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Development of a nighttime cooling model for remote sensing thermal inertia mapping Leckie, Donald Gordon 1980

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DEVELOPMENT OF A NIGHTTIME COOLING MODEL FOR REMOTE SENSING THERMAL INERTIA MAPPING DONALD GORDON LECKIE B.Sc. ( G e o l o g i c a l E n g i n e e r i n g ) U n i v e r s i t y of Manitoba, 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of S o i l S c i e n c e - Remote Sensing) We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August 1980 © Donald Gordon L e c k i e , 1980 by DOCTOR OF PHILOSOPHY i n In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of S o i l Science The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date September 9, 1980. - i i -ABSTRACT The c a p a b i l i t i e s of remote sensing can be u t i l i z e d to map the thermal i n e r t i a of a s u r f a c e . Thermal i n e r t i a i s a property governing the temperature response of a medium to a heat f l u x at i t s s u r f a c e and i s b e n e f i c i a l to g e o l o g i c mapping and s o i l s stud i e s . I t i s hypothesized that a method using only nighttime c o o l i n g w i l l g i ve a simple thermal i n e r t i a model r e q u i r i n g a minimum of input. Albedo and topographic slope and aspect data are not r e q u i r e d . Since l a t e n t heat f l u x i s commonly small at n i g h t the model should be a p p l i c a b l e over s u r f a c e s of v a r y i n g moisture content. The o b j e c t i v e o f t h i s t h e s i s i s to develop a nighttime c o o l i n g model for remote sensing thermal i n e r t i a mapping. Three models (I, I I , and III) are presented. They are based on s o l u t i o n s to the one-dimensional heat conduction equation f o r a s e m i - i n f i n i t e homogeneous s o l i d with i s o t h e r m a l i n i t i a l temperature and time dependent boundary c o n d i t i o n s of heat f l u x at the s u r f a c e . T e s t s of the models on s e v e r a l s o i l types using ground based data i n d i c a t e that a l l three models g i v e meaningful r e l a t i v e r e l a t i o n s h i p s between thermal i n e r t i a s and that model I I I o f t e n y i e l d s accurate q u a n t i t a t i v e r e s u l t s . For the remote sensing implementation of the model ground heat f l u x i s determined as the r e s i d u a l of the energy balance of the s u r f a c e . Thus, a procedure f o r determining net r a d i a t i o n using remotely sensed temperature i s d i s c u s s e d . A l s o , aerodynamic heat t r a n s f e r theory i s used to develop a remote sensing method of e s t i m a t i n g s e n s i b l e heat f l u x . C o r r e c t i o n s f o r the s u r f a c e sublayer are necessary. R e s u l t s f o r vegetated s u r f a c e s are expected to be u n r e l i a b l e . L a t e n t heat f l u x i s assumed to be zero or the average of s e v e r a l s i t e s . T e s t s of these methods using ground based data g i v e good r e s u l t s . An e r r o r a n a l y s i s approach i s used t o e s t i m a t e the e r r o r s r e s u l t i n g from a remote sensing implementation of Model I I I . A i r b o r n e thermal l i n e - s c a n data and ground based m i c r o m e t e o r o l o g i c a l o b s e r v a t i o n s are used to determine t y p i c a l e r r o r s i n the input parameters of the model. E r r o r s i n determining the energy balance components are a l s o analyzed i n d e t a i l . With good i n p u t , model I I I g i v e s r e a s o n a b l e r e s u l t s ( g e n e r a l l y l e s s than 50 pe r c e n t probable e r r o r ) at low thermal -2 -1 -1/2 i n e r t i a s (< 2000 J m C s ). For s u r f a c e s of high thermal i n e r t i a , e r r o r s are l a r g e . The l i m i t a t i o n of the model i s not i n the model i t s e l f , but i n the accuracy of remotely sensed s u r f a c e temperature as determined from thermal i n f r a r e d l i n e - s c a n s u r v e y s . For s u r f a c e s of low thermal i n e r t i a model I I I p r o v i d e s a simple thermal i n e r t i a mapping method which r e q u i r e s a minimum of input and i s a p p l i c a b l e over a wide v a r i e t y of t e r r a i n and ground moisture c o n d i t i o n s . The model i s most s u i t a b l e f o r the i n v e s t i g a t i o n o f s o i l s and may p r o v i d e a u s e f u l model f o r p l a n e t a r y s t u d i e s . - i v -TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i LIST OF APPENDICES ix LIST OF APPENDICES TABLES x i LIST OF APPENDICES FIGURES x i i ACKNOWLEDGEMENTS xiv CHAPTER ONE INTRODUCTION 1 1.1 General Objectives, Approach, and Procedure 1 1.2 D e f i n i t i o n of Thermal Inertia 3 1.3 Benefits of Thermal Inertia Mapping 5 1.4 Review of Existing Models 18 1.5 C r i t e r i a for a Thermal Inertia Model 28 1.6 Objectives 30 CHAPTER TWO NIGHTTIME COOLING MODELS 33 2.1 Introduction 33 2.2 Theory: Solutions of Heat Conduction Equation Applicable to a Nighttime 34 Remote Sensing Thermal Inertia Model (Models I, II, and III) 2.3 Theory: Heat Sharing Approach (Model I) 40 2.4 Application of Models to Real Boundary 44 Conditions 2.5 Testing of Models 54 2.6 Conclusions 60 -v-CHAPTER THREE ESTIMATING ENERGY BALANCE COMPONENTS 3.1 I n t r o d u c t i o n 3.2 Determination of Net R a d i a t i o n 3.3 Determination of S e n s i b l e Heat Flux 3.4 Determination of Latent Heat Flux 3.5 Con c l u s i o n s 63 63 64 65 83 87 CHAPTER FOUR ERROR ANALYSIS 4.1 I n t r o d u c t i o n 4.2 E r r o r i n Net R a d i a t i o n E s t i m a t i o n 4.3 E r r o r i n S e n s i b l e Heat Flux E s t i m a t i o n 4.4 E r r o r i n Latent Heat Flux E s t i m a t i o n 4.5 E r r o r A n a l y s i s of T o t a l E r r o r s 4.6 Co n c l u s i o n s 88 88 89 92 126 127 136 CHAPTER FIVE LITERATURE CITED APPENDICES Appendix I Appendix II Appendix I I I Appendix IV Appendix V CONCLUSIONS D e r i v a t i o n of Equation (2.9) Thermal I n f r a r e d Line-Scan Data f o r Q u a n t i t a t i v e S t u d i e s : An E r r o r A n a l y s i s Use of Polynomial Transformations f o r R e g i s t r a t i o n of A i r b o r n e D i g i t a l Line-Scan Images S p a t i a l V a r i a b i l i t y of A i r Temperature and Wind Speed, A i r Ponding, and Short Term L o c a l Advection : D i s c u s s i o n and Examples R e s u l t s of E r r o r A n a l y s i s of Thermal I n e r t i a Method 139 149 158 162 198 226 244 - v i -LIST OF TABLES Table 1.1 Thermal i n e r t i a of rock types selected from 9 l i t e r a t u r e . Table 1.2 Thermal i n e r t i a of the major s o i l constituents. 13 Table 2.1 Thermal i n e r t i a of a i r . 42 Table 2.2 Average errors in using a linear approximation 51 and using data of sampling times. Table 2.3 Errors due to non-alignment of G curves with 53 sampling times. Table 4.1 Surface roughness lengths for natural non-vegetated surfaces. 97 Table 4.2 Error analysis of SAT method for determining sensible heat flux (H) (wind speed (U) = 2.0 109 m s - 1 ; z Q = 1.0 mm). Table 4.3 Error analysis of SAT method for determining sensible heat flux (H) (wind speed (U) = 3.0 110 m s" 1; z Q = 1.0 mm). Table 4.4 Error analysis of SAT method for determining sensible heat flux (H) (wind speed (U) = 5.0 111 m s ~ l ; z Q = 1.0 mm). Table 4.5 Parameter values and d e f i n i t i o n s for error analysis of SAT method for determining 112 sensible heat flux (H). Table 4.6 Summary of error analysis of SAT method for 113 determining sensible heat flux (H). Table 4.7 The eff e c t of short term l o c a l advection on determination of sensible heat flux (H) using the SAT method. 123 - V I 1 -LIST OF FIGURES F i g u r e 1.1 E f f e c t of p o r o s i t y on thermal i n e r t i a of an 10 "average" rock. F i g u r e 1.2 Thermal i n e r t i a versus p o r o s i t y and percent v o l u m e t r i c water content f o r d i f f e r i n g s o i l 14 c ompositions. F i g u r e 2.1 T y p i c a l n i g h t t i m e course of a c t u a l ground heat f l u x G ( t ) . A l s o i n c l u d e d are the bound- 46 ary c o n d i t i o n s of models I, I I , and I I I . F i g u r e 2.2 Test of models II and I I I f o r a bare peat 55 s o i l . (Surrey d a t a ) . F i g u r e 2.3 Test of model I I I f o r a bare s i l t loam s o i l 57 (Agassiz d a t a ) . F i g u r e 2.4 Thermal i n e r t i a estimates using model I I I 59 f o r rahgeland s i t e s . F i g u r e 3.1 Test of SAT method on a bare s o i l s u r f a c e . 75 F i g u r e 3.2 T y p i c a l temperature p r o f i l e shapes i n a 77 vegetated canopy. F i g u r e 3.3 Test of SAT method on a vegetated s u r f a c e . 79-80 F i g u r e 4.1 S e n s i b l e heat f l u x versus s u r f a c e roughness 94 l e n g t h f o r SAT method. F i g u r e 4.2 Sublayer c o r r e c t i o n f a c t o r (Q) versus s u r f a c e 95 roughness l e n g t h . F i g u r e 4.3 Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s of area A (29/7/78; 99 2300 h) . - v i i i -F i g u r e 4.4 Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s of area A showing a i r 101 ponding (30/7/78; 0400 h ) . F i g u r e 4.5 S e n s i b l e heat f l u x versus a i r - s u r f a c e temp-e r a t u r e d i f f e r e n c e f o r SAT method. F i g u r e 4.6 S e n s i b l e heat f l u x versus wind speed f o r SAT method. F i g u r e 4.7 Sublayer c o r r e c t i o n f a c t o r versus B-^" where B i s the sublayer Stanton number. 103 105 107 F i g u r e 4.8 E r r o r i n s e n s i b l e heat f l u x f o r SAT method 116 versus s u r f a c e roughness l e n g t h . F i g u r e 4.9 E r r o r i n s e n s i b l e heat f l u x f o r SAT method 117 versus a i r - s u r f a c e temperature d i f f e r e n c e . F i g u r e 4.10 E r r o r i n s e n s i b l e heat f l u x f o r SAT method 118 versus wind speed. F i g u r e 4.11 T y p i c a l n o c t u r n a l a i r temperature p r o f i l e with 120 c o l d l a y e r . F i g u r e 4.12 R e s u l t s of e r r o r a n a l y s i s of t o t a l e r r o r s f o r t y p i c a l and good cases under c o n d i t i o n s of d i f f e r e n t ground heat f l u x e s . 132 - i x -LIST OF APPENDICES APPENDIX I DERIVATION OF EQUATION (2.9) 158 APPENDIX II THERMAL INFRARED LINE-SCAN DATA FOR QUANTITATIVE STUDIES: AN 162 ERROR ANALYSIS 1. I n t r o d u c t i o n 162 2. C a l i b r a t i o n R e l a t i o n E r r o r s 164 3. System (Instrument E r r o r s ) 169 4. Atmospheric E r r o r s 176 5. E m i s s i v i t y E r r o r s 179 6. Summary of T o t a l E r r o r s 181 7. C o n c l u s i o n s 194 L i t e r a t u r e C i t e d 196 APPENDIX I I I USE OF POLYNOMIAL TRANSFORMATIONS FOR REGISTRATION OF AIRBORNE DIGITAL LINE- 198 SCAN IMAGES 1. I n t r o d u c t i o n 198 2. Problems of A i r b o r n e Image R e g i s t r a t i o n 199 3. Methods and Procedures 201 4. R e s u l t s 213 5. D i s c u s s i o n and C o n c l u s i o n s 221 L i t e r a t u r e C i t e d 225 -x-APPENDIX IV SPATIAL VARIABILITY OF AIR TEMPERATURE AND WIND SPEED, AIR PONDING, AND SHORT 226 TERM LOCAL ADVECTION : DISCUSSION AND EXAMPLES 1. I n t r o d u c t i o n 226 2. Probable E r r o r i n A i r Temperature Under 227 Ponding C o n d i t i o n s 3. E r r o r s i n A i r Temperature Under Ponding 235 C o n d i t i o n s 4. E r r o r s i n Wind Speed 237 5. Short Term L o c a l Advection 238 6 . S i t e D e s c r i p t i o n 241 L i t e r a t u r e C i t e d 243 APPENDIX V RESULTS OF ERROR ANALYSIS OF THERMAL INERTIA METHOD 244 -x i -LIST OF APPENDICES TABLES T a b l e I I . 1 A n a l y s i s o f p i x e l and blackbody e r r o r f o r seven f l i g h t s . T a b l e I I . 2 E r r o r i n d e t e r m i n i n g a t m o s p h e r i c o f f s e t . T a b l e I I . 3 Summary of e r r o r s ; T r e l a t i o n day c a s e . T a b l e I I . 4 Summary of e r r o r s ; T^ r e l a t i o n day c a s e . T a b l e I I . 5 Summary o f e r r o r s ; T 4 r e l a t i o n n i g h t c a s e . T a b l e I I . f i Summary of e r r o r s ; T r e l a t i o n n i g h t c a s e . T a b l e I I . 7 I n o u t , c o n d i t i o n s , and d e f i n i t i o n s f o r T a b l e s I I . 3 t o I I . 6 . T a b l e I I I . l P o l y n o m i a l t r a n s f o r m s a v a i l a b l e . T a b l e I I I . 2 P o l y n o m i a l t r a n s f o r m s . T a b l e I I I . 3 A c c u r a c y of p o l y n o m i a l t r a n s f o r m s . T a b l e I I I . 4 E f f e c t o f number of RCP's on a c c u r a c y o f p o l y n o m i a l t r a n s f o r m s . T a b l e IV.1 Time-averaged a i r temperature o f s i t e s and d i f f e r e n c e s between the ti m e - a v e r a g e d t e m p e r a t u r e s o f the s i t e s . T a b l e IV.2 Examples o f the d i f f e r e n c e i n a i r temp-e r a t u r e (at 1.5 m) between s i t e s n o t s u s c e p t i b l e and s i t e s s u s c e p t i b l e t o pon d i n g . - x i i -LIST OF APPENDICES FIGURES F i g u r e 1.1 Time dependent boundary c o n d i t i o n of 160 f l u x G ( t ) . F i g u r e II.1 a) E f f e c t i v e energy for a Ge:Hg d e t e c t o r as a f u n c t i o n of s u r f a c e temperature. A l s o 166 given are the T 4 r e l a t i o n for the day case and T r e l a t i o n f o r the night case. b) Change i n e f f e c t i v e energy caused by a one degree temperature change fo r the 166 Ge:Hg d e t e c t o r . F i g u r e II.2 a) E f f e c t i v e energy f o r an InSb d e t e c t o r as a f u n c t i o n of s u r f a c e temperature. A l s o g i v e n are the T r e l a t i o n f o r the day and 167 n i g h t case and T 4 r e l a t i o n f o r the day case. b) Change i n e f f e c t i v e energy caused by a one degree temperature change fo r the InSb 167 d e t e c t o r . F i g u r e II.3 D i f f e r e n c e i n temperature between that d e r i v e d from the T 4 r e l a t i o n and from 170 the T r e l a t i o n . F i g u r e II.4 Temperature e r r o r due to assuming an erroneous e m i s s i v i t y . F i g u r e I I I . l a) Test image (re f e r e n c e image) 182 204 b) Test image (re f e r e n c e and s l a v e image 205 o v e r l a i d and s u b t r a c t e d ) . F i g u r e III.2 a) Procedure f o r s e l e c t i o n of p o l y -nomial transforms. D e l e t i o n of terms 210 from the 29-term p o l y n o m i a l . b) Procedure f o r s e l e c t i o n of p o l y -nomial transforms. A d d i t i o n of terms 211 to the 3-term p o l y n o m i a l . - x i i i -F i g u r e I I I . 3 RMS TCP and RCP e r r o r versus length of polynomial (Polynomials i to x i ; 58 RCP's). F i g u r e III.4 RMS TCP e r r o r versus number of RCP's (Polynomials Y - i i i , Y - i v , and Y - x i i ; 54 TCP* s ) . F i g u r e I I I . 5 RMS TCP e r r o r versus number of RCP's (Polynomials X - i i i , X - i v , and X - x i i ; 54 TCP's). F i g u r e IV.1 Time p l o t s speed (at (02/08/78; F i g u r e IV.2 Time p l o t s speed (at (03/08/78; F i g u r e IV.3 Time p l o t s speed (at (31/08/78; F i g u r e IV.4 F i g u r e IV.5 Time p l o t s speed (at (19/06/78; of a i r temperature and wind 1.5 m) f o r s i t e s of area B 2300 h). of a i r temperature and wind 1.5 m) f o r s i t e s of area B 0400 h). of a i r temperature and wind 1.5 m) f o r s i t e s of area C 2400 h). of a i r temperature and wind 1.5 m) f o r s i t e s of area C 2300 h ) . Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s of area C (01/08/78; 0400 h ) . F i g u r e IV.6 Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s of area C (20/06/78; 0500 h). -xiv-ACKNOWLEDGEMENTS This research was funded by a Natural, Applied, and Health Sciences grant from the University of B r i t i s h Columbia to Dr. P.A. Murtha. Thermal infrared line-scan f l i g h t data ac q u i s i t i o n was p a r t i a l l y funded by a grant provided to Dr. Murtha by Agriculture Canada. The Dept. of Graduate Studies provided greatly appreciated f i n a n c i a l support to the author through University of B.C. Graduate and Leonard S. Klinck Fellowships. I would l i k e to thank Dr. Peter Murtha for his encouragement throughout the study and for providing a home for my interest in remote sensing. Appreciation i s extended to the other members of my committee and Dr. A.K. Mackworth for their technical advice. A special thanks to Dr. T.A. Black without whose assistance this thesis would not be possible and to Dr. R.J. Woodham for providing many helpful suggestions p a r t i c u l a r l y on the image re g i s t r a t i o n problem. Numerous people assisted in the c o l l e c t i o n of f i e l d data. I thank the Agriculture Canada research station at Kamloops, B.C. and in pa r t i c u l a r Dr. A.L. van Ryswyck for their cooperation. Thanks also go to fellow graduate student E.K. Watson and UBC remote sensing technician N.M. Holm for aiding in the f i e l d work. Ms. Holm's drafting was also greatly appreciated. Data to a s s i s t in thi s project were generously made available by M.D. Novak of the Dept. of S o i l Science, University of B.C. and R.J. Williams of the B.C. Ministry of Environment. I wish to thank MacDonald, Dettwiler and Associates Ltd. of -xv-Richmond, B r i t i s h Columbia, f o r p r o v i d i n g generous use of t h e i r computer f a c i l i t i e s , and K.L. Brydges, Dr. R.K. Orth, and R.A. Deane of M.D.A. for t h e i r a s s i s t a n c e with the image r e g i s t r a t i o n study. S p e c i a l thanks and c r e d i t to my wife Marion f o r her encouragement, her word p r o c e s s i n g , and long hours of a s s i s t a n c e throughout the t h e s i s . s -1-Chapter One INTRODUCTION 1.1 GENERAL OBJECTIVES, APPROACH, AND PROCEDURE Thermal i n e r t i a i s a p r o p e r t y of a medium. Knowledge of the thermal i n e r t i a of a medium at the e a r t h ' s s u r f a c e provides i n f o r m a t i o n on the nature of the s u r f a c e which can be b e n e f i c i a l i n the u t i l i z a t i o n of the e a r t h ' s r e s o u r c e s . Remote sensing may be d e f i n e d as the a c q u i s i t i o n of data p e r t a i n i n g to a s u r f a c e or medium by a sensor at some d i s t a n c e from that s u r f a c e and the p r o c e s s i n g and i n t e r p r e t a t i o n of those data i n such a way as to g i v e u s e f u l i n f o r m a t i o n on the nature of that s u r f a c e . The power of remote sensing l i e s i n i t s c a p a b i l i t y to g a i n i n f o r m a t i o n not o b t a i n a b l e by other means or to g a i n i n f o r m a t i o n o b t a i n a b l e by other means but i n a more e f f i c i e n t manner and over l a r g e r s u r f a c e areas. I t i s the g e n e r a l o b j e c t i v e of t h i s study to develop a thermal i n e r t i a model which u t i l i z e s the c a p a b i l i t y of remote sensing to gain u s e f u l i n f o r m a t i o n i n an e f f i c i e n t manner over a l a r g e s u r f a c e area. To f u l f i l l t h i s o b j e c t i v e the model must be simple i n theory and implementation, r e q u i r e a minimum of input, be a p p l i c a b l e over l a r g e s u r f a c e areas, and be v a l i d f o r a wide range of s u r f a c e s , t e r r a i n , and c l i m a t i c c o n d i t i o n s . An a n a l y t i c a l approach d e t a i l i n g the t h e o r e t i c a l and p r a c t i c a l implementation aspects of the method w i l l be used rather than the experimental or " t r y and see" approach o f t e n used -2-i n r e m o t e s e n s i n g s t u d i e s . I n d e v e l o p i n g a new m o d e l t h i s a p p r o a c h i s a d v a n t a g e o u s a s i t s u g g e s t s m o d i f i c a t i o n s t o t h e m o d e l a n d p r o v i d e s a c l e a r p i c t u r e o f t h e t h e o r e t i c a l f o u n d a t i o n s o f t h e m o d e l , t h e m a g n i t u d e a n d c a u s e o f e r r o r s , a n d t h e u l t i m a t e u s e f u l n e s s o f t h e m o d e l . T h e p r o c e d u r e a d o p t e d w i l l now be o u t l i n e d . I n t h e r e m a i n d e r o f t h i s c h a p t e r ( C h a p t e r 1) t h e r m a l i n e r t i a i s d e f i n e d . T h e b e n e f i t s o f t h e r m a l i n e r t i a m a p p i n g a r e d e m o n s t r a t e d a n d c u r r e n t t e c h n i q u e s r e v i e w e d . C r i t e r i a f o r a t h e r m a l i n e r t i a m o d e l a r e e s t a b l i s h e d . C u r r e n t t e c h n i q u e s d o n o t e n t i r e l y m e e t t h e c r i t e r i a . I t i s h y p o t h e s i z e d t h a t a n i g h t t i m e c o o l i n g m o d e l m e e t s t h e c r i t e r i a . T h e d e v e l o p m e n t o f a n i g h t t i m e c o o l i n g m o d e l f o r r e m o t e s e n s i n g t h e r m a l i n e r t i a m a p p i n g w h i c h w i l l m e e t t h e c r i t e r i a i s p r e s e n t e d a s t h e o b j e c t i v e o f t h i s t h e s i s . S p e c i f i c p r o b l e m s a n d o b j e c t i v e s a r e d e f i n e d a n d t h e c o n t e n t s o f t h e t h e s i s a r e o u t l i n e d i n t e r m s o f t h e s e p r o b l e m s a n d o b j e c t i v e s . C h a p t e r 2 i n v e s t i g a t e s p o s s i b l e n i g h t t i m e c o o l i n g m o d e l s a p p l i c a b l e t o r e m o t e s e n s i n g t h e r m a l i n e r t i a m a p p i n g . T h r e e m o d e l s a r e s e l e c t e d a n d t e s t e d w i t h g r o u n d b a s e d d a t a . One m o d e l i s c h o s e n f o r f u r t h e r a n a l y s i s . C h a p t e r 3 d i s c u s s e s m e t h o d s o f d e t e r m i n i n g t h e n e c e s s a r y i n p u t t o t h e m o d e l ( i . e . , a t e c h n i q u e f o r e s t i m a t i n g n e t r a d i a t i o n u s i n g r e m o t e l y s e n s e d s u r f a c e t e m p e r a t u r e , a m e t h o d o f e s t i m a t i n g s e n s i b l e h e a t f l u x i n c o r p o r a t i n g s t a b i l i t y a n d s u b l a y e r c o r r e c t i o n s , a n d p o s s i b l e p r o c e d u r e s o f a p p r o x i m a t i n g l a t e n t h e a t f l u x ) . T h e m e t h o d s a r e a p p l i c a b l e t o r e m o t e s e n s i n g s u r v e y s . T h e e r r o r s i n i m p l e m e n t i n g t h e m o d e l i n a r e m o t e s e n s i n g mode a r e i n v e s t i g a t e d i n C h a p t e r 4. A n e r r o r a n a l y s i s a p p r o a c h i s u s e d . S u p p o r t i n g a n a l y s i s i s -3-obtained from an e r r o r a n a l y s i s of thermal i n f r a r e d l i n e - s c a n data for q u a n t i t a t i v e s t u d i e s presented i n Appendix I I , a study of d i g i t a l a i r b o r n e image r e g i s t r a t i o n by polynomial transform techniques given i n Appendix I I I , and f i e l d experiments i n t o the s p a t i a l and temporal v a r i a b i l i t y o f m i c r o m e t e o r o l o g i c a l parameters d e s c r i b e d i n Appendix IV. Chapter 5 summarizes the procedures and best s u r f a c e and m e t e o r o l o g i c a l c o n d i t i o n s for implementing the nighttime c o o l i n g model developed. A l s o summarized are the l i m i t a t i o n s and p o t e n t i a l of the model. 1.2 DEFINITION OF THERMAL INERTIA Thermal i n e r t i a i s not a fundamental thermal p r o p e r t y as d e f i n e d by heat conduction theory but i s a parameter which has been adopted as i t o f t e n a r i s e s i n s o l u t i o n s of the heat conduction equation. The one-dimensional heat conduction equation i s : where T i s the temperature (K), t i s time ( s ) , z i s d i s t a n c e (m), A i s thermal c o n d u c t i v i t y (W m ^ ) , and C i s the heat c a p a c i t y (J m 3K . T h i s parameter (thermal i n e r t i a ) i s ( p c A ) " ^ 2 or 1/2 -3 (CX) where p i s d e n s i t y (kg m ) and c i s s p e c i f i c heat (J kg "^C *) (pc = C the heat c a p a c i t y ) . The u n i t s o f thermal -2 -1 -1/2 -2 -1 -1/2 i n e r t i a are J m K s o r J m C s . For convenience the u n i t s of thermal i n e r t i a w i l l be a b b r e v i a t e d by TIU (thermal -4-i n e r t i a u n i t s ) . T h i s a b b r e v i a t i o n has been adopted by M i l l e r and Watson (1977) and P r a t t (1980). Although the heat conduction equation has been solved for 1 / 2 numerous cases f o r many years, the parameter (pcX) did not r e c e i v e r e c o g n i t i o n as anything other than a combination of other p r o p e r t i e s u n t i l the 1950's when i t became u s e f u l to workers i n micrometeorology and s o i l s c i e n c e . During t h i s p e r i o d the s t a t u s of the parameter was e l e v a t e d by f o r m a l l y a s s i g n i n g a name to i t . I t was r e f e r r e d to as "co n t a c t c o e f f i c i e n t " (Businger and Buettner, 1961), "conductive c a p a c i t y " ( P r i e s t l e y , 1959), " s o i l product" ( H a l t i n e r and M a r t i n , 1957), and v a r i o u s other terms. 1/2 Using remote sensing techniques (pcX) may be determined. Remote sensing methods of determining e i t h e r heat c a p a c i t y or 1/2 thermal c o n d u c t i v i t y are not a v a i l a b l e . The parameter (pcX) was t h e r e f o r e given the s t a t u s o f a p r o p e r t y of a s u r f a c e or medium and named "thermal i n e r t i a " . The symbol P has been adopted f o r the p r o p e r t y , thermal i n e r t i a , by s e v e r a l remote sensing r e s e a r c h e r s (Watson et a l . , 1971; Kahle, 1977; P r i c e , 1977; P r a t t and E l l y e t t , 1979) and w i l l be used i n t h i s study. For remote sensing purposes thermal i n e r t i a may be thought of as the r e s i s t a n c e to change i n temperature of a medium or s u r f a c e . I t i s a p r o p e r t y governing the temperature response of a medium to an energy f l u x at i t s s u r f a c e . I f two media with equal and is o t h e r m a l i n i t i a l temperature d i s t r i b u t i o n s are subj e c t e d to the same time-varying energy f l u x at t h e i r s u r f a c e s the s u r f a c e of the medium of high thermal i n e r t i a w i l l undergo l e s s temperature change with time than the one of lower thermal -5-i n e r t i a . For the case of a p e r i o d i c a l l y heated homogeneous semi-i n f i n i t e s o l i d the amplitude of the temperature response i s i n v e r s e l y p r o p o r t i o n a l to thermal i n e r t i a . 1.3 BENEFITS OF THERMAL INERTIA MAPPING Although thermal i n e r t i a i s not g e n e r a l l y a pro p e r t y of d i r e c t i n t e r e s t , i t i s a c h a r a c t e r i s t i c p r o p e r t y of a medium and i t i s r e l a t e d to other p r o p e r t i e s which are of primary importance i n s e v e r a l f i e l d s of study. There has been much d i s c u s s i o n of the b e n e f i t s of thermal i n e r t i a a n a l y s i s . S e v e r a l thermal i n e r t i a models have been developed and implemented, but few have been thoroughly t e s t e d f o r s p e c i f i c a p p l i c a t i o n s and b e n e f i t s . A) P l a n e t a r y S t u d i e s The f i r s t remote sensing thermal i n e r t i a model developed and implemented was a p p l i e d to the s u r f a c e temperature v a r i a t i o n o f the s u r f a c e of the moon durin g a l u n a t i o n (Wesselink, 1948; Jaeger, 1953; S i n t o n , 1962). I t y i e l d e d much b e n e f i t . The s u r f a c e of the moon was determined to be a f i n e dust or dust over rock, a f a c t o r important when i t came to l a n d i n g a s p a c e c r a f t on the moon. Thermal i n e r t i a models f o r p l a n e t s and p l a n e t a r y s a t e l l i t e s with very l i t t l e or no atmosphere may be expected to be simple and a c c u r a t e . Remote sensing w i l l p l a y a major r o l e i n -6-increasing man's knowledge of the planets and their resources. Applications to planetary geology may indeed hold the greatest benefits for thermal i n e r t i a analysis in the future. B) Geology The major area of study of thermal i n e r t i a in recent years has been that of geological mapping. Several models have been developed and applied to geologic problems (Watson et a l . , 1971; Lyon, 1976; M i l l e r and Watson, 1977; Kahle, 1977). Thermal i n e r t i a analysis in geologic mapping i s a tool to augment f i e l d mapping and photo interpretation as thermal i n e r t i a i s a property of a medium which cannot be mapped by other means. Its major attribute is that i t has depth penetration and i s not greatly influenced by thin weathering surfaces and coatings or by lichen cover which can obscure c h a r a c t e r i s t i c reflectance patterns. It is predicted that a 0.01 cm lichen cover (Watson, 1973), a 0.1 cm coating of hematite (Miller and Watson, 1977), and a 0.1 cm s o i l layer over rock (Watson, 1973) w i l l have a n e g l i g i b l e e f f e c t on the temperature va r i a t i o n over a day and therefore w i l l have no influence on a diurnal thermal i n e r t i a model (one which uses the amplitude of the diurnal temperature wave). A 10 cm cover of either s o i l or lichen obscures the presence of the underlying rock altogether (Watson, 1973). Rose and Thomas (1968), using a model of van Duin (1954) 1 {Chapter Notes are l i s t e d at end of each chapter}, analyzed a two layered sandy s o i l with layers of d i f f e r e n t porosity and water content (the top dry -7-layer had a thermal i n e r t i a of 960 TIU; the underlying moist layer a thermal i n e r t i a of 1620 TIU). When the top layer was 10 cm thick the lower layer had no ef f e c t on the temperature amplitude . For an upper layer 1.0 cm thick, the r a t i o of the diurnal temperature amplitude for the layered s o i l to that of the moist lower s o i l with no upper layer was approximately 1.1. However, Pratt (1980) indicated that a 1.0 cm thick layer of a dry s o i l (500 TIU) over rock or moist s o i l had a large e f f e c t on the temperature amplitude and estimation of thermal i n e r t i a . He also stated that a 6 cm layer of the dry s o i l w i l l obscure the presence of the underlying surface. Byrne and Davis (1980) also discussed the ef f e c t of layers on thermal i n e r t i a . It may be concluded as Pohn et a l . (1974) did, "when thermal i n e r t i a i s determined from the diurnal temperature change i t i s averaged mainly over the top few centimeters" and also that "depths greater than about 10 cm do not influence the r e s u l t s " . In practice greater depths may, in some cases, be expected to have an influence in terms of moisture supply for the upper layers. Kahle et a l . (1975) demonstrated the benefits of the depth penetration of thermal i n e r t i a analysis by examples from a thermal i n e r t i a image (a d i g i t a l image showing the s p a t i a l d i s t r i b u t i o n of remote sensing thermal i n e r t i a determinations) created for the Pisgah Crater - Lavic Lake area of southern C a l i f o r n i a . The te r r a i n was basalt, cinder cone, and playa (desert lacustrine) material. The image showed that basalt could be detected through a thin layer of sand and that an outwash of ba s a l t i c material a few centimeters thick over a playa gave thermal i n e r t i a values near that of playa or between those of - 8 -p l a y a and b a s a l t . Table 1.1 l i s t s the thermal i n e r t i a of s e v e r a l important rock types ( l a b o r a t o r y or c a l c u l a t e d v a l u e s ) . The problem of rock type i d e n t i f i c a t i o n i s c l e a r by the c l o s e n e s s and o v e r l a p of the thermal i n e r t i a of many of the rock types. The thermal i n e r t i a of rock types i s g e n e r a l l y between 2000 and 4000 TIU and shows an i n c r e a s e with d e n s i t y ( M i l l e r and Watson, 1977). Histograms given by M i l l e r and Watson (1977) demonstrated the thermal i n e r t i a s and v a r i a t i o n s i n thermal i n e r t i a f o r s e v e r a l sedimentary rock types. S i g n i f i c a n t f e a t u r e s are high thermal i n e r t i a of q u a r t z i t e s and a l a r g e d i f f e r e n c e i n thermal i n e r t i a between limestones and d o l o m i t e s . Thermal i n e r t i a changes with p o r o s i t y and water content (Watson et al., 1971; Q u i e l , 1975). F i g u r e 1.1 shows the change i n thermal i n e r t i a of a t y p i c a l rock f o r both a i r and water s a t u r a t e d c o n d i t i o n s . Thermal i n e r t i a data given by Lyon (1975) i n d i c a t e d t hat d i f f e r e n t map u n i t s of the same rock type and f a c i e s w i t h i n a map u n i t o f t e n have d i f f e r e n t thermal i n e r t i a s . I t i s apparent that thermal i n e r t i a a n a l y s i s w i l l a i d i n d i s c r i m i n a t i n g c e r t a i n rock types. A thermal i n e r t i a survey of the Pisgah C r a t e r - L a v i c Lake area of southern C a l i f o r n i a gave "reasonable" thermal i n e r t i a v alues (Kahle et a l . , 1975). Lyon (1976), i n a study on thermal i n e r t i a models, expressed doubt about the accuracy of a b s o l u t e value d e t e r m i n a t i o n s of thermal i n e r t i a . G i l l e s p i e and Kahle (1977), a n a l y z i n g the thermal i n e r t i a survey of the Pisgah C r a t e r - L a v i c Lake area, s t a t e d that "a knowledge o f thermal i n e r t i a o f the s u r f a c e i s not s u f f i c i e n t f o r g e o l o g i c mapping" and t h a t "by i t s e l f , thermal i n e r t i a i s not s u f f i c i e n t to a l l o w Table 1.1. Thermal i n e r t i a s ( J m "2 c " 1 s " 1 / 2 ) of rock types selected from l i t e r a t u r e . Rock Type Thermal I n e r t i a Rock Type Thermal I n e r t i a P e r i d o t i t e Gabbro Basalt Andesite Serpentine Grano d i o r i t e Granite Quartz Monzonite Syenite Rhyolite Obsidian Pumice(dry loose) 3500 1 2300 1 2 2200 , 2250-2650 2 2 2100 , 2250 1 2650 2400 1 3 4 2 2150 , 2400 , 2700 , 3000 2 2450 - 3150 1 1950 1 2 1950 , 1250-2900 1450 1 1 400 Chert Porous Chert Quartzite Sandstone S i l t s t o n e Shale S l a t e Dolomite Limestone Limestone ( s i l i c i f i e d ) Marble C a l c i t e 4000 2 3250 1 2 3100 , 3350 1 3 2250 , 2350 2 2450 1 1400 1 2050 2 4 4 3150 , 3800 , 3950 1 3 4 1900 , 2000 , 3500 2 3000-3600 2350 1 4 3000 Janza (1975, p. 83) Lyon (1975) Carslaw and Jaeger (1959, p. 497) Watson et a l . (1971) - 1 0 -2000 i "—r r POROSITY (°/o) , , E f f e c t of p o r o s i t y on thermal i n e r t i a o f an F i g u r e 1.1. f ^ ^ g e ° % 0 ^ ( a f t e r Watson et a l . , 1971) . -11-i d e n t i f i c a t i o n " . Pohn e t al.. (1974) were s l i g h t l y more o p t i m i s t i c . They analysed a thermal i n e r t i a survey using Nimbus I I I and IV radiometer data over a l a r g e r e g i o n o f Oman, A f r i c a covered by e o l i a n sand, g r a v e l , limestone, dolomite, c h e r t and u l t r a b a s i c igneous rocks. They s t a t e d "even with data of 8-km r e s o l u t i o n , the c o r r e l a t i o n of thermal i n e r t i a with mapped geology shown i s more s t r i k i n g than might have been expected, and c l e a r l y i n d i c a t e s , we b e l i e v e , the u t i l i t y of the general technique". A c o n c l u s i o n was "when thermal i n e r t i a i s a d i a g n o s t i c p r o p e r t y of some m a t e r i a l , i d e n t i f i c a t i o n a l s o can be a c h i e v e d " . A thermal i n e r t i a model implemented over an area of the Raft R i v e r , Idaho ( M i l l e r and Watson, 1977) i n d i c a t e d that u n i t s of l a v a flows, t u f f , and a l l u v i u m c o u l d be d i s c r i m i n a t e d . The major b e n e f i t of thermal i n e r t i a mapping i n g e o l o g i c i n t e r p r e t a t i o n i s as a complementary t o o l to other g e o l o g i c mapping techniques. I t i s p a r t i c u l a r l y v a l u a b l e when other techniques f a i l such as when there are t h i n weathering s u r f a c e s , s u r f a c e c o a t i n g s , l i c h e n cover, or very t h i n l a y e r s of dust or s o i l , or when s p e c t r a l s i g n a t u r e s are not d i s t i n c t . I t can a i d i n i d e n t i f i c a t i o n of rock types and i s u s e f u l for d i s c r i m i n a t i n g u n i t s and d e f i n i n g g e o l o g i c boundaries. C) S o i l s P r a t t and E l l y e t t (1979) echoed the r e s u l t s of r e s e a r c h e r s i n t o the use of thermal i n e r t i a f o r g e o l o g i c mapping. In the c o n c l u s i o n to a study of the v a r i a t i o n of thermal i n e r t i a with m i n e r a l o g i c content, p o r o s i t y , and water content of s o i l s they -12-s t a t e d "mapping of s o i l s i n s e m i - a r i d environments, based upon m i n e r a l o g i c content, i s not d i r e c t l y p o s s i b l e using thermal i n e r t i a mapping techniques". V a r i a t i o n of thermal i n e r t i a with m i n e r a l content i s not l a r g e . On the other hand, p o r o s i t y and moisture content have a very l a r g e e f f e c t on thermal i n e r t i a . P o r o s i t y and moisture content are p r o p e r t i e s u s e f u l i n d e f i n i n g s o i l u n i t s and are r e l a t e d to m i n e r a l o g i c content. • They are important parameters i n themselves. Table 1.2 g i v e s thermal i n e r t i a s of the major s o i l c o n s t i t u e n t s . S o i l s g e n e r a l l y have thermal i n e r t i a s between 500 and 2000 TIU ( M i l l e r and Watson, 1977). P r a t t and E l l y e t t (1979) c a l c u l a t e d thermal i n e r t i a as a f u n c t i o n of m i n e r a l o g i c content, p o r o s i t y , and water content based on work of de V r i e s (1963). For a given p o r o s i t y and zero water content, the change i n thermal i n e r t i a between a c l a y and a sand s o i l i s very small (Figure 1.2a). Organic content g r e a t l y i n f l u e n c e s thermal i n e r t i a . Peat s o i l s have a low thermal i n e r t i a . The e f f e c t of m i n e r a l o g i c content on thermal i n e r t i a i n c r e a s e s when there i s even a small amount of water p r e s e n t . F i g u r e s 1.2b, 1.2c, and 1.2d give the e f f e c t of p o r o s i t y and water content on s o i l s of sand and c l a y composition and F i g u r e 1.2d shows the change of thermal i n e r t i a with water content for a peat s o i l of 80 percent p o r o s i t y . Thermal i n e r t i a v a r i a t i o n s with p o r o s i t y , and e s p e c i a l l y water content, are l a r g e . I f q u a n t i t a t i v e statements are to be made on one of m i n e r a l o g i c content, p o r o s i t y , or water content, the other two parameters must be known. There has been much i n t e r e s t i n the use of remote sensing to e v a l u a t e s o i l m oisture. Schmugge et a l . (1974) i n v e s t i g a t e d the -13-Table 1.2. Thermal i n e r t i a of the major s o i l constituents (de Vries, 1963). S o i l Constituent Thermal In e r t i a Quartz 4200 Clay Minerals 2400 Organic Matter 800 Water 1600 Air 5 -14-1000 r a) Xw« 0 <#> b) POROSITY = 30 % PEAT SOIL POROSITY = 8 0 % Figure 1.2. Thermal i n e r t i a versus porosity and percent volum-e t r i c water content (Xw) for d i f f e r i n g s o i l comp-osi t i o n s (1 i s 100 % sand, 2 i s 50 % sand and 50 % clay, 3 i s 100 % clay, and 4 i s peat with 80 % por o s i t y ) . Figures are derived from data of Pratt and E l l y e t t (1979). Dashed l i n e indicates extrapolation from data for p o r o s i t i e s of 60 % and l e s s . Peat s o i l data are from van Wijk and de Vries (1963). -15-s t u d i e d the u t i l i t y of both a c t i v e and p a s s i v e microwaves. Idso et a l . (1975a) c o r r e l a t e d s u r f a c e albedo with the s o i l water content. Perhaps the g r e a t e s t b e n e f i t of thermal i n e r t i a mapping f o r the study of s o i l s i s i n s o i l water content a n a l y s i s . P r a t t and E l l y e t t (1978) gave an example of s o i l moisture content d e t e r m i n a t i o n using thermal i n e r t i a mapping techniques. Trends i n s o i l moisture were a s c e r t a i n e d . S t u d i e s c o r r e l a t i n g the amplitude of the d i u r n a l s u r f a c e temperature wave i l l u s t r a t e the p o t e n t i a l of thermal i n e r t i a techniques. Idso et a_l. (1975b) and C i h l a r et a l . (1979) showed that the amplitude of the d i u r n a l s u r f a c e temperature of bare s o i l s can be c o r r e l a t e d to s o i l m oisture content f o r a p a r t i c u l a r s o i l . A s i m u l a t i o n of the temperature of bare s o i l s by Rosema (1975a) i n d i c a t e d v a r i a t i o n of temperature amplitudes with moisture l e v e l s . The d a i l y temperature amplitude i s c l o s e l y r e l a t e d to thermal i n e r t i a which i n turn i s s t r o n g l y c o n t r o l l e d by s o i l m o i sture. I t i s e v i d e n t that an accurate thermal i n e r t i a model may be a good method of measuring s o i l moisture. Ambiguities due to e v a p o r a t i o n r a t e , wind speed, a i r temperature, and albedo are l e s s e n e d by a thermal i n e r t i a approach. A l i m i t i n g f a c t o r i s that the p o r o s i t y and s o i l composition must be known; s t u d i e s w i l l have to be s i t e s p e c i f i c . Idso et a l . (1975b) showed that s o i l p r e s s u r e p o t e n t i a l can be r e l a t e d to the amplitude of the d i u r n a l temperature wave and there i s no n e c e s s i t y to determine s o i l type. C i h l a r et a l . (1979) suggested r e l a t i n g the temperature amplitude to a p r e s s u r e p o t e n t i a l r e l a t e d measure of s o i l water c o n t e n t . P r a t t and E l l y e t t (1979), however, expressed -16-some con c e r n over the a p p l i c a b i l i t y o f u s i n g p r e s s u r e p o t e n t i a l measures f o r the d e t e r m i n a t i o n o f s o i l m o i s t u r e s t a t u s by thermal i n e r t i a t e c h n i q u e s . The s o i l water t o s u r f a c e temperature a m p l i t u d e r e l a t i o n s h i p s of Idso e t a l . (1975b) and C i h l a r e t a l . (1979) were r e l a t e d t o the m o i s t u r e c o n t e n t i n the top two t o four c e n t i m e t e r s . The m o i s t u r e c o n t e n t w i l l f l u c t u a t e c o n s i d e r a b l y t h r o u g h o u t the day, e s p e c i a l l y f o r e v a p o r a t i o n from bare s o i l s under h i g h r a d i a t i v e c o n d i t i o n s ( J a c k s o n , 1973; Rose, 1968; Rosema, 1975a). The m o i s t u r e c o n t e n t near one hour b e f o r e noon i s a good e s t i m a t e of the 24-hour average s o i l m o i s t u r e c o n t e n t ; the mean o f the maximum and minimum d a i l y v a l u e s w i l l a l s o g i v e good r e s u l t s (Jackson e t a _ l . , 1976). The maximum w i l l occur i n the v e r y e a r l y morning and the minimum at one t o s e v e r a l hours a f t e r s o l a r noon. Thermal i n e r t i a models u s i n g midday and p r e -dawn tempe r a t u r e s may t h e r e f o r e be expected t o g i v e good e s t i m a t e s of the d a i l y average . A model u s i n g n i g h t t i m e c o o l i n g w i l l g i v e v a l u e s r e l a t e d more c l o s e l y t o the maximum m o i s t u r e c o n t e n t s i n c e at n i g h t the s u r f a c e l a y e r i s b e i n g r e c h a r g e d . D) V e g e t a t i o n The t h e r m a l i n e r t i a o f v e g e t a t e d s u r f a c e s has not been i n v e s t i g a t e d . I t w i l l be p r i m a r i l y r e l a t e d t o the d e n s i t y o f the v e g e t a t i o n and the s t r u c t u r e of the canopy. S u r f a c e t e m p e r a t u r e has been shown to be s t r o n g l y dependent on v e g e t a t i o n d e n s i t y and s t r u c t u r e (Myers and Heilman, 1969). T h i s e f f e c t i s most -17-2 n o t i c e a b l e d u r i n g midday and l e a s t d u r i n g pre-dawn p e r i o d s . Implementations of thermal i n e r t i a mapping by P r i c e (1977) and B e r n i e r et a l . (1980) showed p o t e n t i a l for the use of thermal i n e r t i a f o r v e g e t a t i o n s t u d i e s . Determination of thermal i n e r t i a w i l l be d i f f i c u l t due to problems i n e s t i m a t i n g the s e n s i b l e heat f l u x and evaporation from a vegetated s u r f a c e . E s t i m a t i n g s e n s i b l e heat f l u x by remote sensing techniques i s d i f f i c u l t due to wide ranges of s u r f a c e roughness and zero plane displacements over d i f f e r i n g s u r f a c e s and due to the i n t e g r a t e d nature of temperature measurements by thermal r a d i a t i o n sensors. The e v a p o t r a n s p i r a t i o n w i l l depend on the energy a v a i l a b l e , the s o i l moisture s t a t u s , and the stomatal r e s i s t a n c e of the p l a n t . These may vary g r e a t l y over d i f f e r e n t s u r f a c e s and over short d i s t a n c e s . Byrne et a l . (1979) d i s c u s s e d , with r e f e r e n c e to remote sensing, the environmental c o n d i t i o n s and p h y s i o l o g i c a l processes i n v o l v e d i n c o n t r o l l i n g the temperature of a p l a n t canopy. I t may be expected that e s t i m a t i o n of l a t e n t heat f l u x over an e n t i r e area of a thermal i n e r t i a survey w i l l be d i f f i c u l t and w i l l r e q u i r e a l a r g e number of measurements of ground parameters such as a i r temperature and vapour p r e s s u r e . The thermal i n e r t i a of v e g e t a t i o n i s very low. T h i s l e a d s to another problem. The s u r f a c e temperature w i l l be g r e a t l y i n f l u e n c e d by l o c a l a d v e c t i v e processes of short d u r a t i o n (e.g., Derksen, 1974) . -18-E) O p t i c a l - V i s u a l I n t e r p r e t a t i o n and Automatic D i g i t a l A n a l y s i s Thermal i n e r t i a may be used as another channel of in f o r m a t i o n for both o p t i c a l - v i s u a l a n a l y s i s and mapping, and f o r automatic c l a s s i f i c a t i o n schemes. P r a t t et a l . (1980) a l s o suggest the l a t t e r p o s s i b i l i t y . The absolute value of the thermal i n e r t i a need not be accurate as long as r e l a t i v e thermal i n e r t i a s are c o n s i s t e n t and the s u r f a c e s of i n t e r e s t have c o n t r a s t i n g and c h a r a c t e r i s t i c thermal i n e r t i a s . 1.4 REVIEW OF EXISTING MODELS Th i s s e c t i o n reviews c u r r e n t thermal i n e r t i a models and models r e l e v a n t to thermal i n e r t i a a n a l y s i s . P a r t i c u l a r models are d i s c u s s e d more f u l l y i n order to demonstrate a given type of approach. For d e t a i l e d e x p l a n a t i o n s r e f e r to the l i t e r a t u r e c i t e d . 1.4.1 DESCRIPTION OF MODELS Thermal i n e r t i a models s t u d i e d to date determine the d i u r n a l c y c l e of s u r f a c e temperature. The one-dimensional heat conduction equation (1.1) i s solved s u b j e c t to the boundary c o n d i t i o n of energy balance at the s u r f a c e ( i . e . , energy e n t e r i n g the s u r f a c e must equal that l e a v i n g the s u r f a c e ) . S i n c e t h i s boundary c o n d i t i o n i s not l i n e a r i n temperature, a n a l y t i c s o l u t i o n of the heat conduction equation i s d i f f i c u l t . Three -19-common methods of computing the d i u r n a l s u r f a c e temperature v a r i a t i o n are: Laplace transform techniques, l i n e a r i z e d F o u r i e r s e r i e s , and f i n i t e d i f f e r e n c e methods. M i l l e r and Watson (1977) reviewed these methods and i n v e s t i g a t e d the advantages and disadvantages of each. There are two c a t e g o r i e s of models: those which do not consi d e r s e n s i b l e and l a t e n t heat f l u x i n the energy balance boundary c o n d i t i o n and those which do. Models of the f i r s t category are most a p p l i c a b l e to g e o l o g i c problems. Watson et a l . (1971) and Lyon (1976) developed models based on work by Jaeger (1953). The f l u x at the ground s u r f a c e (G) i s expressed i n terms of a p e r i o d i c f u n c t i o n of s u r f a c e temperature composed of a s e r i e s of step changes i n time. The boundary c o n d i t i o n i s that the f l u x at the ground s u r f a c e i s equal to the net r a d i a t i o n (Rn) at the s u r f a c e . Net r a d i a t i o n i s the t o t a l incoming s o l a r and longwave r a d i a t i o n minus t o t a l outgoing s o l a r ( r e f l e c t e d ) and longwave ( r e f l e c t e d and emitted) r a d i a t i o n . Laplace transform techniques are used. The system i s solved for a given thermal i n e r t i a by i n p u t t i n g a f i r s t approximation of the p e r i o d i c temperature v a l u e s and s o l v i n g for the f l u x . The estimate of the temperature values i s i t e r a t i v e l y improved u n t i l the f l u x converges to that given by the boundary c o n d i t i o n . There are o n l y minor d i f f e r e n c e s between the model of Watson et a l . (1971) and that of Lyon (1976). Both models have been used on simulated data and the model of Watson et a l . (1971) has been implemented using s a t e l l i t e data (Pohn et a l . , 1974). The second category of models i n c l u d e s p r o v i s i o n s f o r s e n s i b l e and l a t e n t heat f l u x i n the boundary c o n d i t i o n s and -20-t h e r e f o r e c o n s i d e r s m e t e o r o l o g i c a l c o n d i t i o n s . These include s o l u t i o n s by: 1) l i n e a r i z e d F o u r i e r s e r i e s techniques ( P r i c e , 1977; P r a t t et a l . , 1980) which i n c l u d e the e f f e c t of temperature dependent processes as a bulk parameter; 2) f i n i t e d i f f e r e n c e methods ( O u t c a l t , 1972; Kahle, 1977; P r a t t , 1980) which consider the m e t e o r o l o g i c a l c o n d i t i o n s i n d e t a i l ; and 3) f i n i t e d i f f e r e n c e methods which a l s o i n c l u d e a water balance as a boundary c o n d i t i o n and c o n s i d e r water t r a n s p o r t i n the s o i l i n d e t a i l (Rosema, 1975a). A requirement of l i n e a r i z e d F o u r i e r s e r i e s s o l u t i o n s i s that the boundary c o n d i t i o n be expressed as a f u n c t i o n of the form A + B Ts where A and B are c o n s t a n t s and Ts i s s u r f a c e temperature. Approximations of formulae for longwave r a d i a t i o n exchange, s e n s i b l e heat f l u x , and l a t e n t heat f l u x are necessary to reduce them to a f u n c t i o n of s u r f a c e temperature of order one. The constant B i s a bulk parameter f o r these energy f l u x processes which i s c o n s i d e r e d a c h a r a c t e r i z a t i o n of "atmospheric" e f f e c t s ( P r i c e , 1977; P r a t t et a l . , 1980). P r i c e (1977) recognized the problems of i n c o r p o r a t i n g m e t e o r o l o g i c a l d e t a i l s i n t o a model and developed a simple model to g i v e " p r o v i s i o n a l 1/2 r e s u l t s " . He used a bulk parameter <* (= B/(P OJ ) , where OJ i s the d i u r n a l frequency) to represent the s u r f a c e temperature dependent terms of the boundary c o n d i t i o n . By assuming <* constant fo r a l l s u r f a c e s , r e l a t i v e values of thermal i n e r t i a can be d e r i v e d . T h i s depends on the constant B being r e l a t e d to the a c t u a l thermal i n e r t i a of the s u r f a c e . P r a t t e t a l . (1980) expressed some doubt on the r e l i a b i l i t y o f t h i s o c c u r r i n g . They gave a d e t a i l e d a n a l y s i s of the l i n e a r i z e d boundary c o n d i t i o n s -21-(A + B Ts) i n terms of formulae for sky r a d i a t i o n , emitted r a d i a t i o n , and s e n s i b l e and l a t e n t heat f l u x . The main l i m i t i n g approximations of the model were s t a t e d as "the simple l i n e a r r e l a t i o n s h i p between a i r and ground temperature, a constant wind speed, constant a i r humidity, and omission of the atmospheric s t a b i l i t y parameters". P r a t t et a l . (1980) e s t a b l i s h e d a c a l i b r a t i o n procedure by which "atmospheric" parameters are d e r i v e d from a c o n t r o l s i t e of known thermal i n e r t i a , albedo, and s u r f a c e temperature change. Other input necessary are maximum incoming s o l a r r a d i a t i o n , average a i r temperature, and maximum and minimum a i r temperature f o r the day. These may be measured at one s i t e w i t h i n the survey a r e a . The albedo and s u r f a c e temperature change at each s i t e must be determined. In some cases slope and aspect might a l s o be r e q u i r e d . The e f f e c t s of e v a p o r a t i o n were not f u l l y analyzed. P r a t t et a l . (1980) s t a t e d that " f u r t h e r r e s e a r c h remains to be done on the i n f l u e n c e of l a t e n t heat t r a n s f e r on the development of s u i t a b l e c a l i b r a t i o n models, e s p e c i a l l y where the r a t e of e v a p o r a t i o n can vary r a p i d l y across a survey scene as a f u n c t i o n o f the s o i l moisture c o n t e n t " . There are a l a r g e number of models of v a r y i n g complexity which p r e d i c t atmospheric and s u r f a c e h e a t i n g , but few have been a p p l i e d to remote sensing s t u d i e s and even fewer a p p l i e d to thermal i n e r t i a a n a l y s i s . O u t c a l t (1972) developed a f i n i t e d i f f e r e n c e model. He c o n s i d e r e d a p p l i c a t i o n s to remote sensing but the model was not s p e c i f i c a l l y designed f o r remote sensing purposes. The model, based l a r g e l y on work by Myrup (1969), simulates d i u r n a l s u r f a c e temperatures by s o l u t i o n of the energy -22-balance equation at the s u r f a c e . The s e n s i b l e and l a t e n t heat f l u x e s are d e s c r i b e d by an aerodynamic form of the heat t r a n s f e r e q u a t i o n s . These are expressed i n terms o f : 1) a i r temperature, a b s o l u t e humidity, and wind speed at a height i n the atmosphere o u t s i d e the he i g h t of d i u r n a l heat wave p e n e t r a t i o n but w i t h i n l o g a r i t h m i c g r a d i e n t s ( t h i s h e ight may be 10 to 300 m); 2) the temperature and ab s o l u t e humidity at the s u r f a c e ; 3) the s u r f a c e roughness. The s o i l heat f l u x and temperature at s e v e r a l depths i n the s o i l are determined by the f i n i t e d i f f e r e n c e form of the heat conduction equation (constant d i f f u s i v i t y ) . The energy balance i s then solved for s u r f a c e temperature by i t e r a t i v e t echniques. The model i s run throughout a d a i l y c y c l e . Kahle (1977) d e s c r i b e d an analogous technique s p e c i f i c a l l y a p p l i e d to thermal i n e r t i a mapping. The e x p r e s s i o n s f o r s e n s i b l e and e v a p o r a t i v e heat f l u x e s are from a mesoscale model of Burke (1945). M e t e o r o l o g i c a l input are a i r temperature, mixing r a t i o of a i r , and wind speed at 1.5 m above the ground. The temperature and mixing r a t i o at the ground s u r f a c e are a l s o r e q u i r e d . Since the mixing r a t i o at the s u r f a c e cannot be measured, i t i s c a l c u l a t e d as the s a t u r a t e d mixing r a t i o at the s u r f a c e temperature m u l t i p l i e d by an a r b i t r a r i l y estimated ground moisture f a c t o r (M) s c a l e d between 0 to 1 from a dry to a s a t u r a t e d s u r f a c e . Solar r a d i a t i o n and downward longwave r a d i a t i o n are from e m p i r i c a l formulae. S o l a r r a d i a t i o n i s a l s o c o r r e c t e d for topographic slope and aspect, and albedo. The model i s run for a wide range of thermal c o n d u c t i v i t i e s (A) and -23-heat c a p a c i t i e s (pc ) of the s o i l ( i . e . , thermal i n e r t i a ) , P albedos, s l o p e s , and asp e c t s . The temperature change (from mid-day to pre-dawn times) i s p r e d i c t e d and a look-up t a b l e created between temperature change, thermal i n e r t i a , albedo, slope, and aspe c t . Midday and pre-dawn images are r e g i s t e r e d to each other, a d i g i t a l t e r r a i n model, and an albedo map. The temperature change, albedo, s l o p e , and aspect are determined for each p i x e l . The thermal i n e r t i a i n the look-up t a b l e corresponding to these four parameters i s the thermal i n e r t i a of the s u r f a c e . A problem of t h i s model i s the use of a mesoscale m e t e o r o l o g i c a l model f o r very l o c a l energy f l u x d e t e r m i n a t i o n s . Kahle (1977) a l s o recognized t h i s problem as w e l l as the m i c r o m e t e o r o l o g i c a l e f f e c t s of s u r f a c e roughness and v a r i a t i o n s i n wind speed, a i r temperature, and mixing r a t i o s over the survey area. The u t i l i t y of both Kahle's and O u t c a l t ' s model i s s e v e r e l y l i m i t e d by the n e c e s s i t y to q u a n t i f y the s u r f a c e vapour c o n d i t i o n . I t cannot be e a s i l y determined remotely unless the s u r f a c e i s s a t u r a t e d and may be very s p a t i a l l y v a r i a b l e . I t s i n f l u e n c e on the temperature p r e d i c t i o n can be l a r g e . Lyon (1976) determined that assuming a water content of zero i n the model l i n g of the temperature of a p l a y a m a t e r i a l by Kahle et a l . (1976) can cause an e r r o r i n the thermal i n e r t i a v a lue determined f o r t h at s u r f a c e of more than 30 p e r c e n t . Use of an estimated ground moisture f a c t o r (e.g., M) i s not v a l i d f o r many s t u d i e s . The models are a l s o l i m i t e d by the problem of r e q u i r i n g much inp u t , which manifests i t s e l f i n the c o s t and complexity of implementing the models. Albedo, s l o p e , and aspect are r e q u i r e d on a p i x e l - b y - p i x e l b a s i s . -24-Rosema (1975b) presented a thermal i n e r t i a model i n c o r p o r a t i n g s t a b i l i t y c o r r e c t e d s e n s i b l e heat and l a t e n t heat f l u x , and proposed that noon-midnight temperature d i f f e r e n c e s be used to gi v e thermal i n e r t i a . The problem o f e s t a b l i s h i n g s u r f a c e water vapour s t a t u s was recognized and a surface temperature s i m u l a t i o n model was developed (Rosema, 1975a, 1975c). I t i n c o r p o r a t e d s o i l moisture t r a n s p o r t and used two boundary c o n d i t i o n s : 1) the moisture balance; and 2) the energy balance. The model i s n e c e s s a r i l y complex and d e t a i l e d as i t i n v o l v e s l i q u i d water and vapour t r a n s p o r t under both temperature and p o t e n t i a l g r a d i e n t s . D e t a i l e d input o f the thermal p r o p e r t i e s and t r a n s p o r t c h a r a c t e r i s t i c s of the s o i l i s necessary. The d e t a i l e d s o i l parameters r e q u i r e d and the complexity of the model negate i t s u s e f u l n e s s as a v i a b l e thermal i n e r t i a mapping model. However, i t i s extremely u s e f u l , as intended, for study of the e f f e c t s of many d i f f e r e n t s o i l , s u r f a c e , and atmospheric parameters which i n f l u e n c e the temperature and energy balance o f a bare s o i l s u r f a c e . A method (Rosema et a_l., 1978), i n c o r p o r a t i n g both temperature and temperature change data, may be u s e f u l i n d e a l i n g with the problem of e v a p o r a t i o n . 1.4.2 ACCURACY OF MODELS The accuracy of the above models and e s p e c i a l l y remote sensing implementations of them have not been s t u d i e d i n d e t a i l . T e s t s a g a i n s t r e a l temperature and thermal i n e r t i a data are -25-l a c k i n g . One d i f f i c u l t y i s that determining thermal i n e r t i a v a l u e s for ground t r u t h i s not easy, e s p e c i a l l y iri s i t u . Lyon (1976) compared the accuracy of s e v e r a l of the models by c o n t r a s t i n g the r e s u l t s of the models given s i m i l a r input. He compared the Watson et a l . (1971), Lyon (1976), and O u t c a l t (1972) models. The comparison gave temperature curves of roughly s i m i l a r shape but with the temperature estimates c o n t r a s t i n g by 3 or 4 C . Correspondence of r e s u l t s from Rosema's (1975a) model with those of the Watson et a l . (1971) and Lyon (1976) models was very poor, although t h i s may have been, i n p a r t , due to assumptions made about model input parameters. The absolute accuracy of the model of Kahle et a l . (1976) and Kahle (1977) was a l s o analyzed by Lyon (1976). A remote sensing implementation o f Kahle's model (Kahle et a l . , 1976) y i e l d e d a thermal i n e r t i a value of between 420 and 1260 TIU f o r a pl a y a m a t e r i a l on a t e s t s i t e i n the Pisgah C r a t e r - L a v i c Lake area of C a l i f o r n i a . Lyon (1976) c a l c u l a t e d , by the method of de V r i e s (1963), the thermal i n e r t i a to be 1420 TIU. Kahle's val u e s may be low due to the assumption of zero moisture content as input i n t o the model. Lyon (1976) concluded h i s study on the e x i s t i n g models, " r e l a t i v e d e t e r m i n a t i o n o f thermal p r o p e r t i e s may be p o s s i b l e by remote s e n s i n g , but a b s o l u t e value d e t e r m i n a t i o n s appear most u n l i k e l y . " M i l l e r and Watson (1977) i n v e s t i g a t e d the accuracy of the three common methods used to c a l c u l a t e the d i u r n a l temperature wave and determine thermal i n e r t i a . A method of comparing the r e s u l t s of the methods with the t h e o r e t i c a l s o l u t i o n for a pure s i n u s o i d f l u x y i e l d e d e r r o r s of 260, 300, and 460 TIU f o r Laplace -26-transform, f i n i t e d i f f e r e n c e , and F o u r i e r s e r i e s algorithms r e s p e c t i v e l y . T o t a l e r r o r s i n a p p l y i n g these methods to models of the f i r s t c ategory were estimated to be of the order of 550 to 750 TIU f o r the three above methods. The a n a l y s i s included estimates of s a t e l l i t e measurement e r r o r , s u r f a c e c o a t i n g e f f e c t s , and " t r a n s i e n t " e f f e c t s such as s e n s i b l e and l a t e n t heat f l u x . The e r r o r s i n the l i n e a r i z e d F o u r i e r s e r i e s methods of P r i c e (1977) and P r a t t et a l . (1980) caused by the v a r i o u s assumptions i n the l i n e a r i z a t i o n of the boundary c o n d i t i o n s have not been f u l l y i n v e s t i g a t e d . P r a t t et a l . (1980), however,showed that a c a l i b r a t i o n procedure employing a c o n t r o l s i t e of known thermal i n e r t i a , albedo, and temperature change can produce good r e s u l t s . Comparison of r e s u l t s of the c a l i b r a t i o n procedure with a f i n i t e d i f f e r e n c e model ( P r a t t , 1980) i n d i c a t e d that e r r o r s r e l a t i v e to the f i n i t e d i f f e r e n c e ^ model w i l l g e n e r a l l y be l e s s than 250 TIU. In most thermal i n e r t i a models and implementations of the models i t i s assumed that s u r f a c e and m e t e o r o l o g i c a l c o n d i t i o n s are constant throughout the survey area. P r a t t (1980) i n v e s t i g a t e d e r r o r s i n t h i s assumption for a f i n i t e d i f f e r e n c e model. Sky r a d i a t i o n and wind speed were assumed constant during the day and a i r temperature was given as a s i n e wave f u n c t i o n expressed i n terms of average a i r temperature and maximum and minimum a i r temperature for a day. E v a p o r a t i o n was ignored. He concluded that v a r i a t i o n i n wind speed and s u r f a c e roughness have major i n f l u e n c e s on the e s t i m a t i o n of thermal i n e r t i a by such techniques. V a r i a t i o n of a i r temperature, sky temperature, and s u r f a c e slope had l e s s e r e f f e c t s . -27-P r a t t (1980) i n d i c a t e d that for s l o p e s of 10 degrees the e r r o r i n thermal i n e r t i a estimate due to assuming zero d i p were 100 and 200 TIU f o r north and south aspects r e s p e c t i v e l y . The thermal i n e r t i a used i n the c a l c u l a t i o n was 1000 TIU. M i l l e r and Watson (1977) c a l c u l a t e d that an e r r o r of approximately 350 TIU i s caused by assuming a f l a t s u r f a c e for a southwest slope of 10 degrees. The e f f e c t of slope and aspect on both temperature and measured albedo were c o n s i d e r e d . S i m u l a t i o n data o f Kahle (1977) showed that aspect has a more important e f f e c t than s l o p e . G i l l e s p i e and Kahle (1977) demonstrated the need for topographic c o r r e c t i o n i n r egions of "moderate to higher r e l i e f " . M i l l e r and Watson (1977) a l s o found topographic c o r r e c t i o n necessary i n order to d i s c r i m i n a t e between two g e o l o g i c u n i t s o f i n t e r e s t . I t appears that major sources of e r r o r are: the e s t i m a t i o n of the e f f e c t s of e v a p o r a t i o n , s p a t i a l e x t r a p o l a t i o n of s u r f a c e and m e t e o r o l o g i c a l c o n d i t i o n s , and the e f f e c t s of topography ( i n areas of moderate and high r e l i e f ) . Another l i k e l y source of e r r o r , p a r t i c u l a r l y i n the models of Watson et a l . (1971), Lyon (1976), and P r i c e (1977), i s the v a r i o u s approximations made concerning the boundary c o n d i t i o n s . There i s a need f o r d e t a i l e d e r r o r a n a l y s i s of the v a r i o u s models under a c t u a l remote sensing implementation c o n d i t i o n s . T h i s i n c l u d e s p r a c t i c a l problems such as e r r o r s due to image m i s r e g i s t r a t i o n , d i s c u s s e d q u a l i t a t i v e l y by M i l l e r and Watson (1977), and e r r o r s i n remotely sensed s u r f a c e temperature data. A l s o needed are comparisons of a c t u a l boundary c o n d i t i o n s to those used by the models, measures of the e f f e c t of t y p i c a l s p a t i a l v a r i a b i l i t y i n m e t e o r o l o g i c a l and s u r f a c e c o n d i t i o n s , and s t u d i e s on the a p p l i c a b i l i t y of the - 2 8 -models over vegetated t e r r a i n . 1.5 CRITERIA FOR A THERMAL INERTIA MODEL The u s e f u l n e s s of a model may be measured by the accuracy of i t s r e s u l t s , the s i m p l i c i t y of implementing i t , the q u a n t i t y of input i t r e q u i r e s , and the range of c o n d i t i o n s over which i t i s a p p l i c a b l e . C o n s i d e r i n g these measures of u s e f u l n e s s and the above d i s c u s s i o n on the b e n e f i t s and c u r r e n t methods of thermal i n e r t i a mapping ( s e c t i o n 1.3 and 1.4), i t i s p o s s i b l e to o u t l i n e c r i t e r i a for a thermal i n e r t i a model. In determining a good model f o r thermal i n e r t i a mapping there are s e v e r a l c r i t e r i a which are d e s i r a b l e . 1) The model should give r e s u l t s u s e f u l f o r q u a n t i t a t i v e a n a l y s i s and/or q u a l i t a t i v e a n a l y s i s . Q u a n t i t a t i v e a n a l y s i s w i l l l i k e l y r e q u i r e the model to y i e l d probable 3 e r r o r s l e s s than approximately 300 TIU. Poorer r e s u l t s w i l l s t i l l p r o vide u s e f u l q u a l i t a t i v e i n f o r m a t i o n . 2) No or l i m i t e d knowledge of the s u r f a c e type i s r e q u i r e d . T h i s i m p l i e s that p r e c i s e values of thermal c o n d u c t i v i t y , s p e c i f i c heat, d e n s i t y , e m i s s i v i t y , albedo, or s u r f a c e roughness need not be known. 3) The i n f l u e n c e of topography should have a minimum e f f e c t on the r e s u l t s . T h i s i m p l i e s that the e f f e c t of topography must be accounted f o r by the model ( r e q u i r i n g use of a d i g i t a l t e r r a i n model, with consequent complexity of implementation) or the model must be independent of - 2 9 -topography. 4) Measurements of m e t e o r o l o g i c a l c o n d i t i o n s are necessary o n l y at a s i n g l e or a few s i t e s . T h i s i m p l i e s that measurements at l o c a l s i t e s may be e x t r a p o l a t e d over the whole r e g i o n of the survey and a l l types of s u r f a c e . 5 ) The model must be a p p l i c a b l e over a wide range of s u r f a c e types and s u r f a c e moisture c o n d i t i o n s . T h i s i m p l i e s that the model should g i v e good r e s u l t s f o r areas of s u r f a c e and c l i m a t i c c o n d i t i o n s ranging from non-vegetated s u r f a c e s i n a r i d regions to vegetated s u r f a c e s i n humid reg i o n s . I n a b i l i t y to s a t i s f y every c r i t e r i o n does not negate but, r a t h e r , l i m i t s the u s e f u l n e s s of the model. I t i s apparent that the c u r r e n t models ( s e c t i o n 1.4) have d i f f i c u l t i e s meeting such s t r i n g e n t c r i t e r i a . Problems a r i s e i n a p p l y i n g the models over evaporating or e v a p o t r a n s p i r i n g s u r f a c e s . Topography must be accounted f o r , thus i n t r o d u c i n g a complex and c o s t l y a n a l y s i s . S p a t i a l s u r f a c e albedo i n f o r m a t i o n i s necessary. Topographic, albedo, and temperature data must be r e g i s t e r e d . The use of approximations i n some of the models and the v a l i d i t y of e x t r a p o l a t i n g m i c r o m e t e o r o l o g i c a l measurements such as a i r temperature, wind speed, and s u r f a c e roughness over d i f f e r e n t regions and s u r f a c e types i n the area of a thermal i n e r t i a study have not been f u l l y e x p l o r e d . The thermal i n e r t i a models p r e s e n t l y a v a i l a b l e , although very u s e f u l , s u f f e r i n v a r y i n g degrees from complexity of implementation, l a r g e number of r e q u i r e d i n p u t s , and l i m i t e d a p p l i c a b i l i t y over a wide range of s u r f a c e and s u r f a c e moisture -30-c o n d i t i o n s . These models use the f u l l d a i l y c y c l e of temperature and energy balance components. 1.6 OBJECTIVES I t i s hypothesized that a model using o n l y n i g h t t i m e c o o l i n g w i l l meet the above c r i t e r i a ( s e c t i o n 1.5). T h e o r e t i c a l l y , the e f f e c t s of topography and albedo are e l i m i n a t e d as there i s no s o l a r r a d i a t i o n input; the model w i l l be a p p l i c a b l e over a wide range of s u r f a c e types and s u r f a c e moisture c o n d i t i o n s ; and the e x t r a p o l a t i o n of m e t e o r o l o g i c a l c o n d i t i o n s can be more r e a d i l y made. The l a t t e r two advantages are due to the g e n e r a l s t e a d i n e s s of the energy balance components at n i g h t . E v a p o r a t i o n and t r a n s p i r a t i o n are extremely v a r i a b l e and d i f f i c u l t to model during the day due to t h e i r dependence on water a v a i l a b i l i t y and the dependence of t r a n s p i r a t i o n on b i o l o g i c a l p r o c e s s e s . At night they vary much l e s s from s u r f a c e to s u r f a c e and with time, and o f t e n make up a small p r o p o r t i o n of the t o t a l energy balance. S o i l heat f l u x i s a more important component of the energy balance at n i g h t and the thermal p r o p e r t i e s of the s u r f a c e media w i l l have a l a r g e i n f l u e n c e on the energy balance. T h i s w i l l be p a r t i c u l a r l y t r u e on c l e a r n i g h t s with low wind speed. The atmosphere becomes very s t a b l e , s e n s i b l e and l a t e n t heat f l u x approach zero, and the ground heat f l u x may be approximated by the net r a d i a t i o n of the s u r f a c e . In such c o n d i t i o n s , procedures f o r e s t i m a t i n g the boundary c o n d i t i o n s for a model are g r e a t l y s i m p l i f i e d . T h e r e f o r e , there i s a p o s s i b i l i t y of developing a n i g h t t i m e remote sensing thermal -31-i n e r t i a mapping model which i s a c c u r a t e , simple, and easy to implement, which r e q u i r e s a minimum of input, and which may be a p p l i e d to a wide v a r i e t y of t e r r a i n . The o b j e c t i v e of the t h e s i s i s to develop a nighttime c o o l i n g model for remote sensing thermal i n e r t i a mapping which w i l l meet the c r i t e r i a s t a t e d i n s e c t i o n 1.5. There are a number of s p e c i f i c problems to be solved and o b j e c t i v e s to be met before i t can be s a i d that a n i g h t t i m e c o o l i n g thermal i n e r t i a model s a t i f i e s the c r i t e r i a c i t e d . 1) What i s the best model to use i n terms of s i m p l i c i t y of implementation and accuracy of r e s u l t s ? V a r i o u s models are developed and t e s t e d i n Chapter 2. 2) What are the best methods of determining the components (net r a d i a t i o n , s e n s i b l e heat f l u x , and l a t e n t heat f l u x ) of the energy balance of a s u r f a c e ? Methods of determining these components are given i n Chapter 3. 3) How w e l l w i l l the models and methods developed f u n c t i o n when implemented i n a remote sensing mode? P a r t i c u l a r problems are: what are the e f f e c t s , on e s t i m a t i n g s e n s i b l e heat f l u x , of e x t r a p o l a t i n g m i c r o m e t e o r o l o g i c a l measurements ( a i r temperature and wind speed) and estimates of s u r f a c e roughness to d i f f e r e n t s u r f a c e s and s p a t i a l areas? what are the e f f e c t s of a i r ponding? what are the e r r o r s i n remotely sensed s u r f a c e temperature? The e r r o r s i n v o l v e d i n remote sensing implementation of the thermal i n e r t i a model and the methods of e s t i m a t i n g net r a d i a t i o n and s e n s i b l e heat f l u x are d i s c u s s e d i n d e t a i l i n Chapter 4. Appendix II analyzes the e r r o r s i n -32-remotely sensed s u r f a c e temperature. 4) Is the small temperature change dur i n g nighttime c o o l i n g l a r g e enough to absorb the e r r o r s inherent i n the model, ground input parameters, and remotely sensed data, and s t i l l y i e l d meaningful thermal i n e r t i a estimates? The t o t a l e r r o r s i n the nighttime c o o l i n g model and the c o n t r i b u t i o n s of the v a r i o u s sources of e r r o r to the t o t a l e r r o r are a l s o d i s c u s s e d i n Chapter 4. 5) What are the best c o n d i t i o n s and procedures under which the model i s most s u c c e s s f u l l y implemented? These are summarized i n Chapter 5. 6) What i s the u l t i m a t e u s e f u l n e s s of the model? P o t e n t i a l a p p l i c a t i o n s are o u t l i n e d i n Chapter 5. 1 I t should be noted that van Duin's model (1954) s l i g h t l y o v e r e s t i m a t e s the d i f f e r e n c e s i n amplitude. 2 T h i s e f f e c t was observed i n f i e l d t e s t s t u d i e s of ground temperature (Barnes PRT-10 i n f r a r e d thermometer) over n a t u r a l g r a s s l a n d t e r r a i n (Lac du Bois rangeland, Kamloops, B r i t i s h Columbia). 3 Probable e r r o r i s a q u a n t i t y such that o n e-half of the e r r o r s w i l l be l e s s than i t and one-half g r e a t e r (Scarborough, 1962). - 3 3 -Chapter Two NIGHTTIME COOLING MODELS 2.1 INTRODUCTION A nighttime remote sensing thermal i n e r t i a mapping method r e q u i r e s a simple model which w i l l p r e d i c t n ighttime s u r f a c e c o o l i n g as a f u n c t i o n of thermal i n e r t i a . An important p r a c t i c a l requirement i s that the model be l i m i t e d to only two remote sensing data a c q u i s i t i o n times. T h i s r e s t r i c t i o n i s based on remote sensing data a c q u i s i t i o n and p r o c e s s i n g c o s t c o n s i d e r a t i o n s . The model must s a t i s f y the one-dimensional heat conduction equation (1.1). I n v e s t i g a t i o n i s t h e r e f o r e undertaken to determine s o l u t i o n s to the heat conduction equation which: 1) best meet a c t u a l i n i t i a l and boundary c o n d i t i o n s ; 2) are f u n c t i o n s - not of the thermal p r o p e r t i e s comprising thermal i n e r t i a , but are simple f u n c t i o n s of thermal i n e r t i a alone; 3) can be modelled and implemented using o n l y two data measurement times. P o s s i b l e s o l u t i o n s to the heat conduction equation a p p l i c a b l e to the a c t u a l i n i t i a l and boundary c o n d i t i o n s are d i s c u s s e d and three probable models (models I, I I , and III) presented. Heat s h a r i n g between the atmosphere and the e a r t h ' s s u r f a c e i s c o n s i d e r e d as an approach f o r a p p l y i n g model I. The heat s h a r i n g approach expresses the thermal i n e r t i a of the s u r f a c e i n terms of the heat added to the s u r f a c e (net r a d i a t i o n -34-(Rn) and l a t e n t heat f l u x (LE)) and the thermal i n e r t i a of the atmosphere (Pa). Since there are problems i n determining Pa, an energy balance approach i s chosen for a p p l y i n g the models. The energy balance approach expresses the thermal i n e r t i a of the s u r f a c e media i n terms of the heat f l u x i n t o the s u r f a c e (G). A d e t a i l e d a n a l y s i s of the boundary c o n d i t i o n of f l u x i n t o the s u r f a c e i s undertaken to determine the measurement times to apply to the models to best s a t i s f y the a c t u a l i n i t i a l and boundary c o n d i t i o n s . E r r o r s i n implementing the models due to non-conformity to a c t u a l boundary c o n d i t i o n s are presented. The models are t e s t e d f o r s e v e r a l n a t u r a l s u r f a c e m a t e r i a l s . 2 . 2 THEORY: SOLUTIONS OF HEAT CONDUCTION EQUATION APPLICABLE TO A NIGHTTIME REMOTE SENSING THERMAL INERTIA MODEL (Models I, I I , and III) S o l u t i o n s to the one-dimensional heat conduction equation for a s e m i - i n f i n i t e homogeneous s o l i d with i n i t i a l c o n d i t i o n s expressed i n terms of temperature d i s t r i b u t i o n and a time dependent boundary c o n d i t i o n of s u r f a c e heat f l u x (G) are a p p l i c a b l e to the n i g h t t i m e c o o l i n g model. P r e v i o u s thermal i n e r t i a models (Watson et a l . , 1971; Kahle, 1977; P r i c e , 1977) have recognized that the heat f l u x boundary c o n d i t i o n may be expressed as a p e r i o d i c f u n c t i o n of time and have i n c o r p o r a t e d the f u l l d a i l y c y c l e of temperature and energy balance. The n i g h t t i m e c o o l i n g model i s r e s t r i c t e d to energy f l u x data a c q u i r e d d u r i n g the n i g h t and t h e r e f o r e the p e r i o d i c boundary c o n d i t i o n cannot be e x p l i c i t l y s t a t e d . Thus the problem must be -35-s o l v e d with a s e t of i n i t i a l c o n d i t i o n s expressed at a time near sunset and boundary c o n d i t i o n s which change dur i n g the n i g h t . The most r e a l i s t i c model i s represented by an i n i t i a l temperature d i s t r i b u t i o n with depth and the s u r f a c e temperature or s u r f a c e f l u x v a r y i n g as a f u n c t i o n of time. Some s o l u t i o n s to t h i s type of problem are presented by Carslaw and Jaeger (1959). The temperature s o l u t i o n s are g e n e r a l l y f u n c t i o n s i n v o l v i n g c o n d u c t i v i t y and d i f f u s i v i t y and not thermal i n e r t i a alone. The i n i t i a l temperature d i s t r i b u t i o n f o r s u r f a c e s of d i s s i m i l a r thermal p r o p e r t i e s w i l l d i f f e r . A s p e c i a l case of the above c o n d i t i o n s p r o v i d e s a promising set of models. The c o n d i t i o n s are: 1) temperature i s constant f o r a l l depths at the time zero (t=0) ; 2) the s u r f a c e heat f l u x (G) i s zero f o r a l l times l e s s than zero and G(t) afterward. Brunt (1932) proposed a model to p r e d i c t n i g h t t i m e c o o l i n g using the above i n i t i a l c o n d i t i o n s and a boundary c o n d i t i o n G(t) equal to a constant (G ) ( i . e . , a s t e p change i n f l u x from zero to o G at time zero and constant t h e r e a f t e r ) . The s o l u t i o n to t h i s o case i s (Brunt,1932; Carslaw and Jaeger, 1959): P = 2 _ _ ^ o _ _ _ 1/2 (Model T) (2.1) /TT [TS - T s J f o f where P i s thermal i n e r t i a , and Ts and T s , are s u r f a c e o f temperature at time zero (t=0) and the f i n a l time (t=t^) r e s p e c t i v e l y . S o i l heat f l u x i s d e f i n e d i n t h i s study as -36-p o s i t i v e upwards toward the s u r f a c e . Brunt's model w i l l be r e f e r r e d to as model I. Ground heat f l u x to the s u r f a c e (G) w i l l o f t e n decrease duri n g the night as a r e s u l t of the su r f a c e c o o l i n g , and r a d i a t i o n heat l o s s from the su r f a c e d e c r e a s i n g . A second model (model II) i s introduced to account for t h i s decrease i n G during the n i g h t . T h i s model assumes isothermal i n i t i a l temperature d i s t r i b u t i o n , a st e p change i n G from zero to G at time t = n/2 ° 0, and G(t) expressed as G + c t for a l l t > 0. The l i n e a r o form of t h i s boundary c o n d i t i o n i s advantageous f o r remote sensing purposes since only two data a c q u i s i t i o n times are re q u i r e d to sol v e for G^ and c i f n i s known. S o l u t i o n s to the heat conduction equation with G of the form at for n = - 1 , 0, and p o s i t i v e i n t e g e r s are given by (Carslaw and Jaeger, 1959): r _ 1 r(n/2 + 1) 1/2 [n + 1] (2-2) [Ts^ - Ts(t)] 3 r(n/2 + 3/2) C where r r e p r e s e n t s the gamma f u n c t i o n and T s ( t ) i s the sur f a c e temperature at time t . Since the heat conduction equation i s l i n e a r , temperature changes from d i f f e r e n t causes may be added n/2 (superposed). The s o l u t i o n f o r G given as a polynomial i n t i s t h e r e f o r e the sum of the s o l u t i o n s obtained from (2.2) f o r each term. With t = t ^ , the f i n a l time of the model, the s o l u t i o n to model II i s thus: .1/2 Ts f] + c T(n/2 + 1) 1/2 [n + 1] T(n/2 + 3/2) C f (2.3) -37-I t w i l l be shown l a t e r ( s e c t i o n 2.4) that a model with n = 2 ( i . e , a l i n e a r change in f l u x with time, ct) g i v e s s a t i s f a c t o r y r e s u l t s and t h e r e f o r e w i l l be used for model II i n f u r t h e r a n a l y s i s . The f i n a l equation for model II i s : P = [Ts - T s J G o ^ + ° ' 7 5 2 C t f / 2 I < M 0 d e l "> <2'4> The i n i t i a l c o n d i t i o n s for models I and II appear u n r e a l i s t i c at f i r s t . However, near sunset the f l u x e s of the energy balance components change r a p i d l y and reverse d i r e c t i o n p a s s i n g through zero. A l s o near sunset the temperature i n the s o i l i s i n t r a n s i t i o n from being c o o l e r than the s u r f a c e temperature to being warmer at depth. Thus, temperature w i l l be c l o s e to i s o t h e r m a l with depth. At the time when G = 0 the temperature very near the s u r f a c e i s constant with depth. The same argument a p p l i e s to the a i r temperature. Although the i n i t i a l c o n d i t i o n s are not p r e c i s e l y met, these models should p r o v i d e meaningful r e s u l t s . Models I and II d e s c r i b e G as a step change at t = 0 . Although G i s changing very r a p i d l y from zero to a maximum or constant value f o r the n i g h t , there i s a t r a n s i t i o n p e r i o d of s e v e r a l hours during which heat f l u x i n c r e a s e s from zero to i t s maximum (the time course of G dur i n g the n i g h t i s g i v e n by F i g u r e 2.1 and i s d i s c u s s e d i n more d e t a i l i n s e c t i o n 2.4). A model (model II I ) i s proposed to a l l e v i a t e some of the problem. I t assumes an i s o t h e r m a l i n i t i a l c o n d i t i o n at t the time when G = 0 o and a l i n e a r i n c r e a s e i n G from t to a time (t„) when G i s near ^ 1 -38-maximum and then assumes a l i n e a r change (decrease) i n G with time to the f i n a l time (t^) of the model. I f the times are chosen c a r e f u l l y , t h i s boundary c o n d i t i o n can provide a reasonable approximation of the a c t u a l time course of the f l u x e s . The boundary c o n d i t i o n G(t) f o r model I I I i s shown i n F i g u r e 2.1 and may be expressed as: G(t) = bt f o r t £ ' (2.5) where b - IG1 - G G ] / [ t 1 - t Q ] <2'6> and G(t) = G x + d t - tj] f o r t > t1 (2.7) where c = [G(t) - G j l / U - t x l ( 2.8) The s u r f a c e f l u x at t i s zero ( i . e . , G = 0) and time i s o o measured from t ( i . e . , t = 0 ) . Heat f l u x e s G, and G(t) are o o 1 measured ground heat f l u x e s at time t and t r e s p e c t i v e l y . The s o l u t i o n for t > t ^ may be obtained by using Duhammel's s u p e r p o s i t i o n i n t e g r a l ( A r p a c i , 1966). T h i s y i e l d s , with t = t (Appendix I ) : P = 0.752 [ b t 3 / 2 - b ( t f - t l ) 3 / 2 + c ( t f - t x ) 3 / 2 ]/[Ts o - Ts f] (2.9) -39-In a p p l y i n g equation (2.9) to a remote sensing thermal i n e r t i a mapping model, i t must be remembered that use of only two times of data a c q u i s i t i o n i s d e s i r a b l e . These times w i l l be t and t ^ , and have a s s o c i a t e d with them measured ground heat f l u x e s G, and G„ and measured s u r f a c e temperatures Ts, and Ts_. Time t I f 1 f o cannot be used for a data a c q u i s i t i o n s i n c e the time of zero f l u x occurs during s u n l i g h t hours and the advantage of a nighttime model i s l o s t . The temperature change between t and t ^ may be expressed as: Ts„ - Ts f = [TSj_ - Ts f] + [Ts o - T s ^ (2.10) o where, by (2.2) with n = 2, a = b, and t = t : T s o - TSj = [0.752 b t * / 2 ] / P (2.11) Using equations (2.9), (2.10), (2.11), (2.6), and (2.8) thermal i n e r t i a may be w r i t t e n as a f u n c t i o n of t , t ^ , and the measured q u a n t i t i e s G , G^, T s ^ ' a n < ^ Ts^. Model I I I i s t h e r e f o r e g i v e n as P = [AG + BG f]/[Ts 1 - Ts f] (Model III) (2.12) where A . 0 . „ 2 , ^ ' 2 . I V l I l l 3 / 2 . e ; « . [ ^ - t , ] ' " ] ,2.13, c l 1 -40-and 0.752 [ t f - tj] 1/2 (2.14) The s o l u t i o n was v e r i f i e d by comparison of c a l c u l a t e d values of P using (2.12), with values determined using (2.2) f o r each term of a l e a s t squares polynomial f i t of the G(t) of model I I I . 2.3 THEORY: HEAT SHARING APPROACH (MODEL I) The theory of s e c t i o n 2.2 i n d i c a t e s that measurements of the ground heat f l u x must be made. Implementation of the models f o r remote sensing thermal i n e r t i a mapping r e q u i r e s methods of e s t i m a t i n g G compatible with remote sensing surveys and using remote sensing data. Heat sharing theory suggests another approach which avoids d i r e c t e s t i m a t i o n of G. Consequently a heat s h a r i n g approach i s i n v e s t i g a t e d f o r model I. The t o t a l a v a i l a b l e energy at the s u r f a c e (net r a d i a t i o n minus l a t e n t heat f l u x ) i s shared between the atmosphere and the ground. Equation (2.1) can be w r i t t e n f o r the atmosphere by r e p l a c i n g G with H , a constant s e n s i b l e heat f l u x i n t o the o o atmosphere, and P with Pa, the thermal i n e r t i a of the a i r . T h i s g i v e s : -41-The energy balance of a s u r f a c e may be w r i t t e n as: H + G = Rn - LE ( 2 ' 1 6 ) where H i s s e n s i b l e heat f l u x ( p o s i t i v e toward s u r f a c e ) , G i s ground heat f l u x ( p o s i t i v e toward s u r f a c e ) , Rn i s net r a d i a t i o n ( p o s i t i v e from s u r f a c e ) , and LE i s l a t e n t heat f l u x of e v a p o r a t i o n or condensation ( p o s i t i v e toward s u r f a c e ) . Using equations (2.1), (2.15), and (2.16), the thermal i n e r t i a of the ground i s as f o l l o w s : 2 [Rn - LE] 1/2 P " F a Vn [Ts c - Ts f] L f (2.17) I t can be seen from (2.17) that e r r o r s i n determining Pa are d i r e c t l y a p p l i e d to P. Table 2.1 g i v e s some va l u e s of Pa f o r atmospheres of v a r y i n g c o n d i t i o n s . The range of Pa's i s very l a r g e . T h e r e f o r e , the e s t i m a t i o n of Pa must be a c c u r a t e . The thermal i n e r t i a of the atmosphere i s given by: Pa = PC pK^ / 2 (2.18) where p i s the d e n s i t y of the a i r , c^ i s the s p e c i f i c heat of a i r and K i s the thermal d i f f u s i v i t y of a i r . For heat s h a r i n g H a n a l y s i s , a constant or F i c k i a n K i s assumed. In f a c t , i n the H lower part of the atmosphere, K i s a f u n c t i o n of h e i g h t ( z ) . H L o g - l i n e a r p r o f i l e a n a l y s i s says that K i s l i n e a r with z. Under H n e u t r a l and near n e u t r a l c o n d i t i o n s K i s g i v e n b y ( P r i e s t l e y , H 1959) : -42-T a b l e 2.1. Thermal i n e r t i a o f a i r (Pa) C o n d i t i o n Pa ( j m " 2 C " 1 s ~ 1 / 2 ) 1 s t i l l a i r 5 s t i r r e d a i r 1 v e r y s t a b l e 400 1 n e u t r a l 4000 1 ve r y u n s t a b l e 40000 2 calm 400 2 windy 10000 1 P r i e s t l e y , 1959. 2 Tanner, 1974. -43-KJJCZ) = ku^z (2.19) where k i s the von Karman constant and i s the f r i c t i o n v e l o c i t y given by: u A = k U ( z ) / l n ( z / z o ) (2.20) where z i s the s u r f a c e roughness length and U(z) the wind speed o at height z. The parameter K does not depend on s u r f a c e temperature or H a i r temperature and t h e r e f o r e has a g r e a t advantage f o r a p p l i c a t i o n to remote sensing thermal i n e r t i a s t u d i e s . However, fo r s t a b l e and u nstable c o n d i t i o n s : V z ) = k u * z [*H V _ 1 (2.21) where $ and $ are s t a b i l i t y f u n c t i o n s f o r heat and momentum H M r e s p e c t i v e l y (Thorn, 1975). The s t a b i l i t y f u n c t i o n s are c h a r a c t e r i z e d by the Richardson number which, i n t u r n , i n the a p p l i c a t i o n c o n s i d e r e d , must be determined using a i r temperature and s u r f a c e temperature. Table 2.1 g i v e s t y p i c a l thermal i n e r t i a s of a i r for s t a b l e and n e u t r a l c o n d i t i o n s . I t i s apparent that a c o r r e c t i o n f o r s t a b i l i t y must be a p p l i e d and the advantage of the heat s h a r i n g approach ( i . e . , not r e q u i r i n g a i r temperature and s u r f a c e temperature) i s l o s t . There are p r a c t i c a l and t h e o r e t i c a l problems i n determining -44-the value of K which w i l l give a c o r r e c t Pa for thermal i n e r t i a H a n a l y s i s . Important f a c t o r s are: the height over which K (z) H should be i n t e g r a t e d to give good Pa estimates, the nature of the i n t e g r a t i o n , the dependence of K (z) on z, the height at which H K (z) becomes n e a r l y constant, and the e f f e c t s of s t a b i l i t y . At H present, there i s no accepted standard procedure for determining K (and thus Pa) which i s a p p l i c a b l e to the thermal i n e r t i a H problem being c o n s i d e r e d . Since a s t a b i l i t y c o r r e c t i o n must be a p p l i e d and t h e r e f o r e s u r f a c e temperature and a i r temperature must be determined, there i s no advantage i n pursuing the heat s h a r i n g approach f u r t h e r . Consequently, an energy balance approach w i l l be used. The ground heat f l u x w i l l be determined as the r e s i d u a l of the other energy balance components ( i . e . , G = Rn - LE - H) which are estimated by techniques compatible with remote sensing surveys (Chapter 3 ) . 2.4 APPLICATION OF MODELS TO REAL BOUNDARY CONDITIONS The course of G with time during the n i g h t i s i n v e s t i g a t e d using measured f i e l d data and a t y p i c a l course i s d e s c r i b e d . The best times of remote sensing data a c q u i s i t i o n are d i s c u s s e d . I t i s important to f i t the i d e a l i z e d boundary c o n d i t i o n s of the models to the t y p i c a l course of G under v a r i o u s r e s t r a i n t s imposed by p r a c t i c a l and t h e o r e t i c a l c o n s i d e r a t i o n s . The magnitude of i n c o n s i s t e n c i e s i n the r e s u l t s of the models due to non-conformity of the boundary c o n d i t i o n s are demonstrated using measured time courses of G from f i e l d data. -45-A) Time Course of G and Data A c q u i s i t i o n Times The course of G with time i s very important s i n c e the thermal i n e r t i a models are based on a boundary c o n d i t i o n of heat f l u x i n t o the ground expressed as a f u n c t i o n of time. The course of G d u r i n g the n i g h t was analyzed f o r s i x t y - f o u r n i g h t t i m e runs obtained f o r v a r i o u s s u r f a c e types. Twenty-three runs were taken from rangeland s i t e s composed mainly of c r e s t e d wheatgrass near Kamloops, B r i t i s h Columbia (Kamloops data).^" Twenty-seven runs of bare s i l t loam s o i l at A g a s s i z , B.C. (Agassiz data) and f o u r t e e n runs for a bare peat s o i l near Surrey, B.C. (Surrey data) were 2 a l s o used. A l l data were for c l e a r n i g h t s . The predominant time course of G was that shown i n F i g u r e 2.1. The slope and form of the segment C to D may vary somewhat. Peak E i s o f t e n l a c k i n g or not very pronounced. O c c a s i o n a l l y the f l u x near B f l u c t u a t e s c o n s i d e r a b l y . S e v e r a l s i m u l a t i o n s of the energy balance of a s u r f a c e (e.g., Rosema, 1975a; Kahle, 1977) a l s o i n d i c a t e the g e n e r a l trend i n G shown by F i g u r e 2.1. The boundary c o n d i t i o n s f o r models I, I I , and I I I appear a p p l i c a b l e for G's i n the form of F i g u r e "2.1. In d e v e l o p i n g standard procedures for implementing these models i n a remote sensing mode, i t i s c r i t i c a l to determine the most a p p r o p r i a t e times to apply the i d e a l i z e d boundary c o n d i t i o n s of the models to best f i t the a c t u a l i n i t i a l and boundary c o n d i t i o n s . Three important f a c t o r s must be c o n s i d e r e d . 1) Only two times of data a c q u i s i t i o n (sampling times) are d e s i r a b l e . 2) The time d i f f e r e n c e between the two data a c q u i s i t i o n 120 100 80 OJ I 60 40 20 0 GO) MODEL I MODEL U MODEL Hr - 4 i I 2 4 6 8 tf t SUNSET I SUNRISE I TIME (HOURS FROM SUNSET) ,t 1F i g u r e 2.1. T y p i c a l n ighttime course of a c t u a l ground heat f l u x G ( t ) . Dotted l i n e i n d i c a t e s course u n c e r t a i n . A l s o included are the boundary c o n d i t i o n s of models I, I I , and I I I . Times and t£ are times f o r model I I I . -47-periods must be as large as possible (this ensures that the change in temperature of the surface i s large). 3)Data acqu i s i t i o n times must be after sunset, and should not be too close to the period of sunset as the sensible and evaporative heat fluxes are often unsteady and d i f f i c u l t to estimate in that period. Also, surface temperature may change rapidly and be sensitive to recent exposure to solar radiation, so that temperature changes for thermal i n e r t i a analysis may be unreliable. In addition, i t must be assumed that there are no major meteorological events (e.g., large scale advection or influx of cloud cover) which w i l l cause large fluctuations or deviations in the course of G during the night. Some generalizations may be made about the time course of G from the analysis of the available ground heat flux data. The times given below are for surfaces of bare s o i l . Times for vegetated surfaces (usually crested wheatgrass sites) were generally one to two hours l a t e r . Point A of Figure 2.1 represents the time of zero ground heat flux. It commonly occurred between 2.5 and 4.5 hours before sunset. S e l l e r s (1965) gives the time of zero heat flux as usually 3 to 4 hours before sunset. The peak flux i s at point B. This generally occurred very near sunset, usually between 1 hour before sunset and 1 hour after sunset. There i s often a sharp decline in G between B and C. Point C i s the point at which a smooth decline in G s t a r t s . For the data considered, point C occurred from 1 to 2 hours after sunset and about 1 to 1.5 hours after B. The smooth decline in G ceases at point D. Sometimes there i s a short duration peak in G - 4 8 -before i t sharply declines and becomes negative. The time at which point D occurred was approximately 5.5 to 8 hours after sunset. This may be extended by 1 to 1.5 hours i f the r i s e in G does not occur. The sharp decrease in G towards negative values usually occurred at or s l i g h t l y (0.5 to 1.5 hours) before sunrise. S e l l e r s (1965) gives the time of zero heat flux as approximately one hour after sunrise. The sample G's were from spring and summer data. There were about eight to nine hours between sunset and sunrise. For s i t e s of greatly d i f f e r e n t thermal properties, the times at which the various points occur may be' d i f f e r e n t or the whole G curve may be shifted somewhat in time. Vegetated surfaces examined generally had G curves shifted by one to two hours. The average time of zero G (point A) was approximately 3.5 hours before sunset for both the peat s o i l examined and the s i l t loam s o i l data. Point A of a drier s i t e of the s i l t loam s o i l occurred s l i g h t l y (approximately 0.5 hours) l a t e r . Times of the other points were s i m i l a r . One should be able to estimate the times of A, C, and D within about 1 to 1.5 hours of the actual time for a wide range of surface material. Due to the expected time lags between the heat flux curve of s i t e s of d i f f e r i n g thermal properties or slope aspects, the data acqui s i t i o n times must be chosen c a r e f u l l y . The two data acqui s i t i o n times should occur on the same position of the G versus time curve in order to give results which can be compared for a l l s i t e s . For example, the f i t of the idealized boundary conditions of models I, II, and III to the actual G w i l l be vastly d i f f e r e n t i f for one s i t e the sample time occurs at point -49-B while for the others i t occurs at point C of their respective G curves. Results w i l l not be comparable. Therefore, times of large changes in G, such as near B and E, should be avoided as data acquisition times. Also, at B, which i s near sunset, G sometimes fluctuates considerably. The atmospheric fluxes may fluctuate, being d i f f i c u l t to estimate. Strong effects of varying sun exposure on d i f f e r e n t topographic slope and aspects may s t i l l be present. To avoid these problems, the best time to acquire data is to sample in segment C to D. Small s h i f t s in the G curve with time w i l l not make a large difference in the resulting thermal i n e r t i a s . This may not give the best f i t of the model to the actual boundary conditions but results w i l l be more consistent between surveys on d i f f e r e n t nights and more comparable between s i t e s . The temperature change between data acquisition times should be as large as possible; thus, times near points C and D should be used. A five-hour time i n t e r v a l i s l i k e l y the maximum one could expect. From the data considered a general rule may be stated: data acqu i s i t i o n times for models I, II, and III should be about 1.5 and 6.5 hours after sunset or perhaps 2 and 7 hours after sunset. Time of zero G for model III may be approximated by a time 3.5 hours before sunset. More precise times may be determined for a given locale by testing t y p i c a l s i t e s of the area of the thermal i n e r t i a survey either before or simultaneously with the survey. Times w i l l also depend on the length of the period between sunset and sunrise. -50-B) Errors Due to Use of Arbitrary Data Acquisition Times Accepting a r b i t r a r y times of data acquisition introduces some errors and r e s t r i c t i o n s to the three models (I, II, and III) proposed. The idealized G for model II must be a step change from zero to the ground heat flux at the f i r s t data a c q u i s i t i o n time and n/2 then be of the form ct . Values of 1 and 2 were investigated 2 for the variable "n". The r value (r = c o r r e l a t i o n c o e f f i c i e n t ) of least squares f i t s to the measured heat fluxes in segments C to D, as- well as the percent errors in thermal i n e r t i a s compared to that given by a polynomial (5th or 8th order) f i t of portion C to D, were approximately equivalent for n = 2 and n = 1. The linear r e l a t i o n i s perhaps somewhat better and w i l l be adopted for further analysis in this study. Errors w i l l occur due to approximating the ground heat flux in the portion C to D by a linear function. Further error w i l l result from approximating this linear, function using only two data a c q u i s i t i o n times, p a r t i c u l a r l y i f these are chosen by a standard procedure such as using the times of 1.5 and 6.5 hours after sunset. These errors are examined using the measured heat 3 fluxes of the Kamloops, Agassiz, and Surrey data. Errors are summarized in Table 2.2. Errors increase greatly as a result of using only two sampling times but the errors are s t i l l generally small. The errors of model II are much smaller than for model I. Additional errors w i l l r e s u l t when the G curves of d i f f e r e n t s i t e s are not aligned in time. The two data points used in implementing the model w i l l not always represent G*s in the same position of the G curve. Assumption of the wrong time for zero G T a b l e 2.2. Average e r r o r s i n using a l i n e a r approximation and usi n g data of sampling t i m e s . 1 Model Data best f i t of C to D p o r t i o n ^ G's at of C s t a r t to D and end p o r t i o n G's at sampling times 3 % e r r o r no. of samples % e r r o r no. of samples % e r r o r no. of samples Kamloops 5.7 ± 4.2 19 9.5 ± 8.7 16 10.0 ± 6.7 16 I A g a s s i z 6.5 ± 4.3 26 10.4 + 7.4 23 9.6 + 7.3 26 Surrey 5.4 + 2.6 7 7.1 + 4.4 7 9.4 + 5.2 14 Kamloops 0.6 + 0.3 19 3.7 + 3.1 20 6.1 + 4.9 20 II A g a s s i z 1.2 + 1.5 25 6.6 + 4.0 23 6.5 + 4.3 26 Surrey 0.6 + 0.3 7 3.7 + 2.2 7 4.7 + 3.1 14 E r r o r s compared a g a i n s t thermal i n e r t i a s c a l c u l a t e d using a model fo r a f i f t h order polynomial f i t o f a p p r o p r i a t e p o r t i o n o f G curve. Percentage e r r o r i s gi v e n as the mean and standard d e v i a t i o n of the a b s o l u t e v a l u e o f the e r r o r expressed as percentage of the average of the polynomial f i t model and model used. Mean of data i n C to D p o r t i o n for model I; l i n e a r best f i t of data i n C t o D p o r t i o n f o r model I I . ^ Sampling times are the average times a f t e r sunset of p o i n t s C and D fo r each s e t of data used. -52-in model III introduces further error. Table 2.3 gives estimates of the size of these errors for s h i f t s in time of various lengths. Also given are errors in model III caused by s h i f t i n g both the sample time and zero time simultaneously. A portion of the errors given may be due to an error of up to 10 percent in the predicted thermal i n e r t i a caused by erroneous input values of 3 surface temperature. Table 2.3 shows that Model III i s less sensitive to s h i f t s in the G curves and sample times than is model II or I. Reasonable choices of sampling times w i l l generally be within the C to D portion of the G curves for most s i t e s . Errors from this source w i l l be small. It may be expected that the major effect of slope aspect on the time course of G w i l l be to s h i f t the time of zero G. A sample time at 1.5 or 2 hours after sunset should be in the C to D portion of the G curve for most s i t e s , p a r t i c u l a r l y for those s i t e s with aspects which are not west. Errors caused by a s h i f t in sample time due to aspect are not expected to be large. Table 2.3 indicates that model III i s not very sens i t i v e to s h i f t s in zero time. Therefore s h i f t s in actual times of zero G due to d i f f e r e n t slope aspect w i l l not cause large errors. It should thus be possible to choose a zero time and sampling times to accommodate most s i t e s of d i f f e r i n g thermal properties and slope aspect. Probable errors in thermal i n e r t i a for model III due to time s h i f t s in the boundary conditions between s i t e s are expected to be less than 20 percent. Tab le 2.3. E r r o r s due to non-alignment of G curves with sampling times (Agassiz d a t a ) . 1 Model s h i f t of 0.5 hour s h i f t 1.0 hour s h i f t 1.5 hour s h i f t % e r r o r ^ no. of samples % e r r o r no. of samples % error no. of samples I sample times 12.3 + 8.6 75 23.0 + 16. 1 50 31.1 + 16 .8 24 II sample times 12.5 + 8.0 75 23.9 + 15. 0 50 32.8 + 15 .4 24 III sample times 6.7 + 4.9 75 12.6 + 6. 2 49 17.7 + 11 .6 24 sample and zero times 9.9 + 5.9 69 14.4 + 9. 0 46 29.3 + 12 .0 23 zero times-* 4.2 + 8.4 162 4.7 + 9. 3 135 6.2 + 12 .7 108 Model s h i f t of 2.0 hour s h i f t 2.5 hour s h i f t 3.0 hour s h i f t % e r r o r no. of samples % e r r o r no. of samples % error no. of samples III zero times-* 9.8 + 20.2 87 12.3 35. 4 54 23.3 48 .1 27 1 T y p i c a l e r r o r s which may occur in implementing the models (errors due to non-alignment of G curves) can be determined by s h i f t i n g the sample times for each s i t e and each n i g h t . These e r r o r s w i l l be repr e s e n t a t i v e of the e r r o r s expected due to sampling on d i f f e r e n t p o r t i o n s of the G curve for d i f f e r e n t s i t e s . The G and temperature at the pre-dawn time i s r e l a t i v e l y s t a b l e . Thus, e r r o r s from s i t e to s i t e due to s h i f t s i n the pre-dawn G should not be s i g n i f i c a n t . Only the post-sunset time i s s h i f t e d . Half hour data for times from 1.5 to 3 hours a f t e r sunset are used. * Percent error i s given as the mean and standard d e v i a t i o n of the absolute value of the er r o r expressed as percentage of the average thermal i n e r t i a s for the two times being compared. 3 Sample times remain constant. -54-2.5 TESTING OF MODELS Figure 2.2 compares results of models II and III for a bare peat s o i l (Surrey data) with theoretical values for a peat s o i l of the same porosity (van Wijk and de Vries, 1963) and measured 2,4 thermal i n e r t i a . Surface temperatures and heat fluxes at 2100 h (t^; approximately 1.5 hours after sunset) and 0300 h (t^) were used. 5 Time of zero flux (t ) for this and subsequent tests of o the model (Agassiz data and rangeland sites) was taken to be the actual time of zero flux (approximately 5.0 hours before t ^ ) . Site #1 was well drained; s i t e #2 poorly drained. Both s i t e s have a porosity of approximately 80 percent. The f i e l d was ploughed between nights 14 and 26. The porosity of s i t e #1 increased to 85 percent after ploughing. Site #3 was located at a compacted s i t e ( t i r e track) near s i t e #1. Its porosity was approximately 80 percent. Model III shows good agreement with the measured and th e o r e t i c a l thermal i n e r t i a s . The trend of model II with moisture content i s correct but the model overestimates thermal i n e r t i a as might be expected from t h e o r e t i c a l consideration of the boundary conditions. Model I gave similar results to model II. The high predicted values of thermal i n e r t i a of models I and II may be explained as follows. Assuming isothermal conditions at the time of zero flux t there w i l l be a surface temperature o change between time t and t , . Models I and II s t a r t at t. o 1 1 (Figure 2.1) and assume isothermal conditions. They w i l l therefore predict a larger temperature change between t^ and t^ than w i l l actually occur. They do not account for the temperature change which has already occurred since the time ( t Q ) -55-2500 2000 i in i O CM Ul z 1500 cc 1000 UJ X I-500 I — THEORETICAL PEAT (porosity 80%) MEASURED _•_,..€>..,..*.. SITE no.1,2,3 MODEL U • , - 0 - , - X " SITE no.1,2,3 MODEL HI - • « > - S I T E no.1,2,3 7 NIGHT OF TRIAL 20 40 60 80 VOLUMETRIC WATER CONTENT (%) F i g u r e 2 . 2 Test of models n and I I I f o r a bare peat s o i l (Surrey d a t a ) . -56-when the temperature d i s t r i b u t i o n was actually isothermal. Inputting the actual measured (smaller) temperature change between time t and t w i l l therefore give a high thermal i n e r t i a estimate. The model results are consistent for a given night. The trend of the thermal i n e r t i a s for the two s i t e s follow very c l o s e l y ( i . e . , for a given night, i f the predicted thermal i n e r t i a is high r e l a t i v e to other nights, i t is high for both s i t e s on that night). The a p p l i c a b i l i t y of the i n i t i a l conditions and course of G of the models w i l l vary from night to night. Consistency of the model i s expected to be greater when thermal i n e r t i a s for the same night are compared. A second test of the model i s for a cultivated s i l t loam s o i l (Agassiz data) with porosity of about 60 to 70 percent. 2,6 Results for model III are shown in Figure 2.3. Site #1 was culti-packed and s i t e #2 disced. The porosity of s i t e #2 i s s l i g h t l y greater. Surface temperature and heat flux measurements were for 2200 h for most nights and 2230 h for nights later in the summer. These times correspond to times between 1.5 and 2.0 hours after sunset. The second sample times were 0400 h and 0430 h respectively. Agreement of Model III with measured thermal i n e r t i a i s good for most nights. Both measured and model III values correspond to theo r e t i c a l values for a s o i l of approximately 25 percent sand and 75 percent clay, and 60 to 70 percent porosity. Model III produces some anomalous thermal i n e r t i a estimates (e.g., nights 21, 26, and 70 to 73). On nights 71 through 73 the estimates for both s i t e s #1 and #2 were high by similar amounts compared to the 2000 h .-I CI i O UJ z < cc U J X 1500 1000 500 , 1 1 THEORETICAL (.75% CLAY, 25% SAND) 60% POROSITY 70% POROSITY • MEASURED SITE no.1 o MEASURED SITE no. 2 - MODEL HE SITE no.1 A MODEL JJE SITE no.2 20 NIGHT OF TRIAL 20' 22 A 20 r - 7 # 0 ^ ^ 6 \ 5 8 4 1 40 " . 66. 72 71 73 21 I I 10 15 20 25 VOLUMETRIC WATER CONTENT (%) 30 35 F i g u r e 2.3. Te s t of model I I I for a bare s i l t loam s o i l (Agassiz d a t a ) . -58-measurement values. On night 70 both si t e s #1 and #2 estimates were s l i g h t l y low. This suggests that there was perhaps some meteorological condition or history of heating for these nights which implies that the i n i t i a l condition assumed by model III (i . e . , isothermal p r o f i l e ) was erroneous. The boundary conditions (heat flux as a function of time) for these cases appear similar to other nights. The factor causing the errors gives a similar error for the two sit e s although they have d i f f e r i n g thermal properties. As in the test of the peat s o i l (Figure 2.2) there is consistency in the model for a given night. The results of models I and II followed the same trend as model III but slopes (P versus volumetric water content) and values of P were larger as was the case for Figure 2.2. Scatter of the data is also considerably more, p a r t i c u l a r l y for Model I. The models were also tested on natural s o i l surfaces. Sites were located on rangeland t e r r a i n . Heat fluxes (G) were measured with heat flux plates and surface temperature with an infrared thermometer. A plot of thermal i n e r t i a determined by model III versus volume f r a c t i o n of water is given (Figure 2.4). The s o i l s generally had a porosity of 50 to 65 percent, although there was often a dry vesicular crust of about 4 mm at the surface. The s o i l moisture used is the average of the top 4 mm layer and a layer from 4 mm to 4 cm with equal weighting to the two layers. Scatter may be due to d i f f e r i n g thermal i n e r t i a s of the s i t e s or differences among the nights used to test the model. Thermal i n e r t i a s for the same s i t e show consistency. The trend of thermal i n e r t i a with moisture content is clear and the values of P are reasonable for the range of s o i l types occurring at the V O L U M E T R I C W A T E R C O N T E N T ( % ) F i g u r e 2.4. Thermal i n e r t i a estimates using model I I I f o r rangeland s i t e s (data p o i n t s i n d i c a t e d by matching symbols other than "•" are data from the same s i t e ) . -60-s i t e s . The a b i l i t y of the models to d i s c r i m i n a t e s u r f a c e s of d i f f e r i n g thermal i n e r t i a s and t h e i r u s e f u l n e s s f o r s o i l moisture s t a t u s s t u d i e s i s e v i d e n t . I t appears, however, that accurate estimates of thermal i n e r t i a using models I and II w i l l r e q u i r e c a l i b r a t i o n v i a t e s t s i t e s of known thermal i n e r t i a . Model I I I appears to g i v e reasonable q u a n t i t a t i v e thermal i n e r t i a s although i t may a l s o r e q u i r e c a l i b r a t i o n i n order to produce c o n s i s t e n t l y good q u a n t i t a t i v e r e s u l t s . 2.6 CONCLUSIONS Test s of the models developed (models I, I I , and III) demonstrate that they p r o v i d e meaningful r e s u l t s . However, r e s u l t s i n d i c a t e that the i n i t i a l and boundary c o n d i t i o n s o f model I and II do not s u f f i c i e n t l y r e p r e s e n t r e a l i t y t o g i v e a c c u r a t e thermal i n e r t i a v a l u e s although they do p r o v i d e u s e f u l r e l a t i v e r e l a t i o n s h i p s of thermal i n e r t i a . Model I I I i s based on reasonable i n i t i a l and boundary c o n d i t i o n s and i s s t i l l r e l a t i v e l y simple. I t i s the l e a s t s e n s i t i v e of the three models to s h i f t s i n the G curves i n r e l a t i o n to the times of data a c q u i s i t i o n . T e s t s of model I I I with ground data i n d i c a t e i t g i v e s c o n s i s t e n t r e l a t i v e r e l a t i o n s h i p s of thermal i n e r t i a and f r e q u e n t l y y i e l d s a c c u r a t e thermal i n e r t i a v a l u e s . However, o c c a s i o n a l s i t e s may g i v e l a r g e e r r o r due to non-conformity of the i d e a l i z e d i n i t i a l and boundary c o n d i t i o n s and data a c q u i s i t i o n times of the model, to the a c t u a l time course o f ground heat f l u x . Required data f o r model I I I a r e ground heat -61-f l u x and s u r f a c e temperature a t two sample times d u r i n g the n i g h t and an e s t i m a t e o f the time o f z e r o f l u x . Use o f model I I I w i t h time of z e r o G a t 3.5 hours b e f o r e sunset and ground heat f l u x and s u r f a c e temperature d a t a a c q u i s i t i o n t imes o f 1.5 and 6.5 hours a f t e r s u n s e t may be expected t o g i v e good r e s u l t s f o r a wide v a r i e t y o f s i t e s i n a s u r v e y a r e a . The s a m p l i n g times and time of z e r o G may be s p e c i f i e d more p r e c i s e l y f o r a p a r t i c u l a r s u r v e y a r e a by t e s t i n g ground heat f l u x a p r i o r i or on the day of t h e s u r v e y . The v a l u e s o f ground heat f l u x , s u r f a c e t e m p e r a t u r e , and time a r e then used i n e q u a t i o n (2.12) t o o b t a i n the t h e r m a l i n e r t i a o f the s u r f a c e . G i v e n a c c u r a t e remote s e n s i n g methods of e s t i m a t i n g ground heat f l u x and s u r f a c e t e m p e r a t u r e , model I I I o f f e r s a good b a s i s f o r the development of a n i g h t t i m e remote s e n s i n g t h e r m a l i n e r t i a mapping method. Data p r o v i d e d by R.J. W i l l i a m s of B.C. M i n i s t r y o f Environment, Resource A n a l y s i s Branch, Kamloops, B.C. S u r f a c e ground heat f l u x i s d e r i v e d from heat f l u x p l a t e s a t 5 cm and an i n t e g r a t i n g thermometer. Data a r e h a l f - h o u r v a l u e s . Data f o r both the A g a s s i z and S u r r e y s i t e s were p r o v i d e d by M.D. Novak, Dept. of S o i l S c i e n c e , U n i v e r s i t y of B r i t i s h C o l umbia. S u r f a c e heat f l u x was c a l c u l a t e d by n u l l - a l i g n m e n t t e c h n i q u e s u s i n g t e m p e r a t u r e p r o f i l e d a t a s t a r t i n g a t a depth o f 7 mm. A g a s s i z d a t a a r e h a l f - h o u r v a l u e s ; S u r r e y d a t a two-hour v a l u e s . -62-For Tables 2.2 and 2.3, A g a s s i z s i t e heat f l u x data are from heat f l u x p l a t e s (at 5 cm) and i n t e g r a t i n g thermometer data; s u r f a c e temperature data are e x t r a p o l a t e d by hand from a 7 mm depth for most runs and 2 mm for the o t h e r s . 4 Surface temperature and thermal i n e r t i a v alues were provided by M. D. Novak of the S o i l Science Department, U n i v e r s i t y of B r i t i s h Columbia. Surface temperatures were determined by c u b i c s p l i n e techniques using temperature p r o f i l e data from a depth of 7 mm downward. The measured thermal i n e r t i a s were c a l c u l a t e d using heat c a p a c i t i e s determined from the volume f r a c t i o n of s o l i d s and water, and thermal c o n d u c t i v i t i e s d e r i v e d from the measured temperature p r o f i l e s and s o i l heat f l u x . The average of the thermal i n e r t i a s c a l c u l a t e d for depths 1.0, 1.5, and 2.0 cm were used. S o i l moisture i s for 0 to 5 cm. ^ Sample times were d i c t a t e d by a v a i l a b i l i t y of data. ^ Surface temperatures and data were provided by M. D. Novak of the Dept. of S o i l S c i e n c e , U n i v e r s i t y of B r i t i s h Columbia. Surface temperatures were c a l c u l a t e d by c u b i c s p l i n e techniques from temperature p r o f i l e data at depths 7 mm and g r e a t e r . Measured thermal i n e r t i a s were determined from heat c a p a c i t i e s d e r i v e d from the volume f r a c t i o n of s o l i d s and water at 1 cm, and thermal c o n d u c t i v i t i e s c a l c u l a t e d using the average heat f l u x e s and temperature g r a d i e n t s f o r the n i g h t i n the top 1 cm. The s o i l moisture at 1 cm i s used. -63-Chapter Three ESTIMATING ENERGY BALANCE COMPONENTS 3.1 INTRODUCTION Thermal i n e r t i a models I, II, and III require that the heat flux into the surface (G) be known for two times during the night. The heat flux into the surface can be determined as the residual of the other energy balance components: net radiation (Rn), sensible heat flux (H), and latent heat flux (LE). The methods used to determine these energy balance components must be commensurate with the c r i t e r i a (section 1.5) for remote sensing thermal i n e r t i a mapping methods. Of primary concern is that limited knowledge of the surface type and a limited number of ground meteorological measurements be required. Net radiation may be estimated using simple procedures. A method of determining Rn from remotely sensed temperature i s b r i e f l y described. Sensible heat flux is governed by complex interactions of air at the surface of a medium and is a function of a i r temperature, wind speed, surface roughness, and surface temperature. Determination of sensible heat flux using remotely sensed temperature is discussed in d e t a i l . Since latent heat flux i s commonly small and not very s p a t i a l l y variable during the night, only approximate methods are required. Several possible approximation procedures are suggested. - 6 4 -3.2 DETERMINATION OF NET RADIATION Net radiation during the night consists of three components: the downward longwave radiation both from the sky and the surroundings, the radiation emitted from the surface, and the longwave radiation reflected upward from the surface. The total emitted radiation E^ is given by the Stephan-Boltzmann Law as: E T = € 0 T s * (3.1) where o i s the Stephan-Boltzmann constant, E i s emissivity, and Ts is surface temperature. The emissivity of a l l locations in a survey area is not generally known ( c r i t e r i a of section 1.5) therefore emissivity must be assumed to be unity or approximate';! by the average value of the expected e m i s s i v i t i e s for the surface types in the survey area. Reflected longwave radiation must be accounted for (Fuchs and Tanner, 1966). If calibrated remotely sensed temperature (Tr) is used as an estimate of surface temperature, an estimate of the t o t a l longwave downward radiation B is made, and an emissivitv of the surface ca i s assumed, net T radiation may be determined by: Rn = Ca 0 Tr 4 - Ca B^ , (3.2) The problem of estimating surface temperature from thermal infrared line-scan data i s discussed in Appendix I I . Instrument (system) error, c a l i b r a t i o n error, atmospheric attenuation, and the e f f e c t of longwave downward radiation and assumed emissivity are considered. Longwave downward radiation for equation (3.2) -65-can be estimated by e i t h e r d i r e c t measurement with ground i n s t r u m e n t a t i o n or by one of s e v e r a l s e m i - e m p i r i c a l formulae for determining longwave r a d i a t i o n (B^) from the atmosphere (Brunt, 1932; Swinbank 1963; Idso and Jackson, 1969; M i t c h e l l et a l . , 1975; P a l l a n d , 1975). I t i s advantageous to measure longwave downward r a d i a t i o n d i r e c t l y , s i n c e the term i s important to the r e s u l t s of the model. I m p l i c i t i n the above a n a l y s i s i s that longwave downward r a d i a t i o n over the e n t i r e survey area i s uniform and equal to that measured or estimated. T h i s w i l l not be the case f o r p i x e l s ( p i c t u r e elements or r e s o l u t i o n c e l l s of the thermal l i n e - s c a n data) with a view f a c t o r , with r e s p e c t to i t s surroundings, which i s not c l o s e to zero. For v a l l e y bottoms some e r r o r may r e s u l t . 3.3 DETERMINATION OF SENSIBLE HEAT FLUX There has been much work i n developing methods to c a l c u l a t e s e n s i b l e heat f l u x u t i l i z i n g ground based i n s t r u m e n t a t i o n . S e v e r a l are a p p l i c a b l e to remote sensing techniques (Fuchs et a l . , 1969; O u t c a l t , 1972), but very few designed p a r t i c u l a r l y f o r remote sensing a p p l i c a t i o n s (Kahle, 1977; S c h i e l d g e , 1978; Soer, 1980). Reasonable c r i t e r i a f o r a remote sensing heat f l u x method are as f o l l o w s : 1) i t uses r a d i o m e t r i c s u r f a c e temperatures; 2) i t accounts f o r atmospheric s t a b i l i t y ; 3) i t i s a p p l i c a b l e on a m i c r o m e t e o r o l o g i c a l s c a l e ; 4) i t r e q u i r e s l i m i t e d ground input ( i . e . , few parameters and few l o c a t i o n s ) ; 5) i t r e q u i r e s l i m i t e d knowledge of the s u r f a c e ( i . e . , the s u r f a c e roughness parameters -66-may be estimated and need not be p r e c i s e l y determined); and 6) i t i s simple. The techniques r e f e r r e d to above assume that the r a d i o m e t r i c s u r f a c e temperature can be simply used i n the aerodynamic heat t r a n s f e r equations. S c h i e l d g e (1978) demonstrates that t h i s assumption can cause s e r i o u s e r r o r . Kahle's method (1977) uses a mesocale atmospheric f l u x model of Burke (1945). I f a method cannot meet c r i t e r i a 4) and 5) i t s u s e f u l n e s s for g e n e r a l remote sensing surveys i s l i m i t e d . For s i t e s p e c i f i c s t u d i e s d e t a i l e d measurements and knowledge of the s u r f a c e are o f t e n a v a i l a b l e and implementation of the model remains u s e f u l . The s i x t h c r i t e r i o n r e c o g n i z e s that use of d i g i t a l thermal i n f r a r e d l i n e - s c a n data w i l l i n v o l v e a l a r g e amount of data p r o c e s s i n g p a r t i c u l a r l y i f s e n s i b l e heat f l u x mapping i s d e s i r e d on a p i x e l by p i x e l b a s i s . The s e n s i b l e heat f l u x method d e s c r i b e d i n t h i s s e c t i o n (3.3) uses an aerodynamic approach and i s developed to s a t i s f y the above c r i t e r i a . The input i s s u r f a c e temperature from a thermal i n f r a r e d l i n e scanner, a i r temperature and wind speed at one height above the s u r f a c e , and s u r f a c e roughness l e n g t h (z^) and zero plane displacement (D). The method, termed the s u r f a c e / a i r temperature (SAT) method i s t e s t e d on a bare and a vegetated s u r f a c e using ground based data. 3.3.1 DEVELOPMENT OF THE SURFACE/AIR TEMPERATURE (SAT) METHOD A) Aerodynamic Equation of S e n s i b l e Heat T r a n s f e r The equation i n the aerodynamic method f o r determining the -67-v e r t i c a l f l u x of s e n s i b l e heat to or from the e a r t h ' s surface i s as f o l l o w s (Thorn, 1975): , Cu(z.)-U(z.)] [T(z 9)-T(z . ) 3 H = pc k 2 — J i 1 - — [ O T 1 (3.3) where p i s the d e n s i t y of a i r , c i s the s p e c i f i c heat of a i r , k P i s von Karman's constant ( =0.4), T(z^) and T(z^) are the temperatures at he i g h t s z^ and z^ r e s p e c t i v e l y , u ( z 2 ^ a n d u ( z ^ ) are the wind speeds at the same r e s p e c t i v e h e i g h t s , and $ and $ H M are the s t a b i l i t y f u n c t i o n s for heat and momentum r e s p e c t i v e l y . A p o s i t i v e value of H i m p l i e s heat i s flowing toward the s u r f a c e , as i s u s u a l l y the case at n i g h t . The s u r f a c e roughness le n g t h (z ) i s d e f i n e d as the height at which the wind speed o e x t r a p o l a t e s to zero when p l o t t e d a g a i n s t the l o g a r i t h m of h e i g h t . Consequently, s u b s t i t u t i n g z^ f o r z^ i n equation (3.3) g i v e s : 2 [T(z )-T(z )] H = pc k U(z ) 1 ° _ [• A T * (3.4) P [ l n ( z 9 / z ) ] 2 which r e q u i r e s measurement of wind speed and a i r temperature at a s i n g l e height only, assuming z and T(z ) can be determined. The o o s t a b i l i t y f u n c t i o n s can be expressed as a f u n c t i o n o f the s t a b i l i t y parameter, known as the Richardson number ( R i ) . The Richardson number for the l a y e r z to z . i n f i n i t e d i f f e r e n c e o 2 form, i s as f o l l o w s (Thorn, 1975) : -68-g CT(z )-T(z )_ Ri 2 ° [z -z 1 <3-5> T U(z„r 2 0 where g i s the g r a v i t a t i o n a l a c c e l e r a t i o n and T i s the mean temperature of the l a y e r which may be approximated by {T(z -)+T(z o) }/2. For s t a b l e c o n d i t i o n s ( u s u a l l y found at n i g h t ) , Thorn (1975) suggests, on the b a s i s of work by Webb (1970), Munn (1966), and Lumley and Panofsky (1964), the f o l l o w i n g semi-e m p i r i c a l r e l a t i o n s h i p : 4»H = $ M = [1-5R1]" 1 (3.6) B) Sublayer C o r r e c t i o n A problem a r i s e s i n mode l l i n g s e n s i b l e heat f l u x when su r f a c e temperature ( r a d i o m e t r i c ) i s used r a t h e r than the t h e o r e t i c a l l y c o r r e c t a i r temperature at Z q , and has been recognized by Fuchs et a l . (1969), C l a r k e (1970), and S c h i e l d g e (1978). The problem i s that at rough s u r f a c e s momentum t r a n s f e r i s enhanced by b l u f f body e f f e c t s , w h i le s e n s i b l e heat t r a n s f e r i s not. T h i s means the r e s i s t a n c e to s e n s i b l e heat t r a n s f e r i s l a r g e r than that of momentum. Consequently, the s e n s i b l e heat f l u x can be w r i t t e n i n terms of T ( o ) , the s u r f a c e temperature, and an a d d i t i o n a l 'sublayer' r e s i s t a n c e ( r C T ) as f o l l o w s : -69-where r , the r e s i s t a n c e to momentum t r a n s f e r , i s given by: M rM = l n ( z 2 / z o ) / u * k ( 3 * 8 ) i n which u^, the f r i c t i o n v e l o c i t y , i s : «* - kU(z ) [ l n ( z /z )Tli>M_1 (3.9) I t i s e a s i l y v e r i f i e d that i n equation (3.4) only r i s present. M The r e s i s t a n c e of the sublayer (r ) can be r e l a t e d to the 5 L sublayer Stanton number (B) developed by Owen and Thomson (1963), who hypothesized that at a rough s u r f a c e , a sublayer develops i n which heat i s t r a n s f e r r e d from the su r f a c e to the main flow above the l a y e r . T h i s heat t r a n s f e r i s p r o p o r t i o n a l to B. Chamberlain (1966, 1968), Thorn (1972), and G a r r a t t and Hicks (1973) d i s c u s s the use of the sublayer Stanton number. The excess r e s i s t a n c e to s e n s i b l e heat t r a n s f e r due to the sublayer i s given by Thorn (1972) as f o l l o w s : rSL = B 1 / u * (3.10) S u b s t i t u t i n g equations (3.8), (3.9), and (3.10) i n t o (3.7) and re a r r a n g i n g g i v e s an ex p r e s s i o n f o r H of the form of equation (3.4) m u l t i p l i e d by a sublayer c o r r e c t i o n f a c t o r Q as f o l l o w s : 7 CT(z )-T(o)D H - pc kZU(z.) rI* H* MD Q P 2 Cln(z./z)D2 z o (3.11) -70-w h e r e : ni + — ! * l L _ r 1 l n ( z 2 / z o ) - 1 (3.12) T e m p e r a t u r e T ( o ) r e p l a c e s T ( Z Q ) i n t h e f i n i t e d i f f e r e n c e e q u a t i o n (3.5) f o r R i . The l a y e r o v e r w h i c h R i i s a p p r o p r i a t e i s s l i g h t l y i n e r r o r as i t d e p e n d s on b o t h z ^ and t h e e f f e c t i v e r o u g h n e s s l e n g t h f o r H. I t c a n be shown t h a t t h e e r r o r i s v e r y s m a l l and t h e r e i s no need t o r e f o r m u l a t e e q u a t i o n ( 3 . 5 ) . An e x p r e s s i o n f o r t h e e f f e c t i v e r o u g h n e s s l e n g t h f o r s e n s i b l e h e a t (z ) (or i t s e f f e c t i v e s o u r c e or s i n k h e i g h t ) c a n H be o b t a i n e d by w r i t i n g t h e t o t a l r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r (r + r ) by a n a l o g y w i t h e q u a t i o n (3.8) as f o l l o w s M SL (Thorn, 1972) : r M + r S L " I n ( z 2 / Z H ) / U * k < 3 ' 1 3 > S u b s t i t u t i n g e q u a t i o n s (3.8) and (3.10) i n t o (3.13) g i v e s : z u = z e ~ k B 1 (3.14) The T ( o ) o f e q u a t i o n (3.7) i s t h e t e m p e r a t u r e a s s o c i a t e d w i t h t h e a c t u a l s o u r c e or s i n k o f h e a t and z i s t h e e f f e c t i v e s o u r c e o r H s i n k h e i g h t when a l o g - l i n e a r t e m p e r a t u r e p r o f i l e r e l a t i o n s h i p i s u s e d t o d e t e r m i n e H . Th e t e m p e r a t u r e a t h e i g h t z o f t h e l o g -l i n e a r t e m p e r a t u r e p r o f i l e i s T ( o ) . When t h e s u r f a c e b e i n g s e n s e d by a r a d i o m e t e r i s t h e a c t u a l s o u r c e o r s i n k f o r h e a t , -71-T(o) equals the r a d i o m e t r i c surface temperature. C) Determining the Value of B ^ to be Used Owen and Thomson (1963) propose a r e l a t i o n s h i p of the f o l l o w i n g form: B 1 = a Cu^h/u]"1 a n (3.15) where a i s a constant for a p a r t i c u l a r roughness, h i s the e q u i v a l e n t sand roughness height (=30 z ), u i s the f r i c t i o n o * v e l o c i t y , u i s the kinematic v i s c o s i t y , a i s the P r a n d t l number, and m and n are c o n s t a n t s . Using data for roughened a r t i f i c i a l s u r f a c e s , they found the f o l l o w i n g r e l a t i o n s h i p : B _ 1 = 1.85 R e A ° - A 5 (3.16) where Re #, the roughness Reynolds number, i s : R e * * u * z /u (3.17) Th e i r a n a l y s i s i s based on the absence of a normal undisturbed (viscous) sublayer due to roughness which occurs f o r Re^ l a r g e r than approximately t h r e e . The parameter a w i l l vary with the p a r t i c u l a r roughness of a s u r f a c e . S e v e r a l e m p i r i c a l f o r m u l a t i o n s f o r B ^ a p p l i c a b l e to 0.44 n a t u r a l s u r f a c e s are as f o l l o w s : 0.23 Re^ * (S c h i e l d g e (1978) 0 45 fo r a grass f i e l d ) , 1.30 Re # * ( G a r r a t t and H i c k s (1973) f o r a 0.45 v a r i e t y of a r t i f i c a l and n a t u r a l s u r f a c e s ) , and 0.13 Re, • * -72-(Monin and Z i l i t i n k e v i c h , 1968). Thorn (1972) suggests the 1/3 e x p r e s s i o n bu^ for vegetated s u r f a c e s . The constant b v a r i e s with the v e g e t a t i o n type (e.g., he found b to be 1.35 f o r a bean c r o p ) . G a r r a t t and Hicks (1973) use a v a r i e t y of data to show that t h e i r e x p r e s s i o n f o r B ^ g i v e s good r e s u l t s f o r Re^ from 5 to 100 f o r most types of s u r f a c e s and i s good for Re^ up to 500 for s o l i d , w e l l spaced roughness elements. For f i b r o u s elements B ^ i s almost constant for Re A > 100. There i s a s l i g h t decrease -1 4 i n B from about 5 to 3 f o r Re^ from 100 to 10 . For Re^ < 5 the Owen and Thomson (1963) form of B 1 does not f i t the data as might be expected from theory. The e x p r e s s i o n : ZH = K / k u * (3.18) where K i s the thermal d i f f u s i v i t y of a i r , i s a b e t t e r r e p r e s e n t a t i o n of the data for Re^ < 5 and g i v e s a good f i t up to an Re^ of approximately 30 to 100. The e x p r e s s i o n (3.18) i s given by Sheppard (1959) and i s d i s c u s s e d by G a r r a t t and Hicks (1973). Using equations (3.14), (3.17), and (3.18): B"1 = [ln(Re f cak)]/k (3.19) I t i s apparent t h a t , f o r accurate d e t e r m i n a t i o n of B ^ and the sublayer c o r r e c t i o n f a c t o r Q, d e t a i l e d knowledge of the nature of the s u r f a c e roughness elements i s necessary. A c c e p t a b l e r e s u l t s may, however, be expected using the f o l l o w i n g g e n e r a l i z a t i o n s . For most s u r f a c e s , e x p r e s s i o n (3.19) w i l l g i v e good r e s u l t s for Re^ up to 100. T h i s w i l l p a r t i c u l a r l y apply to -73-n i g h t t i m e determinations of s e n s i b l e heat f l u x s i n c e wind speed and t h e r e f o r e Re^ i s o f t e n low. For Re # g r e a t e r than 100 large e r r o r s may occur u n l e s s the nature of the s u r f a c e roughness i s taken i n t o account; however, most n a t u r a l s u r f a c e s w i l l have valu e s of B ^ ranging from 5 to 3, d e c r e a s i n g s l i g h t l y with i n c r e a s i n g Re^. For p r a c t i c a l purposes, use of e x p r e s s i o n (3.19) fo r Re^ < 30 and a value of 4.5 f o r Re^ > 30 should g i v e good r e s u l t s . 3.3.2 TEST OF THE SAT METHOD FOR A BARE SURFACE The method was t e s t e d f o r a non-vegetated s u r f a c e using data of Kahle et a_l. (1977) . The s i t e was a bare f i e l d with very sparse f l a t t e n e d m i l o s t u b b l e . Surface roughness was 1 mm. A i r temperature, humidity, and wind speed p r o f i l e s were measured using seven h e i g h t s . A i r temperature and wind speed at 1.5 m was measured by separate instruments l o c a t e d c l o s e to the p r o f i l e measurement system. Surface temperatures were measured with a Barnes P r e c i s i o n Radiometer (PRT-5) at i r r e g u l a r i n t e r v a l s and temperatures at given times i n t e r p o l a t e d by hand drawn c u r v e s . S e n s i b l e heat f l u x e s c a l c u l a t e d by S c h i e l d g e (1978) using the p r o f i l e data are used as standard v a l u e s with which to e v a l u a t e the r e s u l t s of the SAT method. The equations used by S c h i e l d g e are based on work by Businger (1973) and Businger et a l . (1971). The value of H determined from them should be a good e s t i m a t i o n of the a c t u a l s e n s i b l e heat f l u x . To make the SAT method c o n s i s t e n t with that used by -74-Sch i e l d g e a value of 0.35 i s used f o r k, the von Karman constant, and $>.,/$> = 1.35 i s used for n e u t r a l c o n d i t i o n s . Since Businger M n et a l . (1971)^ i n d i c a t e that $ /$ decreases very slowly as — — M H s t a b i l i t y i n c r e a s e s , the above value i s used f o r s t a b l e cases a l s o . The method i s t e s t e d for the time p e r i o d 2100 h to 0500 h on the night of October 13/14, 1976. The r e s u l t s are given i n F i g u r e 3.1. A l s o shown are the r e s u l t s at s e v e r a l stages of refinement of the SAT method. Agreement with the p r o f i l e value i s good. The importance of the sublayer c o r r e c t i o n i s apparent. E x p r e s s i o n (3.19) was used to c a l c u l a t e B ^. The d i f f e r e n c e between the values of H from the SAT and p r o f i l e methods at 2200 and 2300 h i s l i k e l y due to advection of c o o l a i r i n t o the area d u r i n g t h i s p e r i o d . Since s o i l s u r f a c e temperature during t h i s p e r i o d was i n t e r p o l a t e d from measurements taken o u t s i d e the p e r i o d , the e f f e c t of the c o o l e r a i r was not present i n the s u r f a c e temperature data. T h i s i s the reason f o r the low value of H from the SAT method. F i g u r e 3.1 suggests that the SAT method w i l l g i v e r e l i a b l e values of H f o r non-vegetated s u r f a c e s provided that accurate input data i s a v a i l a b l e . 3.3.3 EXTENSION OF THE SAT METHOD TO VEGETATED SURFACES For t a l l roughness elements i t i s necessary to in t r o d u c e a zero-plane displacement d i s t a n c e (D) i n t o equation (3.3) (Monteith, 1973). T h i s i m p l i e s that when wind speed i s p l o t t e d a g a i n s t l n ( z ) i t e x t r a p o l a t e s to zero at the h e i g h t D + z . T h i s o -75-60 r 1 1 1 1 r i i 1 " •>• 1 •— 2200 0000 0200 0400 Time (Oct. 13/14,1976) 3.1. Test of SAT method on a bare s o i l s u r f a c e . The SAT method i s compared with H estimated using an aerodynamic p r o f i l e method and p r o f i l e data (Sc h i e l d g e , 1978). -76-height i s the e f f e c t i v e sink f o r momentum. S i m i l a r l y , the e f f e c t i v e source or sink for heat w i l l be D + z which i s the H h e i g h t to which the l o g a r i t h m i c temperature p r o f i l e e x t r a p o l a t e s to the temperature of the a c t u a l heat source or s i n k . In the case of a bare s u r f a c e , with roughness elements which are not very t a l l (D = 0), the source or sink f o r heat w i l l be the a c t u a l s u r f a c e and T(o) i n equation (3.7) i s the s u r f a c e temperature as would be determined by a radiometer. For vegetated s u r f a c e s the l e v e l of the heat source or sink w i l l be some he i g h t w i t h i n the canopy. The temperature i n d i c a t e d by the remote sensing thermal l i n e scanner w i l l be due to r a d i a t i o n from v a r i o u s p a r t s of the canopy and ground s u r f a c e depending on the s t r u c t u r e and d e n s i t y of the canopy. T h i s w i l l not n e c e s s a r i l y be equal to the temperature of the a i r at the l e v e l of the heat source or s i n k . The sublayer c o r r e c t i o n t h e r e f o r e may not be a p p l i c a b l e . However, Bonn (1977) showed that over g r a s s l a n d , temperatures measured by an a i r b o r n e thermal i n f r a r e d l i n e scanner were e q u i v a l e n t to the a i r temperature at two-thirds the canopy h e i g h t . Since D i s o f t e n taken to be approximately 0.63 of the canopy h e i g h t (Cowan, 1968; Monteith, 1973) and t h i s height i s c l o s e to the l e v e l of the heat source or s i n k , i t may be p o s s i b l e to use scanner temperatures. Fuchs et a l . (1969), however, are p e s s i m i s t i c r e garding the a p p l i c a b i l i t y of using r a d i o m e t r i c temperatures i n equations for c a l c u l a t i n g s e n s i b l e heat f l u x from vegetated t e r r a i n . F i g u r e 3.2 shows the t y p i c a l shapes o f canopy a i r temperature p r o f i l e s d u r i n g the day and n i g h t (e.g., Penman and Long, 1960). Since the radiometer i s o f t e n l i k e l y to view an F i g u r e 3.2. T y p i c a l temperature p r o f i l e shapes i n a vegetated canopy. -78-average height lower than the heat source or sink l e v e l near D, the f l u x e s for both night and day w i l l be underestimates. During the e a r l y morning and e a r l y evening, temperatures w i t h i n the canopy may be n e a r l y i s o t h e r m a l . However, i t i s u n l i k e l y that r a d i o m e t r i c temperatures during the night except p o s s i b l y post-sunset temperatures w i l l y i e l d good s e n s i b l e heat f l u x estimates. 3.3.4 TEST OF THE SAT METHOD FOR A VEGETATED SURFACE In order to t e s t the SAT method on vegetated s u r f a c e s , equations (3.11) and (3.12) were a p p l i e d with the value Z 2 ~ D s u b s t i t u t e d for the height z^. Data of Stenmark and Drury (1970a) were used and the r e s u l t s were compared with the s e n s i b l e heat f l u x determined by v a r i o u s methods for the same s i t e and time (Wesely et a_l., 1969). The s u r f a c e was a l t a fescue grass which was 7 to 8 cm high. The s u r f a c e roughness was approximately 0.90 cm and the zero-plane displacement was between 0.0 and 2.0 cm (Stearns, 1970). Estimates from the SAT method were o f t e n good for the p e r i o d near sunset but were underestimates during the remainder of the n i g h t (Figure 3.3). T h i s i s true even a l l o w i n g that the energy balance (net radiometer, l y s i m e t e r , and s o i l heat f l u x p l a t e s ) c a l c u l a t i o n of H i s considered to give overestimates (Wesely et. a l . , 1969). The underestimate i s a d i r e c t r e s u l t of the temperature p r o f i l e w i t h i n the canopy. The temperature p r o f i l e o f l e a f temperatures w i t h i n the canopy i n d i c a t e d that the i n f r a r e d thermometer was sensing w e l l i n t o the canopy o f t e n lower than a l e v e l of z = 1 cm. T h i s g i v e s a much higher temperature than the temperature 100 «N 2300 0100 Time (May 3/4, 1967 :PST) 0300 i I F i g u r e 3.3a. Test of SAT method on a vegetated s u r f a c e . The SAT method i s compared with H estimated using several other methods (Wesely et a l . , 1969). 125 100 h _ 75 E H5 1900 2100 0100 Time (May 4 / 5 , 1967: PST) 0300 00 o I F i g u r e 3.3b. Test of SAT method on a vegetated s u r f a c e . The SAT method i s compared with H estimated using s e v e r a l other methods (Wesely et a l . , 1969). -81-required by the method. The sublayer correction as might be expected did not improve the estimates of H and in fact made them s l i g h t l y worse. It is apparent that this problem cannot be readily solved in the context of remote sensing determinations of sensible heat flux. The additional d i f f i c u l t i e s in estimating both D and of vegetated surfaces increase doubts as to the usefulness of the procedure for remote sensing purposes. The value of D and Z Q change considerably with vegetation type and canopy structure as well as with wind speed (Monteith, 1973). This point is p a r t i c u l a r l y important since the calculated value of H i s very sensitive to the value of z for the large surface roughnesses o common to vegetative canopies. 3.3.5 NOTE ON BREAKDOWN OF STABLE LAYERS Some concern has been voiced about the use of aerodynamic methods to calculate sensible heat flux for nights of low wind speed and stable conditions (Schieldge, 1978). Businger (1973) discusses such conditions. When the c r i t i c a l Richardson number (Ri ~ 0.2) i s reached transfer of momentum downward by eddy cr fluxes is stopped by buoyancy forces. A layer of laminar or no flow may develop. As wind shear increases and compensating buoyant force does not occur, the l e v e l at which Ri = Ri i s cr lowered and turbulence may reach the ground surface. The momentum of the turbulent layer is then decreased by contact with the surface and the laminar layer may develop again. The process -82-may be c y c l i c a l . Periods are often of the order of one half to one hour (Deacon, 1953). Under these conditions, sensible heat flux may vary considerably with time as the turbulent bursts occur. Calculation of H by aerodynamic methods w i l l depend on the height of instrumentation with respect to the l e v e l at which Ri = Ri and may lead to errors. Averaging atmospheric cr measurements over long periods of time may a l l e v i a t e part of the problem. Bursts of sensible heat flux may, however, resu l t in an instantaneous surface temperature (as measured by a thermal infrared scanner) uncharacteristic of general conditions. Conditions of this nature should c e r t a i n l y be noted when implementing the method. These problems w i l l not occur on windy nights which do not allow a laminar layer to develop. In the calm sit u a t i o n described above, sensible heat flux i s l i k e l y to be small and errors as they influence the energy balance determination of the surface w i l l also be small. 3.3.6 CONCLUSIONS: SENSIBLE HEAT FLUX A method for determining sensible heat flux has been developed s p e c i f i c a l l y for use with remotely sensed surface temperature data. The method requires ground measurements of wind speed and air temperature at one height above the surface, estimates of surface roughness z^, and radiometric surface temperature as measured by a thermal infrared l i n e scanner. These data are then used to evaluate equation (3.11). S t a b i l i t y and sublayer corrections are necessary for the success of the -83-method. When t e s t e d using ground based nighttime data, the method gave good r e s u l t s for a bare s o i l s u r f a c e but was u n s u c c e s s f u l for a vegetated s u r f a c e . S i m i l a r techniques w i l l p robably not be a p p l i c a b l e for most vegetated s u r f a c e s . There are s e v e r a l problems i n v o l v e d i n implementing a s e n s i b l e heat f l u x method in a remote sensing approach. These are: 1) e x t r a p o l a t i n g ground based a i r temperature and wind measurements over the region of a survey; 2) e s t i m a t i n g s u r f a c e roughness parameters; 3) determining s u r f a c e temperature values from thermal i n f r a r e d l i n e - s c a n data; and 4) e v a l u a t i n g how r e p r e s e n t a t i v e instantaneous scanner temperatures are of those corresponding to the long term energy balance. These problems are d i s c u s s e d i n Chapter 4. The method presented, f o r bare s u r f a c e s , p r o v i d e s a t h e o r e t i c a l l y v a l i d , p r a c t i c a l , and accurate b a s i s on which to b u i l d a remote sensing method. 3.4 DETERMINATION OF LATENT HEAT FLUX E s t i m a t i o n of l a t e n t heat f l u x (LE) a p p l i c a b l e to remote sensing i s d i f f i c u l t . Use of aerodynamic methods such as those used for determining H ( s e c t i o n 3.3) r e q u i r e s as input.the water vapour s t a t u s at the s u r f a c e . T h i s cannot be e a s i l y determined by remote sensing techniques. During the day, except p o s s i b l y i n a r i d r e g i o n s , LE can be of the order of the ground heat f l u x (G) or s e v e r a l times l a r g e r . In such cases, e s t i m a t i o n of LE becomes c r i t i c a l to thermal i n e r t i a models and the l a r g e e r r o r s i n e v i t a b l e i n determining LE f o r remote sensing a p p l i c a t i o n s -84-cause the r e s u l t s of such models to be l e s s r e l i a b l e . I t i s for t h i s reason that a nighttime thermal i n e r t i a model i s a t t r a c t i v e . Latent heat f l u x at night over bare and vegetated s u r f a c e s i s commonly very low and a small f r a c t i o n of G. Derksen (1974) s t a t e d that l a t e n t heat f l u x w i l l be low at ni g h t and suggested that for t h i s reason nighttime may be the best time to study d i f f e r e n c e s i n s o i l type and s t r u c t u r e using thermal imagery. S e v e r a l examples of low l a t e n t heat f l u x for a v a r i e t y of d i f f e r e n t s u r f a c e types and m e t e o r o l o g i c a l c o n d i t i o n s were given by S e l l e r s (1965) and Oke (1978). Data of Monteith (1957), Stenmark and Drury (1970a, 1970b), and McNaughton and Black (1973) a l s o i n d i c a t e low nighttime e v a p o r a t i o n . F i e l d data over rangeland t e r r a i n , a bare peat s o i l , and bare s i l t loam s o i l ( d i scussed l a t e r ) a l s o i n d i c a t e low l a t e n t heat f l u x for most n i g h t s . E v a p o r a t i o n w i l l be low and i n some cases condensation w i l l occur. Monteith (1961) d i s c u s s e d condensation on vegetated s u r f a c e s . E vaporation i s i n h i b i t e d by lack of a v a i l a b l e energy, development of a s t a b l e atmosphere, and c o o l i n g of the s u r f a c e which i m p l i e s a l i m i t to the s u r f a c e vapour pressure and thus the vapour pressure g r a d i e n t . The s u r f a c e region of the s o i l i s being r e p l e n i s h e d and much of the flow from depth i s s t o r e d . Stomatal c l o s u r e at night i n c r e a s e s , g r e a t l y r a i s i n g the canopy r e s i s t a n c e of v e g e t a t i o n (Oke, 1978) and ev a p o r a t i o n w i l l be small and not c o n t r o l l e d by the b i o l o g i c a l processes of the v e g e t a t i o n (Monteith, 1956). D i f f e r e n c e s i n LE between s i t e s are a l s o expected to be small i n magnitude. However, duri n g s p e c i a l m e t e o r o l o g i c a l c o n d i t i o n s such as ad v e c t i o n o f d r y or moist a i r i n t o an area, l a t e n t heat f l u x may be l a r g e . Examples were given -85-by data of Monteith (1956), van Bavel and F r i t s c h e n (1964), Rose (1968), and Stenmark and Drury (1970c). Latent heat f l u x i n such cases may be an important component of the energy balance and may vary from s i t e to s i t e . The small i n f l u e n c e of LE for most nighttime cases enables approximate methods of e s t i m a t i n g LE to be used. For thermal i n e r t i a s t u d i e s i n t e r e s t e d i n the r e l a t i v e thermal i n e r t i a values or which are c a l i b r a t e d , the d i f f e r e n c e i n LE among s i t e s w i l l be more important than the a c t u a l value of LE. Approximations such as assuming a zero l a t e n t heat f l u x or an average of s e v e r a l s i t e s are d i s c u s s e d and appear to g i v e r e s u l t s o f s u f f i c i e n t accuracy for thermal i n e r t i a a n a l y s i s . Other p o s s i b i l i t i e s which r e q u i r e f u r t h e r i n v e s t i g a t i o n are the use of a constant Bowen r a t i o for a l l s i t e s or a combination equation approach (e.g., Tanner and Fuchs, 1968) which combines energy balance and aerodynamic methods. For a great many n i g h t s over a l a r g e number of s u r f a c e s , approximation of LE by zero w i l l be v a l i d . A survey of 24 c l e a r n i g h t s of rangeland s i t e s (Kamloops data) i n d i c a t e d t hat 80 percent of the n i g h t s had LE's l e s s than 20 percent of G. Most were much smaller than 20 percent with f l u x e s of zero or l e s s -2 than 5 W m . Vo l u m e t r i c s o i l content to 2.5 cm were g e n e r a l l y between 5 and 10 per c e n t . During the day ev a p o r a t i o n was of the order of G or o c c a s i o n a l l y much l a r g e r . Low nig h t t i m e l a t e n t heat f l u x e s were a l s o i n d i c a t e d by measurements of the author over rangeland s i t e s v a r y i n g from dry bare s o i l to s i t e s of moist v i g o r o u s v e g e t a t i o n . A l l s i t e s had l a t e n t heat f l u x l e s s than 10 -2 -2 W m . Most s i t e s were l e s s than 5 W m . Eight e e n samples from -86-v a r i o u s n i g h t s were used. For a wetter s u r f a c e (Agassiz data) w i t h moisture contents to 5 cm between 18 and 35 percent v o l u m e t r i c , 65 percent of the c l e a r n i g h t s (19 samples) had LE's l e s s than 20 percent of G. For s i t e s of lower moisture content (3 to 6 percent v o l u m e t r i c ) , 80 percent had LE's l e s s than 20 percent of G. Latent heat f l u x e s were determined by Bowen r a t i o methods for these s t u d i e s . Measuring LE over s e v e r a l s i t e s of d i s s i m i l a r s u r f a c e type may y i e l d a good approximation of an average LE which, when a p p l i e d to the thermal i n e r t i a model, w i l l g i ve s a t i s f a c t o r y r e s u l t s . Measurements may use a p r o f i l e method, Bowen r a t i o method, r e s i d u a l energy balance, or other techniques. The number of t e s t s i t e s used w i l l depend on the amount of i n s t r u m e n t a t i o n a v a i l a b l e and the v a r i e t y of s u r f a c e types and the range of moisture contents expected. Since LE i s s m a l l , the magnitude of e r r o r s r e s u l t i n g from t h i s approximation w i l l be s m a l l . The d i f f e r e n c e s i n nighttime l a t e n t heat f l u x between moist and dry s i t e s of the A g a s s i z data d i s c u s s e d above were r a r e l y g r e a t e r -2 than 5 W m . D i f f e r e n c e s i n l a t e n t heat f l u x of rangeland s i t e s -2 were g e n e r a l l y small ( l e s s than or of the order of 5 W m ). The e r r o r s r e s u l t i n g from approximation of LE by zero or an average l a t e n t heat f l u x w i l l vary with the nature of the s u r f a c e s i n a survey area and m i c r o m e t e o r o l o g i c a l c o n d i t i o n s . However, probable e r r o r s i n LE due to using these approximation -2 techniques are expected to be t y p i c a l l y 5 W m . Even i n a worst case s i t u a t i o n , probable e r r o r s i n LE w i l l not l i k e l y be g r e a t e r -2 than 10 W m -87-3.5 CONCLUSIONS Given a c c u r a t e input, the methods of e s t i m a t i n g net r a d i a t i o n , s e n s i b l e heat f l u x , and l a t e n t heat f l u x o u t l i n e d i n t h i s chapter p r o v i d e an accurate estimate of ground heat f l u x . E r r o r s may r e s u l t i f LE i s not c l o s e to zero and at the same time v a r i e s g r e a t l y between s i t e s i n the area of the thermal i n e r t i a survey. These methods are designed to be implemented i n a remote sensing mode and l a r g e l y meet the r e l e v a n t c r i t e r i a f o r thermal i n e r t i a mapping ( s e c t i o n 1.5). Combining the methods of e s t i m a t i n g ground heat f l u x d e s c r i b e d i n t h i s chapter with the thermal i n e r t i a models of Chapter 2, giv e n a c c u r a t e input, w i l l p r o v i d e a t h e o r e t i c a l l y v a l i d , a c c u r a t e , and simple method f o r remote sensing thermal i n e r t i a mapping. -88-Chapter Four ERROR ANALYSIS 4.1 INTRODUCTION In Chapter 2, nig h t t i m e c o o l i n g models (models I, I I , and III) a p p l i c a b l e to remote sensing thermal i n e r t i a mapping were developed. In Chapter 3, s e v e r a l methods of determining the en-ergy balance components necessary as input to the c o o l i n g models were proposed. Given accurate input, model I I I and the methods of determining the energy balance input g i v e good r e s u l t s . T h i s chapter analyzes the e r r o r s i n v o l v e d i n implementing model I I I and the energy balance e s t i m a t i o n methods i n a remote sensing mode a p p l i c a b l e to g e n e r a l thermal i n e r t i a mapping. Key f a c t o r s are the ' e s t i m a t i o n and e x t r a p o l a t i o n of s u r f a c e c o n d i t i o n s (e.g., s u r f a c e roughness and e m i s s i v i t y ) , m i c r o m e t e o r o l o g i c a l c o n d i t i o n s (e.g., a i r temperature and wind speed) and the dete r m i n a t i o n of s u r f a c e temperature from thermal i n f r a r e d l i n e - s c a n d ata. E r r o r s are analyzed f o r the dete r m i n a t i o n of net r a d i a t i o n , s e n s i b l e heat f l u x , and l a t e n t heat f l u x . The t o t a l e r r o r s expected i n implementing the model are estimated. The e r r o r s are de s c r i b e d i n terms of probable e r r o r f o r a worst (poor), t y p i c a l , and good case. Probable e r r o r i s a q u a n t i t y such that one-half the e r r o r s w i l l be l e s s than i t and one - h a l f g r e a t e r (Scarborough, 1962). In support of these e r r o r analyses s e v e r a l f i e l d experiments were conducted over two f i e l d seasons. E i g h t thermal i n f r a r e d -89-l i n e - s c a n surveys were flown over t e s t areas of the Lac du Bois rangeland near Kamloops, B.C. They were flown by the Canada Centre for Remote Sensing F a l c o n fan j e t using a Daedalus (Model 1230) i n f r a r e d l i n e scanner o p e r a t i n g i n the 9.5 to 11.5 um bandpass. Simultaneous ground measurements of surface temperature, a i r temperature, vapour p r e s s u r e , wind speed, net r a d i a t i o n , s o l a r r a d i a t i o n , and ground heat f l u x were recorded. The measurements were taken throughout the d a i l y c y c l e . O v e r f l i g h t s were at s e l e c t e d times during the day and n i g h t . Other f i e l d experiments were conducted to i n v e s t i g a t e the s p a t i a l v a r i a b i l i t y of m i c r o m e t e o r o l o g i c a l parameters and energy balance components and the occurrence of a i r ponding. 4.2 ERROR IN NET RADIATION ESTIMATION In Chapter 3 a procedure f o r e s t i m a t i n g net r a d i a t i o n was proposed. An estimate of s u r f a c e temperature, as determined from c a l i b r a t e d thermal i n f r a r e d l i n e - s c a n data ( T r ) , and measured or e m p i r i c a l l y estimated t o t a l longwave downward r a d i a t i o n (B^) are a p p l i e d t o : Rn = ea o Tr^ - ea B T (4.1) where ea i s the assumed or estimated s u r f a c e e m i s s i v i t y . S i m i l a r procedures were d e s c r i b e d by Watson et a l . (1971), Pease and N i c h o l s (1976), and Kahle (1977). The e r r o r s i n employing t h i s method are dependent upon the accuracy to which the s u r f a c e temperature, s u r f a c e e m i s s i v i t y , and downward longwave r a d i a t i o n -90-can be determined. I t w i l l be assumed that downward longwave r a d i a t i o n i s determined a c c u r a t e l y (by measurement) and i s uniform over the e n t i r e survey area. Surface temperature w i l l u s u a l l y be determined using an estimate of s u r f a c e e m i s s i v i t y (Appendix I I ) . I f the a c t u a l s u r f a c e e m i s s i v i t y (£) i s greater than the assumed e m i s s i v i t y (ea), the temperature estimate w i l l be higher than i f the c o r r e c t e m i s s i v i t y i s used. T h i s w i l l have a compensating e f f e c t of reducing the e r r o r i n net r a d i a t i o n , s i n c e assuming an e m i s s i v i t y lower than the a c t u a l decreases the net r a d i a t i o n c a l c u l a t e d by (4.1). I f e m i s s i v i t y i s known, e r r o r s are dependent only on the e r r o r i n s u r f a c e temperature. For an example of Tr = 10 C, t o t a l -2 downward longwave r a d i a t i o n (B^) = 277 W m (corresponding to an a i r temperature of 10 C (Idso and Jackson, 1969)), and e m i s s i v i t y -2 of 0.95, equation (4.1) y i e l d s a net r a d i a t i o n of 83 W m . A good estimate of temperature e r r o r ( e m i s s i v i t y known) w i l l be 0.5 C, a t y p i c a l estimate approximately 1.0 C, and a poor estimate 1.6 C.^ " These y i e l d probable e r r o r s i n net r a d i a t i o n o f 2.5, -2 4.9, and 7.9 W m r e s p e c t i v e l y f o r the above example. The net r a d i a t i o n of s i t e s c a l c u l a t e d using remotely sensed temperature was compared with ground measurements of net r a d i a t i o n . A t o t a l of 24 s i t e s were t e s t e d i n four n i g h t t i m e thermal i n f r a r e d l i n e - s c a n surveys. The thermal i n f r a r e d l i n e -scan data were c a l i b r a t e d and an atmospheric o f f s e t a p p l i e d to 2 each image. T o t a l longwave downward r a d i a t i o n was estimated by an e m p i r i c a l formula r e l a t i n g a i r temperature to downward longwave r a d i a t i o n (Idso and Jackson, 1969). C l e a r sky c o n d i t i o n s p r e v a i l e d f o r a l l surveys. Ground t r u t h net r a d i a t i o n -91-was measured with a m i n i a t u r e net radiometer. E m i s s i v i t y was assumed to be one. The s i t e s were on rangeland t e r r a i n and v a r i e d from bare s o i l to dense grass s u r f a c e s . The a c t u a l e m i s s i v i t i e s may d i f f e r c o n s i d e r a b l y from one f o r some s i t e s . -2 The average e r r o r s f o r the surveys ranged from 2 to 7 W m . The -2 average e r r o r was 4 W m The method d e s c r i b e d , using remotely sensed surface temperatures and measured or estimated longwave downward r a d i a t i o n to estimate net r a d i a t i o n at n i g h t , g i v e s e x c e l l e n t r e s u l t s . The use of measured downward longwave r a d i a t i o n at one s i t e i n the survey area w i l l g ive good r e s u l t s and avoid d i f f i c u l t i e s which may a r i s e i n using an e m p i r i c a l formula to estimate downward longwave r a d i a t i o n . E m i s s i v i t y estimates should be used i f s u r f a c e e m i s s i v i t y v a r i e s g r e a t l y from one. E r r o r s at s i t e s with a l a r g e p o r t i o n of t h e i r f i e l d of view occupied by surrounding land s u r f a c e s w i l l y i e l d l a r g e r e r r o r s . Minor e r r o r s may r e s u l t from l o c a l short term advection causing s u r f a c e temperature, at the time of the l i n e - s c a n data a c q u i s i t i o n , to be d i f f e r e n t from that o c c u r r i n g d u r i n g average m i c r o m e t e o r o l o g i c a l c o n d i t i o n s . Appendix IV d e f i n e s severe, moderate, and low s u r f a c e temperature e r r o r s (0.8, 0.4, and 0.2 C r e s p e c t i v e l y ) due to short term l o c a l a d v e c t i o n . Such c o n d i t i o n s -2 w i l l r e s u l t i n e r r o r s , f o r most cases, from 4 to 1 W m f o r the severe to low-error cases r e s p e c t i v e l y . For most surveys and n i g h t t i m e c o n d i t i o n s , a reasonable estimate of the t y p i c a l probable e r r o r i n net r a d i a t i o n i s -2 approximately 5 W m . Good r e s u l t s may l e a d to probable e r r o r s of 3 W m~2. -92-4.3 ERROR IN SENSIBLE HEAT FLUX ESTIMATION A method, f o r determining s e n s i b l e heat f l u x , termed the s u r f a c e / a i r temperature or SAT method, has been developed and t e s t e d using ground data ( s e c t i o n 3 . 3 ) . T h i s s e c t i o n examines the e r r o r s i n v o l v e d i n implementing the method i n an o p e r a t i o n a l remote sensing mode. A s e n s i t i v i t y and e r r o r a n a l y s i s approach i s used. S e n s i b l e heat f l u x (H) i s a complicated f u n c t i o n of s u r f a c e roughness l e n g t h ( Z q ) , a i r temperature (Ta), s u r f a c e temperature ( r a d i a t i v e ) (Ts), and wind speed (U). The s e n s i t i v i t y of the SAT method to the parameters s u r f a c e roughness l e n g t h , s u r f a c e - a i r temperature d i f f e r e n c e Td (Td = Ta - T s ) , and wind speed i s demonstrated by p l o t s of s e n s i b l e heat f l u x versus each parameter. An e r r o r a n a l y s i s g i v e s t o t a l e r r o r s i n H using the SAT method for t y p i c a l n i ghttime m i c r o m e t e o r o l o g i c a l c o n d i t i o n s and f o r t y p i c a l cases o f the probable e r r o r s i n the parameters used by the SAT method. A l s o given are the c o n t r i b u t i o n of e r r o r s i n each parameter to the t o t a l e r r o r . S p e c i a l examples demonstrate the change i n t o t a l e r r o r with s u r f a c e roughness, s u r f a c e - a i r temperature d i f f e r e n c e , and wind speed, as w e l l as the e f f e c t s of a i r ponding and short term l o c a l a d v e c t i o n . The l i m i t a t i o n s and optimum c o n d i t i o n s for determining s e n s i b l e heat f l u x using the SAT method are s p e c i f i e d . -93-4.3.1 SENSITIVITY ANALYSIS OF SENSIBLE HEAT FLUX A) S e n s i t i v i t y to z o A major problem in implementing aerodynamic methods of estimating H is the necessity of determining the surface roughness length (z ). The need for accurate values of z for the ' o o SAT method is demonstrated in Figure 4.1 which shows the va r i a t i o n of H with z for various wind speeds and air-surface o temperature differences. Figure 4.2 gives the value of the correction factor Q with respect to z . The change in H with z , o o and thus the error in H due to errors in z , increases with z , o o e s p e c i a l l y for Z q greater than one millimeter. It also increases greatly with wind speed, but only s l i g h t l y with air-surface temperature difference. There are several methods of estimating z for remote o sensing applications. One i s to determine z^ v i a p r o f i l e measurements for the surface types of the survey area. An alternative i s to use tabulated values as found in the l i t e r a t u r e . Empirical formulae may aid t h i s procedure (e.g., Lettau, 1969) . The above methods of estimating z , for remote sensing applications, require that the d i s t r i b u t i o n and nature of surface types be known a p r i o r i , transformed into d i g i t a l map form, and registered to the thermal images. This prerequisite does not meet the c r i t e r i o n (section 1.5), for general remote sensing surveys, that no or only limited knowledge of the surface be required. A technique commensurate with this c r i t e r i o n , but necessarily leading to greater errors, i s to assume some average -94-U = 2.0 ms" - ? i : n Td=8.0; T d = 8 0 | I Td=5.0: / |Td = 5.0 .Td = 2.0 Td = 2.0 0.01 F i g u r e 4.1. S e n s i b l e heat f l u x (H) versus s u r f a c e roughness l e n g t h ( z Q ) f o r SAT method f o r v a r i o u s wind speeds (U) and a i r - s u r f a c e temperature d i f f e r e n c e s (Td). Td i s g i v e n i n degrees C e l s i u s . 1.7 i 1 1 r 0.5 I i 1 " 1 0.01 0.1 1 10 100 Z (mm) o N ' F i g u r e 4.2. Sublayer c o r r e c t i o n f a c t o r (Q) versus s u r f a c e roughness l e n g t h ( z G ) . ( B " l from equation (3.19) fo r Re A < 30 and B - 1 = 4.5 f o r Re^ > 30). -96-value of z r e p r e s e n t a t i v e of the s u r f a c e s expected i n the survey o area. Bare s o i l s commonly have s u r f a c e roughness lengths between 0.01 mm and 20 mm (Table 4.1). Surface roughness length i s commonly an order of magnitude smaller than the a c t u a l height of the s u r f a c e roughness elements (Thorn, 1975). I t i s evident from F i g u r e 4.1 that the s e n s i t i v i t y of H with z i s verv high f o r z o o > 5 mm. T h e r e f o r e , unless z i s known a c c u r a t e l y , the SAT method o should not be used fo r s u r f a c e s with z > 5 mm. A reasonable o estimate of a z f o r bare s u r f a c e s which w i l l allow a wide range o of values of z without extreme e r r o r i s 1.0 mm. For a l a r g e o range of s u r f a c e roughness lengths from very smooth (0.01 mm) to about 5.0 mm ( t h i s range i s c o n s i d e r e d a high z e r r o r (Az ) i n o oH f u r t h e r a n a l y s i s ) and f o r v a l u e s of Td < 5.0 C and values of U < 3.0 m s 1 , e r r o r s i n H due to z w i l l be l e s s than 15 W m -2 o S i m i l a r l y f o r a low z e r r o r Az (z from 0.3 mm to 2.0 mm), o oL o^ -2 e r r o r s i n H w i l l be l e s s than 5 W m (Figure 4.1). I f the s u r f a c e s are known to be smooth, a z estimate of 0.5 mm or l e s s o w i l l g i v e smaller e r r o r s . The z of vegetated s u r f a c e s i s o f t e n g i v e n by the o e x p r e s s i o n z = 0.13h where h i s the height of the canopy o (Monteith, 1973) . Surface roughness lengths w i l l be of the order of s e v e r a l c e n t i m e t e r s . For f o r e s t e d t e r r a i n , z can be as l a r g e o as 1 m. The z of a p a r t i c u l a r canopy w i l l vary with canopy o s t r u c t u r e and d e n s i t y , as w e l l as wind speed. Since H i s h i g h l y s e n s i t i v e to changes i n z f o r the l a r g e v a l u e s of z common to o o v e g e t a t i v e canopies (Figure 4.1) and Z q i t s e l f i s very s e n s i t i v e to the nature of the canopy, a procedure of i n p u t t i n g an assumed average v a l u e of Z q i n t o the equations f o r determining H w i l l Table 4.1. Surface roughness lengths (Z q) for natural non-vegetated su r f a c e s . Surface Type- z (mm) o Source smooth mud f l a t s 0.01 P r i e s t l e y (1959) dry lake bed 0.03 S e l l e r s (1965) sand f l a t s 0.04 Lettau (1969) c a l c u l a t e d smooth bare s o i l 0.1 Oke (1970) smooth bare sandy s o i l 0.4 Fuchs et a l . (1969) deser t 0.3 P r i e s t l e y (1959) sand 0.4 Stearns (1967) bare s o i l s 0.6 Lettau (1969) c a l c u l a t e d 2 bare s o i l 1.0 Schieldge (1978) rough bare s o i l ( disc harrowed) 10.0 Oke (1970) fallow f i e l d 20.0 van Wijk and Borghorst (1963) very smooth ice 0.01 van Wijk and Borghorst (1963) smooth snow (on short grass) 0.05 P r i e s t l e y (1959) snow (on n a t u r a l p r a i r i e ) 1.0 P r i e s t l e y (1959) smooth snow 5.0 van Wijk and Borghorst (1963) Review of l i t e r a t u r e and c a l c u l a t e d . Some wheat se e d l i n g s and stubble d e b r i s . -98-y i e l d l a r g e e r r o r s . S i m i l a r d i f f i c u l t i e s i n e s t i m a t i n g the zero-plane displacement d i s t a n c e (D) of the canopy and the problems of using the r a d i o m e t r i c temperature of the canopy (Chapter 3) f u r t h e r compound the problems of e s t i m a t i n g H f o r vegetated s u r f a c e s . Remote sensing e s t i m a t i o n of H by aerodynamic methods, over vegetated s u r f a c e s , i s t h e r e f o r e not recommended and w i l l not be d i s c u s s e d f u r t h e r . B) S e n s i t i v i t y to A i r and Surface Temperatures The a i r temperature used i n implementing the SAT method w i l l be i n e r r o r f o r some s i t e s due to v a r i a b i l i t y of a i r temperature over a survey area. The s u r f a c e temperature, as measured from a remote sensing p l a t f o r m , w i l l have e r r o r s due to instrument and c a l i b r a t i o n e r r o r , e r r o r s due to atmospheric a t t e n u a t i o n and emission, and e r r o r s due to s u r f a c e e m i s s i v i t y . Appendix IV examines probable e r r o r s i n the a i r temperature used f o r remote sensing surveys. A i r temperatures f o r o p e r a t i o n a l surveys w i l l be gathered from one or s e v e r a l s i t e s w i t h i n the survey area and e x t r a p o l a t e d to the whole survey area. They w i l l be time averaged over a p e r i o d of about one hour. F i g u r e 4.3 shows time t r a c e s of a i r temperature at 1.5 m above s e v e r a l rangeland s i t e s . The s i t e s vary from bare s o i l ( s i t e A-3) to sparse v e g e t a t i o n ( s i t e s A - l and 5) to medium d e n s i t y v e g e t a t i o n cover ( s i t e A-4) to l u s h dense grass ( s i t e A-2). Topography ranges from f l a t l a n d s i t e s ( A - l , 4, and 5) to a h i l l s i d e s i t e (A-3) to a topographic d e p r e s s i o n ( s i t e A-2). S i t e s A-4 and 5 are at e q u i v a l e n t h e i g h t s s l i g h t l y higher (4 m) than s i t e A - l which i s approximately 4 m higher than A-2. E l e v a t i o n d i f f e r e n c e between to E v ' Q UJ UJ 0_ if) Q 2 O UJ CC < cr UJ a UJ 1.0 0.0 25.0 24.0 23.0 22.0 CC < 21.0 v . v •Si, 2 2 4 0 2 2 5 0 2 3 0 0 2 3 1 0 T I M E ( P D T ) 2 3 2 0 2 3 3 0 i vo vo l F i g u r e 4.3. Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s of area A (29/7/78; 2300 h). -100-s i t e A-2 and 3 i s about 20 m. The c l o s e s t s i t e s ( A - l and 2) are 70 m ap a r t , the f u r t h e s t (A-3 and 5) 260 m. Temperature was measured by s i l i c o n diodes mounted i n a psychrometer system c o n s t r u c t e d by the author. The psychrometers had r a d i a t i o n s h i e l d s and were v e n t i l a t e d to a speed of 3.7 m s ^ with a DC fa n . The temperature s i g n a l s from the s i t e s were i n t e r r o g a t e d using a stepping s w i t c h . Each s i t e was scanned at i n t e r v a l s of one to two minutes. Wind speed at 1.5 m was measured with a C a s e l l a anemometer and counter at s i t e A - l . Further examples from d i f f e r e n t s i t e s are given i n Appendix IV. Although at a g i v e n i n s t a n t i n time a i r temperature d i f f e r e n c e s between s i t e s may be up to 1.5 and 2.0 C, the d i f f e r e n c e s i n time averages over p e r i o d s of about one hour are much l e s s . Use of F i g u r e 4.3 and f u r t h e r data of Appendix IV i n d i c a t e s t hat a t y p i c a l probable e r r o r ( r e s u l t i n g from use of the time averaged a i r temperature of one s i t e or the average of s e v e r a l s i t e s as the a i r temperature f o r a l l s i t e s i n a survey area) i s l i k e l y 0.5 C (ATa^). An e r r o r of 1.5 C r e p r e s e n t s a severe or high probable e r r o r (ATa ) f o r non-ponding c o n d i t i o n s which may occur f o r cases where the survey area i s very l a r g e or the temperature data are determined at a d i s t a n c e from the survey area. Ponding of c o o l a i r i n topographic d e p r e s s i o n s can cause a i r temperature d i f f e r e n c e s of 5 C and l a r g e r between s i t e s . F i g u r e 4.4 demonstrates the occurrence of ponding (0350 h to 0400 h) d u r i n g a l i g h t wind p e r i o d . The a i r temperature o f s i t e A-2 best demonstrates t h i s . Ponding g r a d u a l l y develops again a f t e r a wind bur s t (0400 h to 0405 h ) . Ponding begins to develop dur i n g a s h o r t p e r i o d of l i g h t wind (2304 h to 2310 h) i n F i g u r e 4.3. A c fD O *1 3 0) 0) n fD no 0) I—• o > rr CO cn =r o o rt» « H- Q) 3 K-0» rt H- fD •t 3 'O TJ fD O * 3 0) OJ rr M- C 3 n 03 fD — Oi 3 O QJ \ -J * \ -J 3 CO Qj 0) O T3 J> fD O fD O QJ 3" -~ — Oi 3 i-t> 0 cn rr fD (0 AIR TEMPERATURE (C) o C O b b cn o 0) b b 00 b b o T T T" T" T T _1_ o b ro b WIND SPEED (ms"1) - T O T --102-value of 1.5 C temperature d i f f e r e n c e can be used as a l i k e l y probable e r r o r i n a i r temperature for moderate ponding c o n d i t i o n s . The e r r o r i n using s u r f a c e temperatures d e r i v e d from a i r b o r n e thermal i n f r a r e d l i n e - s c a n data i s d i s c u s s e d i n Appendix I I . A t y p i c a l probable e r r o r for a g e n e r a l nighttime remote sensing survey i s 1.5 C (ATs^) (high atmospheric , low e m i s s i v i t y , and intermediate instrument e r r o r ) . Probable e r r o r s w i l l , however, range from 0.5 C (ATs ) f o r a case o f low instrument e r r o r , low atmospheric e r r o r , and known e m i s s i v i t y to 2.1 G ( Ts ) for a case of high instrument e r r o r , high H atmospheric, and high e m i s s i v i t y e r r o r . F i g u r e 4.5 demonstrates the s e n s i t i v i t y of the SAT method to input values of a i r temperature minus s u r f a c e temperature Td (Td = Ta-Ts). The occurrence of s t a b l e atmospheric c o n d i t i o n s g r e a t l y reduces the s e n s i t i v i t y of H with Td, e s p e c i a l l y f o r low wind speeds. The t r a n s f e r of heat and momentum to the s u r f a c e under s t a b l e c o n d i t i o n s i s r e s i s t e d by the occurrence of c o u n t e r a c t i n g upward buoyancy f o r c e s . As the s t r e n g t h of the i n v e r s i o n i n c r e a s e s , the buoyancy f o r c e s i n c r e a s e and the change i n H with Td i s decreased. The e r r o r i n H due to e r r o r s i n Td i n c r e a s e s with wind speed, p a r t i c u l a r l y f o r wind speeds g r e a t e r than about 2.5 m s ^. E r r o r s due to Td g e n e r a l l y decrease with i n c r e a s i n g magnitude of Td. They are very s m a l l when both wind speed i s l e s s than 2.5 m s ~^ and Td i s between 3 and 8 C. These valu e s of Td and wind speed are common d u r i n g nighttime c o n d i t i o n s . There are a l s o s l i g h t i n c r e a s e s i n s e n s i b l e heat f l u x e r r o r s due to Td e r r o r s as z_ i n c r e a s e s . -103-0 2 4 6 8 1 0 12 Td (C) F i g u r e 4.5. S e n s i b l e heat f l u x (H) versus a i r -s u r f a c e temperature d i f f e r e n c e (Td) f o r SAT method (wind speed (U) i n m s " 1 , z Q = 1.0 mm). -104-C) S e n s i t i v i t y to Wind Speed S e n s i t i v i t y of the SAT method to the input value of wind speed i s presented i n F i g u r e 4.6. S e n s i b l e heat f l u x i ncreases almost l i n e a r l y with wind speed. The e r r o r i n H due to an e r r o r i n wind speed i n c r e a s e s g r e a t l y with a i r - s u r f a c e temperature d i f f e r e n c e , while there i s a s l i g h t to moderate i n c r e a s e with i n c r e a s i n g Z q and a very s l i g h t decrease i n e r r o r as wind speed i n c r e a s e s . The magnitude of the e r r o r i n input value of wind speed i s d i s c u s s e d i n Appendix IV. The wind speed used by the model i s a time average over p e r i o d s of approximately one hour. V a r i a b i l i t y i n the time-averaged wind speeds of s i t e s w i l l depend on the topography of the survey area. An e r r o r of 0.15 m s ^ i s used as an example of low probable e r r o r i n wind speed (AU ) . A l a r g e or -1 L high probable e r r o r i s c o n s i d e r e d 0.30 m s (AU ) . E r r o r s may be H l a r g e r , e s p e c i a l l y i f the wind speed i s measured at a s i t e some d i s t a n c e from the survey area. D) S e n s i t i v i t y to Model E r r o r s The model (aerodynamic equations and c o n s t a n t s used to determine s e n s i b l e heat f l u x by SAT method) i t s e l f c o n t a i n s e r r o r s . The major components of the model e r r o r w i l l be due to e r r o r s i n the value of the s t a b i l i t y c o r r e c t i o n (F={<!> <fc } S and M H Q, the sublayer c o r r e c t i o n f a c t o r . E r r o r s i n Q r e s u l t mainly from the e m p i r i c a l d e t e r m i n a t i o n of B \ the r e c i p r o c a l of the sublayer Stanton number. C o n s e r v a t i v e e r r o r s i n F and Q should -105-U (ms _ 1 ) F i g u r e 4.6. S e n s i b l e heat f l u x (H) versus wind speed (U) f o r SAT method ( a i r -s u r f a c e temperature d i f f e r e n c e (Td) i n degrees C; z Q • l»0 nun) . -106-represent the t o t a l model e r r o r , i n c l u d i n g e r r o r s i n other input parameters such as the von Karman constant (k) as w e l l as other approximations of the model. A c o n s e r v a t i v e estimate of probable e r r o r i n the value of B ^ i s 2.5. Most cases w i l l have l e s s e r r o r although, f o r low values of B ^ values which occur at low roughness Reynolds numbers (Re^), the e r r o r may be more. However, t h i s w i l l not r e s u l t i n l a r g e e r r o r s i n H s i n c e H i s a l s o small at low values of Re A. G a r r a t t and Hicks (1973, F i g u r e s 1 and 2) give good i n s i g h t i n t o e r r o r s expected f o r B F i g u r e 4.7 p l o t s Q versus B An e r r o r i n B * of approximately 2.5 w i l l g e n e r a l l y y i e l d an e r r o r of roughly 15 percent of Q. I t i s d i f f i c u l t to estimate the e r r o r i n the s t a b i l i t y c o r r e c t i o n (F) . A reasonable estimate of probable e r r o r i s 0.25 F. An example of s c a t t e r i n experimental F v a l u e s about the t h e o r e t i c a l value used by the SAT method (Thorn et a l . , 1975; F i g u r e 2) i n d i c a t e d that a 25 percent e r r o r i s l i k e l y a c o n s e r v a t i v e estimate of probable e r r o r . However, f o r cases of Ri > 0.1, e r r o r s may be l a r g e r as the e m p i r i c a l r e l a t i o n s f o r the s t a b i l i t y c o r r e c t i o n d e r i v e d by others (e.g., P r u i t t et a l . , 1973; Panofsky, 1963) d e v i a t e from the formula used i n the SAT method (Thorn, 1975). However, i n these more s t a b l e cases, H i s s m a l l and the magnitude of e r r o r s w i l l be s m a l l . The change i n H i s p r o p o r t i o n a l to percentage changes i n F and Q. The magnitude of model e r r o r w i l l t h e r e f o r e be low f o r sm a l l s e n s i b l e heat f l u x e s , but l a r g e f o r l a r g e v a l u e s of H. -107-co tn o c o "55 c <D E - 1 0 B"1 (dimensionless) F i g u r e 4.7. Sublayer c o r r e c t i o n f a c t o r (Q) versus B" 1 where B i s the sublayer Stanton number (from equation (3.12); z 2 = 1.5 m, = 0.41) = 1.0 mm, I -108-4.3.2. ERROR ANALYSIS OF SENSIBLE HEAT FLUX The t o t a l probable e r r o r s i n implementing the SAT method i n a remote sensing mode are determined for a range of common nigh t t i m e m i c r o m e t e o r o l o g i c a l c o n d i t i o n s . For each m i c r o m e t e o r o l o g i c a l c o n d i t i o n , the t o t a l probable e r r o r i s c a l c u l a t e d f o r cases r e p r e s e n t i n g a range of probable e r r o r s i n each of the parameters a f f e c t i n g the method. Tables 4.2, 4.3, and 4.4 give the r e s u l t s for a case of z = 1.0 mm and (Ta + Ts)/2 = o 12 C. R e s u l t s are given for an e r r o r a n a l y s i s with and without model e r r o r . Table 4.5 d e f i n e s the input and output parameters f o r the e r r o r a n a l y s i s . Case 1 r e p r e s e n t s a s i t e s p e c i f i c study i n which Ta, U, and z are known and there i s l i t t l e e r r o r i n s u r f a c e temperature o ( e m i s s i v i t y known, atmospheric and p i x e l e r r o r low). The l e a s t expected probable e r r o r for most g e n e r a l surveys i s g i v e n by case 2. Another example of small e r r o r i s case 3. Case 4 c o n t r a s t s with case 3 to show the e f f e c t of z e r r o r and a l s o r e p r e s e n t s o cases i n which Z q i s known. The t y p i c a l e r r o r f o r most g e n e r a l surveys i s g i v e n by case 5. Case 6 g i v e s the t y p i c a l e r r o r i n H for a survey area with s u r f a c e s of widely v a r y i n g Z q . Case 7 approximates a worst case probable e r r o r . A) Summary of T o t a l E r r o r s Table 4.6 summarizes the e r r o r f o r the cases of low, t y p i c a l , and high e r r o r (cases 2, 5, and 7 r e s p e c t i v e l y ) . T y p i c a l -2 e r r o r s v a r i e d from 6 to 32 W m . The t y p i c a l e r r o r s f o r wind -1 -2 speeds under 3 m s were l e s s than 17 W m . I n low-error cases, Table 4.2. Error analysis of SAT method for determining sensible heat flux (H) (wind speed (U) = 2.0 m s _ 1 ; z Q = 1.0 mm). C o n t r i b u t i o n to R x due to probable e r r o r 5.0 1 2 3 4 5 6 7 no model e r r o r RT7H with model e r r o r 00 ,00 ,00 ,00 ,00 .00 .00 .00 .00 ,00 ,00 - .69 - 1.00 ,00 - .69 ,00 - .20 - .06 - 1.00 .31 .31 ,60 .80 .34 0.0 2.0 3.6 3.0 3.6 6.7 10.3 00 12 ,21 ,18 ,21 ,39 .61 Rp/H 3.9 .34 4.5 .39 4.6 .40 4.5 .39 8.1 .71 8.6 .75 14.5 1.26 5.0 5.3 .31 6.1 .36 5.8 .34 6.1 .36 8.3 .49 11.4 .67 4.8 .33 6.3 .43 6.2 .43 6.8 .47 8.1 .56 11.6 .80 H (W m~2) 17.0 8.0 1 2 3 4 5 6 7 44 ,11 ,11 ,08 .05 1.00 .44 .11 .11 .75 .78 .58 .34 .12 .03 .02 1.5 2.3 4.6 4.5 5.3 6.9 10.8 10 15 ,32 ,31 ,36 ,48 .75 14.5 4 . 3 . E r r o r a n a l y s i s o f SAT method f o r determining s e n s i b l e heat f l u x (H) (wind speed (U) - 3 . 0 m s " 1 ; z Q = 1 . 0 mm). T o t a l E r r o r Td Case C o n t r i b u t i o n to R 2 j due to probable e r r o r no model with model (C) e r r o r e r r o r H T a T T a H T S L T s T T S H UH z o L zoH R I Rp R T / H (W rn"2) 2 . 0 1 _ 1 . 0 0 _ — — — 4 . 0 . 2 2 6 . 6 . 3 7 1 8 . 0 2 . 3 9 — . 3 9 - - - - . 2 2 - 6.4 . 3 6 8 . 3 . 4 6 3 . 3 9 — . 3 9 — - . 0 2 - . 2 0 - 6.4 . 3 6 8 . 3 . 4 6 4 . 4 9 — . 4 9 - - . 0 2 - - - 5 . 3 . 3 2 7 . 8 . 4 3 5 . 0 8 — — . 8 7 — . 0 0 -• . 0 5 - 1 4 . 0 . 7 8 1 4 . 9 . 8 3 6 . 0 6 — — . 6 8 — . 0 0 - - . 2 6 1 5 . 8 . 8 8 1 6 . 7 . 9 3 7 - . 3 0 - - . 5 8 - . 0 1 — . 1 1 2 3 . 7 1 . 3 2 2 4 . 3 1 . 3 5 5 . 0 1 1 . 0 0 _ _ — — 2 . 5 . 0 7 1 1 . 1 . 3 0 3 7 . 0 2 . 1 9 — . 1 9 - — - - . 6 2 - 5 . 7 . 1 6 1 2 . 2 . 3 3 3 . 1 6 — . 1 6 — — . 1 6 - . 5 2 - 6 . 3 . 1 7 1 2 . 5 . 3 4 4 . 3 3 — . 3 3 — — . 3 4 - - - 4 . 3 . 1 2 1 1 . 6 . 3 1 5 . 0 6 — — . 6 9 — . 0 6 - . 1 9 - 1 0 . 3 . 2 8 1 4 . 9 . 4 0 6 . 0 2 — . 2 6 — . 0 2 - - . 7 0 1 6 . 8 . 4 5 1 9 . 9 . 5 4 7 . 1 6 - - . 3 2 - . 0 8 — . 4 4 2 1 . 2 . 5 7 2 3 . 8 . 6 4 8 . 0 1 l - - 1 . 0 0 _ — — 1 . 5 . 0 3 1 4 . 4 . 2 9 4 9 . 0 2 . 0 6 . 0 6 — — - - . 8 8 - 5 . 9 . 1 2 1 5 . 5 . 3 2 3 . 0 4 — . 0 4 — — . 4 2 - . 5 0 - 7 . 7 . 1 6 1 6 . 2 . 3 3 4 . 0 8 _ . 0 8 — — . 8 4 - - - 5 . 4 . 1 1 1 5 . 3 . 3 1 5 . 0 3 — — . 3 4 — . 2 9 - . 3 4 - 9.4 . 1 9 1 7 . 1 . 3 5 6 . 0 1 — — . 1 9 — . 1 6 - - . 6 4 1 2 . 6 . 2 6 1 9 . 0 . 3 9 7 - . 1 1 - - . 1 9 — . 3 5 — . 3 5 1 6 . 9 . 3 5 2 2 . 2 . 4 5 T a b l e 4.4. E r r o r a n a l y s i s o f SAT method f o r determining s e n s i b l e heat f l u x (H) (wind speed (U) = 5.0 m s " 1 ; z Q = 1.0 mm). 9 to Rj due T o t a l E r r o r Td (C) Case C o n t r i b u t i o n to probable e r r o r no model e r r o r with model e r r o r H T a T T a H T s L U L Z O L Z O H R x Rj/H Rp R T/H (W rn"2) 2.0 1 2 3 4 5 6 7 43 43 50 ,10 ,08 1.00 .43 .43 .50 31 ,86 -,65 -- .56 00 00 ,00 ,00 .14 .14 .04 ,00 .27 .13 8.0 12.2 12.2 11.3 24.8 28.6 41.4 .26 .39 .39 .37 .80 .92 1.34 12, 15. 15, 14, 26, 30.0 42.4 .49 .49 .47 .85 .97 1.37 31.0 72.0 -112-Table 4.5. Parameter values and d e f i n i t i o n s for error a n a l y s i s of SAT method for determining s e n s i b l e heat f l u x (H). A l l cases z Q = 1 . 0 mm (Ta + T s ) / 2 = 1 2 C , Ta = 1 2 + T d / 2 , Ts = 1 2 - Td / 2 Parameter Er r o r s ATa-p = 0 . 5 C ; t y p i c a l error ATay = 1 . 5 C ; high error A T s L = 0 . 5 C ; low error ATs<r = 1 . 5 C ; t y p i c a l error A Ts H = 2 . 1 C ; high error £U~L = 0 . 1 5 m s ~ l ; low error A U H = 0 . 3 0 ; m s ~ l ; high error A Z 0 L i s 0 . 3 mm to 2 . 0 mm; low error £ Z 0 H i s 0 . 0 1 mm to 5 . 0 mm; high error Model Errors AQ = 0 . 1 5 Q AF = 0 . 2 5 F De f i n i tions A i n d i c a t e s probable error in a parameter R = probable e r r o r ; the probable error in H where H i s a function of q x, q 2...q n is given by: „ r ? 2 i 1 / 2 i=l where 8H . r . = -r— Aq The r of each parameter(Ta, Ts, u, z Q ) i s derived by applying the probable error in the parameter to Figures 4 . 1 , 4 . 5 , or 4 . 6 . Rj, R-p = probable error in H. Subscript I indicates probable error due to model input only ( i . e . , no model error included); T indicates t o t a l probable error (due to input and model er r o r ) . R/H = r e l a t i v e error in H. Contribution to Rj due to Lqi i s given by r i / R I and represents a measure of the c o n t r i b u t i o n of the error in each parameter to R j . Table 4.6. Summary of error analysis of SAT method for determining sens-i b l e heat flux ( H ) . Total probable error Rj, (due to input and model errors) for cases of low, t y p i c a l , and high errors ( i . e . , cases 2, 5, and 7 of Tables 4.2. 4 .3, and 4.4) . Conditions Low Typical High H Rip R T / H R T R T / H R«P R T / H (W rn"2) 13=2, Td=2 4.5 .39 8.1 .71 14.5 1.26 11.5 U=2, Td = 5 5.3 .31 6.1 .36 11.4 .67 17.0 U=2, Td = 8 4.8 .33 6.8 .47 11.6 .80 14.5 U=3, Td = 2 8.3 .46 14.9 .83 24.3 1.35 18.0 U=3, Td = 5 12.2 .33 14.9 .40 23.8 .64 37.0 U=3, Td=8 15.5 .32 17.1 .35 22.2 .45 49.0 U=5, Td=2 15.2 .49 26.4 .85 42.4 1.37 31.0 U=5, Td=5 25.9 .36 32.0 .45 53.1 .74 72.0 -114--2 -2 e r r o r was l e s s than 26 W m and under 16 W m f o r wind speeds -1 -2 l e s s than 3 m s . For c o n d i t i o n s of low H (< 20 W m ) , low -2 e r r o r s w i l l commonly be between 5 and 10 W m and high e r r o r s -2 between 10 and 25 W m . I n g e n e r a l , e r r o r s may be expected to be -2 between 30 and 90 percent of H and of the order of 0 to 25 W m -2 For most cases at n i g h t , H i s o f t e n l e s s than 20 W m and e r r o r s -2 i n H w i l l u s u a l l y be l e s s than 15 W m , t y p i c a l l y between 5 and -2 10 w m . B) Importance of E r r o r s i n Each Parameter and Model E r r o r s The importance of e r r o r s i n z , Ta, Ts, and U v a r i e s with o the m i c r o m e t e o r l o g i c a l c o n d i t i o n s at the time of the survey. The v a l u e of the c o n t r i b u t i o n of e r r o r s i n each of these parameters to the t o t a l e r r o r (Tables 4.2 to 4.4) g i v e s a measure of the importance of e r r o r s i n each parameter. Low e r r o r s i n z (Az ) o oL do not cause l a r g e e r r o r s i n H f o r wind speeds l e s s than 3 m s Even high e r r o r s i n z (Az ) do not add a l a r g e magnitude to the o oH t o t a l e r r o r f o r wind speeds l e s s than 2 m s . For l a r g e wind speeds, e r r o r s i n z become c r i t i c a l , e s p e c i a l l y f o r cases of o l a r g e Td. However, l a r g e wind speeds and l a r g e Td are g e n e r a l l y not c o m p a t i b l e . i n s t a b l e c o n d i t i o n s . Small a i r and s u r f a c e temperature e r r o r s are of moderate importance i n most cases, but higher e r r o r s are important. At low wind speeds (< 2.5 m s *) and i n t e r m e d i a t e v a l u e s of Td (3 to 8 C) , s e n s i b l e heat f l u x i s f r e e from s i g n i f i c a n t e r r o r s due to Ta and Ts. E r r o r s i n wind speed input v a l u e s are g e n e r a l l y o n l y important at low wind speeds and l a r g e v a l u e s of Td. Large wind speed e r r o r s may r e s u l t i n high e r r o r s i n s e n s i b l e heat f l u x . -115-The magnitude of the e r r o r s due to model e r r o r are p r o p o r t i o n a l to the value of H. The model e r r o r c o n t r i b u t e s a l a r g e p r o p o r t i o n to the t o t a l e r r o r f o r cases of low e r r o r i n the input parameters of the model but only a small p r o p o r t i o n for cases of high input e r r o r . I t l i m i t s the r e l a t i v e t o t a l e r r o r to values g r e a t e r than 30 pe r c e n t . Model e r r o r s are inherent i n any aerodynamic method of determining s e n s i b l e heat f l u x and are not s p e c i f i c to remote sensing a p p l i c a t i o n s . E f f o r t s can, however, be made to reduce e r r o r s i n z and Ts. An approximate map of z may be necessary i n o o survey areas of g r e a t l y v a r y i n g z^. Every e f f o r t should a l s o be made to reduce e r r o r s i n s u r f a c e temperature e s t i m a t i o n from thermal l i n e - s c a n d ata. C) Change of E r r o r s with M i c r o m e t e o r o l o g i c a l C o n d i t i o n s F i g u r e 4.8 shows a decrease i n the t o t a l e r r o r with d e c r e a s i n g z . Surveys over smooth s u r f a c e s i n which the z used o o i n the method i s l e s s than that used i n the above a n a l y s i s (1.0 mm) w i l l have l e s s e r r o r . The e f f e c t of smoother s u r f a c e s (low z estimate) w i l l be p a r t i c u l a r l y important f o r cases of l a r g e o wind speed. The t o t a l e r r o r i n H i s f a i r l y i n s e n s i t i v e to the value of Td at low wind speeds but i n c r e a s e s with Td at l a r g e wind speeds l a r g e l y due to model e r r o r s ( F i g u r e 4.9). T o t a l e r r o r i s very s e n s i t i v e to the magnitude of the wind speed (Figure 4.10). -116-Figure 4.8. Error in sensible heat flux (H) for SAT method versus surface roughness length (z Q) for a case of no error in z Q and a case of error in z Q . For a z Q of 10 n m, the error in H due to z Q given,is that error which re s u l t s from values of z Q of n = n - 0.52 or n = n + 0.30 whichever gives larger errors. Wind speed (U) i s in m s~*. Air-surface temperature (Td) i s 5.0 C centered about 12 C. Model error i s not included. -117-1 1 U = 5.0 a NO MODEL ERROR WITH MODEL ERROR • U = 5.0 U = 3.0 U=3.0 U = 2.0 - « ~ U = 2.0 8 10 Td (C) igure 4.9. Error in sensible heat flux (H) for SAT method versus air-surface temp-erature difference (Td). Wind speed (U) i s in m s - J-; : G = 1.0 mm; Td centered about 12 C. Errors in i n -put parameters are small ( i . e . . case 3, Tables 4.2 to 4.4). -118-± 4 U (ms ) Figure 4.10. Error in sensible heat flux (H) for SAT method versus wind speed (U). Air-surface temperature difference (Td) i s in degrees C and centered about 12 C; z Q = 1.0 mm. Error in input parameters are small ( i . e . , case 3, Tables 4.2 to 4.4). -119-D) S p e c i a l M i c r o m e t e o r o l o g i c a l C o n d i t i o n s l ) A i r Ponding A i r ponding i s a c o n d i t i o n caused by k a t a b a t i c flow of c o o l a i r i n t o topographic d e p r e s s i o n s . The occurrence of ponding i s s t r o n g l y c o n t r o l l e d by both topography and the magnitude of the wind. Appendix IV d i s c u s s e s a i r ponding. Ponded s i t e s may have temperatures as much as 5 C or more lower than non-ponded s i t e s (e.g., F i g u r e 4 . 4 ) . Depressions as l i t t l e as one meter (Oke, 1978) and slopes as small as one degree (Low and G r e i g , 1973) can cause ponding. A i r ponding occurs when wind speeds are low. As ponding occurs, the a i r temperature p r o f i l e above the ponded s i t e w i l l be complex and the SAT method i s not expected to g i v e good r e s u l t s . At some time a f t e r the ponding event i s i n i t i a t e d and the a i r and s u r f a c e temperatures have adjusted, the atmosphere may assume a normal s t a b l e p r o f i l e . Under these c o n d i t i o n s and the low wind speeds a s s o c i a t e d with ponding, s e n s i b l e heat f l u x w i l l be small or zero and the SAT method, even using a i r temperature e x t r a p o l a t e d from non-ponded s i t e s , w i l l o f t e n g i v e a c c u r a t e r e s u l t s . 2)Cold Layer Development of a n o c t u r n a l c o l d l a y e r has been d e s c r i b e d by F l e a g l e and Badgley (1952), Lake (1956), and Oke (1970). The n o c t u r n a l c o l d l a y e r i s d e f i n e d as the l a y e r of a i r at the ground s u r f a c e which i s everywhere c o o l e r than the ground s u r f a c e ( F l e a g l e and Badgley, 1952). The t y p i c a l n o c t u r n a l a i r temperature p r o f i l e with a c o l d l a y e r i s g i v e n i n F i g u r e 4.11. The c o l d l a y e r may be to a height of more than one meter. The h e i g h t of minimum temperature i s commonly of the order of s e v e r a l -120-TEMPERATURE — F i g u r e 4.11. T y p i c a l n o c t u r n a l a i r temp-e r a t u r e p r o f i l e with a c o l d l a y e r . -121-centimeter s ( F l e a g l e and Badgely, 1952) but may be up to 0.5 m (Oke, 1970). The d i f f e r e n c e between the s u r f a c e temperature and the temperature at the height of minimum temperature can be l a r g e r than 3 C (Oke, 1970). The c o l d l a y e r c o n d i t i o n i s most l i k e l y to occur on calm, c l e a r n i g h t s . Use of the SAT method of determining H w i l l not be a p p l i c a b l e . The source or sink f o r heat i s at the height of minimum temperature. Heat t r a n s f e r at the s u r f a c e i s complex. C o n d i t i o n s under which a c o l d l a y e r may develop should t h e r e f o r e be avoided. Development of the c o l d l a y e r depends on wind speed, s u r f a c e roughness, and s t a b i l i t y . Oke(1970) found the maximum wind speeds at 0.25 m f o r which a c o l d l a y e r developed to be: 1) 1.10 m s ^ (1.3 m s 1 at 1.5 m, assuming a n e u t r a l l o g - l i n e a r p r o f i l e ) f o r a smooth bare s o i l , z = 0.1 mm; 2) 0.65 m s * (0.8 at 1.5 m) f o r ° -1 snow, z = 0.5 mm; 3) 0.40 m s (0.6 at 1.5 m) f o r s h o r t g r a s s , ° -1 z = 4.5 mm; and 4) 0.35 m s (0.5 at 1.5 m) f o r a rough bare o s o i l , z = 10 mm. As s t a b i l i t y i n c r e a s e d , the tendency f o r a c o l d o l a y e r to develop i n c r e a s e d . T h e r e f o r e , f o r wind speeds g r e a t e r than 1.0 to 1.5 m s ^ (at 1.5m), i t can be assumed that a c o l d l a y e r w i l l not develop and the SAT method w i l l g i v e a v a l i d r e s u l t . 3)Short Term L o c a l Advection Short term l o c a l a d v e c t i o n may cause the s u r f a c e temperature at the time the thermal l i n e scanner i s viewing the s u r f a c e to be u n r e p r e s e n t a t i v e of the s u r f a c e temperature corresponding to the average m e t e o r o l o g i c a l c o n d i t i o n s over the whole survey area and over the time p e r i o d i n which ground-truth -122-wind and a i r temperature are measured. Appendix IV d e f i n e s a severe or high (ATs ), moderate (ATs w ), and low (ATs ) surface Ha Ma La temperature probable e r r o r due to advection f o r bare s u r f a c e s as 0.8, 0.4, and 0.2 C r e s p e c t i v e l y . Table 4.7 g i v e s examples of the e f f e c t of these e r r o r s . I t may be concluded t h a t , f o r most cases, sho r t p e r i o d l o c a l advection w i l l not have a l a r g e e f f e c t . 4.3.3 SUMMARY: SAT METHOD The SAT method of remote sensing e s t i m a t i o n of s e n s i b l e heat f l u x i s o u t l i n e d below: 1) Obtain thermal i n f r a r e d l i n e - s c a n data. Great care should be taken i n c a l i b r a t i n g the data and o b t a i n i n g ground t r u t h data to estimate the e f f e c t of the atmosphere on s u r f a c e temperature d e t e r m i n a t i o n s . Good estimates of s u r f a c e e m i s s i v i t y should a l s o be made. 2) Obtain ground measurements of a i r temperature and wind speed at 1.5 m above the ground s u r f a c e at one or s e v e r a l s i t e s w i t h i n the survey area. The measurements should be averaged f o r a p e r i o d of approximately one hour centred at the times of the f l i g h t s . C o n d i t i o n s of a i r ponding, sho r t term l o c a l a d v e c t i o n , and development of a s u r f a c e c o l d l a y e r should be noted. 3) An estimate of s u r f a c e roughness l e n g t h must be made. I t i s important to be as accurate as p o s s i b l e . I f the s u r f a c e roughness i n the survey area v a r i e s g r e a t l y , a map of s u r f a c e roughness estimate may be i n c o r p o r a t e d i n t o the a n a l y s i s . 4) The s u r f a c e temperature, a i r temperature, wind speed, and su r f a c e roughness obtained as i n steps 1, 2, and 3 are used to evalu a t e equation (3.11) on a p i x e l - b y - p i x e l b a s i s . T a b l e 4.7. The e f f e c t o f s h o r t term l o c a l a d v e ction on d e t e r m i n a t i o n of s e n s i b l e heat f l u x (H) using the SAT method. ? Total E r r o r , U Td Case C o n t r i b u t i o n to Ri due to probable e r r o r no model with model e r r o r . e r r o r H Ta<i< T s L T s L a T s M a T s H a " L ZoL Ri Rl/H RT R T/H (W m~2) 2 2 1 .40 .40 _ .10 .10 3.2 .27 4.6 .40 11.5 2 .38 .38 .06 — - .09 .09 3.3 .28 4.7 .41 3 .33 .33 — .18 - .08 .08 3.5 .30 4.9 .42 4 .18 .18 - - .56 .04 .04 4.7 .41 5.8 .50 2 5 1 - - - - - - . - - - - - 17.0 3 2 1 . 39 .39 _ _ .02 .20 6.4 .36 8.3 .46 18.0 2 .37 .37 .05 — - .01 .20 6.6 .37 8.4 .47 3 .32 .32 — .18 - .00 .18 7.1 .40 3.8 .49 4 .18 .18 - - .54 .00 .10 9.5 .53 10.9 .60 3 5 1 .16 .16 _ .16 .52 6.3 .17 12.5 .34 37.0 2 .16 .16 .02 — - .16 .52 6.3 .17 12.5 .34 3 .15 .15 .07 - .15 .48 6.5 .18 12.6 .34 4 .11 .11 - - .31 .11 .36 7.5 .20 13.2 .36 5 2 1 .43 .43 _ .00 .14 12.2 .39 15.2 .49 31.0 mJ 2 .40 .40 .08 — — .00 .12 12.7 .41 15.6 .50 3 . 35 . 35 .19 — .00 .11 13.6 .44 16.3 .53 4 .22 .22 - — .49 .00 .07 17.1 .55 19.3 .62 i E r r o r due to s h o r t term l o c a l a d v e c t i o n i s n e g l i g i b l e ( i . e . , c o n t r i b u t i o n to R | i s 0.00). -124-There are c e r t a i n m i c r o m e t e o r o l o g i c a l c o n d i t i o n s under which the best r e s u l t s w i l l be obtained. The model has been developed for and d i s c u s s e d i n terms of n i g h t t i m e s e n s i b l e heat f l u x e s t i m a t i o n and the recommendations below apply to nighttime cases. 1) The s u r f a c e must be non-vegetated. 2) The s u r f a c e roughness length z should be l e s s than 5 mm. o The smoother the s u r f a c e the more accurate the r e s u l t w i l l be. A Z q estimate near 1.0 mm i s good f o r most moderately rough bare s u r f a c e s as i t minimizes e r r o r s f o r s u r f a c e s ranging from very smooth to those with s u r f a c e roughness lengths up to 5 mm. T y p i c a l s u r f a c e roughness lengths f o r v a r i o u s non-vegetated s u r f a c e s are given i n Table 4.1. 3) S t a b l e a i r temperature p r o f i l e s are advantageous. Under s t a b l e c o n d i t i o n s the s e n s i b l e heat f l u x i s much l e s s s e n s i t i v e to e r r o r s i n s u r f a c e and a i r temperature, e s p e c i a l l y at low wind speeds. S t a b l e c o n d i t i o n s are common du r i n g the n i g h t and the model i s developed f o r s t a b l e or n e u t r a l cases. 4) Wind speeds should be l i g h t . Under calm c o n d i t i o n s of s t r o n g s t a b i l i t y , s e n s i b l e heat f l u x i s zero. The SAT method w i l l c o r r e c t l y y i e l d H = 0 f o r a l a r g e range of erroneous i n p u t . For wind speeds l e s s than 2 to 3 m s \ e r r o r s are s m a l l f o r most cases. S t a b l e c o n d i t i o n s found dur i n g the n i g h t tend to suppress wind speeds and l i g h t wind speeds are common. Two problems may a r i s e under c o n d i t i o n s of low wind speeds, a i r ponding and the development of a c o l d l a y e r . A i r ponding may occur i n survey areas with topographic d e p r e s s i o n s , h i l l s , and v a l l e y s . Very poor estimates of s e n s i b l e heat f l u x can occur at -125-ponded s i t e s . T h e r e f o r e , i n some cases the optimum wind speed i s one which i s as low as p o s s i b l e without p e r m i t t i n g a i r ponding to develop. The wind speed r e q u i r e d to suppress a i r ponding w i l l vary g r e a t l y depending on the topography. For s i t e s s t r o n g l y s u s c e p t i b l e to ponding, a wind speed of about 2.0 to 2.5 m s^" appears to e l i m i n a t e ponding (e.g., F i g u r e s 4.3 and 4.4, and Appendix I V ) . The SAT method i s t h e r e f o r e best a p p l i e d over f l a t t e r r a i n without d e pressions which may r e s u l t i n a i r ponding and under c o n d i t i o n s of l i g h t wind. Winds should be strong enough ( g e n e r a l l y g r e a t e r than 1 m s ^) to prevent the development of a c o l d l a y e r . The method w i l l g i v e good r e s u l t s over areas s u s c e p t i b l e to a i r ponding i f wind speeds are strong enough to e l i m i n a t e ponding or i f the temperature p r o f i l e i s that o f normal s t a b l e c o n d i t i o n s (H i s c l o s e to zero and the SAT method, even with a i r temperature input from non-ponded s i t e s , w i l l a l s o y i e l d s mall values of H). 4.3.4 CONCLUSIONS: SENSIBLE HEAT FLUX ERRORS The SAT method i s u s e f u l f o r e s t i m a t i n g s e n s i b l e heat f l u x f o r non-vegetated, smooth (z < 5 mm) s u r f a c e s under s t a b l e o n i g h t t i m e c o n d i t i o n s . Best r e s u l t s are obtained f o r smooth s u r f a c e s and low wind speeds. However, i n survey areas s u s c e p t i b l e to a i r ponding, wind speed should be l a r g e enough to e l i m i n a t e ponding. The e r r o r s i n determining s e n s i b l e heat f l u x using the SAT method depend g r e a t l y upon the m i c r o m e t e o r o l o g i c a l c o n d i t i o n s 1 -126-and errors in the input parameters. For general remote sensing surveys under t y p i c a l nighttime conditions, probable errors in sensible heat flux w i l l be between 30 to 90 percent and w i l l - 2 usually be less than 15 W m . Under special conditions, such as air ponding, severe short term l o c a l advection, development of a cold layer, poor z estimates, or poor surface temperature o estimates, errors in H can be very large. However, i f care i s taken to determine accurate surface temperature from thermal infrared line-scan data and i f good estimates of z are used, o probable errors can generally be less than 40 percent and - 2 t y p i c a l l y between 5 and 10 W m . Errors for s i t e s p e c i f i c studies can be l e s s . The SAT method provides a viable technique for remote sensing estimation and mapping of the sensible heat flux component of the energy balance. 4.4 ERROR IN LATENT HEAT FLUX ESTIMATION A basic premise in the nighttime thermal i n e r t i a method is that latent heat flux at night w i l l be small. Use of the approximation of latent heat flux equal to zero or an average of that measured at several s i t e s i s often s u f f i c i e n t to give reasonable estimates of LE. These approximations are used in further analysis. Chapter 3 has discussed the magnitude and v a r i a b i l i t y of LE over a variety of surfaces. Errors used in the subsequent analysis are derived from these discussions. -127-4.5 ERROR ANALYSIS OF TOTAL ERRORS The thermal i n e r t i a models presented in Chapter 2 (models I, II, and III) require as input the ground heat flux at two times during a night and the temperature change of the surface between these two times. The accuracy of the model w i l l depend on the errors in these input parameters and errors inherent in the models themselves. The model errors are due to approximations of the i n i t i a l and boundary conditions made by the model and vari a t i o n of these conditions between s i t e s . Since model III best f i t s the actual i n i t i a l and boundary conditions and gives the most reasonable results (Chapter 2) the errors in implementing model III are analyzed in d e t a i l . An error analysis of the implementation of model III i s given for three cases. These cases are a worst or poor case (case 1), a t y p i c a l case (case 2), and a good case (case 3). They represent the errors resulting from the range of errors expected in the input parameters to the model and in the model i t s e l f . Errors for the three cases are determined for each of the components of the model. The t o t a l error expected in the resulting thermal i n e r t i a i s then estimated . Errors are discussed in terms of probable error. 4.5.1 DEFINITION OF COMMON CASES OF ERROR IN THE INPUT PARAMETERS A) Ground Heat Flux A method of estimating the energy balance of a surface has been developed. The ground heat flux (G) i s calculated as a -128-r e s i d u a l of the other energy balance components, Rn, H, and LE. The e r r o r i n determining net r a d i a t i o n i s s m a l l . A worst and -2 -2 t y p i c a l - c a s e probable e r r o r of 5 W m i s reasonable; 3 W m w i l l be used as the probable e r r o r f o r the good case. S e n s i b l e heat f l u x i s d i f f i c u l t to estimate. Estimates can be poor. From -2 p r e v i o u s d i s c u s s i o n s , a worst-case e r r o r i s d e f i n e d as 15 W m In most n i g h t t i m e cases, H w i l l be small and e r r o r s w i l l be l e s s , -2 -2 o f t e n between 5 and 10 W m . A t y p i c a l - c a s e e r r o r of 8 W m -2 and a good-case e r r o r of 5 W m are used. Latent heat f l u x i s very d i f f i c u l t to estimate on a s i t e - b y - s i t e b a s i s but i t s magnitude i s o f t e n s m a l l . T y p i c a l and good-case e r r o r s of -2 -2 5 W m are reasonable while an e r r o r of 10 W m i s l i k e l y r e p r e s e n t a t i v e of a worst-case s i t u a t i o n . The t o t a l probable e r r o r s i n G f o r the three cases are t h e r e f o r e : -2 1) case 1 (worst c a s e ) : 18.7 W m -2 2) case 2 ( t y p i c a l c a s e ) : 10.7 W m -2 3) case 3 (good c a s e ) : 7.7 W m The ground heat f l u x of bare s u r f a c e s i s commonly of the order of -2 . . 30 to 90 W m at n i g h t . Under s t a b l e , low wind c o n d i t i o n s , H and LE are small and Rn w i l l be c l o s e to G. T h e r e f o r e , f o r s i t e s not i n f l u e n c e d by a i r ponding or the development of c o l d a i r l a y e r , e r r o r s i n G w i l l be s m a l l . For most s i t e s the good and t y p i c a l case are more r e p r e s e n t a t i v e of the a c t u a l e r r o r s l i k e l y to o ccur. The ground heat f l u x component may t h e r e f o r e be determined with reasonable accuracy. -129-B) Surface Temperature Change Temperature change i s the other input parameter to the thermal i n e r t i a model. Appendix II analyzes the e r r o r i n temperature change as determined from thermal i n f r a r e d l i n e - s c a n d a t a . A worst-case e r r o r of 2.2 C i s reasonable. T h i s i n c l u d e s high instrument e r r o r and high atmospheric o f f s e t e r r o r . A t y p i c a l e r r o r of 1.6 C i s d e f i n e d and corresponds to intermediate instrument and atmospheric e r r o r . The e f f e c t s of e r r o r s due to t y p i c a l p i x e l m i s r e g i s t r a t i o n expected i n the r e g i s t r a t i o n of a i r b o r n e l i n e - s c a n images (Appendix III) are n e g l i g i b l e f o r these cases (Appendix I I ) . A case of good e r r o r i s 0.9 C. T h i s i n c l u d e s m i s r e g i s t r a t i o n e r r o r (Appendix I I ) . The e f f e c t s of s h o r t term l o c a l a d v e c t i o n may i n t r o d u c e a f u r t h e r e r r o r i n temperature change. Appendix IV d i s c u s s e s t h i s problem and d e f i n e s cases of severe (0.8 C), moderate (0.4 C),and low (0.2 C) temperature e r r o r . These w i l l not be a major component of the e r r o r except i n severe cases and when the estimate of temperature i s otherwise good. C) Model E r r o r s The e r r o r i n the model i t s e l f i s due l a r g e l y to the i n i t i a l and boundary c o n d i t i o n s of the model not corresponding to a c t u a l c o n d i t i o n s . I t was p r e v i o u s l y suggested (Chapter 2) that due to t h i s problem the r e s u l t s of the thermal i n e r t i a model may have to be c a l i b r a t e d to g i v e c o n s i s t e n t l y a c curate thermal i n e r t i a v a l u e s . I f a c a l i b r a t i o n procedure i s f o l l o w e d , model e r r o r s w i l l mainly be caused by d i f f e r e n c e s between the i n i t i a l and boundary c o n d i t i o n s of the s i t e s . Such e r r o r s can be analyzed i n -130-terms of changes i n the time chosen as the s t a r t i n g time of the model (time of zero heat f l u x ) and changes i n the remote sensing sampling times. Chapter 2 analyzed these e r r o r s using f i e l d data of G as a f u n c t i o n of time. The percent change i n thermal i n e r t i a for s h i f t s of zero time and sample time was given (Table 2.3) . A good estimate o f the t o t a l probable e r r o r s due to the model may be made. A t y p i c a l case would be a probable e r r o r of about 0.20 P. T h i s e r r o r should account for the e r r o r s r e s u l t i n g from d i f f e r e n c e s i n time of zero f l u x f o r s i t e s of d i f f e r i n g thermal p r o p e r t i e s and slope aspect and from data a q u i s i t i o n times that are not at corresponding p o s i t i o n s of the G curve of each s i t e . A worst-case e r r o r w i l l be i n the order of 0.30 P and a good-case e r r o r 0.15 P. 4.5.2 PROCEDURE FOR CALCULATING TOTAL ERRORS Model I I I g i v e n by equation (2.12) s i m p l i f i e s to the f o l l o w i n g when sample times of t ^ = 5 and t f = 10 hours are used: P = [-17.3 Gj + 100.9 G f ] / [ T s 1 - Ts f] (4.2) In order to s i m p l i f y the f o l l o w i n g e r r o r a n a l y s i s , G^ i s r e l a t e d to G^ by: G f = 0.75 G x (4.3) The value 0.75 i s a reasonable value based on o b s e r v a t i o n s of the -131-ground heat f l u x f i e l d data of s e c t i o n 2.4. T h i s y i e l d s : P = 53.4 G /[Tsj - Ts f] (4.4) The e r r o r s i n and G^ are included as separate e r r o r s ( i . e . , e r r o r a n a l y s i s i s for equation (4.2)) but e r r o r s and thermal i n e r t i a are now only f u n c t i o n s of G., and temperature change. 4.5.3 RESULTS AND DISCUSSION: TOTAL ERRORS Re s u l t s of the e r r o r a n a l y s i s are given i n F i g u r e 4.12. Appendix V pres e n t s a t a b l e g i v i n g the e r r o r a n a l y s i s i n more d e t a i l . The worst-case e r r o r g i v e s very poor r e s u l t s . The t y p i c a l case g i v e s moderately good r e s u l t s f o r low thermal i n e r t i a s . Good r e s u l t s at low thermal i n e r t i a s and moderate to poor r e s u l t s at high thermal i n e r t i a s occur for the good case. E r r o r s i n temperature change are the dominant source of e r r o r . Appendix V g i v e s the percent c o n t r i b u t i o n of e r r o r s i n ground heat f l u x e s t imates, e r r o r s i n temperature change d e t e r m i n a t i o n s , 2 and model e r r o r s to the t o t a l probable e r r o r squared (R ). For thermal i n e r t i a s of the order of 1500 TIU and l a r g e r , the percent c o n t r i b u t i o n to R due to e r r o r s i n temperature change are from 65 to 90 percent, from 10 to 25 percent due to e r r o r s i n G, and u s u a l l y l e s s than 10 percent due to model e r r o r s . For a l l thermal i n e r t i a s , e r r o r s i n the f i n a l ground heat f l u x estimate (G^) are more important than the e r r o r s i n the f i r s t estimate (G^). — — GOOD TYPICAL 500 1500 2500 3500 THERMAL INERTIA (Jm2C~1s*5) 4 . 1 2 . R e s u l t s o f e r r o r a n a l y s i s of t o t a l e r r o r s f o r t y p i c a l and good cases under c o n d i t i o n s of d i f f e r e n t ground heat f l u x e s (G]_ i n W m~2) . T o t a l e r r o r i n thermal i n e r t i a i s giv e n as the r a t i o of probable e r r o r (R) to the thermal i n e r t i a (P). -133-Errors in temperature change contribute greater than 85 percent 2 to the t o t a l R for thermal i n e r t i a s larger than 2500 TIU. For thermal i n e r t i a s less than approximately 1000 TIU, errors in G become important to the t o t a l error (30 to 70 percent of R ). Model error may also contribute s i g n i f i c a n t l y to the t o t a l error (5 to 40 percent of R ). The contribution of temperature change errors decreases as the actual surface temperature change increases. Therefore, as thermal i n e r t i a s increase, t o t a l errors increase. Micrometeorlogical conditions conducive to high temperature change are therefore advantageous. Since during the night an a i r temperature inversion often occurs and conditions for a high evaporative flux are not common ( i . e . , l i t t l e available energy, stable conditions, and a cool surface l i m i t i n g the vapour pressure gradient) i t cannot be expected that H and LE w i l l transfer much energy away from the surface and, indeed, may often add energy to the surface. Most energy w i l l be l o s t by radiation. Hence, calm clear nights are most suitable to night-time surface cooling. Errors in implementing the model w i l l be at a minimum. Temperature change i s large. Although of lesser importance, G w i l l be determined more accurately since H and LE w i l l approximately equal zero and Rn = G. Rn can be determined accurately. If the survey area is free from topographic depressions susceptible to air ponding or i f one i s w i l l i n g to accept possible errors at these s i t e s , calm clear conditions are optimum. The e f f e c t of ponding depends on the severity and duration of the ponding event. Short duration ponding events w i l l not -134-have a l a r g e e f f e c t on the model unless the sampling time occurs d u r i n g one of the events. The s u r f a c e temperature then w i l l be u n r e p r e s e n t a t i v e of the p r e v a i l i n g m i c r o m e t e o r o l o g i c a l c o n d i t i o n s which have occurred during the n i g h t . The boundary c o n d i t i o n s (ground heat f l u x versus time) used by the model w i l l be c o n s i d e r a b l y i n e r r o r for the ponded s i t e i f ponding c o n d i t i o n s are p e r s i s t e n t . An a d v e c t i v e c o o l i n g event of c o n s i d e r a b l e d u r a t i o n has o c c u r r e d . I t i s not i n c l u d e d i n the boundary c o n d i t i o n s of the model. Remote sensing r e s e a r c h e r s have noted c o l d s u r f a c e s i n ponded s i t e s (e.g., Murtha, 1971; Derksen, 1974; Bennett, 1977). A i r temperature at ponded s i t e s may d i f f e r g r e a t l y from that of non-ponded s i t e s (e.g., F i g u r e 4.4). I t i s e v i d e n t that s i g n i f i c a n t changes i n s u r f a c e heat f l u x and temperature may take plac e as a r e s u l t of ponding. The a n a l y s i s of the e f f e c t of short term l o c a l a d v e ction of Appendix IV g i v e s some idea of the p o s s i b l e magnitude of these changes for a bare s u r f a c e . I f c o n d i t i o n s are such that Rn = G, ground heat f l u x e s at the ponded s i t e s w i l l be lower and the r e c o g n i t i o n , i n the boundary c o n d i t i o n s of the model, of i n c r e a s e d G during the a d v e c t i v e (ponding) event i s i n f a c t reduced. Large e r r o r s may r e s u l t . Ponding c o n d i t i o n s should t h e r e f o r e be avoided. Winds of s u f f i c i e n t speed to e l i m i n a t e ponding are r e q u i r e d . However, i n windy c o n d i t i o n s , energy l o s s from the s u r f a c e w i l l be l e s s (H w i l l be towards the s u r f a c e ) . Temperature change w i l l be l e s s . T o t a l e r r o r s i n thermal i n e r t i a may be l a r g e due to a smaller temperature change and l a r g e r , and more s i g n i f i c a n t e r r o r s i n the e s t i m a t i o n of H. The l a r g e e r r o r s i n implementing the model i n a remote -135-s e n s i n g m o d e a r e n o t d u e t o e r r o r s i n t h e m o d e l i t s e l f a n d t h e m e t h o d s o f d e t e r m i n i n g t h e e n e r g y b a l a n c e c o m p o n e n t s n e c e s s a r y t o t h e m o d e l b u t a r e l a r g e l y d u e t o : 1) e r r o r s i n s u r f a c e t e m p e r a t u r e a s m e a s u r e d b y a t h e r m a l i n f r a r e d l i n e s c a n n e r ; a n d 2) t h e e x t r a p o l a t i o n o f m i c r o m e t e o r o l o g i c a l a n d s u r f a c e c o n d i t i o n s o v e r a l a r g e s u r v e y a r e a . T h e e r r o r i n t e m p e r a t u r e c h a n g e , w h i c h i s t h e m o s t i m p o r t a n t s o u r c e o f e r r o r , i s l a r g e l y d u e t o e r r o r s i n t h e d e t e r m i n a t i o n o f a t m o s p h e r i c o f f s e t ( A p p e n d i x I I ) . E r r o r s i n G a r e d u e p a r t l y t o e r r o r s i n L E . I t i s n e c e s s a r y t o t h e m o d e l t h a t L E b e s m a l l o r n o t v a r y g r e a t l y f r o m s i t e t o s i t e . T h i s i s c o m m o n a t n i g h t . E r r o r s i n G a r e , h o w e v e r , l a r g e l y d u e t o e r r o r s i n t h e e s t i m a t i o n o f s e n s i b l e h e a t f l u x . S u r f a c e t e m p e r a t u r e d e t e r m i n a t i o n i s a c r i t i c a l s o u r c e o f e r r o r i n t h e s e n s i b l e h e a t f l u x d e t e r m i n a t i o n . E r r o r s i n s u r f a c e t e m p e r a t u r e a r e m a i n l y d u e t o p o o r e s t i m a t i o n o f t h e a t m o s p h e r i c o f f s e t a n d s u r f a c e e m i s s i v i t y a n d t h e n e e d t o e x t r a p o l a t e a n e s t i m a t e d e m i s s i v i t y o v e r a v a r i e t y o f s u r f a c e t y p e s . T h e o t h e r m a j o r s o u r c e o f e r r o r i n H i s i n t h e e s t i m a t i o n a n d e x t r a p o l a t i o n o f z . P r o b l e m s i n d e t e r m i n i n g G a r e o v e r c o m e o n c a l m , c l e a r o n i g h t s w h e n R n = G a n d H = L E = 0 . H o w e v e r , a i r p o n d i n g i s c o m m o n o n c l e a r , c a l m n i g h t s a n d t h i s m a y l e a d t o e r r o r s . A i r p o n d i n g i n s o m e c a s e s i s a m a j o r l i m i t i n g f a c t o r o f t h e m e t h o d . T h e r a n g e o f v a l u e s o r n o i s e l e v e l i n t h e r m a l i n e r t i a s g i v e n b y t h e m o d e l f o r a s u r f a c e u n i t o f u n i f o r m t h e r m a l i n e r t i a i s e x p e c t e d t o b e l e s s t h a n t h e p r o b a b l e e r r o r g i v e n f o r t h e d e t e r m i n a t i o n o f t h e a c t u a l v a l u e o f t h e r m a l i n e r t i a ( F i g u r e 4.12, A p p e n d i x V ) . E r r o r s i n t h e c o m p a r i s o n o f s i t e s i n a t h e r m a l i n e r t i a s u r v e y a n d i n p a r t i c u l a r c o m p a r i n g s i t e s o f -136-s i m i l a r thermal i n e r t i a are a l s o expected to be l e s s . The major source of e r r o r i n thermal i n e r t i a value i s temperature change. The major source of e r r o r i n temperature change i s the atmospheric o f f s e t . The l a r g e e r r o r i n atmospheric o f f s e t i s due to i n a b i l i t y to determine the o f f s e t a c c u r a t e l y (Appendix II) . The o f f s e t i s not expected to vary g r e a t l y from s i t e to s i t e (except at l a r g e scan angles) and the e r r o r i n the o f f s e t w i l l be almost constant for a l l s i t e s . T h i s w i l l cause s i m i l a r e r r o r s i n p r e d i c t e d thermal i n e r t i a p a r t i c u l a r l y for s i t e s of s i m i l a r thermal i n e r t i a and temperature change. For example, i g n o r i n g atmospheric o f f s e t e r r o r the probable e r r o r i n temperature change for the t y p i c a l case i s approximately 0.9 C.3 T h i s i s e q u i v a l e n t to the temperature change e r r o r f o r the good case. Temperature change e r r o r i s the dominant f a c t o r i n t o t a l thermal i n e r t i a e r r o r and the d i f f e r e n c e i n probable e r r o r i n ground heat f l u x f o r the good versus t y p i c a l case i s s m a l l . T h e r e f o r e , the good-case e r r o r i n the value of thermal i n e r t i a (Figure 4.12) i s an approximation of the noise l e v e l of the model and to some extent the e r r o r i n comparison of s i t e s for the t y p i c a l case. S i m i l a r l y , noise l e v e l and e r r o r i n comparison of s i t e s for the good and worst cases are l e s s . Thus, d i s c r i m i n a t i o n or mapping of u n i t s based on thermal i n e r t i a w i l l be more r e a d i l y made than i s i n d i c a t e d by the e r r o r s g i v e n for determining the a c t u a l value of thermal i n e r t i a . 4.6 CONCLUSIONS The model (model III) g i v e s meaningful r e s u l t s with good -137-input data. The primary problem i n implementing the model i s o b t a i n i n g accurate s u r f a c e temperature data from thermal i n f r a r e d l i n e - s c a n data. M i c r o m e t e o r o l o g i c a l c o n d i t i o n s r e s u l t i n g i n l a r g e temperature changes are advantageous. Non-ponded s i t e s on calm, c l e a r n i g h t s w i l l y i e l d the best r e s u l t s . E r r o r s i n thermal i n e r t i a are l a r g e f o r s i t e s of high thermal i n e r t i a , as a r e s u l t of the small temperature changes at these s i t e s and the consequent l a r g e e f f e c t of temperature change e r r o r s . The magnitude of expected e r r o r s i s too l a r g e to be meaningful i n drawing q u a n t i t a t i v e c o n c l u s i o n s from a thermal i n e r t i a survey. In an i n t e r p r e t i v e mode, such as o p t i c a l - v i s u a l a n a l y s i s of thermal i n e r t i a images for the purpose of a i d i n g g e o l o g i c a l or s o i l s mapping or the d e t e c t i o n of anomalous f e a t u r e s , the method w i l l o f t e n be u s e f u l . For low thermal ( i n e r t i a s ( l e s s than 1500 or 2000 TIU) with good in p u t , the method may give u s e f u l r e s u l t s for q u a n t i t a t i v e a n a l y s i s . Many s o i l s have thermal i n e r t i a s l e s s than 2000 TIU. Thus, the method w i l l be b e n e f i c i a l for s o i l s t u d i e s . T h i s i s p a r t i c u l a r l y t rue of s o i l moisture s t u d i e s of dry to moderately dry s o i l s s i n c e thermal i n e r t i a s are low and change r a p i d l y with moisture content (Chapter 1). Good r e s u l t s are expected f o r most s i t e s p e c i f i c s t u d i e s i n which s u r f a c e roughness and e m i s s i v i t y are known a c c u r a t e l y and the atmospheric o f f s e t of the l i n e - s c a n data can a l s o be determined a c c u r a t e l y . -138-1From Appendix I I : good e r r o r i s e r r o r f o r low p i x e l and atmospheric e r r o r , t y p i c a l e r r o r i s for intermediate p i x e l and atmospheric e r r o r , and poor e r r o r i s l a r g e p i x e l and atmospheric e r r o r . There i s no e m i s s i v i t y e r r o r . 2 Atmospheric o f f s e t r e f e r s to a temperature o f f s e t determined by comparing ground temperature data of t e s t s i t e s to the c a l i b r a t e d remotely sensed temperature. The o f f s e t accounts for atmospheric a t t e n u a t i o n and emission (Appendix I I ) . 3 Assuming the v a r i a t i o n of atmospheric o f f s e t with scan angle i s c o r r e c t e d for or i s small f o r the areas of the survey of i n t e r e s t , the temperature e r r o r s w i l l be mainly due to instrument e r r o r (Appendix I I ) . For the t y p i c a l - e r r o r case the probable e r r o r w i l l be approximately 0.9 C ( i n t e r m e d i a t e p i x e l e r r o r (Table II.5) p l u s m i s r e g i s t r a t i o n e r r o r ) . o -139-Chapter F i v e CONCLUSIONS The p o t e n t i a l of thermal i n e r t i a mapping i n p l a n e t a r y s t u d i e s , as an a i d i n g e o l o g i c a l and s o i l s mapping and i n s o i l moisture s t u d i e s , was demonstrated. I t s u s e f u l n e s s f o r vegetated s u r f a c e s and i n o p t i c a l - v i s u a l and automatic d i g i t a l image c l a s s i f i c a t i o n was d i s c u s s e d . Current thermal i n e r t i a models were reviewed. Some problems i n remote sensing thermal i n e r t i a a n a l y s i s were i d e n t i f i e d . These i n c l u d e problems of the e s t i m a t i o n of l a t e n t heat f l u x , s p a t i a l e x t r a p o l a t i o n of s u r f a c e and m i c r o m e t e o r o l o g i c a l c o n d i t i o n s over the area of a thermal i n e r t i a survey, and the accuracy of the s u r f a c e temperature as measured by a thermal i n f r a r e d l i n e scanner. Current thermal i n e r t i a techniques employ the d i u r n a l c y c l e of temperature and energy balance and t h e r e f o r e must account f o r v a r i a t i o n i n absorbed s o l a r r a d i a t i o n due to s u r f a c e topography and albedo. The models are n e c e s s a r i l y complex. An albedo map and d i g i t a l t e r r a i n model ( i n areas of moderate or high r e l i e f ) must be obtained and r e g i s t e r e d to the temperature data. There i s a need for a thermal i n e r t i a model and technique which i s simple, r e q u i r e s a minimum of input (e.g., no d i g i t a l t e r r a i n model or albedo map), and i s a p p l i c a b l e to s u r f a c e s which may have s i g n i f i c a n t and v a r y i n g moisture content (e.g., non-arid r e g i o n s ) . A technique a p p l i c a b l e over vegetated t e r r a i n would a l s o be of b e n e f i t . The f o l l o w i n g c r i t e r i a d e s i r a b l e f o r a remote sensing -140-thermal i n e r t i a mapping model were presented ( s e c t i o n 1.5). 1) The model should give r e s u l t s u s e f u l for q u a n t i t a t i v e a n a l y s i s and/or q u a l i t a t i v e a n a l y s i s . Q u a n t i t a t i v e a n a l y s i s w i l l l i k e l y r e q u i r e the model to y i e l d probable e r r o r s l e s s than approximately 300 TIU. Poorer r e s u l t s w i l l s t i l l p r o v i d e u s e f u l q u a l i t a t i v e i n f o r m a t i o n . 2) No or l i m i t e d knowledge of the s u r f a c e type i s r e q u i r e d . T h i s i m p l i e s that p r e c i s e values of thermal c o n d u c t i v i t y , s p e c i f i c heat, d e n s i t y , e m i s s i v i t y , albedo, or s u r f a c e roughness need not be known. 3) The i n f l u e n c e of topography should have a minimum e f f e c t on the r e s u l t s . T h i s i m p l i e s that the e f f e c t o f topography must be accounted for by the model ( r e q u i r i n g use of a d i g i t a l t e r r a i n model, with consequent complexity of implementation) or the model must be independent of topography. 4) Measurements of m e t e o r o l o g i c a l c o n d i t i o n s are necessary o n l y at a s i n g l e or few s i t e s . T h i s i m p l i e s that measurements at l o c a l s i t e s may be e x t r a p o l a t e d over the whole re g i o n of the survey and a l l types of s u r f a c e . 5) The model must be a p p l i c a b l e over a wide range of s u r f a c e and s u r f a c e moisture c o n d i t i o n s . T h i s i m p l i e s that the model should give good r e s u l t s for areas of s u r f a c e and c l i m a t i c c o n d i t i o n s ranging from non-vegetated s u r f a c e s i n a r i d r e g i o ns to vegetated s u r f a c e s i n humid r e g i o n s . I t was hypothesized that a model using o n l y nighttime c o o l i n g w i l l meet these c r i t e r i a . S e v e r a l nighttime c o o l i n g models (models I, I I , and III) were developed and t e s t e d using accurate ground based data. -141-Model I I I g i v e s the most accurate and c o n s i s t e n t r e s u l t s . Models I and II and probably model I I I w i l l r e q u i r e some c a l i b r a t i o n procedure i n order to y i e l d c o n s i s t e n t l y accurate thermal i n e r t i a e s t i m a t e s . An energy balance approach i s used to implement the models. T h i s r e q u i r e s that the ground heat f l u x i n the s u r f a c e media be known. Ground heat f l u x (G) i s determined as the r e s i d u a l of the s u r f a c e energy balance components: net r a d i a t i o n (Rn), s e n s i b l e heat f l u x (H), and l a t e n t heat f l u x (LE). Methods of determining these components, a p p l i c a b l e to remote sensing implementation, were developed. Included are: a method of determining Rn using remotely sensed s u r f a c e temperature; a method of determining s e n s i b l e heat f l u x which i n c o r p o r a t e s a s t a b i l i t y and s u r f a c e sublayer c o r r e c t i o n ; and methods of approximating LE. E r r o r s i n implementing model I I I i n an o p e r a t i o n a l remote sensing mode were determined using an e r r o r a n a l y s i s approach. E r r o r s r e s u l t i n g from e x t r a p o l a t i o n of the m i c r o m e t e o r l o g i c a l parameters z , a i r temperature, and wind speed were estimated, o E r r o r s i n the s u r f a c e temperature d e r i v e d from thermal i n f r a r e d l i n e - s c a n data were analyzed i n terms of t h e i r e f f e c t on d e t e r m i n a t i o n o f net r a d i a t i o n , s e n s i b l e heat f l u x , and temperature change of a s i t e . I t was concluded t h a t , although the model and methods of determining the energy balance input components are v a l i d given good input and y i e l d e d good r e s u l t s using ground based data, e r r o r s i n h e r e n t i n the remote sensing implementation of the model l i m i t the u s e f u l n e s s of the model. The major source of e r r o r i s i n the remote sensing d e t e r m i n a t i o n of the temperature change of a s i t e between two -142-times. The main f a c t o r i s the e r r o r i n determining the atmospheric o f f s e t ( i . e . , the e f f e c t of atmospheric a t t e n u a t i o n and e m i s s i o n ) . A smaller e r r o r r e s u l t s from e s t i m a t i n g ground heat f l u x . The major component of the e r r o r i n G i s e r r o r i n s e n s i b l e heat f l u x . The major components of e r r o r s i n H are s u r f a c e temperature d e t e r m i n a t i o n (mainly due to atmospheric o f f s e t and e m i s s i v i t y e r r o r s ) and e s t i m a t i o n of the s u r f a c e roughness length (z ). Wind speed e r r o r s may a l s o be important. E r r o r s i n G w i l l o be small on calm n i g h t s when H = LE = 0 and Rn = G. A i r ponding may cause problems i n determining H at a sampling time or cause an anomalous event of heat f l u x at the s u r f a c e which i s not c o r r e c t l y modelled by the boundary c o n d i t i o n s of the model. E r r o r s i n LE may a l s o c o n t r i b u t e to e r r o r s i n G. C o n d i t i o n s of low LE are common at n i g h t and e r r o r s i n most cases w i l l not be extreme. Model e r r o r s are of r e l a t i v e l y small importance i n the remote sensing implementation of the model. The procedure f o r implementing the n i g h t t i m e thermal i n e r t i a mapping method (model III) i s as g i v e n below. 1) Obtain thermal i n f r a r e d l i n e - s c a n data at 1.5 and 6.5 hours a f t e r sunset. Assume a time of zero f l u x 3.5 hours b e f o r e sunset. The times may be r e f i n e d f o r the p a r t i c u l a r thermal p r o p e r t i e s of the s u r f a c e s of the survey area and f o r the time span between sunset and s u n r i s e . 2) Measure a i r temperature (Ta) and wind speed (U) at 1.5 m above the ground s u r f a c e at one or s e v e r a l s i t e s w i t h i n the survey area. The temperatures and wind speed used i n the model should be approximately one hour averages centered about the time -143-of the f l i g h t s . 3) Measure l a t e n t heat f l u x at one or s e v e r a l s i t e s w i t h i n the survey area. T h i s i s o p t i o n a l depending on the method one uses to approximate LE. 4) Measure longwave downward r a d i a t i o n (B T) at one s i t e i n the survey area. Longwave downward r a d i a t i o n may a l s o be estimated from e m p i r i c a l formulae. A measured value i s recommended. 5) Estimate the s u r f a c e roughness length ( Z q ) , e i t h e r by d i r e c t measurement of s e l e c t e d s i t e s or from values or e m p i r i c a l formulae i n the l i t e r a t u r e . Use of an estimated average for the survey w i l l o f t e n be s u f f i c i e n t . A map of u n i t s of v a r y i n g Z q i s o p t i o n a l , but may be u s e f u l . 6) Estimate s u r f a c e e m i s s i v i t y from the l i t e r a t u r e or measurements. One value may be assumed for the whole survey area or a map of e m i s s i v i t y u n i t s produced. 7) C o r r e c t the thermal i n f r a r e d l i n e - s c a n data for the e f f e c t s of atmospheric a t t e n u a t i o n and emission. T h i s step i s c r i t i c a l , as much of the e r r o r i n implementing the model i s due to the e r r o r i n determining the e f f e c t of the atmosphere on s u r f a c e temperature e s t i m a t e s . One method i s to compare ground based temperature at s e v e r a l s i t e s with temperatures from the thermal i n f r a r e d l i n e - s c a n data f o r the s i t e s and determine an average temperature o f f s e t (Appendix I I ) . T h i s i s then a p p l i e d t o the temperature data to g i v e estimates of s u r f a c e temperature. Well chosen, e a s i l y l o c a t e d , and/or s p e c i a l l y designed and l o c a t e d s i t e s (to permit d e t e r m i n a t i o n of the e f f e c t of scan angle) are advantageous. Great care and perhaps more -144-s o p h i s t i c a t e d techniques should be used i n c o r r e c t i n g for the e f f e c t of the atmosphere on s u r f a c e temperature e s t i m a t i o n . 8) Determine the ground heat f l u x . I t i s estimated as the r e s i d u a l of the energy balance components: net r a d i a t i o n , s e n s i b l e heat f l u x , and l a t e n t heat f l u x . Net r a d i a t i o n i s determined by equation (3.2), s e n s i b l e heat f l u x by equation (3.11), and l a t e n t heat f l u x approximated by zero or an average of the measured value at s e v e r a l s i t e s . 9) R e g i s t e r the two thermal i n f r a r e d l i n e - s c a n images. R e g i s t r a t i o n of a i r b o r n e l i n e - s c a n images i s d i s c u s s e d i n Appendix I I I . Determine the temperature change at each p i x e l . 10) C a l c u l a t e thermal i n e r t i a on a p i x e l - b y - p i x e l b a s i s using equation (2.12). A c a l i b r a t i o n procedure using s i t e s of known thermal i n e r t i a may be used. 11) Produce an image or map of thermal i n e r t i a . Optimum c o n d i t i o n s under which to conduct a thermal i n e r t i a survey using the nighttime model are o u t l i n e d below. 1) E r r o r s i n implementing the model are l e s s f o r survey areas with s u r f a c e s of low thermal i n e r t i a . Temperature changes of s i t e s w i l l be l a r g e and e r r o r s i n temperature change w i l l have a l e s s e r e f f e c t . 2) M i c r o m e t e o r o l o g i c a l c o n d i t i o n s conducive to high heat l o s s from the s u r f a c e and t h e r e f o r e high temperature change are best. C l e a r sky c o n d i t i o n s i n c r e a s e Rn and calm c o n d i t i o n s reduce s e n s i b l e heat f l u x which w i l l add energy to the s u r f a c e at n i g h t . Both c l e a r and calm ( i f no ponding occurs) c o n d i t i o n s are f a v o u r a b l e . -145-3) Topographic and m i c r o m e t e o r o l o g i c a l c o n d i t i o n s f a v o u r a b l e to the development of a i r ponding should be avoided. Calm c o n d i t i o n s are t h e r e f o r e only advantageous on f l a t t e r r a i n or i f e r r o r s at ponded s i t e s are a c c e p t a b l e . Otherwise, steady wind speeds l a r g e enough to suppress ponding are r e q u i r e d . At high wind speeds, however, s e n s i b l e heat f l u x becomes a more important component of the energy balance and i t a l s o becomes more d i f f i c u l t to estimate. Temperature change decreases. E r r o r s i n c r e a s e . Wind speeds j u s t l a r g e enough to e l i m i n a t e ponding are optimum. Often these w i l l be of the order of 2 to 2.5 m s ^ or perhaps l e s s for survey areas of m i l d r e l i e f . 4) Surfaces should be bare. The method of determining s e n s i b l e heat f l u x does not g i v e good r e s u l t s on vegetated s u r f a c e s . The poor r e s u l t s are due to r a d i a t i v e temperature not being r e p r e s e n t a t i v e of the temperature at the l e v e l of the a c t u a l source or sink of heat. I t i s an i n t e g r a t e d value of temperature at v a r i o u s depths i n the canopy depending on the s t r u c t u r e and d e n s i t y of the canopy. A l s o , the method (and a l l aerodynamic methods) of determining s e n s i b l e heat f l u x i s very s e n s i t i v e to values of s u r f a c e roughness l e n g t h (z ) at the l a r g e o v a l u e s of Z q t y p i c a l of vegetated c a n o p i e s . 5) Surface roughness l e n g t h Z q should be l e s s than 5 mm. T h i s requirement i s a r e s u l t of the g r e a t s e n s i t i v i t y of aerodynamic methods of determining H to z at high v a l u e s of z . o o Smoother s u r f a c e s w i l l g i v e b e t t e r r e s u l t s . U n i f o r m i t y of s u r f a c e roughness throughout the survey area w i l l a l s o give b e t t e r estimates of thermal i n e r t i a . 6) Since methods of determining LE f o r remote sensing -146-purposes w i l l be o n l y crude approximations, i t i s important that LE i s a small component of G and/or that i t be uniform throughout the survey area. Latent heat f l u x i s o f t e n low at n i g h t . 7) A s p e c i a l case of low e r r o r w i l l occur for f l a t survey areas (no ponding) on calm, c l e a r n i g h t s . Rn w i l l approximately equal G. E r r o r s i n G w i l l be small as Rn can be determined with good accuracy. The e f f e c t s of the s u r f a c e roughness and problems o f determining LE are not important as H and LE are z e r o . The method w i l l be a p p l i c a b l e even to vegetated s u r f a c e s . The n i g h t t i m e remote sensing thermal i n e r t i a model (model III) does not e n t i r e l y meet the c r i t e r i a f o r thermal i n e r t i a models s t a t e d i n s e c t i o n 1.5. The u s e f u l n e s s of the r e s u l t s f o r q u a n t i t a t i v e a n a l y s i s i s l i m i t e d to s u r f a c e s of low thermal i n e r t i a . The c r i t e r i o n of probable e r r o r l e s s than 300 TIU i s met o n l y f o r s u r f a c e s of low thermal i n e r t i a s (< 1000 TIU) and with good remote sensing i n p u t . U s e f u l q u a l i t a t i v e r e s u l t s may be expected f o r most s u r f a c e s i f remote sensing input i s good. Some l i m i t a t i o n s must be placed on the s u r f a c e types to which the model a p p l i e s i f c r i t e r i o n 2 (no or l i m i t e d knowledge of s u r f a c e type) i s to be met. The s u r f a c e must be non-vegetated and have a roughness l e n g t h l e s s than approximately 5 mm. C r i t e r i o n 3 (independence from topography) and c r i t e r i o n 4 ( e x t r a p o l a t i o n of m i c r o m e t e o r o l o g i c a l measurements) are met f o r non-ponding c o n d i t i o n s and at non-ponded s i t e s . In areas of topography s u s c e p t i b l e to ponding, c r i t e r i a 3 and 4 are met i f wind speed i s high enough to e l i m i n a t e ponding. The higher wind speed, however, reduces the accuracy of the model i n some cases. The -147-model i s a p p l i c a b l e over a wide v a r i e t y of t e r r a i n and moisture c o n d i t i o n s . However, i t i s o n l y v a l i d f o r vegetated s u r f a c e s under s p e c i a l c o n d i t i o n s of H = LE = 0, Rn = G, and no a i r ponding. Conducting thermal i n e r t i a mapping surveys using the nighttime method and temperature data from a g e n e r a l thermal i n f r a r e d l i n e - s c a n m i s s i o n u t i l i z i n g procedures and data commonly a v a i l a b l e to most users, w i l l o f t e n r e s u l t i n moderate to l a r g e e r r o r s in thermal i n e r t i a e s t i m a t i o n . The u s e f u l n e s s of the survey w i l l be l i m i t e d to q u a l i t a t i v e and i n t e r p r e t i v e c o n c l u s i o n s . I f great care i s taken i n determining s u r f a c e temperatures and temperature changes from the i n f r a r e d l i n e - s c a n data, good q u a n t i t a t i v e r e s u l t s w i l l o f t e n be obtained f o r thermal i n e r t i a s l e s s than approximately 2000 or 1500 TIU. The l i m i t a t i o n of the model i s not i n the model i t s e l f but i n the a b i l i t y to determine s u r f a c e temperature and temperature change a c c u r a t e l y . The a b i l i t y to determine temperature change i n r e l a t i o n to the magnitude of the a c t u a l temperature change i s c r i t i c a l . The major problem i n a c c u r a t e measurement of temperature change i s the e r r o r s caused by atmospheric a t t e n u a t i o n and emission. E r r o r s due to the atmosphere and to s u r f a c e e m i s s i v i t y are the main source of e r r o r i n determining s u r f a c e temperature. E r r o r s i n s u r f a c e temperature cause l o s s of accuracy i n the r e s u l t s by i n c r e a s i n g e r r o r s i n both the s e n s i b l e heat f l u x and net r a d i a t i o n component of ground heat f l u x . There i s a need f o r s p e c i a l i z e d procedures f o r determining the e f f e c t of the atmosphere on temperature e s t i m a t e s . Research and development of methods of remotely determining e m i s s i v i t y would -148-be of great advantage. I f these problems are so l v e d , e i t h e r by g r e a t care i n implementing the model or development of new techniques, the l i m i t a t i o n s of the model to s u r f a c e s of low thermal i n e r t i a may, f o r the most p a r t , be dropped. The p o t e n t i a l of the ni g h t t i m e thermal i n e r t i a method as an a i d to s o i l s mapping and as a method of q u a n t i t a t i v e s o i l moisture mapping i s l a r g e . Mapping of g e o l o g i c boundaries and d i s c r i m i n a t i n g between or a i d i n g i n i d e n t i f i c a t i o n of g e o l o g i c u n i t s may a l s o be p o s s i b l e , although the l a r g e r thermal i n e r t i a s of rocks (thus, l a r g e e r r o r s ) reduce the u s e f u l n e s s of the method fo r g e o l o g i c a l a p p l i c a t i o n s . The method i s only u s e f u l f o r vegetated t e r r a i n f o r the s p e c i f i c c o n d i t i o n s of H = LE = 0, Rn = G, and no ponding. These c o n d i t i o n s may, however, be common over l a r g e amounts of f l a t a g r i c u l t u r a l l a n d . The meaning and p o t e n t i a l b e n e f i t s of thermal i n e r t i a a n a l y s i s of vegetated t e r r a i n have not been f u l l y i n v e s t i g a t e d . A p p l i c a t i o n of the model to p l a n e t a r y s t u d i e s o f f e r s promising r e s u l t s . Long p e r i o d s of "nighttime" c o o l i n g (e.g., lunar n i g h t ) , l a r g e temperature changes (e.g., Mars and the Moon), and simple or e a s i l y determined energy balances w i l l r e s u l t i n very u s e f u l estimates of thermal i n e r t i a f o r s e v e r a l p l a n e t a r y bodies. The s i m p l i c i t y and minimum of input to the model are of gr e a t advantage i n p l a n e t a r y a p p l i c a t i o n s . T h e r e f o r e , the model combines the u s e f u l n e s s of the knowledge of the p r o p e r t y thermal i n e r t i a with the c a p a b i l i t i e s of remote sensing to o f f e r b e n e f i t s i n the study of b~th e a r t h and p l a n e t a r y r e s o u r c e s . -149-LITERATURE CITED A r p a c i , V.S. 1966. Conduction Heat T r a n s f e r . Addison-Wesley, London. 550 p. Bennett, R.C. 1977. Use of thermal imagery from an airborne scanning radiometer to d e r i v e the d i s t r i b u t i o n of s u r f a c e temperature. 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APPENDICES -158-Appendix I DERIVATION OF EQUATION (2.9) S o l u t i o n s to the heat conduction equation with time dependent boundary c o n d i t i o n s can be obtained using Duhammel's s u p e r p o s i t i o n i n t e g r a l ( A r p a c i , 1966). Duhammel's s u p e r p o s i t i o n i n t e g r a l g i v e s the temperature T ( z , t ) r e s u l t i n g from a time dependent d i s t u r b a n c e D ( t ) i n a form r e l a t i n g to the s o l u t i o n to a stepwise d i s t u r b a n c e Y ( z , t ) . I t may be giv e n i n the form ( A r p a c i , 1966): ft ,w x ay(z .t-s) . e T(z . t ) - J 0 D(s) ^ ds (1.1) where, f o r the case being c o n s i d e r e d , D(s) i s the stepwise d i s t u r b a n c e (a f l u x (G) at z = 0) and y ( z , t - s ) i s the temperature r e s u l t i n g from a u n i t constant f l u x of d u r a t i o n t - s . The parameter V(z,t-s) (= V(z,t)) i s given as (Carslaw and Jaeger, 1959; constant f l u x c a s e ) : ¥(z , x ) - f { [ S-I ] 1 / 2 e - z 2 / 4 K T - f e r f c ( — ) } U'U where < i s the d i f f u s i v i t y and X i s the thermal c o n d u c t i v i t y , T h e r e f o r e : 1/2 2 MZ.T) WU,tS> . K ...J -Z /iK [t-s] " " S ^ - " * x , 1 ' 2 u - s ] 1 / 2 ( 1 - 3 ) ( -159-1/2 where < / X i s thermal i n e r t i a (P). Consider the time dependent boundary c o n d i t i o n of heat f l u x G(t) given i n the form of F i g u r e 1.1. The f u n c t i o n G(t) may be mathematically d e s c r i b e d as: G(t) - bt 0 £ t £ tl ( I > 4 ) and G(t) = bt - blt-tj] + clt-tj] t > tj (1.5) T h i s boundary c o n d i t i o n i s a time dependent d i s t u r b a n c e as i n equation (1.1) ( i . e . , G(t) = D ( t ) ) . For t > t equation (1.1) i s : T(z.t) - - L ^ { bs[t-s]-1 / 2 e-2M<ft-s] d s + P IT (1.6) O t-bs + cs] [t-s]-1/2 . " ' W - s ] d s } Since the second term a p p l i e s o n l y f o r t > t the l i m i t s of i n t e g r a t i o n f o r the parameter are 0 to t - t ^ . The f i r s t term i s necessary as the d i s t u r b a n c e D(t) = bt has an i n f l u e n c e on temperature f o r a l l times and the l i m i t s of i n t e g r a t i o n are 0 to t . The terms of equation (1.6) are of the form which r e s u l t s from a case with f l u x of the form, f l u x = a t . The s o l u t i o n to an equation of t h i s form i s giv e n by Carslaw and -160--161-Jaeger (1959). For z = 0 i t i s : T(0,t) - 0.752 a t 3 / 2 / P (1 .7 ) T h e r e f o r e the s o l u t i o n to ( 1 .6 ) using ( 1 .7 ) i s : T(0.t) = 0.752 { b t 3 / 2 - b [ t - t l ] 3 / 2 + c [ t - t l ] 3 / 2 > / P d . 8 ) E x p r e s s i n g the thermal i n e r t i a i n terms of the temperature change between time zero (t = 0) and t = t ^ g i v e s : P = 0.752 { b t 3 / 2 - b l t j - t j ] 3 7 2 + c [ t f - t l ] 3 / 2 } / [ T s o - Ts f] ( I . 9 ) where Ts and Ts_ are the temperatures at z = 0 f o r times t = 0 o f and t = t f r e s p e c t i v e l y . -162-Appendix II THERMAL INFRARED LINE-SCAN DATA FOR QUANTITATIVE STUDIES: AN ERROR ANALYSIS 1. INTRODUCTION Thermal i n f r a r e d l i n e - s c a n data have a l a r g e p o t e n t i a l as a data base for s c i e n t i f i c r esearch as w e l l as f o r a p p l i e d i n v e s t i g a t i o n s such as thermal i n e r t i a mapping, s o i l moisture d e t e r m i n a t i o n , e v a p o t r a n s p i r a t i o n e s t i m a t i o n , and v e g e t a t i o n s t r e s s d e t e c t i o n (e.g., G i l l e s p i e and Kahle, 1977; C i h l a r et a l . , 1979; Heilman et a_l., 1970 ; Byrne et a l . , 1979). A p r e r e q u i s i t e f o r such q u a n t i t a t i v e a p p l i c a t i o n s i s that a c curate s u r f a c e temperature values may be d e r i v e d from thermal i n f r a r e d l i n e - s c a n d a ta. D i g i t a l image p r o c e s s i n g and d i s p l a y techniques o f f e r a p r a c t i c a l method of determining s u r f a c e temperatures. The magnitude and source of e r r o r s i n determining. s u r f a c e temperatures from d i g i t a l thermal i n f r a r e d l i n e - s c a n data are i n v e s t i g a t e d with r e s p e c t to the manners in which the data are o f t e n used (e.g., temperature, temperature d i f f e r e n c e between s i t e s , and temperature changes). The purpose of t h i s study i s to determine the e r r o r s expected f o r a g e n e r a l remote sensing thermal i n f r a r e d l i n e - s c a n survey, u t i l i z i n g procedures and data commonly a v a i l a b l e to most u s e r s . E r r o r s i n s u r f a c e temperature, as determined by an i n f r a r e d l i n e scanner, have three components: 1)system (instrument) and c a l i b r a t i o n e r r o r s ; -163-2) atmospheric e r r o r s ; 3) e m i s s i v i t y e r r o r s . These three e r r o r s are analyzed f o r a simulated scanner system. As the scanner's f i e l d o f view sweeps across the ground s u r f a c e , r a d i a t i o n i n the bandpass of the scanner causes a response i n the d e t e c t o r which i s recorded i n analog form. I t w i l l be assumed that the d e t e c t o r output i s l i n e a r with the impulse causing the s i g n a l . T h i s s i g n a l i s then d i g i t i z e d to p i x e l v a l u e s between 0 and 255. The simulated scanner system used i n t h i s study has two i n t e r n a l blackbody r e f e r e n c e sources. T h e i r temperatures are known and may be a d j u s t e d . One i s normally set i n the lower range of expected s u r f a c e temperatures, the other near the higher expected temperatures. During every scan the b l a c k b o d i e s are viewed by the scanner and the s i g n a l recorded and subsequently d i g i t i z e d . These data w i l l be used f o r temperature c a l i b r a t i o n . C a l i b r a t i o n e r r o r i s analyzed f o r an 8-14 um Ge:Hg and a 3-5 um InSb d e t e c t o r . The instrument e r r o r i s analyzed using f l i g h t data of a Daedalus (Model 1230) thermal i n f r a r e d l i n e scanner. Atmospheric a t t e n u a t i o n e r r o r s are determined by comparing a i r b o r n e thermal l i n e - s c a n data with ground t r u t h temperature measurements. A t h e o r e t i c a l approach i s used to i n v e s t i g a t e e m i s s i v i t y e r r o r s and examples of e r r o r s f o r a Ge:Hg d e t e c t o r are presented. The t o t a l e r r o r s a re then simulated f o r t y p i c a l n i g h t and day c o n d i t i o n s . -164-2. CALIBRATION RELATION ERRORS 2.1 THEORETICAL CONSIDERATIONS 4 By the Stephan-Boltzmann Law, E^ = o Ts , i t i s known that the t o t a l r a d i a t i o n E^ emitted from a blackbody s u r f a c e i s p r o p o r t i o n a l to the f o u r t h power of s u r f a c e temperature (Ts), where the p r o p o r t i o n a l i t y constant o i s the Stephan-Boltzmann con s t a n t . The r a d i a t i o n emitted by a s u r f a c e i n the bandpass of a sensor i s found by i n t e g r a t i n g Planck's Law over the bandpass. Planck's Law, the s p e c t r a l d i s t r i b u t i o n of emitted energy, i s given by: EA(.X,Ts) = 2 T r c 2 h X - 5 ( e h c / A k T s - l ) " 1 (II.1) where E ^ ( A , T s ) i s the r a d i a n t energy emitted per u n i t wavelength, c i s the speed of l i g h t , h i s Planck's constant, k i s Boltzmann's constant, Ts i s s u r f a c e temperature, and X i s wavelength. The response of p h o t o n - s e n s i t i v e d e t e c t o r s to emitted r a d i a t i o n i s not a simple r e l a t i o n i n temperature, although Dancak (1979) shows that a r e l a t i o n i n temperature to the f o u r t h power can give a good approximation. The r a d i a t i o n reaching an a i r b o r n e d e t e c t o r i s , however, a l s o i n f l u e n c e d by atmospheric t r a n s m i t t a n c e and emission, as w e l l as the t r a n s m i s s i o n c h a r a c t e r i s t i c s of the scanner o p t i c s . F i n a l l y the s p e c t r a l response curve of the d e t e c t o r i n the bandpass of the system must be c o n s i d e r e d . The energy a v a i l a b l e , from a blackbody s u r f a c e , to produce a s i g n a l i n a d e t e c t o r ( e f f e c t i v e energy) may be given -16 5-as: E = J * 2 [E x(X,Ts)xa(X)To(X) + Ea(X)]R(X)dX (II.2) where \ to A^ i s the bandpass of the system, E^(A,Ts) i s the r a d i a n t energy emitted by a blackbody given by ( I I . 1 ) , R(A) i s the r e l a t i v e d e t e c t o r response, t a ( A ) i s the atmospheric t r a n s m i s s i o n f a c t o r , T o ( ) i s the t r a n s m i s s i o n f a c t o r for the scanner o p t i c s , and Ea( ) i s the energy a r r i v i n g at the det e c t o r due to atmospheric emission and s c a t t e r e d energy. Since the response of the d e t e c t o r i s assumed l i n e a r with impulse causing the response, the output s i g n a l and u l t i m a t e l y p i x e l value may be considered p r o p o r t i o n a l to E. The p r e c i s e c a l i b r a t i o n r e l a t i o n s h i p , t h e r e f o r e , v a r i e s with the type of d e t e c t o r , s p e c t r a l bandpass, the s u r f a c e temperature and temperature range over which c a l i b r a t i o n i s d e s i r e d , and to some extent atmospheric J . . 1 c o n d i t i o n s . 2.2 PRACTICAL EXAMPLES Bastuscheck (1970) c a l c u l a t e s the energy a v a i l a b l e to produce a s i g n a l i n a d e t e c t o r f o r a given blackbody s u r f a c e temperature. T h i s a v a i l a b l e energy i s termed " e f f e c t i v e energy". He a p p l i e s Planck's Law, atmospheric t r a n s m i t t a n c e , and d e t e c t o r response over the bandpass of two d e t e c t o r s , an 8-14 pm mercury doped germanium (Ge:Hg) d e t e c t o r and a 3-5 pm indium antimonide (InSb) d e t e c t o r (Figures I I . l a and I I . 2 a ) . F i g u r e s I I . l b and II.2b g i v e the f i r s t d e r i v a t i v e s of F i g u r e s I I . l a and II.2a -166-F i g u r e II.1. -30 -20 -10 0 10 20 30 40 SURFACE TEMPERATURE (C) (b) a) E f f e c t i v e energy f o r a Ge:Hg d e t e c t o r as a f u n c t i o n of s u r f a c e temperature ( a f t e r Bastuscheck, 1970). A l s o g i v e n are T 4 r e l a t i o n f o r the day case and T r e l a t i o n f o r the ni g h t case. b) Change i n e f f e c t i v e energy caused by a one degree temperature change fM(Ts)} f o r the Ge:Hg d e t e c t o r ( a f t e r Bastuscheck, 1970). -167-0 > 1 i 1 i i i I -20 -10 0 10 20 30 40 SURFACE TEMPERATURE ( C ) (a) • 30 i 1 1 1 i 1 r 0 I 1 1 1 i i i I -20 -10 0 10 20 30 40 SURFACE TEMPERATURE ( C ) (b) F i g u r e I I . 2 . a) E f f e c t i v e energy f o r an InSb d e t e c t o r as a f u n c t i o n o f s u r f a c e temperature ( a f t e r Bastuscheck, 1970). A l s o g i v e n are the T r e l -a t i o n for the day and n i g h t case and T* r e l a t i o n f o r the day case, b) Change i n e f f e c t i v e energy caused by a one degree temp-e r a t u r e change {M(TS)J f o r the InSb d e t e c t o r ( a f t e r Bastuscheck, 1970) . -168-r e s p e c t i v e l y . P r e c i s e c a l i b r a t i o n r e l a t i o n s h i p s are not always a v a i l a b l e and l i n e a r c a l i b r a t i o n s i n temperature (T r e l a t i o n ) and 4 temperature to the f o u r t h power (T r e l a t i o n ) may o f t e n be used. The known s i g n a l s and temperatures of the i n t e r n a l blackbody r e f e r e n c e sources are used to determine the two c o e f f i c i e n t s of these l i n e a r e quations. The e r r o r s i n using these r e l a t i o n s for the Ge:Hg and InSb d e t e c t o r s of F i g u r e II.1 and II.2 are d i s c u s s e d below. Two common c o n d i t i o n s under which thermal i n f r a r e d surveys are obtained may be c a t e g o r i z e d i n t o two cases, a n i g h t and a day case. Two t y p i c a l cases may be d e s c r i b e d : 1) a daytime case with blackbody 1 temperature =10 C and blackbody 2 = 40 C; 2) a ni g h t t i m e case with blackbody 1 temperature = 0 C and blackbody 2 = 20 C. 4 F i g u r e I I . l a a l s o shows the T r e l a t i o n f o r the day case and the T r e l a t i o n for the night case of the Ge:Hg d e t e c t o r . F i g u r e 4 II.2a g i v e s both the T and T r e l a t i o n f o r the day case and the T r e l a t i o n for the night case of the 3-5 um InSb d e t e c t o r . For the daytime 8-14 um Ge:Hg case, the T r e l a t i o n was found to be more 4 accurate than a T r e l a t i o n . E r r o r s are l e s s than approximately 4 0.2 C, whereas using a T r e l a t i o n can lead to e r r o r s of the order of 1.0 C or more. In c o n t r a s t , f o r the n i g h t t i m e case the 4 T r e l a t i o n y i e l d s e r r o r s l e s s than 0.2 C while use of the T r e l a t i o n underestimates temperature by as much as 0.6 C. 4 Use of 3-5 um InSb data c a l i b r a t e d with e i t h e r a T or T r e l a t i o n should not be considered p r e c i s e temperature d a t a . C a l i b r a t i o n 4 r e l a t i o n e r r o r s can be l a r g e . The T r e l a t i o n outperforms the T -169-r e l a t i o n (Figure I I . 2 a ) . Underestimates of over 3.0 C occur for 4 the day case using the T r e l a t i o n as opposed t o 2.0 C for the T r e l a t i o n . For the ni g h t t i m e case maximum e r r o r s are of the order 4 of 2.0 and 1.5 C f o r the T and T r e l a t i o n s r e s p e c t i v e l y . Both are underestimates. F i g u r e II.3 i l l u s t r a t e s the d i f f e r e n c e between using the T 4 and T r e l a t i o n for both night and day cases. C a l i b r a t i o n r e l a t i o n e r r o r s w i l l g e n e r a l l y be smaller for nig h t t i m e images s i n c e i n t e r p o l a t i o n d i s t a n c e between blackbody r e f e r e n c e temperatures i s s m a l l e r . The e r r o r s d e s c r i b e d c l e a r l y demonstrate the d e s i r a b i l i t y o f using a p r e c i s e c a l i b r a t i o n r e l a t i o n or a good approximation f o r the p a r t i c u l a r sensor i n use and the s u r f a c e temperatures expected. Scarpace et a l . (1975) a l s o i n d i c a t e the importance o f using good c a l i b r a t i o n r e l a t i o n s . 3. SYSTEM (INSTRUMENT) ERRORS System e r r o r s are another important f a c t o r i n the det e r m i n a t i o n of accurate s u r f a c e temperatures. Detector and tape n o i s e i n the s u r f a c e temperature s i g n a l , f l u c t u a t i o n i n the backbody r e f e r e n c e s i g n a l s , and i n a c c u r a t e c o n t r o l of blackbody temperatures are expected to be the main o r i g i n o f system e r r o r s . In order to study the e f f e c t of these e r r o r s on s u r f a c e temperature d e t e r m i n a t i o n , a c a l i b r a t i o n procedure must be 4 d e f i n e d . A c a l i b r a t i o n procedure using a T or T r e l a t i o n i s de s c r i b e d b r i e f l y . - 1 7 0 -1 . 2 .9 -.9 Figure II.3. Difference in temperature between that derived from the T 4 r e l a t i o n and from the T r e l a t i o n . Calculated for the t y p i c a l night and day cases (see text) and with blackbody 1 and 2 p i x e l values equal to 25 and 225 respect-i v e l y . -171-3.1 CALIBRATION PROCEDURE The c a l i b r a t i o n equation for the T r e l a t i o n i s as f o l l o w s : Tr = (p - a)/b (II.3) where Tr i s the remotely sensed c a l i b r a t e d s u r f a c e temperature (K), and p i s the (CCT) p i x e l v a l u e . The term "b" i s the slope o f the p i x e l value versus temperature r e l a t i o n . I t i s gi v e n by: b = (p 2 - P^/CTj - Tj) (II.4) where T^ and are the temperatures of blackbody r e f e r e n c e s 1 and 2 r e s p e c t i v e l y and p^ and p^ are the p i x e l values of the corresponding blackbody r e f e r e n c e s . The p i x e l value at Tr = 0, "a" of ( I I . 3 ) , i s given by: a = p - bTj (II.5) 4 The T r e l a t i o n c a l i b r a t i o n equation i s : 11 u Tr =[(p - a)/b] ' (II.6) with b = (p 2 - P L)/( T2 - Ti> (II.7) and a = ? l - bl\ (II.8) -172-C a l i b r a t i o n procedures were i n v e s t i g a t e d using the thermal l i n e - s c a n data from a Daedalus Model 1230 Hg:Cd:Te scanner o p e r a t i n g i n the 9.5-11.5 urn bandpass. The data were flown and processed by the Canada Centre for Remote Sensing (CCRS). The blackbody s i g n a l s were d i g i t i z e d to p i x e l values between 0 and 255 f o r 32 p i x e l s for each blackbody. The c a l i b r a t i o n equations were a p p l i e d f o r each scan l i n e with p^ and p^ the mean blackbody p i x e l v a l u e . Only the c e n t r a l 16 blackbody v a l u e s were used f o r t h i s mean as recommended by CCRS (CCRS T e c h n i c a l Memorandum DPM-TM-79-073). I n s p e c t i o n of the blackbody v a l u e s of s e v e r a l f l i g h t s i n d i c a t e s that the c e n t r a l p i x e l s are more s t a b l e . A procedure which would make the c a l i b r a t i o n process c o m p u t a t i o n a l l y e f f i c i e n t i s to assume the same blackbody p i x e l value for every scan l i n e . T h i s procedure has been used by G i l l e s p i e and Kahle (1977) but should be employed with c a u t i o n . A survey of seven thermal l i n e - s c a n images (Daedalus scanner Model 1230, Canada Centre f o r Remote Sensing) showed that the blackbody value (mean of c e n t r a l 16 blackbody p i x e l s ) g e n e r a l l y v a r i e d by approximately 1.5 to 2.0 p i x e l v a l u e s . An approximation of the mean blackbody p i x e l v a l u e s could be used. Table II.1 (columns 1 and 2) g i v e s i n s i g h t i n t o the v a r i a b i l i t y i n blackbody v a l u e s . However, segments of some images had blackbody values a l t e r e d s i g n i f i c a n t l y from the remainder of the image or had l a r g e r v a r i a t i o n i n the blackbody v a l u e . The s e p a r a t i o n between the maximum and minimum blackbody values for the seven f l i g h t s was t y p i c a l l y between 4 to 25 p i x e l s . A l i n e -b y - l i n e c a l i b r a t i o n procedure i s recommended un l e s s the blackbody values have p r e v i o u s l y been scanned and found to be w e l l behaved. Table II.1. Analysis of p i x e l and blackbody (BB) error for seven f l i g h t s . Data taken from s i x 4-second segments of data 20 seconds apart for each f l i g h t . 1 Results for six consecutive 0.33-second segments are s i m i l a r . Scan rate i s 60 scans per second. Mean 2 of Mean 2 standard deviation of max -min BB value of each of the BB value a l l scan l i n e s 16 BB pi x e l s J r i i g n r (time. date) 1 2 3 BB 1 BB2 BB1 BB2 BB 1 BB2 2300, 10/6/77 3.2 5.2 0.6 0.9 1.4 1.9 2300, 29/7/78 3.3 3.1 0.7 0.5 9.3 1.5 0400, 11/6/77 3 - - 0.6 — 1.6 0400, 30/7/78 3.1 14.5 0.5 2.3 7.4 10.0 0900, 11/6/77 3.0 3.2 0.6 0.5 7.1 1.9 1800, 11/6/77 3.3 3.1 0.7 0.6 8.2 2.2 1430, 29/7/78 2.6 11.9 0.4 3.1 7.2 9.8 1 The reason for the contrasting error of blackbody 1 and 2 for some f l i g h t s i s unknown. 2 Mean of the values for the six segments used. 3 Blackbody 2 was often saturated at p i x e l value 255. -174-I t was determined that the number of p i x e l s to be c a l i b r a t e d c o u l d be reduced by 90 to 95 percent i f a p i x e l value versus 4 temperature lookup t a b l e was cr e a t e d for each scan l i n e . The T r e l a t i o n r e q u i r e d o n l y s l i g h t l y more CPU time than the T r e l a t i o n . 3.2 COMPONENTS OF SYSTEM ERROR An important component of the scanner temperature e r r o r i s the e r r o r i n the p i x e l value (detector and tape n o i s e , which w i l l be r e f e r r e d to as p i x e l e r r o r ) . The magnitude of the e r r o r i s i n v e s t i g a t e d by comparing the values of the same blackbody r e f e r e n c e p i x e l over short p e r i o d s of time dur i n g a f l i g h t . A s i g n i f i c a n t change i s not expected i n blackbody temperature over short p e r i o d s of time and corresponding p i x e l values of every scan l i n e should be the same. Table II.1 (column 3) g i v e s the r e s u l t s for s e v e r a l image segments of seven f l i g h t s . The standard d e v i a t i o n s of the i n d i v i d u a l blackbody p i x e l values are g e n e r a l l y between 1.5 and 10 p i x e l s . The temperature r e s o l u t i o n of the scanner i s given as 0.2 K. A n a l y s i s of d i g i t i z e d f l i g h t data (Table II.1) i n d i c a t e s that the p i x e l e r r o r was not l e s s f o r daytime f l i g h t s as might be expected due to l a r g e r degrees C per p i x e l . Since i t i s the i n t e n t of t h i s Appendix to d e s c r i b e e r r o r s t y p i c a l of o p e r a t i o n a l surveys, a p i x e l e r r o r , as opposed to temperature e r r o r , i s used i n subsequent e r r o r a n a l y s e s . T h i s w i l l l ead to l a r g e r p r e d i c t e d e r r o r s f o r daytime than n i g h t t i m e thermal i n f r a r e d surveys. For example, the degrees C e l s i u s per p i x e l value i s commonly 0.1 and 0.2 f o r n i g h t and day surveys - I r -r e s p e c t i v e l y ; t h e r e f o r e , d e t e c t o r and tape noise can range from 0.15 to .1.0 C for ni g h t surveys and 0.3 to 2.0 C f o r day surveys. The standard d e v i a t i o n of the mean of the c e n t r a l 16 blackbody p i x e l values of each scan l i n e i s a measure of how a c c u r a t e l y the p i x e l values of the blackbodies (p^ and p^) can be determined. The e r r o r , as might be expected, was found to be s i m i l a r to the p i x e l e r r o r . The other major component of system e r r o r i s the blackbody temperatures. I t i s assumed that the temperatures of the blackbody r e f e r e n c e p l a t e s can be set and c o n t r o l l e d to 0.1 C. I t i s f u r t h e r assumed that the blackbody temperatures f o r each scan l i n e are known (to 0.1 C) or a c o r r e c t i o n f o r changes i n blackbody temperature i n c o r p o r a t e d i n the s i g n a l d i g i t i z i n g p r o c e s s . Two other f a c t o r s must a l s o be c o n s i d e r e d . One i s that the blackbody r e f e r e n c e p l a t e s are not p e r f e c t e m i t t e r s and the r a d i a t i o n r e c e i v e d by the d e t e c t o r i s not that o f a blackbody but a greybody. Secondly, the temperature may not be uniform across the blackbody. I f the non-uniformity i s c o n s i s t e n t i n time then i t may be c o r r e c t e d f o r . C o r r e c t i o n s f o r the greybody nature of the blackbody p l a t e s and temperature non-uniformity are assumed to have been made. T y p i c a l probable e r r o r s may be d e r i v e d from the preceding d i s c u s s i o n . Two c o n d i t i o n s of p i x e l e r r o r were used i n subsequent e r r o r a n a l y s i s : p i x e l e r r o r #1 (low p i x e l e r r o r ) had a p i x e l and mean blackbody p i x e l value probable e r r o r of 2 p i x e l s , and p i x e l e r r o r #2 (high p i x e l e r r o r ) had a probable p i x e l e r r o r of 7 p i x e l s . The probable e r r o r i n the temperature of the bla c k b o d i e s was taken to be 0.1 C. -176-Case 7 of Tables II.3 to II.6 (to be d i s c u s s e d i n f u r t h e r d e t a i l i n s e c t i o n 6) g i v e s a d e t a i l e d e r r o r a n a l y s i s o f equations (II.3) and ( I I . 6 ) . T y p i c a l n i g h t and day surveys are considered (Table 1.7). Instrument and tape noise i n s u r f a c e temperature p i x e l value i s the most important e r r o r . E r r o r s i n mean blackbody p i x e l value are a l s o important. E r r o r s f o r day f l i g h t s are l a r g e r due to the t y p i c a l l y wider range of sur f a c e temperatures being measured. 4. ATMOSPHERIC ERRORS The e f f e c t of the atmosphere between the s u r f a c e and sensor i s both a d d i t i v e and m u l t i p l i c a t i v e (equation I I . 2 ) . A simple c o r r e c t i o n procedure for atmospheric e f f e c t i s i n v e s t i g a t e d . An a d d i t i v e atmospheric o f f s e t i s used. Ground t r u t h temperature measurements are compared with c a l i b r a t e d scanner temperatures for the same s i t e . The average d i f f e r e n c e ( o f f s e t ) i s added to the c a l i b r a t e d scanner data. O f f s e t s obtained from s e v e r a l s i t e s w i l l d i f f e r s l i g h t l y . A measure of the e r r o r expected i n ap p l y i n g t h i s type of o f f s e t i s the standard d e v i a t i o n of the mean o f f s e t d e r i v e d from s e v e r a l s i t e s . T able II.2 g i v e s the standard d e v i a t i o n s for f l i g h t s s t u d i e d by the author and s e v e r a l from the l i t e r a t u r e . A t y p i c a l probable e r r o r l i k e l y ranges between 0.4 and 1.5 C. Two cases w i l l be c o n s i d e r e d i n f u r t h e r e r r o r a n a l y s i s : a high 1.3 C (Aatm ), and a low 0.5 C (Aatm ) H ti probable e r r o r . The e r r o r i n measured atmospheric o f f s e t i s l a r g e l y due to problems i n measuring and c o r r e l a t i n g s u r f a c e and a i r b o r n e Table II.2. Error in determining atmospheric o f f s e t (the average standard deviation for the mean of the temperature differences between ground temperature measurements and airborne thermal l i n e scan temperature measurements of the same s i t e ) . F l i g h t number1 standard deviation (C) number of s i t e s 1 1.26 6 2 0.99 6 3 1.47 7 4 0.43 10 5 0.64 9 6 1.07 4 7 1.29 4 8 1.36 5 9 0.79 5 10 1.38 4 11 1.74 « 2 12 0. 58 32 z •i 13 0.60 40 3 - mean = 1.05 F l i g h t s 1 to 5 from data of author (ground-truth temperature measured with a Barnes PRT-10 infrared thermometer), 6 to 11 from data of Heilman et a l . (1976), and 12 and 13 from Schott (1979). 2 From 4 f l i g h t s 8 s i t e s each f l i g h t . 3 From 5 f l i g h t s 8 s i t e s each f l i g h t . -178-measurements. These e r r o r s may be summarized: 1) e x t r a p o l a t i o n of ground measurements to the time of the f l i g h t ( e s p e c i a l l y d u r i n g day f l i g h t s ) ; 2) s h o r t term l o c a l a d v e c t i v e e f f e c t s ; instantaneous scanner temperature may not be c h a r a c t e r i s t i c of the micro-m e t e o r o l o g i c a l c o n d i t i o n s at the time of the ground t r u t h measurements; t h i s e f f e c t can be s e v e r a l degrees for vegetated s u r f a c e s ; 3) e r r o r s i n ground and scanner measurements; 4) e r r o r s i n l o c a t i n g ground t r u t h s i t e s on the thermal imagery. The v a r i a t i o n of the atmospheric o f f s e t due to d i f f e r i n g atmospheric c o n d i t i o n s among s i t e s may cause a d d i t i o n a l e r r o r s . The data o f f l i g h t s 1 t o 5 of Table II.2 were d e r i v e d f o r s i t e s of s l i g h t l y d i f f e r i n g vapour p r e s s u r e ( u s u a l l y < 100 Pa) and a i r temperature (< 2.0 C) , t h e r e f o r e these data have some s i t e v a r i a t i o n i n c o r p o r a t e d i n t o them. There i s a syst e m a t i c change i n the atmospheric o f f s e t with scan angle due to a lengthening of r a d i a t i o n path l e n g t h with i n c r e a s i n g scan angle. Scarpace et a l . (1975) determined, f o r a f l i g h t at 1520 m, that the temperature d e v i a t i o n from a temperature measurement at nadir was 0.1 C at a 30 degree scan angle and up to 0.5 C at a 50 degree scan angle. Temperature d e v i a t i o n from nadir depends on s u r f a c e temperature, atmospheric c o n d i t i o n s , the scanner system, and f l y i n g a l t i t u d e . Schott (1979) a l s o d i s c u s s e d t h i s problem. The atmospheric o f f s e t , t h e r e f o r e , depends on the p o s i t i o n of ground t r u t h s i t e s . S i n c e the ground measurements are compared with c a l i b r a t e d thermal i n f r a r e d data, the atmospheric o f f s e t w i l l -179-a l s o depend on the temperature of the ground t r u t h s i t e s with r e s p e c t to the blackbody temperatures. The atmospheric o f f s e t w i l l have an e f f e c t of reducing the c a l i b r a t i o n r e l a t i o n e r r o r . More s o p h i s t i c a t e d atmospheric c o r r e c t i o n procedures were d e s c r i b e d by Weiss (1971), Shaw and Irbe (1972), Heilman et a l . (1976), Kahle et a l . (1979), and Schott (1979). They may a i d i n reducing e r r o r . S c h o t t ' s method, which i n c l u d e d scan angle c o r r e c t i o n , reduced the standard d e v i a t i o n of the mean o f f s e t for f l i g h t s 12 and 13 of Table II.2 to 0.25 and 0.27 C, r e s p e c t i v e l y . 5. EMISSIVITY ERRORS The energy a v a i l a b l e , from a greybody s u r f a c e , to produce a s i g n a l i n a d e t e c t o r may be given, using ( I I . 2 ) , by: E = J * 2 [e(X)E A(A,Ts)Ta(X)To(X) + Ea(X)]R(X)dX (II.9) + J * 2 [1 - e(X)]B x(X)Ta(X)xo(X)R(X)dX where e(X) i s the e m i s s i v i t y of the s u r f a c e and B^(X) i s the atmospheric downward r a d i a t i o n at the s u r f a c e . Assuming e m i s s i v i t y i s a constant over to (II.9) i s w r i t t e n : E = eE^CTs) + (1 - e)B (11.10) where E (Ts) i s the a v a i l a b l e energy from a blackbody s u r f a c e -180-at Ts given by (II.2) and B i s the energy a v a i l a b l e to produce a s i g n a l i n the d e t e c t o r due to atmospheric downward r a d i a t i o n i f 2,3 a l l atmospheric downward r a d i a t i o n i s r e f l e c t e d . Using F i g u r e I I . l a as the c a l i b r a t i o n r e l a t i o n , E has a s s o c i a t e d with i t a remotely sensed c a l i b r a t e d temperature (Tr) to which the p i x e l value w i l l be mapped. The e r r o r due to e m i s s i v i t y may be given by: Ts - Tr = [Egg(Ts) - E]/M(Ts) ( I I . 11) where M(Ts) i s the 8E/8T at Ts (given by F i g u r e I I . l b ) . Since 8E/9T does not vary g r e a t l y over the small temperature e r r o r s expected, the above approximation i s v a l i d . S u b s t i t u t i n g (11.10) i n t o (11.11) y i e l d s : Ts - Tr = [1 - e ] [ E B B ( T s ) - B]/M(Ts) (11.12) I t may be advantageous to assume a s u r f a c e e m i s s i v i t y other than one. A c o r r e c t i o n to transform the c a l i b r a t i o n r e l a t i o n of F i g u r e I I . l a to one with e=ea ( ea i s the assumed e m i s s i v i t y used i n the c a l i b r a t i o n ) can be d e r i v e d using (11.12) with ea s u b s t i t u t e d for e. T h i s g i v e s an e m i s s i v i t y e r r o r o f : Ts - Tr = [ca - e][E„(Ts) - B]/M(Ts) (11.13) Ca Be where T r e a i s the temperature given by the new (e=ea) c a l i b r a t i o n . The e r r o r i s a l i n e a r f u n c t i o n of the d i f f e r e n c e between a c t u a l and assumed e m i s s i v i t y . The slope i s a f u n c t i o n of the -181-c a l i b r a t i o n r e l a t i o n of the scanner, surface temperature, and B. Figure II.4 gives the temperature error per 0.01 unit of error in emissivity (e-ca) for the 8-14 pm Ge:Hg detector of Figure II.1. Lorenz (1966) and Fuchs and Tanner (1966) discussed errors in infrared thermometry caused by emissivity e f f e c t s . Neglecting emissivity ( i . e . , using £a = l) i s advantageous as knowledge of B is not required. However, this can lead to large errors. If surface type i s known, temperature errors may be reduced by cal c u l a t i n g temperature using standard emissivity values for the surface type and measured or estimated B. Estimating surface emissivity from values as given in the l i t e r a t u r e may be d i f f i c u l t , e s p e c i a l l y i f the surface i s not uniform. Sutherland and Bartholic (1977) described the example of bare s o i l patches in vegetated t e r r a i n . If a single estimate of emissivity i s used for an entire thermal survey area, errors can be large. Three cases w i l l be used in further error analysis: 1) emissivity known; 2) emissivity estimated from standard values (ty p i c a l probable error in emissivity of about 0.02); and 3) emissivity unknown and/or estimated for a survey area of widely varying emissivity (high probable error in emissivity of 0.05). 6. SUMMARY OF TOTAL ERRORS Tables II.3 to II.6 indicate the t o t a l errors expected in surface temperature determination. The conditions for which errors are estimated represent the range of conditions which are S U R F A C E T E M P E R A T U R E (C ) F i g u r e I I . 4 . Temperature e r r o r due to assuming an erroneous e m i s s i v i t y . E r r o r i s g i v e n as the degrees C e r r o r f o r a 0.01 u n i t e r r o r i n e m i s s i v i t y estimate (e-ea) f o r v a r i o u s atmospheric (Ta) and s u r f a c e temperatures. Atmospheric r a d i a t i o n expressed i n terms of a i r temperature by an e m p i r i c a l formula of Idso and Jackson (1969). Table I I . 3 . Summary of errors; T r e l a t i o n clay case. case C o n t r i b u t i o n t o R 2 due to probable e r r o r Totals Pl'P2 T l r T 2 c a l atm L atm H emiss T emiss n R Max Rd 1 1.00 - - -2 — - 1.00 - — 3 — — — .30 - .70 — 4 — — — .06 - - .94 5 — - .74 .26 — 6 — - - - .31 - .69 P i x e l E r r o r #1 7 .65 .33 .02 - • — — — — 8 .65 .32 .02 .01 - — — ™* 9 .32 .16 .01 .01 .50 — — 10 .08 .04 .00 .00 - .88 — 11 .15 .07 .01 .00 .23 — .54 ~* 12 .04 .02 .00 .00 .06 — — .88 13 .06 .03 .00 .00 - .67 .24 -* 14 .03 .01 .00 .00 — .30 — .66 15 .19 .10 .01 .00 — — .70 16 .04 .02 .00 .00 — • — — . 94 P i x e l E r r o r #2 7 .67 .33 .00 — — — 8 .67 .33 .00 .00 — — 9 .61 .31 .00 .00 .08 — 10 .42 .21 .00 .00 - .37 — 11 .52 .26 .00 .00 .07 — . 15 12 .28 .14 .00 .00 .04 — — . 54 13 .38 .19 .00 . .00 - .32 .11 14 .24 .12 .00 .00 - .20 — . 44 15 .55 .28 .00 .00 - — .17 .56 16 .29 .15 .00 .00 .50 1.3 .92 2.0 1.5 2.3 .50 .50 .71 1.4 1.0 2.1 1.6 2.4 .92 2.0 1.7 1.7 1.8 2.2 2.0 2.6 3 9 9 ,6 .50 1.3 1.3 2.4 2.1 3.2 .90 .95 1.5 2.3 2.2 3.4 3.0 4.2 1.7 2.2 2.9 3.0 3.5 4.3 4.2 5.4 5.0 7.5 3.7 4.9 71 8 3 8 1 ,3 .70 .70 1.0 2.0 1.4 2.9 2.3 3.4 1.3 2.8 2.4 2.4 2.5 3.0 2.8 3.7 3.2 4.1 2.7 3.7 4 T a b l e I I . 4 . Summary of e r r o r s ; T r e l a t i o n day case. case C o n t r i b u t i o n t o R 2 due to probable e r r o r T o t a l s p p r p 2 T l f T 2 c a l atm L atm H e m i s s T e m i s s H R Max Rd P i x e l E r r o r #1 7 .66 .33 .01 - -— — — 8 .32 .16 .01 .51 - — — 9 .21 .11 .00 .34 .34 — — — 10 .07 .04 .00 .11 - .78 — — 11 .12 .06 .00 .19 .19 — .44 — 12 .04 .02 .00 .06 .06 — — .83 13 .06 .03 .00 .09 - .61 .21 — 14 .03 .01 .00 .04 - .29 — .63 15 .15 .07 .00 .23 - — .55 — 16 .04 .02 .00 .06 — — — .88 P i x e l E r r o r #2 7 .67 .33 .00 — — — — 8 .61 .31 .00 .08 — — — 9 .57 .29 .00 .07 .07 — — mm 10 .40 .20 .00 .05 - .35 — mm 11 .49 .24 .00 .06 .06 — .15 12 .27 .14 .00 .04 .04 — — .53 13 .35 .18 .00 .05 - .31 .11 14 .22 .11 .00 .03 - .20 — .44 15 .51 .26 .00 .07 — — .16 16 .28 .14 .00 .04 . 54 .49 .70 .86 1.5 1.2 2.1 1.7 2.4 1.0 2.1 1.7 1.7 1. 2, 2, 2, 2.3 2.9 1.9 2.6 .87 1.4 1.9 2.7 2.6 3.8 3.4 4.6 2.1 3.3 2. 3. 3, 4. 4, 5, 8 3 8 6 6 8 5.4 6.6 4.1 5.3 .69 .99 1.2 2 1 3 2 3 1 2 2. 2. 2. 3. 2. 3. 3. 4, 2, 3, 1 6 0 4 4 ,5 ,9 .4 .5 .6 .1 .8 .8 3 1 7 7 T a b l e I I . 5 . Summary o f e r r o r s ; T 4 r e l a t i o n n i g h t c a s e . case C o n t r i b u t i o n to R 2 due to probable e r r o r T o t a l s P Pl'P2 T 1 ' T 2 c a l atm L atm H e m i s s T e r a i s s H R Max Rd 1 _ 1.00 _ — .50 .50 .71 2 — — 1.00 - - 1.3 1.3 1.8 3 _ .44 - .56 - .75 1.1 1.1 4 _ .11 - - .89 1.5 1.9 2.1 5 _ — — .84 .18 - 1.4 1.9 2.0 6 _ — — - - .46 - .54 1.9 2.7 2.7 P i x e l Error #1 .25 .50 .35 7 .62 .31 .07 - - - - — 8 .59 .30 .08 .04 - - - — ;26 .55 .37 9 .12 .06 .02 .01 .79 - - - .54 1.1 .80 10 .02 .01 .00 .00 - .97 - - 1.3 1.9 1.9 11 .06 .03 .01 .00 .40 - .50 - .79 1.6 1.0 12 .02 .01 .00 .00 .11 - - .86 1.5 2.5 2.1 13 .02 .01 .00 .00 - .82 .15 - 1.4 2.4 2.0 14 .01 .01 .00 .00 - .46 - .53 1.9 3.3 2.7 15 .10 .05 .01 .01 mm - .82 - .62 1.1 .87 16 .02 .01 .00 .00 - - - .97 1.4 2.0 2.0 P i x e l Error #2 1.5 1.2 7 .66 .33 .01 — - - - - .86 8 .66 .33 .01 .00 - - — .86 1.5 1.2 9 .49 .25 .01 .00 .25 - - - .99 2.0 1.4 10 .20 .10 .00 .00 - .70 - - 1.6 2.8 2.2 11 .38 .19 .00 .00 .19 - .24 - 1.1 2.6 1.6 12 .16 .03 .00 .00 .09 - - .67 1.7 3.4 2.4 13 .18 .09 .00 .00 - .62 .11 — 1.7 3.4 2.4 14 .11 .05 .00 .00 - .39 - .45 2.1 4.2 3.0 15 .46 .23 .01 .00 - - .30 — 1.0 2.1 1.4 16 .18 .09 .00 .00 — .73 1.6 2.9 2.3 T a b l e I I . 6 . Summary o f e r r o r s ; T r e l a t i o n n i g h t case Case Contr ibution to R2 due to probable error T o t a l s P Pl»P2 T 1 ' T 2 c a l a t m L atm H emisSrp e m i s s H R Max Rd P i x e l E r r o r #1 7 .62 .31 .07 - - - - - .26 .50 .36 8 .31 .16 .07 .49 mm - - - .36 .75 .51 9 .11 .05 .01 .17 .66 - - - .61 1.3 .87 10 .02 .01 .00 .04 - .93 - - 1.4 2.1 1.9 11 .06 .03 .01 .09 .36 - .45 - .83 1.8 1.2 12 .02 .01 .00 .03 .10 - - .84 1.5 2.7 2.2 13 .02 .01 .00 .03 - .79 .15 - 1.5 2.6 2.1 14 .01 .00 .00 .02 - .45 - .52 2.0 3.5 2.7 15 .09 .05 .01 .14 - - .71 - .66 1.3 .94 16 .02 .01 .00 .03 mm - - .94 1.4 2.2 2.0 P i x e l E r r o r #2 .86 1.5 1.2 7 .66 .33 .01 - - - mm — 8 .66 .33 .01 .00 - - - - .86 1.6 1.2 9 .49 .25 .01 .00 .25 - - - 1.0 2.1 1.4 10 .20 .10 .00 .00 - .70 - — 1.6 2.9 2.2 11 .38 .19 .00 .00 .19 - .24 - 1.1 2.6 1.6 12 .17 .08 .00 .00 .09 - ' - .66 1.7 3.5 2.4 13 .18 .09 .00 .00 - .62 .11 - 1.7 3.4 2.4 14 .11 .06 .00 .00 - .38 - .45 2.1 4.3 3.0 15 .46 .23 .01 .00 - - .30 - 1.0 2.1 1.5 16 .18 .09 .00 .00 — — .73 1.6 3.0 2.3 -187-common i n thermal i n f r a r e d surveys. Table II.7 i n d i c a t e s the input and c o n d i t i o n s for each case. They are d e r i v e d from the pre v i o u s d i s c u s s i o n s . The data for the 8-14 pm Ge:Hg scanner were used. Night and day surveys are c o n s i d e r e d . P i x e l e r r o r #1 has small instrument and tape noise ( p i x e l e r r o r ) while p i x e l e r r o r #2 has a high noise l e v e l . For each case, examples of small and l a r g e atmospheric o f f s e t and e m i s s i v i t y e r r o r are 4 co n s i d e r e d . The T r e l a t i o n f o r the day cases and T r e l a t i o n f o r night cases are r e p r e s e n t a t i v e of small c a l i b r a t i o n r e l a t i o n e r r o r s . The T r e l a t i o n day and T r e l a t i o n n i g h t g i v e examples of moderate to l a r g e c a l i b r a t i o n r e l a t i o n e r r o r s . In determining t y p i c a l probable e r r o r s i n c a l i b r a t i o n r e l a t i o n (Acal.) f o r each example, s e v e r a l f a c t o r s are c o n s i d e r e d . The e r r o r s given e a r l i e r ( s e c t i o n 2.2) are the maximum expected ( i . e . , e r r o r s f o r temperatures near the mid-temperature between b l a c k b o d i e s ) . A l s o , i f an atmospheric o f f s e t i s a p p l i e d , t h i s w i l l reduce the e r r o r due to c a l i b r a t i o n r e l a t i o n . T h i r d l y , i f temperatures of d i f f e r e n t s i t e s on the same image are compared, the e r r o r due to c a l i b r a t i o n r e l a t i o n can be q u i t e s m a l l depending on the temperature of each s i t e i n comparison to the bl a c k b o d i e s and each o t h e r . C a l i b r a t i o n r e l a t i o n e r r o r w i l l g e n e r a l l y be l e s s f o r n i ghttime surveys as the temperature d i f f e r e n c e between bla c k b o d i e s i s o f t e n l e s s . 2 The c o n t r i b u t i o n to R (R= probable e r r o r ; see Tab l e II.7) due to each parameter r e p r e s e n t s a measure of the importance o f e r r o r i n that parameter to the t o t a l e r r o r . The probable e r r o r i n measuring temperature d i f f e r e n c e between s i t e s or temperature changes of s i t e s i s in c l u d e d (Rd). Cases 1 t o 6 re p r e s e n t -188-Table II.7. Input, conditions, and d e f i n i t i o n s for Tables II.3 to II.6. A l l cases p - 125 AT X = AT 2 •0.1C Aatm L = 0.5 C, Aatm H = 1.3 C probable error in emissivity i s 0.02 for Aemiss T and 0.05 for Aemiss H P i x e l Error #1 AP • = AP 2 = 2 P i x e l s P i x e l Error #2 AP = Ap x = AP 2 = 7 pi x e l s Day a l l cases T1 = 15 C, T 2 - 55 C Ts = 35 C Ta = 25 C y i e l d s B = 68 W m - 2 Aemiss T = 0.77 C, Aemiss H = 1.9 C (from Figure I I . 4 . ) T r e l a t i o n Acal = 0.05 C T 4 r e l a t i o n Acal = 0.5 C Night al3 cases T x = 0 C, T 2 = 20 C Ts • 10 C Ta = 10 C y i e l d s B = 50 w m~2 Aemiss T = 0.56 C, Aemiss H = 1.4 C (from Figure I I . 4 . ) T r e l a t i o n Acal = 0.25 C T 4 r e l a t i o n A c a l = 0.05 C (continued) - 1 8 9 -T a b l e I I . 7 . ( c o n t i n u e d ) D e f i n i t i o n s A i n d i c a t e s p r o b a b l e e r r o r P r P w P 2 ' = p i x e l v a l u e o f s u r f a c e t e m p e r a t u r e d a t a , a n d b l a c k b o d i e s 1 a n d 2 r e -s p e c t i v e l y T ^ , T 2 = t e m p e r a t u r e o f b l a c k b o d i e s 1 a n d 2 T a = a i r t e m p e r a t u r e a t s c r e e n h e i g h t R = p r o b a b l e e r r o r , t h e p r o b a b l e e r r o r i n T s w h e r e T s i s a f u n c t i o n o f q , ^ 2 ' * ' * ^ n a s g i v e n b y : i = l w h e r e 3Ts . r . = -5— Aq l 3q ± i n Max = 7 r . i-1 1 Rd = p r o b a b l e e r r o r f o r t e m p e r a t u r e d i f f e r -e n c e s ( = /2 R) 2 C o n t r i b u t i o n t o R d u e t o Aq^ ^ i s g i v e n b y : r 2/R 2 a n d r e p r e s e n t s a m e a s u r e o f t h e c o n t r i b u t i o n o f t h e e r r o r i n e a c h p a r a m e t e r t o R . - 1 9 0 -examples of e r r o r s o c c u r r i n g i f i t i s assumed, as i t c o r r e c t l y may be, that the atmospheric o f f s e t e r r o r accounts a l s o for the system (instrument) and c a l i b r a t i o n r e l a t i o n e r r o r of the scanner. The set of a l l cases r e p r e s e n t s the combination of e r r o r s which may occur under the v a r y i n g c o n d i t i o n s and a p p l i c a t i o n s of thermal i n f r a r e d surveys. One of the most common uses of thermal i n f r a r e d surveys i s to determine s u r f a c e temperatures. E r r o r s expected for t h i s a p p l i c a t i o n are given by cases 3 to 6 or 11 to 14 of Tables II.3 through I I . 6 . For s i t e s p e c i f i c s t u d i e s i n which e m i s s i v i t y and atmospheric o f f s e t are known, the e r r o r i s given by case 8. Cases 9 and 10 g i v e e r r o r s f o r s i t u a t i o n s i n which e m i s s i v i t y i s known. Cases 15 and 16 represent the c o n d i t i o n of known atmospheric o f f s e t but assumed e m i s s i v i t y . Temperature e r r o r s f o r daytime images w i l l range from 0.5 to 2.9 C. For gen e r a l remote sensing surveys, a t y p i c a l probable e r r o r i s l i k e l y approximately 1.9 C. E r r o r s f o r nig h t t i m e surveys w i l l g e n e r a l l y be s m a l l e r , ranging from 0.5 to 2.1 C. A t y p i c a l 4 probable e r r o r f o r g e n e r a l surveys i s approximately 1.5 C. E m i s s i v i t y and atmospheric o f f s e t e r r o r s are the most important. Temperature d i f f e r e n c e s ( r e l a t i v e temperature) between s i t e s on the same image are o f t e n r e q u i r e d . For s i t e s with the same e m i s s i v i t y , the probable e r r o r i s given by the Rd of case 7. T h i s i s v a l i d only for s i t e s of s i m i l a r temperature. I f the s i t e s d i f f e r w idely i n temperature, the change i n e m i s s i v i t y e r r o r with temperature ( F i g u r e II.4) w i l l cause an a d d i t i o n a l e r r o r component. For l a r g e temperature d i f f e r e n c e s of approximately 20 C, the a d d i t i o n a l e r r o r component i s about 0.15 -191-C per u n i t e r r o r i n e m i s s i v i t y estimate ( e - e a ) . E r r o r s i n the temperature d i f f e r e n c e s of s i t e s on the same image but with d i f f e r i n g e m i s s i v i t i e s depend on the value of e-ea and the temperature of each s i t e . For an example of e-ea < 0.02 and a d i f f e r e n c e of < 0.02 i n the e m i s s i v i t i e s of the two s i t e s , the magnitude of the e r r o r s due to e m i s s i v i t y vary from 0.0 to about 0.8 C for a 20 C temperature d i f f e r e n c e between s i t e s . S i m i l a r l y , with e-ea i n the range 0.00 to 0.05, and the e m i s s i v i t i e s of the s i t e s not d i f f e r i n g by more than 0.05, the a d d i t i o n a l e m i s s i v i t y e r r o r w i l l vary i n magnitude from 0.0 to 1.9 C. For night images, s u r f a c e temperature d i f f e r e n c e s are not as l a r g e and a d d i t i o n a l e m i s s i v i t y e r r o r s w i l l be l e s s . Temperature d i f f e r e n c e s between s i t e s of s i m i l a r temperature and e m i s s i v i t y w i l l be i n e r r o r by 0.7 to 2.4 C f o r a day image and 0.35 to 1.2 C for a n i g h t image. Good a c c u r a c i e s of t y p i c a l images (temperature and e m i s s i v i t y varying) can be expected to be of the order 1.0 C for day images and 0.5 C f o r n i g h t images. Comparison of temperatures of d i f f e r e n t s i t e s with d i f f e r i n g e m i s s i v i t i e s on separate images can lead to l a r g e e r r o r s . E r r o r s w i l l depend on e-ea and the temperature of each s i t e , as w e l l as the atmospheric c o n d i t i o n s during each f l i g h t . Since the temperature estimates of the d i f f e r e n t s i t e s on separate images are independent, t y p i c a l probable e r r o r s i n temperature d i f f e r e n c e may be g i v e n by the Rd f o r cases 3 to 6 or 11 to 14. E r r o r s f o r comparison of day images w i l l be t y p i c a l l y 1.3 to 4.1 C with a t y p i c a l e r r o r of 2.7 C; f o r n i g h t images 1.0 to 3.0 C, 4 t y p i c a l l y 2.2 C. A f i n a l important a p p l i c a t i o n of thermal i n f r a r e d l i n e - s c a n -192-data i s the i n v e s t i g a t i o n of temperature change of a s i t e i n time. The Rd of cases 9 and 10 give estimates of e r r o r s i n temperature change under c o n d i t i o n s of e q u i v a l e n t temperature and atmosphere. These e r r o r s are l i k e l y good estimates of the e r r o r i n temperature change between two n ighttime surveys taken on the same n i g h t . Surface temperature and atmospheric c o n d i t i o n s o f t e n do not change g r e a t l y during a n i g h t . E r r o r s w i l l be from 0.8 to 2.2 C, a t y p i c a l e r r o r being about 1.6 C. 5 Comparisons of two day images r e q u i r e an e r r o r , i n a d d i t i o n to those of cases 9 and 10, due to the v a r i a t i o n of e m i s s i v i t y e r r o r with s u r f a c e temperature and atmosphere (Figure I I . 4 ) . An a d d i t i o n a l e r r o r of 6 about 0.05 C per u n i t e m i s s i v i t y e r r o r i s reasonable. T h i s w i l l y i e l d probable e r r o r s from 1.0 to 3.2 C, t y p i c a l e r r o r s being approximately 2.2 cJ The a d d i t i o n a l e m i s s i v i t y e r r o r for changes between a day and night image w i l l be high, due to l a r g e s u r f a c e and a i r temperature d i f f e r e n c e s between day and n i g h t . An e r r o r of 0.10 C per u n i t e m i s s i v i t y e r r o r i s reasonable. The range of expected e r r o r s i s t h e r e f o r e 0.9 to 3.2 C. T y p i c a l probable e r r o r s for day-night s i t u a t i o n s are expected to be 7 approximately 2.0 C. A requirement of temperature change measurement i s that the s i t e be a c c u r a t e l y l o c a t e d on both images. The e f f e c t of m i s r e g i s t r a t i o n was t e s t e d on two image segments from each of four n ighttime thermal i n f r a r e d l i n e - s c a n surveys and image segments from three day surveys (three image segments f o r two surveys, two segments for the o t h e r ) . The image segments were 256 x 256 p i x e l s . The t e r r a i n was moderate r e l i e f n a t u r a l rangeland of sparse to dense v e g e t a t i o n . A l l image segments were -193-from one of three t e s t areas. Ground r e s o l u t i o n was between 4 and 5 m. There was no smoothing a p p l i e d to the d i g i t a l data. E r r o r s i n temperature change due to m i s r e g i s t r a t i o n were assumed to be represented by the temperature d i f f e r e n c e s i n p i x e l s a given d i s t a n c e apart on the same image. The average root mean square temperature e r r o r due to m i s r e g i s t r a t i o n s of 1.0, 1.4, 2.0, and 2.8 p i x e l s f o r the above image segments were 0.35, 0.42, 0.49, and 0.59 C r e s p e c t i v e l y for the n i g h t images and 0.98, 1.24, 1.48, and 1.85 C for the day images. Daytime e r r o r s were l a r g e r due to the g r e a t e r v a r i a b i l i t y and range of s u r f a c e temperatures. E r r o r s w i l l depend on the nature of the s u r f a c e s and homogeneity of the s u r f a c e with r e s p e c t to the r e s o l u t i o n of the processed d a t a . A good image-to-image r e g i s t r a t i o n accuracy f o r a i r b o r n e l i n e - s c a n images i s approximately one or s l i g h t l y more than one p i x e l i n both the a l o n g - t r a c k and c r o s s - t r a c k d i r e c t i o n (Appendix I I I ) . A 0.4 C probable e r r o r i n temperature d i f f e r e n c e due to m i s r e g i s t r a t i o n i s reasonable for n i g h t t i m e data. M i s r e g i s t r a t i o n w i l l t h e r e f o r e g e n e r a l l y have onl y a small e f f e c t on the t o t a l probable e r r o r except for cases when e r r o r s i n the other parameters are minimal. T y p i c a l probable e r r o r f o r temperature change duri n g the n i g h t i s t h e r e f o r e approximately 1.6 C. A probable e r r o r of 0.9 C r e p r e s e n t s a case of low e r r o r . A reasonable probable e r r o r r e s u l t i n g from m i s r e g i s t r a t i o n of day images i s 1.2 C. T o t a l probable e r r o r s for temperature changes on two day images are t h e r e f o r e l i k e l y to range from 1.6 to 3.4 C, with t y p i c a l probable e r r o r s of 2.5 C. T y p i c a l probable e r r o r s for day-night temperature changes are expected to be -194-l e s s (approximately 2 . 2 C ) . 7. CONCLUSIONS Good estimates of s u r f a c e e m i s s i v i t y are necessary. E m i s s i v i t y e r r o r s are r e s p o n s i b l e for a l a r g e p o r t i o n of the e r r o r i n s u r f a c e temperature d e t e r m i n a t i o n . E r r o r i n d e t e r m i n a t i o n of the atmospheric o f f s e t was the other major component of e r r o r . I t appears p o s s i b l e t o determine the atmospheric o f f s e t a c c u r a t e l y i f s p e c i a l procedures are used. Accurate ground t r u t h temperature measurement, s i t e l o c a t i o n a i d s , and c o r r e c t i o n f o r change of o f f s e t with scan angle are of primary importance i f a comparison of ground t r u t h and remote sensing temperatures i s used to determine the e f f e c t s of the atmosphere. C a l i b r a t i o n r e l a t i o n e r r o r and system e r r o r s , although not as s i g n i f i c a n t as e m i s s i v i t y and atmospheric o f f s e t e r r o r s , can be important. Choice of a poor c a l i b r a t i o n r e l a t i o n can lead to s e r i o u s e r r o r s . Instrument and tape n o i s e ( p i x e l e r r o r ) as i t a f f e c t s the s u r f a c e temperature p i x e l v alue i s the most important component of system e r r o r s . E r r o r s i n d e t e r m i n a t i o n of s u r f a c e temperature, temperature d i f f e r e n c e s , and temperature changes can be l a r g e . Great care must be taken when using thermal i n f r a r e d l i n e - s c a n data i n a q u a n t i t a t i v e manner. -195-The c a l i b r a t i o n r e l a t i o n s h i p i s the equation mapping detector output (CCT p i x e l values) to temperature. The d i f f e r e n c e between Ea and eEa i s assumed to be s m a l l . B i s approximated i n t h i s study by determining the f r a c t i o n of t o t a l downward r a d i a t i o n i n the bandpass of the scanner using f i g u r e s of atmospheric downward r a d i a t i o n (Kondratyev, 1969; Idso and Jackson, 1968). T h i s i s then a p p l i e d to the t o t a l downward r a d i a t i o n given as a f u n c t i o n of a i r temperature at screen height (Idso and Jackson, 1969). A c o r r e c t i o n for h e m i s p h e r i c a l r a d i a t i o n (Lorenz, 1966) i s a p p l i e d . The atmospheric a t t e n u a t i o n and d e t e c t o r response are in c l u d e d i n a bulk f a c t o r f o r the r a d i a t i o n i n the bandpass. T y p i c a l probable e r r o r f o r g e n e r a l surveys i s d e r i v e d from high atmospheric o f f s e t e r r o r , low e m i s s i v i t y , and i n t e r -mediate p i x e l e r r o r (case 13 i n t e r m e d i a t e p i x e l e r r o r ) . T y p i c a l p robable e r r o r from i n t e r m e d i a t e atmosphere and p i x e l e r r o r . E r r o r corresponds to approximately 10 C change i n s u r f a c e temperature. T y p i c a l probable e r r o r s from i n t e r m e d i a t e atmospheric e r r o r and i n t e r m e d i a t e p i x e l e r r o r p l u s a d d i t i o n a l e m i s s i v i t y e r r o r . The a d d i t i o n a l e m i s s i v i t y e r r o r i s e q u i v a l e n t to one-half the a d d i t i o n a l e r r o r r e s u l t i n g from an i n t e r m e d i a t e e m i s s i v i t y e r r o r . -196-LITERATURE CITED Bastuscheck, C P . 1970. Ground temperature and thermal i n f r a r e d . Photogramm. Eng. 36(8):1064-1072. Byrne, G.F., J.E. Begg, P.M. Fleming, and F.X. Dunin. 1979. Remotely sensed land cover temperature and s o i l water - a b r i e f review. Rem. Sens. Env. 8:291-305. Canada Centre f o r Remote Sensing T e c h n i c a l Memorandum DPD-TM-79-073. 1979. Canada Centre f o r Remote Sensing, Ottawa. C i h l a r , J . , T. Sommerfeldt, and B. Paterson. 1979. S o i l water content e s t i m a t i o n i n f a l l o w f i e l d s from a i r b o r n e thermal scanner measurements. Canadian J . Rem. Sens. 5( l ) : 1 8 - 3 2 . Dancak, C. 1979. Temperature c a l i b r a t i o n of f a s t i n f r a r e d scanners. Photogramm. Eng. 45(6):749-751. Fuchs M. and C B . Tanner. 1966. I n f r a r e d thermometry of v e g e t a t i o n . Agronomy J . 58:597-601. G i l l e s p i e , A.R. and A.B. Kahle. 1978. C o n s t r u c t i o n and i n t e r -p r e t a t i o n of a d i g i t a l thermal image. Photogramm. Eng. 43(8):983-1000. Heilman, J.L., E.T. Kanemasu, N.J. Rosenberg, and B.L. B l a d . 1976. Thermal scanner measurement of canopy temperatures to estimate e v a p o t r a n s p i r a t i o n . Rem. Sens. Env. 5: 137-145. Idso, S.B. and R.D. Jackson. 1968. S i g n i f i c a n c e o f f l u c t u a t i o n i n sky r a d i a n t emittance f o r i n f r a r e d thermometry. Agronomy J . 60:388-392. Idso, S.B. and R.D Jackson. 1969. Thermal r a d i a t i o n from the atmosphere. J . Geophys. Res. 74(23): 5397-5403. Kahle, A.B., D.P. Madura, and J.M. Soha. 1979. P r o c e s s i n g of M u l t i s p e c t r a l Thermal IR Data f o r G e o l o g i c A p p l i c a t i o n s . JPL P u b l i c a t i o n 79-89, J e t P r o p u l s i o n Laboratory, Pasadena, CA. 39 p. Kondratyev, K.Y. 1969. R a d i a t i o n i n the Atmosphere. Academic Press, New York. 912 p. - 1 9 7 -Lorenz, D. 1966. The e f f e c t of the long-wave r e f l e c t i v i t y of n a t u r a l s u r f a c e s on s u r f a c e temeperature measurements using radiometers. J . App. M e t e o r o l . 5:421-430. Scarpace F.L., R.P. Madding, and T. Green I I I . 1975. Scanning thermal plumes. Photogramm. Eng. 41(10):1223-1231. Sc h o t t , J.R. 1979. Temperature measurement of c o o l i n g water d i s c h a r g e d from power p l a n t s . Photogramm. Eng. 45(6): 753-761. Shaw, R.W. and J.G. I r b e . 1972. Environmental adjustments for the a i r b o r n e r a d i a t i o n thermometer measuring water s u r f a c e temperatures. Water Resources. Res. 8:1214-1225. Sutherland, R.A. and J.F. B a r t h o l i c . 1977. S i g n i f i c a n c e of v e g e t a t i o n i n i n t e r p r e t i n g thermal r a d i a t i o n from a t e r r e s t r i a l s u r f a c e . J . App. M e t e o r o l . 16 (8):759-763. Weiss, M. 1971. A i r b o r n e measurements of e a r t h s u r f a c e temperature (ocean and land) i n the 10-12 jam and 8-14 pm r e g i o n s . A p p l i e d Opt. 10:1280-1287. -198-Appendix III USE OF POLYNOMIAL TRANSFORMATIONS FOR REGISTRATION OF AIRBORNE DIGITAL LINE-SCAN IMAGES1 1. INTRODUCTION Accurate image r e g i s t r a t i o n becomes c r i t i c a l i f interpretation techniques such as multitemporal c l a s s i f i c a t i o n , change detection, and thermal i n e r t i a are to be made p r a c t i c a l . Cicone et a l . (1976) studied the e f f e c t of misregistration on multispectral c l a s s i f i c a t i o n techniques applied to a g r i c u l t u r a l surfaces and determined that misregistration on the order of one p i x e l or less can introduce s i g n i f i c a n t errors in c l a s s i f i c a t i o n . Konecny (1976) reviews the mathematical methods of image-to-ground co-ordinate r e g i s t r a t i o n for remote sensing imagery. Few procedures have been developed for or applied to airborne l i n e -scan imagery (Baker and Mikhail, 1975; G i l l e s p i e and Kahle, 1977; Pratt and E l l y e t t , 1978 for d i g i t a l images; and Derenyi, 1974; Kraus, 1978 for fi l m products). Image-to-image r e g i s t r a t i o n of airborne d i g i t a l line-scan data is investigated in t h i s study. Image-to-image r e g i s t r a t i o n i s important as i t i s often the only r e g i s t r a t i o n process necessary. Accurate image-to-map co-ordinate r e g i s t r a t i o n i s not possible in many cases due to i n s u f f i c i e n t number of ground control points (GCP's) avail a b l e on 2 the map or ground truth co-ordinate system. This w i l l occur when the imagery and subsequent interpretation i s at a scale much larger than the ground truth available or in areas devoid of -199-d i s t i n c t c u l t u r a l or n a t u r a l f e a t u r e s f o r ground t r u t h . D i s t o r t i o n s i n l i n e - s c a n imagery depend i n l a r g e p a r t on the motion of the sensor during data a c q u i s i t i o n . I t i s hypothesized t h a t r e g i s t r a t i o n by polynomial transform methods i s s u f f i c i e n t to model image-to-image d i s t o r t i o n . The maximum expected accuracy of r e g i s t r a t i o n a t t a i n a b l e using polynomial transform techniques and the polynomial terms and number of c o n t r o l p o i n t s necessary f o r s u f f i c i e n t accuracy of r e g i s t r a t i o n are important parameters for the development of a r e g i s t r a t i o n p r o c e s s . 2. PROBLEMS OF AIRBORNE IMAGE REGISTRATION R e g i s t r a t i o n of a i r b o r n e l i n e - s c a n images i s a d i f f i c u l t problem due to sy s t e m a t i c scanning d i s t o r t i o n s , viewing p e r s p e c t i v e , and p l a t f o r m i n s t a b i l i t y . Scanner d e s i g n (e.g., scan sweep n o n - l i n e a r i t y ) i n t r o d u c e s v a r y i n g types of d i s t o r t i o n . Panoramic d i s t o r t i o n , which causes the edge of the image to be imaged at smaller s c a l e than the c e n t r e , i s severe i n a i r b o r n e scanners due to a wide f i e l d of view. Surface e l e v a t i o n e f f e c t s can be s i g n i f i c a n t . E r r o r s w i l l be int r o d u c e d i n the l o c a t i o n of c o n t r o l p o i n t s i f chosen at d i f f e r e n t e l e v a t i o n s and poorer r e g i s t r a t i o n w i l l r e s u l t . C o n t r o l p o i n t s of e x c e s s i v e l y high or low topographic p o s i t i o n should t h e r e f o r e be avoided. U n f o r t u n a t e l y , h e i g h t s and de p r e s s i o n s o f t e n p r o v i d e d i s t i n c t c o n t r o l p o s i t i o n s and i n the areas of high r e l i e f the number of c o n t r o l p o i n t s i s r e s t r i c t e d . The i n s t a b i l i t y of the sensing p l a t f o r m i s a major cause of -200-airborne image d i s t o r t i o n and therefore w i l l be a major problem in image r e g i s t r a t i o n . A study for the Canada Centre for Remote Sensing by MacDonald, Dettwiler and Associates Ltd. (1978) undertook to determine a i r c r a f t attitude and position during f l i g h t s of a Convair 580 in order to estimate their e f f e c t on synthetic aperature radar. The e f f e c t of a i r c r a f t motion on an airborne line-scan image may be simulated by introducing a t y p i c a l l i n e scanner to the a i r c r a f t system. A Daedalus (Model 1230) thermal infrared scanner was used in t h i s simulation. It has a resolution of 2.5 m i l l i r a d i a n s and scans at a rate of 60 scans per second (constant angular v e l o c i t y ) . Line-scan data are often e l e c t r o n i c a l l y r o l l s t a b i l i z e d using r o l l data from a scan head mounted gyro and therefore no further r o l l correction i s needed. It was assumed that the scanner data were r o l l s t a b i l i z e d . Analysis of the power spectrum of r o l l , pitch, and yaw of the Convair 580 f l i g h t s showed them to be composed of many frequency components. The larger amplitudes occurred i n the lower frequency components. Frequency components of p i t c h with periods of 10 seconds or more, and frequency components of yaw (and r o l l i f the data were not r o l l s t abilized) with periods greater than 6 seconds would cause image-to-surface co-ordinate d i s t o r t i o n s greater than half a p i x e l with the Daedalus scanner. Lateral notion was generally dominated by a few low frequency components with periods of tens of seconds. Higher frequencies were n e g l i g i b l e . Changes in a i r c r a f t v e l o c i t y had a minor e f f e c t and were of low frequency. A i r c r a f t v e r t i c a l motion also had n e g l i g i b l e e f f e c t over short time periods. The r e s u l t s of the -201-study would vary with the turbulence c o n d i t i o n s , a l t i t u d e , and type of a i r c r a f t but, n e v e r t h e l e s s , they p r o v i d e a s u f f i c i e n t b a s i s f o r e s t i m a t i n g the nature of the a i r c r a f t motion i n r e l a t i o n to image d i s t o r t i o n . I t may be concluded that s i g n i f i c a n t image d i s t o r t i o n can occur over s h o r t d i s t a n c e s and that numerous frequency components c o n t r i b u t e to t h i s d i s t o r t i o n . Determining the a n a l y t i c r e l a t i o n between the image co-o r d i n a t e s of two a i r b o r n e l i n e - s c a n m i ssions i s t e d i o u s . The r e s u l t i n g equations c o u l d not be c o n v e n i e n t l y reduced to a polynomial of a manageable l e n g t h and o r d e r . 3 . M o d e l l i n g a i r c r a f t motion and the r e s u l t i n g d i s t o r t i o n by low order polynomials are not expected to y i e l d s a t i s f a c t o r y r e s u l t s . The approach taken i n t h i s study i s to hypothesize that there are high order p o l y n o m i a l s which are s u f f i c i e n t to model the major components of the image-to-image d i s t o r t i o n over s h o r t a l o n g - t r a c k segments of the images. The order of the polynomial i s expected to depend on the l e n g t h of the segment. Short segments w i l l permit lower order polynomials to be used as approximations of a i r c r a f t motion, thus reducing the polynomial needed i n r e g i s t r a t i o n . There are p r a c t i c a l d i f f i c u l t i e s , however, i n a c q u i r i n g a s u f f i c i e n t number of c o n t r o l p o i n t s as segments become s h o r t . 3 . METHODS AND PROCEDURES In order to t e s t the above h y p o t h e s i s , a study was undertaken to determine the polynomial terms and number of r e g i s t r a t i o n c o n t r o l p o i n t s (RCP's) necessary f o r an ac c u r a t e n o n -r e g i s t r a t i o n of two airborne line-scan images. Registration control points were collected from a test image. The best polynomial transforms of varying lengths were determined and tested using varying numbers of RCP's to calculate the transform c o e f f i c i e n t s . 3.1 TEST IMAGE USED The images used in this study are probably t y p i c a l of most airborne line-scan missions. The images were produced by a mercury cadmium t e l l u r i d e thermal infrared l i n e scanner (Daedalus Model 1230) sensing in the 9.5-11.5 urn band. The scanner operates at 60 scans per second (constant angular v e l o c i t y ) , has o a f i e l d of view of 77 20' and a resolution of 2.5 m i l l i r a d i a n s . At a f l y i n g height of 2225 m above ground l e v e l , ground resolution i s approximately 5.6 m. A scan head mounted gyro was used to e l e c t r o n i c a l l y r o l l s t a b i l i z e the line-scan data. The analogue signal was d i g i t i z e d to 716 pixels/scan l i n e with no panoramic correction applied. The test images are 512 x 512 p i x e l segments of images flown over a f l a t suburban area of North Kamloops, B r i t i s h Columbia, Canada. The missions were flown f i v e hours apart on the night of June 10/11, 1977 by a Falcon fan jet operated by the Canada Centre for Remote Sensing. F l i g h t s were at 2225 m above ground l e v e l at a ground speed of 432 kph for the 4 reference image and 376 kph for the slave image. Winds were calm to l i g h t from the north. Both f l i g h t s were south to north along the same f l i g h t l i n e (the actual f l i g h t paths were at -203-approximately the same heading but o f f s e t by about 120 p i x e l s ) . The centre of the reference image i s o f f s e t approximately 220 pixels west of the actual f l i g h t path centre; the centre of the slave segment 100 pixels west of i t s f l i g h t path centre. The along-track length of 512 pixels is equivalent to 8.5 seconds of f l i g h t time. This time may represent up to one or several cycles of frequency components of r o l l , p i tch, and yaw, contributing s i g n i f i c a n t l y to image d i s t o r t i o n (frequencies of motion are expected to be higher for the Falcon fan jet than those quoted above for the Convair 580). However, i t represents only a f r a c t i o n of a cycle of the major l a t e r a l motion frequencies. Figure I I I . l a i l l u s t r a t e s one of the images to be registered. The two images to be registered have been overlaid and subtracted in Figure I I I . l b . The across-track d i r e c t i o n w i l l be designated the X d i r e c t i o n ; the along-track d i r e c t i o n the Y d i r e c t i o n . The centre of the co-ordinate system used i s indicated as a cross on Figures I I I . l a and I I I . l b . 3.2 CONTROL POINT DESIGNATION The Image Analysis System of MacDonald, Dettwiler and Associates Ltd. of Richmond, B r i t i s h Columbia (Orth et a l . , 1978) was u t i l i z e d to determine control point (CP) image co-ordinates. The system enabled determination of the CP's on a sub-pixel l e v e l by incorporating a zoom c a p a b i l i t y to produce an enlarged (five times in this study) image of a 32 x 32 p i x e l window or chip centred about a control point. The chips of the two images, F i g u r e I I I . l a . T e s t image ( r e f e r e n c e image). The a c r o s s - t r a c k (X) d i r e c t i o n i s from l e f t t o r i g h t on the f i g u r e . -205-Figure I I I . l b . Test image, (reference and slave image overlaid and subtracted). -206-termed the slave and reference, are overlaid and displayed in a f l i c k e r mode. The slave chip may be manually moved about the reference chip u n t i l s a t i s f a c t o r y v i s u a l c o r r e l a t i o n of the control point i s achieved. The p i x e l values of the chips may be normalized or inverted to aid in this process. An automatic c o r r e l a t i o n mode i s also available. For thi s project most control points were manually designated. These procedures give very accurate control point determination. A resampling of control points indicated errors of less than half a p i x e l . One hundred and twelve control points were collected in thi s fashion. Road junctions, roof tops (especially metal roofs), and backyard swimming pools were the chief sources of control points. Control points which yielded large errors when tested with the highest order polynomial were double-checked to make sure an error in control point designation was not made. 3.3 TRANSFORM CALCULATION A program was developed to permit a choice of the polynomial transform to be used. Any combination of the 29 terms given in Table I I I . l may be chosen. The transform c o e f f i c i e n t s are calculated by solving an overdetermined linear least squares f i t of the r e g i s t r a t i o n control points to determine the best f i t polynomial transform between the images. Due to the high powers of the terms and the accuracy required, a very stable algorithm i s necessary. It was found that a method using a Householder Transformation (Businger -207-Table I I I . l . Polynomial transforms a v a i l a b l e (any combination of 29 terms below). 1 Y Y 2 Y 3 Y* Y5 X XY XY 2 XY3 XY 4 XY 5 X 2 X 2Y X 2Y X 2 Y 3 X2Y* X3 X 3Y X 3Y X 3 Y 3 x3Y4 X 4 X 4Y X*Y X*Y 3 xV X 5 X 5Y -208-and Golub, 1971) to transorm the matrix of XY terms to upper triangular form and then solving by back substitution worked well. Correction vectors were computed from the residuals. Increased accuracy was obtained by applying the correction vectors to the solution and recomputing i t e r a t i v e l y . 3.4 POLYNOMIAL SELECTION Polynomials of varying length are given in Table III.2. They represent the optimum polynomial transform of that length. The optimum polynomial was defined as the one containing the terms which gave the lowest sum of squares residuals determined from the RCP's. Preference was given to polynomials with terms of lower degree i f the difference in the sum of the squares residuals was not s i g n i f i c a n t . 5 Polynomials x to v i i were derived by progressively deleting the least s i g n i f i c a n t terms 6 from the 29-term polynomial (polynomial x i ) . The short polynomials ( i i to v i i ) were constructed by adding the most s i g n i f i c a n t term to a three-term polynomial (polynomial i ) . Figure III.2a and III.2b summarize these procedures. One hundred RCP's were used in a l l cases. Although strong conclusions cannot be drawn from a single r e g i s t r a t i o n example, there are several points to note in the progression to larger polynomials. 1) The X transform i s much more e a s i l y f i t by short polynomials with terms of low degree than i s the Y transform. 2) For the X transforms, powers of X are the most -209-T a b l e I I I . 2 . Polynomial transforms. Term Polynomial No. of Terms Terms of Polynomial Added Y - i X-1 3 3 1, Y , X 1, Y , X TTT Y - i l X - i i A A 1. V, X, Y^ 1, Y , X . X 2 Y 2 X 2 Y - l l i X - i i i X - i i i a 5 5 5 1. Y , X , Y f p YX 1. Y , X , X 2 , X ^ 1, Y , X , Y Z , X Z X 3 Y2 Y - i v X - i v 6 6 1 , Y , X , Y ' , YX, YJX 1, Y , X, Y , XS X J Y3X Y 2 Y-v X-v 7 7 1. Y , X , Y l . Y X , X*. Y^X 1, Y . X . Y , X 2 , X 3 , Y ^ &2 Y - v i X - v i 8 8 1. Y , X . Y 2 YX, Y 2 X , X3 , Y3X 1, Y , X, Y 2 . X 2 , X 3 , x\ Y ^ X 2 Y 2 X X« Y - v l i X - v i i 9 9 — i v — X YTf—5TZ v5 Y^X^ .Y - 'X ' ' , Y^X" 1, Y , X , YX, X , l , i - V S , V A Y 2 1, V , X , Y 2 , x 2 , x J , Y ^ x - . r , Y X Y 2 X 2 Y - v i i i X - v i i i 13 13 U Y , X , Y X , X ^ , ^ X . i V , XJ, Y*. Y ' X < v3v VJY2 Y 3 X * I. V. X , Y 2 , Y X , X 2 . YX2, X 3 . Y 2 X 2 , Y X 3 , X*. Y X * , Y * X 2 -Y - i x X - i x 18 17 1. Y , X, YX , X 2 , Y i . Y^X YXZ, Y S Y ^ , » J . Y * X \ 3 x 2 , Y 2 x 3 , Y X * , Y 3 X 3 , Y« X 3 , Y3 X* 1, Y , X , YX . X2 Y 2 X , Y X 2 . X\ Y*, Y 3 X , Y 2 X 2 , Y X 3 , X*. Y X ^ , Y * X 2 , Y2x<>. Y * X 3 -Y-x X-x 20 22 : ~ — 5 — y y — 7 2 — v 3 Y ^ X , YX^, Y ' 1 , YX->, Y " X , WK W>:&\ Y * k 2 . Y 3 X 3 , Y*x3, Y 3 X 4 , i^ X t y Y2 YX X 2 Y X 2 . X 3 , Y * , Y 3 X , Y 2 X 2 , 'X?; Y A X , ^ / ^ ; ? X i , Y * X 2 . Y 3 X 3 , Y 2 X * , Y ^ X 3 " , Y^X* -Y - x i , X - x i 29 a l l terms of Table 1. Y - x i l , X - x l i 6 1, Y , X , Y 2 , Y X , X2 (quadratic) X - x i i i , Y - x l i i ( b i l i n e a r ! A I 1, Y , X , YX --210-d r i f t . KM.--I (significant tern polynomial test aifcniflcancc lot remaining terns include tern into r-olynoxial Include tern into polynomial test terns of high degree previously deleted .Ye*. refine polynoaial test significance of renaming terns of a low degree* -Yes- include terms of low degree deleted ln refinement stage refined polynoaial for use in testing transform • Yes-1 Terns of low degree not deleted «t this ateg*. 2 Terns of low degree only deleted at this refinement (tag*. This bla.es the polynonlal towards one with tema of low degree vnleta wil l b* compu-tationally -ore efficient ln the actual regi.tration process. Tern, of high degree nay account for distortion which nay alao be accounted for by terns of low degree. F i g u r e I I I . 2 a . Procedure f o r s e l e c t i o n o f p o l y -nomial t r a n s f o r m s . D e l e t i o n of terms from the 29-term p o l y n o m i a l . -211-expanded polynomial for use in testing transform Y e s — J 3. Terms of a low degree favoured over terms of high degree i f difference not s i g n i f i c a n t . F i g u r e III.2b. Procedure for s e l e c t i o n of nomial transforms. A d d i t i o n of to the 3-term p o l y n o m i a l . p o l y -terms -212-s i g n i f i c a n t . 3) The Y transform i n v o l v e s more complicated r e l a t i o n s h i p s of X, Y, and XY terms. 4) At polynomials v i and v i i (8 and 9 terms) the Y transform has no d e f i n i t e best form. Many high degree XY terms had almost e q u i v a l e n t s i g n i f i c a n c e . I t was found that p o l y n o m i a l Y - v i i which was d e r i v e d from r e d u c t i o n of the 29-term polynomial gave the most accurate nine-term Y p o l y n o m i a l . By c o n t r a s t , the X transform was not ambiguous. Polynomial X - v i i d e r i v e d from the expansion of polynomial i was e x a c t l y the same as polynomial X - v i i d e r i v e d from r e d u c t i o n of the 29-term p o l y n o m i a l . 5) Polynomial x c o n t a i n s o n l y terms with an f - v a l u e g r e a t e r than 1.0 (most g r e a t e r than 2 or 3) with r e s p e c t to p o l y -nomial x. A l l terms d e l e t e d had f - v a l u e s l e s s than 1.0. Remaining terms of polynomial i x have s i g n i f i c a n c e g r e a t e r than 3.95 with r e s p e c t to i x . T h i s i s the f i v e percent c o n f i d e n c e l e v e l . A l l terms d e l e t e d had f - v a l u e s l e s s than 3.95. Most of the terms d e l e t e d to produce these p o l y -nomials were of high degree. Many other high degree terms were of c o n s i s t e n t l y low s i g n i f i c a n c e throughout the a n a l y s i s . T h i s would suggest that i n c l u s i o n of terms of yet higher degree would not r e s u l t i n any s i g n i f i c a n t i n c r e a s e i n accuracy. -213-4. RESULTS 4.1 LENGTH OF POLYNOMIAL The results given in Table III.3 indicate that the higher order polynomials with large number of RCP's resulted in sub-p i x e l r e g i s t r a t i o n accuracy. 7 The r e g i s t r a t i o n accuracy of a polynomial can be measured in several ways. The root mean square (RMS) error of the RCP's may be used. An independent measure of accuracy is the RMS error of control points not used to derive the polynomial c o e f f i c i e n t s (TCP's). RMS error does not always give a good indication of the d i s t r i b u t i o n of the error. Therefore, Table III.3 l i s t s the RMS error of the TCP's, the mean of the absolute value of the errors (mean TCP e r r o r ) , and the error below which 85 percent of the TCP's occurred (derived from frequency d i s t r i b u t i o n s of the e r r o r s ) . Figure III.3 plots the RMS TCP and RMS RCP error of polynomials i to x i versus number of terms in the polynomial. RCP's and TCP's for a l l cases were evenly di s t r i b u t e d about the image and were the same for each polynomial tested. This attempted to ensure that there was no p o s i t i o n a l bias in the r e s u l t s . There are several points to be extracted from the data with the caveat that the data are derived from only one r e g i s t r a t i o n example. 1)Registration accuracy dropped sharply for short low order polynomials. Polynomials of 5 or 6 terms (polynomials iv and i i i ) are the shortest polynomials to give s a t i s f a c t o r y r e s u l t s . -214-T a b l e I I I .3. Accuracy of p o l y n o m i a l t r a n s f o r m s . Nuaber of Tenia Nunber of RMS RCP Error Nucber of RMS T CP Error Mean TCP Error 852 Threshold Polynoaial Y X RCP'a Y X TCP't 7 X Y X T X >1 2° 29 .67 .760 .597 25 1.17 .672 .949 .553 58 .638 .544 54 1.50 .852 1.11 .662 1.9 1.2 25 1.31 .800 1.02 .630 29 0.00 0.00 83 25.5 ' 9.60 11.3 6.19 20.0 12.0 25 16.7 6.81 8.87 5.08 X 20 22 87 .734 .603 25 1.08 .674 .906 .554 58 .707 .570 54 1.24 .781 .934 .584 1.8 1.1 25 1.07 .716 .840 .555 29 .176 .452 83 1.37 1.11 1.00 .649 1.5 1.8 25 1.32 1.01 1.02 .797 l i i a 17 87 .807 .632 25 1.06 .675 .886 .551 56 .712 .597 54 1.25 .760 .935 .572 1.7 1.1 25 1.08 .687 .836 .554 29 .883 .508 83 1.33 .915 .993 .648 1.7 1.1 25 1.31 .805 1.03 .655 vill 13 13 87 .376 .677 25 1.17 .718 ~l'."o'2" .583 58 .726 .658 54 1.26 .771 .983 .5»6 1.8 l . l 25 1.08 .755 .349 .623 29 .686 .366 83 1.23 .871 .963 .666 1.6 1.4 25 1.21 .864 .980 .637 vll 9 9 87 .915 .715 25 1.19 .778 1.02 .618 53 .604 .698 54 1.16 .804 .935 .662 1.6 1.1 25 1.03 .788 .825 .634 29 .850 .618 83 1.12 .835 .910 .684 1.4 1.3 25 1.10 .871 .872 .735 vl 8 8 87 1.01 .730 25 1.05 .757 .868 .642 58 .899 .702 54 1.20 .322 .936 .646 1.8 1.1 25 1.08 .805 .853 .692 29 1.03 .619 83 1.07 .870 .874 • .679 1.5 1.2 25 1.10 .853 .900 .726 7 7 87 1.01 . .750 25 1.09 .835 .900 .697 58 .903 .705 54 1.16 .868 .956 .693 1.7 1.1 25 1.03 .844 .816 .722 29 1.04 .641 83 1.03 .860 .845 .677 1.4 1.1 25 1.03 .896 .837 .754 ly 6 6 87 1.02 .771 25 1.09 .883 .897 .741 58 .904 .726 54 1.17 .891 .962 .693 1.7 1.3 25 1.03 .896 .815 .766 29 1.04 .651 83 1.03 .928 .848 .711 1.4 1.4 25 1.03 .969 .833 .796 Y-Ul , 5 5 87 1.05 1.01 25 1.16 1.20 .914 .929 X-lIla 56 .909 1.04 54 1.20 1.12 .991 .835 1.7 1.6 25 1.07 1.23 .827 .970 29 1.06 1.02 S3 1.08 1.17 .864 .923 1.6 1.6 25 1.11 1.28 .832 .936 11 4 4 87 1.62 1.11 25 1.70 1.33 1.31 1.07 -58 1.55 1.09 54 1.76 1.26 1.42 1.00 1.7 1.7 25 1.74 1.34 1.33 1.03 29 1.71 1.10 83 1.61 1.25 1.29 .967 2.3 2.0 25 1.67 1.36 1.29 1.07 1 3 3 87 2.15 10.4 25 2.10 10.5 1.75 8.77 33 2.00 10.7 54 2.29 10.2 1.85 8.77 3.5 15.6 25 2.10 10.5 1.74 8.86 29 2.20 11.3 83 2.16 10.1 1.76 8.76 3.2 13.4 25 2.15 10.4 1.73 8.83 i l l ft 6 87 1.04 .998 25 1.16 1.15 .934 .886 58 .906 1.02 54 1.22 1.09 .937 .850 1.7 1.7 25 1.07 1. IB .821 .396 29 1.03 .935 83 1.15 1.15 .893 .892 1.6 1.6 25 1.13 1.21 .843 .897 xllt 4 * 37 1.75 10.4 25 1.54 10.5 1.16 J.75 59 1.65 10.7 54 1.75 10.3 1.34 8.79 2.7 15.0 25 1.48 10.5 1.13 8.90 29 1.81 11.3 83 1.75 10.1 1.41 8.73 2.6 15.6 25 1.69 10.3 .984 8.76 - 5 87 - T869 25 - 2724 : Uii 58 - .861 54 - 2 . 01 - 1.44 - 2 .8 25 - 2.30 - 1.68 25 - .'21 83 - 2.11 - 1.31 - 3.3 25 - 2.34 - 1.67 2.5 r 2.0 </> _i UJ X g 1.5 cc cc UJ (0 S oc 1.0 • Y Polynomial RMS TCP Error • — X Polynomial RMS TCP Error o—• Y Polynomial RMS RCP Error • X Polynomial RMS RCP Error I tn l > —«o ' • • • o 10 F i q u r e I I I . 3 . RMS TCP and RCP e r r o r ve i to x i ; 58 RCP's). 15 20 25 30 NUMBER OF TERMS rsus lenqth of polynomial (Polynomials -216-2) Polynomial X-iv gives markedly better results than a six-term quadratic equation (polynomial X - x i i ) . 3) Better results for the X transform are obtained by using progressively longer polynomials, although a maximum i s reached beyond the 17-term polynomial (X-ix). The sign i f i c a n c e of terms deleted to derive polynomial X-ix were a l l less than the five percent confidence l e v e l . 4) Although the sum of the squares RCP error decreased with the number of terms for the lengthy Y transformations, the TCP error did not decrease. This l a t t e r point suggests that perhaps there i s some complexity of d i s t o r t i o n or random d i s t o r t i o n in the Y d i r e c t i o n which cannot be e a s i l y accounted for by use of polynomial expressions derived from RCP's. A factor may be that the added complexity of the Y d i s t o r t i o n causes the number of RCP's to influence the errors more than the X transform, thus n u l l i f y i n g the expected increase in accuracy for the longer polynomials. Another p o s s i b i l i t y i s that the RMS RCP error is not always the best c r i t e r i o n for selecting the polynomial terms which w i l l give the best r e g i s t r a t i o n accuracy. This l a t t e r suggestion may be p a r t i c u l a r l y true for the short polynomials (less than six terms) as their errors seem to be somewhat unstable. For example, polynomial X - i i i produced abnormally high TCP errors. It was found that polynomials X - i i i a , despite having s i g n i f i c a n t l y greater RCP errors, had much lower TCP errors. Also, the polynomials were chosen using 100 RCP's. For short polynomials they may not represent the best polynomial derived from only a few RCP's. -217-The maximum expected accuracy using polynomial transforms i s approximately an RMS error of 1.0 p i x e l with 85 percent of errors l i k e l y less than 1.5 pixels for the Y transform, and ah RMS error of 0.75 pixels with 85 percent less than 1.0 for the X transform. The X transform was consistently easier to f i t , and gave lower RMS RCP error and r e g i s t r a t i o n (RMS TCP) error. This might be expected in l i g h t of a i r c r a f t motion and scanner c h a r a c t e r i s t i c s . The major d i s t o r t i o n s in the X dir e c t i o n are panoramic and r o l l d i s t o r t i o n . Since r o l l i s previously corrected for by an electronic r o l l s t a b i l i z a t i o n system, i t i s reasonable to assume that the polynomial required in the X di r e c t i o n w i l l be simpler than the Y transformation which must account for pitch (perhaps with a large high frequency distortion) and yaw which has i t s major e f f e c t on the Y displacement. 4.2 NUMBER OF REGISTRATION CONTROL POINTS The number of r e g i s t r a t i o n control points was also found to be important. The TCP errors given in Table III.3 for the 25-TCP case of the 87-,58-,and 29-RCP transforms i l l u s t r a t e the ef f e c t s of the number of RCP's on r e g i s t r a t i o n accuracy. The same TCP's were used in each case. Errors in the X transform decrease with increasing number of control points. The e f f e c t i s more noticeable for the lengthy polynomials. The increase in TCP error of Figure III.3 for the long polynomials i s an a r t i f a c t of th i s e f f e c t . High order polynomials of Table III.3 using only 29 -218-RCP's gave high e r r o r s . The e f f e c t s of the number of RCP's and le n g t h of polynomial are c l o s e l y r e l a t e d . The b e n e f i t s of a l a r g e number of terms i s n u l l i f i e d u nless s u f f i c i e n t RCP's are a v a i l a b l e . Table III.4 and F i g u r e s III.4 and I I I . 5 demonstrate the e f f e c t of the numbers of RCP's on polynomials of 5 and 6 terms. A marked i n c r e a s e i n e r r o r occurs with l e s s than approximately 25 RCP's. T h e r e f o r e , f o r the examples examined, not u n t i l the number of RCP's i s s e v e r a l times the number of terms does the a d d i t i o n of f u r t h e r c o n t r o l p o i n t s become unimportant. Another f a c t o r i n v o l v e d i n determining the number of c o n t r o l p o i n t s necessary f o r good r e g i s t r a t i o n , besides p r o v i d i n g adequate redundancy f o r the l e a s t squares s o l u t i o n , i s the s c a l e of d i s t o r t i o n or s i z e of area i n f l u e n c e d by a giv e n d i s t o r t i o n . C o n t r o l p o i n t s must be s u f f i c i e n t i n number and w e l l enough d i s t r i b u t e d to account for a l l regions of d i s t o r t i o n . Use of a small number of c o n t r o l p o i n t s , although g i v i n g s a t i s f a c t o r y r e g i s t r a t i o n over much of the image, may le a d to e x c e s s i v e l y l a r g e e r r o r s i n a few areas of the image. A d d i t i o n of f u r t h e r c o n t r o l p o i n t s t h e r e f o r e reduces the maximum r e g i s t r a t i o n e r r o r . The number of TCP's with l a r g e e r r o r s (> 2.0 p i x e l s ) and the s i z e of these e r r o r s i n c r e a s e d as fewer RCP's were used. T h i s trend may be seen i n the 85 percent t h r e s h o l d v a l u e s of Table I I I . 4 . L o c a l i z e d areas of high d i s t o r t i o n were v i s i b l e on the images. Some c o n t r o l p o i n t s gave c o n s i s t e n t l y l a r g e e r r o r s . Most c o u l d be d i r e c t l y t r a c e d to p o i n t s which were double-checked f o r accuracy but were i n l o c a l e s noted as areas of severe d i s t o r t i o n . T h e r e f o r e , even f o r good poly n o m i a l s with l a r g e numbers of Table III.4 Number of Terms Polynomial Y X E f f e c t of number of RCP's on accuracy of polynomial transforms i v Y - i i i , X - i i i a x i i Number of RCP's 58 29 24 18 12 9 6 RMS RCP Error Y X .904 1.04 1.07 .817 .840 .566 0.00 .726 .651 .693 .504 .504 .368 0.00 58 29 24 18 12 9 6 .909 1.06 1.08 .875 .841 .605 0.00 58 29 24 18 12 9 6 .906 1.03 1.06 .839 .723 .428 0.00 1.02 .985 1.03 .898 .931 .600 0.00 54 54 54 54 54 54 54 .786 54 1.20 1.12 .981 .728 54 1.21 1.18 1.00 1.07 54 1.21 1.17 1.00 .962 54 1.25 1.23 1.01 .953 54 1.25 1.34 1.02 .743 54 1.25 1.46 1.02 .564 54 2.02 1.96 1.71 .885 .917 .927 .954 1.04 1.12 1.66 1.22 1.28 1.26 1.32 1.47 1.40 2.02 1.09 1.14 1.13 1.19 1.31 1.38 6.11 .987 1.02 1.01 1.04 1.12 1.11 1.71 .850 .869 .867 9.07 1.02 1.04 5.02 1.7 1.7 1.7 1.9 1.8 2.1 3.0 1.7 1.8 1.8 1.9 2.1 2.2 3.0 Number of TCP's RMS TCP 1 Error Mean TCP Error 85% Threshold Y X Y X Y X 54 1.17 .894 .962 .693 1.6 1.3 54 1.13 .944 .945 .699 1.6 1.3 54 1.17 .951 .972 .709 1.7 1.3 54 1.45 .994 1.18 .775 2.0 1.5 54 1.23 1.02 1.01 .770 1.9 1.5 54 1.63 1.08 1.33 .802 2.4 1.6 54 2.02 1.36 1.71 1.07 3.0 2.0 1.7 1.8 1.8 2.0 2.2 2.3 2.9 1.7 1.7 1.7 1.8 2.1 2.2 8.1 I ro I—1 10 -220-V7, 16 - 1.5 | in Z l.< o £ 1.3 1 UJ " 1.2 j s 1 . 1 1 . Polynomial Y—iii polynomial Y-iv — • — Polynomial Y-xii 10 .81 l_ O F i g u r e III.4. 10 20 3° NUMBER OF RCP's 40 60 60 RMS TCP e r r o r versus number of RCP's (Poly-nomials Y - i i i , Y - i v , and Y - x n ; 54 TCP s ) . NUMBER OF RCP' F i q u r e I I I . 5 . RMS TCP e r r o r versus number of RCP's (Poly-nomials X - i i i , X - i v , and X - x i i ; 54 T C P ' s ) . -221-control points, there w i l l s t i l l be small l o c a l i z e d areas of poor r e g i s t r a t i o n . Errors of some points in the areas of high d i s t o r t i o n were as high as 3.0 p i x e l s , whereas the maximum error of points not in distorted areas was generally less than 2.0 pi x e l s except when a low number of RCP's was used. Over terrain with r e l i e f , additional l o c a l i z e d errors w i l l occur due to topography. Polynomial transforms derived from control points w i l l not register images as accurately. 5. DISCUSSION AND CONCLUSIONS The results obtained represent close to the best r e g i s t r a t i o n accuracy attainable by applying polynomial transform methods over large image segments. F l i g h t conditions were good with calm to l i g h t winds and turbulence was not excessive. The frequency and magnitude of platform motion, however, may be less for f l i g h t s at higher alt i t u d e s and with larger a i r c r a f t . There were a large number of accurately determined, and well dis t r i b u t e d r e g i s t r a t i o n control points. The t e r r a i n was f l a t . The number and accuracy of the control points used i s far above what would be expected over natural t e r r a i n . Analysis of the importance of terms in polynomials of lengths near 29 terms showed that longer polynomials with terms of higher degree would not contribute s i g n i f i c a n t l y to the r e g i s t r a t i o n . There may be some advantage in correcting for additional systematic d i s t o r t i o n s before determination of the transform -222-c o e f f i c i e n t s , as was done i n the case of r o l l . For example, c o r r e c t i o n for panoramic d i s t o r t i o n would a i d i n reducing the complexity of the polynomial r e q u i r e d . B e t t e r r e s u l t s might be expected over a smal l e r segment of the image p a r t i c u l a r l y i n the al o n g - t r a c k (Y) d i r e c t i o n , i f s u f f i c i e n t number of c o n t r o l p o i n t s are a v a i l a b l e . D i s t o r t i o n caused by lower frequency components of a i r c r a f t motion have the l a r g e s t e f f e c t on image d i s t o r t i o n . The d i s t o r t i o n w i l l be e a s i e r to model over s h o r t e r segments s i n c e o n l y a p a r t o f a c y c l e w i l l have to be modelled. In the implementation of a r e g i s t r a t i o n procedure there w i l l be compromises between the s i z e of the segments to be r e g i s t e r e d , the l e n g t h of polynomials, and the accuracy a t t a i n e d . Large images w i l l have to be r e g i s t e r e d i n a piecewise manner. The optimum combination i s yet to be i n v e s t i g a t e d . The transform one uses f o r image-to-image r e g i s t r a t i o n depends on computational t r a d e - o f f s made to produce a given r e g i s t r a t i o n accuracy. The a n a l y s i s procedures presented are a p p l i c a b l e to r e g i s t r a t i o n of many forms of remote sensing data. The complexity and nature of terms of the po l y n o m i a l s , the number of RCP's, and the r e g i s t r a t i o n accuracy d e r i v e d from a n a l y s i s of the images used i n t h i s study are probably t y p i c a l of those found f o r most a i r b o r n e image-to-image r e g i s t r a t i o n of images of s i m i l a r s i z e obtained using s i m i l a r scanning systems and a i r c r a f t , and over f l a t t e r r a i n . There appears to be a t h r e s h o l d o f d i m i n i s h i n g r e t u r n i n both the number o f polynomial terms and the number of r e g i s t r a t i o n c o n t r o l p o i n t s r e q u i r e d f o r the r e g i s t r a t i o n of a i r b o r n e l i n e - s c a n images b y polynomial transform techniques. For the t e s t images used,the t h r e s h o l d s -223-were at about 5 or 6 terms and 25 r e g i s t r a t i o n control points ( i . e . , 25 RCP's for the polynomials of 5 or 6 terms). B i l i n e a r and quadratic polynomials were not optimum and did not y i e l d good r e s u l t s . Despite a complex t h e o r e t i c a l r e l a t i o n of d i s t o r t i o n in airborne line-scan images, simple polynomials can give accurate r e s u l t s . The polynomial judged to be the best for the images tested was a five-term Y transform (polynomial Y - i i i ; Y = a + Y + 2 X + Y + YX) and a six-term X transform (polynomial X-iv; X = a + 2 2 3 Y + X + Y +X + X ) using approximately 25 RCP's. This leads to a Y r e g i s t r a t i o n error of approximately 1.2 p i x e l s RMS error and an X r e g i s t r a t i o n error of 1.0 p i x e l s . The minimum expected error on a p p l i c a t i o n of polynomial transform methods to airborne line-scan r e g i s t r a t i o n was in the order of 1.0 p i x e l s (RMS error) i n the Y along-track d i r e c t i o n and 0.75 pixels (RMS error) in the X across-track d i r e c t i o n . * Condensed version of t h i s Appendix presented at the Sixth Canadian Symposium on Remote Sensing, Halifax, N.S., May 21-23, 1980. 2 A control point (CP) i s an e a s i l y recognizable l o c a t i o n on a reference image which may also be located on a second image or given precise cartographic co-ordinates. A co n t r o l point has two sets of co-ordinates. I f i t consists of the l i n e and p i x e l co-ordinates of an image -224-and c a r t o g r a p h i c c o o r d i n a t e s , i t i s r e f e r r e d to as a ground c o n t r o l p o i n t (GCP) and i s used i n image-to-map r e g i s t r a t i o n . I f the c o o r d i n a t e s are the c o o r d i n a t e s of a r e f e r e n c e image and some other image, the CP i s r e f e r r e d to as: 1) a r e g i s t r a t i o n c o n t r o l p o i n t (RCP) i f i t i s used i n the d e t e r m i n a t i o n of the image-to-image r e g i s t r a t i o n transform c o e f f i c i e n t s , or 2) a t e s t c o n t r o l p o i n t (TCP) i f i t i s not used i n the d e t e r m i n a t i o n of the r e g i s t r a t i o n transform but i s used to t e s t the accuracy of the r e g i s t r a t i o n . The term l e n g t h i n r e f e r e n c e to polynomial transforms means the number of terms with non-zero c o e f f i c i e n t s . Reference image i s the image to which the second image, termed the s l a v e image, i s to be r e g i s t e r e d . The degree r e f e r s to the sum of the power of Y and of X. i The s i g n i f i c a n c e of a term to a polynomial was determined by the s t a t i s t i c a l f - v a l u e which r e s u l t s when that term i s d e l e t e d from the polynomial (Croxton and Crowley, 1955). The f - v a l u e i s determined by: (SSE - SSE_)/(M-P) m P SSE /(N-M) m where: SSE = sum of the squares e r r o r of RCP's due to the polynomial ( s u b s c r i p t m denotes o r i g i n a l p olynomial being t e s t e d a g a i n s t , p the polynomial to be t e s t e d ) ; M-P = degrees of freedom (change i n number of terms); M = number of terms i n o r i g i n a l p o l y n o m i a l ; P = number of terms, polynomial to be t e s t e d ; N-M = degrees of freedom; N = number of RCP's used to c a l c u l a t e transform (= 100). R e g i s t r a t i o n accuracy i s the accuracy expected i n the a c t u a l r e g i s t r a t i o n p r o c e s s . In t h i s study i t i s c o n s i d e r e d best represented by the accuracy to which t e s t c o n t r o l p o i n t s (TCP's) are r e g i s t e r e d by a polynomial t r a n s f o r m . A l s o termed TCP e r r o r . -225-LITERATURE CITED Baker, J.R. and E.M. M i k h a i l . 1975. Geometric A n a l y s i s and R e s t i t u t i o n of D i g i t a l M u l t i s p e c t r a l Scanner Data A r r a y s . LARS Information Note 052875. Laboratory f o r A p p l i c a t i o n s of Remote Sensing, Purdue U n i v e r s i t y , West L a f a y e t t e , IN. Businger, P. and G.H. Golub. 1971. L i n e a r l e a s t squares s o l u t i o n s by Householder t r a n s f o r m a t i o n s . Iri Bauer, F.L. (ed.). L i n e a r A l g e b r a . S p r i n g e r - V e r l a g , New York. Cicone, R.C., W.A. M a l i l a , J.M. Gleason, and R.F. Nalopka. 1976. E f f e c t s of m i s r e g i s t r a t i o n on m u l t i s p e c t r a l r e c o g n i t i o n . Proc. Symp. Machine P r o c e s s i n g of Rem. Sens. Data. West L a f a y e t t e , IN. pp.4Al-4A17. Croxton, F. and D. Cowden. 1955. A p p l i e d General S t a t i s t i c s . P r e n t i c e - H a l l , Engelwood C l i f f s , New J e r s e y . 693 p. Derenyi, E. 1974. P l a n i m e t r i c accuracy of i n f r a r e d l i n e scan imagery. Canadian Surveyor 28:247-254. G i l l e s p i e , A.R. and A.B. Kahle. 1977. C o n s t r u c t i o n and i n t e r p r e t a t i o n of a d i g i t a l thermal i n e r t i a image. Photogramm. Eng. 43(8):983-1000. Konecny, G. 1976. Mathematical models and procedures f o r the geometric r e s t i t u t i o n o f remote sensing imagery. XIII Congress of the I n t . Soc. of Photogramm. Comm. I I I . H e l s i n k i , F i n l a n d . Kraus, K. 1978. R e c t i f i c a t i o n of m u l t i s p e c t r a l scanner imagery. Photogramm. Eng. 44(4):453-457. MacDonald, D e t t w i l e r and Assoc. 1978. Yaw Compensated S y n t h e t i c Array Radar F i n a l Study Report. MacDonald, D e t t w i l e r and Assoc. L t d . r e p o r t p u b l i s h e d f o r the Canadian Centre f o r Remote Sensing, Ottawa. Orth, R.K., F. Wong, and J.S. MacDonald. 1978. The p r o d u c t i o n of 1:125,000 maps of p r e c i s i o n r e c t i f i e d and r e g i s t e r e d LANDSAT imagery using the MDA image a n a l y s i s system: i n i t i a l r e s u l t s . 12th I n t ' l Symp. on Rem. Sens, of Env. Man i l a , P h i l l i p i n e s . pp. 2163-2176. P r a t t , D.A. and C D . E l l y e t t . 1978. Image r e g i s t r a t i o n f o r thermal i n e r t i a mapping, and i t s p o t e n t i a l use f o r mapping of s o i l moisture and geology i n A u s t r a l i a . Proc. 12th I n t ' l Symp. on Rem. Sens, o f Env., M a n i l a , P h i l i p p i n e s , pp. 1207-1217. -226-Appendix IV SPATIAL VARIABILITY OF AIR TEMPERATURE AND WIND SPEED, AIR PONDING, AND SHORT TERM LOCAL ADVECTION : DISCUSSION AND EXAMPLES 1. INTRODUCTION Most remote sensing implementations of thermal i n e r t i a models r e q u i r e e x t r a p o l a t i o n of m i c r o m e t e o r o l o g i c a l measurements of a i r temperature and wind speed over l a r g e areas. The occurrence of a i r ponding and sho r t term l o c a l a d v e c t i o n cannot g e n e r a l l y be accounted f o r by thermal i n e r t i a models, and leads to e r r o r s i n thermal i n e r t i a e s t i m a t e s . The s p a t i a l v a r i a b i l i t y of a i r temperature and wind speed, the magnitude of a i r temperature change duri n g ponding, and the e f f e c t s of s h o r t term l o c a l a d v e c tion on s u r f a c e temperature are important f a c t o r s i n the e r r o r a n a l y s i s of a thermal i n e r t i a method. In order to i n v e s t i g a t e these f a c t o r s , f i e l d experiments over two seasons were conducted. Test s i t e s f o r these i n v e s t i g a t i o n s were l o c a t e d on the Lac du B o i s rangeland near Kamloops, B.C. Psychrometers were designed and c o n s t r u c t e d , and a system to i n t e r r o g a t e the s i g n a l s from a s p a t i a l a r r a y of psychrometers was developed. The s i g n a l s were t r a n s m i t t e d by wire to a c e n t r a l r e c o r d i n g s t a t i o n . A i r temperatures were measured by s i l i c o n diodes mounted i n the psychrometers. The psychrometers were s h i e l d e d and v e n t i l a t e d at 3.7 m s ^ using a DC f a n . The temperature at each s i t e was measured every one to -227-two minutes using a stepping switch and recorder. Wet bulb measurements were also taken to examine latent heat flux (section 3.4). Wind speeds were measured with a Casella anemometer and counter. A l l measurements were at 1.5 m above the ground surface. The psychrometers of Figures IV.5 and IV.6 were not ventilated; and wind speeds were estimates. Sites of each psychrometer for the experiments were chosen to test the s p a t i a l v a r i a b i l i t y over similar and dissimilar surface types and topographic position. Particular sets of sites were located to examine air ponding. Detailed descriptions of the s i t e s are given in section 6. 2. PROBABLE ERROR IN AIR TEMPERATURE UNDER NON-PONDING CONDITIONS The va r i a t i o n of air temperature over s i t e s of similar and di s s i m i l a r surface types and topographic position can be seen in Figures IV.1 to IV.6 and also Figures 4.3 and 4.4. Only the non-ponded cases w i l l be considered in this section. The maximum temperature difference between s i t e s at a given time ranges between 0.5 to 2.0 C. The differences in time-averaged temperatures for the periods of the figures, however, are much less (Table IV.1). Also given are the a i r temperatures at the Kamloops airport weather station (see section 6). Airport temperatures are at the same height above the ground and for an equivalent time period (from average hourly screen temperature data). Typical a i r temperature errors which w i l l occur by using one s i t e as the temperature for the whole survey area are given F i a u r e IV 2. Time p l o t s of a i r temperature and wind speed (at 1.5 m) f o r s i t e s y of area B (03/08/78; 0400 h). 2330 i 1— 2340 2350 0000 0010 TIME ( P D T ) 0020 0030 F i q u r e IV.3. Time p l o t s of a i r temperature and wind speed (at 1.5 m) F i g u r e i v . 3 . ^ ^ ^ c ( 3 1 / 0 8 / 7 8 . 2 4 0 o h) . < 0350 0400 0410 0420 0430 TIME (PDT) 0440 0450 0500 I to F i g u r e IV.4. Time p l o t s of a i r temperature and wind speed (at 1.5 m) fo r s i t e s of area C (01/08/78; 0400 h). 21.0 O 10.0 0 4 4 0 0 5 0 0 0 5 2 0 TIME (PDT) 0 5 4 0 F i g u r e IV.6. Time p l o t s of a i r temperature (at 1.5 m) f o r s i t e s of area C (20/06/78; 0500 h). Wind speeds estimated to be 1.0 to 1.5 m/s. -234-Table IV.1. Time-averaged a i r temperature of s i t e s and di f f e r e n c e s between the time-averaged temp-eratures of the s i t e s . Date/ Time S i t e Mean Mean Temp. Temp.(C) D i f f e r e n c e ! (C) 29/7/78 A - l 23.4 0.6 2300 A-2 22.9 0.9 A-3 23.5 0.5 A-4 24.3 0.8 A-5 24.1 0.7 A i r p o r t 2 23.0 0.7 02/8/78 B-l 22.9 0.5 2300 B-2 22. 8 0. 5 B-3 22.1 1.0 A-4 23.5 0.9 A i r p o r t 23.5 0.7 03/8/78 B- l 18.4 0.1 0430 B-2 18.5 0.1 B-3 18.3 0.2 Ai r p o r t 16.3 2.1 20/6/78 C - l 19.2 0.3 0000 C-2 18.7 0. 3 C-3 18.9 0.2 C-4 19.0 0.2 A i r p o r t 18.6 0.4 20/6/78 C - l 8.5 0.2 0500 C-2 8.9 0 . 3 C-3 8.5 0.2 C-4 8.7 0.2 Airpor t 12.9 4.3 01/8/78 C - l 19.8 0.7 0000 C-2 20.4 0.5 C-3 21.0 0.9 C-4 20.2 0.5 A i r p o r t 21.2 0.9 01/8/78 C - l 14.9 0.2 0430 C-2 15.3 0.3 C-3 15.0 0.2 C-4 15.1 0.2 A i r p o r t 16.7 1.6 JMean temperature d i f f e r e n c e placed opposite s i t e from which d i f f e r e n c e of other s i t e s (Airport s i t e not included) i s taken. Mean d i f f e r e n c e i s given as the nean of the absolute values of the d i f f e r e n c e s . Kamloops a i r p o r t . -235-as the mean difference between s i t e s in Table IV.1. It is reasonable to assume that a t y p i c a l probable error for non-ponding cases (ATa T) w i l l be about 0.5 C. A value of 1.5 C is l i k e l y a severe or high probable error (ATa ) representative of cases where the survey area is very large or the temperature data are measured at a distance from the survey area, as in the case of the airport data of Table IV.1. 3. ERRORS IN AIR TEMPERATURE UNDER PONDING CONDITIONS Air temperature va r i a t i o n due to a i r ponding may be determined using f i e l d data measured at si t e s where ponding may occur. Table IV.2 gives examples. The degree to which ponding develops depends greatly on wind speed, the duration of calm or l i g h t wind conditions, and the pa r t i c u l a r topography of the s i t e s . Oke (1978) states that height differences less than one meter may resu l t in air ponding. Low and Greig (1973) report the occurrence of katabatic flow on slopes as low as one degree. Air temperature differences between ponded s i t e s and non-ponded s i t e s can be large, sometimes in excess of 10 C (Lawrence, 1958; Anstey et a l . , 1959; Hovecar and Martsolf, 1971; Derksen, 1974). The effects on thermal images have been noted in many studies (eg., Murtha, 1971; Derksen, 1974; Bennett, 1977). Comparison of air temperature versus time plots of d i f f e r e n t s i t e s (Figures IV.1 to IV.6) also aids in the interpretation of the magnitude of air ponding. Topography indicates that s i t e s B-3 and C - l should tend to be susceptible to ponding. Ponding i s not strongly evident T a b l e IV.2. Examples o£ the d i f f e r e n c e i n a i r t e m p e r a t u r e ( a t 1.5 m) between s i t e s n o t s u s c e p t i b l e ( h i g h s i t e ) and s u s c e p t i b l e (low s i t e s ) t o p o n d i n g . D - l A - l A - l A - l B - l A - l B - l A - l B - l A - l A - l A - l K-1 E-2 E - l E-2 E - l E-2 D-2 A-2 A-2 A-2 B-4 A-2 B-4 A-2 B-4 A-2 A-2 A-2 E-2 E-3 E-2 E-3 E-2 E-3 9/6/77 9/6/77 10/6/77 13/7/77 13/7/77 14/7/77 14/7/77 14/7/77 14/7/77 15/7/77 18/6/78 10/6/78 20/7/77 20/7/77 21/7/77 21/7/77 22/7/77 22/7/77 0330 0000 0000 2330 0000 0430 0500 2300 2330 0400 0500 0530 2300 2300 0430 0430 0430 0500 1.9 0.7 0.6 •0.5 0.4 5.1 1.2 -0.1 0.0 0.4 -0.2 0.2 0.8 1.6 0 1 -0 0 3 3 2 2 0.5 - 1.03 0.5 - 1.03 1.03 1.0 - 1.5 1.0 - 1.5 calm calm 2.0 2.0 calm 5.0 5.0 0.0 0.5 0.5 0.5 0.5 - 1.0: - l . o : - l . o : - 1.5: - 1.0 calm ^Approximate. 2 Temperature o f hig h s i t e minus temperature o f low s i t e ; temperatures measured with an Assmann v e n t i l a t e d psych-rometer up to s e v e r a l minutes apart. E s t i m a t e s . -237-fo r the wind speeds ( u s u a l l y g r e a t e r than 1 m s of the examples. The group of s i t e s of F i g u r e 4.3 and 4.4 were s p e c i f i c a l l y chosen to i n v e s t i g a t e a i r ponding. They are d e s c r i b e d i n Chapter 4 and demonstrate that ponding can cause l a r g e a i r temperature changes over short p e r i o d s of time. Severe d i f f e r e n c e s i n temperature between ponded and non-ponded s i t e s may occur. In moderate ponding c o n d i t i o n s (moderate due to topography or winds p a r t i a l l y suppressing ponding), t y p i c a l probable e r r o r s in using the a i r temperature of a non-ponded s i t e f o r that of a ponded s i t e may be given as approximately 1.5 C, the same e r r o r (ATa ) as assumed when e x t r a p o l a t i n g a i r temperature over l a r g e d i s t a n c e s . 4. ERRORS IN WIND SPEED Using an averaged wind speed over a p e r i o d of one hour should e l i m i n a t e most e r r o r s i n e x t r a p o l a t i n g ' winds from one re g i o n of the survey area to another. Wind speeds at n i g h t are g e n e r a l l y low. Some e r r o r s may remain due to v a r y i n g topographic exposure to the wind. The magnitude of these d i f f e r e n c e s w i l l depend on the s p e c i f i c topography. Over t e r r a i n where topographic exposure may be a problem and wind speed e r r o r s may be high, i t w i l l be assumed t h a t probable e r r o r s i n wind speed fo r any given s i t e w i l l be approximately 0.3 m s ^ (AU ). A low -1 H . probable e r r o r of 0.15 m s i s assumed f o r most s i t u a t i o n s (AU ). E r r o r s due to e x t r a p o l a t i o n over l a r g e d i s t a n c e s may be extreme. Comparison of f i e l d data from rangeland s i t e s to data from a nearby m e t e o r o l o g i c a l s t a t i o n (Kamloops a i r p o r t ) y i e l d e d -238-hour l y average wind speeds c o n t r a s t i n g by 0.0 to 2.5 m s ; most -1 between 0.0 and 1.5 m s 5. SHORT TERM LOCAL ADVECTION Short term l o c a l a d v e c t i v e events w i l l cause s u r f a c e temperature to be u n r e p r e s e n t a t i v e of average long term m i c r o m e t e o r o l o g i c a l c o n d i t i o n s . Thus, the instantaneous s u r f a c e temperature as measured by a thermal i n f r a r e d l i n e scanner w i l l y i e l d net r a d i a t i o n , s e n s i b l e heat f l u x e s , and temperature changes (between f l i g h t s ) which w i l l g i v e erroneous r e s u l t s i f a p p l i e d to thermal i n e r t i a d e t e r m i n a t i o n . Time p e r i o d s f o r such events are d e f i n e d i n t h i s study as being l e s s than twenty minutes and u s u a l l y of the order of, or l e s s than, f i v e to ten minutes. Large s c a l e a d v e c t i o n i s not of concern i n t h i s matter. The thermal i n e r t i a model being i n v e s t i g a t e d i s based on the assumption that there i s no l a r g e f l u c t u a t i o n i n the energy balance components and no major m e t e o r o l o g i c a l events (eg., l a r g e s c a l e a d v e c t i o n or i n f l u x of c l o u d cover) which w i l l cause the time course of ground heat f l u x (G) to change g r e a t l y d u r i n g the p e r i o d of the survey. Such cases w i l l cause s e r i o u s e r r o r i n the model. Watson et a l . (1971) s i m u l a t e the e f f e c t of l a r g e s c a l e a d v e c t i o n events on t h e i r model and show that i t can be l a r g e . The time t r a c e s of a i r temperature and wind speed i n F i g u r e s IV.1 to IV.6 and F i g u r e s 4.3 and 4.4 g i v e some i n d i c a t i o n of the magnitude of changes i n wind speed and a i r temperature with time. The most s t r i k i n g changes occur d u r i n g s h o r t p e r i o d s o f a i r -239-ponding. Changes i n a i r temperature can be severe and r a p i d . S e n s i b l e heat f l u x w i l l o f t e n be zero or upward at the onset of ponding. A severe a d v e c t i v e event for non-ponding c o n d i t i o n s may be represented by a 2 m s ^ change i n wind speed and a 2.0 C change i n a i r temperature. Moderate a d v e c t i v e events may be d e s c r i b e d by changes i n wind speed o f 1 m s ^ and a i r temperature of 1.0 C. F i g u r e IV.3 i s a good example. Short term advection of a l t e r n a t i n g warm and c o o l a i r bodies through the area was p h y s i c a l l y n o t i c e a b l e to the experimenter. Advective events causing small changes i n s u r f a c e temperature may be d e f i n e d as those with wind speed changes of 0.5 m s ^ and a i r temperature changes of 1.0 C. The magnitude of t y p i c a l changes i n s u r f a c e temperature due to l o c a l s h o r t term a d v e c t i v e events i s d e s i r e d . An approximate method of determining these e r r o r s d u r i n g the n i g h t i s f e a s i b l e . Model I (Chapter 2) i s used. A change i n ground heat f l u x equal i n magnitude to that caused by the a d v e c t i v e event i s added to the boundary c o n d i t i o n at time t ^ . The change i n temperature a f t e r a given time due to t h i s impulse i s given by: T s a - T 8 n - i r ^ - ^ l 1 7 2 ( I V - X ) where T s a i s the temperature at time t f o r boundary c o n d i t i o n s c o n t a i n i n g the a d v e c t i v e impulse, Ts i s the temperature at t without the a d v e c t i v e impulse, P i s the thermal i n e r t i a , and Gc i s the change i n ground heat f l u x due to the a d v e c t i v e event. Gc i s approximated by the change i n s e n s i b l e heat f l u x due to the a d v e c t i v e event. Estimates o f s u r f a c e temperature change using t h i s approximate method w i l l be s u f f i c i e n t f o r use i n an e r r o r -240-a n a l y s i s . A t y p i c a l case w i l l be analyzed i n order to g i v e estimates of the magnitude of the s u r f a c e temperature change caused by an a d v e c t i v e event. The t y p i c a l case i s : z =1.0 mm, a wind speed -1 ° of 2.5 m s , a i r - s u r f a c e temperature d i f f e r e n c e s of 3 to 7 C, and P = 2000 TIU. Largest temperature d i f f e r e n c e s between a d v e c t i v e and non-advective c o n d i t i o n s occurred f o r a severe wind speed and temperature i n c r e a s e . Using the above procedure d i f f e r e n c e s , over p e r i o d s of 20 minutes for the case of severe a d v e c t i o n d e s c r i b e d above, can be over 1.0 C. For moderate a d v e c t i v e c o n d i t i o n s (moderate a d v e c t i v e event d e s c r i b e d above) over 20 minutes temperature d i f f e r e n c e s w i l l g e n e r a l l y be l e s s than 0.5 C; f o r the small a d v e c t i o n case d i f f e r e n c e s w i l l be l e s s than 0.25 C. Five-minute a d v e c t i v e event e r r o r s w i l l g e n e r a l l y cause s u r f a c e temperature changes < 0.6 C f o r the severe, < 0.3 C for the moderate, and < 0.2 C for the small a d v e c t i v e event. The f o l l o w i n g are t h e r e f o r e reasonable probable e r r o r s i n the instantaneous scanner -temperature due to s h o r t term l o c a l a d v e c t i v e events : 0.8 C for a severe event (high probable e r r o r , A T s H a ) 0.4 C for a moderate event (moderate probable e r r o r , A T s u ), and 0.2 for a small event (low probable e r r o r , ATs ). Ma La These values are t y p i c a l of bare s u r f a c e s and are supported by the g e n e r a l s t a b i l i t y of bare s u r f a c e temperature readings as measured i n the f i e l d . However, vegetated s u r f a c e s , due to t h e i r low thermal i n e r t i a , w i l l r e a d i l y change temperature as a r e s u l t o f a d v e c t i v e events. T h i s was i l l u s t r a t e d by f i e l d data i n which temperature readings v a r i e d by over 1.0 C depending on wind c o n d i t i o n s . Derksen (1974) r e p o r t s changes i n temperature of 2 C -241-for wind b u r s t s over g r a s s l a n d during the day. 6. SITE DESCRIPTION A l l s i t e s were on n a t u r a l rangeland t e r r a i n of moderate r e l i e f near Kamloops, B r i t i s h Columbia.^" There were three main study areas. Area A was d e s c r i b e d i n Chapter 4 ( s e c t i o n 4.3.1). F i g u r e IV.1 and IV.2 are data from area B. S i t e B - l i s a s p a r s e l y (20 percent cover) vegetated s i t e c e n t r a l to the other s i t e s . S i t e s B-2 and B-3 are s p a r s e l y vegetated s i t e s with some sage brush. S i t e B-3 i s at a lower e l e v a t i o n than the other s i t e s and s i t u a t e d i n a broad v a l l e y . S i t e A-4 i s a s i t e a l s o used f o r F i g u r e s 4.3 and 4.4. I t s cover i s medium (30 to 60 percent cover) d e n s i t y grass s p e c i e s . S i t e s B-3 and A-4 were f u r t h e s t apart (approximately 370 m). Wind speed was measured at s i t e B - l . Study area C had three s i t e s of medium d e n s i t y grass s p e c i e s and sparse to medium d e n s i t y sage brush (C-2, 3, and 4). S i t e C-1 was bare s o i l . I t was a l s o s i t u a t e d i n a d e p r e s s i o n 1 to 8 m lower than other s i t e s . Wind speed was measured at s i t e C - l . A d i s t a n c e of 250 m separated the s i t e s f u r t h e s t apart (C-2 and C-4) . Group D (Table IV.2) contained a h i l l s i d e s i t e (D-1) of medium d e n s i t y grass and a v a l l e y bottom s i t e (D-2) of dense t a l l g r a s s . . E l e v a t i o n d i f f e r e n c e was about 15 m; d i s t a n c e apart -242-approximately 80 m. Group E (Table IV.2) had a s i t e of medium d e n s i t y v e g e t a t i o n on a k n o l l ( E - l ) , a s i t e of dense v e g e t a t i o n on a f l a t l a n d area (E-2) and a s i t e (E-3) o f l u s h dense v e g e t a t i o n i n a hollow. E l e v a t i o n d i f f e r e n c e s between E - l and E-2, and E-2 and E-3 were about 2 to 3 meters and s e p a r a t i o n between them about 20 to 40 m. The Kamloops a i r p o r t i s s i t u a t e d i n a r i v e r v a l l e y at an e l e v a t i o n 230 to 400 m below and a d i s t a n c e of 3 to 6 km from the study areas. M i c r o m e t e o r o l o g i c a l c o n d i t i o n s are expected to be q u i t e d i f f e r e n t from the rangeland s i t e s . Experimental s i t e s were on the Lac du B o i s rangeland, Kamloops, B.C. V e g e t a t i o n was n a t u r a l grass s p e c i e s and some areas of sage brush ( a r t e m i s i a t r i d e n t a t a ) . The predominant grass s p e c i e s were bluebunch wheatgrass (agropyron spicatum) and needle-and-thread ( s t i p a  comata). Kentucky b l u e g r a s s (poa p r a t e n i s ) was dominant i n topographic d e p r e s s i o n s . L o c a l changes i n e l e v a t i o n were up to 100 m. The v e g e t a t i o n and t e r r a i n was f u r t h e r d e s c r i b e d by Watson (1977). The c l i m a t e , v e g e t a t i o n , and s o i l s of the area were d i s c u s s e d by van Ryswyk et a l . (1966). \ -243-LITERATURE CITED Anstey, T.H., G.M. Weiss, A.W. Watt, J.C. Wilcox, and P.N. Sprout. 1959. R e l a t i o n of s o i l , temperature and topography to f r u i t growing i n Summer land, B r i t i s h Columbia. Can. J . P l a n t Science 39:297-315. Bennett, R.C. 1977. Use of thermal imagery from an ai r b o r n e scanning radiometer to d e r i v e the d i s t r i b u t i o n of s u r f a c e temperature. S i x t h B r i t i s h Columbia S o i l S cience Workshop Report. B r i t i s h Columbia M i n i s t r y of A g r i c u l t u r e , V i c t o r i a , B.C. pp. 49-54. Derksen, W.J. 1974. Thermal i n f r a r e d p i c t u r e s and the mapping of m i c r o c l i m a t e . Neth. J . A g r i c . S c i e n c e 22:119-132. Hovecar, A., and J.D. M a r t s o l f . 1971. Temperature d i s t r i b u t i o n s under r a d i a n t f r o s t c o n d i t i o n s i n a c e n t r a l P e nnsylvania v a l l e y . A g r i c . M e t e o r o l . 8:371-383. Lawrence, E.N. 1958. Temperature and topography on r a d i a t i o n n i g h t s . M e t e o r o l . Mag. 87:71-78. Murtha, P.A. 1971. F r o s t pockets on thermal imagery. F o r e s t r y C h r o n i c l e 47(2):79-81. Watson, E.K. 1977. A remote sensing based m u l t i l e v e l rangeland c l a s s i f i c a t i o n f o r the Lac-du-Bois rangelands, Kamloops, B r i t i s h Columbia. Masters T h e s i s , Dept.of S o i l S cience, U n i v e r s i t y o f B r i t i s h Columbia, 85 p. Watson, K., L.C. Rowan, and T.W. O f f i e l d . 1971. A p p l i c a t i o n of thermal m o d e l l i n g i n the g e o l o g i c i n t e r p r e t a t i o n of IR images. Proc. 7th I n t ' l Symp. on Rem. Sens, of Env., Ann Arbor, MI. pp.2017-2041. van Ryswyck, A.L., A. McLean, and L.S. Marchand. 1966. The c l i m a t e , n a t i v e v e g e t a t i o n and s o i l s of some g r a s s l a n d s at d i f f e r e n t e l e v a t i o n s i n B r i t i s h Columbia. Can. J . P l a n t S c i ence 46:35-50. -244-Appendix V RESULTS OF ERROR ANALYSIS OF THERMAL INERTIA METHOD Tc W m"2 (C) Case C o n t r i b u t i o n to R 2 due to probable e r r o r T o t a l E r r o r Tc Model R/P 500 20 2.33 1 2 3 .73 .63 .73 24 34 ,24 ,03 ,03 .03 950 590 400 1.92 1.18 .79 40 4.67 1 .68 .23 .09 500 .99 2 .58 .31 .11 330 .62 3 .66 .21 .13 210 .42 60 7.00 1 .61 .20 .19 350 .70 2 .52 .28 .24 220 .44 3 .57 .18 .25 150 .30 80 9.34 1 .54 .18 .29 280 .56 2 .44 .24 .32 180 .35 3 .47 .16 .37 120 .25 100 11.7 1 .46 .15 .39 240 .48 2 .30 .20 .43 150 .31 3 .39 .13 .48 108 .22 20 .78 1 .25 .74 .01 4230 3.28 2 .17 .82 .01 3400 2.27 3 .25 .74 .01 2020 1.35 40 1.56 1 .24 .72 .03 2490 1.66 2 .17 .80 .03 1720 1.15 3 .24 .71 .05 1030 .69 1500 60 2.33 1 .23 2 .16 3 .23 .70 .77 .67 ,07 ,07 ,10 1700 1170 710 1.13 .78 .47 80 3.11 1 .22 2 .15 3 .21 .66 .74 .62 12 11 ,17 1310 900 550 87 ,60 ,37 100 3.89 1 .21 2 .14 3 .19 .62 .69 .57 17 16 ,24 1080 740 460 72 49 ,31 (continued) -245-Appendix V (continued) P G W m~2 Tc (C) Case C o n t r i b u t i o n probable to R 2 due to e r r o r T o t a l E r r o r G TC Model R R/P 2500 20 .47 1 .11 .89 .00 12500 5.00 2 .07 .93 .00 8900 3.56 3 .11 .89 .00 5120 2.05 40 .93 1 .11 .88 .01 6280 2.51 2 .07 .92 .01 4470 1.79 3 .11 .87 .02 2580 1.03 60 1.40 1 .11 .86 .03 4230 1.69 2 .07 .90 .03 3000 1.20 3 .10 .85 .05 1740 .70 80 1.87 1 .10 .84 .06 3210 1.28 2 .07 .88 .05 2280 .91 3 .10 .82 .08 1330 .53 100 2. 33 1 .10 .82 .08 2610 1.04 2 .07 .87 .07 1850 .74 3 .10 <.79 .12 1090 .44 3500 20 .33 1 .06 .94 .00 23820 6.81 2 .04 .96 .00 17130 4.89 3 .06 .94 .00 9750 2.79 40 .67 1 .06 .94 .01 11940 3.41 2 .04 .96 .01 8580 2.45 3 .06 .93 .01 4900 1.40 60 1.00 1 .06 .93 .02 8000 2.29 2 .04 .95 .02 5750 1.64 3 .06 .92 .03 3290 .94 80 1.33 1 .06 .91 .03 6040 1.73 2 .04 .94 .03 4340 1.24 3 .06 .90 .04 2490 .71 100 1.67 1 .06 .90 .05 4870 1. 39 2 .04 .92 .04 3490 1.00 3 .06 .88 .07 2020 .58 120 2.00 1 .05 .88 .07 4100 1.17 2 .04 .91 .06 2940 .84 3 .05 .85 .10 1710 .49 (continued) -246-Appendix V (continued) D e f i n i t i o n s P = thermal i n e r t i a ( j m~ 2C~ 1s~ 1 /' 2) = ground heat f l u x at f i r s t sample time Tc = temperature change, between the two sample times A i n d i c a t e s probable e r r o r case 1 = worst (poor) case: AG 1 = 18.7 W m ATc = 2 Amodel = 0.30 P .2 C case 2 = t y p i c a l case : G = 10.7 W m"2 ; ATc = 1.6 Amodel = 0.20 P 1 C case 3 = good case: AG X = 7.7 W m~2 ; ATc = 0.9 C Amodel = 0.15 P R = probable e r r o r ; the probable e r r o r i n P where P f u n c t i o n of q ^ q 2 , . . . q n i s given by: i s a r X 2 ,-/2 i=l where r. = « Aq. i 9 q i M i R/P = r e l a t i v e e r r o r 2 2 C o n t r i b u t i o n to R 2 due to hq^ i s given by r±/K r e p r e s e n t s a measure of the c o n t r i b u t i o n of the e r r o r each parameter i s R. and i n 

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