indicates heavily overcast conditions. The large bias i s again due to the inadequate handling of cloud a b s o r p f by the model. The preceding analyses have shown that for clear skies model underestimation in spring, f a l l and summer i s influenced by the pattern of s i g n i f i c a n t underestimation occurring during the middle part 237 of the day due to the adjustment of the regression c o e f f i c i e n t s to compensate for the overestimations during the early and l a t e r parts of the day. In winter an additional factor i s the lack of temporal representativeness of the regression c o e f f i c i e n t s . Under partly cloudy conditions the dominant pattern i s model underestimation. The patterns of over and underestimates in the dirunal regime shown for the clear sky analyses are also evident. The radiation estimates under partly cloudy conditions are further affected by l i m i t a t i o n s in the developmental data used to define the partly cloudy regime in the Hay and Hanson (1978) model. A f a i r l y small sample of observations was used to define the partly cloudy regime. This w i l l l i m i t the predictive power of the equation under these conditions. A larger v- sample is needed to capture the v a r i a b i l i t y inherent in the partly cloudy s i t u a t i o n . The developmental data set i s weighted toward the summertime. This means that conditions occurring at other times of the year would not be f u l l y represented by the c o e f f i c i e n t s . Therefore, a p p l i c a t i o n of the model outside the summertime context should introduce further bias in the model's performance. Variations in the model's behaviour for the three overcast „ days r e f l e c t l i m i t a t i o n s in the model's formulation (e.g. an inadequate treatment of cloud absorption) and differences in the radiation conditions on the three days. Conditions on day 105/80 and to a lesser extent day 263/79 suggest that an additional factor may be the effects of changing s u n - s a t e l l i t e azimuth angle under some cloud regimes. 238 6.5 Daily S t a t i s t i c s In a manner sim i l a r to both the Gautier et a l . (1980) and the revised Tarpley (1979) models (Sections 5.5 and 4.5) v e r i f i c a t i o n of the revised Hay and Hanson (1978) model was extended to "include d a i l y and longer time periods. The model was evaluated for periods of cl e a r , partly cloudy and overcast days and for the sample of nine (9) days. V e r i f i c a t i o n s t a t i s t i c s at the twelve stations were combined to y i e l d an average d a i l y value for each period. " The average s t a t i s t i c s for the four periods are l i s t e d in Table 6.4. From a comparison of the results in Table 6.4 with the hourly results in Table 6.2 we note the following: there i s an overall decrease i n the Root Mean Square Error and Mean Bias Error for the clear partly cloudy and overcast days. The model underestimates the measured radiation under clear and partly cloudy conditions but overestimates the measured for the overcast period. Model underestimation i s s t i l l evident for the total sample of nine (9) days. For this sample the RMS error of 9.8% i s smaller than that observed f o r the pa r t l y cloudy and overcast days r e f l e c t i n g the influence of the much smaller error observed for the clear day period. The average d a i l y MBEs calculated for the revised Hay and Hanson (1978) model are larger than those observed for both the revised Tarpley (1979) and the Gautier et a l . (1980) models. A s i m i l a r pattern of s i g n i f i c a n t overestimation for the overcast days i s observed for the three models. The underestimation observed for the clear sky condition is in contrast to the s l i g h t overestimation by the Gautier et a l . (1980) model and the unbiased performance of the revised Tarpley (1979) model for the same period. The s i g n i f i c a n t underestimation for the partly 2 3 9 Table 6.4 Daily Average V e r i f i c a t i o n S t a t i s t i c s from the Application of  the revised Hay-Hanson (1978) model to periods of clear, p a r t l y  cloudy and overcast days and to an a l l day sample.of nine (9) days. Units HJm^day'. Array size;5x5.  PERIOD N MBE RMSE MBE% •RMSE% r MEAN MEAN OBS CAL CLEAR 48 0.992 1.130 -4.70 5.80 .997 19.5 18.6 DAYS P/CLOUDY 24 -1.050 1.310 -11.80 14.70 .848 8.9 7 9 DAYS OVERCAST 36 0.772 1.210 15.90 25.00 .814 4.9 5 6 DAYS 9-DAY 108 -0.386 1.200 ,-3.20 9.80 .993 12.3 11.9 SAMPLE 240 cloudy period is in contrast to the s l i g h t underestimation by the Gautier et a l . (1980) model and the unbiased performance of the revised Tarpley (1979) for the same period. The ov e r a l l decrease i n the RMS error for the d a i l y analysis is again consistent with the expected behaviour of this s t a t i s t i c with increased length of averaging period. The RMS error of 6% for the cle a r sky period i s s i m i l a r to that obtained for both the Gautier et a l . (1 980) and the revised Tarpley .(1979) models. Daily averaging for the Hay and Hanson (1978) model results in the greatest reduction in the average RMS error observed for the three models under clear sky conditions. The RMS error for the partly cloudy period is smaller than that observed for the revised Tarpley (1979) model but larger than the RMS error for the Gautier et al . (1980) model. The RMS error for the overcast period is smaller than for the other two models. The average d a i l y RMS error of 9.8% for the total sample is smaller than the hourly average (23%) quoted in Table 6.2. These d a i l y and hourly s t a t i s t i c s are s i m i l a r to those quoted by Hay and Hanson (1978). The d a i l y average RMS error of (9.8%) i s larger than for the Gautier et a l . (1980) model but smaller than for the revised Tarpley (1979) model for the same sample of nine (9) days. The smaller average RMS errors observed for the d a i l y and longer time periods are due to the cancel l a t i o n of random er r o r s . The greater reductions observed for the Hay and Hanson (1978) model i s c l e a r l y related to the pattern of short term over and underestimates described in e a r l i e r sections (6.3.1 and 6.4). 241 6.6 Implications of Model Accuracy for Spatial Sampling Requirements In an approach s i m i l a r to that used for the revised Tarpley (1979) model (Section 4.6) and the Gautier et a l . (1980), (Section 5.6) we evaluated the model for the network as a unit to determine the sampling distances congruent with the accuracy of the model. Values for the Grouse Mountain location are excluded to ensure comparability between our present results and those of Hay (1981) given in Table 1.1. In spite of our limited sample of nine (9) days,-some preliminary statements can be made on the basis of our r e s u l t s . The average RMS error for the hourly values i s ±22.9% of the mean measured global r a d i a t i o n . The average RMS error decreases to ±9.4% of the mean measured global radiation for d a i l y values. The data in Table 1.1 indicate than an error of 22.9% for hourly values is associated with a sampl ing distance of approximately 30 km. This suggests that, on average, to maintain hourly estimates to within ±22.9% we are limited to a spa t i a l resolution of approximately 30 km. At shorter distances the variations due to errors in the c a l c u l a t i o n procedures would be of the same magnitude as the actual variations in solar r a d i a t i o n . An in t e r p o l a t i o n technique based on measured data would be the more appropriate method at these shorter distances. An error of 9.4% for d a i l y values i s associated with a sampling distance of approximately 15' km. At distances less than 15 km an i n t e r p o l a t i o n technique should be used i f the desired error i s less than ±9.4% for d a i l y values. For hourly values the sampling resolution provided by the revised Hay and Hanson (1978) model i s id e n t i c a l to that provided by the revised Tarpley (1979) model but less than that 242 provided by the Gautier et a l . (1980) model. For d a i l y values the sampling resolution is better than that provided by the revised Tarpley (1979) model and s l i g h t l y worse than the resolution provided by the Gautier et a l . (1980) model. 6.7 Summary and Conclusions Implementation of the Hay and Hanson (1978) model to the present study area has shown that the regression c o e f f i c i e n t s developed for the t r o p i c a l A t l a n t i c are inappropriate for a mid-latitude location such as the study area. This i s due to the large bias introduced e s p e c i a l l y under p a r t l y cloudy and overcast conditions. Smaller bias under clear skies are related to the cancel l a t i o n of overestimates from the early and l a t e r parts of the day by underestimates occurring around the middle of the day. Revised regression c o e f f i c i e n t s lead to a s i g n i f i c a n t reduction in the bias observed for the partly cloudy and overcast conditions. Under clear skies the model's performance at many stations i s e f f e c t i v e l y unbiased, but there is an increase in the RMS errors. The pattern of over and underestimates observed for the i n i t i a l a p p l i c a t i o n i s maintained with increased underestimation. This can be attr i b u t e d to the smaller value of the (b) c o e f f i c i e n t and p a r t i a l l y explains the behaviour of the Mean Bias and RMS err o r s . The diurnal regime of over and underestimates observed for the clear sky is also present under the partly cloudy and overcast conditions but additional factors such as the inadequate handling of cloud absorption influence the errors in the estimated r a d i a t i o n . 243 Evaluations of the model's s e n s i t i v i t y to changes i n the spat i a l averaging configuration reveal that the model i s i n s e n s i t i v e to such changes under a l l conditions but e s p e c i a l l y under clear skies. The i n s e n s i t i v e behaviour for p a r t l y cloudy and overcast conditions is in contrast to that observed for the revised Tarpley (1979) model. Seasonal assessments of the model indicate that the dominant model behaviour for c l e a r and partly cloudy conditions i s underestimation. The pattern of underestimation for the clear sky i s influenced by consistent underestimates occurring under high radiation inputs for spring, f a l l and summer and'by the additional influence of inappropriate c o e f f i c i e n t s in the wintertime. The behaviour of the model under partly cloudy conditions is influenced by the diurnal regime of over and underestimates observed for the clear sky and additional factors such as a limited developmental data set that i s weighted toward summertime conditions and multiple r e f l e c t i o n from the sides of clouds that would increase the measured * r a d i a t i o n but cannot be incorporated i n the simple l i n e a r formulation of the model. Although underestimation is observed at some stations on day 197/80, the dominant pattern is for model overestimation on the overcast days. This we a t t r i b u t e in part to inadequate handling of cloud absorption and the probable influence of the changing s u n - s a t e l l i t e azimuth angle. Daily assessments lead to the expected decreases «in the RMS errors f o r c l e a r , partly cloudy and overcast conditions. The average d a i l y RMS error of ±6% for c l e a r skies represents the greatest decrease 244 for the three models r e s u l t i n g from an increase in the averaging period. In terms of the mesoscale s p a t i a l v a r i a b i l i t y in the solar irradiance resolvable by t h i s p a r t i c u l a r model, hourly and d a i l y RMS errors of ±22.9% and ±9.4% r e s t r i c t the model to s p a t i a l resolutions of approximately 30 km and 15 km r e s p e c t i v e l y . Improvements to the Hay and Hanson (1978) model would include: 1) the use of a larger sample to generate regression c o e f f i c i e n t s ( e s p e c i a l l y for the partly cloudy condition); 2) the need to account for the effects of cloud absorption; 3) consideration of the effects of changing azimuth angle been sun and s a t e l l i t e under overcast conditions; and 4) the use of a b i - d i r e c t i o n a l model to address the diurnal regime of over and underestimates for the clear sky s i t u a t i o n . 245 CHAPTER SEVEN 7. Conclusions The current study has a two-fold purpose; 1) to provide a comparison of three models which use s a t e l l i t e data to calculate solar r a d i a t i o n incident upon the earth's surface and 2) to attempt to characterize the mesoscale s p a t i a l v a r i a b i l i t y in observed global solar r a d i a t i o n which may be resolved using s a t e l l i t e data. The a b i l i t y to conduct a mesoscale study such as the present i s la r g e l y dependent on the accuracy to which the s a t e l l i t e imagery can be earth-located. In Chapter Two we present a technique which we believe makes i t possible to earth-locate the s a t e l l i t e imagery to within ±1.25 km in the East-West d i r e c t i o n and ±1.71 km in the North-South d i r e c t i o n . However, as we have demonstrated in Section 2.8, t h i s accuracy may be compromised by non-linearity across the image in the East-West d i r e c t i o n . This feature w i l l be of special importance under partly cloudy skies when the ra d i a t i o n conditions w i l l be s p a t i a l l y heterogeneous. Determinations of pixel s i z e has led us to the conclusion that the resolution of the s a t e l l i t e data for our study area i s on the order of 0.82 km in an East-West drection and 1.67 km in a North-South d i r e c t i o n r e s u l t i n g in a sin g l e pixel area of 1.37 sq km. This finding contradicts the usually quoted resolution of 2 km (Hay 1981, Gautier et a l . , T980) for locations near 50° North l a t i t u d e . I n i t i a l a p p lications of the models (Chapters Four, Five and Six) reveal the s u p e r i o r i t y of the Gautier et a l . (1980) model over the other two models in estimating the hourly radiation under c l e a r , partly 246 cloudy and overcast summer conditions and also reveal s i g n i f i c a n t bias e s p e c i a l l y in the Hay and Hanson (1978) formulation, necessitating r e v i s i o n of the o r i g i n a l regression c o e f f i c i e n t s . The need for new regression c o e f f i c i e n t s highlights a major disadvantage in most regression models ( i . e . the lack of general a p p l i c a b i l i t y in the spa t i a l dimension). The revised c o e f f i c i e n t s lead to reduced bias in both models and improved predictions under some conditions. In the case of Tarpley's (1979) model, improvements occur under c l e a r skies and some anomalies under partly cloudy and overcast conditions are removed but the s i m i l a r i t y between the performance of the model with o r i g i n a l and revised c o e f f i c i e n t s (for the partly cloudy and overcast conditions) i s the feature of note. In the case of the Hay and Hanson (1978) model the revised c o e f f i c i e n t s lead to s i g n i f i c a n t improvements under partly cloudy and overcast conditions but the model predictions deteriorates for the clear sky anal y s i s , due to the influence of a smaller (b) c o e f f i c i e n t . The superior performance of the Gautier et a l . (1980) model i s only maintained for the partly cloudy and overcast conditions with the revised Tarpley (1979) model giving better estimates under clear sky conditions. Estimates from the revised Tarpley (1979) model are also better than those from the revised Hay and Hanson (1978) model. Summary s t a t i s t i c s showing the average performance of the three models on an hourly and d a i l y basis for periods of c l e a r , partly cloudy and overcast days and for the total sample of nine (9) days are l i s t e d i n Table 7.1. Several features are to be noted: 1) the superior performance of the Gautier et a l . (1980) model for hourly estimates under partly cloudy and overcast conditions and for the nine Table 7.1 Summary S t a t i s t i c s Showing the Average Performance of the Three Models  on an Hourly (Numbers in Parentheses) and Daily Basis for Periods of  Clear, Partly Cloudy and Overcast Days and for the Nine (9) Days. '. -2 ^1 " : Z? i Units: MJ m day for d a i l y values K J m hr for hourly values. PERIOD N MBE RMSE MBE% RMSE% R MEAN MEAN OBS CAL CLEAR DAYS 48-(553) 0. (44. 514 6) GAUTIER ET AL. (1980) 0.839 (120.8) 2. (2. 63 63) 4. (7. 3 1) ( . 999 . 993) 19. (1693. <j 5 7) 20. (1738. 0 4) PARTLY CLOUDY DAYS 24 (300) -0. (-21. 27 6) 1. 07 (264.2) -3. (-3. 02 02) 11. (37. 95 0) ( .751 .919) 8 . (714. 9 6) 8. (693. 7 0) OVERCAST DAYS 36 (438) 0. (46. 570 9) 1. 28 (165.8) 11. (11. 73 73) 26. (41. 4 5) ( .743 .836) 4. (399. 9 4) 5. (446. 4 3) TOTAL SAMPLE 9-DAYS 108 (1291) 0. (30. 358 0) 1. 06 (178.3) 2. (2. 9 9) 8. (17. 6 4) ( . 995 .982) 12. (1027. 3 1) 12. (1057. 6 1) Table 7.1 (continued) PERIOD N MBE RMSE MBE% RMSE% R MEAN OBS MEAN CAL CLEAR DAYS PARTLY CLOUDY DAYS 48 (553) 24 (300) HAY AND HANSON (1978) =1 0.922 1.13 - T 7 T (-80.0) (224.8) (-4.7) -1.05 1.31 -11.8 (-84.4) (288.1) (-11.8) 5.8 (13.3) 14.7 (40.3) . 997 (.986) .848 (.908) 19.5 (1693.7) 8.9 (714.6) 18 .6 (1613.7) 7.9 (630.2) OVERCAST TOTAL SAMPLE 9-DAYS 36 0.772 1.21 15.9 25.0 .814 (438) (63.5) (217.3) (15.9) (54.4) (.702) 108 -0.386 1.2 -3.2 9.8 .993 (1291) (-32.4) (238.7) (-3.2) (23.2) (.972) 4.9 (399.4) 12. 3 (1027.1) 5.6 (462.9) 11. 9 (994.7) Table 7.1 (continued) PERIOD N MBE RMSE MBE% RMSE% R MEAN MEAN OBS CAL TARPLEY (1979) CLEAR DAYS 48 (553) -0. (-33. 389 8) 0. (117. 794 4) -2. (-2. 0 0) 4. (6. 07 93) ( . 998 . 993) 19. (1693. 5 7) 19. (1659. 1 9) PARTLY CLOUDY DAYS 24 (300) 0. (13. 173 8) 2. (383. 13 4) 1. (1. 94 94) 23. (53. 9 7) ( .642 .834) 8. (714. 9 6) 9. (728. 1 4) OVERCAST DAYS 36 (438) 0. (49. 602 5) 1. (215. 69 3) 12. (12. 4 4) 34. (53. 9 9) ( .642 .711) 4 . (399. 9 4) 5. (448. 5 9) TOTAL SAMPLE 9-DAYS 108 (1291) 0 . (5. 066 5) 1. (236. 5 2) 0. (0. 54 54) 12. (23. 2 0) ( . 987 . 967) 12. (1027. 3 1) 12. (1032. 3 6) 250 day sample; 2) the s i g n i f i c a n t decreases in RMS errors (for a l l conditions) for the three models when the averaging time is increased; 3) the fact that increases in the averaging period have a greater b e n e f i c i a l e f f e c t on the Hay and Hanson (1978) model than on the other two models such that the d a i l y RMS errors are quite s i m i l a r to those for the Gautier et a l . model but smaller than those for the revised Tarpley (1979) model (except for clear days when the RMS error for the three models are quite s i m i l a r ) and 4) the substantial increases in the average Mean Bias Errors and RMS errors when the models are evaluated for partly cloudy and overcast conditions, indi c a t i n g the i n a b i l i t y of the three models to properly describe the radiation behaviour under these conditions. In terms of temporal and s p a t i a l averaging changes, the model by Hay and Hanson (1978) proves to be less s e n s i t i v e to changes in spatial averaging than the Tarpley (1979) model. Responses to a change in the array s i z e seem to depend not on the size of the array, but on the radi a t i o n c h a r a c t e r i s t i c s represented in the array. Within the l i m i t s imposed by the small sample size of the present study, the results with the Gautier et a l . (1980) model would tend to refute the claim that an 8 x 8 pixel array p a r t i a l l y accounts for discrepancies between s a t e l l i t e and pyranometric measurements (Gautier et a l . , 1980). Tarpley's (1979) model has been shown to be more s e n s i t i v e to changes in temporal averaging than the Gautier et a l . (1980) model p a r t i c u l a r l y under cloudy and overcast conditions. Evaluations of the mesoscale s p a t i a l v a r i a b i l i t y in hourly and d a i l y values of the global solar irradiance, though based on a very small sample of nine (9) days s t i l l allow us to propose some 251 tentative conclusions. We believe that our results provide i n i t i a l support for the Hay (1981) assertion that s a t e l l i t e data i s best suited to resolve the mesoscale s p a t i a l v a r i a b i l i t y in solar r a d i a t i o n . In this context, the degree of resolution appears to be dependent upon the model considered and the nature of the data analysed. The average hourly and d a i l y RMS errors (±17.1% and ±8.2%) for the Gautier et a l . (1980) model l i m i t s us to s p a t i a l resolutions of 15 km and 12 km respectively. A larger sample (19) days with a greater proportion of partly cloudy and overcast days leads" to a reduction in the sp a t i a l r e s o l u t i o n . Larger RMS errors for the revised Hay and Hanson (1978) and the revised Tarpley (1979) models also r e s u l t i n reduced s p a t i a l resolutions. Finer resolutions in the mesoscale spatial v a r i a b i l i t y of the global solar irradiance can only be attained by improving model accuracy. This leads us to a consideration of possible areas of improvement in the three models. Invariably the shortcomings in model formulation are revealed under partly cloudy and overcast conditions. R e a l i s t i c a l l y , a solution to the complexities of the ra d i a t i o n behaviour under these conditions may never be r e a l i z e d . However, we have a mandate to make our parameterizations as accurate as possible. For the Gautier et a l . model we h i g h l i g h t two major areas where improvements are necessary: 1) Improved s p e c i f i c a t i o n of a cloud threshold and 2) Determination of cloud absorption. The cloud threshold i s a c r u c i a l feature of the Gautier et a l . (1980) model and therefore requires a rigorous and objective s p e c i f i c a t i o n . ' To date th i s has not been attempted. The use of the IR data provided by the VISSR 252 sensor may be part of t h i s s o l u t i o n . Cloud absorption to t h i s point has been determined as a l i n e a r function of cloud brightness where brightness i s used as an indicator of thickness. This parameterization is limited by the fact that clouds of d i f f e r e n t thicknesses may have s i m i l a r brightness. At present we can o f f e r no objective solution to this question. Inclusion of the effects of aerosols is another area for improvement. For the Hay and Hanson (1978) model suggested improvements include 1) the need for a larger more seasonally representative developmental data set, with greater s p e c i f i c a t i o n of the partly cloudy region of the r e l a t i o n s h i p 2) under clear skies the use of a b i - d i r e c t i o n a l reflectance model may help to solve the consistent pattern of over and underestimates in the diurnal regime and 3) the need to account e x p l i c i t l y for the e f f e c t s of cloud absorption. Improvements in the Tarpley (1979) model also involve the use of a larger more seasonally representative data set. In addition, cloud absorption needs to be e x p l i c i t l y treated within the model's framework. Results from the overcast and partly cloudy days support t h i s conclusion. The use of the r a t i o ( * n ^ ) 2 needs to be reconsidered in the l i g h t of some of the results presented i n Chapter Four. 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Journal of the Atmospheric Sciences, 19, 182-188. 260 APPENDIX A . l 261 LIST OF SYMBOLS NOTATION B B0 BB Et F0 F!, F. lA [G [0 [R s sc "m C c l d mcr UPPER CASE ROMAN cloud albedo predicted minimum brightness normalized predicted minimum brightness observed minimum brightness equation of time instantaneous shortwave flux at the top of the atmosphere the i t h value of the estimated and observed tr a n s l a t i o n s r e s p e c t i v e l y energy absorbed in the atmosphere energy absorbed at the surface solar constant energy r e f l e c t e d back at space irradiance at the surface irradiance at the surface under cloudy conditions mean target brightness mean cloud brightness increment i n brightness that w i l l bring about a change i n absorption dimensionless counts counts counts minutes _2 W m pixels W m W m" 1353 W m W m c W m~2 W m"2 counts counts counts -2 262 L L s LAT LST MBE N N r N 2, N 3 RMSE REFL RNORM SC SR SW+ T r T d THR longitude of standard meridian st a t i o n longitude l o c a l apparent time l o c a l standard time mean bias error number of observations/sample size number of pixels in c l e a r , p a r t l y cloud and cloudy categories res p e c t i v e l y root mean sqaure error pixel brightness normalized reflectance measured s a t e l l i t e brightness normalized s a t e l l i t e measured brightness f l u x measured at the s a t e l l i t e atmospheric transmittance dewpoint temperature cloud threshold LOWER CASE ROMAN a, b, c, d, e regression c o e f f i c i e n t s a(u-|), a(u 2) absorption c o e f f i c i e n t s f o r water vapour paths sun and s a t e l l i t e r e s p e c t i v e l y degrees degrees hours hours pixels/KJ m"2/MJ m"2 dimensionless dimensionless pixels/KJ m"2/MJ m"2 counts dimensionless counts dimensionless W m"2 W o c o s e ) °F counts KJ m"2 dimensionless 263 a(u') slant path water vapour absorption a(u-j b), a ( u 2 b ) absorption below cloud level f o r sun and s a t e l l i t e path respectively a(u-j t), a ( u 2 t ) absorption above cloud level f o r sun and s a t e l l i t e path respectively abs cloud absorption az sun's azimuth from south d day number which runs from 0(Jan. 1) to 364 (Dec. 31) h hour angle of the sun l a t s t a tion l a t i t u d e m opt i c a l airmass n cloud f r a c t i o n u p r e c i p i t a b l e water u' slant path of p r e c i p i t a b l e water z st a t i o n elevation dimensionless dimension! ess dimensionless dimensionless degrees dimensionless degrees degrees dimensionl ess dimensionless cm cm m a B, B 1 6 e LOWER CASE GREEK surface albedo scattering c o e f f i c i e n t s f o r beam and d i f f u s e r a d i a t i o n r e s p e c t i v e l y d e c l i n a t i o n of the sun in degrees sol a r zenith angle azimuth angle between sun and s a t e l l i t e dimensionless dimensionless degrees degrees degrees 264 l a t i t u d e and seasonally adjusted constant atmospheric transmittance transmission due to water vapour absorption transmission due to water vapour scattering transmission due to Rayleigh s c a t t e r i ng dimensionless dimensionless dimensionless dimensionless dimensionless 265 APPENDIX A.2 266 To determine the zenith and azimuth angles of the s a t e l l i t e we make the following assumptions: 1) The earth i s s p h e r i c a l ; 2) The s a t e l l i t e sub-point i s over the equator and i s constant ( i . e . perfect geosynchronous o r b i t ) ; 3) No correction i s made f o r atmospheric r e f r a c t i o n . The cosine of the s a t e l l i t e azimuth $ i s given by: /Tan ^ , - Tan 6 , f l 0 , i cos ip = (A.2.1) T*"2 • + <T§H>2 + 1 the cosine of the s a t e l l i t e zenith angle i s given by: cos e = p c o s ^ cos 6 - 1 ( A 2 2 ) p 2 - 2 'p C O S <f> C O S 6 + 1 <J> = st a t i o n l a t i t u d e 6 = s a t e l l i t e subpoint longitude - st a t i o n longitude or axis distance radius of earth _ semi-maj of o r b i t (for geosynchronous o r b i t s , p i s always very close to 6.612). Equations used in the Gautier et a l . (1980) model to cal c u l a t e water vapour absorption c o e f f i c i e n t s above and below a cloud layer. u l a = W f r a c ' u l ' u2a = W f r a c ' u2 {k.2.z\) a ( u ^ ) computed with Yamamoto's equations a(u'lb) = a(u ]) - a(u l a) (A.2.4) a u 2 b = u 2 (1 - W f r a c) (A.2.5) 267 ( u 2 t ) = a ( u u + u 2 t ) - a ( u u ) ( u 2 a ) c = a ( u l + u 2 } " a ( u l " u2b> (A.2.6) (A.2.7) 268 APPENDIX A.3 c 269 The following notation applies to the Tables included i n the appendices. N = number of observation MBE = Mean Bias Error RMSE = Root Mean Square Error MBE% = r e l a t i v e Mean Bias Error RMSE% = r e l a t i v e Root Mean Square Error r = c o r r e l a t i o n c o e f f i c i e n t OBS. = Observed Radiation ( d a i l y t o t a l at the station) CAL. = Calculated Radiation ( d a i l y t o t a l at the station) DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE34 RMSE% R GRSMT 28887 32 179 16 -205 70 266 37 - 10 23 13 24 0 998 NRTHMT 28787 3001 3 16 -76 65 165 23 -4 09 8 81 0 995 BCHYDRO 28887 30453 16 -97 90 187 19 -5 14 9 84 0 996 LANGA 28749 30437 16 - 105 52 186 91 -5 55 9 83 0 995 VANAIR 28738 30527 16 - 1 1 1 78 183 86 -5 86 9 64 0 996 FERRY 28220 30428 16 - 137 99 208 53 -7 26 10 97 0 992 PITMED 28609 29086 16 -29 81 124 54 - 1 64 6 85 0 995 MISSHAB 28795 29551 16 -47 22 141 76 -2 56 7 68 0 995 ABBLIB 28542 292 17 16 -42 2 1 151 79 -2 31 8 31 0 992 ABBA IR 28562 29302 16 -46 29 167 19 -2 53 9 13 0 989 LANGLEY 28589 29403 16 -50 86 127 56 -2 77 6 94 0 995 CLISTN 28387 30310 16 -120 17 207 00 -6 34 10 93 0 995 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 2234 5522 11 -298 84 389 70 -59 53 77 63 0 919 NRTHMT 49 13 668 1 11 -160 72 262 2 1 -26 46 43 17 0 875 BCHYDRO 6184 8028 11 -167 63 228 13 -22 97 31 26 0 945 LANGA 8637 7937 1 1 63 69 260 87 8 83 36 16 0 866 VANAIR 10288 9003 11 1 16 77 195 48 14 27 23 88 0 955 FERRY 5182 9606 11 -402 17 539 29 -46 05 61 76 0 802 PITMED 6839 8097 11 - 1 14 31 221 62 - 15 53 30 1 1 0 875 MISSHAB 9120 9945 11 -75 02 303 64 -8 30 33 58 0 897 ABBLIB 9548 9453 1 1 8 64 296 07 1 Ol 34 45 0 905 ABBAIR 957 1 10094 1 1 -47 56 340 12 -5 18 37 06 0 875 LANGLEY 7 157 8626 11 - 133 56 280 70 - 17 03 35 79 0 800 CLISTN 5784 10001 11 -383 35 . 493 89 -42 16 54 32 0 859 DAY 197/80(OVERCAST) STATION CAL . OBS . N MBE RMSE MBE°/. RMSE54 R GRSMT 1322 5764 15 -296 10 359 55 -77 06 93 57 0 420 NRTHMT 4765 5399 16 -39 62 1 16 76 - 1 1 74 34 60 0 831 BCHYDRO 5703 5504 16 12 43 96 25 3 61 27 98 0 898 LANGA 8888 5882 16 187 88 232 77 51 1 1 63 32 0 94 1 VANAIR 9373 6866 16 156 70 187 67 36 52 43 73 0 946 FERRY 1423 6197 16 -298 43 366 14 -77 05 94 53 0 487 PITMED 8422 6281 16 133 85 232 43 34 10 59 21 0 804 MISSHAB 9023 671 1 16 144 46 199 88 34 44 47 65 0 932 ABBLIB . 9898 8621 16 79 80 168 58 14 81 31 29 0 905 ABBAIR 10174 7542 16 164 52 226 07 34 90 47 96 0 892 LANGLEY 7891 6968 16 57 68 160 49 13 24 36 85 0 885 CLISTN 2230 6556 16 -270 38 316 59 -65 99 77 26 0 631 Table A.3.1: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Appl i c a t i o n of the Or i a i n a l Tarpley (1979) Model to Three Days in Summer Representing (3) Sky Conditions. Units: KJ n r 2 hr-1 Array Size: 5 x 5 DAY 121/80(CLEAR) STATION CAL . OBS. N MBE RMSE MBE% RMSE% R GRSMT 23374 26 122 12 -228 . 95 292 . 72 - 10 . 52 13 . 45 0 .991 NRTHMT 23667 262 19 13 - 196 . 30 231 . 17 -9 . 73 1 1 . 46 0 .994 BCHYDRO 23687 26152 13 - 189 . 64 2 16 .31 -9 . 43 10 . 75 0 . 997 LANGA 23645 25974 13 - 179 . 16 200 . 72 -8 .97 10 .05 0 .996 VANAIR 23966 25763 13 - 138 . 19 174 .95 -6 . 97 8 .83 0 .996 FERRY 24066 26334 13 - 174 . 46 224 .04 -8 .61 1 1 .06 0 .998 PITMED 23787 254 17 13 - 125 . 4 1 143 . 56 -6 . 4 1 7 . 34 0 .998 MISSHAB 23228 253 19 13 - 160 . 85 19S . 75 -8 . 26 10 . 20 0 .992 ABBLIB 23482 25355 13 - 144 . 1 1 188 . 53 -7 . 39 9 .67 0 .994 ABBAIR 23842 25409 13 - 120 . 57 153 . 53 -6 . 1 7 7 .86 0 .997 LANGLEY 23520 25344 13 - 140 . 36 178 . 74 -7 . 20 9 . 17 0 .992 CLISTN 23335 2596 1 13 -202 .03 22 1 .08 - 10 . 12 1 1 .07 0 .997 DAY 160/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3166 65 10 14 -238 .82 307 .88 -5 1 . 36 66 . 2 1 0 .910 NRTHMT 7730 9954 14 - 158 .83 333 .99 -22 . 34 46 . 98 0 . 963 BCHYDRO 8260 8379 14 -8 . 5 1 280 . 38 - 1 .42 46 .84 0 .920 LANGA 1 1701 92 14 14 177 .61 48 1 . 24 26 . 99 73 . 12 0 .848 VANAIR 12457 10708 14 124 . 9 1 404 . 36 16 . 33 52 . 87 0 .908 FERRY 7 184 1 1398 14 -300 .94 558 . 56 -36 . 97 68 .61 0 .868 PITMED 8979 8728 14 17 . 9 1 288 . 16 2 . 87 46 . 22 0. 918 MISSHAB 10635 9446 14 84 . 88 365 . 83 12 . 58 54 . 22 0. 860 ABBLIB 10393 96 14 14 55 . 61 336 . 51 8 . 10 49 00 0. 845 ABBA IR 10256 9282 14 69 . 58 394 . 60 10. '50 59 . 52 0. 780 LANGLEY 8291 7687 14 43 . 18 233. 46 7 . 86 42 . 52 0. 9 10 CLISTN 9224 10460 14 -88 . 28 354 . 36 -11. 82 47 . 43 0. 937 DAY 105/80(0VERCA ST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 2439 2381 12 4 . 77 ' 159 . 72 2 . 40 80. 49 0. 633 NRTHMT 3785 4223 10 -43 . 80 223. 57 - 10. 37 52 . 95 0. 877 BCHYDRO 5156 4885 12 22 . 50 156 . 68 5. 53 38 . 49 0. 91 1 LANGA 7345 4955 12 199 . 16 27 1 . 03 48 . 23 65. 64 0. 88 1 VANAIR 8231 5907 12 193 . 60 288 . 4 1 39 . 33 58 . 59 0. 896 FERRY 3065 7064 12 -333 . 29 407 . 69 -56 . 61 69 . 25 0. 973 PITMED 6304 4623 12 140. 08 195 . 80 36 . 36 50. 82 0. 925 MISSHAB 6782 402 1 12 230. 10 295 . 84 68 . 67 88 . 28 0. 777 ABBLIB 693 1 4015 12 243 . 00 298 . 59 72 . 62 89 . 23 0. 845 ABBA I R 7662 4280 12 28 1 . 86 336 . 96 79 . 02 94 . 47 0. 833 LANGLEY 6377 47 18 12 138 . 22 175 . 04 35 . 15 44 . 52 0. 942 CLISTN 3432 5153 12 - 143 . 43 226 . 28 -33 . 40 52 . 69 0. 94 3 Table A.3.2: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Original Tarpley (1979) Model to three days in Spring Representing (3) Sky Cover Conditions. Units: KJ m-2 hr-1. Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS . N GRSMT 304 18 32 179 16 NRTHMT 30426 30013 16 BCHYDRO 30468 30453 16 LANGA 30398 30437 16 VANAIR 30405 30527 16 FERRY 30204 30428 16 PITMED 30361 29086 16 MISSHAB 304 14 ' 29551 16 ABBLIB 30329 292 17 16 ABBA IR 30337 29302 16 LANGLEY 30364 29403 16 CLISTN 30275 30310 16 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N GRSMT 3267 5522 1 1 NRTHMT 6095 668 1 1 1 BCHYDRO . 6799 8028 1 1 LANGA 8692 7937 1 1 VANAIR 1 1363 9003 1 1 FERRY 6744 9606 1 1 PITMED 6089 8097 1 1 MISSHAB 9272 9945 1 1 ABBLIB 9420 9453 1 1 ABBAIR 10273 10094 1 1 LANGLEY 7249 8626 1 1 CLISTN 6948 10001 1 1 DAY 197/80(0VERCAST) MBE RMSE MBE% RMSE% R - 1 10 07 155 52 -5 47 7 73 0 999 25 79 77 18 1 38 4 1 1 0 998 0 94 8 1 80 0 05 4 30 0 998 -2 42 76 03 -0 13 4 00 0 998 -7 62 62 03 -0 40 3 25 0 999 - 14 01 70 44 -0 74 3 70 0 998 79 74 1 12 72 4 39 6 20 0 997 53 95 93 20 2 92 5 05 0 998 69 46 12 1 72 3 80 6 67 0 996 64 68 136 58 3 53 7 46 . 0 994 60 05 82 14 3 27 4 47 0 999 -2 16 85 24 -0 1 1 4 50 0 998 MBE RMSE MBE% RMSE% R -205 01 373 56 -40 84 74 42 0 87 1 -53 34 260 05 -8 78 42 8 1 0 823 -111 69 2 16 26 - 15 30 29 63 0 920 68 64 382 53 9 51 53 02 0 759 2 14 53 32 1 73 26 21 39 31 0 923 -260 15 464 73 -29 79 53 22 0 790 -182 55 302 65 -24 80 4 1 12 0 849 -61 23 44 1 07 -6 77 48 78 0 779 -2 94 4 18 52 -0 34 48 70 0 802 16 30 379 66 1 '78 4 1 37 0 865 -125 2 1 332 62 - 15 97 42 4 1- 0 677 -277 56 451 74 -30 53 49 69 0. 8 18 STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3581 5764 15 - 145 5 1 ' 208 74 -37 87 54 32 0 770 NRTHMT 5535 5399 16 8 50 97 30 2 52 28 83 0 874 BCHYDRO 6486 5504 16 61 33 129 57 17 83 37 66 0 891 LANGA 9 188 5882 16 206 66 280 87 56 22 76 40 0 829 VANAIR 87 12 6866 16 1 15 39 159 95 26 89 37 28 0 909 FERRY 3275 6 197 16 - 182 66 229 07 -47 16 59 14 0 925 PITMED 7738 628 1 16 91 12 169 42 23 2 1 43 16 0 833 MISSHAB 8244 67 1 1 16 95 80 139 93 22 84 33 36 0 932 ABBLIB 9304 862 1 16 42 70 2 10 89 7 92 39 14 0 807 ABBAIR 8797 7542 16 78 47 122 78 16 65 26 05 0 938 LANGLEY 7430 6968 16 28 89 1 16 47 6 63 26 74 0 893 CLISTN 4093 6556 16 - 153 93 180 29 -37 57 44 00 0 932 Table A.3.3: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Revised Tarpley (1979) Model to Three Days in Summer Representing (3) Sky Cover Conditions. Units-KJ nr'? h r - 1 . Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 30426 32 179 16 - 109 53 157 94 -5 45 7 85 0 999 NRTHMT 30433 30013 16 26 23 76 18 1 40 4 06 0 998 BCHYDRO 30470 30453 16 1 01 84 49 0 05 4 44 0 998 LANGA 30378 30437 16 -3 72 82 13 -0 20 4 32 0 998 VANAIR 30395 30527 16 -8 19 68 69 -0 43 3 60 0 999 FERRY 30220 30428 16 - 13 01 72 23 -0 68 3 80 0 998 PITMED 30383 29086 16 81 09 1 14 54 4 46 6 30 0 997 MISSHAB 304 65 29551 16 57 1 1 100 33 3 09 5 43 0 997 ABBLIB 30352 29217 16 70 89 123 80 3 88 6 78 0 996 ABBAIR 30360 29302 16 66 07 138 24 3 6 1 7 55 0 994 LANGLEY 30388 29403 16 6 1 57 85 37 3 35 4 65 0 999 CLISTN 30277 303 10 16 -2 04 86 17 -0 1 1 4 55 0 998 DAY 239/80(P/CL0UDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3 16 1 5522 11 -2 14 63 380 44 -42 76 75 79 0 873 NRTHMT 5774 668 1 1 1 -82 5 1 257 45 - 13 58 42 39 0 822 BCHYDRO 65 14 8028 11 - 137 58 232 92 - 18 85 31 92 0 9 19 LANGA 7910 7937 11 -2 43 293 89 -0 34 40 73 0 797 VANAIR 9535 9003 1 1 48 30 182 15 5 90 22 25 0 94 1 FERRY 5368 9606 11 -385 23 501 46 -44 1 1 57 42 0 825 PITMED 5977 8097 11 - 192 7 1 309 77 -26 18 42 08 0 845 MISSHAB 8153 9945 11 - 162 99 355 60 - 18 03 39 33 0 846 ABBLIB 8236 9453 11 - 1 10 59 289 70 - 12 87 33 7 1 0 899 ABBAIR 8252 10094 11 -167 4 3 323 73 - 18 25 35 28 0 889 LANGLEY" 6687 8626 11 -176 29 33 1 8 1 -22 '48 42 31 0 734 CLISTN 6059 10001 11 -358 36 489 53 -39 42 53 84 0 844 DAY 197/80(0VERCAST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3505 5764 15 - 150 6 1 2 13 4 1 -39 19 55 54 0 767 NRTHMT 5432 5399 16 2 06 ' 97 10 0 61 28 77 0 872 BCHYDRO 6 1 19 5504 16 38 40 9 1 77 1 1 16 26 68 0 930 LANGA 7864 5882 16 123 86 162 44 33 69 44 19 0 913 VANAIR 8039 6866 16 73 32 132 94 17 09 30 98 0 897 FERRY 2891 6 197 16 -206 66 247 49 -53 35 63 90 0 951 PITMED 7574 6281 16 80 87 165 26 20 60 42 10 0 834 MISSHAB 7926 67 1 1 16 75 92 1 19 73 18 10 28 54 0 944 ABBLIB 8580 862 1 16 -2 56 163 64 -0 48 30 37 0 871 ABBAIR 8439 7542 16 56 1 1 12 1 03 1 1 90 25 68 0 922 LANGLEY 7252 6968 16 17 75 108 95 4 08 25 02 0 907 CLISTN 4023 6556 16 - 158 3 1 184 16 -38 64 44 94 0 939 Table A.3.4: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Revised Tarpley (1979) Model to Three Days in Summer Based on a Flux Averaging Approach. Units: KJ nr^ hr-1. Array Size: 5 x 5 DAY 183/80(CLEAR) STATION GRSMT NRTHMT BCHYDRO LANGA VANAIR FERRY PITMED MISSHAB ABBLIB ABBAIR LANGLEY CLISTN CAL . OBS . N MBE RMSE MBE% RMSE% R 30365 32 1 79 16 -113 . 36 162 . 54 -5 . 64 8 .08 0 .998-304 22 30013 16 25 . 55 75 . 50 1 . 36 4 .03 0 .998 30439 30453 16 -0 .92 82 .69 -0 .05 4 . 34 0 .998 30353 30437 16 -5 . 23 75 . 49 -0 . 27 3 . 97 0 . 998 30400 30527 16 -7 . 9 1 63 . 39 -0 . 4 1 3 . 32 0. 999 302 1 1 30428 16 -13 .53 68 . 60 -0. 7 1 3 .61 0. 998 3038 1 29086 16 80 . 94 115 . 39 4 . 45 6 . 35 0. 997 30376 29551 16 5 1 . . 57 88 32 2 . 79 4 . 78 0. 998 30333 292 17 16 69 . 72 120. 67 3 . 82 6. 6 1 0. 996 30372 29302 16 66 . 84 138 . 82 3 . 65 7 . 58 0. 994 30368 29403 16 60. 32 82 . 31 3 . 28 4 . 48 0. 999 30334 30310 16 1 . 47 80. 5 1 0. 08 4 . 25 0. 998 DAY 197/'8O(0VERCAST) STATION GRSMT NRTHMT BCHYDRO LANGA VANAIR FERRY PITMED MISSHAB ABBLIB ABBAIR LANGLEY CLISTN CAL . OBS . N MBE RMSE MBE% RMSE% R 4 164 5764 15 - 106 . 66 220 .03 -27 . 76 57 . 26 0 .58 1 591 1 5399 16 3 1 .97 149 .50 9 . 47 44 . 30 0 . 756 5972 5504 16 29 . 20 63 . 97 8 . 49 18 . 60 0 . 960 8406 5882 16 157 . 76 209 .91 42 .92 57 . 10 0 .845 8646 6866 16 1 1 1 . 26 156 . 78 25 . 93 36 . 54 0 .909 3573 6 197 16 - 164 .02 242 . 35 -42 . 35 62 . 57 0. . 703 7677 628 1 16 87 . 30 169. 84 22 , 24 43 . 27 6. 829 8122 67 1 1 16 88 . 18 137 . 73 2 1 . 02 32 . 83 0. 928 8844 8621 16 13 . 92 142 . 86 2 . 58 26. 51 0. 913 9586 7542 16 127 . 75 285 . 89 27 . •10 60. 65 0. 729 7360 6968 16 24 . 48 1 16 . 17 5 . 62 26 . 67 0. 897 4124 6556 16 - 152 . 00 178 . 29 -37 . 10 43 . 51 0. 936 DAY 239/80(P/CLOUDY) STATION GRSMT NRTHMT BCHYDRO LANGA VANAIR FERRY PITMED MISSHAB ABBLIB ABBAIR LANGLEY CLISTN CAL . OBS . N MBE RMSE MBE% RMSE% R 3370 5522 1 1 - 195 . 57 ' 368 . 67 -38 .96 73 . 44 0 .845 6013 668 1 1 1 -60 . 75 248 . 88 - 10 .00 40 . 98 0 . 834 748 1 8028 1 1 -49 . 68 262 . 39 -6 .81 35 .95 0 . 883 8449 7937 1 1 46 . 52 331 . 48 6 . 45 45 .94 0 . 799 10526 9003 1 1 138 . 4 1 256 . 5 1 16 .91 31 . 34 0. 938 6233 9606 1 1 -306 58 48 1 . 2 1 -35 . 1 1 55 . 1 1 0. 799 6029 8097 1 1 - 187 . 94 311 . 14 -25 .53 42 . 27 0. 829 9249 9945 1 1 -63. 29 448 24 -7 , 00 49 . 58 0. 77 1 9518 9453 1 1 5. 90 4 19 . 62 0. 69 48 . 83 0. 799 9474 10094 1 1 -56 . 38 433. 82 -6 . 14 47 . 28 0. 795 6990 8626 1 1 - 148 . 73 336. 68 - 18 . 97 42 . 93 0. 692 6395 10001 ' 1 1 -327 . 86 511. 99 -36 . 06 56 . 31 0. 778 ro -P=> Table A.3.5: Hourly V e r i f i c a t i o n S t a t i s t i c s for Tarpley (1979) Model to Three Days the Individual Stations from the Application of the Revised in Summer Based on a 3 x 3 Array. Units: KJ m-2 h r ~ l DAY 121/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 24585 26 122 12 - 128 .08 2 19 . 37 -5 . 88 10 .08 0 .990 NRTHMT 25333 262 19 13 -68 . 15 12 1 .81 -3 . 38 6 .04 0 .997 BCHYDRO 25333 26152 13 -63 .01 107 . 35 -3 . 13 5 . 34 0 .998 LANGA 25323 25974 13 -50 .06 82 . 27 -2 .51 4 . 12 0 . 998 VANAIR 25457 25763 13 -23 .52 102 .02 - 1 . 19 5 . 15 0 .996 FERRY 25490 26334 13 -64 •91 138 . 33 -3 . 20 6 .83 0 .997 PITMED 25379 254 17 13 - 2 .93 53 . 8 1 -0 . 15 2 . 75 0 .999 MISSHAB 25039 25319 13 -2 1 . 57 9 1 . 96 - 1 . 1 1 4 . 72 0 .996 ABBLIB 25165 25355 13 - 14 . 59 100 . 13 -0 . 75 5 . 13 0 .996 ABBAIR 25470 25409 13 4 . 68 80 . 54 0 . 24 4 . 12 0 . 998 LANGLEY 25266 25344 13 -6 .02 69 . 46 -0 . 3 1 3 . 56 0 .997 CLISTN 25205 2596 1 13 -58 . 13 98 . 23 -2 . 9 1 4 . 92 0 . 997 DAY 276/79(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE7, R GRSMT 14755 16278 10 - 152 . 30 177 . 89 -9 . 36 10 . 93 0 . 997 NRTHMT 14734 1498 1 10 -24 . 65 87 . 89 - 1 .65 5 .87 0 .995 BCHYDRO 14764 14833 10 -6 . 95 99 . 14 -0 .47 6 . 68 0 . 994 LANGA 14697 15 143 10 -44 . 60 74 . 17 -2 .95 4 . 90 " 0 998 VANAIR 14697 14936 10 -23 . 89 8 1 . 23 - 1 .60 5 . 44 0. 998 FERRY 14242 151 16 10 -87 .42 124 .98 -5 . 78 8 . 27 0. 998 PITMED 14702 14593 10 10 . 94 9 1 27 0 . 75 6 . 25 0. 997 MISSHAB 14678 15007 10 -32 . 84 106 . . 6 1 -2 . 19 7 . 10 0. 988 ABBLIB 1468 1 149 14 10 -23 . 22 89 . 23 - 1 . 56 5 . 98 0. 994 ABBAIR 14669 15131 10 -46 . 14 112. 06 -3 . 05 7 . 4 1 0. 989 LANGLEY 14682 14600 10 8 . 27 92 . 75 0. 57 6 . 35 0. 993 CLISTN 14632 15151 10 -5 1 . 93 127 . 78 -3 . 43 8 . 43 0. 985 DAY 025/80(CLEAR ) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 6114 7378 7 -180. 62 ' 190. 83 -17. 14 18 . 1 1 0. 985 NRTHMT 6232 688 1 7 -92 . 81 1 18 . 63 -9 . 44 12 . 07 0. 983 BCHYDRO 62 15 6494 7 -39 . 87 119. 2 1 -4 . 30 12 . 85 0. 952 LANGA 6151 6942 7 -112. 91 153 . 93 -11. 39 15 . 52 0. 963 VANAIR 6033 6770 7 - 105 . 16 124 . 65 - 10. 87 12 . 89 0. 974 FERRY 6163 6837 7 -96 . 29 106 . 23 -9. 86 10. 88 0. 989 PITMED 6 166 7454 7 - 183 . 89 199 . 12 - 17 . 27 18 . 70 0. 982 MISSHAB 6486 7533 8 -130. 93 203 . 37 - 13 . 90 2 1 . 60 0. 975 ABBLIB 6192 7354 7 - 165 . 94 176 . 1 1 - 15 . 80 16 . 76 0. 996 ABBAIR 6144 7524 7 - 197 . 16 203 . 44 - 18 . 34 18 . 93 0. 998 LANGLEY 6489 7198 . 8 -88 . 66 165 . 82 -9 . 85 18 . 43 0. 980 CLISTN 6163 7304 7 - 163 . 05 167 . 66 - 15 . 63 16 . 07 0. 999 Table A.3.6: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Revised Tarpley (1979) Model to Three Clear Days. Units: KJ nr? hr-1. Array Size: 5 x 5 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE"/. RMSE0/ R GRSMT 3267 5522 1 1 -205 .01 373 . 56 -40 . 84 74 . 42 0 .871 NRTHMT 6095 668 1 1 1 -53 . 34 260 .05 -8 . 78 42 .81 0 . 823 BCHYDRO 6799 8028 1 1 - 1 1 1 . 69 2 16 . 26 - 15 . 30 29 . 63 0 920 LANGA 8692 7937 1 1 68 .64 . 382 . 53 9 .51 53 .02 0 759 VANAIR 1 1363 9003 1 1 2 14 . 53 32 1 . 73 26 21 39 . 3 1 0 923 FERRY 6744 9606 1 1 -260 . 15 464 73 -29 79 53 . 22 0 790 PITMED 6089 8097 1 1 - 182 55 302 65 -24 80 4 1 . 1 2 0 849 MISSHAB 9272 9945 1 1 -6 1 23 44 1 07 -6 77 48 78 0 779 ABBLIB 9420 9453 1 1 -2 94 4 18 52 -0 34 48 70 0 802 ABBAIR 10273 10094 1 1 16 30 379 66 1 78 4 1 37 0 865 LANGLEY 7249 8626 1 1 - 125 2 1 332 62 - 15 97 42 4 1 0 677 CLISTN 6948 10001 1 1 -277 56 45 1 74 -30 53 49 69 0 8 18 DAY 160/80(P/CL0UDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 4 147 6510 14 - 168 78 290 69 -36 30 62 52 0 925 NRTHMT 101 16 9954 14 1 1 59 286 75 1 63 40 33 0 956 BCHYDRO 10572 8379 14 155 6 1 333 62 26 17 55 74 0 918 LANGA 13499 92 14 14 306 04 540 91 46 50 82 19 0 866 VANAIR 14285 10708 14 255 48 433 26 33 40 56 65 0. 926 FERRY 8012 1 1398 14 -24 1 86 644 09 -29. 7 1 79 12 0. 756 PITMED 10443 8728 14 122 50 249 34 19 . 65 39 99 0. 953 MISSHAB 1 1748 9446 14 164 . 38 396 1 1 24 . 36 58 . 7 1 0. 873 ABBLIB 11763 96 14 14 153 . 49 3 14. 17 22 . 35 45 . 75 0. 910 ABBAIR 1 1092 9282 14 129 . 3 1 402 . 73 19 . 51 60. 75 0. 792 LANGLEY 91 14 7687 14 101 . 9 1 273 . 07 18 . 56 49 . 73 0. 889 CLISTN 1 1520 10460 14 75 . 72 366 . 12 10. 13 49. OO o. 94 1 Table A.3.7: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Revised Tarpley (1979) Model to Two Partly Cloudy Days. Units: KJ n r 2 hr-1. Array Size: 5 x 5 DAY 263/79(0VERCAST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3138 3888 9 -83 . 33 209 .99 - 19 . 29 48 . 6 1 0 . 533 NRTHMT 3750 2897 9 94 . 73 278 . 36 29 . 42 86 . 47 -0 .016 BCHYDRO 3655 2864 9 87 . 88 208 .91 27 .61 65 .64 0 . 189 LANGA 4237 4050 9 20 . 76 144 . 64 4 .61 32 . 14 0 . 855 VANAIR 4468 4 189 9 30 . 92 167 . 16 6 .64 35 .91 0 . 857 FERRY 2 166 4090 9 -2 13 . 70 288 . 33 -47 .03 63 . 45 0 . 905 PITMED 3143 2323 6 136 . 78 160 . 40 35 . 33 4 1 .43 0 .838 MISSHAB 4036 3037 9 1 1 1 .00 145 . 19 32 .90 43 .03 0 . 890 ABBLIB 4121 2890 9 136 . 84 157 . 4 1 42 . 62 49 .02 0 . 889 ABBAIR 4256 3172 9 120 .44 14 1 . 50 34 . 17 40 . 14 0 .914 LANGLEY 3794 2683 9 123 . 37 150 . 26 4 1 .38 50 . 40 0 . 885 CLISTN 2806 4345 9 -17 1 .03 276 . 46 -35 .43 57 . 26 0 . 758 DAY 105/80(0VERCAST) STATION CAL . OBS. N MBE RMSE MB E % RMSE% R GRSMT 38 1 1 2381 12 1 19 . 18 160 . 23 60 .06 80 . 74 0 .693 NRTHMT 4 187 4223 10 -3 . 53 289 . 69 -0 . 84 68 .60 0 . 809 BCHYDRO 54 13 4885 12 43 . 99 226 .92 10 .80 55 . 74 0 . 849 LANGA 6697 4955 12 145 . 18 278 . 16 35 . 16 67 . 36 0. 820 VANAIR 8473 5907 12 2 13 . 76 272 .05 43 .42 55 . 26 0. 937 FERRY 4531 7064 12 -206 . 15 300 .82 -35 .02 51 . 10 0. 975 PITMED 5916 4623 12 107 . 73 150 .01 27 .96 38 . 94 0. 909 MISSHAB 6377 402 1 12 196 3 1 262 . 27 58 . 58 78 . 27 0. 709 ABBLIB 7079 4015 12 255 . 26 316 77 76 . 28 94 . 66 0. 729 ABBAIR 7620 4280 12 278 . 33 364 . 92 78' .03 102 . 31 0. 662 LANGLEY 6066 47 18 12 112. 33 180. 13 28 . 57 45 . 8 1 0. 923 CLISTN 44 18 5 153 12 -6 1 . 23 287 . 78 - 14 . 26 67 . 02 0. 896 DAY 197/80(OVERCAST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3581 5764 15 - 145 . 51 208 . 74 -37 . 87 54 . 32 0. 770 NRTHMT 5535 5399 16 8 . 50 97 . 30 2 . 52 28 . 83 0. 874 BCHYDRO 6486 5504 16 61 . 33 129 . 57 17 . 83 37 . 66 0. 891 LANGA 9188 5882 16 206 . 66 280. 87 56 . 22 76. 40 0. 829 VANAIR 8712 6866 16 115. 39 159 . 95 26 . 89 37 . 28 0. 909 FERRY 3275 6197 16 • - 182 . 66 229 . 07 -47 . 16 59. 14 0. 925 PITMED 7738 628 1 16 9 1 . 12 169 . 42 23 . 21 43 . 16 0. 833 MISSHAB 8244 67 1 1 16 95 . 80 139 . 93 22 . 84 33 . 36 0. 932 ABBLIB 9304 862 1 16 42 . 70 210. 89 7 . 92 39 . 14 0. 807 ABBAIR 8797 7542 16 78 . 47 122 . 78 16 . 65 26 . 05 0. 938 LANGLEY 7430 6968 16 28 . 89 1 16 . 47 6 . 63 26 . 74 0. 893 CLISTN 4093 6556 16 • - 153 . 93 180. 29 -37 . 57 44 . OO 0. 932 Table A.3.8: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Appl i c a t i o n of the Revised Tarpley (1979) Model to Three Overcast Days. Units: KJ nr? h r " 1 . Array Size: 5 x 5 DAY 239/8O(P/CL0UDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 229G 5522 11 -293 23 379 62 -58 42 75 63 0 922 NRTHMT 6076 6681 1 1 -54 99 355 6.1 -9 05 58 55 0 838 BCHYDRO 72 17 8028 1 1 -73 7 1 261 7 1 - 10 10 35 86 0 924 LANGA 8505 7937 11 5 1 63 223 47 7 16 30 97 0 899 VANAIR 10706 9003 11 154 7G 225 69 18 91 27 57 0 954 FERRY 5069 9606 11 -4 12 40 513 45 -47 22 58 80 0 849 PITMED 6997 8097 1 1 -99 93 208 75 - 13 58 28 36 0 884 MISSHAB 9535 9945 1 1 -37 35 293 01 -4 13 32 4 1 0 897 ABBLIB 9725 9453 1 1 24 77 278 20 "2 88 32 37 0 907 ABBAIR 9466 10094 1 1 -57 13 347 28 -6 23 37 84 0 87 1 LANGLEY 7569 8626 1 1 -96 16. 263 57 - 12 26 33 61 0 803 CLISTN 5862 10001 11 -376 26 489 97 -4 1 38 53 89 0 857 Table A.3.9: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Tarpley (1979) Model to a Partly Cloudy Day in Summer Using a Modified Threshold. Units: KJ m_2 h r - 1 . Threshold: B + 28 counts C O 279 APPENDIX A.4 DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 32121 32 179 16 -3 57 94 65 -0 18 4 7 1 0 998 NRTHMT 3 1 174 30013 16 72 53 1 13 62 3 87 6 06 0 997 BCHYDRO 31310 30453 16 53 56 89 82 2 81 4 72 0 999 LANGA 31597 30437 16 72 47 136 74 3 8 1 7 19 0 995 VANAIR 31507 30527 16 6 1 30 1 15 56 3 2 1 6 06 0 996 FERRY 30794 30428 16 22 86 89 60 1 20 4 7 1 0 997 PITMED 30940 29086 16 1 15 88 152 72 6 37 8 40 0 998 MISSHAB 30844 29551 16 80 83 100 67 4 38 5 45 0 999 ABBLIB 30453 292 17 16 77 25 109 22 4 23 5 98 0 998 ABBAIR 30908 29302 16 100 34 148 38 5 48 8 10 0 997 LANGLEY 30752 29403 16 84 29 12 1 89 • 4 59 6 63 0 998 CLISTN 30360 30310 16 3 16 100 66 0 17 5 31 0 997 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 5901 5522 1 1 34 52 240 59 6 88 47 93 0 906 NRTHMT 6618 6681 11 -5 74 2 1 1 76 -0 95 34 86 0 865 BCHYDRO 7325 8028 1 1 -63 88 163 34 -8 75 22 38 0 951 LANGA 7579 7937 11 -32 56 257 73 -4 51 35 72 0 850 VANAIR 8892 9003 11 - 10' 12 144 19 - 1 24 17 62 0 963 FERRY 8235 9606 11 -124 65 280 13 - 14 27 32 08 0 909 PITMED 6280 8097 11 -165 14 255 48 -22 44 34 71 0 889 MISSHAB 8471 9945 11 - 134 02 306 60 - 14 82 33 91 0' 902 ABBLIB 8842 9453 1 1 -55 54 222 93 -6 46 25 94 0 942 ABBAIR 8917 10094 11 -106 99 273 01 - 1 1 66 29 75 0 910 LANGLEY 6596 8626 11 - 184 54 322 46 -23 '53 4 1 12 0 766 CLISTN 7940 10001 11 - 187 39 333 34 -20 6 1 36 66 0 908 DAY 197/80(0VERCAST) STATION CAL . OBS. N MBE RMSE MBE% RMSE% R GRSMT 5919 5764 15 10 36 132 09 2 70 34 38 0 81 1 NRTHMT 6093 5399 16 • 43 38 ' 90 26 12 85 26 75 0 917 BCHYDRO 6424 5504 16 57 46 89 33 ' 16 70 25 96 0 945 LANGA 6756 5882 16 54 62 63 4 1 14 86 17 25 0 987 VANAIR 6684 6866 16 - 1 1 34 78 95 -2 64 18 40 0 956 FERRY 6134 6197 16 -3 95 100 00 - 1 02 25 82 0 929 PITMED 6354 6281 16 4 56 161 40 1 16 4 1 12 0 745 MISSHAB 6707 67 1 1 16 -0 30 120 56 -0 07 28 74 0 9 17 ABBLIB 7094 862 1 16 -95 45 170 67 - 17 72 31 68 0 929 ABBAIR 6893 7542 .16 -40 52 128 65 -8 60 27 29 0 902 LANGLEY 6350 6968 16 -38 63 98 62 -8 87 22 64 0 932 CLISTN 5813 6556 16 -46 43 8 1 01 - 1 1 33 19 77 0 960 Table A.4.1: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from'the Application of the Gautier et a l . (1980) Model to Three Days in Summer Representing (3) Sky Cover Conditions. Units: KJ m"^  h r - l . Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS. N MBE RMSE MBE% RMSE% R GRSMT 32892 32 179 16 44 . 57 16 1 .42 2 . 22 8 .03 0 .994 NRTHMT 31898 30013 16 1 17 . 78 196 . 56 6 . 28 10 . 48 0 .991 BCHYDRO 32013 30453 16 97 . 48 159 . 25 5 . 12 8 . 37 0 .995 LANGA 32075 30437 16 102 . 34 201 .68 5 . 38 10 .60 0 .988 VANAIR 32072 30527 16 96 . 56 175 . 5 1 5 .06 9 . 20 0 .992 FERRY 3 1356 30428 16 58 .03 168 . 56 3 .05 8 .86 0 . 990 PITMED 31409 29086 16 145 .24 202 .04 7 . 99 1 1 . 1 1 0 .992 MISSHAB 31482 2955 1 16 120 . 69 177 .88 6 . 53 9 . 63 0 .993 ABBLIB 31 177 29217 16 122 . 46 185 .09 6 . 7 1 10 . 14 0 . 992 ABBAIR 3 1436 29302 16 133 . 35 2 10 .09 7 . 28 1 1 47 0 . 990 LANGLEY 31398 29403 16 124 7 1 194 . 20 6 . 79 10 57 0 .992 CLISTN 30807 303 10 16 3 1 07 163 . 70 1 64 8 64 0 .991 DAY 239/8O(P/CL0UDY ) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 57 15 5522 1 1 17 54 228 95 3 49 45 61 0 907 NRTHMT 6378 6681 1 1 . "27 55 2 16 23 -4 54 35 60 0 863 BCHYDRO 7151 8028 1 1 -79 73 180 95 - 10 93 24 79 0 939 LANGA 7454 7937 1 1 -43 9 1 259 02 -6 09 35 90 0 848 VANAIR 8845 9003 1 1 - 14 39 142 57 - 1 76 17 42 0 964 FERRY 8123 9606 1 1 - 134 84 298 42 - 15 44 34 17 0 888 PITMED 6020 8097 1 1 - 188 78 265 91 -25 65 36 13 0 888 MISSHAB 8286 9945 1 1 - 150 83 298 46 - 16 68 33 o r 0 912 ABBLIB 8643 9453 1 1 -73 56 2 18 25 -8 56 25 40 0 950 ABBAIR 8733 10094 1 1 - 123 7 1 276 33 - i d 48 30 1 1 0 915 LANGLEY 6489 8626 1 1 -194 28 322 77 -24 77 4 1 16 0 779 CLISTN 792 1 10001 1  - 189 1 1 325 05 -20 80 35 75 0 917 DAY 197/80(0VERCAST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 5832 5764 15 4 56 138 38 1. 19 36 01 0 790 NRTHMT 6025 5399 16 39. 13 97 4 1 1 1 60 28 . 86 0 895 BCHYDRO 6369 5504 16 54 . 00 102 02 15 70 29 . 65 0. 916 LANGA 6670 5882 16 49. 24 67 72 13. 39 18 . 42 0. 975 VANAIR 6635 6866 16 - 14 . 4 1 87 . 90 -3 . 36 20. 49 0. 938 FERRY 6001 6 197 16 - 12 . 29 99 . 03 -3 . 17 25 . 57 0. 932 PITMED 6346 6281 16 4 . 07 163 . 95 1 . 04 4 1 . 77 0. 738 MISSHAB 6660 67 1 1 16 -3 . 22 125 . 00 -0. 77 29. 80 0. 903 ABBLIB 7 127 862 1 16 -93. 39 175 . 18 - 17 . 33 32 . 51 0. 924 ABBAIR 6907 7542 16 -39. 66 136 . 15 -8 . 4 1 28 . 89 0. 883 LANGLEY 6261 6968 16 -44 . 19 107 . 60 - 10. 15 24 . 7 1 0. 915 CLISTN 5796 6556 16 -47 . 5 1 86 . 88 -11. 59 2 1 . 20 0. 945 Table A.4.2: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Gautier et a l . (1980) Model to Three Days in Summer Based on a Pixel Averaging Approach. , Units: KJ hr"'. Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS . GRSMT 32232 32 179 NRTHMT 31131 30013 BCHYDRO 31 155 30453 LANGA 31G65 30437 VANAIR 31554 30527 FERRY 30355 30428 PITMED 3 1085 29086 MISSHAB 31279 2955 1 ABBLIB 30510 292 17 ABBAIR 30887 29302 LANGLEY 307 16 29403 CLISTN 30848 303 10 DAY 239/8O(P/CL0UDY) STATION CAL . OBS . GRSMT 5695 5522 NRTHMT 6450 668 1 BCHYDRO 7053 8028 LANGA 7593 7937 VANAIR 8725 9003 FERRY 8475 9606 PITMED 6283 8097 MISSHAB 84 16 9945 ABBLIB 8645 9453 ABBAIR 8894 10094 LANGLEY 6728 8626 CLISTN 804 1 10001 DAY 197/8O(0VERCAST) STATION CAL . OBS . GRSMT 5985 5720 NRTHMT 6038 531 1 BCHYDRO 6260 5434 LANGA 6702 5802 VANAIR 6659 6788 FERRY 6153 604 2 PITMED 6262 6 164 MISSHAB 6569 6604 ABBLIB 6984 8553 ABBAIR 6752 7462 LANGLEY 6383 6874 CLISTN 6013 6448 N MBE RMSE MBE% RMSE% R. 16 3 . 33 83 . 83 0 . 17 4 . 17 0 .998 16 69 . 88 103 . 82 3 . 73 5 . 53 0 . 998 16 43 . 87 90 . 79 2 . 30 4 . 77 0 .998 16 76 . 77 1 12 . 69 4 .04 5 . 92 0 .997 16 64 . 2 1 108 . 16 3 . 37 5 .67 0 .997 16 -4 . 53 84 85 -0 24 ' 4 . 46 0 997 16 124 97 154 52 6 87 8 . 50 0 999 16 108 00 126 40 5 85 6 84 0 999 16 80 77 1 13 03 4 42 6 19 0 998 16 99 02 139 37 , 5 4 1 7 6 1 0 998 16 82 05 1 18 6 1 4 46 6 45 0 998 16 33 6 1 89 1 1 1 77 4 70 0 998 N MBE RMSE MBE% RMSE% R 1 1 15 72 244 1 7 3 13 48 64 0 927 1 1 -20 99 194 36 -3 46 32 00 0 889 1 1 -88 64 1 74 67 - 12 15 23 93 0 959 1 1 -3 1 24 253 64 -4 33 35 15 0 850 1 1 -25 26 1 17 97 -3 09 14 4 1 0. 980 1 1 - 102 82 253. 90 -11. 77 29 08 0. 928 1 1 - 164 92 259. 77 -22 . 4 1 35 . 29 0. 878 1 1 - 139 . 00 298 . 28 - 15 . 37 32 . 99 0. 908 1 1 -73 . 45 226 . 38 -8 . 55 26 . 34 0. 940 1 1 - 109 . 12 276. 42 -11. 89 30. 12 0. 909 1 1 - 172 . 59 290. 1 1 -22 . 01 36 . 99 0. 828 1 1 - 178 . 20 32 1 . 54 - 19. 60 35 . 37 0. 919 N MBE RMSE MBE% RMSE% R 13 20 39 ' 150 31 4 63 34 16 O 615 14 5 1 88 1 18 28 13 68 3 1 18 0 796 14 58 96 107 58 15 19 27 72 0 831 14 64 28 97 65 15 51 23 56 0 904 14 -9 24 1 19 89 - 1 91 24 73 0 818 14 7 92 127 03 1 84 29 43 0 837 14 7 00 179 30 1 59 40 73 0 594 14 -2 52 143 1 1 -0 53 30 34 0 844 14 -112 08 192 35 - 18 35 31 48 0 85 1 14 -50 70 155 98 -9 51 29 26 0 759 14 -35 06 1 17 39 -7 14 23 91 0 826 14 -3 1 07 101 84 -6 75 22 1 1 0 829 Table A.4.3: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Gautier et a l . (1980) Model to Three Days in Summer Based on an 8 x 8 Array. Units: KJ m-2 h r - 1 DAY 121/80(CLEAR ) STATION CAL . OBS . N GRSMT 26G16 26 122 12 NRTHMT 26482 262 19 13 BCHYDRO 26493 26152 13 LANGA 26938 25974 13 VANAIR 275 12 25763 13 FERRY 27595 26334 13 PITMED 27 170 25417 ' 13 MISSHAB 25760 253 19 13 ABBLIB 26247 25355 13 ABBAIR 26648 25409 13 LANGLEY 26881 25344 13 CLISTN 26066 259G1 13 DAY 276/79(CLEAR) MBE RMSE MBE% RMSE% R 4 1 . 19 157 . 14 1 . 89 7 . 22 0 . 990 20 . 26 99 . 79 1 . oo 4 . 95 0 .995 26 . 22 87 . 6 1 1 . 30 4 . 35 0 . 997 74 . 16 101 . 32 3 . 7 1 5 .07 0. .998 134 . 55 157 .68 6 . 79 7 . 96 0 996 97 01 137 .93 4 . 79 6 .81 0. 996 134 . 84 150 . 76 6 . 90 7 . 7 1 0. 998 33 . 92 142 .89 1 . 74 7 . 34 0. 990 68 . 6 1 144 . 50 3 . 52 7 41 0. 992 95 . 33 157 . 86 4 . 88 8 . 08 0. 992 118. 18 17 1. 36 6 . 06 8 . 79 0. 993 8 . 1 1 65 . 23 0. 4 1 3 . 27 0. 998 STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 16123 16278 10 - 15 . 54 83 . 20 -0 . 95 5 . 1 1 0 .995 NRTHMT 15379 1498 1 10 39 .82 123 .63 2 .66 8 . 25 0 . 990 BCHYDRO 154 14 14833 10 58 .06 129 . 47 3 .91 8 . 73 0 . 989 LANGA 15172 15143 10 2 . 84 143 .89 0 . 19 9 .50 0 . 976 VANAIR 15291 14936 10 35 58 135 . 72 2 . 38 9 .09 0. .986 FERRY 14693 151 16 10 -42 . 29 95 .90 -2 . 80 6 . 34 0. .994 PITMED 15522 14593 10 92 . 90 132 . 13 6. 37 9 . 05 0. 994 MISSHAB 15118 15007 10 1 1 . 16 106 . 42 0. 74 7 . 09 0. 988 ABBLIB 15238 149 14 10 32 . 45 113. 48 2 . 18 7 . 61 0. 987 ABBAIR 15095 15131 10 -3 . 60 125 . 04 -o! 24 8 . 26 O. 984 LANGLEY 15199 14600 10 59 . 96 122 . 28 4 . 1 1 8 . 38 0. 990 CLISTN 15065 15151 10 -8 . 65 112. 43 -0. 57 7 . 42 0. 988 DAY 025/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 7064 7378 7 -44 . 87' 76 . 36 -4 . 26 7 . 24 0 .988 NRTHMT 7220 688 1 7 48 .43 98 . 70 4 .93 10 .04 0 .975 BCHYDRO 7273 6494 7 1 1 1 .31 155 .43 12 .00 16 . 75 0 .954 LANGA 7073 6942 7 18 . 70 86 .09 1 .89 8 .68 0. .979 VANAIR 6709 6770 7 -8 . 72 133 . 75 -0 .90 13 .83 0. 905 FERRY 7247 6837 7 58 .51 73 . 66 5. .99 7 . 54 0. 991 PITMED 7078 7454 7 -53 60 121 . 38 -5 . 03 1 1 . . 40 0. 952 MISSHAB 7062 7533 8 -58 .88 91 . 66 -6 . 25 9. 73 0. 998 ABBLIB 7084 7354 7 -38 . 56 87 . 62 -3 . 67 8 . 34 0. 990 ABBAIR 6709 7524 7 -116. 52 15 1. 20 - 10. 84 14 . 07 0. 990 LANGLEY 7183 7198 8 - 1 . 90 54 . 58 -0. 2 1 6 . 07 0. 997 CLISTN 7123 7304 7 -25 . 86 61 . 03 -2 . 48 5. 85 0. 989 Table A.4.4: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Appl i c a t i o n of the Gautier et a l . (1980) model to Three Clear Days. Units: KJ nr? h r - 1 . Array Size: 5 x 5 DAY 263/79(0VF.RCAST ) STATION CAL . OBS. N GRSMT 4029 3888 9 NRTHMT 3289 2897 9 BCHYDRO 3270 2864 9 LANGA 3515 4050 9 VANAIR 3921 4 189 9 FERRY 361 1 4090 9 PITMED 2264 2323 6 MISSHAB 2804 3037 9 ABBLIB 295 1 2890 9 ABBAIR 2855 3 172 9 LANGLEY 2797 2683 9 CLISTN 3806 4345 9 DAY 105/8O(0VERCAST) STATION CAL. • OBS . N GRSMT 6428 2381 12 NRTHMT 5457 4223 10 BCHYDRO 6834 4885 12 LANGA 7298 4955 12 VANAIR 79 1 1 5907 12 FERRY 7456 7064 12 PITMED 5858 4623 12 MISSHAB 5964 402 1 12 ABBLIB 6174 4015 12 ABBAIR 6432 4280 12 LANGLEY 627 1 47 18 12 CLISTN 7049 5 153 12 DAY 197/80(OVERCAST) STATION CAL . OBS . N GRSMT 5919 5764 15 NRTHMT 6093 5399 16 BCHYDRO 6424 5504 16 LANGA 6756 5882 16 VANAIR 6684 6866 16 FERRY 6 134 6 197 16 PITMED 6354 6281 16 MISSHAB 6707 67 1 1 16 ABBLIB 7094 862 1 16 ABBAIR 6893 7542 16 LANGLEY 6350 6968 16 CLISTN 5813 6556 16 MBE RMSE MBE% RMSE0/. R 15 . 70 140 . 19 3 .64 32 . 45 0 .657 4 3*. 57 195 . 54 13 . 53 60 . 74 0 . 200 45 .08 1 10 . 97 14 . 17 34 . 87 0 . 779 -59 . 43 178 . 1 1 - 13 . 2 1 39 . 58 0 . 785 -29 .84 168 . 55 -6 . 4 1 36 . 2 1 0 .879 -53 . 2 1 138 . 30 - 1 1 . 7 1 30 .43 0 .927 -9 . 8 1 95 . 24 -2 . 53 24 .60 0 .677 -25 . 83 1 10 .08 -7 .66 32 .63 0 .88 1 6 . 8 1 91 . 74 2 . 12 28 . 57 0 . 854 -35 .23 78 . 20 -9 .99 22 . 19 - 0 .950 12 . 67 93 .69 4 . 25 3 1 .42 0 . 872 -59 88 201 . 36 - 12 . 40 4 1 . 7 1 0 .774 MBE RMSE MBE% RMSE% R 337 . 22 396 . 57 169 .94 199 .85 0 .751 123 . 46 233 . 35 29 . 24 55 . 26 0 .886 162 . 40 2 13 . 12 39 . 89 52 . 39 0 .940 195 . 22 235 . 47 47 . 28 57 .02 0 . 94 1 166 . 96 209 . 53 33 . 92 42 . 56 0. .972 32 . 6 1 192 . 27 5 . 54 32 .66 0. .970 102 . 89 1 28 . 73 26 . 7 1 33 .42 0. 962 16 1 . 92 -205. . 50 48 . 32 6 1 . 32 0. 863 179 . 88 2 14. 01 53 . 76 63 . 96 0. 904 179 . 36 237 . 31 50'. 29 66 . 53 0. 86 1 129. 42 16 1 . 8 1 32 . 91 4 1 . 15 0. 948 157 . 97 226 . 82 36 . 79 52 . 82 0. 935 MBE RMSE MB E % RMSE0/ R 10. 36 132 . 09 2 . 70 34 . 38 0. 81 1 43 . 38 90. 26 12 . 85 26 . 75 0. 917 57 . 46 89. 33 16 . 70 25 . 96 0. 945 54 . 62 63 . 4 1 14'! 86 17 . 25 0. 987 -11. 34 78 . 95 -2 . 64 18 . 40 0. 956 -3 . 95 100. OO - 1 . 02 25. 82 0. 929 4 . 56 161 . 40 1 . 16 4 1 . 12 0. 745 -0. 30 120. 56 -0. 07 28 . 74 6. 917 -95 . 45 170. 67 - 17 . 72 31 . 68 o. 929 -40. 52 128 . 65 -8 . 60 27 . 29 0. 902 -38 . 63 98 . 62 -8 . 87 22 . 64 0. 932 -46 . 43 8 1 . 01 -11. 33 19. 77 0. 960 Table A.4.5: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Gautier et a l . (1980) Model to Three Overcast Days. Units: KJ rt r 2 h r " 1 . Array Size: 5 x 5 DAY 239/8O(P/CL0UDY) STATION CAL . OBS . N MBE RMSE MB E % RMSE0/ R GRSMT 5901 5522 1 1 34 . 52 240 . 59 6 . 88 47 . 93 0 .906 NRTHMT GG 18 668 1 1 1 -5 . 74 2 1 1 . 76 -0 . 95 34 . 86 0 .865 BCHYDRO 7325 8028 1 1 -63 . 88 163 . 34 -8 . 75 22 . 38 0 .95 1 LANGA 7579 7937 1 1 -32 . 56 257 . 73 -4 .51 35 .72 0 .850 VANAIR 8892 9003 1 1 - 10 . 12 144 . 19 - 1 . 24 17 .62 0 .963 FERRY 8235 9606 1 1 - 124 . 65 280 . 13 - 14 . 27 32 .08 0 .909 PITMED 6280 8097 1 1 - 165 . 14 255 . 48 -22 . 44 34 . 7 1 0 .889 MISSHAB 847 1 9945 1 1 - 134 .02 306 . 60 - 14 .82 33 .91 0 .902 ABBLIB 8842 9453 1 1 -55 . 54 222 . 93 -6 . 46 25 .94 0 .942 ABBAIR 8917 10094 1 1 - 106 . 99 273 .01 - 1 1 .66 29 . 75 0 .910 LANGLEY 659G 8626 1 1 - 184 54 322 . 46 -23 . 53 4 1 . 12 0 . 766 CLISTN 7940 10001 1  - 187 . 39 333 . 34 -20 .61 36 .66 0 . 908 DAY 1GO/80(P/CLOUDY) STATION CAL . OBS . N M B E RMSE MBE'/ RMSE0/ R GRSMT 7934 6510 14 101 77 2 14 47 2 1 89 46 13 0 905 NRTHMT 101 10 9954 14 1 1 14 296 36 1 57 4 1 68 0 963 BCHYDRO 10139 8379 14 125 67 285 22 2 1 00 47 65 0 933 LANGA 10424 9214 14 8G 44 249 77 13 13 37 95 0 957 VANAIR 1 1236 10708 14 37 73 263 85 4 93 34 50 0 972 FERRY 1 1562 1 1398 14 1 1 73 4 15 78 1 44 51 07 0 900 PITMED 8433 8728 14 -2 1 10 2 13 49 -3 39 34 24 0 978 MISSHAB 91 10 9446 14 -24 01 167 15 -3 56 24 77 0 969 ABBLIB 9087 9G 14 14 -37 67 24 1 16 -5 49 35 12 0 917 ABBAIR 8585 9282 14 -49 77 32 1 87 -7 51 48 55 0 857 LANGLEY 8389 7G87 14 50 16 220 26 9 14 40 12 0 925 CLISTN 1 1292 10460 14 59 43 260 83 7 95 34 91 0 962 DAY 261/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MB E % RMSE% R GRSMT 6652 14 900 8 - 103 1 .04 1 196.59 -55.36 64 . 25 0. 880 NRTHMT 4899 G425 8 -190. 73 27 1 . 53 -23 . 75 33. 81 0. 939 BCHYDRO 4784 6994 8 -27G . 27 383 . 24 -3 1 . 60 43 . 84 0. 932 LANGA 33 14 3465 7 -2 1. 53 108 . 4 1 -4 . 35 2 1 . 90 0. 919 VANAIR 284 1 2820 7 2 . 93 62 . 91 0. 73 15 . 61 0. 985 FERRY 4 304 4268 9 4 . 01 93 . 30 0. 84 19 . 67 0. 895 PITMED 59 19 6306 7 -55 . 33 158 . 25 -6 . 14 17 . 57 0. 947 MISSHAB 7862 9010 7 - 163 . 9G 374 . 43 - 12 . 74 29 . 09 0. 932 ABBLIB 5 149 5093 7 7 . 99 186 . 12 1 . 10 25. 58 0. 75 1 ABBAIR 5277 5791 7 -73 . 39 178 . 43 -8 . 87 2 1 . 57 0. 872 LANGLEY 4588 5 152 7 -80. 50 191 . 75 - 10. 94 26 . 05 0. 989 CLISTN 4992 4905 8 10. 84 70. 02 1 . 77 1 1 . 42 0. 967 Table A.4.6: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Gautier et a l . (1980) Model to Three Partly Cloudy Days. Units: KJ m-2 hr-1. Array Size: 5 x 5 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 590G 5522 1 1 34 92 240 43 6 96 47 .90 0 906 NRTHMT 6624 6681 1 1 -5 25 2 1 1 88 -0 86 34 88 0 865 BCHYDRO 733 1 8028 1 1 -63 3 1 163 10 -8 68 22 35 0 951 LANGA 7585 7937 11 -32 02 257 90 -4 44 35 74 0 850 VANAIR 8900 9003 1 1 -9 4 1 144 16 - 1 15 17 61 0 962 FERRY 824 1 9606 11 - 124 07 279 90 - 14 2 1 32 05 0 909 PITMED 6285 8097 11 - 164 69 255 13 -22 37 34 66 0 889 MISSHAB 8477 9945 11 - 133 46 306 44 - 14 76 33 89 0 902 ABBLIB 8849 9453 1 1 -54 87 222 88 -6 39 25 94 0 942 ABBAIR 8954 10094 11 -103 67 273 49 - 1 1 30 29 80 0 909 LANGLEY 6602 , 8626 11 - 184 02 322 19 -23 47 4 1 08 0 766 CLISTN 7946 10001 11 -186 84 332 97 -20 55 36 62 0 908 Table A.4.7: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of the Gautier et a l . (1980) Model to a Partly Cloudy Day Using a Modified Threshold. Units: KJ rrr 2 h r " 1 . Array Size: 5 x 5 ro Co CTl 287 APPENDIX A.5 DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 3 1G93 32 179 16 -30 . 35 204 . 75 - 1 . 5 1 10 . 18 6 .997 NRTHMT 31297 3001 3 16 80 . 25 177 . 37 4 . 28 9 . 46 0 . 996 BCHYDRO 31253 30453 16 49 .95 189 . 17 2 .62 9 . 94 0 . 996 LANGA 30834 30437 16 24 . 78 180 . 37 1 . 30 9 . 48 O .995 VANAIR 30660 30527 16 8 . 3 1 172 . 3 1 0 . 44 9 .03 0 .997 FERRY 31928 30428 16 93 . 75 14 1 . 72 4 .93 7 . 45 0 .998 PITMED 30698 29086 16 100 . 79 16 1 .51 5 . 54 8 . 88 O .997 MISSHAB 30616 29551 16 66 . 58 159 . 27 3 .60 8 . 62 0 . 997 ABBLIB 304 27 292 17 16 75 .60 158 . 33 4 . 14 8 .67 0 .996 ABBAIR 30403 29302 16 68 . 78 160 . 13 3 . 76 8 . 74 0 .995 LANGLEY 30660 29403 16 78 . 55 155 . 57 4 . 27 8 . 47 0 . 998 CLISTN 31561 303 10 16 78 . 18 181 .02 4 . 13 9 .56 0 . 996 DAY 239/80(P/CL0UDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 1 1597 5522 1 1 552 34 629 60 1 10 .04 125 43 0 749 NRTHMT 12385 6681 1 1 5 18 56 546 00 85 38 89 89 0 914 BCHYDRO 13040 8028 1 1 455 65 478 49 62 44 65 57 0 953 LANGA 13273 7937 1 1 485 10 523 06 67 23 72 49 0 910 VANAIR ' 14277 9003 1 1 479 44 5 14 92 58 58 62 9 1 0 934 FERRY 14075 9606 1 1 406 31 499 36 46 53 57 18 0 869 PITMED 12239 8097 1 1 376 59 425 46 5 1 16 57 80 0 863 MISSHAB 13508 9945 1 1 323 91 380 67 35 83 42 10 0 940 ABBLIB 13821 9453 1 1 397 15 455 73 46 22 53 03 0 931 ABBAIR 13340 10094 1 1 295 1 1 4 14 73 32 16 45 20 0 884 LANGLEY 12810 8626 1 1 380 30 455 28 48 49 58 06 0 794 CLISTN 13932 10001 1  357 36 436 45 39 3 1 48 00 0 928 DAY 197/8O(0VERCAST) STATION CAL . OBS . N MBE ' RMSE MBE% RMSE'/. R GRSMT 1457 1 5764 15 587 . 15 633 69 152 . 80 164 . 91 0. 809 NRTHMT 15409 5399 16 625 . 63 683 . 46 185 . 39 202 . 53 0. 855 BCHYDRO 15620 5504 16 632 25 684 . 26 183 . 78 198 . 89 0. 96 1 LANGA 15846 5882 16 622 . 79 67 1 . 38 169 . 4 1 182 . 63 • 0. 916 VANAIR 15780 6866 16 557 . 15 599 . 98 129 . 84 139. 82 0. 899 FERRY 15157 6 197 16 559 . 97 626 . 2 1 144 . 57 16 1. 67 0. 7 18 PITMED 157 1 1 G281 16 589 . 43 648 . 92 150. 16 165. 31 0. 828 MISSHAB. 15829 67 1 1 16 569 . 88 6 11. 06 135. 86 145 . 68 0. 940 ABBLIB 16243 862 1 16 476 . 39 514 . 68 88 . 42 95 . 52 0. 932 ABBAIR 16067 7542 16 532 . 86 575 . 68 113. 05 122 . 1 3 0. 917 LANGLEY 15472 6968 16 53 1 . 48 584 . 75 122 . 04 134 . 27 0. 91 1 CLISTN 15152 6556 16 537 . 22 582 . 55 131 . 1 1 142 . 17 0. 929 Table A.5.1: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of the Original Hay and Hanson (1978) Model to Three Days in Summer Representing (3) Sky Cover Conditions. Units: KJ m-2 hr-'. Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE GRSMT 30600 32 179 16 -98 64 27 1 38 NRTHMT 29999 30013 16 -o 87 212 75 BCHYDRO 29932 30453 16 -32 6 1 24 1 25 LANGA 29296 30437 16 -7 1 34 252 79 VANAIR 29031 30527 16 -93 46 255 20 FERRY 30957 30428 16 33 06 133 81 PITMED 29090 29086 16 0 26 175 28 MISSHAB 28965 2955 1 16 -36 60 205 15 ABBLIB 28678 292 17 16 -33 70 191 33 ABBAIR 28642 29302 16 -4 1 31 193 65 LANGLEY 29031 29403 16 -23 22 191 74 CLISTN 30400 303 10 16 5 6 1 209 60 DAY 239/8O(P/CL0UDY) STATION CAL . OBS . N MBE RMSE GRSMT 4913 5522 1 1 -55 34 242 42 NRTHMT 6109 6681 11 -51 99 307 46 BCHYDRO 7 103 8028 1 1 -84 07 280 02 LANGA 7457 7937 1 1 -43 64 326 32 VANAIR 8982 9003 11 - 1 97 192 92 FERRY 8675 9606 1 1 -84 62 348 25 PITMED 5887 8097 11 -200 86 3 17 90 MISSHAB 7994 9945 11 - 177 37 318 36 ABBLIB 8351 9453 11 -100 13 2 14 83 ABBAIR 8494 10094 11 - 145 43 344 58 LANGLEY 6753 8626 11 -170 27 339 7 1 CLISTN 8457 10001 11 - 140 32 342 08 DAY 197/80(0VERCAST ) STATION CAL . OBS. N MBE RMSE GRSMT 5231 5764 15 -35 53 • 229 93 NRTHMT 64 13 5399 16 63 38 173 12 BCHYDRO 6734 5504 16 76 83 202 70 LANGA 7077 5882 16 74 69 180 55 VANAIR 6976 6866 16 6 91 2 14 92 FERRY 6030 6197 16 - 10 47 154 74 PITMED 6872 6281 16 36 96 244 46 MISSHAB 7051 67 1 1 16 2 1 23 208 68 ABBLIB 7679 862 1 16 -58 86 232 82 ABBAIR 74 12 7542 16 -8 07 225 96 LANGLEY 6508 6968 16 -28 75 168 73 CLISTN 602 2 6556 16 -33 38 199 95 MBE% RMSE% R -4 . 90 13 . 49 0 .996 -0 .05 1 1 . 34 0 . 994 - 1 . 7 1 12 .68 0 . 994 -3 . 75 13 . 29 0 .993 -4 . 90 13 . 38 0 . 996 1 . 74 7 .04 0 .998 0 .01 9 . 64 0 .997 - 1 . 98 1 1' . 1 1 O .997 - 1 .85 10 . 48 0 997 -2 . 26 10 . 57 0 996 - 1 . 26 10 . 43 0 997 0 . 30 1 1 .06 0 995 MBE% RMSE% R - 1 1 02 48 29 0 867 -8 56 50 62 0 748 - 1 1 52 38 37 0 834 -6 05 45 23 0 752 -0 24 23 57 0 935 -9 69 39 88 0 798 -27 29 43 19 0 782 -19 62 35 2 1 0 909 - 1 1 65 25 00 0 960 -15 '85 37 55 O 869 -21 71 43 32 0 712 -15 43 37 62 0 874 MBE% RMSE% R -9 25 59 84 0. 295 18 78 51 30 0. 607 22 33 58 92 0. 534 20 32 49 1 1 0. 681 1 . 61 50. 09 0. 574 -2 . 70 39 . 95 0. 783 9 . 42 62 . 28 0. 317 5. 06 49. 75 0. 668 - 10. 92 43 . 21 o. 760 - 1 . 7 1 47 . 94 0. 553 -6 . 60 38 . 74 0. 713 -8 . 15 48 . 80 0. 486 ro 00 CO Table A.5.2: Hourly V e r i f i c a t i o n S t a t i s t i c s for the Individual Stations from the Application of a Revised Hay and Hanson (1978) Model to Three Days in Summer Representing (3) Sky Cover Conditions. Units: KJ m~2 hr-1. Array Size: 5 x 5 DAY 183/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 30487 32 179 16 - 105 . 7 1 282 . 83 -5 . 26 14 .06 0 . 995 NRTHMT 30033 30013 16 1 . 22 2 12 . 42 0 .06 1 1 . 32 0 . 994 BCHYDRO 297S9 30453 16 -42 .77 249 . 18 -2 . 25 13 .09 0 .994 LANGA 29192 30437 16 -77 . 8 1 252 . 32 -4 .09 13 . 26 0 . 994 VANAIR 28965 30527 16 -97 . 6 1 260 .43 -5 . 12 13 .65 0 . 995 FERRY 30957 30428 16 33 .05 132 .01 1 . 74 6 .94 0 .998 PITMED- 291 19 29086 16 2 . 10 167 . 57 0 . 12 9 . 22 0 . 997 MISSHAB 28826 29551 16 -45 . 32 2 13 . 37 -2 . 45 1 1 . 55 0 . 997 ABBLIB 28678 292 17 16 -33 . 70 194 .66 - 1 . 85 10 .66 0 .997 ABBAIR 28642 29302 16 -4 1 . 30 195 .47 -2 . 25 10 . 67 0 . 996 LANGLEY 29039 29403 16 -22 . 76 191 . 12 - 1 . 24 ' 10 . 40 0 .997 CLISTN 30445 303 10 . 16 8 .43 206 .44 0 . 44 10 . 90 0 . 995 DAY 197/8010VERCAST) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 587 1 5764 15 7 . 12 252 . 58 1 85 65 . 73 0. 296 NRTHMT 6678 5399 16 79. 90 183 . 05 23 . 68 54 . 24 0. 599 BCHYDRO 6513 5504 16 63 . 0 1 177 . 37 18 . 32 5 1 . 56 0. 59 1 LANGA 7020 5882 16 7 1 . 13 178 . 25 19 . 35 48 . 49 0. 666 VANAIR 6872 6866 16 0. 4 1 212 . 62 0. 10 49 . 55 O. 572 FERRY 6387 6 197 16 1 1 . 83 165 . 09 3 . 05 42 . 62 0. 761 PITMED 6801 6281 16 32 . 53 244 . 73 8 . 29 62 . 35 O. 314 MISSHAB 6652 67 1 1 16 -3 . 72 22 1 . 92 -0. 89 52 . 91 0. 606 ABBLIB 7497 8621 16 -70. 22 239. 47 - 13 . 03 44 . 44 0. 756 ABBAIR 8017 7542 16 29 . 69 2 12. 79 6. 30 45 . 14 O. 629 LANGLEY 6344 6968 16 -39 . 03 175 . 85 -8 . 96 40. 38 0. 700 CLISTN 595 1 6556 16 -37 . 79 191 . 60 -9 . 22 46 . 76 O. 519 DAY 239/80(P/CLOUDY) STATION CAL . OBS . N MBE RMSE MBE% RMSE% R GRSMT 5 113 5522 1 1 -37 . 19 242 . 65 -7 . 4 1 48 . 34 0 . 856 NRTHMT 6141 6681 1 1 -49 . 10 297 .90 -8 .08 49 .05 0 . 760 BCHYDRO 7349 8028 1 1 -6 1 . 69 27 1 . 6 1 -8 . 45 37 . 22 0 . 843 LANGA 7378 7937 1 1 -50 . 78 337 . 4 1 -7 .04 46 . 76 0. . 733 VANAIR 9176 9003 1 1 15 . 65 192 . 57 1 .91 23 . 53 0. .934 FERRY 8476 9606 1 1 - 102 . 73 382 . 4 1 -11 . 76 43 79 0. 758 PITMED 5727 8097 1 1 -2 15. .43 330 . 58 -29 . 27 44 .91 0. 772 MISSHAB 7964 9945 1 1 - 180. 16 330 09 - 19 93 36 . 51 0. 896 ABBLIB 8592 9453 1 1 -78 . 25 200 77 -9. 1 1 23 . 36 0. 96 1 ABBAIR 8491 10094 1 1 - 145 . 70 345 . 80 - 15 . 88 37 . 68 0. 867 LANGLEY 6702 8626 1 1 - 174 . 97 343 . 54 -22 . 31 43 . 8 1 0. 706 CLISTN 8474 10001 1 1 - 138 . 85 337 . 81 - 15 . 27 37 . 16 0. 878 Table A.5.3: Hourly V e r i f i c a t i o n S t a t i s t i c s f o r the Individual Stations from the Application of a Revised Hay and Hanson (1978) Model Based on a Spatial Averaging Approach. Units: KJ m-2 hr-1. Array Size: 3 x 3 DAY 121/80(CLEAR) STATION CAL . OBS . N MBE RMSE MBE % RMSE% R GRSMT 24253 26122 12 - 155 . 76 3 10 . 48 -7 . 16 14 . 26 0 . 994 NRTHMT 24G72 262 19 13 - 1 19 .02 262 . 53 -5 .90 13 .02 0 . 996 BCHYDRO 244GO 26152 13 - 130 . 2 1 267 .67 -6 .47 13 .31 0 ,995 LANGA 23906 25974 13 - 159 .08 274 .63 -7 .96 13 . 75 0 . 996 VANAIR 24038 25763 13 - 132 . 68 274 . 10 -6 . 70 13 .83 0 . 996 FERRY 26333 26334 13 -0 . 1 1 201 .65 -0 .01 9 . 95 0 .996 PITMED 24099 254 17 13 - 101 .42 235 . 36 -5 . 19 12 .04 0 . 997 MISSHAB 23117 253 19 13 - 169 . 38 284 .09 -8 . 70 14 . 59 0 . 986 ABBLIB 23533 25355 13 - 140 . 14 277 . 10 -7 . 19 14 . 2 1 0 . 991 ABBAIR 23659 25409 13 - 134 . 6 1 256 . 36 -6 .89 13 . 12 0 .994 LANGLEY 23945 25344 13 - 107 .63 220 .98 -5 . 52 1 1 . 33 0 . 995 CLISTN 2499 1 2596 1 13 -74 .60 187 .54 -3 . 74 9 . 39 o . 999 DAY 276/79(CLEAR) STATION CAL . OBS. N MBE RMSE MBE% RMSE% R GRSMT 14824 16278 10 - 145 . 43 246 . 14 -8 . 93 15 . 12 o. 996 NRTHMT 14502 1498 1 10 -47 . 86 203 . 57 -3 . 19 13 . 59 0. 995 BCHYDRO 14488 14833 10 -34 . 55 212 . 01 -2 . 33 14 . 29 0. 985 LANGA 14025 15143 10 -111. 8 1 231 . 02 -7 . 38 15 . 26 0. 991 VANAIR 1394 1 14936 10 -99. 45 245 . 47 -6 . 66 16 . 44 0. 994 FERRY 1509 1 15116 10 -2 . 52 149 . 32 -0. 17 9. 88 0. 998 PITMED 14205 14593 10 -38 . 76 22 1 . 54 -2 . 66 15 . 18 ' 0. 997 MISSHAB 14016 15007 10 -99 . 05 219. 2 1 -6 . 60 14 . 61 0. 994 ABBLIB 14030 14914 10 -88 . 38 2 14. 98 -5 . 93 14 . 42 o. 995 ABBAIR 139 18 15131 10 -12 1. 25 239 . 20 -8 . 01 15 . 81 0. 993 LANGLEY 14 116 14600 10 -48 . 38 205 . 89 -3 . 3 1 14 . 10 0. 997 CLISTN 14937 15151 10 -2 1. 38 178 . 1 1 - 1 . 4 1 1 1 . 76 0. 985 DAY 025/80CCLEAR) STATION CAL . OBS. N MBE RMSE MBE% RMSE% R GRSMT 6224 7378 7 - 164 .94 ' 200 .68 - 15 .65 19 .04 0 .993 NRTHMT 6321 6881 7 -80 . 1 1 159 .09 -8 . 15 16 . 18 0 .987 BCHYDRO 6 181 6494 7 -44 . 75 157 . 84 -4 .82 17 .01 0 . 968 LANGA 6038 6942 7 -129 .05 198 .64 - 13 .01 20 .03 0 .983 VANAIR 5795 6770 7 -139 .23 193 . 10 - 14 . 40 19 . 97 0. 899 FERRY 7 103 6837 7 37 .97 49 . 7 1 3 .89 5 . 09 0. 995 PITMED 6 148 7454 7 - 186 . 49 243 . 63 - 17 .51 22 . 88 0. 940 MISSHAB 6075 7533 8 -182 . 32 232 . 68 - 19. 36 24 . 7 1 0. 989 ABBLIB 6040 7354 7 - 187 . 62 239. 17 - 17 . 86 22 . 77 0. 994 ABBAIR 5946 7524 7 -225 . 40 267 . 44 -20. 97 24 . 88 0. 996 LANGLEY 6 172 7 198 8 - 128 . 28 182 . 03 - 14 . 26 20. 23 0. 988 CLISTN 6600 7304 7 - 100. 55 145 . 87 -9. 64. 13. 98 0. 986 Table A . 5 . 4 : Hour ly V e r i f i c a t i o n S t a t i s t i c s f o r the Ind i v i dua l S t a t i o n s from the A p p l i c a t i o n o f a Rev ised Hay and Hanson (1978) Model to Three C lea r Days at D i f f e r e n t Times o f the Year . U n i t s : KJ m-2 hr- I . A r ray S i z e : 5 x 5 DAY 239/80(P/CL0UDY) STATION CAL . OBS . N GRSMT 4913 5522 1 1 NRTHMT 6 109 668 1 1 1 BCHYDRO 7 103 8028 1 1 LANGA 7457 7937 1 1 VANAIR 8982 9003 1 1 FERRY 8675 9606 1 1 PITMED 5887 8097 1 1 MISSHAB 7994 9945 i 1 ABBLIB 835 1 94 53 ABBAIR 8494 10094 1 \ LANGLEY 6753 8626 1 1 CLISTN 8457 10001 1 1 DAY 160/80(P/CL0UDY) STATION CAL . OBS . N GRSMT 6275 6510 14 NRTHMT 9125 9954 14 BCHYDRO 8907 8379 14 LANGA 904 3 92 14 14 VANAIR 9807 10708 14 FERRY 10975 1 1398 14 PITMED 6772 8728 14 MISSHAB 7282 9446 14 ABBLIB . 7698 9614 14 ABBAIR 7047 9282 14 LANGLEY 6667 7687 14 CLISTN 10293 10460 14 MBE RMSE MBE% RMSE% R -55 34 242 42 - 1 1 02 48 29 0 867 -5 1 99 307 46 -8 56 50 62 0 748 -84 07 280 02 - 1 1 52 38 37 0 834 -43 64 326 32 -6 05 45 23 0 752 - 1 97 192 92 -0 24 23 57 0 935 -84 62 348 25 -9 69 39 88 0 798 200 86 3 17 90 -27 29 43 19 0 782 177 37 3 18 36 - 19 62 35 2 1 0 909 100 13 2 14 83 - 1 1 65 25 OO 0 960 145 43 344 58 - 15 85 37 55 0 869 170 27 339 7 1 -2 1 7 1 43 32 0 7 12 140 32 342 08 - 15 43 37 62 0 874 MBE RMSE MBE% RMSE% R - 16 78 208 34 -3 61 44 81 0 887 -59 20 266 89 -8 33 37 54 0 97 1 37 72 262 30 6 30 43 82 0 935 - 12 22 239 44 - 1 86 36 38 0 954 -64 37 298 65 -8 42 39 05 0 959 -30 20 409 20 -3 7 1 50 26 0 895 139 69 2 18 96 -22 4 1 35 12 0 977 154 58 2 17 67 -22 91 32 26 0 973 136 92 254 1 1 - 19 94 37 00 0 936 159 58 349 17 -24 67 52 67 0 867 -72 85 249 64 - 13 27 45 47 0 910 - 1 1 92 275 69 - 1 60 36 90 0 95 1 Table A.5.5: Hourly Verification Statistics for the Individual Stations from Hay and Hanson (1978) Model to Two Partly Cloudy Days in Spring Array Size: 5 x 5 the Application of a Revised and Summer. Units: KJ nr 2 hr"l. DAY 263/79(0VERCAST ) STATION CAL . OBS . N GRSMT 4209 3888 9 NRTHMT 3733 2897 9 BCHYDRO 3776 2864 9 LANGA 4 146 4050 9 VANAIR 4662 4 189 9 FERRY 4431 4090 9 PITMED 2317 2323 6 MISSHAB 3320 3037 9 ABBLIB 3504 2890 9 ABBAIR 3379 3172 9 LANGLEY 3266 2683 9 CLISTN 4515 4345 9 DAY 105/80(0VERCAST) STATION CAL . OBS . N GRSMT 6001 2381 12 NRTHMT 5601 4223 10 BCHYDRO 6721 4885 12 LANGA 72 18 4955 12 VANAIR 7872 5907 12 FERRY 7549 7064 12 PITMED 5606 4623 12 MISSHAB 5678 402 1 12 ABBLIB 58 16 4015 12 ABBAIR 6 187 4280 12 LANGLEY 6 146 47 18 12 CLISTN 7080 5153 12 DAY 197/80(OVERCAST) STATION CAL . OBS . N GRSMT 5231 5764 15 NRTHMT 64 13 5399 16 BCHYDRO 6734 5504 16 LANGA 7077 5882 16 VANAIR 6976 6866 16 FERRY 6030 6197 16 PITMED 6872 628 1 16 MISSHAB 7051 67 1 1 16 ABBLIB 7679 862 1 16 ABBAIR 74 12 7542 16 LANGLEY 6508 6968 16 CLISTN 6022 6556 16 MBE RMSE MBE% RMSE% R 35 . 73 257 . 87 8 . 27 59 . 70 0 . 302 92 . 87 305 . 48 28 . 85 94 .89 -0. . 302 101 . 36 2 19 . 44 31 .85 68 . 95 0. .051 10 .66 239 . 79 2 . 37 53 . 29 0. 2 15 52 . 52 224 .61 1 1 . 28 48 . 25 0. 694 37 . 94 169 . 74 8 . 35 37 . 35 0. 870 -o. . 89 119 . 88 -0 . 23 30 . 97 0. 208 31 . 55 170. 28 9 35 50. 47 0. 582 68 . 23 165 . 13 2 1 . 25 51 . 43 0. 479 22 . 95 131 . 26 6 . 51 37 . 24 0. 723 64 . 76 146 . 19 2 1 . 72 49 . 03 0. 752 18 . 85 269 . 04 3 . 90 55 . 73 0. 37 1 MBE RMSE MBE% RMSE% R 301 .65 365 . 89 152 .01 184 . 38 0 . 528 137 . 8 1 243 .31 32 .64 57 .62 0 .923 152 . 92 201 . 55 37 . 56 49 . 5 1 0 .937 188 . 58 240 . 43 45 .67 58 . 23 0 . 923 163 . 7 1 245. .69 33 . 26 49 .91 0 .946 40 . 40 195 .45 6 . 86 33 . 20 0. .963 8 1 . 92 127 . 35 2 1 . 27 33 .06 0. .921 138 .06 227 . 80 4 1 . 20 67 . 98 0. 687 150 05 2 18. 38 44 . 84 65 . 26 0. 812 158 . 92 273 . 4 1 44 ! 56 76 . 65 0. 702 1 19 . 01 200. 56 30. 27 51 . 01 0. 853 160. 57 233 . 1 1 37 . 39 54 . 28 0. 936 MBE RMSE MBE% RMSE% R -35 . 53' 229 . 93 -9 . 25 59 . 84 0 . 295 63 . 38 173 . 12 18 . 78 51 . 30 0 607 76 . 83 202 . 70 22 . 33 58 .92 0 . 534 74 . 69 180 . 55 20 . 32 49 . 1 1 0 681 6 . 9 1 2 14 92 1 .61 50 .09 0. 574 - 10 .47 154 , . 74 -2 . 70 39 .95 0. 783 36 . 96 244 . 46 9 . 42 62 . 28 0. 317 2 1 . 23 208 . 68 5 . 06 49 . 75 0. 668 -58 . 86 232 . 82 - 10. 92 43 . 2 1 0. 760 -8 . 07 225. 96 - 1 . 7 1 47 . 94 0. 553 -28: ' 75 168 . 73 -6 . 60 38 74 0. 7 13 -33 . 38 199 . 95 -8 . 15 48 . 80 0. 486 Table A.5.6: Hourly Verification Statist ics for the Individual Stations from the Application of a Revised Hay and Hanson (1978) Model to Three Overcast Days. Units: KJ m-2 hr-1. Array Size: 5 x 5 294 APPENDIX A.6 / 295 C* PROGRAM TO IMPLEMENT TARPLEY'S INSOLATION MODEL, USING * C* WINDOWS ON UNIT 8 PRODUCED BY 'NAVSYS'. * C*************************^ M.D.R. 8/81 * C REAL*8 STAT,DAT 1 COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) C DO 4 I = 1, 1 2 C DO 3 J=1,24 C DO 2 K=1,2 C DATA(I,J,K)=-9.00 C 2 CONTINUE C 3 CONTINUE C 4 CONTINUE WRITE(6,100) READ(5,101) IMODE IF(IM0DE.EQ.2) GOTO 10 5 CALL AVGS(NAVTIM,STAT,XLAPT,XMEAN,XMED,XMOD,X1,X2,X3,&7,&99 WRITE(7,102) NAVTIM,STAT,XMEAN,XMED,XMOD,X1,X2,X3 GOTO 5 7 WRITE(7,102) NAVTIM,STAT WRITE(6,104) NAVTIM,STAT GOTO 5 C C THIS BLOCK PROCESSES STAGE 2 OF MODEL: C 10 WRITE(6,105) READ(5,106) NSTAT DO 15 I=1,12 15 NIMG(I)=0 20 CALL AVGS(NAVTIM,STAT,XLAPT,XMEAN,XMED,XMOD,X1,X2,X3,&40,&44 CALL B(X1,X2,X3,ISTNUM(STAT),BMEAN) CALL CLD(BMEAN,XN,XICLD) CALL STORE(NSTAT,NAVTIM,STAT,XLAPT,BMEAN,XN,XICLD,XMEAN) GOTO 2 0 4 0 WRITE(6,104) NAVTIM,STAT GOTO 20 C C RELATE TO KNOWN RADIATION, ETC C 44 CALL SPAN(NSTAT) CALL CALC(NSTAT,IDY) C CALL PDATA(IDY) 99 STOP C c 100 FORMAT(' ENTER PROCESSING STAGE') 101 FORMAT(11) 102 FORMAT(I10,2X,A8,3F6.2,3F9.5) 103 FORMAT(I 10,2X,A8,2F7.2,F7.3,2F7.2) 104 FORMAT(' ** WARNING" MISSING DATA FOR ',I10,2X,A8) 105 FORMAT(' HOW MANY STATIONS?') 106 FORMAT(12) END C C 296 SUBROUTINE STORE(NSTAT,NAVTIM,STAT,XLAPT,BMEAN,XN,XICLD,XME Q*********************************************************** C* STORE INTERMEDIARY SATELLITE RELATED VALUES IN ARRAY 'DAT 1 ' . c**************************************************************** C REAL*8 STAT,DAT 1 INTEGER ISTN/0/ COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) C ISTN=ISTNUM(STAT) NIMG(ISTN)=NIMG(ISTN)+1 IMG=NIMG(ISTN) DAT1(1,IMG,ISTN)=NAVTIM DAT 1 (2,IMG,ISTN)=STAT DAT 1(3,1MG,ISTN)=XLAPT DAT 1 (4,1MG,ISTN)= BMEAN DAT 1 (5,IMG,ISTN)=XN DAT 1 (6,IMG,ISTN)=XICLD DAT 1(7,IMG,ISTN)=XMEAN RETURN END C C SUBROUTINE SPAN(NSTAT) r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C* COMPUTE THE TIME SPAN FOR WHICH THE IMAGES IN 'DAT 1 ' ARE * C* VALID. * c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C REAL*8 DAT1 COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) DO 50 IS=1,NSTAT DATI(8,1,IS)=DAT1(3,1,IS)-0.25 DAT 1 (9,NIMG(IS),IS)=DAT1 (3,NIMG(IS),IS)+0.25 NI=NIMG(IS) DO 30 IM=2,NI DAT 1 (8,IM,IS) = (DAT1 (3,1M-1 ,IS)+DAT1 (3,1M,IS))/2.0 DAT 1 (9,IM-1 ,IS)=DAT1 (8,IM,IS) 30 CONTINUE DO 40 IM=1,NI IF((DAT 1(9,1M,IS)-DAT1(8,IM,IS)).LE.1.0) GOTO 35 DAT1(10,IM,IS)=1.0 GOTO 40 35 DAT1(10,IM,IS)=0.0 4 0 CONTINUE 50 CONTINUE C DO 60 IS=1,NSTAT C NI=NIMG(IS) C WRITE(2,100) ((DAT1(I,J,IS),1=1,10),J=1,NI) C60 CONTINUE RETURN 100 FORMAT(F11.0,2X,A8,2F7.2,F7.3,2F7.2,3F7.2) END 297 C C SUBROUTINE CALC(NSTAT, I DY) Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C* RELATE THE SATELLITE DATA TO KNOWN RADIATION DATA. * C REAL*8 DAT1 REAL K(12),ZS(12)/1128.0,11*0.0/ REAL BOS(12)/65.20,68.78,69.95,74.89,76.16,62.86,75.08,76.0 *76.23,77.55,74.52,65.71/ REAL BB(12) COMMON/RDATA/DAT1 (10,40,12),NIMG(12),DATA(12,24 , 2) ID=DAT1(1,1,1)/l0000 WRITE(6,103) ID READ(5,104) W C C READ IN THE RADIATION VALUES C 5 READ(4,100,END= 9 9) IDY,HR,(K(I),I = 1 ,12) IHR = HR + 0.5 XKJ=K(11) K(11)=K(10) K(10)=K(9) K(9)=K(8) K(8)=K(7) K(7)=K(6) K(6)=K(5) K(5)=K(4) K(4)=XKJ NAVTIM=IDY*10000 HRII = HR-0.5 CALL SGEOM(NAVTIM,HRII,XLPT,X1,X2,X3) ZA=ARCOS(X1)*180.0/3.14159 COSX=X1 IF(COSX.LT.-O.O) COSX=0.0 TST=HR-1.0 C C FOR EACH STATION, COMPUTE THE WEIGHTED AVERAGED VALUES C 'XNBAR', 'XICBAR', AND 'XIMBAR' IF POSSIBLE. C DO 50 IS=1,NSTAT C WRITE(6,222) IS C 222 FORMAT(' DOING STN # ',15) CALL B(X1,X2,X3,IS,BB(IS)) W2=W IF(IS.EQ.1) W2=W*0.8 B0=BOS(IS) XM=EXP(-ZS(IS)/8243.0)/(COSX+0.15/(93.885-ZA)**1.253) PSIR=0.972-0.0826*XM+0.00933*XM*XM PSIWA=1.0-0.077*(W2*XM)**0.3 PSIWS=1.0-0.00225*W2*XM PSI=PSIR*PSIWA*PSIWS C WRITE(2,101) DAT1(2,1,IS),IDY,HR,K(IS),PSI,BB(IS),B0 C C FIRST MAKE SURE WE HAVE IMAGES THAT EARLY 298 C AND IF SO, FIND FIRST IMAGE PARTIALLY WITHIN HOUR OF INTERES C IF(DAT 1(8,1,1S).GT.TST+0.001) GOTO 25 NI=NIMG(IS) DO 10 IM=1,NI IM1=IM IF(DAT1(9,IM,IS).GT.TST+0.001) GOTO 15 10 CONTINUE 15 IF((DAT1(10,IM1,IS).EQ.1.0).OR.(DAT1(9,IM1,IS).LE.TST)) GO C TO 25 C IF EVERYTHING OK, COMPUTE WEIGHTED AVERAGES C STOPPING IF ANYTHING GOES WRONG (IE: MISSING DATA) C TFRAC=0 XNBAR=0 XICBAR=0 XIMBAR=0 20 IF(IM1.GT.NIMG(IS)) GOTO 25 T1=DAT1(8,IM1,IS) T2=DAT1(9,IM1,IS) C C 'TIME' IS AMOUNT OF TIME FROM THIS IMAGE WHICH IS WITHIN C THE HOUR OF INTEREST C TIME=AMIN1(T2,HR)-AMAX1(T1,TST) FRAC=TIME/(T2-T1) TFRAC=TFRAC+FRAC XNBAR=XNBAR+FRAC*DAT1(5,IM1,IS) XICBAR=XICBAR+FRAC*DAT1(6,IM1,IS) XIMBAR=XIMBAR+FRAC*DAT1(7,IM1,IS) C WRITE(2,73) TFRAC,FRAC,XNBAR,XICBAR,XIMBAR C -7 3 FORMAT(2X,2F6.3,3F9.3) IM1=IM1+1 IF(T2.LT.HR) GOTO 2 0 XNBAR=XNBAR/TFRAC XICBAR=XICBAR/TFRAC XIMBAR=XIMBAR/TFRAC QS=PRDCT(COSX,PSI,XNBAR,BB(IS),XIMBAR,B0,XICBAR) IF(QS.LT.O.O) QS=0.0 C WRITE(2,102) QS,XNBAR,XICBAR,XIMBAR,COSX IHR=HR+0.5 DATA(IS,IHR,1)=K(I S) DATA(IS,IHR,2)=QS CALL STATS(K(IS),QS,IS,.FALSE.) GOTO 50 C C MISSING DATA MESSAGE C 25 WRITE(2,105) DATA(IS,IHR,1)=K(I S) DATA(IS,IHR,2)=_9000.0 C C LOOP TO NEXT STATION C 50 CONTINUE C 299 C LOOP TO NEXT HOUR C GOTO 5 C99 CONTINUE 99 CALL STATS(0.0,0.0,0,.TRUE.) RETURN 100 FORMAT(1X,I4,2X,F7.3,/,8(F9.3,5X),/,3(F9.3,5X),F9.3) 101 FORMAT(' ',2X,A8,I5,F6.2,4F9.3) 102 FORMATC+' , 58X, 4F9. 3 , F6 . 3 ) 103 FORMAT(' INPUT W FOR ',16) 104 FORMAT(F5.2) 105 FORMAT('+',58X,'*** MISSING DATA ***1) END C C C FUNCTI ON PRDCT (COSX, PS'I , XN , BB , XIM, B0 , XI CLD) Q************************************************************* C* COMPUTE THE PREDICTED INSOLATION * Q********* ************************ **************************** c IF(XN.GE.0.4) GOTO 5 PRDCT=- 1 94 . 3.7 + 3872 .48*COSX+1 07 . 65*PSI-224 . 6*XN-97 . 7 0* * (XIM/BB)**2 GOTO 15 5 IF(XN.GT.0.9999) GOTO 10 PRDCT=-199.28+4 04 7.97*COSX~329.34*XN*(XICLD/B0)**2 GOTO 15 10 PRDCT=-49.77+2187.16*COSX-168.8*(XICLD/B0)**2 15 RETURN END C C C SUBROUTINE AVGS(NAVTIM,STAT,XLAPT,XMEAN,XMED,XMOD,X1,X2,X3, Q***************************************************** ******** C* COMPUTE MEAN, MEDIAN, AND MODE OF WINDOWS ON UNIT 8, AS * C* WELL AS SOME GEOMETRICAL VALUES. * Q************************************************************* c c C 'VALS' HOLDS OBSERVED PIXEL VALUES C 'COUNT' IS USED FOR COMPUTATION OF MODE C 'STAT' HOLDS STATION NAME C 'CLAT' IS APPROXIMATE CENTER LATITUDE OF AREA C COMMON BLOCK 'WIND' IS FILLED WITH WINDOW INFO. C INTEGER VALS(2500),COUNT(64),NTIM/0/ REAL*8 STAT COMMON/WIND/VALS,NR,NC C C READ IN WINDOW INFORMATION, ZERO COUNTERS, AND COMPUTE C THE GEOMETRICAL PARAMETERS (IF NOT ALREADY DONE FOR THIS C TIME) --C X1=COS(SOLAR ZENITH ANGLE) = COS(Z) C X2=SXN(Z)*COS(SUN-SATELLITE AZIMUTH) = SIN(Z)*COS(SSA) 300 C X3=SIK(Z)*COS(SSA)**2 C READ(8,100,END=44) NR,NC,NAVTIM,STAT SUM=0 DO 10 1=1,40 COUNT(I)=0 10 CONTINUE IF(NAVTIM.EQ.NTIM) GOTO 15 NTIM=NAVTIM CALL SGEOM(NAVTIM,0.0,XLPT,Y1,Y2,Y3) XLAPT=XLPT X1=Y1 X2=Y2 X3 = Y3 C C NOW COMPUTE THE MEAN, MEDIAN, AND MODE OF THE PIXEL VALUES C C FIRST READ IN THE WINDOW, THEN CHECK THE VALUES C AO PIXEL VALUE INDICATES MISSING DATA C 15 DO 20 1=1,NR JS=(I-1)*NC+1 JE=JS+NC-1 READ(8,101) (VALS(J),J=JS,JE) 2 0 CONTINUE NVALS=NR*NC DO 2 2 K=1,NVALS IF((VALS(K).LT.12).OR.(VALS(K).GT.256)) RETURN 1 22 CONTINUE C C NOW TALLY INTO 'COUNT' FOR COMPUTATION OF MODE C DO 2 5 1=1,NVALS IV=VALS(I) COUNT(IV/4)=COUNT(IV/4)+1 SUM=SUM+IV 2 5 CONTINUE C C SORT 'VALS' FOR EASY SPOTTING OF MEDIAN VALUE C 'SSORT' IS MTS FORTRAN LIBRARY ROUTINE C CALL SSORT(VALS,NVALS,1) C C COMPUTE MEAN, MEDIAN, AND FINALLY MODE C XMEAN=SUM/NVALS XMED=VALS(NVALS/2+1) MAX=COUNT(1) DO 30 1=2,40 IF(COUNT(I).LE.MAX) GOTO 30 MAX=COUNT(l ) IMAX=I 30 CONTINUE XMOD=IMAX*4 RETURN 44 RETURN 2 301 100 FORMAT(14,4X,13,1 10,14X,A8) 101 FORMAT(80I3) END C C C SUBROUTINE B(X1,X2,X3,NS,B1) C* RETURN THE PREDICTED TARGET CLEAR BRIGHTNESS. C REAL A(12)/37.366,39.093,41.230,40.321,40.159,42.322,40.129 *39.549,41.444,41.036,39.255,39.255/ REAL BAB(12)/40.292,43.348,42.435,50.447,52.738,29.589,50.8 *52.292,49.775,52.343,51.297,38. 183/ REAL C(12)/7.483,8.596,9.849,8.653,9.368,2.770,7.900,6.001, *5.324,5.518,8.135,7.792/ REAL D(12)/15.115,12.865,10.817,10.161,8.986,2.568,9.065, * 12.640,12.336,10.905,10. 1 57,7.157/ B1=A(NS)+BAB(NS)*X1+C(NS)*X2+D(NS)*X3 C WRITE(6,10) NS,BB C10 FORMAT(' PROCESSING FOR',14,' BB*,F7.2) RETURN END C C c SUBROUTINE CLD(BMEAN,XN,XICLD) C* COMPUTE THE CLOUD AMOUNT AND MEAN CLOUD BRIGHTNESS. * c INTEGER VALS(2 500),INC1/12/,INC2/20/ REAL CTHRES/0.4/ COMMON/WIND/VALS,NR,NC N2 = 0 N3 = 0 SCLD=0.0 T1=BMEAN+INC1 T2=BMEAN+INC2 NVALS=NR*NC C C DECIDE WHICH INTERVAL EACH PIXEL INTENSITY BELONGS IN C AND INCREMENT APPROPRIATE COUNTER. ALSO KEEP SUM OF C CLOUDY PIXEL VALUES (IE: THOSE >T1) FOR COMPUTATION C OF XICLD. C DO 10 1=1,NVALS IV=VALS(I) IF(IV.LE.T1) GOTO 10 IF(IV.GT.T2) GOTO 5 N2=N2+1 GOTO 10 5 N3=N3+1 SCLD=SCLD+IV 302 10 CONTINUE C SCLD=SCLD~N3*BMEAN C C COMPUTE XN AND, IF APPROPRIATE, XICLD C XN=(0.5*N2+N3)/FLOAT(NVALS) XICLD=0.0 IF((XN.LT.0.4).OR.(N3.EQ.0)) RETURN XICLD=SCLD/N3 RETURN END C C c FUNCTION FTM(NVTIM) Q*************************************************************** C* FLOATING POINT TIME IN HRS. * Q************************** ******** ***************************** IT=MOD(NVTIM,10000) FTM=IT/100+MOD(IT,l00)/60.0 RETURN END C C C SUBROUTINE SGEOM(NAVTIM,XPT,XLAPT,X1,X2,X3) REAL CLAT/49.217/,SATAZ/13.633/ C C FIRST COMPUTE DECLINATION, SOLAR ZENITH ANGLE, AND SOLAR C AZIMUTH FOR THIS TIME. C RADDEG= 3.14159/180.0 C1=279.457* RADDEG C2=0.985647*RADDEG YR=NAVTIM/l0000000 JD=MOD(NAVTIM,10000000)/10000 IF(XPT.EQ.O.O) GOTO 5 XLAPT=XPT GOTO 10 5 Tl=FTM(NAVTIM) X=AINT((YR-65)*365.251)+JD+TI/24.0 G=C1+C2*X X=X/365.2422 C C 'EQ' IS EQUATION OF TIME VALUE C EQ=(-102.5-0.142*X)*SIN(G)+(-429.8+.033*X)*COS(G)+596.5*S *-2.0*COS(2*G)+4.2*SIN(3*G) + 1 9..3*C0S(3*G)-12.8*SIN(4*G) EQ=EQ/3600 C C 'XLAPT' IS LOCAL APPARENT TIME IN HOURS (CONSTANTS VARY C WITH CENTRAL LOCATION) C XLAPT=TI-8.0+EQ-10.8/60.0 IF(XLAPT.LT.0.0) XLAPT=XLAPT+2 4.0 10 HA=15.0*(XLAPT-12.0)*RADDEG 303 PSI=2*3.14159*(JD-1)/365.0 C C 'DEC IS SOLAR DECLINATION C DEC=0.006918-0.399912*COS(PSI)+0.070257*SIN(PSI) * -0.006758*COS(2*PSI)+0.000907*SIN(2*PSI)-0.002697*COS(3*Psi * +0.001480*SIN(3*PSI) C C 'COSZ' AND 'SINZ' ARE SIN AND COS OF SOLAR ZENITH ANGLE C 'CA' IS COS OF SOLAR AZIMUTH C COS Z = SIN(CLAT* RADDEG)*SIN(DEC)+COS(CLAT*RADDEG)*COS(DEC)* SINZ=SIN(ARCOS(COSZ)) CA=(SIN(CLAT*RADDEG)*COS Z-SIN(DEC))/(COS(CLAT*RADDEG)* SIN z) C C -NOW COMPUTE THE 3 PARAMETERS OF INTEREST C SIGN=1 IF(XLAPT.LT.12.0) SIGN=-1 SSA=ABS(SATAZ*RADDEG-SIGN*ARCOS(CA)) X1=COSZ X2=SINZ*COS(SSA) X3=X2*COS(SSA) RETURN END C C c FUNCTION ISTNUM(STAT) REAL*8 STAT,NAMES(12)/'GRSMT ','NRTHMT ','BCHYDRO ', * 'LANGA ','VANAIR ','FERRY ','PITMED ',*MISSHAB ', * 'ABBLIB ','ABBAIR ',*LANGLEY ','CLISTN '/ LOGICAL EQCMP DO 10 1=1,12 J = I IF(EQCMP(8,STAT,NAMES(I))) GOTO 15 IF(I.EQ.12) WRITE(6,100) 10 CONTINUE 15 ISTNUM=J RETURN 100 FORMAT(' HELLLP!... THERE IS AN UNKNOWN STATION SOMEWHERE') END C C C SUBROUTINE STATS(O,C,IS,PFLAG) C******************************************************** C* KEEP TRACK OF SUMS AND SQUARES AND PRODUCTS OF DATA * C* VALUES 'O' AND 'C GIVEN FOR STATION NUMBER 'IS'. * C* THESE SUMS ARE PRINTED ALONG WITH MBE AND RMSE FOR * C* ALL STATIONS WHEN CALLED WITH 'PFLAG'=.TRUE. (UNIT 9)* c******************************************************** c REAL*8 SC(12)/12*0.0/, SO(12)/12*0.0/,SCO(12)/12*0.0/, 1 SC2(12)/l2*0.0/,S02(12)/12*0.0/,STNAME,STN LOGICAL PFLAG 304 C C C C C C INTEGER N(12)/I 2*0/ IS IT PRINTOUT TIME? IF(O.LT.O.O) RETURN IF(C.LT.O.O) C=0.0 IF(PFLAG) GOTO 20 SC(IS)=SC(IS)+C SO(lS)=SO(lS)+0 SC2(IS)=SC2(IS)+C*C S02(IS)=S02(IS)+0*0 SCO(IS)=SCO(IS)+0*C N(IS)=N(IS)+1 RETURN JUMP HERE TO PERFORM PRINTOUT ON UNIT 9 20 WRITE(2,100) DO 30 1=1,12 RMSE=DSQRT((SC2(I)-2*SCO(IX+S02(I))/N(I)) RMBE=(SC(I)-SO(I))/N(I) STN=STNAME(I) RMBEP=(SC(I)-SO(I))/SO(l)*100.0 RMSEP=RMSE/(SO(l)/N(I ) ) * 100.0 R=(N(I)*SCO(I)-SC(I)*SO(I))/DSQRT((N(I)*SC2(I)-SC(I )*SC(I 1 • *(N(I)*S02(I)-SO(l)*SO(l))) RSQ=R*R WRITE(2,101) STN,SC(I),SO(I),SC2(I),S02(I),SCO(l),N(I) 1 ,RMBE,RMSE,RMBEP,RMSEP,R,RSQ 3 0 CONTINUE RETURN 100 FORMAT(' 1 VERIFICATION ANALYSIS'/,'+' ,/ 1'STATION SC SO SC2 S02 SCO', 2' N MBE RMSE MBE% RMSE% R RSQ' 101 FORMAT(A8,5F10.0,1 5,3X,4F8.2,2F8.5) END C C c FUNCTION STNAME(IS) Q***************************************** C* RETURN THE NAME OF THE STATION WITH NUMBER 'IS' * " " " : * * * * * * * * * * * * * * * * * * * * REAL*8 STNAME,NAMES(12)/'GRSMT ','NRTHMT ','BCHYDRO * 'LANGA ','VANAIR ','FERRY ',*PITMED ','MISSHAB * 'ABBLIB ',*ABBAIR ','LANGLEY ','CLISTN '/ STNAME=NAMES(IS) RETURN END SUBROUTINE PDATA(IDY) REAL K(12),OX(2 4),CY(2 4),OY(2 4) REAL*8 DAT1,NAME INTEGER NS(12) COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) 305 Q*************************** C DATA STATEMENT SIGNIFIES PLOT REQUEST BY STATION I F * C "STN NUMBER" IS ZERO THEN NO PLOT.* Q****************************************** *************** DATA NS/0,0,3,0,0,0,0,0,0,0,0,0/ C C DO PLOT FOR EACH STATION C DO 80 11 = 1 , 12 C C CHECK I F PLOT REQUESTED FOR THIS STATION C I F ( N S ( I I ) . E Q . 0 ) GO TO 80 Q* ***************** • C SET UP X/Y AXES* Q**********************************: CALL PLOT(10.0,2.0,3) CALL PLOT(2.0,2.0,2) CALL PLOT(2.0,6.0,2) C C PLACE TICKS AND NUMBERS ON Y AXIS C Y=6.0 DO 10 1=1,4 CALL PLOT(1.8,Y,2) RN=Y-2.0 CALL NUMBER(1 .5,Y,0.2,RN,+90.0,- 1) Y=Y-1.0 CALL PLOT(2.0,Y,3) 10 CONTINUE Q************************************************* ************ C PLACES TICKS AND NUMBERS ON X AXIS* Q* *********************************************************** * X=2.0 DO 15 I=1,9 CALL PLOT(X,1.8,2) RN=(X-2.0)*3.0 CALL NUMBER(X,1.5,0.2,RN,0.0,-1) X=X+1.0 CALL PLOT(X,2.0,3) 15 CONTINUE C LABEL AXES C CALL SYMBOL(4.0,0.8,0.25,'LOCAL APPARENT TIME',0.0,19) CALL SYMBOL(0.8,4.0,0.25,'IRRADIANCE',+90.0,10) C*********************************************************** C PLOT CODES FOR LINES* CALL PLOT(8.0,5.0,3) CALL PLOT(10.0,5.0,2) CALL SYMBOL(8.0,5.0,0.10,11,0.0,-1) CALL SYMBOL(10.0,5.0,0.10,11,0.0,-1) CALL SYMBOL(10.5,5.0,0.25,'OBSERVED',0.0,8) CALL PLOT(8.0,4.5,3) CALL DASHLN(0.2,0.2,0.2,0.2) 306 CALL PLOT(10.0,4.5,4) CALL SYMBOL(8.0,4.5,0.10,12,0.0,-1) CALL SYMBOL(10.0,4.5,0.10,12,0.0,-1) CALL SYMBOL(10.5,4.5,0.25,'CALCULATED',0.0,10) Q************************************************************ C PLACE DATA IN PLOTTING ARRAYS.* £*********• DO 18 1=1,24 IF(DATA(II,1,1).GT.0.0)OY(I)=DATA(II,I,1)/l000.0+2.0 OX(l)=(I)/3.0+2.0 IF(DATA(II,1,2).GT.0.0)CY(I)=DATA(II,I,2)/1000.0+2.0 18 CONTINUE Q************************************************************* C PLOT STATION NAME AND DAY.* Q************************************************************* DAY=FLOAT(IDY) NAME=DAT1(2,1,11) CALL SYMBOL(8.0,4.0,0.25,NAME,0.0,8) CALL NUMBER(8.0,3.5,0.25,DAY,0.0,-1) Q* * *********************************************************** C PLOT OBSERVED DATA.* Q* *********************************************************** * CALL PLOT(2.0,2.0,3) IP = 4 DO 19 1=1,24 IF(OYd).LT.O.O) GO TO 195 CALL PLOT(OX(I),OY(I),IP) CALL SYMBOL(OX(I),OY(I),0.10,1 1 , 0.0,- 1 ) IP = 2 GO TO 19 195 IP = 3 19 CONTINUE Q************************************************************* C PLOT CALCULATED DATA.* Q************************************************************* CALL PLOT(2.0,2.0,3) IP = 4 DO 20 1=1,24 IF(CY(I).LT.0.0) GO TO 30 CALL PLOT(OX(I),CY(I),IP) CALL SYMBOL(OX(I),CY(I),0.10,12,0.0,-1) IP = 4 GO TO 20 30 IP = 3 20 CONTINUE Q**************************************************** C RESET ORIGIN Q**************************************************** CALL PLOT(12.0,0.0,-3) C RETURN FOR NEXT STATION.* Q****** 80 CONTINUE C READY FOR END OF RUN C**************************************************** CALL PLOTND RETURN END 308 C PROGRAM TO IMPLEMENT GAUTIER'S MODEL C REAL KDOWN REAL*8 STAT,DAT1 COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) C C READ NUMBER OF STATIONS C READ(5,106) NSTAT 106 FORMAT (12) C C INITIALISE IMAGE COUNTERS C DO 15 1=1,12 15 NIMG(I)=0 C C INPUT IMAGE FILE AND DETERMINE ANGLES C 20 CALL INPUT(NAVTIM,STAT,XLAPT,X1 ,X2 ,X3 ,&40 , 6.44 ) C C CALCULATE ABSORPTIVITIES C CALL ABS(X1,ISTNUM(STAT)) C C CALCULATE MINIMUM BRIGHTNESS FOR GIUVEN ANGLES C CALL B(X1,X2,X3,ISTNUM(STAT),BMIN) IBMIN=BMIN+0.5 C C EXTRATERRESTRIAL RADIATION C RK0=1353.0*X1 C C DETERMINE NORMALISED REFLECTANCE C RMIN=RNORM(BMIN,X1) C C DETERMINE MINIMUM REFLECTANCE C RMIN=RMIN*RK0 C C DETERMINE MINIMUM BRIGHTNESS THRESHOLD C CALL THRESH(X1,X2,X3,RMIN,ALBS,THR) C C CALCULATE THE SOLAR FLUXES C CALL CALC(X1,X2,X3,THR,ALBS,KDOWN,NCLDY,XDOWN0,XDOWNC) CLOUDY=NCLDY C C STORE THE DATA BY STATION AND IMAGE C CALL STORE(NSTAT,NAVTIM,STAT,XLAPT,BMIN,CLOUDY,KDOWN,XDOWN0 *XDOWNC) C C RETURN FOR MORE IMAGES 309 C GO TO 20 C C MISSING DATA DROP OUT 40 WRITE(6,104) NAVTIM,STAT GO TO 20 C C DETERMINE THE TIME COVERAGE OF THE IMAGE C 44 CALL SPAN(NSTAT) C C MERGE THE SATELLITE AND MEASURED RADIATION DATA C CALL MERGE(NSTAT,IDY) C CALL PDATA(IDY) 99 STOP 104 FORMAT(' ** WARNING-MISSING DATA FOR ',I10,2X,A8) END C C C INPUT - INPUTS IMAGE AND CALCULATES SUN AND SATELLITE ANGLES C C SUBROUTINE INPUT(NAVTIM,STAT,XLAPT,X1,X2,X3,*,*) INTEGER VALS(2500),COUNT(64),NTIM/0/ REAL*8 STAT COMMON/WIND/VALS,NR,NC C C INPUT IMAGE SIZE, TIME AND STATION NAME C READ(8,100,END=44) NR,NC,NAVTIM,STAT C WRITE(6,5) NAVTIM 5 FORMAT(' NAVTIM - ',112) IF(NAVTIM.EO.NTIM) GO TO 15 NTIM=NAVTIM C C DETERMINE L.A.T. AND ANGLES C CALL SGEOM(NAVTIM,0.0,XLPT,Y1,Y2,Y3) XLAPT=XLPT X1=Y1 X2 = Y2 X3 = Y3 C C INPUT BRIGHTNESS COUNTS C 15 DO 20 1=1,NR JS=(I-1)*NC+1 JE=JS+NC-1 READ(8,101) (VALS(J),J=JS,JE) 20 CONTINUE NVALS=NR*NC DO 22 K=1,NVALS C C CRUDE QUALITY CONTROL OF DATA 310 C IF((VALS(K).LT.12).OR.(VALS(K).GT.256)) RETURN 1 22 CONTINUE C C NORMAL RETURN C RETURN C C END OF FILE RETURN C 44 RETURN 2 100 FORMAT(I 4,4X,I 3,1 10,14X,A8) 101 FORMAT(8013) END C c C ABS - INPUTS PRECIP. WATER AND CALCULATES ABSORPTIVITIES C C SUBROUTINE ABS(COSZZ,IS) REAL ZS(12)/1128.0,11*0.0/ COMMON/ATTEN/ABSUN,ABSAT,ABSUNT,ABSUNB,BREFL,ABSATT, *ABSATB,DENOM LOGICAL FIRST/.TRUE./ C C IF FIRST PASS - READ IN PRECIP. WATER C IF(.NOT.FIRST) GOTO 23 FIRST=.FALSE. WRITE(6,20) 20 FORMATC- INPUT PRECIP.WATER IN MM.') READ(5,21) W 21 FORMAT(F4.1) WRITE(6,22) W 22 FORMAT(' PRECIP.WATER=' , F4 . 1 ) C C NOTE:WE CURRENTLY IGNORE HEIGHT OF GROUSE C C C ******* TEMPORARY FIX ********* C 23 COSZ=COSZZ IF(COSZZ.LT.O.O) COSZZ= 0.009 ZED=ARCOS(COSZZ)*l80.0/3.14159 C IF(ZED.GT.90.0) ZED=90.0 C C KASTEN'S AIR MASS ALGORITHM C AMSUN=EXP(-ZS(lS)/8243.0)/(COSZZ+0.l5/((93.885-ZED)**1.253) U1 = W IF(ZS(IS).EQ.1128.0) U1=U1*0.8 C WRITE(6,31) Ul 31 FORMAT(F9.3) ABSUN=YAM(U1,AMSUN) C C 57.7 IS ZENITH ANGLE FOR SATELLITE AT VANCOUVER LATITUDE 311 C AMSAT=EXP(-ZS(IS)/8243.0)/(COS(57.7*3.14159/180.0)+0.l5/((93.8 *85-57.7)**1.253)) C WRITE(6,204) AMSUN,AMSAT 204 FORMAT( 1AMSUN=',F9.3,2X,'AMSAT=',F9.3) C C "YAM" IS FUNCTION - SEE BELOW C ABSAT=YAM(U1,AMSAT) U2=U1*0.3 ABSUNT=YAM(U2,AMSUN) ABSATT=YAM(U2,AMSAT) U2=U1*0.7 ABSUNB=YAM(U2,AMSUN) C C "COUL" IS FUNCTION - SEE BELOW C BREFL=COUL(COSZ) C WRITE(6,28) AMSUN,AMSAT,BREFL,ABSUN,ABSAT ABSATB=YAM(U2,AMSAT) RETURN 28 FORMAT(5F9.3) END C C C THRESH - DETERMINES THE .MINIMUM BRIGHTNESS THRESHOLD C C SUBROUTINE THRESH(X1,X2,X3,RMIN,ALBS,THR) COMMON/ATTEN/ABSUN,ABSAT,ABSUNT,ABSUNB,BREFL,ABSATT, *ABSATB,DENOM RK0=1353.0*X1 DEN0M=(1.0-BREFL)*(l.0~ABSUN)*(1.0"ABSAT)*(1.0-0.076) ALBS=(RMIN-RK0*BREFL)/(DENOM*RK0) ALBP=ALBS+0.00566/X1 THR=RK0*(BREFL+ALBP*DENOM) C WRITE(6,38) DENOM,ALBS,ALBP,RMIN,THR RETURN 38 FORMAT( 5F12.3) END C C C RNORM - CONVERTS COUNTS TO NORM REFLECTANCE C USES 2ND ORDER POLYNOMIAL C C FUNCTION RNORM(BMIN,X1) RIR=0.00154+BMIN*0.000166+BMIN*BMIN*0.0000137 RNORM=RIR/X1 C WRITE(6,66) BMIN,X1,RIR,RNORM 66 FORMAT(' BMIN,X1,RIR,RNORM ',4F8.3) RETURN END C 312 C C CALC - DETERMINES THE SOLAR FLUXES C CLEAR ("KDOWNO") C CLOUDY ("KDOWNC") C AVERAGE ("KDOWN") C C SUBROUTINE CALC(XI,X2,X3,THR,ALBS,KDOWN,NCLDY,KDOWNO,KDOWNC INTEGER VALS(2500) COMMON/ATTEN/ABSUN,ABSAT,ABSUNT,ABSUNB,BREFL,ABSATT, *ABSATB,DENOM COMMON/WIND/VALS,NR,NC REAL KDOWNO,KDOWNC,KDOWNT,KDOWN,REFL(2500) C C I N I T I A L I S E C NCLEAR= 0 NCLDY=0 KDOWNC=0.0 NVALS=NR*NC AVALBS=0.0 RKO = 1353.0 * XI KDOWN0=0.0 KDOWNT=0.0 IF(RKO.GT.O.O) GO TO 1 GO TO 50 1 DO 2 0 1=1,NVALS C C CONVERT COUNTS TO NORM REFLECANCE TO REFLECTED FLUX C RVAL = V A L S ( I ) REFL(I)=RNORM(RVAL,X1)*RK0 C C C COMPARE WITH THRESHOLD C C 15 CONTINUE I F ( R E F L ( I ) . G E . T H R ) GO TO 10 C C COUNT NUMBER OF CLEAR PIXELS C NCLEAR=NCLEAR+1 C GO THROUGH CLEAR SKY CALCULATION ONLY ONCE C IF(I C P . E Q . 9 8 7 6 ) GO TO 20 ICP=9876 RK0=X1*1353.0 KDOWN0=RK0*(1.0"BREFL)*(1.0~ABSUN)*(1.0+ALBS*0.076) C WRITE(6,5) KDOWNO,RKO,ALBS,NCLEAR,REFL(I),THR 5 FORMAT(' KDOWN 0,RK 0,ALBS,NCLEAR,REFL(I),THR '/ 1 3F8.2,I5,2F8.2) C GO TO 15 GO TO 20 C C CLOUDY PIXEL CALCULATIONS C 313 10 NCLDY=NCLDY+1 IF(NCLDY.EQ.25) KDOWN0=RK0*( 1 .0~BREFL)*(1 .0-ABSUN)*'( 1 . 0-ALB* 1.076) C WRITE(6,4) NCLDY,REFL(I),THR 4 FORMAT(' NCLDY,REFL(I),THR ',I5,2F8.2) C C CALCULATE CLOUD ABSORPTIVITY ("CLABS") C STEP*(RK0-THR)/20.0 CLABS=((REFL(I)-THR)/STEP)* 0.01 C C CALCULATE CLOUD ALBEDO ("CLREFL") C PHI=RK0*(1.0-BREFL)*(1.0-ABSUNT)*(1.0-ABSUNB)*ALBS* 1(1.0-ABSATT)*((1.0-CLABS)**2)*(1.0"ABSATB)*(1.0-0.076) C WRITE(6,720) ALBS,ABSUNB,ABSATB ' 720 FORMAT(3F9.3) BETA=RK0*(1.0-BREFL)*(1.0-ABSUNT)*(1.0-0.076)*(1.0-ABSATT) GAMMA=RK0*BREFL SW=REFL(I) GEE=BETA-2.0*PHI EFF=(GAMMA+PHI-SW)*PHI C WRITE(6,240) RKO,BREFL,ABSUN,ABSAT,ABSUNT,ABSUNB, C *ABSATT,ABSATB,CLABS,PHI,BETA,GAMMA,GEE,EFF,REFL(I), C *THR,VALS(I) 240 FORMAT(' ' ,8F10.3/8F10.3,1 6) ROOT=SQRT(GEE*GEE~4.0*EFF) C WRITE(6,600) ROOT 600 FORMAT(' ROOT= ',F9.3) GEE=GEE*(- 1 .0) C WRITE(6,700) GEE 700 FORMAT( 'GEE= ',F9.3) ALBCL1=(GEE+ROOT)/(2.0*PHI) ALBCL2=(GEE-ROOT)/(2.0*PHI) C WRITE(6,60) ALBCL1,ALBCL2 60 FORMAT(' CLOUD ALBEDO:(+VE ROOT),(-VE ROOT)= ',2F8.3) CLREFL=ALBCL1 IF(CLREFL.GT.0.85) CLREFL=0.85 C C C DETERMINE SOLAR FLUX IN CLOUDY PIXEL C KDOWNT=RK0*(1.0~BREFL)*(1.0~ABSUNT)*(1.0~CLREFL)*(1.0-CLABS *(1.0-ABSUNB) KDOWNC=KDOWNC+KDOWNT C WRITE(6,8) RKO,I 8 FORMAT(' RKO,I ',F8.1,I4) 20 CONTINUE ICP = 0 IF(NCLDY.GT.0) GO TO 40 C C IF NO CLOUD ACTUAL SOLAR = CLEAR SKY VALUE C KDOWN=KDOWNO C WRITE(6,48) KDOWN 3 1 4 GO TO 50 C C IF CLOUD IN IMAGE THEN WEIGHT FOR CLEAR AND C CLOUDY PIXELS C 4 0 KDOWN=(KDOWNO *NCLEAR+RDOWNC)/(NCLEAR+NCLDY) C WRITE(6,48) KDOWN,KDOWNO,KDOWNC,NCLEAR,NCLDY 50 RETURN 48 FORMAT(' KDOWN ',3F9.3,2I5) END C C C STORE - STORES ALL DATA BY IMAGE AND STATION C C SUBROUTINE STORE(NSTAT,NAVTIM,STAT,XLAPT,BMIN,CLOUDY,KDOWN, *XDOWN0,XDOWNC) REAL*8 STAT,DAT 1 REAL KDOWN,XDOWN0,XDOWNC INTEGER ISTN/0/ COMMON /RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) ISTN=ISTNUM(STAT) NIMG(ISTN)=NIMG(ISTN)+1 IMG=NIMG(ISTN) DAT 1 (1 ,IMG,ISTN)=NAVTIM DAT1(2,IMG,ISTN)=STAT DAT 1 (3,IMG,ISTN)=XLAPT DAT 1 (4,IMG,ISTN)=BMIN DAT 1 (5,IMG,ISTN)=CLOUDY DAT 1 (6,IMG,ISTN)=KDOWN DAT 1(7,IMG,ISTN)=XDOWN0 C WRITE(6,7 10) BMIN,CLOUDY,KDOWN,XDOWN0 RETURN 710 FORMAT( *BMIN' ,F6.2,2X,'CLOUDY * ,F9.3,2X, 'KDOWN' ,F9.3,2X, *'XDOWN0',F9.3) END C C C MERGE - FOR EACH HOUR MERGE MEASURED RADIATION AND C CALCULATED (SATELLITE) SOLAR VALUES C FOR APPROPRIATE IMAGES C C SUBROUTINE MERGE(NSTAT,IDY) REAL*8 DAT 1 REAL K(12),ZS(12)/1128.0,11*0.0/,KDBAR,KD0BAR,CLDBAR COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) ID=DAT1(1,1,1)/l0000 WRITE(2,106) ID C C READ IN MEASURED RADIATION DATA C 5 READ(4,100,END= 9 9) IDY,HR,(K(I),I = 1,12) WRITE(2,107) IHR=HR+0.5 XKJ=K(11) 315 K(11)=K(10) K(10)=K(9) K(9)=K(8) K(8)=K(7) K(7)=K(6) K(6)=K(5) K(5)=K(4) K(4)=XKJ NAVTIM=IDY*10000 HRII=HR-0.5 C C CONVERT TIME TO L.A.T. C CALL SGEOM(NAVTIM,HRII,XLPT,X1,X2,X3) ZA=ARCOS(X1)*180.0/3.14159 TST=HR-1.0 DO 50 IS=1,NSTAT C C WRITE MEASURED DATA C C WRITE(2,101) DAT 1 (2,1 ,IS),IDY,HR,K(IS) IF(DAT 1(8,1,1S).GT.TST+0.001) GO TO 25 NI=NIMG(IS) DO 10 IM=1,NI IM1=IM IF(DAT 1 (9,IM,IS).GT.TST+0.001) GOTO 15 10 CONTINUE 15 IF( ( DAT 1 (.1 0, IM1 , IS) .EQ. 1 . 0) .OR. (DAT 1 (9 , IM1 , IS) .LE.TST) ) GOTO TFRAC= 0 25 KDBAR= 0 KD0BAR=0 CLDBAR=0 2 0 IF(IM1.GT.NIMG(IS)) GO TO 2 5 T1=DAT1(8,IM1,IS) T2=DAT1(9,IM1,IS) C WRITE(6,90) T1,T2 90 FORMAT( 'T1=',F9.3,2X,'T2=1,F9.3) C C DETERMINE WEIGHTS FOR EACH IMAGE IN HOUR C TIME=AMIN1(T2,HR)-AMAX1(T1,TST) FRAC=TIME/(T2-T1) TFRAC=TFRAC+FRAC C C WEIGHT THE CALCULATED VALUES C KDBAR=KDBAR+FRAC*DAT1(6,IM1,IS) KD0BAR=KD0BAR+FRAC*DAT1(7,IM1,IS) CLDBAR=CLDBAR+FRAC*DAT1(5,IM1,IS) C WRITE(6,98) TFRAC,FRAC,KDBAR,KD0BAR,CLDBAR IM1=IM1+1 IF(T2.LT.HR) GO TO 20 C C CONVERT TO KJ M~2 HOUR-1 C KDBAR=KDBAR/TFRAC*3.60 316 KDOBAR=KDOBAR/TFRAC* 3.60 CLDBAR=CLDBAR/TFRAC C C WRITE CALCULATED VALUES FOR HOUR AND STATION C IF(KDBAR.LT.0.0) KDBAR=0.0 C WRITE(2,102) KDBAR,KDOBAR,CLDBAR IHR=HR+0.5 DATA(IS,IHR,1)=K(IS) DATA(IS,IHR,2)=KDBAR CALL STATS(K(IS),KDBAR,IS,.FALSE.) GO TO 50 25 WRITE(2,105) DATA(IS,IHR,1)=K(IS) DATA(lS,IHR,2)=-9000.0 50 CONTINUE GO TO 5 99 CALL STATS(0.0,0.0,0,.TRUE.) RETURN 100 FORMAT(1X,I4,2X,F7.3,/,8(F9.3,5X),/,3(F9.3,5X),F9.3) 101 FORMAT(' ',2X,A8,I5,F9.2,F12.3) 106 FORMAT('1 GAUTIER''S MODEL (FLUX AVERAGING) ~ ',15,/, , . ) 107 FORMAT(' ') 105 FORMAT(' + ' ,37X, ' *** MISSING DATA ***') 102 FORMAT('+',37X,3F9.3) 98 FORMAT(2X,2F6.3,2X,3F9.3) END C C C YAM - YAMAMOTO'S ABSORPTIVITlES FOR WATER VAPOUR C C FUNCTION YAM(UT,AM) C C CONVERSION TO CONVERT FROM MM TO CORRECT UNITS C EG.IF RELATIONSHIP FOR U IN CM THEN C USE U=UT/10.0. C U=(UT/10.0) * AM C WRITE(6,60) UT, U, AM 60 FORMAT(' ', 3F 10.3) IF(U.GT.0.5) YAM=0.099*U**0.34 IF(U.LE.0.5) YAM=0.14*U**0.44 RETURN END FUNCTION COUL(COSZ) C C C C COUL - COULSON'S BEAM REFLECTANCE C c c ZEDD=ARCOS(COSZ)*180.0/3. 14159 3 1 7 COUL=0.0467563+0.0014173*ZEDD-O.00005258*(ZEDD**2)+ *0.000000651*(ZEDD**3) IF(COUL.LT.0.046) COUL=0.046 RETURN END C C SUBROUTINE SPAN(NSTAT) £************************************************************** C* COMPUTE THE TIME SPAN FOR WHICH THE IMAGES IN 'DAT1' ARE * C* VALID. * C************************************************************** C REAL*8 DATI COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) DO 50 IS=1,NSTAT DAT 1(8,1,IS)=DAT1(3,1,IS)-0.25 DAT 1 (9,NIMG(IS),IS)=DAT1(3,NIMG(IS), I S)+0.25 NI=NIMG(IS) DO 30 IM=2,NI DAT 1(8,IM,IS)=(DAT1 (3,IM-1 ,IS)+DAT1(3,IM,IS))/2.0 DAT 1(9,IM-1 ,1S)=DAT1(8,1M,IS) 30 CONTINUE DO 40 IM=1,NI IF((DATI(9,IM,IS)~DAT1(8,IM,IS)).LE.1.0) GOTO 35 DAT 1 (10,IM,IS) = 1 .0 GOTO 4 0 3 5 DAT 1 (10,IM,IS) = 0.0 4 0 CONTINUE 50 CONTINUE DO 60 IS= 1,NSTAT NI=NIMG(IS) C WRITE(9,100) ((DAT1(I,J,IS),1=1,10),J=1,NI) 60 CONTINUE RETURN 100 FORMAT(F11.0,2X,A8,2F7.2,F7.3,2F7.2,3F7.2) END C C C C SUBROUTINE B(X1,X2,X3,NS,B1) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C* RETURN THE PREDICTED TARGET CLEAR BRIGHTNESS. * c************************************************************** C C C MINIMUM BRIGHTNESS COEFFS BASED ON ANALYSIS C OF CLEAR IMAGES IN ALL SEASONS (NO SNOW) C REAL A(12)/37.366,39.093,41.230,40.321,40.159,42.322,40.129 *39.549,41.444,41.036,39.255,39.255/ ' REAL BAB(12)/40.292,43.348,42.435,50.447,52.738,29.589,50.8 *52.292,49.775,52.343,51.297,38.183/ REAL C(12)/7.483,8.596,9.849,8.653,9.368,2.770,7.900,6.001, 318 *5.324,5.518,8.135,7.792/ REAL D(12)/15.115,12.865,10.817,10.161,8.986,2.568,9.065, * 12.640,12.336,10.905,1 0.157,7. 157/ B1=A(NS)+BAB(NS)*X1+C(NS)*X2+D(NS)*X3 C WRITE(6,10) NS,BB C10 FORMAT(' PROCESSING FOR' ,14, ' BB',F7.2) RETURN END C C C C FUNCTION FTM(NVTIM) Q*************************************************************** C* FLOATING POINT TIME IN HRS. * Q* ************************************************************* * IT=MOD(NVTIM,10000) FTM=IT/100+MOD(IT,100)/60.0 RETURN END C C C C SUBROUTINE SGEOM(NAVTIM,XPT,XLAPT,X1,X2,X3) REAL CLAT/49.217/,SATAZ/16.1/ C FIRST COMPUTE DECLINATION, SOLAR ZENITH ANGLE, AND SOLAR C AZIMUTH FOR THIS TIME. C RADDEG=3.14159/180.0 C1=279.457*RADDEG C2=0.985647*RADDEG YR=NAVTIM/10000000 JD=MOD(NAVTIM,10000000)/l0000 IF(XPT.EQ.O.O) GOTO 5 XLAPT=XPT GOTO 10 5 TI=FTM(NAVTIM) X=AINT((YR-65)*365.251)+JD+Tl/24.0 G=C1+C2*X X=X/365.2422 C C 'EQ' IS EQUATION OF TIME VALUE C EQ=(-102.5-0.142*X)*SIN(G)+(-429.8+.033*X)*COS(G)+596.5*S *-2.0*COS(2*G)+4.2*SIN(3*G) + 19.3*COS(3*G)- 12.8*SIN(4*G) EQ=EQ/3600 C C 'XLAPT' IS LOCAL APPARENT TIME IN HOURS (CONSTANTS VARY C WITH CENTRAL LOCATION) C XLAPT=TI-8.0+EQ-10.8/60 . 0 C WRITE(6,60) NAVTIM,JD,TI,EQ,XLAPT 60 FORMATC NAVTIM,JD,TI,EQ,XLAPT ' ,I 10,1 4,3F10.4) 319 IF(XLAPT.LT.0.0) XLAPT=XLAPT+2 4.0 10 HA=15.0*(XLAPT-12.0)*RADDEG PSI=2*3.14159*(JD-1)/365.0 C C 'DEC IS SOLAR DECLINATION C C C •C * * DEC=0.006918-0.399912*COS(PSI)+0.070257*SIN(PSI) -0.006758*COS(2*PSI)+0.000907*SIN(2*PSI)-0.002697*COS(3*P +0.001480*SIN(3*PSI) " L U b U P r ! r ? ? Z T * A S ! ' S I N Z ' A R E S I N A N D C 0 S 0 F SOLAR ZENITH ANGLE C 'CA' IS COS OF SOLAR AZIMUTH COS Z = SIN(CLAT*RADDEG)* SIN(DEC)+COS{CLAT*RADDEG)*COS(DEC)* SINZ=SIN(ARCOS(COSZ)) CA=(SIN(CLAT*RADDEG)*COSZ-SIN(DEC))/(COS(CLAT*RADDEG)*SI N C NOW COMPUTE THE 3 PARAMETERS OF INTEREST C SIGN=1 IF(XLAPT.LT.12.0) SIGN=-1 SSA=ABS(SATAZ*RADDEG-SIGN*ARCOS(CA)) XI=cosz X2=SINZ*COS(SSA) X3=X2*COS(SSA) C - WRITE(6,25) NAVTIM,X1,XLAPT 2 5 FORMAT(' NAVTIM,X1 ,XLAPT' , I 12 , 2F10.3 ) RETURN END C c c c C ISTNIM - ASSIGNS NUMBER TO STATION NAME C C FUNCTION ISTNUM(STAT) REAL*8 STAT,NAMES(12)/'GRSMT ','NRTHMT ','BCHYDRO ', * 'LANGA ','VANAIR ','FERRY ','PITMED ','MISSHAB ', * 'ABBLIB ','ABBAIR ','LANGLEY ','CLISTN '/ LOGICAL EQCMP DO 10 1=1,12 J=I IF(EQCMP(8,STAT,NAMES(I))) GOTO 15 IF(I.EQ.12) WRITE(6,100) 10 CONTINUE 15 ISTNUM=J RETURN 100 FORMAT(' HELLLP!... THERE IS AN UNKNOWN STATION SOMEWHERE') END SUBROUTINE STATS(O,C,IS,PFLAG) C* KEEP TRACK OF SUMS AND SQUARES AND PRODUCTS OF DATA * C* VALUES 'O' AND 'C GIVEN FOR STATION NUMBER 'IS'. * C* THESE SUMS ARE PRINTED ALONG WITH MBE AND RMSE FOR * C* ALL STATIONS WHEN CALLED WITH 'PFLAG'=.TRUE. (UNIT 9)* 320 C C C C C c 20 30 1 00 101 0/, SO(12)/12*0.0/,SCO(12)/12*0 0/,S02(12)/12*0.0/,STNAME,STN 0/, REAL*8 SC(12)/12*0 1 SC2(12)/12*0 LOGICAL PFLAG INTEGER N(12)/l2*0/ IS IT PRINTOUT TIME? IF(O.LT.O.O) RETURN IF(C.LT.O.O) C=0.0 IF(PFLAG) GOTO 20 SC(IS)=SC(IS)+C SO(IS)=SO(lS)+0 SC2(IS)=SC2(IS)+C*C S02(IS)=S02(IS)+0*0 SCO(.IS)=SCO(IS)+0*C N(IS)=N(IS)+1 RETURN JUMP HERE TO PERFORM PRINTOUT ON UNIT 2 WRITE(2,100) DO 30 1=1,12 RMSE=DSQRT((SC2(I)-2*SCO(I)+S02(I))/N( I )) RMBE=(SC(I)-SO(I))/N(I) STN=STNAME(I) RMBEP=(SC(I)-SO(l))/SO(l)*100.0 RMSEP=RMSE/(SO(I)/N(I ) ) * 100.0 R=(N(I)*SCO(I)-SC(I)*SO(I))/DSQRT((N(I)*SC2(I)-SC(I)*SC(I 1 *(N(I)*S02(I)-SO(l)*S0(I))) RSQ=R*R WRITE(2,101) STN,SC(I),SO(l),SC2(I),S02(I),SCO(I),N(I) 1 ,RMBE,RMSE,RMBEP,RMSEP,R,RSQ CONTINUE RETURN FORMAT(' 1 VERIFICATION ANALYSIS'/, '1' , / 1'STATION SC 2' N MBE FORMAT(A8,5F10.0,I5 END SO SC2 RMSE MBE% 3X,4F8.2,2F8.5) S02 RMSE5 SCO' , RSQ' C C C FUNCTION STNAME(IS) C* RETURN THE NAME OF THE STATION C*: REAL*8 STNAME,NAMES(12)/'GRSMT * 'LANGA ','VANAIR ','FERRY * 'ABBLIB ','ABBAIR ','LANGLEY STNAME=NAMES(IS) RETURN END ','NRTHMT ','PITMED ' BCHYDRO 'MISSHAB 'CLISTN '/ 321 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * PROGRAM TO I M P L E M E N T H A Y / H A N S O N I N S O L A T I O N M O D E L , U S I N G * C * WINDOWS ON U N I T 8 PRODUCED BY ' N A V S Y S ' . * C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * M . D . R . 8/81 * C R E A L * 8 S T A T , D A T 1 C O M M O N / R D A T A / D A T 1 ( 1 0 , 4 0 , 1 2 ) , N I M G ( 1 2 ) , D A T A ( 1 2 , 2 4 , 2 ) C DO 4 I = 1 , 1 2 C DO 3 J = 1 , 2 4 C DO 2 K = 1 , 2 C D A T A ( I , J , K ) = - 9 . 0 0 C 2 C O N T I N U E C 3 C O N T I N U E C 4 C O N T I N U E W R I T E ( 6 , 1 0 0 ) R E A D ( 5 , 1 0 1 ) IMODE I F ( I M 0 D E . E Q . 2 ) GOTO 10 5 C A L L A V G S ( N A V T I M , S T A T , X L A P T , X M E A N , X M E D , X M O D , X 1 , X 2 , X 3 , & 7 , & 9 9 W R I T E ( 7 , 1 0 2 ) N A V T I M , S T A T , X M E A N , X M E D , X M O D , X 1 , X 2 , X 3 , GOTO 5 7 W R I T E ( 7 , 1 0 2 ) N A V T I M , S T A T W R I T E ( 6 , 1 0 4 ) N A V T I M , S T A T GOTO 5 C C T H I S BLOCK P R O C E S S E S S T A G E 2 OF M O D E L : C 10 W R I T E ( 6 , 1 0 5 ) R E A D ( 5 , 1 0 6 ) N S T A T DO 15 I = 1 , 1 2 15 N I M G ( I ) = 0 20 C A L L A V G S ( N A V T I M , S T A T , X L A P T , X M E A N , X M E D , X M O D , X 1 , X 2 , X 3 , & 4 0 , & 4 C A L L C A L I B ( X 1 , X M E A N , X I R ) C A L L S T O R E ( N S T A T , N A V T I M , S T A T , X L A P T , X I R , X 1 ) GOTO 20 40 W R I T E ( 6 , 1 0 4 ) N A V T I M , S T A T GOTO 20 C C R E L A T E TO KNOWN R A D I A T I O N , E T C C 44 C A L L S P A N ( N S T A T ) C A L L C A L C ( N S T A T , I D Y ) C C A L L P D A T A ( I D Y ) 9 9 S T O P C C 100 F O R M A T C E N T E R P R O C E S S I N G S T A G E ' ) 101 F O R M A T ( I I ) 102 F O R M A T ( I 1 0 , 2 X , A 8 , 3 F 6 . 2 , 3 F 9 . 5 ) 103 F O R M A T ( I 1 0 , 2 X , A 8 , 2 F 7 . 2 , F 7 . 3 , 2 F 7 . 2 ) 104 F O R M A T ( ' * * W A R N I N G - - M I S S I N G DATA FOR ' , I 1 0 , 2 X , A 8 ) 105 F O R M A T C HOW MANY S T A T I O N S ? ' ) 106 F O R M A T ( 1 2 ) E N D C 322 C C SUBROUTINE STORE(NSTAT,NAVTIM,STAT,XLAPT,XIR,X1) r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ~ * : * x x x x C* STORE INTERMEDIARY SATELLITE RELATED VALUES IN ARRAY 'DAT 1 C REAL*8 STAT,DAT1 INTEGER ISTN/0/ COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) C ISTN=ISTNUM(STAT) NIMG(ISTN)=NIMG(ISTN)+1 IMG=NIMG(ISTN) DAT 1 (1 ,IMG,ISTN)=NAVTIM DAT 1 (2,1MG,ISTN)= STAT DAT 1 (3,IMG,ISTN)=XLAPT DAT 1 (4,IMG,ISTN)=XIR DAT 1(5,1MG,ISTN)=X1 DAT 1 (6,IMG,ISTN) = 0.0 DAT 1 (7,IMG,ISTN) = 0.0 RETURN END C C c SUBROUTINE SPAN(NSTAT) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C* COMPUTE THE TIME SPAN FOR WHICH THE IMAGES IN 'DAT 1' ARE C* VALID. r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c REAL*8 DAT1 COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) DO 50 IS=1,NSTAT DAT 1(8,1,IS)=DAT1(3,1,IS)~0.25 DAT 1 (9,NIMG(IS),IS)=DAT1(3,NIMG(IS),IS)+0.25 NI=NIMG(IS) DO 30 IM=2,NI DAT 1(8,IM,IS) = (DAT1(3,IM~1,IS)+DAT1(3,IM,IS))/2.0 DAT 1(9,IM-1 ,IS)=DAT1(8,IM,IS) 30 CONTINUE DO 40 IM=1,NI IF((DAT 1 (9,IM,IS)-DAT1(8,IM,IS)).LE.1.0) GOTO 35 DATI(10,IM,IS)=1.0 GOTO 4 0 35 DAT1(10,IM,IS)=0.0 40 CONTINUE 50 CONTINUE C DO 60 IS=1,NSTAT C NI=NIMG(IS) C WRITE(2,100) ( (DAT1 (I , J,IS) , 1 = 1 , 1 0) , J=1 ,'NI ) C60 CONTINUE RETURN 100 FORMAT(F11.0,2X,A8,2F7.2,F7.3,2F7.2,3F7.2) END 323 C C C SUBROUTINE CALC(NSTAT,IDY) C* RELATE THE SATELLITE DATA TO KNOWN RADIATION DATA. C C C C C C C C C C C REAL*8 DAT1 REAL K(12),ZS(12)/1128.0,11*0.0/ REAL BOS(l2)/65.34,68.82,69.99,74.84,76.16,62.79,75.13,76.1 *77.57,76.27,74.61,65.77/ REAL BB(12),TRAN(12) COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) ID=DAT1(1,1,1)/l0000 : ACOEFF=0.788 BCOEFF=-1.078 WRITE(6,103) ID READ(5,104) W READ IN THE RADIATION VALUES 5 READ(4,100,END=99) IDY,HR,(K(I),I=1,12) IHR = HR + 0.5 XKJ = K(11 ) K(11)=K(10) K(10)=K(9) K(9)=K(8) K(8)=K(7) K(7)=K(6) K(6)=K(5) K(5)=K(4) K(4)=XKJ NAVTIM=IDY*10000 HRII = HR-0.5 CALL SGEOM(NAVTIM,HRII,XLPT,X1,X2,X3) ZA=ARCOS(X1)*180.0/3.14159 COSX=X1 IF(COSX.LT.O.O) COSX=0.009 TST=HR-1.0 FOR EACH STATION, COMPUTE THE WEIGHTED AVERAGED VALUES 'XNBAR', 'XICBAR', AND 'XIMBAR' IF POSSIBLE. DO 50 IS=1,NSTAT WRITE(6,222) IS 222 FORMAT(' DOING STN # ',15) B0=0.0 BB(IS)=0.0 TRAN(IS)=K(IS) C • WRITE(6,19) TRAN(IS) 19 FORMAT(F9.3) 324 PSI=0.0 IF(PSI.GT.1.0) PSI=1.0 WRITE(2,101) DAT 1(2,1,1S),IDY,HR,K(IS),PSI,BB(IS),BO FIRST MAKE SURE WE HAVE IMAGES THAT EARLY AND IF SO, FIND FIRST IMAGE PARTIALLY WITHIN HOUR OF INTERES IF(DAT1(8,1,IS).GT.TST+0.001) GOTO 25 NI=NIMG(IS) DO 10 IM=1,NI IM1=IM IF(DAT 1 (9,IM,IS).GT.TST+0 . 00 1 ) GOTO 15 I CONTINUE ) IF((DAT1(10,IM1,IS).EQ.1.0).OR.(DAT 1 (9,1M1 ,IS).LE.TST)) G IF EVERYTHING OK, COMPUTE WEIGHTED AVERAGES STOPPING IF ANYTHING GOES WRONG (IE: MISSING DATA) TFRAC=0 XIRBAR=0.0 TRAN=0.0 IF(I'M1 .GT.NIMG(IS) ) GOTO 25 T1=DAT1(8,IM1,IS) T2=DAT1(9,IM1,IS) 'TIME' IS AMOUNT OF TIME FROM THIS IMAGE WHICH IS WITHIN THE HOUR OF INTEREST TIME=AMIN1(T2,HR)-AMAX1(Tl,TST) FRAC=TIME/(T2-T1) TFRAC=TFRAC+FRAC XIRBAR=XIRBAR+FRAC*DAT1(4,IM1,IS) WRITE(2,73) TFRAC,FRAC,XIRBAR 3 FORMAT(2X,2F6.3,3F9.3) IM1=IM1+1 IF(T2.LT.HR) GOTO 20 XIRBAR=XIRBAR/TFRAC XEXT=4870.799*COSX. TRAN(IS)=TRAN(IS)/XEXT QS=ACOEFF*XEXT+BCOEFF*XEXT*XIRBAR IF(QS.LT.O.O) QS=0.0 WRITE(2,102) QS,XIRBAR,TRAN(IS),COSX IHR=HR+0.5 DATA(IS,IHR,1)=K(IS) DATA(IS,IHR,2)=QS CALL STATS(K(IS),QS,IS,.FALSE.) GOTO 50 MISSING DATA MESSAGE WRITE(2,105) 325 DATA(IS,IHR,1)=K(IS) DATA(IS,IHR,2)=-9000.0 C C LOOP TO NEXT STATION C 50 CONTINUE C C LOOP TO NEXT HOUR C GOTO 5 C99 CONTINUE 99 CALL STATS(0.0,0.0,0,.TRUE.) RETURN 100 FORMAT(1X,I4,2X,F7.3,/,8(F9.3,5X),/,3(F9.3,5X),F9.3) 101 FORMAT(' ',2X,A8,I5,F6.2,4F9.3) 102 FORMAT(' + ' ,58X,4F9.3,F6.3)' 103 FORMAT(' INPUT W FOR ',16) 104 FORMAT(F5.2) 105 FORMAT('+',58X,'*** MISSING DATA ***') END C C C SUBROUTINE AVGS(NAVTIM,STAT,XLAPT,XMEAN,XMED,XMOD,X1,X2,X3, Q************************************************************* C* COMPUTE MEAN, MEDIAN, AND MODE OF WINDOWS ON UNIT 8, AS * C* WELL AS SOME GEOMETRICAL VALUES. * Q* *********************************************************** * c c C 'VALS' HOLDS OBSERVED PIXEL VALUES C 'COUNT' IS USED FOR COMPUTATION OF MODE C 'STAT' HOLDS STATION NAME C 'CLAT' IS APPROXIMATE CENTER LATITUDE OF AREA C COMMON BLOCK 'WIND' IS FILLED WITH WINDOW INFO. C INTEGER VALS(2500),COUNT(64),NTIM/0/ REAL*8 STAT COMMON/WIND/VALS,NR,NC C C READ IN WINDOW INFORMATION, ZERO COUNTERS, AND COMPUTE C THE GEOMETRICAL PARAMETERS (IF NOT ALREADY DONE FOR THIS C TIME) --C X1=COS(SOLAR ZENITH ANGLE) = COS(Z) C X2=SIN(Z)*COS(SUN-SATELLITE AZIMUTH) = SIN(Z)*COS(SSA) C X3=SIN(Z)*COS(SSA)**2 C READ(8,100,END=44) NR,NC,NAVTIM,STAT SUM=0 DO 10 1=1,40 COUNT(I)=0 10 CONTINUE IF(NAVTIM.EQ.NTIM) GOTO 15 NTIM=NAVTIM CALL SGEOM(NAVTIM,0.0,XLPT,Y1,Y2,Y3) XLAPT=XLPT 326 X1 =Y 1 X2=Y2 X3=Y3 C C NOW COMPUTE THE MEAN, MEDIAN, AND MODE OF THE PIXEL VALUES C C FIRST READ IN THE WINDOW, THEN CHECK THE VALUES C AO PIXEL VALUE INDICATES MISSING DATA C 15 DO 20 1=1,NR JS=(I-1)*NC+1 JE=JS+NC-1 READ(8,101) (VALS(J),J=JS,JE) 2 0 CONTINUE NVALS=NR*NC DO 22 K=1,NVALS IF((VALS(K).LT.12).OR.(VALS(K).GT.256)) RETURN 1 2 2 CONTINUE C C NOW TALLY INTO 'COUNT' FOR COMPUTATION OF MODE C DO 2 5 1=1,NVALS IV=VALS(I) COUNT(IV/4)=COUNT(IV/4)+1 SUM=SUM+IV 2 5 CONTINUE C C SORT 'VALS' FOR EASY SPOTTING OF MEDIAN VALUE C 'SSORT' IS MTS FORTRAN LIBRARY ROUTINE C CALL SSORT(VALS,NVALS,1) C C COMPUTE MEAN, MEDIAN, AND FINALLY MODE C XMEAN=SUM/NVALS XMED=VALS(NVALS/2+1) MAX=COUNT(1) DO 30 1=2,40 IF(COUNT(I).LE.MAX) GOTO 30 MAX=COUNT(I) IMAX=I 30 CONTINUE XMOD=IMAX*4 RETURN 4 4 RETURN 2 100 FORMAT(I 4,4X,I 3,1 10,14X,A8) 101 FORMAT(80I3) END C C c FUNCTION FTM(NVTIM) (2************************************ C* FLOATING POINT TIME IN HRS. * r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * IT=MOD(NVTIM,10000) 327 FTM=IT/100+MOD(IT,100)/60.0 RETURN END SUBROUTINE SGEOM(NAVTIM,XPT,XLAPT,X1,X2,X3) REAL CLAT/49.217/,SATAZ/16.1/ FIRST COMPUTE DECLINATION, SOLAR ZENITH ANGLE, AND SOLAR AZIMUTH FOR THIS TIME. RADDEG=3.14159/180.0 CI =279.457*RADDEG C2=0.985647*RADDEG YR=NAVTIM/10000000 JD=MOD(NAVTIM,10000000)/10000 IF(XPT.EQ.O.O) GOTO 5 XLAPT=XPT GOTO 10 TI=FTM(NAVTIM) X=AINT((YR-65)*365.251)+JD+Tl/24.0 G=C1+C2*X X=X/365.2422 'EQ' IS EQUATION OF TIME VALUE EQ=(-102.5-0.142*X)*SIN(G)+(-429.8+.033*X)*COS(G)+596.5*S *-2.0*COS(2*G)+4.2*SIN(3*G)+l9.3*COS(3*G)-12.8*SIN(4*G) EQ=EQ/3600 'XLAPT' IS LOCAL APPARENT TIME IN HOURS (CONSTANTS VARY WITH CENTRAL LOCATION) XLAPT=TI-8.0+EQ-10.8/60.0 IF(XLAPT.LT.0.0) XLAPT=XLAPT+24.0 HA=15.0*(XLAPT-12.0)*RADDEG PSI=2*3.14159*(JD-1)/365.0 'DEC IS SOLAR DECLINATION DEC=0.006916-0.399912*COS(PSI)+0.0702 57*SIN(PSI) * -0.0067 58*COS(2*PSI)+0.000907*SIN(2*PSI)-0.002697*COS(3*Psi * +0.001480*SIN(3*PSI) 'COSZ' AND 'SINZ' ARE SIN AND COS OF SOLAR ZENITH ANGLE 'CA' IS COS OF SOLAR AZIMUTH COS Z = SIN(CLAT*RADDEG)*SIN(DEC)+COS(CLAT*RADDEG)*COS(DEC)* SINZ=SIN(ARCOS(COSZ)) CA=(SIN(CLAT*RADDEG)*COSZ"SIN(DEC))/(COS(CLAT*RADDEG)*SIN NOW COMPUTE THE 3 PARAMETERS OF INTEREST SIGN=1 IF(XLAPT.LT.12.0) SIGN=-1 328 SSA=ABS(SATAZ*RADDEG-SIGN*ARCOS(CA)) X1=C0SZ X2=SINZ*C0S(SSA) X3=X2*COS(SSA) RETURN END C C C FUNCTION ISTNUM(STAT) REAL*8 STAT,NAMES(12)/1GRSMT ','NRTHMT ','BCHYDRO ', * 'LANGA ','VANAIR ','FERRY ','PITMED ','MISSHAB ', * 'ABBLIB ','ABBAIR ','LANGLEY ','CLISTN '/ LOGICAL EQCMP DO 10 1=1,12 J = I IF(EQCMP(8,STAT,NAMES(I))) GOTO 15 IF(I.EQ.12) WRITE(6,100) 10 CONTINUE 1 5 ISTNUM=J RETURN 100 FORMAT(' HELLLP! .. . THERE IS AN UNKNOWN STATION SOMEWHERE') END C C C SUBROUTINE STATS(O,C,IS,PFLAG) Q******** ************************************ C* KEEP TRACK OF SUMS AND SQUARES AND PRODUCTS OF DATA * C* VALUES 'O' AND 'C GIVEN FOR STATION NUMBER 'IS'. * C* THESE SUMS ARE PRINTED ALONG WITH MBE AND RMSE FOR * C* ALL STATIONS WHEN CALLED WITH 'PFLAG'=.TRUE. (UNIT 9)* c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c REAL*8 SC(12)/12*0.0/, SO(12)/12*0.0/,SCO(12)/12*0.0/, 1 SC2(12)/12*0.0/,S02(12)/12*0.0/,STNAME,STN LOGICAL PFLAG INTEGER N(12)/l2*0/ C C IS IT PRINTOUT TIME? C IF(O.LT.O.O) RETURN IF(C.LT.O.O) C=0.0 IF(PFLAG) GOTO 20 SC(IS)=SC(IS)+C SO(lS)=SO(lS)+0 SC2(IS)=SC2(IS)+C*C S02(lS)=S02(lS)+0*0 SCO(IS)=SCO(lS)+0*C N(IS)=N(IS)+1 RETURN C C JUMP HERE TO PERFORM PRINTOUT ON UNIT 9 C 20 WRITE(2,100) DO 30 1=1,12 329 RMSE=DSQRT((SC2(I)-2*SC0(I)+S02(I))/N(I)) RMBE=(SC(I)-SO(l))/N(l) STN=STNAME(I) RMBEP=(SC(I)-SO(l))/SO(l )* 100.0 RMSEP=RMSE/(S0(I)/N(I))*100.0 R=(N(I)*SCO(I)-SC(l)*SO(l))/DSQRT((N(I)*SC2(I)-SC(I)*SC(I 1 *(N(I)*S02(I)-SO(l)*SO(l))) RSQ=R*R WRITE(2,101) STN,SC(I),SO(I),SC2(I),S02(I),SCO(l),N(I) 1 ,RMBE,RMSE,RMBEP,RMSEP,R,RSQ 30 CONTINUE RETURN 100 FORMAT(' 1 VERIFICATION ANALYSIS'/,'!',/ 1'STATION SC SO SC2 S02 SCO', 2' N MBE RMSE MBE% RMSE% R RSQ' 101 FORMAT(A8,5F10.0,1 5,3X,4F8.2,2F8.5) END C C C FUNCTION STNAME(IS) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C* RETURN THE NAME OF THE STATION WITH NUMBER 'IS' * Q***************************** ********************** c REAL*8 STNAME,NAMES(12)/'GRSMT ','NRTHMT ','BCHYDRO ', * 'LANGA ','VANAIR ','FERRY ','PITMED ','MISSHAB ', * 'ABBLIB ','ABBAIR ','LANGLEY ','CLISTN '/ STNAME=NAMES(IS) RETURN END C C SUBROUTINE CALIB(X1,XMEAN,XIR) C £************************************************* C*PERFORMS CALIBRATION OF SATELLITE DATA * (2************************************************* DIMENSION RTAB(256),ATAB(256) DATA PI/3.14159265/ DATA SSC/303.487/ DATA SC1,SC2/1389.1,1365.0/ DATA A, B/0.683,2.340/ DATA LDAY,LHMS/0,0/ IF(LDAY.EQ.9) GO TO 2 LDAY=9 SOLCO=SC2/SC1 XMAX=SSC/PI DO 1. 1 = 1 ,256 IC=I-1 IC=IFIX(((lC/4.0)**2/4064.6)*254.0+0.5) K=IFIX((IC-8)/A+0.5) FTLAM=40*K+20 XRAD=0.0067759*FTLAM IF(XRAD.LT.O.O) XRAD=0.0 IF(XRAD.GT.XMAX) XRAD=XMAX 330 RTAB(I)= PI*SOLCO*XRAD ATAB(I)=RTAB(I)/(SOLCO*SSC ) 1 CONTINUE C WRITE(6,7) ATAB 7 FORMAT(' ',12F7.2) 2 IMEAN=XMEAN+0.5 XIR = 0.0 IF(X1.GT.0.001)XIR=ATAB(IMEAN) / X1 C WRITE(6,5) XIR,XMEAN,X1 5 FORMAT(3F10.4) RETURN END Q****************************** SUBROUTINE PDATA(IDY) REAL K(12),OX(24),CY(24),OY(24) REAL*8 DAT 1,NAME INTEGER NS(12) COMMON/RDATA/DAT1(10,40,12),NIMG(12),DATA(12,24,2) Q********************************** ********************** C DATA STATEMENT SIGNIFIES PLOT REQUEST BY 'STATION IF* C "STN NUMBER" IS ZERO THEN NO PLOT.* Q* ********* * ******************************** DATA NS/1,2,3,4,5,6,7,8,9,10,11,12/ C C DO PLOT FOR EACH STATION C DO 80 11=1,12 C C CHECK IF PLOT REQUESTED FOR THIS STATION C IF(NS(II).EQ.0) GO TO 80 Q************ ************************************* ********* C SET UP X/Y AXES* Q****************************************** **************** CALL PLOT(10.0,2.0,3) CALL PLOT(2.0,2.0,2) CALL PLOT(2.0,6.0,2) C C PLACE TICKS AND NUMBERS ON Y AXIS C Y=6.0 DO 10 1=1,4 CALL PLOT(1.8,Y,2) RN=Y-2.0 CALL NUMBER(1.5,Y,0.2,RN,+90.0,- 1) Y=Y-1.0 CALL PLOT(2.0,Y,3) 10 CONTINUE *** **** X=2.0 DO 15 1=1,9 CALL PLOT(X,1.8,2) RN=(X-2.0)*3.0 CALL NUMBER(X,1.5,0.2,RN,0.0,-1) 331 X=X+1.0 CALL PLOT(X,2.0,3) 15 CONTINUE C LABEL AXES C CALL SYMBOL(4.0,0.8,0.25,'LOCAL APPARENT TIME',0.0,19) CALL SYMBOL(0.8,4.0,0.25,'IRRADIANCE',+90.0,10) C*********************************************************** C PLOT CODES FOR LINES* c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CALL PLOT(8.0,5.0,3) CALL PLOT (-10.0,5.0,2) CALL SYMBOL(8.0,5.0,0.10,11,0.0,-1) CALL SYMBOL(10.0,5.0,0. 10, 1 1 ,0.0,-1) CALL SYMBOL(10.5,5.0,0.25,'OBSERVED',0.0,8) CALL PLOT(8.0,4.5,3) CALL DASHLN(0.2,0.2,0.2,0.2) CALL PLOT(10.0,4.5,4) CALL SYMBOL(8.0,4.5,0.10,12,0.0,-1) CALL SYMBOL(10.0,4.5,0.10,12,0.0,-1) CALL SYMBOL(10.5,4.5,0.25,'CALCULATED',0.0,10) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C PLACE DATA IN PLOTTING ARRAYS.* r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DO 18 1=1,24 IF(DATA(II,I,1).GT.0.0)OY(I)=DATA(II,I,l)/l000.0+2.0 OX(I)=(I)/3.0+2.0 IF(DATA(II,1,2).GT.0.0)CY(I)=DATA(II,1,2)/1000.0+2.0 18 CONTINUE c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C PLOT STATION NAME AND DAY.* c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DAY=FLOAT(IDY) NAME=DAT1(2,1,11) CALL SYMBOL(8.0,4.0,0.25,NAME,0.0,8) CALL NUMBER(8.0,3.5,0.25,DAY,0.0,-1) Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C PLOT OBSERVED DATA.* r ; * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CALL PLOT(2.0,2.0,3) IP = 4 DO 19 1=1,24 IF(OY(I).LT.0.0) GO TO 195 CALL PLOT(OX(l),OY(l),IP) CALL SYMBOL(OX(I),OY(I),0.10,11 ,0.0,- 1) IP = 2 GO TO 19 195 IP = 3 19 CONTINUE C************************************************************* C PLOT CALCULATED DATA.* C************************************************************* CALL PLOT(2.0,2.0,3) IP = 4 DO 20 1=1,24 332 IF(CY(I).LT.O.O) GO TO 30 CALL PLOT(OX(I),CY(I),IP) CALL SYMBOL(OX(I),CY(I),0.10,12,0.0,-1) IP = 4 GO TO 20 30 IP = 3 20 CONTINUE C RESET ORIGIN r * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * x x CALL PLOT(12.0,0.0,-3) C RETURN FOR NEXT STATION.* Q******************* ********************************* 80 CONTINUE Q*************************************** ************* C READY FOR END OF RUN Q***************************************** *********** CALL PLOTND RETURN END C c 

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