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The analysis of wet snow effects on earth-space transmission systems Hulays, Rafeh Ahmad 1998

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T H E ANALYSIS OF W E T SNOW EFFECTS ON EARTH-SPACE TRANSMISSION SYSTEMS  by RAFEH AHMAD HULAYS M A . S c . (EE), University of British Columbia, 1992 B.Sc. (EE), Monmouth College, 1988  A T H E S I S IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY  in  T H E F A C U L T Y O F G R A D U A T E STUDIES THE DEPARTMENT OF ELECTRICAL ENGINEERING  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A June, 1998  © Rafeh Ahmad Hulays, 1998  In presenting this thesis in partial fulfilment  of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  F/<^4Wcx^^  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  f«£r 7v  Co—(><*M*r  f^^«^'t)  Abstract  Abstract: In this thesis, the effect of precipitation, including the melting layer, on wireless communication systems is examined. A melting layer model is developed which is based upon the most reliable data, available from radar and direct observations. The dominant factor in determining the level of reflectivity and attenuation in the melting layer is found to be the initial-snow density; its average and bounding limits are derived in this work. Several meteorological models (Awaka, Capsoni and ITIJ-R models) are used to develop interference models. These models are modified to include the effect of the melting layer. The results show that while the melting layer can on occasion affect interference, its role overall is small and offers little improvement in interference prediction. A notable exception is at the lower frequencies (f = 4 GHz) where melting layer enhancement of interference is quite noticeable. It is also concluded that it is unlikely the melting layer will be included into standard interference prediction techniques because of the high computation cost it entitles. While the effect of the melting layer on interference prediction is found to be minimal, the opposite is true for the ice/snow region situated above the melting layer, especially at high frequencies. Variations in modelling this region yielded substantial variations, in the predicted interference levels.  University of British Columbia  ii  Electrical Engineering  Abstract  Interference effect is also examined on low-gain fixed and mobile terminals. This effect is found to be debilitating for reliable communication. This makes frequency sharing between services a risky proposition. The Awaka meteorological model is also used as a basis to develop an attenuation model for satellite links that include the effect of the melting layer. Several links, which are part of the Advanced Communications Technology Satellite (ACTS) program, are used to study the effect of the melting layer on total attenuation. It is shown that, while the melting layer's effect is small at low latitudes, the opposite is true at high latitudes. Simplified models for specific, total and excess attenuation in the melting layer are developed and simple procedures to use these models in conjunction with traditional rainonly attenuation models are presented. These were shown to agree well with predictions, using the rigorous approach. This study is relevant for the new generation of satellite networks geared toward the delivery of low-cost bandwidth to small businesses and homes.  University of British Columbia  in  Electrical Engineering  Table of Contents  Table of Contents ABSTRACT......  .' H  TABLE OF CONTENTS  IV  LIST OF FIGURES  XI  LIST OF TABLES  XX  LIST OF SYMBOLS  XXII  ACKNOWLEDGMENTS  XXVIII  CHAPTER 1: INTRODUCTION 1.1  1  INTERFERENCE  2  /. / . /  I n t e r f e r e n c e mechanisms  2  /. 1.2  I n t e r f e r e n c e due to h y d r o m e t e o r scatter  4  1.2  FADING  .•  1.2.1  M echanisms  1.2.2  Excess  6  of attenuation  '.  a t t e n u a t i o n due to m e l t i n g l a y e r .  7 9  1.3  MODELING THE MELTING LAYER  10  1.4  S U M M A R Y O F THESIS O B J E C T I V E S  11  1.5  THESIS O U T L I N E  11  CHAPTER 2: INTERFERENCE MODELING 2.1  14  INTERFERENCE THEORY  14  2. /. /  Radar equation  14  2.1.2  Simplified radar equation  16  2.2  W I D E - B E A M A N T E N N A RADIATION PATTERN  17  2.3  G E O M E T R I C A L C O N S IDERATIONS  20  2.4  METEOROLOGICAL PARAMETERS  21  2.4.1  General rain-cell structure  2.4.2  Vertical structure of precipitation  2.5  ;  23  ATTENUATION AND SCATTERING MODELING OF HYDROMETEORS  2.5.1 2.6  The w e i g h e d i n t e r p o l a t i o n model STRUCTURE OF SIMULATION  2.6.1  24  25 !  C o n s t a n t i n t e g r a t i o n steps  University of British Columbia  2/  26  26  iv  Electrical Engineering  Table of Contents  2.6.2  Flexible integration steps  26  2.6.2.1  Region I: rain region  26  2.6.2.2  Region II: melting-snow region  29  2.6.2.3  Regions III and IV: ice and snow regions  29  2.6.3  29  COST 210 Integration  2.7  I M P O R T A N T F E A T U R E S IN T H E S I M U L A T I O N P R O G R A M  C H A P T E R 3: INTERFERENCE 3.1  COST 210/ITU  MODELS  32  INTERFERENCE MODEL  [4,5]  3.1.1  Rain-cell structure  3.1.2  Modified rain-cell structure  3.1.3  Procedure for obtaining the CDF of transmission loss Case 1: no melting layer  3.1.3.2  Case 2: with melting layer present  3.2  .....33  33  3.1.3.1  3.1.4  31  35 36  :  36 37  Software CAPSONI'S  3D  38  INTERFERENCE MODEL  [5,6,7]  3.2.1  Rain-cell structure  3.2.2  Procedure for obtaining the CDF of transmission loss  40  40 42  3.2.2.1  Case 1: no melting layer  42  3.2.2.2  Case 2: with melting layer present  46  3.2.3. Software..  47  3.3  C A P S O N I ' S 3D  3.4  A W A K A - M 0 D I F 1 E D 3D  3.5  COMMENTS ON THE INTERFERENCE MODELS  C H A P T E R 4:  INTERFERENCE M O D E L WITH A GAUSSIAN RAIN-RATE DISTRIBUTION I N T E R F E R E N C E M O D E L [8]  48 49  5 1  SPATIAL STRUCTURE OF PRECIPITATION  54  4.1  PRECIPITATION TYPES  54  4.2  CLIMATIC REGIONS  55  4.3  HORIZONTAL STRUCTURE OF PRECIPITATION  56  4.4  4.3.1  Rain rate statistics  56  4.3.2  Radar rain maps  57  4.3.3  Simulated rain maps  58  V E R T I C A L STRUCTURE OF PRECIPITATION  59  C H A P T E R 5: MODELING PRECIPITATION: MELTING-SNOW LAYER  64  5.1  MELTING PARTICLES S HAPES  66  5.2  MODELING THE MELTING PARTICLES  67  5.2.1  Concentric sphere representation  67  5.2.2  Composite sphere representation  69  University of British C o l u m b i a  v  Electrical Engineering  Table of Contents  5.2.3  Spongy particle model  69  5.3  M E L T I N G PROCESS  72  5.4  T H E F A L L SPEED OF M E L T I N G PARTICLES  75  5.5  P A R T I C L E - S I Z E DISTRIBUTION  80  5.6  I N I T I A L - S N O W D E N S I T Y A N D P E A K R E F L E C T I V I T Y IN T H E B R I G H T B A N D  83  5.6.1  Radar Observations of Bright Band Peak Reflectivities  5.6.2  Derivation of Initial-snow Densities  5.6.3  Initial-Snow Densities and Discussion  85 88 91  5.7  THICKNESS OF T H E M E L T I N G L A Y E R  94  5.8  HEIGHT OF THE MELTING L A Y E R  99  5.9  P E R C E N T A G E O F T I M E T H E M E L T I N G L A Y E R IS P R E S E N T  100  5.10  D U R A T I O N ( H O R I Z O N T A L DIMENSIONS) O F T H E B R I G H T B A N D  103  5.11  FREQUENCY DEPENDENCE OF BRIGHT B A N D REFLECTIVITY AND ATTENUATION  103  CHAPTER 6: ATTENUATION  AND  SCATTERING  PROPERTIES  OF PRECIPITATION  Ill  6.1  INTRODUCTION  111  6.2  M E S C A T T E R I N G M E T H O D FOR SPHERES A N D C O N C E N T R I C SPHERE  112  6.2. J  Scattering theory  112  6.2.2  Simulation  6.2.3  Concentric sphere representation  114  6.2.4  Composite sphere representation  115  6.2.5  Spongy particle model  116  6.2.6  Comparison of different models  117  114  6.3  A R T I F I C I A L DIELECTRIC M O D E L FOR A T T E N U A T I O N [89]  118  6.4  A R T I F I C I A L D I E L E C T R I C M O D E L FOR S C A T T E R I N G [77]  120  6.5  E X T E N S I O N O F T H E POWER L A W (AR ) M O D E L T O A T T E N U A T I O N IN T H E M E L T I N G L A Y E R  122  6.6  S C A T T E R I N G A N D A T T E N U A T I O N PROPERTIES A B O V E T H E M E L T I N G L A Y E R  123  6.7  OBSERVATIONS  124  B  CHAPTER 7: APPLICATION  TO MICROWAVE  127  LINKS  7. /  EVALUATION OF THE WEIGHED INTERPOLATION APPROACH  127  7.2  EVALUATION OF MODELS WITH RESPECT TO EXPERIMENTS  130  7.3  STATISTICAL STUDY OF THE CHARACTERISTICS OF INTERFERENCE  135  / 35  7.3.1  Transmission loss dependence on melting layer presence  7.3.2  Rain height variation versus median rain height  137  7.3.3  Transmission loss dependence on rain height  /40  7.3.4  Transmission loss dependence on modeling the ice/snow region  7.3.5  Transmission loss dependence on frequency  University of British Columbia  vi  / 43 / 45  Electrical Engineering  Table of Contenis  7.3.6  P o l a r i z a t i o n effect  7.3.7  Effect  on transmission  o f the i n i t i a l - s n o w density  7.4  INTERFERENCE  7.5  DISCUSSION AND CONCLUSION  loss on interference  SENSITIVITY TO ATTENUATION  C H A P T E R 8: INTERFERENCE  147 148  AND SCATTERING  148 149  ON MOBILE SATELLITE LINKS DUE TO HYDROMETEOR  SCATTER  153  8.1  DESCRIPTION  154  8.2  SIMULATIONS  154  8.3  DISCUSSION AND CONCLUSION  159  C H A P T E R 9: EFFECT OF THE MELTING LAYER ON FADING IN SATELLITE COMMUNICATIONS  166  9.1  THE POWER-LAW (AR ) DATA FITTING PROCEDURE B  166  9.2  REFLECTIVITY AND ATTENUATION IN RAIN  167  9.3  TOTAL ATTENUATION IN THE MELTING LAYER  168  9.3. J  Method ofComputation  9.3.2'  a R r e p r e s e n t a t i o n o f total  9.3.3  Excess  9.3.4  Numerical Simulation  9.4  b  168 attenuation  169  Attenuation  172 172  SIMULATION OF ATTENUATION ON AN EARTH-SPACE LINK USING A MODIFIED AWAKA RAIN-  CELL MODEL  175  9.4.1  M e t h o d o f computation  175  9.4.2  Numerical Simulations  176  9.5  SIMPLE TECHNIQUES TO INCLUDE MELTING LAYER EFFECTS ON EXISTING RAIN-ATTENUATION  MODELS  182  9.5.1  M e t h o d II: simplified  9.5.2  M e t h o d I: simplified  procedure  182  procedure  / 82  C H A P T E R 10: DISCUSSION AND CONCLUSION 10.1  185  MELTING LAYER MODELING  186  10.1.1  D e r i v a t i o n o f an i n i t i a l - s n o w density  relationship  10.1.2  A new r e l a t i o n s h i p between  coefficient  10.1.3  A revised  10.1.4  The a R specific  10.1.5  G e n e r a l remarks  10.2  melting b  INTERFERENCE  10.2.1  the drag  University of British Columbia  o f snow  a n d i n i t i a l - s n o w density  l a y e r thickness a t t e n u a t i o n model  186 186  extension  to the melting  layer  187 187  MODELING  New interference  / 86  187  models  187  vii  Electrical Engineering  Table of Contents  10.2.2  Influence of the melting layer on interference prediction  10.2.3  Influence of the ice/snow region on interference prediction  10.2.4  General remarks  10.2.5  Interference effects on mobile terminal  10.3  ATTENUATION  188 189 189 189 190  MODELING  new attenuation model  10.3.1  A  10.3.2  Influence of the melting layer on attenuation for satellite communications  10.3.3  Simplified attenuation models  10.3.3.1  The aR  10.3.3.2  Simplified procedures to add  10.4  CONCLUDING  b  190 190 190  total attenuation formulation  191  melting layer effect into rain-only attenuation model statistics ..191 191  COMMENTS  CHAPTER 11: RECOMMENDATIONS FOR FUTURE WORK 11.1  192 192  MELTING LAYER MODELING 11.1.1  A novel melting-layer reflectivity and attenuation models  //. 1.2  Initial-snow density derivation (Reflectivity vs Scattering cross section)  11.1.3  Water content above the melting layer  11.1.4  Duration statistics of the melting layer  11.2  194 194  / 94  INTERFERENCE MODELING  195  11.2.1  Attenuation and Scattering models  11.2.2  Ice/snow region modeling  11.2.3  Radar simulations  11.2.4  A simplified interference model  11.3  /92  195 195 195 195  FADE MODELING  •  196  Further improvements on the simplified procedures to add the effect of the melting layer into  11.3.1  rain-only attenuation model statistics  / 96  11.3.2  Radar simulations  11.3.3  E x p e r i m e n t a l v e r i f i c a t i o n of the f a d i n g m o d e l  11.4  CONCLUDING  196 / 96  REMARKS  197  REFERENCES  198  APPENDIX A -.^INTERFERENCE MODELS A. 1  SOFTWARE AND HARDWARE REQUIREMENTS  209 '•  210  A. 1.1  Software requirements  210  A.1.2  Hardware requirements  210  A. 1.2.1  ITU-R program  210  A.1.2.2  Awaka program  210  A. 1.2.3  Capsoni program  211  University of British Columbia  viii  Electrical Engineering  Table of Contents  RUNNING THE INTERFERENCE PROGRAMS  A.2  A.l.I  211  Program inputs  211  A.2.1.1  paramet  A.2.1.2  ml_pr  A.2.1.3  hfr_dh  212  A.2.1.4  hydro  213  A.2.1.5  rain_pr  213  A.2.2  211 :  212  Program outputs  214  A.2.2.1  interm  214  A. 2.2.2  CDF.m  214  A.2.3 A. 3  Compiling and running the models  215  PROGRAM LISTINGS  216  A.3.1  3D (Awaka, Capsoni) model (ddd_mod.f)  216  A.3.2  ITU-R model (itu_mod.f)  A.3.3  Awaka driver program (drive_a.f)  216  A.3.4  Capsoni driver program (drive_c.f)  216  A. 3.5  ITU-R driver program (drive_i.f)  216  2 /6  APPENDIX B: SCATTERING AND ATTENUATION MODELS AND PROGRAMS B. l  M E SCATTERING METHOD FOR SPHERES AND CONCENTRIC SPHERE  B. l.l  Scattering theory  217 217  217  B.l.2  Simulation  220  B.l.3  Software  22/  B. l.3.1  Software and Hardware Requirements  223  B.1.3.1.1  Software requirements  223  B.1.3.1.2  Hardware requirements  223  B.l.3.2  Running the Mie Scattering Programs  B.l.3.2.1  223  Program inputs  223  B.l.3.2.1.1  frequ  223  B.l.3.2.1.2  hydro  224  B. 1.3.2.2  Program output  225  B.1.3.2.3  Compiling and running the models  225  B. 1.3.3  Generating the Precipitation Attenuation and Scattering Characteristics  225  B.l.3.3.1  Inputs  226  B.l.3.3.2  Outputs  226  B.1.3.3.3  Compiling and running the program  226  B.l.3.4  Program Listings  B.l.3.4.1  mie_co.f  B.l.3.4.2  mie_cp.f  B.l.3.4.3  mie_sp.f  University of British Columbia  227 227 •.  227 227  ix  Electrical Engineering  Table of Contents  B.1.3.4.4  comp.f  227  B.1.3.4.5  mp.f  227  B.1.3.4.6  atten.f  227  B.l.3.5  Input file Listings  228  ARTIFICIAL DIELECTRIC MODEL FOR ATTENUATION [26]  B.2  228  B.2.1  Artificial dielectric model for melting layer attenuation (ADM 1)  229  B.2.2  Artificial dielectric model for melting layer attenuation (ADM 2)  230  ARTIFICIAL DIELECTRIC MODEL FOR SCATTERING [77]  B. 3  A P P E N D I X C : COST210  231  + MODIFICATIONS LINKS SIMULATIONS  234  C. l  C O S T 210 EXPERIMENTAL LINKS SIMULATIONS  234  C. 2  OTHER SIMULATIONS  235  A P P E N D I X D: ATTENUATION MODEL D. 1  253  SOFTWARE AND HARDWARE REQUIREMENTS  253  D.I.I  Software requirements  254  D.l.2  Hardware requirements  254  D.2  RUNNING THE ATTENUATION PROGRAM  D.2.1  Program inputs  D.2.1.1 D.2.1.2  General parameters Meteorological  D.2. i .3  Mie scattering and attenuation  D.2.2  254 file files  254 255 files  Program outputs  255  255  D.2.2.1  interm  255  D.2.2.2  CDF.m  255  D.2.3 D.3  254  Compiling and running the models  PROGRAM LISTINGS  256 257  D.3.1  driver.f  257  D.3.2  fade.f  257  University of British C o l u m b i a  x  Electrical Engineering  List of Figures  List of Figures Figure 1-1:  Long term interference mechanisms [5]  3  Figure 1-2:  Short term (enhanced) propagation mechanisms [5]  4  Figure 1-3:  Paths of possible interference (Solid line: wanted paths, dashed lines: unwanted paths) [5]  5  Figure 1-4:  Occurrence of large melting layer attenuation associated with high melting layer reflectivity, 29 July, 1980. Dotted curve shows calculated attenuation if melting layer treated as rain up to the nose of bright band [25]  10  Figure 2-1:  General interference geometry.  15  Figure 2-2:  The radiation pattern of the wide-beam antenna showing the main lobe, side lobe and the normalized sum of the two.  19  Figure 2-3:  The figure contrasts the size and rain distribution of rain rate in the rain cell for the ITU-R (dashed), gaussian (dotted), and the exponential (solid) rain-cell model for R = 6.0 mm/h and R = 12.5 mm/h. For the gaussian and the exponential rain-cells, we use the mean roll-off rate r .  22  Figure 2-4:  The following figure shows how rain maps of arbitrary shape and distribution could be modeled as the one (fictitious) cylindrical cell required by this model. The radius of the fictitious cell is chosen so that all of the relevant rain is in the cell. The numbers in the figure indicate rain rates in mm/h.  23  Figure 2-5:  Vertical attenuation and reflectivity profiles in a rain cell.  24  Figure 2-6:  Rain-cell showing the four different integration regions.  28  Figure 3-1:  Rain rate, attenuation and scattering profiles in both the ITU-R raincell and the Capsoni-exponentially decaying rain-cell.  34  Figure 3-2:  ITU-R rain cell vs. modified ITU-R rain cell.  35  Figure 3-3:  Flow-chart diagram for the modified ITU-R interference model driver program.  38  Figure 3-4:  Comparison between fitted (solid) and original (dashed)CDFs of rain rate.  44  Figure 3-5:  Domain of mobility of a rain-cell divided into a grid of squares. Interference calculations are done with the rain-cell centered in each square. While cell 1 will definitely cause interference, cell 2 may, under the right conditions, cause interference.  45  M  M  0  Figure 3-6:  Flow-chart diagram for the modified Capsoni interference model  University of British Columbia  xi  48  Electrical Engineering  List of Figures  driver program. Figure 3-7:  Diameters of the rain cells with exponential and gaussian rain rate distributions, assuming a rain-rate threshold, at the edge of the cell, /?, . = 0.4 mm/h.  Figure 3-8:  The Capsoni-derived rate of decay of rain rate (solid) is contrasted to the Awaka-suggested rate (dashed).  Figure 4-1:  Reflectivity field (dBz) from a tropical cyclone observed with a Cband radar on Taiwan. The coastline is shown [42].  Figure 4-2:  (a) Average normalized profiles as a function of slant distance and relative to the three maximum ranges indicated, (b) Same as (a) but relative to the standard deviation [35].  60  Figure 4-3:  Two views of the radar bright band: at the left a vertical profile of reflectivity and Doppler velocity as measured with vertically pointing radar; at right a PPI map at 8° elevation on which the melting layer appears as a bright ring at about 12 miles [53],  62  Figure 4-4:  Relative occurrence of the 5 classes of vertical reflectivity profiles stratified by the reflectivity of rain near the surface. The number of hours a given reflectivity (±1 dB) was observed in Montreal over the vertically pointing radar is shown at the top [9].  63  Figure 5-1:  Calculated drop shapes of 13 water drops. The numbers indicate the equivolumetric sphere radius [54].  Figure 5-2:  Melting-snow particle representations: (a) concentric sphere representation (b) Composite sphere representation (c) percolating composite sphere representation (spongy particle model)  68  Figure 5-3:  Backscattering and attenuation profiles in the melting layer contrasting the different levels obtained using the percolating and non-percolating particle representations, (f = 20.0 GHz and R = 12.5 mm/h.) [77].  72  Figure 5-4:  Different melting profiles in the melting layer  73  Figure 5-5:  Reflectivity profile of the melting layer for the concentric (solid) and the percolating composite sphere model (dotted) with p = 0.1 g/cm,/ = 9.344 GHz, R = 12.5 mm/h. and for S,, S , S profiles.  50  51 58  66  74  3  s  2  3  Figure 5-6:  McGill's observed averaged reflectivity profiles of the bright band [9].  Figure 5-7:  Relation between snowflake fall velocity (V ) and the ratio of snowflake mass to snowflake cross-sectional area (M/S) obtained at Nagaoka city prefecture, Japan ...  76  Figure 5-8:  Drag coefficient of snowflakes as a function of snowflake density...  77  T  Figure 5-9* 6  '  75  79  Terminal velocity of snowflakes as a function of snowflake density for the different drag coefficient curves displayed in figure 5-8 are contrasted to equ. 5.21 (o).  University of British Columbia  xii  Electrical Engineering  List of Figures  Figure 5-10:  Effect of velocity profiles on specific attenuation in the melting layer (Kharadly's attenuation model) [81].  80  Figure 5-11:  Comparison between Marshall and Palmer negative exponential function (equation 5.27) and their measurements and those of Laws and Parsons [65].  82  Figure 5-12:  Scatter plot of bright band peak reflectivity Z j (crosses) and Snow reflectivity just above the bright band Z versus rain reflectivity just below the bright band Z^ using data from the wind profiler ...  84  e(peak  e(snow)  ram)  Figure 5-13:  Attenuation profile of the melting layer with S =(h/H) , p = 0.05, 0.1, 0.2, 0.3, 0.7 g/cm , f = 20.0 and 30.0 GHz using the concentric sphere model. 2  2  85  s  3  Figure 5-14:  Figure 5-15:  Figure 5-16:  Average reflectivity enhancement in the melting layer is plotted against rain reflectivity Delft radar (x), Chilbolton radar interpolation (+), Cost 205 proposed average, Goddard proposed average, ... Peak reflectivities in the melting layer are plotted against rain reflectivity for McGill's Noaa (partial) data set (+), Chilbolton data set (o) and Delft data set (x) ...  89  90  Radar averaging effect is seen for a melting layer at/= 0.9 GHz and R = 0.25 mm/h. Reflectivity profile of the melting layer is shown in solid for an initial-snow density of 0.1 g/cm while the effect of averaging is seen for radar resolutions of 30 meters (x), 60 meters (+) and 100 meters (o) ... 3  Figure 5-17:  Reflectivity increase versus initial particle density for our model (solid) is compared to those derived by Russchenberg (—) and D'Amico (-.-.-) for rain rates of 1 mm/h. (+), 5 mm/h. (o) and 20 mm/h. (x).  92  Figure 5-18:  Derived (average and bounding) effective initial-snow densities versus rain rate for both the spongy-sphere representation (solid) and the concentric-sphere representation (dashed).  93  Figure 5-19:  A comparison between McGill's bright band thickness for 20, 50 and 80 % exceedance, A R C data (o) and Klassen's bright band thickness.  96  Figure 5-20:  Schematic drawing illustrating how bright band top, bottom and thickness are extracted [9].  97  Figure 5-21:  Ratio of melting layer to bright band thickness {l/f(p )J as a function of initial-snow density.  98  s  Figure 5-22:  Vertical reflectivity profiles at 9.344 GHz for different rain rates using the spongy sphere model with the S melting profile.  99  Vertical reflectivity profiles at 9.344 GHz for different rain rates using  100  2  Figure 5-23:  University of British Columbia  xiii  Electrical Engineering  List of Figures  the concentric sphere model with the S melting profile. 3  Figure 5-24:  Rain height distribution relative to its median value [5].  101  Figure 5-25:  Simulated statistics of the presence of the melting layer (Equation 5.38).  102  Figure 5-26:  Class presence percentage versus rain rate. Equation 5.40 (dotted) and 3.42 (solid) are compared to observed values (+)•••  104  Figure 5-27:  Vertical reflectivity (a) and attenuation (b) profiles at 10.0 GHz for various rain rates using the concentric sphere model with the S melting profile ( ) and the spongy sphere model with the S melting profile (—).  106  Vertical reflectivity (a) and attenuation (b) profiles at 20.0 GHz for various rain rates using the concentric sphere model with the S melting profile ( ) and the spongy sphere model with the S melting profile (—).  107  Vertical reflectivity (a) and attenuation (b) profiles at 40.0 GHz for various rain rates using the concentric sphere model with the 5, melting profile ( ) and the spongy sphere model with the S melting profile (—).  108  Vertical reflectivity (a) and attenuation (b) profiles at 60.0 GHz for various rain rates using the concentric sphere model with the Sj melting profile ( ) and the spongy sphere model with the S melting profile (—).  109  Vertical reflectivity (a) and attenuation (b) profiles at 100.0 GHz for various rain rates using the concentric sphere model with the S, melting profile ( ) and the spongy sphere model with the S melting profile (-—).  110  (a) Concentric sphere with inner radius a and an outer radius a . (b) One-sphere representation obtained by either setting a to zero or a, =a  113  Figure 6-2:  Snow permittivity as a function of initial snow density, (a) spheroidal air inclusions in an ice matrix ( ), (b) spherical air inclusions in an ice matrix ( ), (c) spheroidal ice inclusions in an air matrix (—) and (d) spherical ice inclusions in an air matrix (•-•-).  115  Figure 6-3:  Attenuation and backscattering profiles of the melting layer for various melting particle models (A, B l , B2, C l , C2), with p, = 0.1 g/cm\/= 20.0 GHz, R = 5.0 and S = (h/H) .  117  Figure 6-4:  Rain-induced attenuation. The artificial dielectric model are in solid while the Mie scattering values are crossed [26].  118  Figure 6-5:  Melting layer-induced attenuation . The 1st artificial dielectric model without correction factor (—), with the correction factor ( ), the 2nd model without correction factor ( ) and with the correction  119  3  2  Figure 5-28:  3  2  Figure 5-29:  2  Figure 5-30:  2  Figure 5-31:  2  Figure 6-1:  s  m  s  m  2  University of British Columbia  xiv  Electrical Engineering  List of Figures  factor (-.-.-) are compared to the Mie scattering results (+). Figure 6-6:  A D M 1 attenuation model with the correction factor for p = 0.1 ( ), 0.2 (—), and 0.3 g/cm ( ) are compared to Mie scattering results s  119  3  .. (+)• Figure 6-7:  Scattering geometry of a rain particle due to an incident electromagnetic wave.  120  Figure 6-8:  Rain-induced backscatter. The artificial dielectric model is shown as a solid curve while the Mie scattering values are crossed [85].  1 2 1  Figure 6-9:  Melting layer-induced backscatter. The Artificial dielectric models are compared to Mie scattering computations (+) for different initial-snow densities.  121  Figure 6-10:  A comparison of the attenuation profile of the melting layer between the extended kR (see equation 6.12) model (solid) and Mie scattering computations (crosses) for^= 10 GHz.  126  Figure 7-1:  Summary of the deviation of transmission loss (in dB) from original ITU-R model for different frequencies and for all links.  129  Figure 7-2:  Average deviation from experimental values for (a) short links (Gz, L1-L8, D1-D6), (b) long links (Bf, Bb, Ff, Fb) and (c) all links (Gz, L1-L8, D1-D6, S1-S5, Bf, Bb, Ff, Fb) for 0.001, 0.01 0.1 and 1.0% of times ...  133  Figure 7-3:  The differences in dB between the Cost 210 predicted interference levels and those predicted in this work for all links at 0.001, 0.01, 0.1 and 1 % of time.  134  Figure 7-4:  This figure contrasts interference levels simulated, with the modified Awaka model for the B f link, using the full rain-height distribution and the median rain height.  139  Figure 7-5:  This figure contrasts interference levels simulated, with the modified Awaka model for the B f link (h = 3.0 km,/= 4.0 km), using the full rain-height distribution and the median rain height.  139  Figure 7-6:  Transmission loss levels are generated as a function of median rain height for the Bf link (f= 4.0 GHz).  141  Figure 7-7:  Transmission loss levels are generated as a function of median rain height for the B f link (/= 11.2 GHz).  141  Figure 7-8:  Transmission loss levels are generated as a function of median rain height for the Bf link (f= 20.0 GHz).  142  Figure 7-9:  Transmission loss levels are generated as a function of median rain height for the B f link (/= 40.0 GHz).  142  Figure 7-10:  Transmission loss levels are generated as a function of median rain height for the Bf link (/= 11.2 GHz).  143  0  FRm  University of British Columbia  xv  Electrical Engineering  List of Figures  Figure 7-11:  Transmission loss levels are generated as a function of median rain height for the L1-L8 links.  144  Figure 7-12:  Transmission loss levels are generated as a function of frequency for the Bf link (h = 2.1 km).  145  Transmission loss levels are generated as a function of frequency for the Bf link (h = 3.0 km).  146  Transmission loss levels are generated as a function of frequency for the Bf link (h = 3.9 km).  146  Figure 7-15:  Transmission loss levels as a function of polarization angle for the L3 link.  147  Figure 7-16:  Transmission loss levels as a function of polarization angle for the D1 link.  150  Figure 7-17:  Bistatic scattering cross section of melting-snow.  151  Figure 7-18:  Transmission loss statistics for the Bf link (f = 4 GHz).  151  Figure 7-19:  Interference prediction for the DI link using Kharadly's attenuation model (solid) and the power law attenuation model (dashed).  152  Figure 8-1:  Interference geometry between an earth transmitting station and a receiving mobile terminal.  155  Figure 8-2:  System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for f = 4 GHz and different rain rates calculated using the original Awaka model (solid) and the modified model (dashed).  Figure 8-3:  System 2 (35 dB gain and 2.8 degrees half-beamwidth): Interference level relative to desired signal level for f = 4 GHz and different rain rates calculated using the original Awaka model (solid) and the modified model (dashed).  Figure 8-4:  System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for f = 20 GHz and different rain rates calculated using the original Awaka model (solid) and the modified model (dashed).  Figure 8-5:  System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for R = 10 mm/h and different frequencies calculated using the original Awaka model (solid).  162  Figure 8-6:  Interference level relative to desired signal level statistics f o r / = 4 GHz and different d distance (km) calculated using the original ITUR model (solid) and the modified FTU-R model (dashed).  163  Interference level relative to desired signal level statistics f o r / = 10 GHz and different d, distance (km) calculated using the original ITUR model (solid) and the modified ITU-R model (dashed).  163  FRm  Figure 7-13:  FRm  Figure 7-14:  FRm  161  161  162  tr  Figure 8-7:  r  University of British Columbia  xvi  Electrical Engineering  List of Figures  Figure 8-8:  Interference level relative to desired signal level statistics for/= 20 GHz and different d, distance (km) calculated using the original ITUR model (solid) and the modified ITU-R model (dashed).  164  Interference level relative to desired signal level statistics for/= 30 GHz and different d distance (km) calculated using the original ITUR model (solid) and the modified ITU-R model (dashed).  164  Figure 8-10:  Probability of bit error for binary and four-phase PSK and DPSK [95].  165  Figure 9-1:  Excess attenuation in the melting layer for variable initial snow density and f = 20.195, 27.5, 45, 75 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( as a function of rain rate.  173  r  Figure 8-9:  tr  Figure 9-2:  )  Excess attenuation in the melting layer for variable initial snow density and f = 20.195 , 27.5, 45 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( as a function of percentage of time, assuming the melting layer occurring 100 % of the time.  173 )  Figure 9-3:  Excess attenuation in the melting layer for variable initial snow density and f = 20.195 , 27.5, 45 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( ) as a function of percentage of time, with the melting layer occurring at appropriate percentage of time.  174  Figure 9-4:  Fading geometry and domain of mobility of rain-cells.  176  Figure 9-5:  Comparison of melting layer-induced attenuation using the concentric ( ) and spongy ( ) sphere models. The rain-only attenuation is in solid.  178  Figure 9-6:  Attenuation statistics for the Fairbanks, A K , site. (Rain only: -100%: , Method I: — , Method II: -.-.-.-).  179  Figure 9-7:  Attenuation statistics for the Vancouver, BC, site. (Rain only: -100%: , Method I: — , Method II: -.-.-.-).  Figure 9-8:  Attenuation statistics for the Clacksburg, M D , site. (Rain only: M L -100%: , Method I: — , Method II: -.-.-.-).  Figure 9-9:  Attenuation statistics for the B C site f o r / = 45, 75 and W0 GHz (Rain only: , M L -100%: , Method I: — , Method II: -.-.-.-).  181  Figure 9-10:  Comparison of the total attenuation using Method II statistical method in section 9.4 ( ) and the simplified method ( ). The rain-only attenuation is in solid.  183  Figure 9-11:  Comparison of the total attenuation using Method I statistical method in section 9.4 (—) and the simplified method (......). The rain-only attenuation is in solid.  184  Figure C - l :  Transmission loss characteristics for the Gz link.  238  University of British  Columbia  xvii  , ML , ML ,  179  180  Electrical Engineering  List of Figures  Figure C-2:  Transmission loss characteristics for the LI link.  238  Figure C-3:  Transmission loss characteristics for the L2 link.  239  Figure C-4:  Transmission loss characteristics for the L3 link.  239  Figure C-5:  Transmission loss characteristics for the L4 link.  240  Figure C-6:  Transmission loss characteristics for the L5 link.  240  Figure C-7:  Transmission loss characteristics for the L6 link.  241  Figure C-8:  Transmission loss characteristics for the L7 link.  241  Figure C-9:  Transmission loss characteristics for the L8 link.  242  Figure C-10:  Transmission loss characteristics for the DI link.  242  Figure C - l l :  Transmission loss characteristics for the D2 link.  243  Figure C-12:  Transmission loss characteristics for the D3 link.  .243  Figure C-13:  Transmission loss characteristics for the D4 link.  244  Figure C-14:  Transmission loss characteristics for the D5 link.  244  Figure C-15:  Transmission loss characteristics for the D6 link.  245  Figure C-16:  Transmission loss characteristics for the SI link.  245  Figure C-17:  Transmission loss characteristics for the S2 link.  246  Figure C-18:  Transmission loss characteristics for the S3 link.  246  Figure C-19:  Transmission loss characteristics for the S4 link.  247  Figure C-20:  Transmission loss characteristics for the S5 link.  247  Figure C-21:  Transmission loss characteristics for the Ff link.  248  Figure C-22:  Transmission loss characteristics for the Fb link.  248  Figure C-23:  Transmission loss characteristics for the Bf link.  249  Figure C-24:  Transmission loss characteristics for the Bb link.  249  Figure C-25:  Transmission loss characteristics for the Bf link (h GHz).  Figure C-26:  Transmission loss characteristics for the Bf link (h GHz).  Figure C-27:  Transmission loss characteristics for the Bf link (h 11.2 GHz).  Figure C-28:  Transmission loss characteristics for the Bf link (h 20.0 GHz).  Figure C-29:  Transmission loss characteristics for the Bf link (h  FRm  FRm  FRm  FRm  University of British Columbia  FRm  xviii  = 3.0 km,/=4.0  250  = 3.0 km,/=8.0  250  = 3.0 km,/=  251  = 3.0 km,/=  251  = 3.0 km,/=  252  Electrical Engineering  List oi' Figures  30.0 GHz). Figure C-30:  Transmission loss characteristics for the Bf link {h 40.0 GHz).  University o f British C o l u m b i a  FRm  xix  = 3.0 km,/=  252  Electrical Engineering  List of Tables  List of Tables Table 4-1:  Comparison between measured rainfall statistics and models for the Vancouver region.  57  Table 5-1:  A summary of the different stages of melting of a particle. The two melting processes considered are the percolating particle process (denoted by the subscript f) and the non-percolating process (denoted by the subscript m).  71  Table 5-2:  Drop-size distribution and associated velocities in rain for different precipitation rates [61].  Table 6-1:  Coefficients for the calculation of specific rain attenuation.  125  Table 7-1:  Rain rate and melting layer statistics.  128  Table 7-2:  Summary of the errors (in dB) between the simulated results for the above links and their experimental values.  132  Table 7-3:  Summary of the difference (in dB) between the transmission loss predicted by the original and modified models for the Bf link with/ = 4, 8, 11.2, 20 and 40 GHz.  136  Table 7-4:  Summary of the difference (in dB) between the transmission loss predicted by the original and modified models for the links in table C1.  138  Table 8-1:  Loss due to atmospheric gases on an earth-space link for terminals with 20° elevation angle [94].  158  Table 9-1:  The fitted a and b parameters for reflectivity and attenuation results for different frequencies. Set I refers to the fitted parameters using the whole range of rain rates (up to 300 mm/h) while Set II is optimized for the region where the melting layer exists (up to 50 mm/h).  168  Table 9-2:  The fitted a and b parameters for total attenuation in the melting layer as a function of frequency for different initial-snow density (concentric-sphere model).  170  Table 9-3:  The fitted a and b parameters for total attenuation in the melting layer as a function of frequency for different initial-snow density (spongysphere model).  171  Table 9-4:  The rain rate statistics for the relevant regions in which the above terminals lie.  177  Table 9-5:  Relevant parameters for the ACTS propagation terminals under investigation.  177  Table C-l:  Experimental links parameters summary.  236  University of British Columbia  xx  Electrical Engineering  List of Tables  Table C-2:  Measured transmission loss statistics for the different experimental links.  237  Table C-3:  Rain rate statistics for the different experimental sites described in table C - l .  237  University of British Columbia  xxi  Electrical Engineering  List of Symbols  List of Symbols 3D model  Three dimensional model. It refers to the meteorological models by Capsoni and Awaka.  ACTS  Advanced Communications Technology Satellite.  ARC  Alberta Research Council.  BER  Bit Error Rate.  CCIR  The International Radio Consultative Committee  C O S T 210  A European Union project to study the influence of the atmosphere on interference between radio communication systems at frequencies above 1 GHz.  CRC  Communications Research Centre.  ITU-R  Group R of the International Telecommunication Union, formerly known as the CCIR.  ML  Melting Layer.  McGill  McGill university  S  Melting degree of a particle = volume of water/total volume.  VPR  Vertically Pointing Radar.  VSAT  Very Small Aperture Terminal.  a  f  Radius of the melting-snow particle using the spongy-sphere model.  a  m  Radius of the melting-snow particle. Representative particle radius of a fictitious medium assuming equi-size particles.  a  R  Radius of the rain particle.  a  s  Radius of the snow core in the concentric-sphere representation.  d  c  Diameter of the ITU-R rain-cell.  dr  Integration step.  dV  Common volume element.  e  A parameter determined from initial-snow density according to equation 3.19.  f  Frequency in GHz.  f,.  Characteristic (or resonant) frequency of the particle.  University of British Columbia  xxii  Electrical Engineering  List of Symbols  f(p )  Correction factor of melting layer thickness = H/B (for the spongy-sphere model).  h|  Heights from sea level of antenna 1.  h  Heights from sea level of antenna 2.  s  2  <2 n  >  hR  Rain height in km.  F  FRm  n  Spherical hankell function of the second kind of order n.  g  Median rain height in km. - Gravitational constant. - Low frequency value of polarizability of a particle.  <g >  Intergrand of g .  g  e  Effective value of polarizability at high frequencies.  g  r  Receiver antenna normalized radiation pattern.  2  2  g(  Transmitter antenna normalized radiation pattern.  g'(6)  Normalized radiation pattern for the main beam.  g"(9)  Normalized radiation pattern for the secondary beam.  j  Spherical Bessel function of the first kind of order n.  n  l  The distance that the signal traverses the rain-cell on the receiver side.  r  1,  The distance that the signal traverses the rain-cell on the transmitter side.  m  Particle mass.  p(i,j)  Percentage of rainfall, with rain rate Rj, coming from raindrops of radius a^  r  Distance from the center of the rain-cell where the rain rate decays to R /e.  0  M  <r >  The average rate of decay of rain rate in the cell.  r  i 2  Distance between antennas 1 and 2.  r  cff  Effective radius of earth = 8500 km.  r  m  Scale length of rain attenuation.  r  max  Radius of a 3D/Awaka rain-cell.  0  v  m  Melting-particle velocity.  v  R  Rain-particle velocity.  University of British Columbia  xxiii  Electrical Engineering  List of Symbols  v  s  Snow-particle density.  x  a  Volume fraction of air in the melting-snow particle.  x  Melted mass fraction in the melting-snow particle (spongy-sphere representation).  r  X|  Volume fraction of ice in the melting-snow particle. It can also be the percentage of occurrence of the melting layer as a function of rain rate.  x  Probability of the melting layer occurring as a function of rain rate.  m  x | m  p  Probability of the melting layer occurring as a function of position,  z  Reflectivity in unit surface/unit volume.  A  Cross sectional area of the particle.  ADM1  First artificial dielectric model for attenuation.  ADM2  Second artificial dielectric model for attenuation.  A  a  Attenuation coefficient due to air gases.  A  p  Attenuation coefficient due to precipitation.  B  Thickness of the bright band.  CF1  Correction factor for A D M 1 (equation 4.24).  CF2  Correction factor for A D M 2 (equation 4.25).  C  d  Drag coefficient.  C  d m  Drag coefficient of the melting-snow particle.  C , d  Drag coefficient of the rain particles.  C  d s  Drag coefficient of snow.  CV  Common volume.  F(M")  Correction factor for the artificial dielectric model for scattering.  F(n)  Correction factor for the artificial dielectric model for scattering.  F(S)  Correction factor for the artificial dielectric model for scattering.  F(9,<t>)  Correction factor for the artificial dielectric model for scattering.  G"  Receiver antenna gain.  G,  Transmitter antenna gain.  r  University of British Columbia  xxiv  Electrical Engineering  List of Symbols  Q  Total scattering cross section of a particle.  s  Q,  Total cross section of a particle.  H  Thickness of the melting layer.  K  Number of dBs the secondary beam is below the main beam.  L  Transmission loss.  Ni  Number of particles of the ith rain-drop size per unit volume (number density).  N  m  Number density of the melting particles.  N  R  Number density of rain.  N(r ,R )  The number of cells in a km for a unit A R  N(R )  Spatial density of cells with peak rain rate R for all r .  P(L,AL)  Histogram of the probability density function of transmission loss.  P(L>L')  Cumulative distribution function of transmission loss.  P(r )  Probability of occurrence of r .  P,  Average received power.  P,  Transmitted power.  P"'(R)  Third derivative of P(R).  R  Rain rate in mm/h.  0  M  M  0  R  m a x  2  M  and A(ro/<ro>).  M  0  0  Maximum rain rate where the melting layer may exist or the minimum rain rate where the melting layer cannot exist.  R,  Distance between dV and the receiver antenna.  R,  Distance between dV and the transmitter antenna.  R  M  Maximum rain rate in the rain-cell.  R  s  Threshold where rain rate below it is assumed zero.  S|  (h/H), where H is the thickness of the melting layer, and h is the depth into the melting layer from the top of the melting layer.  S,  " (h/H)  S  (h/H)  3  2  3  University of British Columbia  xxv  Electrical Engineering  List of Symbols  S  4  (h/H)  S  sal  The degree of melting S where the melting-snow particle (spongy-sphere representation)  05  saturates with water, i.e. all air is forced out of the particle. V  a  Volume of air in the melting-snow particle.  V  f  Volume of the melting-snow particle using the spongy-sphere representation.  Vi  Volume of ice in the melting-snow particle.  V  R  Volume of the rain drop.  V  s  Volume of snow in the melting-snow particle.  V  w  Volume of water in the melting-snow particle.  Ze  Equivalent reflectivity factor = 101og(z).  a  Specific attenuation, dB/km.  a'  Double-sided half-beamwidth of the main beam.  a"  Double-sided half-beamwidth of the secondary beam.  a  Average attenuation (section 5.5.2).  a v g  excess  Total attenuation in the melting layer.  oc |  Total attenuation in the melting layer.  a  Backscatter of a particle.  tola  o  avg  Average scattering cross section (section 5.5.2).  p  a  Density of air = 0.0012929 g/cm .  p  f  Density of the melting-snow particle using the spongy-sphere representation.  3  pi  Density of ice = 0.917 g/cm .  p  s  Density of snow.  p  w  Density of water = 1 g/cm .  3  3  ^  Permittivity of free space.  e,  Elevation angle of transmitter 1.  University of British Columbia  xxvi  Electrical Engineering  List of Symbols  Z^  Elevation angle of transmitter 2.  e  Average permittivity of the melting-snow particle.  avg  e  m  Average permittivity of the melting-snow particle.  e  s  Snow permittivity.  r|  - Antenna loss factor.  r|  n  - Derivative of j . n  - Skin depth of water.  5  - Angle between earth center and the two terminals. Xq  Free space wavelength.  0  3 dB half-beamwidth.  h  (9,<)))  Polar and azimuth angles.  Y  Total attenuation along the transmitter and receiver paths.  r  R  AS  Attenuation outside the core cell of the ITU-R rain-cell. A subdomain the domain of mobility.  University of British Columbia  xxvii  Electrical Engineering  Acknowledgment  Acknowledgment  I would like to express my appreciation to Dr. M.M.Z. Kharadly who has provided me with much needed support, supervision and suggestions throughout the course of my studies at UBC. I also would like to thank Dr. R.L. Olsen of the Communications Research Centre for his assistance and helpful advice. I would like to thank Ms. A . Chan and I. Dommel for help proofreading my thesis. I would like to thank all my friends and teachers since everyone of them left their mark on my life and character. Finally, I would like to thank my mother, father , brothers and sisters, whom I long to see soon and whose advice and encouragement have always been with me. This research was supported by the Communications Research Centre, Industry Canada.  University of British Columbia  xxviii  Electrical Engineering  Chapter 1: Introduction  Chapter 1: Introduction  In designing radio systems, an engineer has to ensure satisfactory performance and immunity from interference with other systems. Performance is determined by the availability time of the system, Bit Error Rate (BER), fade margins etc. Installing a radio link may cause interference with other links, services and networks that exist in the vicinity and use the same frequency range. At microwave and millimeter-wave frequencies, precipitation affects the performance of a wireless communication terminal, and could cause undesired interaction with other terminals. Scattering by precipitation particles results in signal fading and attenuation. This can increase the BER, arid if the attenuation is severe enough, decrease the availability time of the system. Also, scattered signals may interfere with the operation of another terminal. Precipitation effects tend to be the dominant interference mechanism at lower percentages of time (associated with the high levels of undesirable interference and attenuation) and thus clearly need to be understood and analyzed. Models are available for calculating attenuation (e.g., [l]-[3]) on a communications link and interference (e.g., [4]-[8]) between systems due to precipitation. These models, however, do not consider some aspects of precipitation such as the effect of the meltingsnow layer. This may have been due to the difficulty in modeling some of these effects,  University of British Columbia  1  Electrical Engineering  Chapter 1: Introduction  which is especially true for the melting layer. Recently, however, considerable attention has been given to modeling the melting-snow layer, and previously unavailable data about its microphysics and statistics are currently available (e.g., [9]-[14]). It is now possible to attempt to determine its contribution to interference and attenuation. However, one finds that this information is not consistent or that little work has been done to correlate the measured data to models. Even when this is done, the measurements themselves are not consistent because of the small ensemble of experimental data used. In this work, we review and evaluate the literature relevant to the melting layer and devise models using the most reliable data. This model is then used to examine the effect of the melting-snow layer on both interference and fading, and whether such effects warrant changes in international regulations, set forth by the Radio Communication Sector of the International Telecommunication Union (ITU-R). 1.1 Interference 1.1.1 Interference mechanisms  Microwave interference can be caused by several mechanisms, of which one or more may occur at the same time [4]. Eight principal interference mechanisms, detailed below, have been outlined by the ITU-R. The first four are long term and continuous (figure 1-1) and the others are short term (figure 1-2): a) Line-of-sight: Line-of-sight interference can occur when a terminal is in the propagation path of another terminal. b) Diffraction: The diffraction of a part of a transmitted signal due to terrain can cause the diffracted signal to interfere with other links operating at the same frequency.  University of British Columbia  2  Electrical Engineering  Chapter 1: Introduction  Troposcattar  Figure 1-1:  Long term interference mechanisms [5]  c) Tropospheric scatter: In the troposphere, signal energy is scattered due to the nonuniform structure of the atmosphere. This mechanism defines the background interference level for longer paths (e.g., 100-150 km). d) Scatter from terrain and buildings: The scattering of transmitted signals by terrain and buildings is not a problem at this time, but expected to become one as the number of terminals increases. e)  Enhanced line-of-site: Line-of-site interference could be further enhanced by multipath and focusing effects.  f)  Surface super-refraction and ducting: Under suitable conditions, this mechanism can cause part of the signal to propagate parallel to the surface of the earth and thus increase the potential for interference. This is most important over water and in flat coastal land areas.  University of British Columbia  3  Electrical Engineering  Chapter 1: Introduction  Figure 1-2: Short term (enhanced) propagation mechanisms [5]  g) Elevated layer reflection and refraction: This interference mechanism is due to layers in the atmosphere at heights up to a few hundred meters. h) Hydrometeor scatter: Hydrometeor scatter interference is the dominant interference mechanism for small percentages of time. In this work, interference due to hydrometeor scatter is examined in general, but only one element of this problem, namely that due to the melting layer, is dealt with in depth.  1.1.2 Interference due to hydrometeor scatter Microwaves are scattered by hydrometeors such as rain, snow, melting-snow and ice particles causing interference between communication links operating at the same frequency and sharing a common volume (figure 1-3). It is thus necessary to estimate the level of this interference in the design of microwave communication links.  University of British Columbia  4  Electrical Engineering  Chapter 1: Introduction  Figure 1-3: Paths of possible interference (Solid line: wanted paths, dashed lines: unwanted paths) [5]  Previous work on modeling interference produced two widely accepted models, the ITUR/COST 210 model [4,5] and the 3D/Capsoni model [5,6] (The 3D/Awaka model [8] is a variant of the Capsoni model). These two models differ in their description of the size and spatial distribution of precipitation in the rain cell. The latter model seems to be more realistic, albeit substantially more complex. Both models have the same vertical profile of the rain cell. Both assume a rain medium with constant scattering and attenuation up to the rain height. Above the rain height, an ice/snow region is assumed, with a reflectivity roll-off of 6.5 dB/km and zero attenuation. In reality, there exists a region of high reflectivity (and attenuation) between these two regions, where snow melts into rain. This region is referred to as the melting-snow layer. It is also frequently referred to as the bright band by radar meteorologists. Neither of the two models accounts for the effect of the melting layer on interference.  University of British Columbia  5  Electrical Engineering  Chapter I: Introduction  The effect of the melting layer on interference has only recently been the subject of serious investigation ([14]-[19]), showing that the melting-snow layer can indeed affect interference levels. Having shown that the presence of the melting-snow layer, in or near the center of the common volume, can be a significant factor in interference prediction, it becomes important to statistically determine this effect on microwave systems. The primary objective of this work is to develop a melting-layer model, based upon the most reliable data, and use this model to estimate the its effects on interference between microwave communication systems. The effect of the melting layer on interference (and other parameters such as rain height, frequency, choice of attenuation model, etc.) was examined using three rain-cell and meteorological models. The first is based upon the COST 210/fTU-R rain-cell, the second is based upon Capsoni rain-cell model and the third upon the Awaka-modified Capsoni rain-cell. These will be discussed in detail in the subsequent chapters. Statistics based upon radar observations, such as mean melting layer thickness, percentages of time the melting layer is present and initial-snow density, are incorporated into the model. Significant modifications to the above-mentioned models were required in order to include the effect of the melting layer.  1.2 Fading An engineer determines the system requirements such as transmitted power level, receiver and transmitter gains in order to deliver the minimum power necessary for detection. The power levels received by an antenna can be increased by either increasing the transmitted power level or increasing the gain of the transmitting antenna, which means increasing  University of British Co l u m b i a  6  Electrical Engineering  Chapter 1: Introduction  the antenna sizes. Both options add to the cost of a system, affecting its economic viability. Increasing the power transmitted from a satellite or increasing the gain of its antenna means a larger satellite and this adds significantly to the cost of the delivery of satellite systems. Increasing the cost of earth-based subscriber units can affect the level of penetration of a service. This is especially true if the system aims to provide service to a significant number of subscribers. Satellite links are designed so that the power level received by a terminal is at least several dBs above the minimum signal level necessary for detection. This is due to the extra attenuation on the signal caused by precipitation. This power margin is determined by the availability requirements placed upon the system and the climatic region where the terminal is located. A higher power margin adds to the overall system cost. It is therefore desirable to have the lowest power margin possible for a predetermined acceptable performance (Availability time and BER).  1.2.1 Mechanisms of attenuation A wave propagating in the atmosphere experiences  losses due to absorption and  scattering effects by the particles populating the atmosphere. The different contributions to the propagation losses are summed up in [21] as follows: a) Attenuation by atmospheric gases: The level of attenuation due to atmospheric gases, such as oxygen and water vapor, depends mainly on frequency, elevation angle, altitude above sea level and water vapor density. At frequencies below 10 GHz, its effect could normally be neglected.  University of British Columbia  7  Electrical Engineering  Chapter 1: Introduction  b) Attenuation by rain, other precipitation and clouds: This is by the far the most serious atmospheric effect on microwave links. Water droplets scatter and absorb radiation at a significantly higher rate than other particles in the atmosphere. Rain effects become especially severe at wavelengths approaching the size of the water droplets. This is especially true for frequencies above 10 GHz. The losses due to rain effects can be as high as few tens of dBs. It is then necessary that enough extra power (power margin) be transmitted to overcome the maximum attenuation caused by rain to guarantee an acceptable availability time (e.g., 99.99% of time) c) Focusing and defocusing: The regular decrease of refractive index with height causes ray-bending. Defocusing loss is usually small (less than 0.5 dB) and could be neglected, except at very low elevation angles (< 5°) d) Decrease in antenna gain due to wave-front incoherence: This is caused by turbulence in the antenna path which causes a degree of phase incoherence. This will produce the appearance of gain reduction. e) Scintillation and multipath effects: This is caused by small-scale irregularities in the refractive index. In the absence of rain, it is unlikely that scintillation can cause serious degradation to system performance except if the fade margin is in the order of a few dBs or if the region of operation at low latitudes. f) Attenuation by sand and dust storms: This fade mechanism might present problems in arid or semi-arid areas. Most of these contributions depend on frequency, geography and elevation angle. Increasing the operating frequency tends to increase the severity of each of the above effects. At terminals operating at elevation angles above 10°, only gaseous attenuation,  University of British Columbia  8  Electrical Engineering  Chapter 1: Introduction  precipitation attenuation and possibly scintillation need be considered. In this work, only attenuation due to precipitation is considered, emphasizing the contribution of the melting-snow layer. 1.2.2  Excess attenuation due to melting layer  In previous studies (e.g., [22]-[24]) on terrestrial links, signal fading which could not be accounted for by using rain-attenuation models only, occurred when a melting layer was present in the propagation path. Melting layer attenuation has also been observed on earth-space links. Excess attenuation of 3-7 dB was observed for a 28.56 GHz earth-space link (figure 1-4) [25]. Despite the above evidence to the importance of melting layer in earth-space communications, the melting-layer presence has not received adequate attention in the formulation of attenuation prediction models. In very small aperture terminals (VSAT), the fade margin is very limited; and for mobile and hand held receivers, it can be in the order of few dBs and the presence of a melting layer may seriously inhibit good system performance. In this work, we examine the excess attenuation introduced by the presence of the melting layer on satellite communication links, especially at high frequencies. A power-law formulation for total attenuation in the melting layer in terms of rain rate is devised and the excess attenuation caused by the melting layer is examined. The melting layer is incorporated into an attenuation model, based upon the Awaka/3D meteorological model, to generate fade statistics and simplified procedures are developed to incorporate  University of British Columbia  9  Electrical Engineering  Chapter 1: Introduction  Q I  •  •  •  -  08-19  I  I  1  08 20  1—I  l — J 1  1  08-21  1  »  1  '  ' »•  08 22  TIME ( E S T . ) Figure 1-4: Occurrence of large melting layer attenuation associated with high melting layer reflectivity, 29 July, 1980. Dotted curve shows calculated attenuation if melting layer treated as rain up to the nose of bright band [25]  the effect of melting layer attenuation into fading simulations generated by rain-only fade models.  1.3 Modeling the melting layer In order to determine the melting layer contribution to interference and fading, it is essential to estimate melting layer attenuation and scattering, as accurately as possible. These, in turn, depend on the microphysical properties of the melting layer. Extensive models are available for this purpose. However, these models depend on parameters that are not very well understood and they are computationally intensive. Simple models that simulate the behavior of the melting layer are adapted for this work (e.g., [26]). Radar observations of the melting layer [9]-[14] are examined to obtain the appropriate melting  University of British Columbia  10  Electrical Engineering  Chapter 1: Introduction  layer statistics such as melting profile, thickness, initial-snow density, probability of the presence of the melting layer. However, as stated earlier, one finds that this information is not consistent or that there is little work done to correlate the measured data to models. Even when this is done, the measurements themselves do not provide a reliable measure of the bright band because of the small ensemble of experimental data used. We will examine this in chapter 5 in great details. In this work, we review and evaluate the literature relevant to the melting layer and devise models using the most reliable data. 1.4 Summary of thesis objectives  The main objectives of the work in this thesis may be summarized as follows: 1. Develop a model for the melting-snow layer and for melting-layer attenuation and scattering, using the most reliable information available. 2. Include the melting-snow layer in interference models. 3. Study the effect of the melting-snow layer on interference. 4. Study interference due to precipitation in general. 5. Predict excess attenuation on earth-space links due to the melting-layer using the layer's model and the meteorological model utilized in this work. 6. Make recommendations to regulatory agencies with regard to the effect of the melting layer on interference and fading. 1.5 Thesis outline  University of British Columbia  11  Electrical Engineering  Chapter 1: Introduction  The treatment of the bistatic radar equation, used to calculate interference due to precipitation induced scatter, and the procedure used in its evaluation is presented in chapter 2. In- order to evaluate the effect of the melting layer on interference, several procedures are discussed. Chapter 3 presents the different meteorological models used to describe statistics and properties of rain. The two models that gained acceptance, the ITU-R/COST210 and Capsoni (and its variants), are used here. Modifications to the models, in order to include the effect of melting-snow layer scattering and attenuation, are examined. Chapter 4 discusses the macroscopic properties of precipitation, such as vertical structure, rain rate distributions, climate regions, different rain-cell representations. Chapter 5 examines the microphysics of precipitation, especially the melting layer. These include descriptions of the drop shapes, composition, melting process, velocity, particlesize distribution. Statistics of the melting layer, such as thickness, initial-snow density, percentage of presence of the melting layer, are either described or derived. Chapter 6 deals with the various scattering and attenuation models for rain, melting layer and the region above the melting layer. These include Mie scattering for both sphere and concentric sphere [27], artificial dielectric scattering [28][29] and attenuation models, power-law (ai^) attenuation model [30] and its extension to the melting layer etc. In chapter 7, the validity of the different prediction techniques is ascertained in comparison with experimental data. The effect of the melting layer on interference is examined in detail. Other parameters such as rain height, modeling of the ice/snow region, frequency, etc. are also examined. In chapter 8, the effect of hydrometeor scatter on low gain mobile terminals is examined.  University of British Columbia  12  Electrical Engineering  Chapter 1: Introduction  In chapter 9, the effect of the melting layer on fading in earth-space links is examined. The total attenuation in the melting layer is found to obey a power-law (aR ) relationship. b  The a and b parameters are derived for a wide range of frequencies. These relationships are used to include the effect of the melting layer into rain-only attenuation models. Also the Awaka meteorological model is used to predict the attenuation statistics on satellite links with and without the presence of the melting layer. Discussions and conclusions based upon the previous chapters are presented in chapter 10 and recommendations for future research in chapter 11. In this work, we used FORTRAN 77 for the development of the models. Matlab was also used for some analysis and data manipulation.  University of British Columbia  13  Electrical Engineering  Chapter 2: Interference Modeling  Chapter 2: Interference Modeling  The theory of interference between two links, sharing a common volume and operating at the same frequency, is similar to that of a radar. The bistatic radar equation formulation is simple. Its numerical evaluation, on the other hand, is not, and depending on the scattering volume and properties, it can be extremely difficult. This is especially true in the case of interference on microwave links, where the scattering and attenuation paths are both long and variable. The comparable sizes of individual scatterers with the wavelength (Mie scattering regime) make it extremely difficult to evaluate the scattering and attenuation characteristics of the media. Simplifications to the radar equation are then necessary in order to obtain simulation results in a reasonable time. In this chapter, the analysis required to solve the interference problem is introduced (section 2.1). Furthermore, issues that enter into the evaluation of the radar equation are discussed, such as antenna parameters (section 2.2), link geometry (section 2.3), rain properties (section 2.4) and simulation techniques (sections 2.5 and 2.6). For a thorough description of the simulation program, refer to appendix A.  2.1  Interference Theory  2.1.1 Radar equation  University of British Columbia  14  Electrical Engineering  Chapter!: Interference Modeling  rain c e l l narrowbeam  h o r i z o n ray  At  A, Figure 2-1: General interference geometry.  Consider the interference geometry shown in figure 2.1. The signal received by an antenna A , from an electromagnetic wave radiated from an antenna A, and scattered by r  precipitation, is given by [31]:  P ^ X-G,G rj r rg,(yf)g,{(pWe,<f» r  r  l  lr  y(R„R ).dV r  where P , P , G, g, 77, R, X, }(R ,R ), t  r  t  r  (2.1)  0(6,(fs) are, in order, the transmitted power, the  average received power, antenna gain, the antenna normalized radiation pattern, the antenna loss factor ( 0 < TJ< 1), the distance from dV to the antenna, the wavelength, the propagation loss due to precipitation and atmospheric gases along the total path (R + R,.), t  the bistatic scattering cross section in the direction of 0 and tp, respectively. Subscript t denotes transmitter and r denotes receiver. University of British Columbia  15  Electrical Engineering  Chapter 2: Interference M o d e l i n g  2.1.2 Simplified radar equation Evaluating the above equation is time consuming, and it is desirable to simplify it without significantly affecting the accuracy of the prediction technique. This is done by using the narrow beam approximation to the antenna with the smaller beam in the common volume. Since the transmitter and receiver parameters are interchangeable, it will be assumed that the antenna with the narrow beam approximation has the subscript '1', and the other antenna has the subscript 2'. Equation 2.1 may be re-written as follows:  Tr  dx?  riR,R ).dv  RJRI  J  (2.2)  t  Using the narrow beam approximation, gj= 1, dV = S dr, (where S is the 3 dB circular a  a  cross section perpendicular to the main beam axis), and S =nR 6\ l4, 1  u  h  equation 2.2  becomes: P A G G,e? ,ri ri  r g {<p)<T(e,<j»  2  r  =  ]  l  P,  i  2  dr  256K  J  1  where d  th  2  r  R\  is the 3 dB half-beamwidth of the narrow beam antenna. The propagation loss y  is given by:  -0.1 I A <Jr-0A  A„dr  r  >=10  »  "  (2-4)  where A , A , /,, l are the attenuation coefficient due to precipitation (given in dB/unit a  r  length), the attenuation coefficient due to atmospheric gases (given in dB/unit length), the distance the electromagnetic wave traverses the rain cell on the transmitter side, and the distance the electromagnetic  wave traverses the rain cell on the receiver side,  respectively. University of British Columbia  16  Electrical Engineering  Chapter 2: Interference Modeling  The bistatic cross section 0(6, tp) is given by: ""2*  "max  o-(0,0) = 5 V . ( 0 , 0 , a ) = J/2(fl)<T .(0,0,a)da (  |  0.  (2.5)  0  where n{a) is the raindrop-size distribution, a is the equivalent radius of a spherical raindrop, and o{6,<p,a) is the bistatic cross section of a drop of radius a in the direction of (6, <p). From equation 2.3 it can be seen that, in order to evaluate interference levels, we need to quantitatively describe the following quantities: 1. Transmitting and receiving antenna parameters such as gain, frequency, and polarization (chapter 2). 2. Coordinates and direction of the antennas (chapter 2). 3. Macroscopic rain structure and properties such as rain rate statistics, rain-cell structures etc. This will be covered in chapter 4 . 4. Microscopic rain structure such as particle distribution, composition, density etc. This will be covered in chapter 5. 5. Precipitation scattering and attenuation modeling. This will be covered in chapter 6. 6. Gaseous attenuation. This includes attenuation due to atmospheric gases and water vapor in air. This will not be discussed in detail. It would suffice to say that the formulations provided in [32] are used.  2.2 Wide-beam Antenna Radiation Pattern The main lobe of the wide-beam antenna is represented by a gaussian-shaped pattern:  g (9) = e  (2.6)  1  2  University of British Columbia  17  Electrical Engineering  Chapter 2: Interference Modeling  where g' {6) is the gain at angle 9 from the main axis and a' is the double-sided half2  2  power beamwidth. The secondary lobe is accounted for b y assuming a gaussian-shaped pattern, with a larger double-sided half-power beamwidth (a'2) and a gain of K dB (a default value of -17 dB is recommended [5]), below the main lobe: g';(G) = \ Q  0 A K  xe-  4 ] n 2  ^  (2.7)  ) 1  The radiation pattern is thus given by:  8i(0)-  i io  0 1 K  +  If the double-sided half-beamwidth of the secondary lobe is not available, it is recommended that it be determined from the main lobe double-sided half-beamwidth according to [5]: a"=l.5^xa;_  (2.9)  A typical antenna radiation pattern is shown in figure 2-2. In evaluating equation 2.3, there is a need to obtain the average gain at each integration step dr. If dr is small and if the wide-beam antenna gain vary little in dr, it may be assumed constant. This may not always be true, especially if dr is large, the antenna has a high gain, and is located relatively close to the common volume. In this case, it is necessary to integrate the gain function in equation 2.8. Thus the average gain between 9/ and 9 , angles between the edges of dr and the main lobe axis of the wide beam antenna, 2  is given by: Int[g (9)]= | g ( ^ - r | [ e x p ( - a ^ ) + A e x p ( - a c ? ) ^ 2  2  2  2  0,  University of British Columbia  2  (2.10)  0,  18  Electrical Engineering  C h a p t e r ! : Interference Modeling  Angle away from main beam axis of the antenna (degrees)  Figure 2-2: The radiation pattern of the wide-beam antenna showing the main lobe, side lobe and the normalized sum of the two.  where: a, =(41n2)/(a )  2  2  a  2  =(41n2)/«)  A = 10 r = i / ( i  2  01K  +  io  0  ,  K  )  The average gain in dr, <g (dr)>, is then given by: 2  \ (d)ie - \ {eye gl  g2  if sign{6 ) = sign(d ) 2  x  (2.11)  < g {dr) >= 2  \g (dpl0 + \g (eyi0 2  2  if  signidjtsignfa)  l o  This is given as a function of the error function, erf, as: University of British Columbia  19  Electrical Engineering  Chapter 2: Interference Modeling  V"2  1  < g (dr) >= 2  e r / ( - 0 7^7) + er/(-0, ^7)1  A[C//(-0  2  v^7)+ erf(-0 V^7)  (2.12)  t  2  if ,y<gn(0 )*5igrt(0|) 2  2.3 Geometrical Considerations The narrow-beam antenna is taken as the origin of the system, with the x axis in the direction from the narrow beam antenna to the wide beam antenna. The z axis is normal to earth at the origin. The angle £ is defined as the angle between the earth center and the two terminals. Then,  6=^  (2.13)  rad  is the distance between the two terminal and r g is the effective radius of earth  where r  i2  e  {r = 8500 km). Because the normal (normal to the ground sphere) at antenna 2 (wide eff  beam antenna) is tilted by an angle S relative to the normal at antenna 1, the elevation angle of antenna 2 (£?) becomes: (2.14)  £ = sin~'(cos£ sine) +sine cos<5>) t2  2  2  where e is the elevation angle relative to the earth normal at antenna 2. The coordinates 2  of antenna 2 are given by: (2.15)  2  V  where hi and h are the heights from sea levels of antenna 1 and antenna 2, respectively. 2  University of British Columbia  20  Electrical Engineering  Chapter 2: Interference Modeling  Meteorological Parameters  In this section, we cover only the items that are necessary for the understanding of the materials in-the next two chapters. A more detailed treatment of the nature of precipitation is given in chapters 4 and 5. General rain-cell structure The rain-rate distribution inside the rain cell can theoretically take any shape. However, in this work we consider only three rain cell models that are cylindrical in shape. The first is the COST210/TTU-R model which assumes a constant rain rate in the rain cell. The Capsoni rain-cell model assumes a symmetrical rain-cell with an exponentially decaying rain rate from a maximum (Rm) at the centre of the rain cell [33] (figure 2-3): R(x,y)=R(r)=R cJ—)  (2.16)  M  where r is the distance from the center of the rain-cell center, and r the distance from the 0  center of the rain-cell where the rain rate decays to RM/C. The Lane-Stutzman rain-cell model assumes a symmetrical gaussian variation of rain rate in the cell (figure 2-3): 1  R{x,y) = R{r)=R  M  (2.17)  exp  For r = r , the rain rate is also R^/e for the gaussian rain distribution. Figure 2-3 presents 0  a comparison between the rain-rate distributions in the rain-cell using the above models. The figure shows that the gaussian rain-cell has, on average, a faster decay rate than the exponential model. At the center of the rain-cell (r = 0), it provides a smooth continuous transition which seems to be more realistic than the sharp discontinuity of the exponential  University of British Columbia  21  Electrical Engineering  Chapter 2: Interference Modeling 15r  15r  Distance from the center of the rain-cell in km Figure 2-3: The figure contrasts the size and rain distribution of rain rate in the rain cell for the ITU-R (dashed), gaussian (dotted), and the exponential (solid) rain-cell model for R - 6.0 mm/h and R - 12.5 mm/h. For the gaussian and the exponential rain-cells, we use the mean roll-off rate M  M  model. For low rain rates (R=5 mm/h), the rTU-R model is a fraction of the size of the exponential and Gaussian model (This is due to the odd characteristic of the average rate of decay, discussed in chapter 3, at lower rain rates). At higher rain rates (e.g., R=12.5 mm/h), they become more comparable in size. Other rain-rate distributions, including non-symmetrical distributions, can easily be used with simple modifications to the rain-rate module in the simulation program. Rain-cells that are arbitrary in shape and rain-rate distribution can also be simulated. The regions in the rain-cell where no rain exists can either be modeled as regions of very low rain rates  1  In [6] and [7], Capsoni et.al. used R(x,y) = R exp(-r/r ) M  0  for the gaussian distribution. This is  obviously a mistake. University of British Columbia  22  Electrical Engineering  C h a p t e r ! : Interference M o d e l i n g  Figure 2-4: The following figure shows how rain maps of arbitrary shape and distribution could be modeled as the one (fictitious) cylindrical cell required by this model. The radius of the fictitious cell is chosen so that all of the relevant rain is in the cell. The numbers in the figure indicate rain rates in mm/h.  (e.g., R - 0.0001 mm/h) or zero rain rates  (figure 2-4). This characteristic can be  especially useful in modeling interference (or fading on a microwave path) if rain-map data are available. In this work, however, we only study rain-cells with axially symmetric characteristics (ITU-R, exponential, and gaussian rain-cell models).  2.4.2 Vertical structure of precipitation Rain (with constant reflectivity and attenuation) is assumed to occur in the rain cell up to the bottom of the melting layer where attenuation and reflectivity are assumed to constantly change with height. Above the melting layer, an ice and dry-snow region, where reflectivity decays at a constant rate and attenuation is negligible, is present (see figure 2-5). It is not a necessary condition, however, for the melting layer to exist at all  University of British C o l u m b i a  23  Electrical Engineering  Chapter 2: Interference Modeling Ice + Snow + Clouds  0 C isotherm  Melting snow  f > rs c O 3  rt  r: <  -a  Rain  c  Figure 2-5: Vertical attenuation and reflectivity profiles in a rain cell.  time. The bright band is usually associated with stratiform rain (low level rain rate), and hardly even present when and where convective rain predominates (high rain rates). 2.5  Attenuation and Scattering Modeling of Hydrometeors  For an integration element dr in the melting-layer region, there are potentially two states, one where rain is present and the other where melting-snow is present. For a rain-cell with a constant rain rate and for small rain cells (ITU-R model), it is possible to assume that the melting layer is present in all the rain-cell (with the probability of occurrence x , m  the probability of presence of the melting layer for a the given rain rate, as presented in chapter 3) or not present (with probability l-x ). m  For rain-cells where rain rate varies and where the cells are large (e.g., Capsoni rain-cell model), the above assumption is no longer valid. The rain rates in the rain-cell change from a maximum at the center of the rain-cell, as high as 400 mm/h, to a rain rate as low  University of British Columbia  24  Electrical Engineering  Chapter 2: Interference Modeling  as few tenths of mm/h at the edge of the cell. Associated with each of these rates is a different percentage of presence of the melting layer. In this case, several approaches to include the effect of the melting layer scattering and attenuation into the interference model were examined. Some were discounted because of the prohibitive computing cost, others because of the lack of needed measured data and still others because, despite their promise, lacked a physical foundation and because they did not offer measurable advantage over the approach followed here. This approach obtains the transmission loss by interpolating between the transmission loss calculated using rain-only model and those assuming a 100% presence of the melting layer. This interpolation is weighed by the probability of occurrence of the melting layer at the different percentages of time. We shall call this approach as the weighed interpolation model. The weighed interpolation model In this method, the transmission loss is calculated independently for two cases. In the first case, the melting layer is assumed not to exist. In the other case, the melting layer is assumed to exist at all times. It will become obvious that since the melting layer exists only part of the time that the actual transmission loss will be between the two above stated cases. It will also become obvious that since the melting layer is present almost all of the time at the higher percentage of time, the actual interference will follow that of the second case while at the lower percentage of time it will follow that of the first case. In between, the actual transmission loss lies between the two. A method has been developed to obtain the transmission loss statistics using those calculated in the first and second case  University of British Columbia  25  Electrical Engineering  Chapter 2: Interference Modeling  and the probability of presence of the melting layer. At x% of time, the transmission loss p % is given by: x  L( %)  ^r,iinlx%)  =  X  where L  ,  mm(x%)  +  ml(x%) (An/U%) ~ ^V«i'n(.t%) )  (2-18)  X  L i( %), are the transmission losses in dB for case 1 and case' 2, m  X  respectively. x„,i %) is the probability of melting layer presence at x% of the time. This is (x  the probability that a melting layer exist for the rain rate corresponding to x% of time. The weighed interpolation approach was also tested using power ratio (instead of power ratio in dB) and was found to give similar results.  2.6 Structure of Simulation Meteorology-based interference models, such as the ITU-R and Capsoni model, require the evaluation of the radar equation 2.3 over many rain-cells. This is especially true for the Capsoni model where rain-cell movement and different rain-rate decay in the rain-cell significantly increase the number of interference scenarios. With the inclusion of rainheight statistics, over 100,000 distinct interference cases are possible with some rain cells spanning hundreds of kilometers (e.g., RM = 400 mm/h and r = 20 km). In the evaluation 0  of equation 2.3, two integration procedures are examined.  2.6.1 Constant integration steps In this method, a constant integration step, dr, is supplied. The smaller the integration step, the higher is the accuracy of the simulation but also the higher is the simulation time. This method is found to be appropriate for the ITU-R model, where rain-cells are small and significantly fewer scenarios need to be considered (no movement of cells and no rain-rate decay in the cells). For the Capsoni model, rain-cells can be in the order of hundreds of kilometers. A constant dr has to be small enough to accommodate small rainUniversity of British Columbia  26  Electrical Engineering  Chapter 2: Interference Modeling  cells. However, for large rain-cells (with large r ), the integration procedure is found to be 0  unduly slow. Flexible integration steps In this method, the integration step, dr, is a variable that depends on rain-rate decay in the rain cell (r ) and on the vertical position of dr (Obviously, dr cannot be very large near 0  the melting layer). In the rain-cell, we observe four distinct regions (figure 2-6). The first is the rain region (Region I), the second is the region where the melting layer may exist (Region IT), the third is the region of ice and snow directly above the melting layer (Region DI), and the fourth is the ice and snow region on top (Region IV). The scattered signals in region DI may be severely attenuated by rain and melting-snow while those of region IV are not. The transition height between regions DT and IV is dependent on the geometry of the interference scenarios and on the discretion of the user. Region I: rain region Region I is vertically homogeneous with scattering and attenuation constant with height. Scattering and attenuation variation occur, though, in the horizontal plane with the varying rain rate. Let dr be the distance between two points where a variation of AR in rain rate exists. Consider these two points are at distances ri and r from the center of the 2  rain-cell. Therefore the rain rates at these two points are:  Hn) =  RM exp  V 0 J  R(r )= R 2  (2.19 a,b)  r  M  exp  V o ) r  The difference in rain rate between the two points is then given by:  University of British Columbia  27  Electrical Engineering  Chapter 2: Interference Modeling  Region IV  Region III  0 degree isotherm  Region II  Region I  Figure 2-6: Rain-cell showing the four different integration regions.  dR=R{ri)-R(r )=R„ 2  expf^l-expf^]  (2.20)  For the exponential rain-cell, the maximum rain-rate variation occurs at the center of the rain-cell. If r is set to zero, then the rain-rate variation , AR, is given by: t  „  A/? =  dR  ,  (2.21)  = l-exp!  and the distance dr between the two points, where AR variation in rain rate occurs, is given by:  dr = r-, 7 c o s £ =  'o COS£  f AR n lm 1 — A  \  (2.22)  where £ is the elevation angle along the propagation path.  University of British Columbia  28  Electrical Engineering  C h a p t e r ! : Interference Modeling  The desired level of rain rate variation is used to determine the integration steps as a function of the maximum rain rate in the rain-cell and the decay rate r , which in turn 0  determines the accuracy of the calculations. The wide-beam antenna characteristics and location should also be considered when choosing AR. The larger the variation in the wide-beam antenna gain at the common volume is, the smaller AR should be. Even though equation 2.22 is derived for the exponential rain-cell model, it may also used with the gaussian model. 2.6.2.2 Region II: melting-snow region In region II, with the melting layer present, the above method may not be fully appropriate. In this case, dr, as calculated for Region I, may be too large that the vertical variations in the melting layer may not be adequately accounted for. In this case, the melting-layer thickness and the path elevation angle need also to be considered. The integration step, dr, is then given by: A D A  (  = _ ^ L l n 1cose V  d r x  ^ M  R  J  dr = HI (ns'xne)  (2.23)  2  dr - min(dr ,dr ) {  2  where £ is the elevation angle along the propagation path, H is the thickness of the melting layer, and n is the minimum acceptable integration steps in the melting layer. 2.6.2.3 Regions D3 and IV: ice and snow regions In Regions DI and TV, the integration steps are calculated in a similar fashion to region D. Since the reflectivity in the ice/snow region decay at a constant rate of several dBs/km, the accuracy of the prediction depends on how large the integration steps are. Region ID  University of British Columbia  29  Electrical Engineering  Chapter 2: Interference M o d e l i n g  is more critical than region IV since the scattered signal may pass through rain and the melting layer and be attenuated while the scattered signal in region IV is hardly attenuated. The maximum vertical integration steps allowed dh = drxsine {dhm, dh/v for regions III and IV, respectively) are used to control the accuracy of calculations. Therefore:  cos£  ^  R  M  j  dr =dh/s'm£  (2.24)  2  dr - min(drj ,dr ) 2  While the scattered signal in region UI may suffer significant attenuation levels when traversing the rain and melting-snow media just below it, the scattered signal in region IV does not. The transition height between regions HI and IV is dependent on the geometry of the interference scenarios and on the discretion of the user. The user can either set a pre-determined region III thickness or the transition could occur at the height where the attenuation along the scattered path becomes negligible.  2.6.3 COST 210 Integration The Cost 210 model [5] does not use the same numerical approach to evaluating the radar equation. They divide the rain cell into four regions. In each of these regions, the radar equation is evaluated in a combination of analytical derivation and approximation. While this may be reasonably successful with the original rain cell - where scattering and attenuation are either constant or are changing linearly - it is not when a melting layer is present. Also, it was decided to avoid methods that have significant approximations. Unfortunately, this meant paying a heavy computational cost. The same is true for the Awaka/Capsoni model. University of British Columbia  30  Electrical Engineering  Chapter 2: Interference Modeling  2.7 Important Features in the Simulation Program The program developed for interference predication is described in detail in appendix A. However, there are notable characteristics that stand out and need emphasizing. These are: 1. Flexibility: The program accommodates  different  attenuation,  scattering, and  microphysical models. New models can easily be introduced as they become available. 2.  Any shape of rain-cell with any type of rain rate distribution (including rain maps) could be introduced.  3.  Accuracy: Accuracy increases with running time (or running time increases with more accuracy)  4. Dynamic: The program can be made more accurate as more advanced computers enter the market by varying the integration steps. However, this program should only be seen as a research tool and not a field prediction model. It is unlikely that it would be adopted by the ITU-R as the standard prediction model because of its long computation time and its large memory requirements. This may change, however, with the advent of better and faster computers.  University of British Columbia  31  Electrical Engineering  Chapter 3: Interference Models  Chapter 3: Interference Models  In chapter 2, we discussed the theory necessary to evaluate interference for a single raincell. A procedure was presented to calculate the interference, with and without the presence of a melting layer and a program was developed to implement the procedure. This program, the engine of the simulation, will be used by the different meteorological models, discussed here, to calculate the interference levels over many rain cells. In this chapter, the meteorological models, that generate the different locations, characteristics (rain rate, rain rate distribution, rain height, rain-cell diameter, etc ...) and frequency of occurrence of rain cells, are introduced. For each rain-cell, with its unique location and characteristics, the interference level is calculated. A histogram of the different interference levels and their associated probabilities is made and interference statistics are generated. Several meteorological models are presented. The COST 210 model [5], adopted by the ITU-R as the standard model [4] to predict interference statistics for microwave links, is discussed in section 3.1. This model, however, may not be wholly realistic since it assumes scattering to exist in a cell, with a constant rain rate, located at the center of the common volume. Another model, that is based on radar observations, the Capsoni model [6, 7, 34], is presented in section 3.2. This model considers rain cells where the rain rate  University of British Columbia  32  Electrical Engineering  Chapter 3 : Interference Models  decreases exponentially, from a maximum at the center of the rain cell. The average rate of decay of rain rate is derived using radar data, and a variation around this average is assumed. The rain cell is allowed to move in a domain of mobility to simulate the movement of rain cells. The same procedure is applied to rain cells with gaussian, rather than exponential, rain rate distribution. This is discussed in section 3.3. The Capsoni model assumes that only cells with rain rates 5 mm/h and higher exist. The -  Awaka-modified Capsoni model [8] extends this to rain-cells with maximum rain rates just above (14 mm/h (section 3.4). Also, the average rate of decay of rain rates in the rain cell is used, instead of the full distribution. This assumption considerably simplifies the computational requirements of the model. In section 3.5, the different interference models are discussed and in chapter 7 and 8, examples are presented comparing the different levels of interference prediction offered by the different models.  3.1 COST 210/ ITU Interference Model [4,5] It is necessary here to distinguish between the COST 210 meteorological model (which is described in sections 3.1.1 and 3.1.2 and the methods in which the radar equation is evaluated, which are described in chapter 2. In the rest of this work, we use the same assumptions as COST 210 regarding the cell size, rain height, rain distribution, reflectivity and attenuation profiles but we use different attenuation and scattering models and different means to evaluate the radar equation to eliminate, as much as possible, approximations and to be able to accurately include the melting layer effect.  3.1.1 Rain-cell structure  University of British C o l u m b i a  33  Electrical Engineering  Chapter 3: Interference Models  COST 210/ITU-R RAIN-CELL  CAPSONI-TYPE RAIN-CELL R a i n rate i  Figure 3-1: Rain rate, attenuation and scattering profiles in both the ITU-R rain-cell and the Capsoni-exponentially decaying rain-cell. The rain-cell center is assumed to be centered at the intersection of the main-beam antenna axes (i.e., center of the common volume) (figure 2-1). Scattering is assumed to occur within a fixed cylindrical rain cell of constant rain rate (figure 3-1). The diameter of the cell depends on the rain rate, R, according to:  d = 3.3R~°  km  m  c  (3.1)  On the other hand, attenuation is assumed to occur inside and outside the rain cell. Inside the rain cell, attenuation is constant (figure 3-1). Outside the rain cell, the attenuation T , H  between the edge of the rain cell and a point at distance d, is given by:  l-e- d/r  r =A B  / (  ,  dB/km  (3.2)  COS£  where r , the scale length for rain attenuation, is given by: m  University of British Columbia  34  Electrical Engineering  Chapter 3: Interference Models  Ice& Snow  FR  h  Melting Laver  Rain  Rain  ITU-R rain cell  Modified ITU-R rain  Figure 3-2: ITU-R rain cell vs. modified ITU-R rain cell.  r =600/J" 10"'* *" as  km  +,)  m  (3.3)  where e is the elevation angle, and A is the specific attenuation for rain in the cell. R  Equation 3.2 is valid if the whole path is located below the rain height, h . If only part of FR  the path (between distances d and d from the edge of the rain cell) is below the rain x  2  height, the attenuation T becomes: R  -</, lr„,  _  - J , / r ,  r =Ar-  dB/km  R  (3.4)  COS£  For the portions of the propagation path that are above h , zero attenuation is assumed. FR  3.1.2 Modified rain-cell structure The rain cell, described above, has been modified to include the effect of the meltingsnow layer. The top of the melting layer is assumed to maintain the same rain height, h , FR  University of British Columbia  35  Electrical Engineering  Chapter 3: Interference Models  as the original model. In effect, the melting layer is displacing part of the rain medium (figure 3-2). The diameter of the melting layer is assumed to be the same as that of the rain below it. Outside the rain cell, melting-layer attenuation is assumed to decrease at the same rate as rain attenuation. Since the specific attenuation varies with height in the melting layer, equation 3.2 becomes:  r =-^-  dB/km  R  (3.5)  COS£  where A  mi  is the average specific attenuation in the melting layer, inside the cell, for the  same height as the region between d and d , i  M  and n is the number of integration steps.  Equation 3.5 states that melting layer statistics for the core-cell are extended into the attenuation-only outer region. This may not be valid, however, since it implies that the statistics of the presence of the melting layer in the outer region is independent of the point rain rate and is only dependent on the presence of the melting layer in the core-cell. 3.1.3  Procedure for obtaining the CDF of transmission loss  While the study of interference levels as a function of rain rates, rain heights, frequency, etc ... is important, system designers are interested in the cumulative effect of all of these parameters (interference statistics). The procedures used to obtain interference (or transmission loss) statistics, with and without the presence of a melting layer, are presented below. 3.1.3.1 Case 1: no melting layer  University of British Columbia  36  Electrical Engineering  Chapter 3: Interference Models  The cumulative distribution function of transmission loss due to hydrometeor's scatter may be calculated in the following steps: 1. For each combination of rainfall rate and the rain height, the transmission loss is calculated. With n discrete rain rates and m discrete rain heights, there are mxn interference scenarios. 2.  Rain rate and rain height statistics are assumed to be independent. Therefore, the probability of the occurrence of an interference scenario P = P x P tj  l  jt  where P is the s  probability of occurrence of rainfall rate /?,, and P. is the probability of occurrence of rain height h  FR  .  3. The transmission loss values, calculated for each of the above-mentioned interference scenarios, are arranged in ascending order and their corresponding probabilities are summed up to generate the cumulative distribution function. In order to avoid large quantization errors, it is recommended that at least 10-20 rain rate values be involved in the calculations. 3.1.3.2 Case 2: with melting layer present With the melting layer present, the procedure to calculate the cumulative distribution of transmission loss is very similar to the one presented in case 1. However, in this case, 2x mxn interference scenarios are possible. The multiplication by 2 accounts for the two scenarios, when the melting layer is and is not present. Therefore, the probability of the occurrence of an interference scenario P = P x P xx tj  and Pa = P<x PjX(\-x ) m  University of British Columbia  l  }  m  when the melting layer is present,  when the melting layer is not present, where x  m  37  is the rain rate-  Electrical Engineering  Chapter 3: Interference Models  Input:  Computation of L for: i- Melting layer not present ii- Melting layer present  General parameters Rain heights statistics Rainfall rate statistics Melting layer statistics  ;i  r  T  Calculate: Receiver coordinates Receiver elevation angle Locations of rain-cells  Histogram of L for: i- Melting layer not present ii- 100% ML present iii- x % ML present  Calculate: Melting layer thickness % of presence of melting layer  Cumulative distribution of L for: i- Melting layer not present ii- 100% ML present iii- x % ML present  i I  .  '  i- Rain only ii- 100% ML present iii- x % ML present  j  |  j  i  Calculate: Cumulative statitics for  F i g u r e 3-3:  *  j ! !  Output  Flow-chart diagram for the modified ITU-R interference model driver  program.  dependent probability of presence of the melting layer. The rest of the procedure is the same as in case 1.  3.1.4 Software The above procedures are implemented in a program. The program calculates the parameters for the different interference scenarios. These parameters are used by the main interference module, discussed in chapter 2, to compute the transmission loss for each interference scenario. Figure 3-3 shows a simplified flow chart of the program. The program can generally be divided into:  University of British Columbia  38  Electrical Engineering  Chapter 3: Interference Models  1. Reading input parameters such as rain-rate statistics, link geometry, characteristics,  melting  layer profile, thickness  and initial-snow  antenna  density, and  attenuation and scattering models to be used, rain height variations, etc. 2. Calculating the parameters for the different cases of interference and their associated probabilities of occurrence. 3. Invoking the main program for transmission loss calculation. 4.  Finding the transmission-loss cumulative distribution function.  5. Generating output. The general parameters for the interference geometry are contained in the "parameters" file. The rain statistics are contained in the "rain_pr" file, the rain-height statistics in the "hfr_dh" file, and the percentages of presence of the melting layer as function of rain rate in the "ML_pr" file. Samples of these files are given in appendix A. Steps 2 and 4 are explained above in sufficient detail. Several types of output files are available. The most important are "list2" and "CDFs". The file "list2" provides the interference levels versus rain rates for different rain heights. The file "CDFs" contains the transmission loss cumulative distribution function for: 1- Melting layer has a zero probability of occurrence (original ITU-R model). 2- Melting layer has a 100 % probability of occurrence below  R , and zero probability above R , where R max  max  inax  is the  maximum rain rate the melting layer can exist. 3- Melting layer has x % probability of m  occurrence. The input and output files, along with the simulation software, are presented in detail in appendix A.  University of British Columbia  39  Electrical Engineering  Chapter 3: Interference Models  3.2 Capsoni's 3D Interference Model [5,6,7] 3.2.1 Rain-cell structure The original model assumes rain cells with the same vertical structure as the ITU-R model. A rain medium with constant attenuation and reflectivity, vertically, is assumed to extend up to the 0°C isotherm. Reflectivity, in the ice and snow region above the 0°C isotherm, is assumed to decay at a rate of 6.5 dB/km and attenuation is assumed zero. The horizontal distribution of rain is modeled through a population of synthetic rain cells. The properties of these cells are deduced from radar-derived rain maps in the Po river valley, Italy [34,35]. Two  rain-cell models are developed: the biaxial and monoaxial rain-cell models. The  biaxial model accounts for the spatial asymmetry of rain cells by modeling them as ellipses. This model is not considered, however, because it requires significantly more computation time than the monoaxial model, especially for interference applications. The monoaxial model assumes rotational symmetry of rain rate in the rain cell. Also the rain rate is assumed to monotonically decay, exponentially, as we move away from the center of the rain cell, where the maximum rain rate occurs. The maximum rain rate in any rain cell was found to be no less than 5 mm/h and, on average, the rain rate distribution in the rain cell is well represented by (figure 3-1): R(r) = R exp(-r/r ) M  (3.6)  0  where r is the distance from the center of the rain-cell center, R is the peak rain intensity M  at the center of the rain-cell, and r is the distance from the center of the rain-cell where 0  the rain rate decays to R /e (Referred to as the rate of decay of rain in the cell in this M  work. The higher r is, the lower the rate of decay of rain in the cell is). 0  University of British Columbia  40  Electrical Engineering  Chapter 3: Interference Models  For a given peak rainfall rate, a distribution of r is observed. The probability of this 0  distribution is found to obey: exp(-r / < r >) 0  Q  (3.7)  <r > Q  where <r > is the average rate of decay of the rate in the cells, which is found to be 0  related to the maximum rain rate in the rain-cell according to: -10  <r  (3.8)  +  >=1.7  Q  -0.26  v  j  w  v  ;  u  Even though the above quantities are derived from radar observations at one specific site, the authors of the model argue that equations 3.6-3.8 are valid for any location. The difference, from one location to another, is the probability of occurrence of the different rain cells (with different peak rain rates and rates of decay). Using equation 3.7. the cells joint spatial density N(r ,R ) 0  M  (the number of cells in km and for unit ARM and 2  A(ro/<r >)) is given by: 0  N(r ,R ) = N(R )exxi 0  M  M  (3.9)  V o> < r  where N(R ) is the spatial density of cells with peak rain rate R , for all r . M  M  0  The only unknown in equation 3.9 is N(R ). Using the concept of area probability, N(R ) M  M  is related to the local rain intensity cumulative distribution function P(R):  4n<r > 0  (3.10)  P"(R)  R  M  University of British Columbia  41  Electrical Engineering  Chapter 3: Interference Models  where P  is the third order derivative of the P(R) with respect to ln(R). In order to  guarantee a positive N(R ), P(R) function is modeled by means of log power law of the M  form:  R.,. ^  r  P(R) = P  n  In  V  R  (3.11)  J  where P , /?„.„•„ and n are parameters found by fitting equation 3.11 to the discrete rain rate 0  cumulative distribution function. Using equation 3.11 into 3.10 yields:  P n(n-\)(n-2)  R..  0  4n<r (R )>  2  0  M  R  M  V  "  R  \«-3  (3.12)  J  Using this model, rain rate statistics are then the only meteorological information needed to obtain the population of synthetic rain-cells necessary to model interference in any region.  3.2.2 Procedure for obtaining the CDF of transmission loss #  In section 3.2.1, a general description of the model was given, fn this section, the procedure to obtain the cumulative distribution function of transmission loss is examined in detail. 3.2.2.1 Case 1: no melting layer The different rain cells are characterized by the maximum rain rate at the center of the rain cell (R ) and the rate at which the rain rate decay (r ). In order to limit the M  0  computation time, 100 rain cells, corresponding to  \0xR and 10xr , are used to calculate M  0  interference, with:  5<R  M  <0.SR  0.6 < rn  aSin  < 20  (3.13.a)  mm/h.  (3.13.b)  km  University of British Columbia  42  Electrical Engineering  Chapter 3: Interference Models  F o r each o f the rain cells, it is necessary to truncate the radial distance at a certain distance  r , where max  the rain rate b e y o n d this point is assumed to be zero. T h e rain-rate  threshold (R ) s h o u l d be l o w enough that it can not cause significant interference (R s  s  0.4 m m / h is r e c o m m e n d e d ) . T h e radius o f the rain-cell is then g i v e n by:  R. ^  f  (3.14)  R, M  V  J  In order to determine the number o f cells w i t h R  M  peak rainfall and r rate o f decay, u s i n g 0  equations 3.9 a n d 3.12, it is necessary to evaluate the u n k n o w n parameters in equation 3.11. T h e parameters  R  , P , n are  aiin  0  determined by fitting  P(R)  to the e x p e r i m e n t a l l y  o b t a i n e d point rain rate statistics. T h e parameter  ^«sin  -  4  x  ^ m a x  R  usm  w  a  s  is chosen large enough to a v o i d negative rain-rate values. A value o f suggested [8], w i t h  R  asin  < 4 0 0 m m / h , where / ?  m a x  is the largest rain  rate g i v e n i n the p o i n t rain-rate statistics d i s t r i b u t i o n . T h e quantities n a n d P are obtained u s i n g linear regression, as f o l l o w s : 0  Jc = l n  a  sin  and  y = \n(P)  (3.15)  v R JJ Then  n=  N  (3.16)  and  University o f British C o l u m b i a  43  Electrical Engineering  Chapter 3: Interference Models  % Time Figure 3-4: Comparison between fitted (solid, o) and original (dashed, +) CDFs of rain rate.  exp  (3.17)  N  where N is the number of points in the rain rate statistics distribution. Figure 3-4 shows close agreement between the fitted and original rain rate cumulative distribution function. Moving rain-cells can assume any position relative to the center of the common volume. For each rain cell, a "domain of mobility" could be defined as a circle of radius r  max  centered at the ground projection of the center of the common volume. For simplicity, the use of a square box of size 2  was suggested [5] (figure 3-5) with coupling between  two independent communication systems occurring only when the intersection point of the main beam axes of the antennas is closer to the cell peak than r . max  The above assumption is valid only for interference scenarios with small common volume. If the common volume is large (A wide-beam receiving antenna located at a large distance from the common volume, as in the case of the Chilbolton-Baldock link),  University of British C o l u m b i a  44  Electrical Engineering  Chapter 3: Interference Models  Figure 3-5:  Domain of mobility of a rain-cell divided into a grid of squares. Interference calculations are done with the rain-cell centered in each square. While cell 1 will definitely cause interference, cell 2 may, under the right conditions, cause interference.  rain cells outside the "domain of mobility" could contribute to interference (figure 3-5). This is especially true for the small domains of mobility associated with the smaller rain cells. A rain-cell is assumed to have the same probability to be at any location inside the domain of mobility. The domain of mobility is subdivided into nxn squares. The transmission loss is calculated with,the center of the rain-cell located at the center of each square (subdomain). The size of the subdomains should be small enough to guarantee statistical stability. A subdomain, of surface area AS and side AX, is suggested [5]: AA = - r l n ( l - A / ? )  (3.18)  0  with AR = 0.4, corresponding to 40% rain rate variation. The transmission loss is computed for every combination of r , RM and position in each 0  subdomain, and probabilities assigned to each combination. The histogram of the probability density function of transmission lpss is then given by:  University of British Columbia  45  Electrical Engineering  Chapter 3: Interference Models  P(L,AL) = j£j£ff(L,A£,/^  where H(L,AL,R' ,r^) M  (3.19)  is the histogram of the number of rain-cell positions generating a  calculated transmission loss  L'KLKL'+AL.  The last step in the procedure consists of integrating the probability density function to obtain the cumulative distribution of transmission loss:  P(L>L')=Y P{L,M)AL  (3.20)  J  L=L'  Equation 3.19 does not consider the variation of rain height around its median. In this work, a variable rain height is considered. The histogram of the probability density function of transmission loss, for the whole rain-height distribution, is obtained by generating different histograms for each rain height (HFR) using equation 3.19. These are multiplied by the probability of occurrence of the corresponding rain heights and summed. In other words equation 3.19 becomes:  m.AL) =  xff>(L.AL,/4,/M  jt=j i=i ./=i  (3-21) < r  o(^w)>  where f(h R) is the probability density function of the rain-height distribution and n is the F  h  number of samples in the distribution. 3.2.2.2 Case 2: with melting layer present In this work, the melting layer is introduced by substituting the scattering and attenuation values in the original model with those of the melting layer, where appropriate. In this case, the procedure to obtain the interference statistics is the same as in case 1. Essentially, equations 3.19 and 3.21 remain the same with the appropriate substitution of  University of British C o l u m b i a  46  Electrical Engineering  Chapter 3: Interference Models  H. (L, AL.R'  M  ,rg j for the original model with the those obtained for the modified model.  The transmission loss statistics are calculated as in section 2.5.2 by a weighed average of transmission loss calculated for the two cases when the melting layer is present at all time and the case when the melting layer is not considered. 3.2.3 Software The above procedures are implemented in a program. The program calculates the parameters for the different interference scenarios. These parameters are used by the main interference module, discussed in chapter 2, to compute the transmission loss for each interference scenario. Figure 3-6 shows a simplified flow chart of the program. The program can generally be divided into: I. Reading input parameters such as rain-rate statistics, link geometry, antenna characteristics, melting layer profile, thickness and initial-snow density, and attenuation and scattering models to be used, rain height variations, etc. 2. Calculating the parameters for the different cases of interference such as diameter, rain rate distribution and location along with their associated probabilities of occurrence. 3. Invoking the main program for transmission loss calculation. 4. Finding the transmission loss cumulative distribution function. 5. Generating output. The general parameters for the interference geometry are contained in the "parameters" file. The rain statistics are contained in the "rain_pr" file, the rain-height statistics in the  University of British Columbia  47  Electrical Engineering  Chapter 3: Interference Models  "hfr_dh" file, and the percentages of presence of the melting layer as function of rain rate in the "ML_pr" file. Samples of these files are given in appendix A. Steps 2 and 4 are presented in sufficient detail in section 3.2.1-3.2.3. Two types of output files are generated, both are discussed in details in appendix A. They are "interm" and ''CDF.m". The file "interm" provides the individual interference levels for each of the interference scenarios along with the probability of occurrence, rain rate, rate of rain decay, rain height and cell location. The file "CDF.m" contains the cumulative distribution function of transmission loss for the following relevant cases: 1- Melting layer has a zero probability of occurrence (original model). 2- Melting layer has a 100 % probability of occurrence below R  inax  and zero probability above R„ . ULK  The transmission  loss for an x% of presence of melting layer is then calculated by taking the weighed average of the above two cases as discussed in chapter 2.  3.3 Capsoni's 3D Interference Model with a Gaussian Rain-rate Distribution The procedure to calculate interference statistics using rain cells with a gaussian rain rate distribution is essentially the same as in the Capsoni-exponential rain-cell model. However, the different rain-rate distribution in the rain cell forces a change in its radius, which becomes: r.max =  (3.22)  r J-\n(R /R ) 0  S  M  The spatial density of cells with peak rain rate RM, for all r , also changes to: 0  P n(n-\) 0  2/r < r >  2  0  R  2K <r >  2  M  University of British Columbia  n  48  RM  In  R,'a sin  m-2 (3.23)  Vw J A  Electrical Engineering  Chapter 3: Interference Models  Input: General parameters Rain heights statistics Rainfall rate statistics Melting layer statistics  Computation of L for: i- Melting layer not present ii- Melting layer present, 100 % iii- Melting layer present, x % a  Calculate Histogram of L for: i- Melting layer not present ii- 100% ML present  Parameters P, Rosin, n  Receiver coordinates Receiver elevation angle  Calculate:  Cumulative distribution of L for: i- Melting layer not present ii- 100% ML present  <Tf>  Coordinates of the rain-cells  Calculate: N(r ,RJ Parameters fed into interference module 0  Output  Figure 3-6: Flow-chart diagram for the modified Capsoni interference model driver program.  Because of the smaller size of the gaussian rain-cell (figure 3-7), its domain of mobility is smaller than that of the exponential rain-cell. Also the smaller size of the rain cell decreases the computational requirements of the model.  3.4 Awaka-Modified 3D Interference Model [8] Awaka introduced two major modifications to the Capsoni's 3D model. The first was to extend the model to rain cells of maximum rain rate lower than 5 mm/h. The second was to introduce an average rate of decay of rain, <ro>, in the rain cells that is valid for rain rates between 0.4 mm/h and 500 mm/h:  University of British Columbia  49  Electrical Engineering  Chapter 3: Interference Models  Exponential-Equ.3.8  E C  '§ 10  Exponential-Equ.3.24  1  to Gaussian-Equ.3.24  CD  Y  c  '5  CC  Gaussian-Equ.3.8 10* 10  10°  10' Rain rate in mm/h  10'  10°  Figure 3-7: Diameters of the rain cells with exponential and gaussian rain rate distributions, assuming a rain-rate threshold, at the edge of the cell, R = 0.4 mm/h. x  <r >= 0  10-1.51og, R 0  (3.24)  M  ln(/? /0.4) M  An examination of figure 3-8 shows that Awaka's rates of decay follow closely the ones derived by Capsoni for rain rates above 10 mm/h while at the same time avoiding the excessive behavior around 5 mm/h and the exclusion domain below it. The effect of which can be seen in figure 3-7, where the radii of both rain-cells agree closely at higher rain rates and diverge significantly around 5 mm/h rain rates. A distribution of rates of decay around the average <r#> is not considered. The assumption of a constant <r > in 0  the rain cell reduces computation time considerably.  The equivalent to equations 3.19 and 3.21 for the Awaka procedure are, respectively:  University of British Columbia  50  Electrical Engineering  Chapter 3: Interference Models  10'  10  10°  10  —  10"  —  10°  1  •  —  —  .  —  —  10 Rain rate in mm/h  10  1  .  —  2  I  10  3  Figure 3-8: The Capsoni-derived rate of decay of rain rate (solid) is contrasted to the Awaka-suggested rate (dashed).  P(L,AL)  = ZJ H(L,  AL, R'  P(L,AL) =^ ^ H ( L A L , R l  i M  ) x AS x N(R' ) x AR  (3.25)  L  M  ) x A S x N ( R  M  i M  ) x A R  i M  M  xf(h  k F R  )xAh  F R  (3.26)  3.5 Comments on the Interference Models Many models, or variations of the models, have been presented thus far to calculate interference statistics between communication links. They are: 1. COST 2 lO/ITU-R model •  Original model: no melting layer is assumed present  •  Modified model: with melting layer assumed present  2. Capsoni model (exponential rate of decay)  University of British Columbia  51  Electrical Engineering  Chapter 3: Interference Models  •  No rain height variations => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present  •  With rain height variations => Original model: no melting layer is assumed present Modified model: with melting layer assumed present  3. Gaussian-Capsoni model •  No rain height variations => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present  •  With rain height variations => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present  4.  Awaka-exponential model •  No rain height variations => Original model: no melting layer is assumed present =s> Modified model: with melting layer assumed present  •  With rain height variations => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present  5. Awaka-Gaussian model •  No rain height variations  University of British Columbia  52  Electrical Engineering  Chapter 3: Interference Models  => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present •  With rain height variations => Original model: no melting layer is assumed present => Modified model: with melting layer assumed present  There is a total of 18 different models. However, because of computational constraints, only the ITU-R and the Awaka-type models are used on a regular basis. Calculations of the Capsoni treatment is shown to illustrate the validity of the Awaka model. Also the exponential and gaussian models are shown to give similar results. Rain height distributions are used in all calculations, except where stated otherwise. Then the two models, that need to be considered in detail, are the ITU-R model (with and without the melting layer) and the Awaka-exponential model (with and without the melting layer).  University of British Columbia  53  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  Chapter 4: Spatial Structure of Precipitation  In order to model the effect of precipitation, it is necessary to understand both the macroscopic and microscopic properties of storms. The macroscopic characteristics of precipitation include the size, distribution and movement of rain-cells, as well as rain-rate profiles in the cell and rain height. The microscopic characteristics of precipitation include type, composition, size, distribution, density, and shape of the precipitation particles. In the previous chapters, we made assumptions about the structure of precipitation. In the next two chapters, it is hoped that these assumptions will become clearer. In this chapter, the macroscopic properties of precipitation are examined, while the next chapter deals with the microscopic aspect of precipitation. The two chapters are by no means exhaustive. They are intended to provide the necessary elements to the understanding of this work and supporting the modeling effort. Precipitation types  Two categories of rain are of primary interest: convective rain and stratiform rain. Convective rain occurs in cells, typically in weather fronts. It produces intensive rainfall rates of short duration (few minutes to one hour) and limited extent (few kilometers University of British Columbia  54  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  radius). In North America, most rainfall exceeding 25 mm/h is the result of convective storms [36]. Stratifrom rainfall covers a wide area (few tens to hundreds kilometers), lasting typically more than one hour with rain rates lower than 25 mm/h. The meltingsnow layer is usually associated with stratiform rainfall. However, the melting layer has been observed in structures with significantly higher rain rates (~ 40 mm/h.) [9]. Precipitation can also take the form of hail, sleet, graupel, freezing rain and snow. From a propagation point of view, rain and melting snow have the greatest effect on microwave communications links.  4.2. Climatic regions According to the rTU-R , the world may be divided into 15 distinct rain climates (A-Q). An average annual cumulative statistics of the rainfall was assigned to each climate [37]. The Global model [28] by Crane divides the world into polar (Zones A and B), temperate (Zones C and D), subtropical (Zones E and F) and tropical (Zones G and H) climatic regions. Each of these regions is subdivided into two zones. Zones B and D are subsequently subdivided into regions B l , B2 and DI, D2, respectively. For example, Vancouver is located in rTU-R rain climatic zone D, rain zone B l , near zone C, in Crane's classification. The long-term statistics (10 years) for the Vancouver region are given in table 4.1 [39]. These agree with the ITU-R zone D statistics. However, a recent study [40] revealed a dissimilar behaviour from that of the traditional zones description for rain rates percentages of 1 and over. rTU-R classifications of climatic regions are constantly being revised as more data and studies become available. For  University of British Columbia  55  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  example, a recent study [41] showed that the rTU-R climatic division of Saudi Arabia is inadequate and proposed a new climatic map. When using rain-rate statistics in propagation modeling, it is preferable to use the measured long-term statistics for the region; ITU-R and Global model rain rate distributions should only be used when such measured statistics are not available.  4.3. Horizontal structure of precipitation 4.3.1. Rain rate statistics Rain-rate statistics are perhaps the single most important variable for meteorologists and microwave propagation workers. Point rain rate is needed for many propagation models. Rain rate exceedance percentages (shown in table 4-1) are the percentage of time that a given rain rate will be equaled or exceeded. Rain rate statistics can be long term, yearly, seasonal and monthly. Worst month statistics are becoming ever more important for work in microwave propagation since much of the heavy rain tends to occur during a relatively short period in the year. The distribution of precipitation in both space and time can vary significantly. In thunderstorms, rainfall rates may vary by tens of millimeters per hour from minute to minute over a few hundred meters [42]. Even in stratiform rainfall, where rain rate is assumed to be widespread and uniform, there are considerable variations. Because of these variations, it is not possible to place a network of rain gauges with enough density to cover a large area. This is especially true if it is desirable to detect precipitation over water; it then becomes necessary to use radar to extrapolate the point rain rate.  University of British Columbia  56  Electrical Engineering  Chapter 4: Spatial Structure or'Precipitation  Percent of Year 0.001 0.002 0.003 0.005 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1.0 2.0 5.0  ITU-R, Zone: D (mm/h.) 42.0 -  Crane, Zone B l (mm/h.) 45.0 34.0  Crane, Zone C (mm/h.) 78.0 62.0  29.0  -  -  -  22.0 15.5 11.0  41.0 28.0 18.0  19.0 -  13.0 8.0 -  -  -  6.4 4.2 2.8  11.0 7.2 4.8 2.7 1.8 1.1 0.5  4.5  -  -  1.5 1.0 0.5 0.2  ,2.1 -  Measured (mm/h.) 31.0 25.0 21.0 16.5 13.0 10.0 8.6 7.6 6.0 5.0 4.2 3.5 2.5 -  Table 4-1:  Comparison between measured rainfall statistics [39] and models [28] for the Vancouver region.  4.3.2. Radar rain maps Rainfall  distribution could be mapped  using radar reflectivity (preferably  dual  polarization). Such maps could be used to model attenuation and interference between links. They would also give a better understanding of the way precipitation occurs. A typical radar reflectivity map of precipitation is shown in figure 4-1. However, radar accuracy in estimating rain rate from reflectivity is limited. For single polarization radar, differences in the order of 50% have been recorded between radar and raingauge measurements. With dual polarization radar, the difference decreases to 22% [43]. For instantaneous rain rate measurements, the error can be significantly larger. These errors may be due to the canting angle of rain drops, signal fluctuation from precipitation [44], variability of the Z-R (reflectivity to rain rate) relationship due to the variation in the drop-size distribution [45] and the type of precipitation measured (rain, ice, snow, melting-snow etc.).  University o f British C o l u m b i a  57  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  Figure 4-1: Reflectivity field (dBz) from a tropical cyclone observed with a C-band radar on Taiwan. The coastline is shown [42].  A disadvantage of using radar reflectivity maps in propagation modeling are excessive run time and storage required for computer simulation. Also radars, with suitable data collection and storage, are very expensive. State-of-the art weather radars are not available in many countries, and even in countries like Canada, they cover only a fraction of the country. In these areas, point rain rate measurements using rain gauges are the only means of collecting rain statistics.  4.3.3. Simulated rain maps A rain cell is defined as any continuous region where rain rate exceeds a certain threshold. Rain-cell structure statistics are very important in modeling fading and interference caused by precipitation. This is especially true for earth-satellite systems  University of British C o l u m b i a  58  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  operating at frequencies above 10 GHz where the fade margin is small. Rain-cell structure statistics are also important in the proposed measurements of rain rate from space, where a satellite footprint may contain one or more rain cells [46] [47]. Rain-cell characteristics (such as rain-cell diameters, distribution of the rain rates in the rain cell) have been investigated by many researchers using rain gauge networks (e.g.,[33] [48]) and radar (e.g., [34] [35] [46] [49]) which resulted in several rain-cell models. These show that, statistically, the diameter of intense rain areas decreases as the average rain rate within the area increases. In this work, we will deal primarily with two models as discussed in chapter 3. The first is the rTU-R rain-cell model. This model is an extension of the Misme and Finbel rain-cell model [50]. It assumes a cell with a circular radius r and a constant rain 0  rate. The second model (referred to as the Capsoni-type rain-cell model) is perhaps more realistic. It is based on radar data near Milano, Italy. Averaged profiles of radar-derived rain rate maps show cell-like behavior in rain with the rain-rate maximum at the center of the cell. The rain rate decreases with increasing distance from the center of the cell (figure 4-2)  and is best represented by an exponentially decaying function [34]  [35]. A  distribution of the different rate of decay of rain rates in the rain cells was derived using the radar observations. Another model of rain rate variation in the cell, a gaussian variation with axial symmetry, was derived using rain gauge data [33]. These three raincells are the basis of the interference models used in this work.  4.4. Vertical structure of precipitation  University of British Columbia  59  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  JL  CONDITIONAL STANDA.10 DEVIATION t  i  i  i  5*00 3M0 WOO NEGATIVE OISTANCE FROM MAXIMUM  I 0 ^  (<>)  B„-20mm/h  1(00 3600 5(00 H POSITIVE DISTANCE FROM MAXIMUM  Figure 4-2: (a) Average normalized profiles as a function of slant distance and relative to the three maximum ranges indicated, (b) Same as (a) but relative to the standard deviation [35].  Precipitation particles are produced by two principal mechanisms. The first mechanism is called the coalescence process, where water particles in the clouds, falling at different velocities, collide and merge together forming larger particles. When the water particles are large enough they fall in the form of rain precipitation [51]. The second process is called the ice-crystal mechanism, and it can give rise to the presence of the melting layer. Much of the rain reaching the earth's surface (outside the tropics) results from melted-snow particles. The snow particles are aggregates of ice crystals. The ice-crystal mechanism occurs in a mixed phase cloud where both ice crystals and supercooled water are present at subfreezing temperatures. Initially, aerosol particles  University of British Columbia  60  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  act to form ice crystals. The ice crystals grow, by water vapor deposition, while the supercooled water droplets evaporate. Once they reach the size of a few hundred microns, ice crystals aggregate together through collision to form snowflakes. Large crystals and snowflakes can grow by colliding with supercooled water droplet in a process called accretion or riming. Snowflakes that fall through the 0°C level melt and collapse into rain droplets. However, if temperatures are low enough, the snowflakes may reach the ground. This is especially true at high latitudes [52]. At cloud tops, where temperatures are well below 0°C, small ice crystals of low concentration give rise to low radar reflectivity. As the particles descend through the cloud, the temperature increases and the particles aggregate and multiply, the radar reflectivity increases. This increase in reflectivity continues as ice particles aggregate into snowflakes, usually up to the top of the melting layer. The melting-snow layer is the region below the 0°C isotherm in which precipitation changes from snow to rain (figure 4-3). Higher reflectivity is observed from this layer. The reasons for the increased reflectivity are: (1) that the dielectric constant of water is many times higher than that of ice [53], (2) the large size of the particles and (3) the higher concentration of the melting-snow particles relative to those of rain drops. Radar observations show the rain medium to be statistically vertically homogeneous up to the edge of the melting layer (i.e., the reflectivity exceeded for specified percentage of time is constant). The rain medium is well understood and particle size distributions are available for different rain types. Using a vertically pointing radar, Fabry et al. [9] grouped the resultant vertical reflectivities into five classes:  University of British Columbia  61  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation Otmm  tMoc*t.  mftmc  Figure 4-3:  Two views ofthe radar bright band: at the left a vertical profile of reflectivity and Doppler velocity as measured with vertically pointing radar; at right a PPI map at 8° elevation on which the melting layer appears as a bright ring at about 12 miles [53].  1.  Low level rain: where precipitation forms directly in the liquid form and no melting is present.  2.  Rain with bright band: by far the most common profile in temperate latitudes and at low to moderate rain rates.  3. Rain from compact ice: no clear bright band can be found and the profile undergoes a quick transition from the rain regime to the ice regime 4. Showers: This type of echo is the convective counterpart of the class 1 echo. Here too, the precipitation forms directly in the water phase. 5. Deep convection: exclusively generated by thunderstorms and squall lines. Although frozen precipitation often occurs in deep convection, there is never a clear reflectivity signature associated with melting.  University of British Columbia  62  Electrical Engineering  Chapter 4: Spatial Structure of Precipitation  The rate of occurrence of these different classes as a function of rain rate for the Montreal area is given in figure 4-4 [9]. Only in the second class is the bright band observed.  10  20  30  40  Equivalent reflectivity factor of rain (dBZ)  50  Figure 4-4: Relative occurrence of the 5 classes of vertical reflectivity profiles stratified by the reflectivity of rain near the surface. The number of hours a given reflectivity (±1 d B ) was observed in Montreal over the vertically pointing radar is shown at the top [9],  University of British C o l u m b i a  63  Electrical Engineering  Chapter 5: Modeling Precipitation: Melting-Snow Layer  Chapter 5: Modeling Precipitation: Melting-Snow Layer  Precipitation is a medium formed by an ensemble of particles of different sizes. The scattering properties of a medium are determined by the scattering properties of the individual particles and by the number of particles of a certain size in a unit volume of the scattering medium. The scattering properties of individual particles depend on their size, shape and composition. Precipitation particles come in many forms; They include snow (from loose snow to ice), melting snow and rain particles. These different types of particles have different permittivity and sizes and hence different scattering properties. In snow, different snow densities translate. into different permittivities and hence different scattering properties. For melting-snow, different representations of the melting particle yield particles of different sizes, densities, composition, and hence scattering properties. The scattering property of the ensemble of particles per unit volume is also a function of the number of particles of a certain size in the unit volume (number density). The number density depends on the size, type, density and velocity of the particle and on rainfall rate. With higher rain rates, there are usually more and larger particles present. For snow and melting-snow particles, density and composition determine the size and fall speed of the  64  Chapter 5: Modeling Precipitation: Melting-Snow Layer  particles and hence the number density. In melting-snow, different stages of melting translate into continuously differing particle sizes and velocities and hence different number densities. This chapter examines the microphysics necessary to model the scattering and attenuation characteristics of the melting-snow layer, the snow region immediately above it, and the rain medium below. A model is devised for the melting layer based upon the best available data. Snow and rain will be modeled as the limiting values of melting snow particles (snow is present at the top of the melting layer while rain is at the bottom). The shape, composition and melting process of melting particles are discussed in sections 5.1, 2 and 3, respectively. The fall speed and the particle-size distribution (number density) are discussed in sections 5.4, and 5. The effect of initial-snow density on the strength of the bright band in the melting layer is discussed in section 5.6. The thickness and height of the melting layer determine the type and composition of the precipitation particles with height and hence the scattering properties of precipitation with height. This is discussed in sections 5.7 and 5.8. The percentage of time the melting layer is present, and its size and duration are discussed in sections 5.9 and 5.10. It will be seen that different assumptions about the properties of the melting particles lead to different attenuation and scattering levels in the melting layer. Hence, observations of the bright band (e.g., [9]-[14] and [53]-[75]) are used to examine the validity of the different assumptions used, where possible, and to estimate parameters, such as the initial-snow density, where appropriate. In this chapter all attenuation and scattering computations are done using the Mie scattering theory, except where it is explicitly stated otherwise (refer to chapter 6).  65  Chapter 5: Modeling Precipitation: Melting-Snow Layer  0,25 a50  0.75  1.00  1.25  1.50  = oOOOO  Figure 5-1: Calculated drop shapes of 13 water drops. The numbers indicate the equivolumetric sphere radius [54]. 5.1  Melting Particles Shapes  Studies of the shape and orientation of rain drops have been made by many investigators (e.g., [53]-[58]). These studies show that water droplets smaller than 0.2 cm in diameter are spherical in shape. Particles above 0.2 cm in diameters start to deform into oblate spheroids with a flattened base. However, it could be seen from figure 5-1 that the deformation of the particles is not severe up to a equivalent rain diameter of 0.35 cm. The melting-snow layer is seen in section 5.9 to occur during low intensity rain (mostly below 30 mm/h.). For rain rates of 40 mm/h and under, it can be readily shown, from table 5-5, that the number of rain particles with diameters over 0.35 cm is not significant. This is also true for a rain rate of 150 mm/h, where most of the rainfall (72.1 %) comes in the form of rain drops 0.35 cm or under in diameter (Table 5.5 in section 5.5). For melting layer considerations, it is not only possible to model rain drops as spheres but also desirable since analytical solutions are not readily available for the deformed raindrop-shapes.  66  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  A recent study [28] modeled rain and melting-snow particles as oblate and prolate spheroids. It showed that when the deformation is not severe, the spheroid representation does not offer significantly different results from the sphere representation. The sphere representation is thus adequate for scattering and attenuation purposes.  5.2 Modeling the Melting Particles Melting occurs because a temperature difference exists between the melting particles and surrounding air. Usually, melting is assumed to occur from the outside of the particle, with the water forming a shell around the snow core (e.g., [15] [17] [26] [29] [77]-[80] ). A more recent approach assumes that the water forming on the outside of the melting particle percolates to the inside, forming a mixture of air, water, and ice (e.g., [11]-[13] [17] [76] [77] [78]). The application of these two models lead to significantly different attenuation and scattering properties in the melting-snow layer.  5.2.1 Concentric sphere representation In this representation, the melting particles are modeled as concentric spheres with a layer of water of permittivity e surrounding a dry snow core of permittivity e (e.g., [26]). w  s  Melting is assumed to occur on the outside of the particle and all the melted water is assumed to remain there. A representation of a melting particle with a radius a and a m  snow core of radius a is given in figure 5-2a. Breakup and coalescence in the melting s  layer are ignored (i.e., there is a one to one correspondence between the initial snow particles and the resultant rain particles) and conservation of mass is assumed (i.e., we ignore vapor condensation on the melting particle). Then, a melting particle with a melting degree 5 ( S is defined as the ratio of volume of water to the total volume in the melting particle. S changes between 0 for snow at the top of the melting layer to 1 for rain  67  Chapter 5: Modeling Precipitation: Melting-Snow Layer  am Ice/Water/Snow  (b)  (a)  IceAVater/Snow  (c)  Figure 5-2: Melting-snow particle representations: (a) concentric sphere representation (b) Composite sphere representation (c) percolating composite sphere representation (spongy particle model)  at the bottom of the melting layer) has a radius a calculated, using conservation of mass m  property, from: (5.1)  a... -  where a is the radius of the resultant rain drop and p is the initial-snow density of the R  s  particle. The initial snow particle's radius can be found by setting S to zero in equation 5.1. The snow core radius in the melting particle is obtained from: a =aJl-S)"  (5.2)  3  s  In order to calculate the average permittivity of snow, it is necessary to obtain the volume fractions of ice (x,) and air (x ) in snow (refer to chapter 6). These are given as a function a  of air, ice and snow densities as:  Pi ~ Ps P, ~ Pa  (5.3a,b)  Ps - Pa Pi - Pa  68  Chapter 5: Modeling Precipitation: Melting-Snow Layer  where p , p , p are the snow core density (also the initial-snow density), the air density s  a  t  (0.0012929 g/cm ) and the ice density (0.917 g/cm ), respectively.' 3  3  5.2.2 Composite sphere representation It is possible to treat the melting particle as a sphere composed of an air-ice-water mixture with an average permittivity,£ (figure 5-2.b), e.g. [84]. It will be seen in chapter 6 that in m  order to evaluate the average permittivity, it is necessary to calculate the volumes of water (V ), ice (V,) and air (V ) in the melting particle. Since the mixture composition of this w  a  particle-model is the same as the concentric sphere model, the water volume (V, ) in the v  particle is given by: (5.4)  where V , V m  s  are the melting particle volume and the snow core volume in the  concentric-sphere representation, respectively. Using the conservation of mass and volume properties, the volume of ice and air in the snow core of the concentric sphere particle are calculated from:  V, =  (5.5)  Pi -Pa  v,=v -v = x  t  V,(Pt-P.)  (5.6)  iPi-Pa)  where the snow volume in the melting particle is given by: (5.7)  5.2.3 Spongy particle model  69  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  The melting particle is modeled as a sphere where water is assumed to progressively percolate inside the snow as the particle falls, until saturation is reached, i.e., when all the air in the mixture has been removed (figure 5-2c), e.g. [12, 13]. It is readily seen that this particle will have a smaller size and hence higher density than the above melting particle representations. The density of the melting particle is given by:  for l - x ,  p.=-^—  P  1-*,  w  (5.8)  1  P, = x p f  + (l - x, )p,  w  for 1 - x < f  where x is the melted mass fraction and p f  w  is the density of water. The total mass of the  melting particle is equal to the mass of the resultant rain particle (ignoring the mass of the air fraction). Hence, 3  3  3  a —a a x, = ^ ^ = ^ S a a R  (5.9)  R  The radius of the new melting particle (a ) is then given by: t  f.  y/3  The water volume (V ) in the melting particle can be readily obtained from: w  V =x xV w  f  (5.11)  R  where V is the resultant rain-drop volume. Using the properties of mass and volume R  conservation, the volume of ice is obtained using:  v  _P„(v*-v*)+P.{v»-Vf)  (  Pi -Pa  70  5  1  2  )  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  Degree of  Drop Size  melting (S) ^  (cm)  Drop Size  Drop Density  Drop Density  a  Pm  Pi  Cl  f  m  (cm)  (g/cm ) 3  Terminal Velocity  Terminal Velocity  v  V  f  m  (g/cm ) 3  (m/s)  (  m/s)  Number of Drops per Unit Volume  Number of Drops per Unit Volume  N  N  f  m  (m' )  (nv )  3  3  (snow) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (rain)  0.27 0.22 0.19 0.17 0.16 0.15 0.14 0.14 0.13 0.13 0.125  0.269 0.210 0.177 0.155 0.137 0.129 0.127 0.127 0.126 0.125 0.125  0.10 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1.00  2.07 2.90 3.61 4.21 4.75 5.25 5.72 6.17 6.60 7.01 7.41  0.10 0.21 0.35 0.53 0.77 0.92 0.94 0.96 0.98 0.99 1.00  2.07 3.09 4.08 • 5.14 6.35 7.05 7.16 7.25 7.31 7.37 7.41  354 251 203 174 154 140 128 119 111 105 99  354 237 180 143 1 16 104 102 101 100 100 99  Table 5-1:  A summary of the different stages of melting of a particle. The two melting processes considered are the percolating particle process (denoted by the subscript f) and the non-percolating process (denoted by the subscript m).  where V is the total volume of the melting particle. Since the density of air is negligible, f  it is ignored; hence: V,=^(V -V ) A R  (5.13)  W  The volume of air in the mixture can be readily obtained by subtracting the ice and water volumes from the total volume of the melting particle. Hence, a  V  = f V  - ( K  +  V  (5.14)  i )  Table 5-1 contrasts the size, speed, density and the number density of the melting particles as a function of melting ratio S for the percolating and non-percolating models. The larger size and greater concentration of the non-percolating model suggest higher attenuation and reflectivity levels than those obtained using the percolating model. This can be seen in figure 5-3.  71  Chapter5: Modeling Precipitation: Melting-Snow Layer  Specific attenuation (dB/km) ' K  :  :  . „ . . , , „ ,,. Backscattenng cross secdon (cm^cro^)  v  » -  i: c  a  :  > »  a a  D  3  Figure 5-3:  Backscattering and attenuation profiles in the melting layer contrasting the different levels obtained using the percolating and non-percolating particle representations, (f = 20.0 GHz and R = 12.5 mm/h.) [77].  5.3 Melting Process The melting process of snowflakes affects the composition, speed, and number density of particles and hence the scattering and attenuation properties of the melting layer. Researchers in the field have developed rigorous methods to describe the melting process and the melting rate (e.g., [60] [61]). However, melting of snowflakes is a complex process involving many factors. These include air temperature, relative humidity of the air, snowflake size and density, as well as vertical air velocity. Such factors are hard to predict and sometimes difficult to measure. Accordingly, an accurate evaluation of the rate at which melting occurs may not be possible. Alternate approaches have been developed by making reasonable assumptions. One such approach uses an average melting rate which is independent of particle size, introducing the concept of a "melting  72  Chapter 5: Modeling Precipitation: Melting-Snow Layer  0.2 0.4 0.6 0.8 Melting ratio S defined as (water volume)/(total volume) Figure 5-4:  Different melting profiles in the melting layer ( u \  05  5,=  ,  ^4  =  h: normalized depth into the melting layer from the top H: thickness of the melting layer  profile" [16][26][81]. This profile is simply the functional relationship between the degree of melting (5) and the position of the melting particles in the melting layer, S being defined as the ratio of the melted volume (water) to the total volume of the melting particle. Some of the profiles used in this work are shown in figure 5-4 where (S) is shown as a function of depth in the melting layer. The melting profiles considered are:  S, =  . s  3  .  (5.15)  S = 4  where H is the thickness of the melting layer and h is the depth into the melting layer, from the top of the melting layer. The effect of these profiles on scattering in the melting layer is shown in figure 5-5 for both the concentric sphere and the spongy sphere models. Figure 5.5 shows that dramatically different reflectivity characteristics are obtained using  73  Chapter 5: Modeling Precipitation: Melting-Snow Layer  different melting profiles. The validity of any of these profiles depends on how well it agrees with observations. The observed averaged reflectivity profiles of the bright band at 9.685 GHz (obtained with McGill VPR ) are shown in figure 5-6 [9]. These show that the 1  peak reflectivity is near the center of the bright band. A comparison of the two figures (figures 5-5 and 5-6) shows that the reflectivity profile generated using the percolating sphere model for the melting profile S agrees well with the observed profiles. 2  Since the attenuation and scattering models used in the interference model are based on a concentric representation of the melting-now particle, it is desirable to obtain a melting profile for the concentric-sphere representation to approximate the observed reflectivity profile of the bright band. With the exception of the discontinuity at the bottom of the 54 52  0.2 0.4 0.6 0.8 normalized depth into the melting layer (h/H) Figure 5-5:  Reflectivity profile of the melting layer for the concentric (solid) and the  percolating composite sphere model (dotted) with p - 0.1 g / c m \ / = 9.344 GHz, R = s  12.5 mm/h. and for S S , S h  2  3  profiles.  We make use of VPR reflectivity profiles despite some of the doubt casted on the accuracy of the radar. This can be done since we are interested here in relative, rather, than absolute values. 1  74  Chapter 5: Modeling Precipitation: Melting-Snow Layer 1500  10 20 30 Equivalent reflectivity factor (dBZ)  40  Figure 5-6: McGill's observed averaged reflectivity profiles of the bright band [9].  melting layer between the rain and melting layer reflectivities, the S profile is appropriate 3  for the concentric sphere model.  ,  5.4 The Fall Speed of Melting Particles The rigorous way for obtaining melting-particle velocity (v ) is by applying the force m  balance equation: mg=0.5p C Av  (5.16)  2  a  d  m  where g is the gravitational constant, m is the particle mass, p is the air density, Cd is the a  drag coefficient and A is the cross sectional area of the particle. The drag coefficient of melting snow is not known but can be extrapolated from that of snow and the resultant rain.  75  Chapter 5: Modeling Precipitation: Melting-Snow Layer  Figure 5-7: Relation between snowflake fall velocity (V ) and the ratio of snowflake mass to snowflake cross-sectional area (M/S) obtained at Nagaoka city prefecture, Japan in (a) January 1978 and (b) January 1979 [60]. T  The drag coefficient of rain can easily be obtained by using rain drop velocities in the above equation. Rain particles terminal velocities are very well known and are given [62] as: v =9.32 x [l - exp(-a„ / 0.885) R  1147  ]  (5.17)  where a is the radius of the rain drop. These values closely agree with other workers R  (e.g., [63]). For snow, a drag coefficient ranging from 0.6 to 1.2, depending on the shape and density of the snow particles, was used in [60]. However an inspection of the results in figure 5-7 shows that the drag coefficient can be higher than 1.2 and lower than 0.6. Klassen [11] suggested a formula relating the drag coefficient of snow to its density. The drag coefficient he uses ranges from 0.5 for ice particles to 1 for soft snow according to: Q , = l-0.5e  (5.18)  76  Chapter 5: Modeling Precipitation: Melting-Snow Layer  1.4r  r=0.025  r=0.3b"  10 Snow density  Figure 5-8: Drag coefficient of snowflakes as a function of snowflake density with rain particle radius, r, as parameter.  where e is a parameter determined from the initial-snow density (p ), between 0.005 and s  0.9 g/cm : 3  p , = (0.005)^(0.9)'  (5.19)  Equation 5.18 underestimates the observed values given in [60]. Also, since the drag coefficient for rain varies between 1.192 and 0.45, the drag coefficient for some rain particles is higher than that of snow. Although the effect of drag coefficient on velocity is small (except for very large variations of Cds - equation 5.16), the following modification to the above formula is then proposed to bring it more into line with the observed values in figure 5-8: C = L3-0&e  (5.20)  ds  77  Chapter 5: Modeling Precipitation: Melting-Snow Layer Using experimental data, Magano and Nakamura [64] derived a formula relating snow velocity to its size and density: v =S.S[(p -p )2aJ  (5.21)  2  s  s  a  where a is the radius of the snowflake. The above formula is used to derive the drag s  coefficient for snow and is found to provide consistent values for initial-snow densities higher than 0.03 g/cm . A density of 0.1 g/cm is used as the cutoff dry-snow density 3  3  where the formula applies. The drag coefficient for particles with intermediate densities (Q. ) is found by interpolating between Q = 1.37 for p = 0.03 and those calculated for ?  s  rain {CM): s  a  0M  A  a—a  r  (1.37 -C )+1.37  (5.22)  DR  B  where a .03 is the radius of the snow particle of density p = 0.03 g/cm corresponding to a 3  s  0  rain particle of radius a and drag coefficient CJR. Ay =3 was found to provide the most R  satisfactory performance (figure 5-8 and 5.9). For snow flakes with densities lower than 0.03 (a very rare occurrence directly above the melting layer), the drag coefficient is assumed to stay constant at 1.37. The snow velocities as a function of density are compared to those obtained from equation 5.21 (figure 5-9), which are in turn obtained from observations. The drag coefficient formula suggested by Klassen (equation 5.18) and its modification (equation 20) are also shown in figure 5-8 and its effect on snow velocity in figure 5-9. Figure 5-9 shows that the snow velocity devised in this work (solid lines) gives good agreement with those based on observations (o) and they are in better agreement than those used by Klassen. Russenberg [96], uses a constant drag coefficient of 1.2 for snow 78  Chapter 5: Modeling Precipitation: Melting-Snow Layer •  14i  •  •  '  '  '  ' • ' I 1  Snow density F i g u r e 5-9: Terminal velocity of snowflakes as a function of snowflake density for the different drag coefficient curves displayed in figure 5-8 are contrasted to equ. 5.21 (o).  flakes of all densities, an assumption inferior to ours especially for high initial-snow density particles. For the melting-snow particles, the drag coefficient (C</ ) may be obtained by m  interpolating between the values for rain and those of snow [60]: C =  "~ _  C  dm  a  S  C  (a -a  JR  m  j(S=0)  )C  s(S=0)  +  (5.23)  ds  R  a  Alternatively, an estimate of the melting particle velocity, based on the particle velocities at the top and bottom of the melting layer, may be used. In [26][81], a "velocity profile" was proposed: v  =v +(v -v,)sin (*5/2)  (5.24)  >  m  v  /?  where v , VR, V are the snow, rain and melting particle velocities, respectively. s  S  79  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  Spociile  A t t c n i « t i o n , dI3/lcjm  1  2.00  t  1 .SO  1  i  \  ^\ *" •  / ' ' ' '  —  1.20  —  1  ]  —  P = 0.10 g/cm  3  —  s  / , ' ' '  \ \ \  i eo 1.40  1  /  Y-1.5 Y - 1.0  —  Y-0.5  —  / \ /  —  .oo —  —  o.so  —  0.60  s  0.40  2  —  / ^ s  3  0.20  1  1  i  I  0.40  O.GO h / H  Figure 5-10: Effect of velocity profiles on specific attenuation in the melting layer (Kharadly's attenuation model) [81]  /= 12 GHz, R = 4 mm/h. <h}  fh}  . s  3  In [26] and [81], a constant initial-snow density of 1.5 m/sec was used since the model was developed for low-density snow. As shown in figure 5-9, it is not valid for higher density snow. Since we are using the scattering and attenuation models from [26] and [81] in our interference calculations, these models were revised to reflect the velocities devised here and represented in equation 5.22. Varying the parameter y in equation 5.24 has little effect on the predicted attenuation in the melting layer (figure 5-10) and is taken as unity in this work. Very little difference was found to occur in results obtained using equations 5.23 and 5.24. This suggests that, while the initial-snow velocity is important in melting layer attenuation and scattering calculations, the velocity profile is not.  5.5 Particle-Size Distribution  80  Chapter 5: Modeling Precipitation: Melting-Snow Layer Rain rate (Rj mm/h.) Diameter (D, - cm) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7  0.25  1.25  28.0 50.1 18.2 3.0 0.7  10.9 37.1 31.3 13.5 4.9 1.5 0.6 0.2  2.5  5.0  12.5  25.0  50.0  100.  150.  Percent of the total rain volume (P(ij)) 7.3 4.7 2.6 1.2 1.7 27.8 20.3 11.5 7.6 5.4 18.4 32.8 31.0 24.5 12.5 25.4 19.0 22.2 23.9 19.9 7.9 11.8 17.3 19.9 20.9 3.3 5.7 10.1 12.8 15.6 4.3 8.2 1.1 2.5 10.9 6.7 0.6 LO 2.3 3.5 0.2 2.1 0.5 1.2 3.3 0.3 0.6 1.1 1.8 0.2 1.1 0.5 0.3 0.5 0.2  1.0 4.6 8.8 13.9 17.1 18.4 15.0 9.0 5.8 3.0 1.7 1.0 0.7  1.0 4.1 7.6 11.7 13.9 17.7 16.1 11.9 7.7 3.6 2.2 1.2 1.0 0.3  V;  (m/s) 2.06 4.03 5.4 6.49 7.41 8.06 8.53 8.83 9.00 9.09 9.13 9.14 9.14 9.14  Table 5-2: Drop-size distribution and associated velocities in rain for different precipitation rates [61].  The rain medium consists of water drops of different sizes. In order to relate attenuation and scattering to an actual rain rate, it is necessary to know the number of rain particles of a certain size per unit volume. Representative distributions were obtained by Laws and Parsons [65]. These are given in table 5-2 as presented in Medhurst [63]. The number of particles of ith rain-drop size per unit volume (Ni) is given by: N. = f ' 48;rxl0 a, v,.  (5.25)  R j P a  3  3 3  where Rj is the rain rate in mm/h, paj) is the percentage of rainfall, with rain rate Rj, coming from rain drops of radius a,-. Marshall and Palmer used their measurements and those of Laws and Parsons [66] and proposed a negative exponential function: /V. = n(a)da = N^da  (5.26)  where a is the rain drop radius in mm, n(a)da is the number per unit volume of rain drops of sizes between a-(da/2) and a+(da/2), N = 1.6 x 10 and A = 8.2R' . R is the rain 4  0  rate in mm/h.  81  021  Chapter 5: Modeling Precipitation: Melting-Snow Layer  -DISTRIBUTION -LAWS  FUNCTION  AND P A R S O N S  - M A R S H A L L - P A L M E R MEASURED  icr'  z  0  3  4  5  0(mm)  Figure 5-11: Comparison between Marshall and Palmer negative exponential function (equation 5.26) and their measurements and those of Laws and Parsons [65].  Laws and Parsons and Marshall and Palmer relations agree well (figure 5-11) and even though the studies are over 50 years old, they are still considered to be typical of the average drop-size distribution for rain [67]. Unlike rain, the melting layer is essentially vertically inhomogeneous. It is bounded by dry snow from above and rain from below. In between, the precipitation particles suffer a progressively increasing degree of melting as the particles fall through the layer. The fall velocity of the partially-melted particles depends on the degree of melting [26]. As a consequence, the number of particles per unit volume (number density) at any particular height within the layer will be a function of position in the layer. Therefore, in general, the size distribution of the melting-snow particles will be different from height to height, and from that of the resultant rain. In the melting layer, the concentration of particles per unit volume (N ) is calculated from: mi  N  Ri  X  V  Ri  =  N  n u  X  v m  (5.27)  i  82  Chapter 5: Modeling Precipitation: Melting-Snow Layer where v/?, and v , are the rain and melting particle velocities, respectively. The above m  equation states that, given a box in the melting layer, the number of particles entering from the top of the box is equal to the number of particles leaving the bottom of the box. For simplicity, the above equation ignores the effect of coalescence and breakup in the melting layer. The information regarding the importance of these phenomena in the melting layer is contradictory. Some studies (e.g., [68]-[70]) conclude that coalescence above the melting layer and breakup below it, have no significant consequences. However, other researchers (e.g., [61] [71]-[73]) have reached a different conclusion. Others (e.g., [74]) found evidence that breakup occurs only part of the time. Certain models (e.g., the artificial dielectric model) treats rain and the melting layer as a fictitious medium composed of equi-size representative particles where the radius a is m  given by [26]: (5.28)  The representative radius in the rain medium is obtained by setting S to unity in the above equation. The velocity of the representative rain particle is obtained from equation 5.17 or by interpolating the velocity values in table 5-2 and the number of representative drops in this fictitious rain medium is obtained from:  The number of representative drops per unit volume in the melting layer is then obtained from equation 5.27 and the velocity is obtained from equation 5.24. 5.6 Initial-Snow Density and Peak Reflectivity in the Bright Band  83  Chapter 5: Modeling Precipitation: Melting-Snow Layer 50  ^ 40k  N  CQ T3  S30k  N  o  >20'  N 10 <  0  30  20  10  0  Z  e ( r a j n )  40  (dBZ)  Figure 5-12: Scatter plot of bright band peak reflectivity Z  (crosses) and Snow reflectivity (dots) just above the bright band Z versus rain reflectivity just below the bright band Z using data from the wind profiler. Here we are only concerned about peak reflectivity data. The solid line is the median peak reflectivity given in [9]. t{peak)  elsnow)  e(rain)  Different types of hydrometeor particulates can be found above the melting layer. These range from loose snow at a density of 0.005 g/cm , to pure ice at a density of 0.9 g/cm 3  3  [11]. The density of these particulates plays an important role in melting-layer attenuation and scattering, since it directly affects the size, velocity and concentration of particles in the melting layer. Higher initial-snow densities mean smaller and faster particles with less concentration in space, and hence weaker reflectivity. Conversely, the lower density snow (e.g., powder snow) produces slow falling particles with high concentration of particles in unit space and this means strong reflectivity in the melting layer. The change in initialsnow density would seem to explain explains the wide scatter in peak and snow reflectivities noted in radar observations of the bright band [9] [11] [96] (figure 5-12). An  84  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  O.OO  0.20  0.60  0.-4O  O.SO  l.OO  h/H  Figure 5-13: Attenuation profile ofthe melting layer with S =(h/H) , p = 0.05, 0.1, 2  2  s  0.2, 0.3, 0.7 g/cm , f = 20.0 and 30.0 GHz using the concentric sphere model. 3  example of the effect of initial-snow density on attenuation profiles is shown in figure 513. Reflectivity profiles have the same dependence on initial-snow density. We  stated above that the initial-snow density can determine the attenuation and  reflectivity levels (referred to as strength) of the bright band in the melting layer. Conversely, it should be possible to determine an "effective" initial-snow density for a given peak bright-band reflectivity value. Using the models developed in this work for melting layer reflectivity in conjunction with peak reflectivity observations, it is possible to derive an "effective" initial-density average for snow immediately above the melting layer. We refer to it as "effective" initial-snow density since we have no mean of verifying that, physically, it is the initial-snow density. It simply gives the desirable reflectivities.  5.6.1 Radar Observations of Bright Band Peak Reflectivities  85  Chapter 5: Modeling Precipitation: Melting-Snow Layer A review of literature revealed four radar-data sets of peak reflectivities that may be used to obtain an average peak reflectivity in the bright band and hence an average effective initial-snow density. These are: •  A set from the 3 GHz Chilbolton radar with 100 meters vertical resolution [96] [97].  •  A set from the 3.315 GHz Delft radar with 30 meters vertical resolution [11].  •  A set from McGill's VPR (9.4 GHz) radar with 15 meters vertical resolution [9].  •  A set from McGill's N O A A (0.912 GHz) radar with 60 meters vertical resolution [9].  The Delft and Chilbolton radar sets are too small to derive a meaningful average peak reflectivity, considering the large scatter in the values of peak reflectivities observed. This can be seen in figure 5-15 where the Chilbolton, and Delft radar sets are plotted along with a partial set from McGill's NOAA's radar. The Delft and Chilbolton radar sets will be used however to evaluate the average peak reflectivity derived, its characteristics and to examine the consistency of the observations. Both the V P R and NOAA's sets are large enough for a meaningful average peak reflectivity derivation. Reference [9] contains two figures showing different peak bright band reflectivities as a function of reflectivity of the rain just below the bright band. NOAA's radar set figure is reproduced here in figure 5-12. It also shows a median peak reflectivity curve (solid) as derived in [9]. However a visual inspection shows that such a curve does not represent peak reflectivity behavior, especially for rain reflectivity above 25 dBz. It was therefore necessary to derive a new median curve. The V P R data have been accorded a low value in our derivation of the average peak reflectivity because of problems observed with the radar during the measurement period. This may be due to the effect of the wetting of the radome, which has been shown to  86  Chapter 5: Modeling Precipitation: Melting-Snow Layer  cause severe degradation in the noise figure of an X-band receiver [106]. For a water sheet thickness of 0.01 inch, a 9 dB noise figure degradation is observed. Also due to the nature of the figure, it was impossible to evaluate the accuracy of their derived median peak reflectivity curve (the figure was not in the form of scatter plot but used color coding to indicate densities. Also we had available to us photocopies). As we were not able to get the data points from McGill and we could not count on the reliability of the observations due to the problems encountered by the V P R radar, the McGill's V P R data are mentioned for the sake of completion only. On the other we succeeded in reading a considerable part of the N O A A scatter plot which is shown in figure 5-12. The same was done for the Delft data while the Chilbolton data set was kindly provided by Dr. Goddard of the Rutherford Appleton Laboratory, U K . A procedure was developed to guarantee an accurate reproduction of the data points in the figures. This was achieved by scanning a figure, loading it into a graphic program, setting the mouse pointer to a cross, pointing to the different data points and recording the coordinate. The same is done for the axes of the figure and the different grid points on the axes. A program was written that uses the axes coordinate to rotate, translate and map the data points into the correct coordinates, yielding the original scatter plot. A surprising accuracy of under 0.1 dB was achieved. The different data sets are shown in figure 5-15. The interpolations for the different data sets excess peak reflectivities (the excess peak reflectivity = peak reflectivity minus rain reflectivity) are shown in figure 5.14 along with McGill's N O A A and VPR-derived median curves. For the N O A A data sets three interpolation curves were derived: the first is for the lower part of the data set (o), the second is for the upper part of the data set (o) and another for the whole set (  87  ). The  Chapter 5: Modeling Precipitation: Melting-Snow Layer McGill's NOAA-derived average reflectivity (....) is clearly seen to underestimate the newly derived averages. Klassen's data (x) are at the lower end for very low rain rates and at the higher end for higher rain rates. Due to the small data sample, this can be seen as representative of a specific event rather than describing a typical one. The McGill-derived VPR median curve (-.-.-) is consistent with their Noaa derived set. If one to assume the same error in the methodology of deriving the median, it is safe to assume that a newderived peak reflectivity curve for the VPR will be close to the one derived for Noaa. We believe that the average reflectivity enhancement curve chosen here is the best one possible and is closer to observations than the ones chosen by McGill-derived Noaa average, VPR, COST 205 [97] and Goddard [96]. Because of the large dispersion of peak reflectivity data, bounding curves were established at average + 2 and ± 4 dBs from the devised average. The new average peak reflectivity curve along with the other bounding curves are compared with the different data sets in figure 5.15. These are in turn contrasted to Goddard curve (—) and Cost 205 derived curve (-.-). Radar averaging was examined to determine whether it did reduce the measured peak reflectivity. The results showed that for radar resolutions of up 60 meters, very minimal effect was observed (figure 5-16) and at 100 meters this effect becomes important only for rain rates of under 1 mm/h. 5.6.2  Derivation of Initial-snow Densities  The density of snow above the melting layer is derived by comparing calculated melting layer peak reflectivities using the Aden and Kerker formulation [27], for different initialsnow densities, with the observed values. However, as was mentioned earlier, different models give rise to different reflectivity levels in the melting layer. This meant that an  88  Chapter 5: Modeling Precipitation: Melting-Snow Layer  20 r 18-  .i  *  i  0  5  10  i  i  i  1  15 20 25 • 30 Rain reflectivity in dBz  1  35  1  40  1  45  Figure 5-14: Average reflectivity enhancement in the melting layer is plotted against rain reflectivity Delft radar (x), Chilbolton radar interpolation (+), Cost 205 proposed average, Goddard proposed average, McGill's VPR (-.-.-), McGill's Noaa's derived average (....), our Noaa's radar (o and —) and the chosen average reflectivity enhancement (middle solid line). Also shown are the ± 2 and +. 4 dB bounding curves.  effective initial-snow density need to be found for each model. In this work, we utilize two models in deriving the initial-snow density since they are the most commonly used. These are the concentric sphere model and the spongy (percolating) particle model where the average permittivity of the melting-snow particle is calculated using the Bohrenextended Maxwell theory (See sections 6.2.4 and 6.2.5). For each of these models, the appropriate melting profile (S2 for the spongy particle model and S3 for the concentric sphere model), discussed in section 5.4, is used. The velocity of the melting-snow particles is calculated using the rigorous treatment in section 5.4 along with the drag coefficients in equations 5.22 and 5.23. Laws and Parsons rain  89  Chapter 5: Modeling Precipitation: Melting-Snow Layer  Figure 5-15: Peak reflectivities in the melting layer are plotted against rain reflectivity for McGill's Noaa (partial) data set (+), Chilbolton data set (o) and Delft data set (x). Also shown are the devised average and its bounds (solid). These are compared to the other averages devised by other investigators: COST 205 (-.-.-)and Goddard (—).  drop distribution is assumed. In these calculations, conservation of mass is assumed. In other words, condensation, evaporation and coalescence are ignored. Using the above models and correcting for attenuation in the bright band, the reflectivity enhancement in the bright band was calculated for different initial-snow densities and rain rates for both the concentric sphere and the spongy particle models. The results of these models are compared to derivations done by Russchenberg [98] and D'amico [99] in figures 5-17. This show that for the spongy sphere model, our results are largely in line with those of D'amico except at the very low and very high end of initial-snow densities. It is also noted that Russchenberg results diverges from ours. The D'Amico model uses the same representation as our spongy spherical melting particle (albeit with different  90  Chapter 5: Modeling Precipitation: Melting-Snow Layer 35  10' 0  1  1  50  1  '  '  1  1  1  i  100 150 200 250 300 350 400 450 Height from the top of the melting layer in meters  I  Figure 5-16: Radar averaging effect is seen for a melting layer at/= 0.9 GHz and R = 0.25 mm/h. Reflectivity profile of the melting layer is shown in solid for an initial-snow density of 0.1 g/cm while the effect of averaging is seen for radar resolutions of 30 meters (x), 60 meters (+) and 100 meters (o). These are in turn contrasted to the melting layer profile for an initial-snow density of 0.15 g/cm (dashed). 3  3  velocity and melting profile parameters) meanwhile Russchenberg's melting particles are modeled as spheroids. As expected, the concentric sphere model (not shown in figure 5-17) yields higher reflectivity enhancement for the same rain rate, frequency and initial-snow density. The initial-snow density is derived for the average reflectivity enhancement observed and the ± 2 and ± 4 dBs bounding curves for both the spongy and the concentric sphere models. This can be seen in figure 5.18. 5.6.3 Initial-Snow Densities and Discussion The derived average and bounding reflectivity enhancement curves are compared with models computations for different initial snow densities for the concentric sphere and spongy model and the resultant initial-snow densities are shown in figures 5-18.  91  Chapter 5: Modeling Precipitation: Melting-Snow Layer  20  Initial-snow density in g/cm 3 A  Figure 5-17: Reflectivity increase versus initial particle density for our model (solid) is compared to those derived by Russchenberg (—) and D'Amico (-.-.-) for rain rates of 1 mm/h. (+), 5 mm/h. (o) and 20 mm/h. (x).  For the spongy sphere representation, most observed data are bounded between the calculated curves using 0.06 and 0.4 g/cm initial-snow density. The (observed) adjusted 3  median peak excess reflectivity is rain rate dependent with most values lying between the calculated peak reflectivity curve of 0.1 and just under 0.25 g/cm , with a maximum at 3  about 2.0 mm/h. The initial-snow densities derived using the concentric-sphere model are higher. The variation of initial snow densities as a function of rain rate (the bell curve shape) can be seen in all of the data sets we examined. This may be, according to [106], expected since at low rain rates, there is little aggregation of crystals and consequently the densities of particles should be high. At very high rain rates one should expect more riming (capture of supercooled cloud droplets by snow) if the high rates are associated with  92  Chapter 5: Modeling Precipitation: Melting-Snow Layer  10"  —  o-  1  ' I  - — b -  ~ - © -*  — -©•«. 05  ; /  -  •=-=&-  c •o 3 o c  1  -&  <^=-  10  9T  Is  './/:  - "9-  //  'c  o>-^ H " Si  10"  10"  -C3T>^  i  1  10"'  10"  i  10  Rain rate in mm/h.  Figure 5-18:  Derived (average and bounding) effective initial-snow densities versus rain rate for both the spongy-sphere representation (solid) and the concentric-sphere representation (dashed).  strong vertical motions, resulting in high density particles. If the high rates are due to aggregation and the subsequent decrease in fall velocity, then the density should be very low. Russchenberg, et. al. [101] uses radar data to derive the initial-snow density in the same manner as we do in this work. It should be noted however that Russchenberg publication is dated more than two years after the work at UBC have been done. He concludes that "because the model is based on one effective particle with average properties, it is expected that small values of mass density do not apply; radar reflectivity measurements can be simulated with a value of mass density larger than 0.1 g/cm ." For example, at 2 3  mm/h, he derives an initial-snow density of 0.15 g/cm and at 1.5 mm/h, he derives an initial-snow density of 0.3 g/cm . These derived values are within the range of our 3  93  Chapter 5: Modeling Precipitation: Melting-Snow Layer  derived curves and indeed the first point coincides with our initial-snow densitiy for spongy model. However, Russchenberg uses a very limited amount of data ("two separate vertical scans" of the Chilbolton radar). Zhang [102] [103] [104] takes an initial-snow density minimum of 0.257 g/cm . Many of 3  the initial snow-density values he uses in his work are higher. He bases this number on the summary of Oguchi [67]. An examination of Oguchi paper shows that Zhang took the number out of context. Oguchi uses Matsumoto and Nishitsuji work [105] to classify snow into dry, slightly moist snow, Moist snow, wet snow and watery snow. Oguchi's summary show clearly that at 0°C initial-snow density is 0.09 g/cm . The 0.257 g/cm 3  3  quoted by Zhang as the initial-snow (dry snow formed of ice and air) is indeed wet snow, which is defined as the melted state of moist snow. Figure 5-18 and the lowest initialsnow density possible (as well as the multitude of other radar data shown here) shows clearly that snow densities above the melting layer can be much lower than the minimum proposed by Zhang.  5.7 Thickness of the Melting Layer The melting layer thickness depends on many factors such as the temperature profile in the melting layer, the size and density of the melting-snow particles, water vapor saturation, etc ... Many, if not all, are hard to predict. It thus becomes necessary to make appropriate estimates of the melting layer thickness. By observing the bright band, Klassen [12] found a strong correlation between the thickness of the melting layer and rain rate. He developed the following formula for bright band thickness, B (km): £ = 100xz 0  (5.30)  17  94  Chapter 5: Modeling Precipitation: Melting-Snow Layer  where z is the reflectivity df rain below the bright band and is given by: Z=  400xR  14 .  (5.31)  with R the rain rate in mm/h. Using significantly more observations of the bright band, the McGill group [9] estimated a larger bright band thickness than that predicted by the Klassen formula. Their results, plotted in figure 5-19, showing bright band thickness versus rain rate, were used to derive the 20%, 50% and 80% bright band thickness exceedance percentage (The derivation was done by Dr. Rod Olsen of the Communications Research Centre for the 20 and 80 % exceedance formulae. The 50 % exceedance formulae was culled directly out of the McGill report [9]). These formulas are given as follows:  5 = 146.4xz  for high z.  20% exceedance  (5.32.a)  for low z  20% exceedance  (5.32.b)  fl = 210.0x0.085  for high z.  50% exceedance  (5.33.a)  7i = 140.0xz  017  for low z  50% exceedance  (5.33.b)  B = 193.9 xz°  161  for high z.  80% exceedance  (5.34.a)  B=2S7.0Xz  Q0788  for low z  80% exceedance  (5.34.b)  0104  fl=164.0Xz  00975  (  z  It can be seen that each of the above formulae has two parts, one for low z, the other for high z. When working with the above formulae, the higher of the two values of thickness of the bright band calculated from each part is chosen. Figure 5-19 shows that McGill's bright band thickness is considerably larger than that observed by Klassen. Alberta Research Council bright band observations [14] are almost identical to McGill's median bright-band thickness for high z. For lower z values they are higher (near the 80 %  95  Chapter 5: Modeling Precipitation: Melting-Snow Layer 1200  000 cS Fabri 50%  Fabri 20%  200  Klassen  0 10-1 L  10°  Rain rate in mm/h.  Figure 5-19: A comparison between McGill's bright band thickness for 20, 50 and 80 % exceedance, ARC data (o) and Klassen's bright band thickness.  percentage exceedance curve). In both cases (low and high z values), they are significantly higher than those derived by Klassen and agree more with McGill's results. Since McGill's observations are based on a considerably larger database and agree better with those observed by the Alberta Research Council, we consider them to be the more representative. The top and the bottom of the melting layer were determined from the maximum curvature in log(Z ) [9], where Z is the equivalent reflectivity factor. This is the region of e  e  maximum curvature in the top and the bottom of the bright band (figure 5-20) . When examining the reflectivity of the melting layer, using the spongy representation of the melting-snow particle with S melting profile (figure 5-7), it is noted that the bottom of 2  the bright band, as defined above (maximum curvature), occurs approximately one third  96  Chapter 5: Modeling Precipitation: Melting-Snow Layer  7  co  . Z in snow 30 m from • bright band top (dBZ) e  ^-e (snow)  m  e  B: Bright band thickness (m)  Z  Z at the peak of the bright band e  e  (peak)  CO  c 7  A;  . Z in rain 150 m from the • bright band bottom (dBZ) e  (rain)  Increasing equivalent reflectivity factor (Z ) e  Figure 5-20:  Schematic drawing illustrating how bright band top, bottom and thickness are  extracted [9].  of the way up from the bottom of the melting layer, at the saturation point of the meltingsnow particle (equation 5.8). At the saturation point, S is derived as: S  =•  Ps  -  1  (5.35)  The saturation point is seen to depend only on the initial-snow density of the meltingsnow particle. Using the melting profile of the spongy-sphere model, the new thickness of the melting layer is given by: H=  (5.36)  Bxf(p )=BxpS s  Si  The function f(p ) is given in figure 5-21. The figure shows that as the initial-snow s  density increases, a larger correction of the melting layer thickness is required. This is  97  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  0  Initial-snow density (g/cm 3) A  Figure 5-21: Ratio of bright band to melting layer thickness {l/f(p )} as a function of initial-snow density. s  because the melting-snow particle saturates with water faster at higher densities. The melting particle will have similar size and properties as rain droplets. Using the S3 melting profile in equation 5.15, we observe that the maximum curvature at the top of the bright band for the concentric sphere representations is 0.15 of the way from the top of the melting layer; hence the new thickness is given by H = ].18xB. It should be noted that the initial-snow density only affects the relative location of the bottom of the bright band and only for the spongy particle. The location of the top of the bright band depends on the melting profile which is determined by other parameters such as the temperature gradient.  98  Chapter 5: Modeling Precipitation: Melting-Snow Layer  a-1000-1500' -10  ' 0  1  1—^ 10  '  ^ 20  '  L 30  1  1 40  50  Equivalent reflectivity factor (dBZ) Figure 5-22: Vertical reflectivity profiles at 9.344 GHz for different rain rates using the spongy sphere model with the S melting profile. 2  Figures 5-22 and 5-23 show the calculated reflectivity profiles of the bright band at 9.344 GHz for both the spongy and concentric sphere representations. These figures are attenuation corrected and are seen to agree reasonably well with the averaged profiles observed by McGill's V P R (figure 5-6). The percolating model, with the S? melting profile, provides the best fit to observations. For the concentric sphere representation, an Sj melting profile seems to be appropriate.  5.8 Height of the Melting Layer The height of the top of the melting layer is taken to be the rain height  (HFR).  The median  rain height (h ) is a variable that depends on the location and the prevailing weather FRm  systems in this location. If the exact median rain height is not available, a median value  99  Chapter 5: Modeling Precipitation: Melting-Snow Layer  '2 -1000h _-| 5QQ I -10  1  U  1  I  40  I  1  U  10 20 30 Equivalent reflectivity factor (dBZ)  I  0  I  I  I  I  50  Figure 5-23: Vertical reflectivity profiles at 9.344 GHz for different rain rates using the concentric sphere model with the S melting profile. 3  may be assumed to depend on the station's geographic latitude, 0 (in degrees), as proposed by Leitao et al. [75]: = - - 0.07266 5  f  h  Rm  8  km  (5.37)  Rain height varies around the median rain height (figure 5-24) [5].  5.9 Percentage of Time the Melting Layer is Present The melting-snow layer is not present at all times when precipitation occurs. The meltingsnow layer tends to be present at lower rain rates where stratiform rain dominates, and to be absent at higher rain rates where convective type rain dominates. Until very recently, no statistics were available about the percentage of time the melting layer (bright band) is present at a certain rain rate. In previous work, we assumed that the melting layer existed 100% of the time for rain rates below R  max  time above this value [15].  100  - 30.0 mm/h, and 0% of  Chapter 5 : M o d e l i n g Precipitation: Melting-Snow Layer  Probability,  14  | Median  -1.5  -1.0  -.05  0  ii  +0.5  + 1.0  +15  +2.0  Height difference, km  Figure 5-24: Rain height distribution relative to its median value [5].  In this work, as an initial approach, a formula has been devised to test the sensitivity of interference to variations in the percentage of time the melting layer is present. It assumes that the melting layer occurs 100% of the time for lower rain rates and 0% of time above a specified maximum rain rate R^,  with the percentage (xi) of its presence at  intermediate values given by:  X; =  100 x 1-  R;  (5.38)  V Rmax J Figure 5-25 shows the percentage of presence of the melting layer as a function of rain rate, with b as parameter. The recent availability of data [9] on the percentage of presence of the melting layer as a function of rain reflectivity (figure 4-4) made it possible to determine the correct values of the parameters in equation 5.38. The results in figure 4-4 were converted into percentages versus rain rates and are given in figure 5-26. These show that for very low rain rates (rain rates below 0.125 mm/h.), significant percentage of the rain arriving at the  101  Chapter 5: Modeling Precipitation: Melting-Snow Layer  100  £  90h  V)  £  80  CL  w  1  7 0  |  50  •I  40  « o> 60 c  <5 § 20 CD  °- 10 0  5  10  15  20  25  30  Rain rate in mm/h. Simulated statistics of the presence of the melting layer (Equation 5.38).  Figure 5-25:  earth surface comes directly from clouds in the liquid form. This forces a change to equation 5.38: f  i ~  X  \+2  X  X  1-  \Z ^ m  where Z„ Z  T O U  ./  2  j  — JC, x 1-  z,.  Y  (5.39)  V^max.l J J  and Zm^, are the equivalent reflectivities associated with /?,-, R .i max  2  and  R»uix.2, respectively. The equivalent reflectivity Z is obtained by taking Wxlogio(z), where z is obtained using equation 5.32. R j ltUiX  is the maximum rain rate where class 1 presence  percentage is consequential, Rma.x.2 is the maximum rain rate where class 2 presence percentage is consequential, x\ is the percentage of occurrence of class 1 for the lowest recorded rain rate and X/+2 the percentage of occurrence of class 1 and 2 combined. For an explanation of the different classes refer to chapter 2. Briefly, the classes of interest here are class 1 (low level rain with no bright band) and class 2 (rain with bright band). For the McGill data, the parameters in equation 5.39 were found to be:  102  Chapter 5: Modeling Precipitation: Melting-Snow Layer  6 = 4.2 c=1.8 ^,=03  mm/h.  = 50 mm/h.  R  mM2  _ {42.143  R,<R  '  (5 maxA  40)  lo  X ]  _ {93.393 X ] + 2  l.o  R <R ^ t  *,>*  m  m a  ,  2  2  Another equation was derived for the percentage of presence of the melting layer for the McGill data as a function of rainfall rate. This is given by: x, =x exp(-fl,/20.)-x, exp(-/?./ 0.2)  (5.41)  I+2  Both equations (5.39 and 5.41) are compared to the McGill data in figure 5.25. Equation 5.39 agrees better with the observations. 5.10  Duration (Horizontal dimensions) of the Bright Band  The duration of the bright band may be important in modeling the effect of the melting layer on interference, using the 3D model. Firstly, the 3D model rain cell has a relatively large diameter, and it may be unrealistic to assume that the melting layer exists all across the rain cell. Secondly, the percentage of presence of the melting layer depends on the rain rate. The rain rate depends also on the location within the rain cell; the likelihood of presence of the melting layer increases on the edges of the rain-cell. However, no statistics are found in literature about the duration of the bright band and thus no duration statistics could be utilized. 5.11  Frequency Dependence of Bright Band Reflectivity and Attenuation  103  Chapter 5: Modeling Precipitation: Melting-Snow Layer  100 H  1  1  •  + +  Classl + Class2  .+  10"  Rain rate in mm/h. Figure 5-26: Class presence percentage versus rain rate. Equation 5.40 (dotted) and 3.42 (solid) are compared to observed values (+) (seefigure4-4).  Calculated reflectivity and attenuation profiles in the melting layer are given in figures 526 to 5-31 for both the concentric-sphere model and the spongy sphere model. The calculations were done using Aden and Kerker [27] formulations (See chapter 6). The assumptions made are those discussed in this chapter and deemed to be the most appropriate. These are: 1 • Laws and Parsons rain drop size distribution. 2. Conservation of mass properties of particles; no condensation, evaporation, breakup or coalescence. 3. The rigorous treatment of the velocity of particles. The drag coefficient of the melting-snow particles is given in equation 5.23 and for snow by equation 5.22. 4. The derived average initial-snow densities for the concentric and spongy sphere models in figure 5-18. 104  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  5. The derived melting-layer thickness, adjusted for different initial-snow densities as in equation 5.36. 6. The appropriate recommended melting profiles (52 for the spongy sphere model and S3 for the concentric sphere model). These figures show that while the melting layer always causes increased attenuation when compared with rain, this is not true for reflectivity at the higher frequencies (e.g., f = 40, 60, 100 GHz). At these frequencies, there is indeed a reduction in reflectivity as compared with rain. It may also be seen from these figures that the concentric sphere and the spongy sphere representations give, overall, similar attenuation behavior up to 100 GHz and similar reflectivity behavior up to 40 GHz. It is noted that for all frequencies and rain rates examined, the position of the peak reflectivity in the melting layer is lower than that of attenuation. As frequency increases, the reflectivity peak moves toward the bottom of the melting layer. At the same time, the peak reflectivity decreases and for frequencies above 40 GHz, it is noted that the melting layer's reflectivity diminishes below that of rain. On the other hand the melting layer continue to exhibit excess attenuation. This automatically means that at very high frequencies, the melting layer acts to decrease the interference level.  105  Chapters: Modeling Precipitation: Melting-Snow Lay  University of British Columbia  106  Electrical Engineering  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Lave  University of British Columbia  107  Electrical Engineering  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  o  CL OO  c  ID  -a o £ ID ID  -C CL  C  o 00  c  ea  3  O  (LU)  doi pueq }L|6uq  CO >  IUOJJ  IL|6I9H X o o  •§5 X l w C  00  c  'S  .2 u ID  ti <o CB  J S  CO  ID  w  >  1 CO  C  > CS I  (LU)  University of British Columbia  do; pueq iL|6uq  108  in  LUOJI  }L|6;eH  E  "Q.  00  e  o  CL  TJ C CO  Electrical Engineering  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Layer  o  I— Q. 30  to  •a  o a.  00  c  '5 3  o (LU)  do) pueq ii|6uq  LUOJJ  iu6i9H  >  <2 N  X  a o ©  VO  w -=  1  u  2 *o  £  XI  00  ' C .2 13  c \C u E  tJ  <U  C3  —' <U  1  > £ <U  Q.  CO O  >> 00  12. > ..  o  (LU)  University of British Columbia  do; pueq ;u6uq  109  LUOJJ  </! 1>  -S  ;Lj6|eH  Electrical Engineering  Chapter 5: M o d e l i n g Precipitation: Melting-Snow Lay  University of British Columbia  110  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  Chapter 6:  Attenuation and Scattering Properties of Precipitation  6.1 Introduction In this chapter, scattering and attenuation theory of rain and melting particles is presented. Mie scattering methods for concentric spheres [82] (section 6.2) are especially suited to the concentric-sphere model: The composite and spongy-sphere model, where the particle is a mixture of air, water and ice, requires a solution of the sphere problem (which is simpler to solve than the concentric sphere problem). However, in order to calculate the scattering and attenuation properties, it is essential to obtain the average permittivity of the water-ice-air mixture in the melting-snow particle. Two methods, to calculate the average permittivity of a water-ice-air mixture, are presented. The first is the MaxwellGarnett theory for spherical inclusions [83] and the second, which is thought to be more appropriate, is the Bohren-extended theory for spheroidal inclusions [84]. Similarly, the permittivity of dry snow could be calculated (dry snow is a mixture of air and ice). Mie scattering computation, especially for the concentric sphere, is quite complex and computer intensive. It is therefore necessary to employ simplified attenuation and scattering models in the interference model because of the large number of scattering arid attenuation computations required. For interference calculations, Kharadly's artificial  University of British Columbia  111  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  dielectric attenuation and scattering models for the melting layer are used (e.g., [17] [18] [26] [28] [76] [85]) (see section 6.3). Other simplified models for attenuation such as the aR relationship for rain [86] and its extension to the melting layer (see section 6.5), are h  also used. Finally, a discussion of the scattering and attenuation properties of the ice/snow region above the melting layer is presented in section 6.6.  6.2 Mie scattering method for spheres and concentric sphere 6.2.1 Scattering theory A melting particle is either represented by a sphere with average permittivity e or a two m  concentric spheres particle, with a layer of water of permittivity e surrounding a dry w  snow core of permittivity e . The scattering of a plane electromagnetic wave due to two s  concentric spheres, with different refractive indices, has been treated by Aden and Kerker [82]. The total cross section Q,, the total scattering cross section Q , and the backscattering s  cross section o~b of a sphere of radius a with a concentric spherical shell of thickness a s  a  m  (figure 6-1) and whose refractive indices are Nj (N,=Je^)  and N  2  (N^ = y[e~),  respectively, are:  (6.1)  (6.2)  2  (6.3)  ^4S-')"(2»+'i(«:-»i) k~ „_,  University of British Columbia  112  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  Concentric sphere representation  One-sphere representation  (a)  (b)  F i g u r e 6-1: (a) Concentric sphere with inner radius a and an outer radius a,„. (b) Onesphere representation obtained by either setting a to zero or a =a . s  s  s  m  where the propagation constant for free space k = 2n/A\ w i t h X as the wavelength. T h e values o f a  and b  are calculated by a p p l y i n g the boundary c o n d i t i o n s to the w a v e  s  n  n  equations i n the three distinct m e d i a (inner sphere, outer sphere and free space), w h o s e full f o r m u l a t i o n is g i v e n i n A p p e n d i x B . T h e attenuation, in d B , for one particle o f size i, is then given by:  a =-4.343x10 Q, •  (6.4)  s  i  and that o f a w h o l e d i s t r i b u t i o n o f m particle-sizes o f number density TV,- is:  (6.5)  -Ja,.=-4.343xl0 Xa,^ 5  a  T h e scattering for the w h o l e distribution o f particles is obtained in a s i m i l a r fashion:  (6.6)  m  T h e equivalent reflectivity is obtained f r o m the scattering values u s i n g :  Z = 10 l o g , t  A  m +2 2  4  7T  2  5  m  University of British Columbia  i  (6.7)  x 10  -1  113  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  6.2.2  Simulation  A multiple precision program [87] [88] was modified and used to implement the Mie scattering equations. The details of this implementations are given in Appendix B.  6.2.3 Concentric sphere representation The attenuation (a ) and backscattering (<j ) contribution, from a size-distribution of m  m  particles at a certain point in the melting layer, are given by:  »,„=£«,,„•"„„•  (6-8)  /=/  n  ^ = Iff^™ where a  m  and a  (6.9) m  are the specific attenuation and .backscattering values in the melting  layer, respectively, a  mi  and rj ; are the individual particle contributions and N m  mi  is the  number of particles of size / per unit volume. The inner and outer radii (a , a,„) are t  calculated using the treatment in section 5.2.1 and number of particles per unit volume in the melting layer (N ) according to section 5.5; using the rigorous velocity treatment mi  (equations 5.16, 5.22 and 5.24). The average permittivity of the initial snow particle and the snow core is calculated using either the Maxwell Garnett theory of spherical air inclusions in an ice matrix or the Bohren modified Maxwell-Garnett theory of ellipsoidal air inclusion in an ice matrix. These two methods are described below. The volume percentage of ice and air in the snow mixture is given in equation 5.3. The permittivity of snow versus density for a frequency of 20 GHz is given in figure 6-2. In this case, it is found that, using air inclusions in an ice matrix, yields results comparable to known snow permittivity values.  University of British Columbia  114  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  0.1  0.2  0.3  0.4 0.5 Snow density  0.6  0.7  0.8  0.9  Figure 6-2: Snow permittivity as a function of initial snow density, (a) spheroidal air inclusions in an ice matrix ( ), (b) spherical air inclusions in an ice matrix ( ), (c) spheroidal ice inclusions in an air matrix (—) and (d) spherical ice inclusions in an air matrix ( • - • - ) .  6.2.4 Composite sphere representation If an average permittivity e  av  of the melting particle can be obtained, the scattering and  attenuation properties of the melting-snow particle can be obtained from the simpler single sphere representation. The average permittivity of the water-ice-air mixture, forming the melting particle, is calculated from Maxwell-Garnet theory for spherical inclusions [83] or the Bohren-extended theory for spheroidal inclusion [84]. The average permittivity, in either case, is calculated in two steps. The first assumes that the ice inclusions are in a water matrix and the second step assumes that the air inclusions are in the ice-water matrix. Using the Maxwell-Garnet theory, the average permittivity ( £ ) of a av  University of British Columbia  115  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of. Precipitation  medium consisting of spherical inclusions, with permittivity  in a matrix, with a  permittivity £,„, is calculated from:  £...,  = 1+-  ^ [e -e )/{e +2e ) i  i  m  i  m  )  (6.10)  where x, is the inclusion fraction relative to the whole volume. Using the Bohrenextended theory, the average permittivity of the sphere is given by:  £....  ='  l-x x,f3 i+  (6.11)  2£„  B  =  -  log i ~m  £  J  £  The volumes of water (v ), ice (v ) and air (v ) in the melting particle are calculated using w  (  a  equations 5.4-5.6. The radius of the sphere and the number of particles per unit volume is the same as for the concentric sphere representation. 6.2.5  Spongy particle model  The melting particle is modeled as a spherical particle where water is assumed to progressively percolate inside the snow as the particle falls, until saturation is reached, i.e., when all the air in the mixture has been removed [11]. The average permittivity of the sphere is calculated by either of the above methods. The volumes of water (v ), ice lv  (vi) and air (vj in the melting particle are calculated using equations 5.8-5.14. It could be readily seen that this particle will have a smaller size and hence higher density than the above melting particle representation. The sizes of the melting-snow particle ( aj ), for different S, are given by equations 5.8-5.10. Using the same treatment as the above two representations, the number of particles per unit volume is calculated.  University of British Columbia  116  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation A  •  x  X (£ UI3/£ U13) U0IJ33S SSOJ3 gUUSUEOSipBg V  V  Figure 6-3: Attenuation and backscattering profiles of the melting layer for various melting particle models (A, B1, B2, C l , C2), with p = 0.1 g/cm ,/= 20.0 GHz, R = 5.0 3  s  and S = (h/Hf. 6.2.6 Comparison of different models Results are obtained for rain rates between 0.1 and 40 mm/h and frequencies between 1100 GHz. These show that attenuation and scattering cross section values vary significantly from one model to the other. Figure 6-3 shows the different scattering and attenuation profiles in the melting layer obtained using the concentric sphere model (model  A), composite-non-percolating  sphere  (model  B | , B2) and the  spongy  (percolating) sphere representations (model C i , C ). Subscript 1 indicates that the average 2  permittivity of the melting particle was calculated using the Maxwell-Garnett theory while subscript 2 is used for the Bohren-extended theory. The difference in computed levels using the different models testifies to the difficulty to modeling melting snow in the melting layer.  University of British Columbia  117  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation 1000|  i 100^  0.0001-1  1  1—  1  1  1—  1  10  1—i 1000  100  FREQUENCY (GHz)  Figure 6-4: Rain-induced attenuation. The artificial dielectric model are in solid while the M i e scattering values are crossed [26].  6.3 Artificial dielectric model for attenuation [89] Since Mie computations of attenuation and scattering are computer intensive, simplified models are used to simulate scattering and attenuation in rain and melting-snow. The artificial dielectric model treats rain as an artificial dielectric medium composed of onesize particles of equivalent radius a and number density N [26]. The model is explained in detail in Appendix B. Two forms of the model (ADM1 and ADM2), explained in detail in Appendix B, were applied to the melting-snow layer [89]. Both give satisfactory results for a wide range of rain rates, 0.25-150.0 mm/h, for rain and 0.25-30 mm/h for melting snow and frequencies of 1-1000 GHz for rain and 1-40 GHz for melting snow (figures 6-4 and 6-5). Even though these models are developed for an initial snow density of 0.1 g/cm , it is shown 3  that they are satisfactory for p = 0.l, 0.2, 0.3 g/cm (Fig. 6.6). 3  s  University of British C o l u m b i a  118  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  0.2 0.4 0.6 0.8 Normalized depth into the melting layer (h/H) Figure 6-5: Melting layer-induced attenuation . The 1st artificial dielectric model without correction factor (—), with the correction factor ( ), the 2nd model without correction factor ( ) and with the correction factor (-.-.-) are compared to the Mie scattering results (+).  0.2 0.4 0.6 0.8 Normalized depth into the melting layer (h/H) Figure 6-6: ADM1 attenuation model with the correction factor for p = 0.1 ( (-—), and 0.3 g/cm ( ) are compared to Mie scattering results (+). s  ),0.2  3  University of British Columbia  119  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  6.4 A r t i f i c i a l dielectric model for scattering [77] The model treats rain as an artificial dielectric medium composed of one-size particles of equivalent radius a  m  [85].The equivalent, or representative, radius is the same one in the  attenuation model in section 6.3. Under the effect of an external electric field, the drop may be represented by an equivalent dipole of moment (figure 6-7). The theory and formulations are presented in detail in Appendix B. Two forms of the model were later extended to melting-snow particles [17]. The above models are found to give satisfactory results for a wide range of rain rates, 0.25-150.0 mm/h, for rain and 0.25-30 mm/h for melting snow and frequencies of 1-1000 GHz for rain and 1-40 GHz for melting snow (figures 6-8 and 6-9). Even though these  Z  lr,e,*| A  / i  r /  Figure 6-7: Scattering geometry of a rain particle due to an incident electromagnetic wave.  University of British Columbia  120  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  .x 10  0.2 0.4 0.6 0.8 Normalized depth into the melting layer (h/H)  0  Figure 6-9: Melting layer-induced backscatter. The Artificial dielectric models are compared to Mie scattering computations (+) for different initial-snow densities. models are developed for an initial snow  density of 0.1 g/cm , it is shown that they^  are satisfactory for p = 0.1, 0.2, 0.3 g/cm  (figure 6-9).  s  University of British Columbia  3  3  121  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  6.5 Extension of the power law (aR ) model to attenuation in the melting layer b  Specific attenuation in rain is found to follow a power law relation [86]: a  = kR  (6.12)  p  R  where k and B are function of frequency. The model is further refined and variations due to elevation angle, half-beamwidth and polarization are introduced [90]. The kR formula for specific attenuation in rain is extended to specific attenuation in the fi  melting layer [15] . The specific attenuation in the melting layer is given as the kR 1  0  multiplied by a factor a ( ,s> that account for the effect of the melting layer. Hence, the n  R  specific attenuation in the melting layer (a ) is: m  a„=a xkR  (6.13)  p  n[RS)  a„(R,s) is the normalized effect of the melting-snow layer. This is found to be well represented by: f A f 5 e " * ° ' + M,5 "'e" u H  | S  ai  l  "•<"> = {,  6j5  " +M,e"^+l J  0<S < 1  "  .....  s=i  (  '  with, hA . y 0.003 , „ 0.0002 I(R) = O , 2 •. r> , r, n,0.003 , „ 0.0002 2(R) 0 1 2 D  M  C  +  C  R  D  + C  rfilSnM  R  V U . u a , u ;  D  M  =D  +  D  R  + D  R  where M = -1, a = 1.7, b, = 20, b = 6,b = 230 and a ft a,, a , Co, C,, C , DQ, D,, D 3  2  2  3  2  2  are frequency dependent parameters listed in table 6-1 for a melting layer with initial  Tim Vlaar performed the datafittingon the attenuation calculations done by Dr. Kishk using the above formulae (6.13 and 6.14) I supplied for initial snow density of 0.1 g/cm . This has been included in the Master thesis. 1  2  University of British Columbia  122  Electrical Engineering  2  Chapter 6: Attenuation and Scattering Properties of Precipitation  snow density 0.1 g/cm , and using the same velocity profile assumption made in the artificial dielectric models and initial snow velocity of 1.5 m/s. The formula gives reasonable values for a frequency range between 1-100 GHz and rain rates between 0.25-150 mm/h. for rain and 0.25-50 mm/h. for melting snow. Figure 6-10 shows that the model agrees well with Mie scattering computed results. Different frequency-dependent parameters can be generated using different initial snow densities. However, a simple method to obtain the attenuation levels in the melting layer for other initial snow density, using those obtained for p = 0.1 g/cm is presented. The attenuation 3  s  in the melting layer for any initial snow density p is given by: s  a ^kR {a \ -\)f(p )  + kR  p  m  n prQX  where a \  n p  (6.16)  p  5  is the normalized effect of the melting-snow layer for p = 0.1 g/cm , f(p ) 3  = 0 1  s  s  is a function of the initial-snow density. It should be noted that/(pj depends also on rain rate and frequency but an average f(p ) = 1, 0.6, 0.45 for p = 0.1, 0.2, 0.3 g/cm was 3  s  s  found to give satisfactory results. Scattering and attenuation properties above the melting layer Rayleigh scattering is generally assumed for the ice/snow region above the melting layer. At the zero degree isotherm, the scattering cross section is assumed to be that of Raleigh scattering due to rain and is given by: 2  _K ^  w  ~  isotherm) ~~  £„+2  University of British Columbia  zxlO"  m /m  18  2  123  3  (6.17)  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  where e = complex relative permittivity of water, X is the wavelength, and z is the sum w  of the six powers of the diameters of all hydrometeors per unit volume. The magnitude of z is given by [91]: z = 400/?  14  mm /m 6  (6.18)  3  where R is the rain rate. Reflectivity is assumed to decay as we move higher into the ice/snow region. There is no general agreement on the reflectivity roll-off for the ice/snow region above the melting layer. Previous work (e.g., [5]) used 6.5 dB/km reflectivity roll-off. A recent submission to ITU-R [92] suggests that this needs to be changed to 4.0 dB/km for the lower part of the ice/snow region. Attenuation in the ice/snow region is negligible and is usually assumed zero. This assumption, used in all propagation models examined, may not be fully correct as super cooled water droplets may exist above the zero degree isotherm. Observations As we have seen, the simplified attenuation and scattering models used in this work agree relatively closely with those calculated using the rigorous formulation. The attenuation model is found to agree very closely with the ITU/COST 210 and the rigorous calculations. The scattering model can, however, diverge in some cases up to 2.5 dB. However, it is the small differences in specific attenuation between the models that are more important and can significantly affect predicted total attenuation and interference levels. For example, at 40 GHz and rain rate of 25 mm/h, a difference of 0.4 dB/km in specific attenuation may be seen as trivial. But for a path of 15 km, a 6 dB difference is observed.  University of British Columbia  124  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  ov -tf  ON o  ON  VO VO tf  VO  o  vo  VO VO Ov"  3  tf  o o  OV •tf  vo tf tf vd  8  vo  vO  tf rtf  o  ci +  3 VO  Ov  o  d  CN  •tf  OV d  3  Ov  C3  c 'a  CN  d  ^5 in  vo OO  Ov  Ov Ov  IS  VO CO  —  CN o  tf  vd  CO  Ov tf oi vd  CN  ^3-  Ov  o Ov  c  O  o  Ov CN VO  O  OlO CN Ov I  VO OV  3  cn in t—  u o  vo o oo  o cn  m Ov • t f Ov r - CN d d  tf-  CN  u  vO cu  z o  o ov  Ov d  o tf r-  co  tf"  rOV tf  in  VO CN  CN  o  VD  vp  O cn  in m  University of British Columbia  O tf  O vd  o tf  vo  q oo  H Ov  Sip  O  o tf  o  o tf  o  Ov i n vO — VO  q Ov  o d  q CN  125  O tf CN 0 0 — in — r~  O Ov —: d  cn vo  IS  o  tf Ov d  o o O , cn cn  d tf  d -  Electrical Engineering  Chapter 6: Attenuation and Scattering Properties of Precipitation  18  Normalized depth into the melting layer (h/H)  Figure 6-10: A comparison of the attenuation profile of the melting layer between the extended IcR" (see equation 6.12) model (solid) and Mie scattering computations (crosses) for f= 10 GHz.  University of British Columbia  126  Electrical Engineering  Chapter 7: Application to Microwave Links  Chapter 7: Application to Microwave Links  In this chapter, the validity of the interference models discussed earlier is evaluated in section 1 and then compared to experimental results in section 2. In section 3, the effect of the different regions of precipitation and their modeling on interference are examined with special emphasis on the melting-snow layer.  7.1 Evaluation of the weighed interpolation approach In chapter 2, the melting layer was shown to exist only part of the time. In order to account for the effect of the melting layer in interference modeling, the weighed interpolation approach was proposed. In this approach, the interference level is generated by a melting layer presence probability-weighed interpolation between the cumulative distribution functions for transmission losses of the original model (no melting layer presence is assumed) and the modified model (the rain cell is modified to include the presence of the melting layer). The Awaka and Capsoni rain-cell models are horizontally non-homogeneous, meaning that the rain rate in the rain cell is not constant. Indeed, the rain rate may be over 100 mm/h. at the center of the rain cell and a fraction of a mm/h at the edges. Since the melting layer presence probability is dependent on rain rate, it is extremely difficult (and  127  Chapter 7: Application to Microwave Links  % Time 0.001 0.003 0.01 0.03 0.1 0.3 1.0  Rain Rate (mm/h.) 62.0 42.0 26.3 15.0 8.3 4.3 1.9  % melting layer present 0.0 0.0 21.6 44.0 61.2 74.5 83.8  Table 7-1: Rain rate and melting layer statistics  indeed may not be possible) to verify the above interpolation approach using a rigorous approach with the Awaka and Capsoni models. The ITU-R model, on the other hand, assumes rain cells that are small in diameter with constant rain rates. It is therefore possible to assume that when the melting layer is present, it is present in all of the rain cell. The transmission loss calculated for the ITU-R model, discussed in chapter 3, is a statistically valid approach to include the effect of the melting layer (although the rain cell itself is artificial). The modified ITU-R model (referred to here as Mod. ITU-R1) is used to examine the validity of the weighed interpolation approach (referred to here as Mod. ITU-R2) for variants of the ChilboltonBaldock COST 210 link (See Appendix C for details about the link). The comparison was done for different rain heights {h =2.\, 3.0 and 3.9 km) and FR  frequencies (/'= 4, 8, 11.2, 20, 30 and 40 GHz). Figures 7-l(a-f) presents a summary of the transmission losses deviation from that of rain for (a) the model where the melting layer is assumed present at all time (ML, 100%), (b) Mod. ITU-R 1 (c ) Mod. ITU-R2 and the deviation between Mod. ITU-R 1 and ITU-R2 models, at different percentages of times. It is seen that, overall, the two models give comparable results for all frequencies and % of times except at f = 4 GHz and 0.001% of the time. The reason for this could be seen in  128  Chapter 7: Application to Microwave Links  table 7-1 where at 0.001% of the time, the rain rate is 62 mm/h and the probability that a melting layer occur is zero. The weighed interpolation approach forces the transmission loss at 0.001% of the time to be that of rain. While this is not an issue when the melting  f = 11.2 GHz Summary  Overall Summary  • ML, 100% • ML, 100% • Mod. ITU-R1 • Mod. ITU-R1  • Mod. ITU-R2  D Mod. ITU-R2 0(Mod. ITU-R1 • Mod. ITU-R2)  0(Mod. ITU-R1 Mod. ITU-R2)  (d)  (a)  f = 20 GHz Summary  f = 4 GHz Summary • ML, 100%  B M L , 100%  • Mod. ITU-R1  • Mod. ITU-R1  • Mod. ITU-R2  OMod. ITU-R2  • (Mod. ITU-R1 Mod. ITU-R2)  0(Mod. ITU-R1 Mod. ITU-R2)  (e)  (b)  f = 40 GHz Summary  f = 8 GHz Summary 1.8 1.6  !  1.4 1.2 1 0.8 0.6  0.4 0.2 0  • ML, 100%  • ML, 100%  • Mod. ITU-R1  • Mod. ITU-R1  OMod. ITU-R2  • Mod. ITU-R2  0(Mod. ITU-R1 • Mod. ITU-R2)  • (Mod. ITU-R1 • Mod. ITU-R2)  fl  •1 =9JLJ  -+  o 6 (f)  (C)  Figure 7-1: Summary of the deviation of transmission loss (in dB) from original ITU-R model for different frequencies and for all links.  129  Chapter 7 : Application to Microwave Links  layer causes a degradation in the interference level, it is definitely one when the melting layer enhances the interference level as can be seen in f = 4 GHz. It is suggested that it may be appropriate, in these cases, to increase our estimate of the interference level. Overall, the difference between the two models is under 0.25 dB, which is negligible, especially when contrasted to the 2.5 dB difference between M L (100%) interference levels and those generated using rain-only cells.  7.2 Evaluation of Models with respect to Experiments In order to ascertain the validity of the interference models discussed in the previous chapters, they are compared to several experimental links, part of the COST 210 project in Europe to study interference [5]. Both the original (without the consideration of the effect of the melting layer) and modified (considering the effect of the melting layer) ITU-R and Awaka models are used in the comparison, with 6.5 dB reflectivity roll-off in the ice/snow region above the melting layer. The parameters of the links are listed in table C-1, their measured transmission loss in table C-2, and rain rate statistics in table C-3 and they consist of short and long paths, different frequencies and gain, different common volume heights, etc. Some are measured interference statistics (all links with the exception of S1-S5) while others are radar simulations (S1-S5). The simulated transmission loss cumulative distribution functions for the different links, using the original and modified ITU-R and Awaka models, are compared to the experimental results in figures C1-C24. The relative performance of each of these models, in comparison to the measured results, is shown in table 7-2 for 0.001, 0.01, 0.1 and 0.1 percentages of time. This is simply the difference between the simulated and the measured values at the above mentioned percentages of time. The average error in dB is  130  Chapter 7: Application to Microwave Links  also shown in table 7-2 for the short links (Gz-D6), long links (Sl-Fb) and all links. These are also shown in figures 7-2 (a, b, c). It can be seen from figures 7-2 (a, b, c) that, on average, the interference levels predicted here are within 2 dB of the original rTU-R model in the COST 210 report. While on average is small, we note from figures 7-3 that for certain links the difference can be in the order of several dBs. The differences are attributed to the different scattering and attenuation models used. They may also be due to the simplifications made in the COST 210 implementation (such as dividing the scattering medium into large segments) in order to reduce the computational requirements. The effect of different of choosing different scattering and attenuation models will be discussed in 7.4. It may be noted from the above results that, on average, the presence of the melting layer in the prediction model improves the accuracy of prediction for both the ITU-R and the Awaka models for the above links. However, this improvement is quite small, in the order of 0.25 dB and may be seen as statistically insignificant. Although initially, the melting layer may not be viewed here as a significant factor in improving interference prediction, it will be seen that it plays an important role in interference prediction at the lower frequencies. Unfortunately, no transmission loss statistics are known to have been collected at low frequencies. Since both prediction models tend to overestimate the value of interference, especially for links where the common volume is deep inside the ice/snow region (e.g., L5-L8 and D3D6), for -6.5 dB/km reflectivity roll-off, the use of a -4.0 dB/km roll-off can only increase the predicted interference level and therefore increase the error. It can be concluded that  131  Chapter 7; Application to Microwave Links oo oi  VO  vd  11  o o d  — r d ov o ov  CN  tf d  OV tf  tf  vo  CN  3  p  Ov  2 M  8  tf  5  3  o o d  8  ^- in  o •tf  OV CN — CO  tf  00  o  Ov VO  Ov  8  vd  Ov rn  VC  tf  ov  d  3 tf o  Ov vd  tf  VD  5  Ov Ov  d  d  oo  co Ov  CN  d  tf oo  o tf  00 •^3  •tf  3  tf ov  Ov  o  o o  as o  vO  =|g CN CN CN tf tf  vO  00  tf CN  —• I  s  CN tf  o tf  Ov ov  Ivolvo O Ov vo vol  CN  VO  tf  hJ  University of British Columbia  Q  132  vo  a  a CN CO  U  Electrical Engineering  Chapter 7: Application to Microwave Links  Short Links  O ITU-R • modified ITU-R • COST 210 0ITU-COST21O B Awaka • Modified Awaka  il 0.001  0.01  0.1  1  %Time  Long Links  D ITU-R • modified ITU-R • COST 210 HITU-COST210 S Awaka • Modified Awaka  (b)  All Links  Figure 7-2: Average deviationfromexperimental values for (a) short links (Gz, L1-L8, D1-D6), (b) long links (Bf, Bb, Ff, Fb) and (c) all links (Gz, L1-L8, D1-D6, S1-S5, Bf, Bb, Ff, Fb) for 0.001, 0.01 0.1 and 1.0 % of times. The models shown are (i) the ITU-R model, as implemented in this work (ii) the modified ITU-R model, as implemented in this work to include the effect of the melting layer, (iii) The Cost 210 report calculations, (iv) the difference between our calculations and those of the cost 210, (v) the Awaka model, as implemented in this work and (vi) the modified Awaka model, as implemented in this work to include the effect of the melting layer.  University of British Columbia  133  Electrical Engineering  Chapter 7: Application to Microwave Links  0.001 %  Gz  U  L2  13  L4  15  L6  L7  U  01  02  D3  04  D5  D6  S1  S2  S3  S4  SS  Bf  Bb  Ff  Fb  Links  Figure 7 - 3 : The differences in dB between the Cost 210 predicted interference levels and those predicted in this work for all links at 0.001, 0.01, 0.1 and 1% of time.  University of British Columbia  134  Electrical Engineering  Chapter 7: Application to Microwave Links  the use of -6.5 dB/km roll-off, with these models, is more appropriate for interference prediction.  7.3 Statistical Study of the Characteristics of Interference In section 7.2, the interference models were tested against experimental results and they were found to be, overall, satisfactory. In this section, simulations for a wide range of interference cases are examined. Any or all of the ITU-R and Awaka models (original and modified) are used to generate the interference levels for the parameters under study. When appropriate, the prediction levels obtained by the two models are compared. The presence of the melting layer in the interference calculations is always assumed, except when stated otherwise. Appropriate melting layer presence percentages and initialsnow density values (as a function of rain rate) are used in the simulations. The results of simulations will be introduced in the following sections as appropriate.  7.3.1  T r a n s m i s s i o n loss d e p e n d e n c e o n m e l t i n g l a y e r p r e s e n c e  The presence of the melting layer was shown in [15] to affect interference level prediction under appropriate circumstances. One observes that, for the same link, the melting layer can either enhance or degrade interference levels, depending on the rain rate, rain-cell location, rain height, etc. ... The transmission loss cumulative distribution function, which is of interest to system designers, gives the overall effect of the melting layer. Simulations are done for a wide range of frequencies (f = 4.0, 8.0, 11.2, 20.0, 30.0 and 40.0 GHz) for the Chilbolton-Baldock path (Bf). The results of the simulations are given in figures C-25 to C-30 and their summary in table 7-3 for 0.001, 0.01, 0.1 and 1.0 percentages of time. These show that at lower frequencies (f = 4 and 8 GHZ), the melting  University of British Columbia  135  Electrical Engineering  Chapter 7: Application to Microwave Links  0.01  0.001 Awaka ITU-R f=4.0GHz,h = 2.1km 0 f=4.0GHz,h = 3.0km 0 f= 4.0 GHz, h = 3.9km 0 f=8.0GHz,h = 3.0 km 0 f= 11.2 GHz. h = 3.0km 0 f =20.0 GHz, h = 3.0 km 0 /= 30.0 GHz, h — 3.0 km 0 f = 40.0 GHz, h = 3.0km 0 FRm  FRm  FRm  FRm  FRm  FRm  FRm  FRm  2.3 1.9 1.4 0.1 0.0 0.0 0.0 0.0  1.0  0.1  Awaka  ITU-R  Awaka  ITU-R  Awaka  ITU-R  1.0 1.0 0.9 0.4 0.1 -0.5 -0.5 -0.4  2.3 2.3 2.2 0.8 -0.2 -1.0 -1.2 -1.0  2.2 2.2 1.8 1.2 0.3 -0.7 -1.0 -0.8  3.0 3.3 2.6 1.5 0.1 -1.0 -3.1 -3.7  1.4 1.4 1.3 1.2 0.9 0.0 -0.5 -0.5  -0.3 0.1 1.9 0.1 0.1 -2.9 -5.3 -4.1  Table 7-3: Summary of the difference (in dB) between the transmission loss predicted by the original and modified models for the Bf link with/= 4, 8, 11.2, 20 and 40 GHz.  layer increases the interference level, while at the higher frequencies (> 20 GHz) it decreases it. This decrease at the higher frequencies is due to the high attenuation levels that the signal experiences in the melting layer. Not only does the scattering from the melting layer experience those attenuation but also the scattered signal from the ice/snow region directly above it. It may also be seen from chapter 5 that at high frequencies, the melting layer exhibits excess attenuation but gives a reduction in the scattering level. At intermediate frequencies (f = 11.2 GHz), the overall effect of the melting layer on interference is minimal. This is due to two factors. The first is that, in a scenario in the distribution, the increase in scattering due to the melting layer is removed due to melting layer attenuation. The second is that the number of scenarios, where the increase in interference caused by the melting layer, are balanced by those scenarios where the melting layer causes a decrease in interference levels. When f = 4 GHz, we note interference enhancement, due to the presence of the melting layer, in the 2-3 dB range, especially at the 0.01 and 0.1 percentages of time. This means that the presence of the melting layer may increase interference levels between 60-100%. This in turn requires system designers to increase the coordination distance by 25-40% (the interference power is proportional to the square of the distance between the two  University of British Columbia  136  Electrical Engineering  Chapter 7: Application to Microwave Links  terminals) to maintain the same system performance. When varying the location of the melting layer relative to the center of the common volume, we continue to observe a significant melting layer effect. It can, therefore, be concluded that the melting layer influences interference statistics even when the median rain height is outside the common volume, i.e. for a wide range of link geometries. It is worth noting that while both the Awaka and the ITU-R models show a reduction in interference levels, when the melting layer is present, at high frequencies, the reduction in the Awaka model is marginal. Since the Awaka model better represents the rain medium, we may readily assume that it is able to better reflect the effect of the melting layer. Hence, at higher frequencies, the presence of the melting layer may be discounted in interference prediction. Table 7-4 shows the summary of the difference in dB between the original and modified models for the 24 COST 210 links. It is seen that on average, a small difference exists between the original and modified models prediction. The difference is, on average, less than 1.5 dB for all percentages of time and under 1.0 dB for most of them.  7.3.2 Rain height variation versus median rain height In this section, we consider the scenario where the 0°C isotherm is fixed. This is an important scenario since at a specific month or a season of the year, the 0°C isotherm may vary little around its median value. This is also useful in reducing the computational requirements of the model by considering just one rain height rather than the full rainheight distribution. The Bf link is, arbitrarily, used to test the model sensitivity to variations around its median rain height. Figure 7-4 contrasts the simulated interference with the full rain height distribution with the simulated results assuming only a median  University of British Columbia  137  Electrical Engineering  Chapter 7: Application to Microwave Links  rain height. The figure shows that a small difference exists This is also seen to be the case at 4.0 GHz and h  FRm  between the two scenarios.  - 3.0 km, shown in figure 7-5, for  both the original and modified models.  Gz LI  Awaka 0.0  0.1  Awaka 0.0  0.0  -0.0  ITU-R  0.0  1.0  0.1  0.01  0.001  ITU-R 0.5  Awaka 0.6  1.2  Awaka 0.3  ITU-R  ITU-R 0.4  0.0  -0.1  0.0  -0.0  0.0  -0.0 -0.4  -0.4  L2  0.0  -0.1  0.0  L3  0.0  -0.3  -0.1 -0.1  -0.4  -0.6  -0.4  -0.4 -0.4  L4  0.0  0.0  -0.0  -0.5  -0.6  0.0  -0.4  -0.4  L5 L6  0.0  0.0  -0.0  -0.5  -1.0  0.0  -0.4  -1.3  0.0  0.0  -0.2  -0.1  -1.1  -0.5  -0.5  -0.9  L7  0.0  -0.1  -0.7  -1.0  -1.1  -0.6  -0.8  L8  0.0  -0.5  -0.8  DI  -0.0  -0.7  -0.2  -0.8 0.0  -0.4  0.0  -0.3 -0.3  -0.8 -0.9  -0.5  -0.6 -0.3  D2  0.0  0.0  -0.0  -0.3  -1.0  0.5  -1.1  -1.1  D3 D4  0.0  0.0  -0.0  0.0  -1.5  -0.1  -1.0  -1.6  0.0  0.0  -0.0  -0.3  -1.0  -0.7  -0.5  -1.0  -1.5  -0.7 -0.8  -1.0  -0.8  -1.0  D5  0.0  0.0  -0.0  D6  0.0  0.0  -0.0  0.0  -0.7  -0.6  -0.5  -1.2  SI  0.0  0.0  0.0  0.0  0.0  0.0  0.1  -0.6  S2  0.0  -0.3  -0.0  0.0  -0.2  -0.3  0.2  -0.7  S3  0.0  -0.1  -0.0  -0.3  -0.7  -0.3  0.2  -0.2  S4  0.0  0.0  -0.0  -0.2  -0.8  -0.6  -0.1  -1.3  S5  0.0  0.0  -0.0  -0.1  -0.4  -1.0  -0.4  -1.1  0.0  0.0  -0.1  0.0  0.2  0.5  0.9  -1.1  -0.2  -0.3  0.1  -0.4  0.5  -0.5  -0.9  -0.6  0.0  -0.5  0.0  -0.9  0.0  -0.7  Bf Bb Ff  0.0 0.0  0.0 -0.1 0.0  0.0  -1.0  0.0  0.0  Fb  0.0  Average  0.0  Table 7-4:  Summary ofthe difference (in dB) between the transmission loss predicted by the  0.1  0.1  0.4  0.6  0.5  0.5  0.7  original and modified models for the links in table C - l .  University of British Columbia  138  Electrical Engineering  Chapter 7: Application to Microwave Links  -130  Figure 7-4: This Figure contrasts interference levels simulated, with the modified Awaka model for the Bf link, using the full rain-height distribution ( ) and the median rain height ( —).  -130  i  •  •  •  —  i  •  i  % Time Figure 7-5: This figure contrasts interference levels simulated, with the modified Awaka model for the Bf link (h - 3.0 km,/= 4.0 km), using the full rain-height distribution ( ) and the median rain height ( — -). The bottom curves are those for the original model and the top curves for the modified model FRm  University of British Columbia  139  Electrical Engineering  Chapter 7: Application to Microwave Links  7.3.3 Transmission loss dependence on rain height In this section, the transmission loss dependence on the median rain height is examined. Figures 7-7, 7-8, 7-9 and 7-10 show the simulated interference levels, with and without the presence of the melting layer, for 0.001, 0.01, 0.1 and 1.0 % of the time as a function of median rain height. Predictably, we see melting layer enhancement for both the ITU-R and the Awaka models at 4 GHz, a small effect at 11.2 GHz and some signal strength degradation at higher frequencies. This degradation can be especially seen for the ITU-R model at the higher percentages of time. It may also be noted that at 4 GHz, the interference level increases with increased median rain height. This is because the rain region has higher reflectivity than the ice/snow region and because the attenuation at 4 GHz is small. For the Awaka model, the maximum interference level is found to occur at 3.5 km median rain height, 0.5 km above the center of the common volume, after which the interference level drops. For the ITU-R model, the interference level continues to increase with rain height, leveling off near 3.5 km median rain height. At 11.2 GHz, the increase in interference levels with median rain height becomes smaller, especially at the lower percentages of times. At higher frequencies, 20 to 40 GHz, one notes that the interference level suffers a reduction with increasing rain height. At these frequencies, the ice/snow region causes significantly more interference than rain. This is because, at high frequencies, rain severely attenuates the signal while no such attenuation occurs in the ice/snow region. It may be noted that the interference level, as a function of rain height, of Awaka model is quite flat compared to those predicted by the ITU-R model. This is due to the size of the rain-cell and its movement, providing more diverse interference scenarios with the signal, at times, traversing only the ice/snow region.  University of British Columbia  140  Electrical Engineering  Chapter 7: Application to Microwave L i n k s  Awaka model  ITU-R model  Rain height (hFR), km  Rain height (hFR), km  Figure 7-6: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of median rain height for the Bf link (f= 4.0 GHz) for the ITU-R and Awaka models (original: , modified: —  ITU-R model  Awaka model  -160  2  3  2  4  3  4  Rain height (hFR), km  Rain height (hFR), km  Figure 7-7: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of median rain height for the Bf link (/'= 11.2 GHz) for the ITU-R and Awaka models (original: modified: ).  University of British Columbia  141  Electrical Engineering  Chapter 7 : Application to Microwave Links  ITU-R model  Awaka model -130 m  H  H  ^ _ -x  co-140  X --)t g - ^ —  o  1-150 w  'E C/)  i-160  •170  1  2 3 Rain height (hFR), km  4  -170  1  2 3 Rain height (hFR), km  4  Figure 7-8: Transmission loss 0.001  (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of median rain height for the Bf link (f= 20.0 GHz) for the ITU-R and Awaka models (original: , modified: ).  Figure 7-9: Transmission loss 0.001  (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of median rain height for the Bf link (f= 40.0 GHz) for the ITU-R and Awaka models (original:. , modified: ).  University of British Columbia  142  Electrical Engineering  Chapter 7: Application to Microwave Links  7.3.4 Transmission loss dependence on modeling the ice/snow region In section 7.3.3, it was determined that the contribution of scattering from the ice/snow region to interference is dominant at high frequencies and low rain heights. It is then surprising that the modeling of the ice/snow region has not received more attention. The current models use a reflectivity roll-off in the ice/snow region of 6.5 dB/km. A recent study suggested that this may not be appropriate and suggested the use of a reflectivity roll-off of 4.0 dB/km [19]. Figure 7-10 shows the interference levels obtained for the Bf link as a function of rain height using both 6.5 and 4.0 dB/km reflectivity roll-offs. It can be seen that when the center of the common volume is deep inside the ice/snow region, strong variations exist between the predicted levels using the two roll-off levels. The deeper the common volume is into the ice/snow region, the more important is the need for accurate modeling  Modified ITU-R model  Modified Awaka model  -120  -120 CQ TJ  -160  1  2 3 Rain height (hFR), km  4  -160  1  2 3 Rain height (hFR), km  4  F i g u r e 7-10: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of median rain height for the Bf link (/= 11.2 GHz) for the modified ITU-R and Awaka models (-6.5 dB/km roll-off: , -4.0 dB/km roll-off: —-).  University of British Columbia  143  Electrical Engineering  Chapter 7: Application to Microwave Links  Modified Awaka model  Modified ITU-R model  -110  -110  (0-120  -120  CQ  C/>  o -130  1-130  ~<^<  .  CO  "E  -140  co  §-140 -150  1  2  -150  3 4 CV height, km  1  2 3 4 CV height, km  5  7-11: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*)% of times levels are generated as a function of median rain height for the L1-L8 links for the modified ITlf-R and Awaka models (-6.5 dB/km roll-off: , -4.0 dB/km roll-off: —-). Figure  of the ice/snow region. This can also be seen in figure 7-11 where the different L links (L1-L8) are shown as a function of common volume height. In general, and especially for shorter links, the difference between the interference levels predicted using the two reflectivity roll-offs is given by (6.5-4.0)*(/i v - h ), where hcv is the height of the C  FRm  common volume (hcv > h ). The above equation is intuitive; the first part is simply the FRm  difference in the reflectivity roll-off and the second part is the depth of the C V , relative to the rain height, in the ice/snow region. Because of the importance of the ice/snow region in interference prediction, especially at high frequencies, a thorough study of this region using radar data is needed. The inclusion of reflectivity roll-off statistics into the interference model should also be investigated. This may, however, substantially increase the computational requirements in interference modeling.  University of British Columbia  144  Electrical Engineering  Chapter 7: Application to Microwave Links  7.3.5 Transmission loss dependence on frequency The operating frequency is perhaps the most important determinant of how precipitation interacts with the signal. The higher the frequency of operation, the more complex the interaction and the more the signal is affected by the precipitation medium. Figures 7-12 to 14 show the 0.001, 0.01, 0.1 and 1.0 % of times interference level exceedances for the B f link for h  FRm  = 2.1, 3.0 and 3.9 km as a function of frequency. The  figures show that, initially, the interference level increases with frequency, reaching a maximum near 11.2 GHz. This is due to the enhanced reflectivity of rain with higher frequency. Beyond this point the interference levels decay with increasing frequency. This is due to the increased attenuation in rain which obliterate any reflectivity enhancement due to higher frequency. This is especially true at higher median rain heights (e.g., 3.9 km), where the signal is exposed to longer paths in rain where attenuation is found.  Awaka model  ITU-R model  Frequency, GHz  Frequency, GHz  Figure 7-12: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of frequency for the Bf link (h = 2.1 km) for the ITU-R and Awaka models (original: , modified: ). FRm  University of British Columbia  145  Electrical Engineering  Chapter 7: Application to Microwave Links  ITU-R model  Awaka model  -170  0  10 20 30 Frequency, GHz  40  10 20 30 Frequency, GHz  0  Figure 7-13: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of frequency for the Bf link (h = 3.0 km) for the ITU-R and Awaka models (original: , modified: ). FRm  Awaka model  ITU-R model  -130  •130 —«i-140 -150 -160  -170 '0  10 20 30 Frequency, GHz  -170 40 0  10 20 30 Frequency, GHz  Figure 7-14: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of frequency for the Bf Ymk.(h - 3.9 km) for the ITU-R and Awaka models (original: , modified: ). FRm  University of British Columbia  146  Electrical Engineering  Chapter 7: Application to Microwave Links  7.3.6 Polarization effect on transmission loss T h e L 3 l i n k , described  in table C - l , is used to study the effect  o f p o l a r i z a t i o n on  interference l e v e l p r e d i c t i o n . T h e L 3 transmitter is p o i n t i n g vertically (elevation angle = 90 degrees) w i t h a h o r i z o n t a l p o l a r i z a t i o n (polarization angle = 90 degrees).. C h a n g i n g the p o l a r i z a t i o n o f the transmitter between 0-180 degrees (0 and  180 degrees being  the  vertical p o l a r i z a t i o n ) , the predicted interference levels are g i v e n for 0.001, 0.01, 0.1 and 1.0% o f times u s i n g the I T U - R m o d e l (figure 7-15). T h i s figure shows that a m a x i m u m interference l e v e l is achieved at the horizontal p o l a r i z a t i o n w h i l e the interference l e v e l at the vertical p o l a r i z a t i o n is about 10 d B lower. T h e reason for this drastic change can be seen in figure 7-17. W h e n the signal p o l a r i z a t i o n direction is at or near the direction o f the r e c e i v i n g antenna (9 = 0 in figure 7-17), little scattering is generated and hence l o w  Figure 7-15: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of polarization angle for the L3 link for the ITU-R model (original: , modified: ).  University of British Columbia  147  Electrical Engineering  Chapter 7: Application to Microwave Links  interference levels are observed. When the signal polarization direction is away from the direction of the receiver, as in the case of the D l link (figure 7-16), little change due to polarization effect is observed. This effect is not accounted for in the original ITU-R and Awaka models and may be of importance in coordination distance prediction.  7.3.7 Effect of the initial-snow density on interference Figure 7-18 shows that the interference level predicted by the modified models is dependent on the choice of the initial-snow density. While a constant 0.1 g/cm initial3  snow density overestimates the effect of the melting layer, obtained using the appropriate initial-snow density relationship, a constant 0.2 g/cm  3  initial-snow density gives  acceptable levels. 7.4 Interference Sensitivity to Attenuation and Scattering An often ignored part of interference predication is the sensitivity of interference to variations in specific attenuation and scattering. This variation may be the result of different atmospheric conditions such as wind direction and velocity, type of rain (stratiform or convective) or it can be due to the use of different scattering and attenuation models. According to [106], there may be up to 2.5 dB difference between the simplified scattering model, the ITU scattering model and exact calculations (Mie scattering). At low rain rates and frequencies, variations in scattering are more important than those of attenuation. The opposite is true for high frequencies. For links where the common volume is well within the rain median, a small variations in attenuation can mean significant changes in  University of British C o l u m b i a  148  Electrical Engineering  Chapter 7 : Application to Microwave Links  the predicted interference level. For example, at 30 GHz and rain rates of 50 mm/h, an attenuation variation of 0.2 dB/km may be seen as minute but traversing a distance of 15 km, the variation in interference is 3 dB. An example can be seen in figure 5-19 where two different attenuation models are used to predict the interference levels for the DI link. The attenuation models are the power law model and the simplified Kharadly model. The two predictions vary by 2 dB, despite the fact that the two models give almost identical specific attenuation. More work needs to be done on the sensitivity of interference to attenuation and scattering modeling as well as variations due to meteorological factors. However this is outside the scope of this work. The work of Kharadly [28] on oblate spheroid and the divergence of attenuation and scattering values from sphere and depending on the canting angle of particles is especially relevant.  7.5 Discussion and Conclusion The melting layer is seen to affect interference level prediction, with better results obtained using the modified models. However, this improvement is quite small to justify the increased computation cost and time entitled in adding the melting layer to the rainonly scatter interference models, especially at high frequencies where its effect is seen as minimal. The melting layer is seen to increase interference at lower frequencies, and marginally decrease it at higher frequencies. At intermediate frequencies, the melting layer effect is quite small.  University of British Columbia  149  Electrical Engineering  Chapter 7: Application to Microwave Links  The ice/snow region is seen to be more important in determining interference levels, especially at higher frequencies and at the higher latitudes where the 0°C isotherm is low and where the common volume is deep inside the ice/snow region. The choice of scattering and interference models can affect the resultant interference level. This is especially true for attenuation at higher frequencies. Small variation in specific attenuation may translate into several dBs variations in interference prediction. Interference prediction is also seen to depend on the choice of the meteorological model. Differences in the order of several dBs are observed between the Awaka and ITU-R model.  -120  1351  •  0  •  ' —  50 100 150' Polarization angle, degree  Figure 7-16: Transmission loss 0.001 (+), 0.01 (x), 0.1 (o) and 1.0 (*) % of times levels are generated as a function of polarization angle for the DI link for the ITU-R model (original: , modified: ).  University of British Columbia  150  Electrical Engineering  3.0-  Chapter 7: Application to Microwave Links  H-plane  * = exact X  96°  T  x  = model  x  x  X  x  X  /  i t 1  /  /  $  = 180°  ; *2.0  = 90°  \  /  X  \  *  \  model  \  x  \ \  1.0-  x  \\  \  *  i  3.0  \  1v ° x  X  /  fi n  *  t\  *'  1 1  — r l — *  i  1  0=0°  J  •  X.' * . ' '  * -X.  1.0-  X -  '  (9 = 0°  2.0-  E-plane  •  forward  3.0-  Figure 7-17: Bistatic scattering cross section of melting-snow (m /m ) for 1.25 mm/h rain rate, 9.6 GHz 2  3  frequency and 0.02 melting ratio. The direction of the signal polarization is at 9 = 0 ° .  -120  CQ  1-130-  % Time Interference > Ordinate Figure 7-18: Transmission loss statistics for the Bf link (f = 4 GHz). The interference levels obtained using the modified model, with appropriate initial-snow density relationship (-.-.-.), are compared to those obtained using 0.1 g/cm ( -) and 0.2 g/cm ( ) initial-snow densities. The original (rain-only) model 3  3  is in solid  University of British Columbia  151  Electrical Engineering  Chapter 7: Application to M i c r o w a v e Links  Figure 7-19: Interference prediction for the DI link using Kharadly's attenuation model (solid) and the power law attenuation model (dashed).  University of British Columbia  152  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  Chapter 8: Interference on Mobile Satellite Links due to Hydrometeor Scatter  Not long ago, mobile satellite communications systems were very rare and spectrum was in abundance. Today, we face an ever increasing use of mobile satellite communication and we are facing a crowded spectrum. Within the next few years, dozens of satellite systems for mobile and fixed (aimed at business applications, utilizing low gain small antennas) applications are being launched with frequencies ranging from under 1 GHz to 45 GHz. The recent debate (and indeed vicious fight) over spectrum between the newly emerging Local Multipoint Distribution System (LMDS) and satellite services in the United States is a case in point. It is therefore important to examine whether it is possible for terrestrial and satellite services to share the same frequency band. This chapter does not claim to have resolved the issue but is meant to raise some of the problems that arise with frequency sharing.  University of British Columbia  153  Electrical Engineering  Chapter 8: Interference on M o b i l e Satellite Links Due to Hydrometeor Scatter  While interference caused by precipitation has received considerable attention for fixed services applications, there appears to be little done on its effect on mobile systems. In this chapter, we estimate the interference level at a typical mobile terminal in the vicinity of a transmitting earth station.  8.1  Description  The geometry considered assumes a high directivity earth station with 30 degree elevation angle transmitting to a satellite (satellite 1). A mobile terminal, which receives a geostationary satellite (satellite 2) signal at the same frequency, is in the vicinity of the earth station and has an elevation angle of 25 degrees (figure 8-1). The mobile antenna has a gain of 28.5 dB and half-beamwidth of 7.2 degrees; its sidelobes are assumed to have a gaussian shape with half-beamwidth 3.675 times that of the main lobe at -17 dB (See equations 2.6-2.12). The rain rate statistics considered are those of the forward Chilbolton-Baldock (CB) link [5]. The rain height is set at 3.0 km and the reflectivity above the 0°C isotherm decreases at a rate of 6.5 dB/km. For the sake of simplicity, a flat earth is considered. The geostationary satellite is located 35786 km (typical) from the mobile antenna. It is therefore possible to assume a constant elevation angle for the mobile terminal, as long as its domain of mobility does not exceed a few hundred kilometers.  8.2  Simulations  University of British Columbia  154  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  Receiving Station  d t r  Transmitting Station  Figure 8-1: Interference geometry between an earth transmitting station and a receiving mobile terminal.  In this chapter, we examine the interfering signal strength (caused by precipitation) versus the desired signal arriving from the satellite. This is a more satisfactory approach than examining the absolute interference values since, because of the lower powers involved, these mobiles are more susceptible to interference. Also the interference to desired signal ratio corresponds to the signal to noise ratio (SNR) which provides the systems engineer with a better means to predict the bit error rate of the system. The interference signal, P'interference, due to precipitation is calculated according to the treatment in chapters 2, 3 and 4. The scattering formulations are those of the artificial dielectric model (section 6.4) and the attenuation formulations are those of the meltinglayer extended power-law (aR ) formulations (section 6.5). The initial-snow density used b  University of British Columbia  155  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  in this chapter are those derived in chapter 5 and presented in figure 5-17. The desired signal arriving from a satellite is calculated according to: ^ ^ f f i P +G , - ^ , where  EIRP  (8.1)  is the Effective Isotropic Radiated Power of the satellite transmitter,  GR  is  the gain of the receiving antenna, Lrmai is the total loss encountered by the satellite's signal arriving at the mobile antenna. The effective isotropic radiated power (EIRP) is defined by:  EIRP  =  P (6,<fi)  (8.2)  lgl  where P, is the available antenna input carrier power from the transmitter power amplifier, including circuit coupling losses and antenna radiation losses, and g,(9<p) is the transmitting antenna gain function in the angular direction (9, <p) of the receiver. The total loss is a combination of path loss, losses due to atmospheric gases and precipitation. Therefore the total losses are given by: L„ r  Hd  = 92.45+ 201og, S +20 l o g / + L + L 0  l 0  a  (8.3)  p  where S is the distance between the mobile and the satellite in kilometers, 92.45 is a constant for 201og 4^/?/c in kilometers and gigahertz, / is the frequency in gigahertz, 10  L  u  is the attenuation due to atmospheric gases and L  p  is the attenuation due to  precipitation. An examination of the above equation reveals that the total path loss contains three parameters that are frequency dependent. These are 20logio(f), gaseous losses and losses  University of British Columbia  156  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  due to precipitation. Attenuation on earth-space links due to precipitation are discussed in detail in chapter 9. However, at this point in the study, they are ignored. We will, however, return briefly to them at a later point in this chapter: The gaseous losses for different frequencies at an elevation angle of 25° are given in table 8-1 [94]. The difference between the interference and desired signal levels in dBm is given by:  P (interference/desired)  P(interference) ' P'(desired) (8.4)  Therefore the difference in transmission losses is given by:  L(iiuerjerence/desired) ~ L(desired) ' L(inierference)  (8.5)  Assuming the satellite's effective isotropic radiated power (EIRP) to be 30 dBW and that of the transmitting station 80 dBW,'the interference signal level, relative to that of the desired signal, is calculated for different mobile-terminal and rain-cell positions, for a range of rain rates between 0.25 and 30 mm/h with the melting layer present, and up to 150 mm/h for rain only. A frequency range between 1.0 and 40.0 GHz is examined. The analysis is divided into two parts. In the first part, the interference analysis is done for a fixed Awaka-type rain cell with and without the presence of the melting layer (figures 8-2 to 8-5). In the second part, a statistical transmission loss is calculated and examined using the ITU-R and modified ITU-R model (figures 8-6 to 8-9).  For long separations, the main lobe of the earth station and mobile terminal's antennas intersect either high into the rain structure, where reflectivity is very weak, or above it, where reflectivity is assumed zero. However, it is observed that the interference level  University of British Columbia  157  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  Frequency (GHz)  Path Loss in dB  4.0 10.0 15.0 20.0 30.0 40.0  0.1 0.4 0.6 1.6 2.0 3.1  Table 8-1: Loss due to atmospheric gases on an earth-space link for terminals with 25° elevation angle [94].  remains strong. Because of the low gain of the receiving antenna, the gain of the side lobes is relatively high. Therefore, much of the interference is due to the coupling between the main lobe of the earth station and the side lobes of the mobile terminal. This is seen in figure 8-2, while the interference level in figure 8-3 is seen to decrease rapidly with distance for a higher gain antenna (35 dB gain and 2.8 degrees half-beamwidth). For a higher gain mobile antenna, the sidelobes diminish rapidly reducing the coupling between the two antennas. It is also observed that the melting layer at 10 and 20 mm/h enhances the interference and thus degrading the S/N ratio at/= 4 GHz. Its effect becomes very small at 20 GHz (figure 8-4) and indeed at higher frequencies, the melting layer acts to decrease the interference level. This can be further seen in the statistical treatment in figures 8-6 to 8-9. It is also noted in figure 8-5 that P(i mesired) increases with frequency; however, they are almost nter1  identical at 30 and 40 GHz. Please note that the interference level decreases between 30 and 40 GHz due to severe rain attenuation. This decrease is, however, being offset by the increasing path loss experienced by the satellite's signal as the frequency increases.  University of British Columbia  158  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  8.3  Discussion and Conclusion  The results of these computations show that the interference signal could be significantly higher than the desired signal. For example, at a rain rate of 30 mm/h and a frequency of 20 GHz, the interference signal is about 4 dB higher than the desired signal at a separation distance of 80 km from the transmitting station; the two signals are about equal at 120 km and, at 160 km, the interference level falls to approximately 4 dB below the desired signal level. Generally, the interference level increases with increasing rain rate or frequency. Also, the presence of a melting layer tends to enhance interference at lower frequencies and to decrease it at higher frequencies. It is also found that interference coupling is mainly due to the sidelobe characteristics of the receiving antenna and, with higher receiver antenna gain (e.g., 35 dB gain and 2.8 degrees half-beamwidth), the interference level is sharply reduced, 43 dB below the desired signal. This result indicates a need for more accurate modeling of the receiver antenna radiation pattern. The interference  problem could be mitigated with the use of spread spectrum  communications since (i) the spread spectrum signal makes use of the underutilized part of the spectrum, (ii) before detection, the interfering signal is spread over a wide bandwidth. Also the interference from a spread spectrum signal to other systems tends to be minimal because of its lower transmitted power per bandwidth in Hertz (a 2 dB improvement in the S/N ratio corresponds to a significant increase in system performance - 10 fold improvement in the bit error rate in figure 8-10).  University of British Columbia  159  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  In the above figures we discussed system performance in view of the interference to the desired signal level. However, for an actual system it is desirable to operate several dBs above the noise floor. For a probability of bit error of 10", a signal to noise ratio of a 1  4  least 8 dB is required (figure 8-10) [95] for two and four phase PSK modulation and 10 dB for four phase-DPSK modulation. This further increases the requirements on the coordination distance. For system 1, and at 20 GHz, it is necessary to maintain 150 km separation to achieve a signal to interference ratio of 10 dB, 99.99 % of the time (figure 8-8). In addition, and depending on the frequency, the mobile link will suffer from degradation due to hydrometeor attenuation. This will further dictate an increase in the coordination distance. Hydrometeor-induced fading on earth-space systems is discussed in chapter 9. Operating the mobile terminal within a 150 km region from the earth station results in an increase of the bit error rate and as the mobile terminal moves closer to the earth station, its desired signal is exceeded by the interference signal resulting in the loss of the link.  In conclusion, for a viable mobile communication in the presence of strong interferers, the suppression of the sidelobes is essential.  1  Frame error will be significantly higher. A frame can consist of hundreds or thousands of bits.  University of British Columbia  160  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  Figure 8-2: System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for f = 4 GHz and different rain rates calculated using the original Awaka model (solid) and the modified-model (dashed).  50  distance between earth station and receiver, km Figure 8-3: System 2 (35 dB gain and 2.8 degrees half-beamwidth): Interference level relative to desired signal level for f = 4 GHz and different rain rates calculated using the original Awaka model (solid) and the modified model (dashed).  University of British Columbia  161  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  distance between earth station and receiver, km Figure 8-4: System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for f = 20 GHz and different rain rates calculated using the original Awaka model (solid) and the modified model (dashed).  10  10 10 distance between earth station and receiver, km  10  Figure 8 - 5 : System 1 (28.5 dB gain and 7.2 degrees half-beamwidth): Interference level relative to desired signal level for R = 10 mm/h and different frequencies calculated using the original Awaka model (solid).  University of British Columbia  162  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  Figure 8-6: Interference level relative to desired signal level statistics f o r / = 4 GHz and different d, distance (km) calculated using the original ITU-R model (solid) and the modified ITU-R model (dashed). r  -30 10" L  10"  10  10  % of time  Figure 8-7: Interference level relative to desired signal level statistics f o r / = 10 GHz and different d,r distance (km) calculated using the original ITU-R model (solid) and the modified ITU-R model (dashed).  University of British Columbia  163  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter 20  15 .20 km CQ  -o  10  • • - *  •^C—-\!  : 50 km  CO Q  ra -5  80 km "  CO  >^J_00 km '  c -15  '  '''^•;TS\;  J  'S.  -20  -  :  15C km  *' " "T^x •  -25 10"  10  10  : \  10  % of time  Figure 8-8: Interference level relative to desired signal level statistics for f -20 GHz and different d distance (km) calculated using the original ITU-R model (solid) and the modified ITU-R model (dashed). lr  10  10 % of time  Figure 8-9: Interference level relative to desired signal level statistics f o r / = 30 GHz and different d, distance (km) calculated using the original ITU-R model (solid) and the modified ITU-R model (dashed). r  University of British Columbia  164  Electrical Engineering  Chapter 8: Interference on Mobile Satellite Links Due to Hydrometeor Scatter  0  2  4 6 8 10 SNR per bit. y (dB)  12  14  ft  Figure 8-10: Probability of bit error for binary and four-phase PSK and DPSK [95].  University of British Columbia  165  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communications  Chapter 9: Effect ofthe Melting Layer on Fading in Satellite Communication  In this chapter, the attenuation properties of the melting layer are examined. The total and excess attenuation caused by melting layer presence are derived. The original Awaka/Capsoni meteorological model is modified and used to generate attenuation statistics for satellite links (with and without the melting layer). This model is, in turn, applied to different cases, specifically three links of a NASA-sponsored propagation experiment to measure fade statistics in the Ka band. Simple procedures to include the contribution of melting layer attenuation into rain-only attenuation models are discussed. For the sake of simplicity, we refer to backward reflectivity as simply reflectivity in this chapter. 9.1 T h e power-law (aR* ) data fitting procedure 3  It has been observed that reflectivity and attenuation properties may obey a loglog relationship [86]. Fitting reflectivity (z) and attenuation (a) into an aR relationship, b  where R is the rain rate in mm/h, has been discussed in chapter 6. The parameters a and b  University of British Columbia  166  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communications  are given by the forms in equations 3.17 and 3.16, respectively. The variables x and v are given by: (9.1)  x = \n(R) y = \n(a)  or  y  = ln(z)  (9.2)  9.2 Reflectivity and attenuation in rain  Attenuation and reflectivity values for rain are generated using the Mie scattering programs developed in this work (chapters 5 and 6) using the assumptions made in section 5.11 about the microphysics of the melting layer. The attenuation and reflectivity, as a function of rain rate, are fitted into aR equations, where a and b are frequency b  dependent parameters. Therefore the reflectivity ZR (mm /m ) and attenuation a (dB/km) 6  3  R  of rain are given by: h  R  z  a =aR" b  R  a  mmV  (9.3)  dB/km  (9.4)  3  z =a R -  Two sets of a and b frequency-dependent parameters are generated. The first set (set I) is generated using rain rates up to 300 mm/h. The other (set II) was generated, using rain rates up to 50 mm/h, to optimize accuracy in the rain rate region where the melting layer is likely to be present. The parameters a and b are listed in table 9-1, from which a noticeable difference can be seen between the two sets.  University of British Columbia  167  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite  Set II: Short rain rate disti i b u u u n Reflectivity Attenuation  Set I: Whole rain rate distribution Attenuation Reflectivity  Freq. GHz  dal  bal  a  zl  Communications  bu  b  boa  z2  1.00  6.4353E-05  0.90230  355.507  1.38622  6.6243E-05  0.87899  307.807  1.44600  2.00  2.7016E-04  0.93477  352.306  1.38205  2.7211E-04  0.91268  305.821  1.44157  4.00  1.3249E-03  1.03402  340.822  1.36674  1.2486E-03  1.01780  298.891  1.42454  6.00  3.7637E-03  1.11493  339.174  1.39790  3.3497E-03  1.11881  289.390  1.44908  7.00  5.6241E-03  1.13041  346.461  1.41878  4.9777E-03  1.14290  289.168  1.47610  8.00  7.9153E-03  1.13561  355.638  1.43472  7.0278E-03  1.15390  292.868  1.49829  10.00  0.01374  1.13084  374.585  1.45206  0.01240  1.15499  306.697  1.52436  12.00  0.02116  1.12077  391.393  1.45461  0.01940  1.14588  323.622  1.53142  15.00  0.03518  1.10830  409.545  1.43919  0.03283  1.13176  348.131  1.51775  20.00  0.06661  1.09278  417.367  1.38229  0.06324  1.11474  375.135  1.45726  25.00  0.10852  1.07304  400.391  1.30537  0.10485  1.09603  379.497  1.37523  30.00  0.16042  1.04685  365.998  1.22192  0.15851  1.06990  363.725  1.28629  35.00  0.22075  1.01736  321.918  1.14031  0.22351  1.03763  333.584  1.19774  40.00  0.28720  0.98760  274.797  1.06560  0.29760  1.00247  295.432  1.11305  45.00  0.35714  0.95919  229.275  1.00053  0.37759  0.96713  254.535  1.03418  50.00  0.42814  0.93306  188.158  0.94569  0.46016  0.93367  214.645  0.96315  55.00  0.49780  0.90974  152.470  0.90012  0.54186  0.90345  177.648  0.90125  60.00  0.56415  0.88914  122.455  0.86166  0.61979  0.87667  144.830  0.84740  65.00  0.62651  0.87090  98.115  0.82817  0.69300  0.85287  117.232  0.79985  70.00  0.68509  0.85479  78.933  0.79866  0.76191  0.83173  94.987  0.75801  75.00  0.73981  0.84078  63.948  0.77342  0.82658  0.81324  77.341  0.72264  80.00  0.78985  0.82884  52.122  0.75329  0.88599  0.79744  63.207  0.69488  85.00  0.83407  0.81883  42.569  0.73834  0.93867  0.78422  51.635  0.67443  90.00  0.87180  0.81054  34.687  0.72743  0.98364  0.77335  42.017  0.65919  95.00  0.90323  0.80373  28.170  0.71902  1.02091  0.76453  34.060  0.64702  100.00  0.92914  0.79808  22.881  0.71234  1.05129  0.75736  27.606  0.63745  Table 9-1: The fitted a and b parameters for reflectivity and attenuation results for different frequencies. Set I refers to the fitted parameters using the whole range of rain rates (up to 300 mm/h) while Set II is optimized for the region where the melting layer exists (up to 50 mm/h).  9.3 Total Attenuation in the Melting Layer 9.3.1 Method of Computation The total attenuation that an electromagnetic wave suffers while traversing the meltingsnow layer is calculated using a numerical integration of the specific attenuation over the path in the melting layer. (9.5)  where a is the specific attenuation in the melting-snow layer (dB/km), £ is the elevation m  angle of propagation, H is the thickness of the melting layer, and h is the position in the melting layer.  University of British Columbia  168  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communications  At times, it is desirable to obtain normalized values of melting layer attenuation, independent of both the elevation angle and the melting layer thickness. In this case, we set H to 1 km for all rain rates and the elevation angle to e = 90°. From the normalized values of total attenuation, it is possible to derive the total attenuation for any elevation angle, using any melting layer thickness model. 9.3.2 aR representation of total attenuation b  The total attenuation in the melting layer, for various initial-snow densities for both the concentric-sphere and the percolating-sphere (spongy) model, are derived according to equation 9.5 and are found to fit the aR power-law relationship. The parameters a and b b  are shown in table 9-2 for the concentric-sphere model and table 9-3 for the spongysphere model, assuming 1 km melting layer thickness and a 90° elevation angle. The true total attenuation value could then be obtained from:  .  a  olal  = Hx  (9.6)  lOOOx stnf  where H is the true thickness of the melting layer and e is the elevation angle. We have already established earlier that initial-snow density varies with rain rate. Unfortunately using variable densities (densities as a function of rain rate) does not yield an acceptable fit of the total attenuation into an aR equation. However, it is found that a constant b  initial-snow density with rain rate yields acceptable agreement between the calculated data and the fitted results. Therefore, in order to calculate the total attenuation in the melting layer at a given rain rate, it is necessary to interpolate between the different a and b coefficients obtained for the different initial-snow densities.  University of British Columbia  169  Electrical Engineering  Chapter 9: Effect of I lie Melting Layer on Fading in Satellite Communications  tvo  s  s  3  *n o ci  E  s  s  ©  s  o ci  s  s  3  tn  vo Ov  ©  3  3 o o  S  VO  3  o vd  •s. m o  Ov  ci  3  3  3  3  II  VO  3  s  ii  3 Tf Ov  3 wv Ov O  3  3  Ov  ci  3  o  \4\  Ov  ci  Ov  Ov  O  ci  3  3  U5  University of British Columbia  170  Electrical Engineering  Chapter 9: bffect of the Melting Layer on Fading in Satellite Communications  © rs rs OO r~ Os OS v t 00 Vt 50 Tf OS N Tf OS 10 Vt Vl 3" vt •+ O N 0N © rs c i T f s o f - OS rs *S 00 00 ON ON ON ON OS OS OS q q q © © © © © © ci ci ci  7\  g  -O  u in  o m o <3 o o ©  ©  NO  © © ©  00 00 00 © ON © 00 O N d d vo  pn  g  eu  o m  g in  g  -c  w  E  -c  •&  *n fS  O  o pn  g  •o  ON  ON  ON  ©  d  d  ©  NO  00 Vt SO © ON Tf so V l ON Vt r~ CS pn © © © o © © d ci  o  ON  g  ON  ON  ON  ON  ©  ©  ©  ©  1 Fre  ci  ON  ON  d  ©  d  ©  o r*o o o d  ON  cs ON CO © © CS cs © © © ©  vo <S © © ©  f S CO ON O N NO r NC 00 Vt © © ON  ON  ON  d  ©  ©  00 o © © d  Vl © T f cs c i cn © © o d ©  NO  ON  ON  ©  NO  ON  ON  ©  ©  ©  ©  d  ©  ON  m CS r-  Cl  © ©  a  .H a  c © o  NO  © © ©  rs  Os  d  d  t> Os SO o cn OS rs Tf d d  3  d  d  OS  SO Tf 00 OS m Tf 00 00 SO s o  r-  Os  O  OS  OS  OS  vt Os  2  d  d  d  d  vt  o Tf o q  rs o r00 00 OS 00 OS  d  OS  o rr~ r~  OS  SO  cn v t so Os  d  d  d  oo rs o o T? d  r s Tf m Vt rs Tf T f Vt d d  vt OS  d  OS OS Tf 00 v t r - cs OS pn 00 Os v t Os OS OS so r> Os 00 t> s o v t Tf OS  OS  OS  OS  Os  OS  d  d  d  d  d  d  ON  ON  ON  d  ©  ©  ©  00 rs 00 cn  Os  Os  Os  ON  d  d  d  ©  OS  OS  cs rs  rs  d  d  vt  r> 00 rs r>P - Os cn OS oc 00 o d  ci  d  SO  9 t>  Vi cs ON cs  © o © rpn r s Vt o o o C l Tf v t SO t> ON  OS  Os  Os  OS  ©  O  o  O  o  o  pn pn 00 S £ o OS OS co o o o  0C  ©  .—  OS vt  m  o  rs v t  University of British Columbia  OS  Os  OS  d  d  d  _  00 Tf rs d  00 rs OS Os 00 rs m d d  §  c s Tf r~ cn SO rs 00 SO Tf v t o d  vt  PS  cn Tf o pn  SO  OS  OS  OS  Os  SO vt OS  o  o  o  ©  o  m  rs 00 cn T T O  vt 00 00 v t Tf oc rs 00 Tf o ot OS cn OS Tf 00 v t SC r » f> 0( o o o o a  o  00 rf © 00  ?  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SO  cn Tf  r-  N 00  cs OS rs r 00 o rs 00 cn OS Tf TH r s rs cn d d d d  Tf  Tf cn pn OS rs rs o o OS OS o q q O d  r» o CN p 00 SO 00  ON  NO  C"l  so  Cl  NO  00 cs © d  d  cs SO m Tf rn Vt vt ro o i—* ci d d d  ©  cn rr> cs ON  vt  rs r-» r f Tf r - Os Vt t> r - 00 Tf O cs r s cs C J Vt SO  © © ©  w  »-H ©*  d  ON  © © d  g  ON  vt OS  ON  m T  ^  OS  00 Vt c i ON 00 Cl 00 © cs © © © © © ©  ^< Os  t>  ON  00 O N NO c i p i vt cs Tf NO T f r> cs o ©  CS  pn  © © ©  d  — OS 00 00 Vt CS t> SO cs o SO SO SO r f v t r> 00 O O N Os Os OS O q © c i O d ~~  cs CN Tf  NO CS  3-  Vt o  d  ©  O  © © ©  ON  ©  •s  NO  VO NO VO  o  pn  ON  00 CM c i VO NO 00 <N T f cs o ©  ON  <3  r» Os T 00 Tf pn r s r CS 3 SO o cs pn © o o O d ci o d d Cl  m  rTf SO Tf O q q  00 00 m o Os SO vt o o vt o rs rs cn d d d  -  d  Ml  •  ©  so oo r o SO vt  •s OS  00 o d d  00 r f O N s o Vt 00 pn 00 rs © 00 CS SO Tf v t rs 00 vo * f CO 00 rs Os o rs cs r f Vt 00 rs rs ON O N O N Os OS OS p q q c i c i d © d ©  cs Vt c i Vt NO r f CS tn  o © o d  pn  ON  ON  cs ©  V© </»  ©  ©  3 r~  es 00 Tf o vo Vt v t CS cn v s © o o o © o © d CJ  o o  p- © Tf 00 o o © o o d ©  ©  ON (S  © 00 CS ON c i © © © © © NO  c i (S c i © Vt O N 00 V I cs © © © © © © d © d ©  ON  -ft  v» rf © © d  o o © d  o  Tr  cn  r~ rs SO rs q  3  TT t>  ON Cl  ON  vo Vt Vt cs NO cs ON vo O N 00 00 oc oc oc © © © Cl  © 1-* 00 Tf  Vt r> c  Cl NO  c  ON  Vt Tf 00 00 ©  ON  rVt Tf  Tf 1  © Vl © Vl © Vl © r - 00 oc ON ON  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  9.3.3 Excess Attenuation The excess attenuation due to the melting layer is obtained by subtracting rain attenuation over the same path from the total attenuation. (9.7)  where a  is the specific attenuation in rain. It is also possible to derive the excess  R  attenuation due to the melting layer using the aR representations for the total attenuation b  in the melting layer and the specific attenuation in rain. (9.8) where aR is the normalized attenuation in the melting layer given in dB/km and kR is the b  1  specific attenuation of rain given in dB/km. 9.3.4 Numerical Simulation Examples of excess attenuation due to the melting layer are shown in figures 9-1 to 9-3 for a terminal with an elevation angle of 29.4 degrees and for 20.195, 27.5, 45 and 100 GHz channels. The first two frequencies are those of the Advanced Communications Technology Satellite (ACTS) discussed further in section 9.4.2. These figures show that the concentric and the percolating-sphere model give fairly comparable results. It is worth noting that the two models give closer predictions as frequency increases. They also show that excess attenuation due to the melting layer increases with higher rain rates and higher frequencies.  University of British Columbia  172  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  401  401  Rain rate (mm/h)  Rain rate (mm/h)  Figure 9-1: Excess attenuation in the melting layer for variable initial snow density and f = 20.195 , 27.5, 45, 75 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( ) as a function of rain rate.  % Time  % Time  Figure 9-2: Excess attenuation in the melting layer for variable initial snow density and f = 20.195 , 27.5, 45 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( ) as a function of percentage of time, assuming the melting layer occurring 100 % ofthe time.  University of British Columbia  173  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  % Time  % Time  Figure 9-3:  Excess attenuation in the melting layer for variable initial snow density and f = 20.195 , 27.5, 45 and 100 GHz using both the concentric-sphere model ( ) and the spongy-sphere model ( ) as a function of percentage of time, with the melting layer occurring at appropriate percentage of time.  Figure 9-2 displays the excess attenuation as a percentage of time, assuming that the melting layer is present 100% of the time. In order to obtain the excess attenuation as a percentage of time, assuming appropriate melting layer occurrences, the exceedance percentages are multiplied by the probability the melting layer occur at each percentage of time, xa%), (The cumulative distribution function for point rain rate gives the rain rate at each percentage of time. Using the melting layer presence statistics, discussed in chapter 5, X(i%) is derived). These are displayed in figure 9-3. The data points in figure 9-3 are simply those of figure 9-2 shifted toward the lower exceedance percentage points by their associated individual probability of presence of the melting layer. These will be used later to discuss simple procedures to add the effect of the melting layer to attenuation statistics derived using classical rain-only attenuation models.  University of British Columbia  174  Electrical Engineering  Chapter 9: Effect ofthe Melting Layer on Fading in Satellite Communication  9 . 4 Simulation of Attenuation on an Earth-Space Link Using a Modified Awaka Rain-Cell Model In this section, the Awaka meteorological model is used to study the effect of the melting layer on attenuation statistics. Two types of statistical approaches are used to include the effect of the melting layer in the overall statistics. The first (Method I), like the modified ITU-R model for interference, assumes that the melting layer either exists or it does not along the whole propagation path. These two distinct cases are assigned their respective probability of occurrence. The second method (Method II)  an artificial medium with  attenuation properties the average of those for rain and melting snow, scaled to appropriate probability of occurrences.  9.4.1 Method of computation Specific attenuation values along the propagation path are summed to generate the total attenuation. Since attenuation above the 0°C isotherm is negligible and can be assumed zero, only the regions below 0°C isotherm need be considered.  The meteorological treatment for the Capsoni and Awaka models, discussed in chapter 6, is used here with one notable exception: the domain of mobility in the interference scenario is centered around the center of the common volume. In this case, no common volume exists to center the domain of mobility around. The domain of mobility of a rain cell is taken as the area where the existence of a rain cell may cause fading on a link. Figure 9-4 shows a typical domain of mobility. The size of the domain of mobility in the y direction is determined solely by the size of the rain cell. This is also true in the - x direction. In the + x direction (direction of propagation), the length of the domain is  University of British Columbia  175  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  2r,  T  Figure 9-4: Fading geometry and domain of mobility of rain-cells.  determined by both the size of the rain cell, the height of the 0°C isotherm, and the terminal's elevation angle. In the y direction, the domain of mobility is 2r  max  x direction, it is r  tnax  elevation angle. If h  FR  long and in the + x direction is  + h /sin£, FR  long. In the -  where £ is the'  is negative, no computation is necessary since attenuation above  the 0°C isotherm is assumed zero. 9.4.2 Numerical Simulations The above described attenuation model is applied to three of the seven different sites of the ACTS propagation experiment in North America, with Vancouver being one of the considered sites. The purpose of the experiment is to investigate the feasibility of using the Ka-band (27.5/20 GHz) spectrum for satellite communications, which will depend, on how accurately the rain-fade statistics and fade dynamics can be predicted. This is especially important for very small aperture terminal (VSAT) operation, where the fade  University of British Columbia  176  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  margin is at a premium. The three sites, we apply the above model to predict the attenuation statistics for, has their elevation angles, latitudes, 0°C isotherm heights, rain regions, and associated rain rate statistics, listed in table 9-4 and 9-5. The three cases are chosen to correspond to low (AK), medium (BC) and high (MD) elevation angles. Attenuation statistics generated with the melting layer present (melting layer present 100 % of time for rain rates below 40 mm/h and 0 % otherwise, method I and II), and with no melting layer are shown for each site. Due to the lack of availability of melting layer statistics for the different regions, those derived from McGill's university observations are used. The attenuation levels predicted using the concentric and the spongy-sphere models are very similar, hence only the spongy-sphere model need be used to predict fading for the rest of this chapter. This is shown for the Vancouver site in figure 9-5. Figures 9-6 to 9-8 show the attenuation statistics for the three considered sites for the two % time 1 0.3 0.1 0.03 0.01 0.003 0.001  Rain rate (mm/h) K C D 0.7 2.1 1.5 2.8 4.5 ' 4.2 12 5 8 9 13 23 15 19 42 26 29 70 100 42 42  Table 9-5: The rain rate statistics for the relevant regions in which the above terminals lie. Site Fairbanks, AK, USA Vancouver, BC, Canada Clarcksburg, MD  CCIR Rain Zone C D K  Latitude (North) Deg. 64.86 49.26 39  Median Rain height (km) 1.0912 2.2237 2.9686  Path elevation angle (degree) 7.96 29.26 38.91  Table 9-4: Relevant parameters for the ACTS propagation terminals under investigation.  University of British Columbia  177  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  BC site, f = 20.195 GHz  B  % Time  C  s i t e  > = f  2  7  5  G  H  z  % Time  Figure 9-5: Comparison of melting layer-induced attenuation using the concentric ( and spongy ( ) sphere models. The rain-only attenuation is in solid.  )  ACTS frequencies (20.195 and 27.5 GHz). It is seen that for the terminal with high elevation angle (MD), the effect of the melting layer is minimal on the overall attenuation statistics. On the other hand, this effect increases with lower elevation angles (BC and A K ) . This is due to three factor: The first is that lower elevation angles translate into longer paths in the melting layer and hence enhanced melting layer contribution. The second is that the terminal with the high elevation angle lies in a region where high rain rates, and low melting layer probability of occurrence, are dominant. The third is that in the region where the high elevation terminals lie (low latitudes), the 0°C isotherm is quite high, meaning that most of fading will come from attenuation due to rain. In the northern regions, the 0°C isotherm is low and this translates into a smaller rain contribution to the total fade.  University of British Columbia  178  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  AK site, f = 20.195 GHz 30  1  AK site, f = 27.5 GHz  '  \ \  \  '• \  •.  \  '.  \  3 20  \ \  •.  \  c  '•.  \  q  '•.  \  '.  \ \  C  —  '-. \  -  .  CD  3 10  0  •  ^  \  N  X  -2  10  %Time  10  10  u  % Time  Figure 9-6: Attenuation statistics for the Fairbanks, AK, site. (Rain only: 100%: , Method I: —-, Method II: -.-.-.-).  Figure 9-7: Attenuation statistics for the Vancouver, BC, site. (Rain only: 100%: , Method I: — , Method II: -.-.-.-).  University of British Columbia  179  ,ML •  , ML -  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  60  MD site, f = 20.195 GHz 60  -2  10  % Time  0  10°  MDsite, f = 27.5 GHz  -2  10  10  % Time  Figure 9 - 8 : Attenuation statistics for the Clacksburg, M D , site. (Rain only: 100%: , Method I: — , Method II: -.-.-.-).  0  , ML -  It is also noted that method I and II give different attenuation prediction, with Method I consistently higher. It is argued here that Method I is more appropriate for the higher elevation angles (20 degrees and above) since the propagation path in the melting layer is relatively short and it is reasonable to assume that when the melting layer is present, it is present along the full path. For very low elevation angles, as in the case of Alaska, this may not hold. This can be seen in the excessive attenuation predicted using method I. In this case method II may well be more appropriate. The melting layer effect is quite small at the higher percentages of time, while at lower percentages of time the melting layer adds several dBs to fading. For high latitude regions, where low rain rates are dominant and where the propagation path is longer, this  University of British Columbia  180  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  effect may be important, especially for low fade margin systems. The effect of the melting layer, on the other hand, is deemed unimportant in relation to the total fading statistics at the lower latitudes. Yearly variations of rain rates, in these cases, can be far more important than the attenuation enhancement given by the presence of the melting layer. Figure 9-9 shows predicted attenuation levels for 45, 75 and 100 GHz for a terminal at the Vancouver site. These show that the contribution of the melting layer at these frequencies is quite high.  f = 45 and 100 GHz  f - 7 5 GHz  % Time  % Time  Figure 9-9: Attenuation statistics for the BC site for/= 45, 75 and 100 GHz (Rain only: ML -100%: , Method I: — , Method II: -.-.-.-).  University of British Columbia  181  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  9.5 Simple Techniques to Include Melting Layer Effects on Existing Rainattenuation Models It may be desirable to add the effect of melting layer attenuation to existing rain attenuation models without performing the extensive computations done in section 9.4. Procedures, to combine the attenuation statistics generated by conventional rain-only models with the excess attenuation due to the melting layer, for method I and n, are developed.  9.5.1 Method II: simplified procedure At each exceedance % point, the rain attenuation generated by a rain attenuation model and the melting layer excess attenuation are scaled to appropriate melting layer percentages and are summed. The total attenuation level is then obtained by adding a scaled level of excess attenuation to the rain attenuation statistics as in: + e(i%)  =  where  A,(i%  a  h  X  (i%)  x  A(,%), a a%) and r  e  X(\%)  are the total, rain, excess melting layer attenuation  value and the melting layer probability of occurrence for the.ith% exceedance percentage.  9.5.2 Method I: simplified procedure Rain attenuation statistics are generated assuming that the attenuation medium consists of rain particles only. We know, however, that a melting layer exists a certain percentage of time. For these cases, where the melting layer exists, the excess attenuation due to the melting layer (figure 9-2) is added to the rain attenuation at each % of time. Two sets of attenuation cumulative distribution functions are then present. The first is the original, assuming rain-only media. The other is the adjusted function assuming presence of the melting layer. The probabilities of occurrence for different attenuation levels, in  University of British Columbia  182  Electrical Engineering  Chapter 9: Effect of the Melting Layer on Fading in Satellite Communication  each curve, are obtained. Using the rain rate cumulative distribution function, a rain rate is assigned to each of these points. The two sets of probabilities, with and without the melting layer, are multiplied by X(j%) for the set where the melting layer is present and 1xa%) for the original set. The two sets of data are then summed to generate the combined cumulative distribution function. Figures 9-10 and 9-11 show a comparison between the simplified procedures, used to include the effect of the melting layer, and the exhaustive treatment offered in section 9.4. These figures show that they give very close values for all % points except at 0.001% of time for method II. In this case, the simplified procedures underestimates the effect of the melting layer with respect to the exhaustive treatment.  f = 20.195 GHz  f = 27.5 GHz  % Time  % Time  Comparison of the total attenuation using Method II statistical method in section 9.4 ( ) and the simplified method ( ). The rain-only attenuation is in solid.  Figure 9-10:  University of British Columbia  183  Electrical Engineering  Chapter 9: Effect of the M e l t i n g Layer on Fading in Satellite Communication  f = 27.5 G H z  f = 20.195 G H z  . \® \  \  V.  V, V v.  10  % Time  •  -2  % Time  10*  Figure 9-11: Comparison of the total attenuation using Method I statistical method in section 9.4 (-—) and the simplified method (  ). The rain-only attenuation is in solid.  University of British Columbia  184  Electrical Engineering  Chapter 10: Discussion and Conclusion  Chapter 10: Discussion and Conclusion  In this chapter, a discussion of the thesis work is given and a summary of its conclusions is presented. The chapter is divided into four sections. The first is about the modeling of the melting layer, the second is about interference modeling, the third is about fading, and concluding remarks are in the. fourth section. In this work, several important conclusions and procedures have been developed as well as several unexpected results were encountered. Perhaps, the most important one is the severe effect of the melting layer on fading for low-elevation (and low rain height) earth-space links similar to those encountered in the high latitudes (section 9.4). With the tools developed here for the rigorous and simplified prediction of melting-layer contribution, an engineer has available in his/her hand the tools necessary to estimate the total attenuation and thus the power and margin needed for proper operation. Another important conclusion of this work is that frequency sharing between different services is fraught with dangers, especially for the new broadband services with their low gain terminals. The small effect of the melting layer on interference (excepting at the lower frequencies), was unexpected, as we anticipated a stronger influence on interference prediction. A s well, the ice/snow region seems to be far more important in interference level prediction.  University of British Columbia  185  Electrical Engineering  Chapter 10: Discussion and Conclusion  The tools developed in this work for both interference and fading can be used to further study the effect of interference and fading on communications links. Also, we developed in this work the most comprehensive melting-snow layer model, based upon the most reliable data available.  10.1  Melting Layer Modeling  A model of the melting-snow layer is developed in this work. Some of the elements of the model come from other workers such as Klassen. Other elements are either developed or extended here. The details of the melting-snow model are available in chapter 5. Here, only the novel elements of the model are mentioned. 10.1.1 Derivation of an initial-snow density relationship Using measured peak reflectivity [9], along with the Mie scattering programs developed in this work, the snow density above the melting layer is derived. The initial-snow density is found to depend on rain rate. A relationship between the initial-snow density and rain rate is developed for both the concentric-sphere model and the spongy-sphere model. 10.1.2 A new relationship between the drag coefficient of snow and initial-snow density Using the work of Magano and Nakamura [64], a relationship between the drag coefficient of snow and initial snow density is developed and is given by equation 5.22. 10.1.3 A revised melting layer thickness The bright-band thickness is shown to underestimate the thickness of the melting layer, assuming a spongy-sphere representation of the melting-snow particle. A formula relating the thickness of the melting layer to bright band thickness is developed and is given by equation 5.37.  University of British Columbia  186  Electrical Engineering  Chapter 10: Discussion and Conclusion  10.1.4 The aR specific attenuation model extension to the melting layer b  The aR specific attenuation model for rain is extended to the melting layer for different h  initial-snow densities and is found to give acceptable results. This model is only valid for the concentric-sphere representation of the melting-snow particles (chapter 6). A version of the aR could be derived for the spongy-sphere model. However, it is not considered necessary h  for this work. 10.1.5 General remarks 1. In this work, a comprehensive melting-layer model is developed. Mie scattering programs are developed to compute attenuation and backscatter for a wide range of frequencies and rain rates for the concentric, composite and spongy-sphere representations of melting snow. These are given in appendix B. *  2. Melting profile: the use of  melting profile for the spongy-sphere particle and S3  melting profile for the concentric-sphere particle is found to provide reflectivity profiles similar to observations [9]. 3. Percentage of melting layer presence: general formulae (two of them) for the percentages of presence of the melting layer are developed. These are found to agree well with observations.  Interference Modeling  10.2  10.2.1 New interference models The melting layer is introduced into the rTU-R, Capsoni and Awaka models. Simulation software for these models are developed for both the original (with no melting layer) and  * The S melting profile was also recommended in [9] since it gives peaks that are located near the center of the bright band. 3  University of British Columbia  187  Electrical Engineering  Chapter 10: Discussion and Conclusion  modified (with melting layer) models. Techniques to include the effect of the melting layer in rain-cells and in the general meteorological models are developed. The simulation programs developed are flexible in that rain-cells of different shapes and distributions, different scattering and attenuation properties, and different melting layer models could easily be introduced. The simulations are, however, computer intensive and should only be regarded as research tools. They are not suitable for field use. It takes between 2-4 days, depending on the link parameters, for a SPARC 5 to generate the interference statistics using the Awaka model. The software for the models are given in appendix A.  10.2.2 Influence of the melting layer on interference prediction The presence of a melting layer in interference prediction models is found to affect the accuracy of prediction very little for the COST 210 data base. Improvements of only 0.5-1 dB are observed. This has been attributed to the high frequency of the links. This work showed that at these frequencies, the melting layer has little or no effect on interference prediction. It also showed that the inclusion of the melting layer increases interference levels by few dBs. It is recommended that the rTU-R prediction procedures be amended to include this effect at lower frequencies. At the very least, the current predicted interference levels should be increased by 2.5-3.5 dB at/= 4 GHz and 2 dB at/= 8 GHz. Overall, the melting layer is found to increase interference at the lower frequency range (1-8 GHz), marginally decrease it at the higher frequency range (above 20 GHz) and have little effect at intermediate frequencies (11.2 GHz).  University of British Columbia  188  Electrical Engineering  Chapter 10: Discussion and Conclusion  10.2.3 Influence of the ice/snow region on interference prediction The ice/snow region is found to play a very important role in interference prediction, especially at the higher frequencies and when the common volume is deep inside the ice/snow region. It is therefore desirable that better models for this region are used. Indeed, the ice/snow region is more important than the melting layer in interference modeling. 10.2.4 General remarks 1. Frequency: the operating frequency is shown to first increase the interference level and then decrease it when attenuation becomes very high. 2. Polarization: the direction of polarization of a linearly polarized wave is shown to significantly influence interference level prediction. A spot, in or near the direction of the transmitter's polarization where terminals could be installed with reduced risk of interference, is observed. 3. Rain height distribution: the use of full rain height distribution offered little difference in interference prediction to the scenarios when only the median rain height is used. 4. The predicted interference level can be sensitive to variations in attenuation and scattering. This is especially true for attenuation at the higher frequencies when the common volume of the link is within the rain medium. 10.2.5 Interference effects on mobile terminal Low gain mobile terminals are found to suffer serious performance degradation due to interference due to precipitation. In order to effectively operate in the presence of strong interferes, the side lobes of the mobile must be suppressed. Otherwise, the mobile terminal's availability will suffer.  University of British Columbia  189  Electrical Engineering  Chapter 10: Discussion and Conclusion  10.3  Attenuation Modeling  10.3.1 A new attenuation model The Awaka meteorological model is modified and used to develop an attenuation prediction software. To this effect the domain of mobility is changed and two methods to include the effect of melting layer attenuation are introduced. The total attenuation due to precipitation (with and without the melting layer) is calculated for some of the ACTS propagation terminals. 10.3.2 Influence of the melting layer on attenuation for satellite communications It is shown that for terminals with high elevation angles, the effect of the melting layer is minimal on the overall fading statistics. On the other hand, this effect increases with lower elevation angles. It is also shown that method I and II give different attenuation predictions, with method I consistently higher. The melting layer effect is quite small at the higher percentages of time. The opposite is true at the lower % of time with the melting layer adding several dBs to attenuation. For the high latitude regions, where low rain rates are dominant and where the propagation paths in the melting layer are long, this effect may be important, especially for low fade margin systems. The effect of the melting layer, on the other hand, is deemed unimportant in relation to the total fading statistics at the lower latitudes. Yearly variations of rain rates, in these regions, can be far more important than the degradation caused by the presence of the melting layer. 10.3.3 Simplified attenuation models Simple models for total and excess attenuation in the melting layer are developed and are found to agree well with extensive methods. Simplified procedures are developed to include these simple models into classical rain-only attenuation models. University of British Columbia  190  Electrical Engineering  Chapter 10: Discussion and Conclusion  10.3.3.1 The aR total attenuation formulation b  An aR^ formulation for total attenuation in the melting layer is developed for both the concentric and spongy-sphere models using the Mie scattering software discussed in chapter 6 and appendix B for the melting-snow layer model developed in chapter 5. 10.3.3.2 Simplified procedures to add melting layer effect into rain-only attenuation model statistics  Two methods to add the effect of the melting layer (using the above simple atf attenuation model) to rain-only attenuation models are developed. A comparison between the simplified procedures and the exhaustive treatment, used to include the effect of the melting layer, showed that they give very close values except at 0.001% of time in method n (see section 9.5). It is expected that the above two procedures could be used to alter current attenuation prediction models to account for the melting layer. It is also expected that they could provide the regulatory bodies with simple means to account for melting layer attenuation in fade models on earth-space links. 10.4  Concluding Comments  A melting-snow model is developed in this work. The melting layer is then introduced into the ITU-R, Capsoni and Awaka meteorological models for interference prediction. The effect of the melting layer on interference is studied. The Awaka meteorological model is also used to predict total attenuation on earth-space links and the effect of the melting layer is also studied. Simple models are also developed to account for the effect of the melting layer on fading. No such models are deemed possible for interference at this juncture.  University of British Columbia  191  Electrical Engineering  Chapter 11: Recommendations for Future Work  Chapter 11: Recommendations for Future Work  In this chapter, recommendations for future research work on melting layer microphysics, interference and fading are provided. These recommendations stem from observations in this work and they are by no means comprehensive.  11.1 Melting Layer Modeling Although the McGill University observation [9] were used as the source of melting layer presence statistics, the applicability of these measurements to other regions needs to be verified. Further examination of initial-snow densities in different atmospheric conditions is also necessary.  11.1.1  A novel melting-layer reflectivity a n d attenuation models  In this section, we propose the development of a new formulation to estimate specific reflectivity and attenuation is the melting layer based upon multi-parameter radar data and the melting-layer attenuation  and reflectivity  models developed in this thesis.  Specifically, given: 1. Rain reflectivity (corresponds to a rain rate)  University of British Columbia  192  Electrical Engineering  Chapter 11: Recommendations for Future Work  2. Peak reflectivity 3. Location of the peak 4. Thickness of the bright band 5. Location of the melting-snow particle saturation point (this thesis) 6. Scaling of the above relationships (1-5) to different frequencies (using the models developed in this thesis - this could be verified with simultaneous observations from two radars using two different frequencies) the shape of reflectivity in the bright band could be approximated through a form of interpolation. Estimating specific attenuation in the melting layer is a different story because no observations of peak attenuation are available. Peak attenuation in the melting layer do not coincide with those of peak reflectivity. However, using the models developed in this work, relationships between peak attenuation and reflectivities and peak attenuation and reflectivity relative positions in the melting layer could easily be derived, the shape of attenuation in the bright band could be approximated through a form of interpolation. It should be noted however that the above model will only be valid for attenuation and back-reflectivity. Bistatic reflectivity is not considered here but may be possible to estimate if the forward reflectivity relationship is developed and interpolation between the forward and back reflectivities is done.  University of British Columbia  193  Electrical Engineering  Chapter 11: Recommendations for Future Work  11.1.2 Initial-snow density derivation (Reflectivity vs Scattering cross section) In this work, the average peak reflectivities in the bright band were used to derive the initial-snow density relationship. It is recommended that a full distribution of initial-snow densities, corresponding to the full distribution of reflectivities, be derived. It may also be argued that the use of average reflectivity is not appropriate when obtaining the average initial-snow density and that the use of the average scattering cross section is more appropriate. Since reflectivity is a logarithmic function of scattering, the average reflectivity will, in general, be lower than the average scattering. Therefore the initial-snow density derived using the average reflectivity is higher than those that will be obtained using the average scattering cross section and therefore we may be underestimating the effect of the melting layer. The initial-snow density distribution may also be derived from the snow reflectivities directly above the melting layer. 11.1.3 Water content above the melting layer In this work, when the melting layer is present, all precipitation is assumed to come in the form of snow or ice. It was therefore possible to assume that attenuation above the 0°C isotherm is zero. This may not be the case if supercooled water droplets exist. 11.1.4 Duration statistics of the melting layer Statistics about the presence of the melting layer have been collected by the McGill university. An equally important parameter that has been ignored is the duration and dimension of the melting layer. For example, because of lack of any information about the duration or dimension of the melting layer, there is no clear way to establish when University of British Columbia  194  Electrical Engineering  Chapter 11: Recommendations for Future Work  method TJ in section 9.5.1 ceases to be valid, since this method assumes that, when the melting layer exists, it does so along the full path.  11.2 Interference Modeling  11.2.1 Attenuation and Scattering models More attention needs to be paid to the sensitivity of interference prediction to variations in the attenuation and scattering used (see chapter 6). This is especially true for attenuation at the higher frequency range. In this work, we dealt cursorily with this issue and more work needs to be done to ascertain the validity of these models. 11.2.2 Ice/snow region modeling The ice/snow region was shown to be very important in the prediction of interference levels, especially at higher frequencies and when the common volume is deep inside the ice/snow region. It is recommended that future interference studies include the ice/snow reflectivity statistics.  11.2.3 Radar simulations Radar reflectivity or rain rate maps, when available, should be used to predict interference on microwave systems or to test the prediction models for different rain regions.  11.2.4 A simplified interference model The current interference models (chapter 4), developed here, are computer intensive and are unlikely to be adopted by the ITU-R body. In order for the ITU-R to adopt recommendations to the effect of the melting layer, a more computer-efficient model  University of British Columbia  195  Electrical Engineering  Chapter 11: Recommendations for Future Work  needs to be developed. Another option is to develop simplified procedures to add the effect of the melting layer into rain-only models.  11.3 Fade Modeling  11.3.1 Further improvements on the simplified procedures to add the effect of the melting layer into rain-only attenuation model statistics The simplified procedures developed here (section 9.4), to add the effect of the melting layer into rain-attenuation only models, give excellent agreement with the rigorous models except at the lower percentages of time where more work is needed. 11.3.2 Radar simulations Radar reflectivity and rain rate maps, where available, should be used to predict fading on microwave systems or to test the prediction models for different rain regions.  11.3.3 Experimental verification of the fading model The fading models were developed and melting layer effect was examined. However, these were not compared to actual measurements. It is recommended that, in order to compare the model's predication with actual measurements and to extract  the  contribution of the melting layer from the total fade, the following experiment be conducted: 1. One or more receivers, such as the ACTS terminal at UBC, measuring signal level from a satellite(s) at multiple frequencies.  i  2. One or more high-resolution radar scanning along the receive path.  University of British Columbia  196  Electrical Engineering  Chapter 11: Recommendations for Future Work  3. A mountain running parallel to the main beam axis of the receivers where one can mount meteorological equipment to do measurements such as type of snow, rain rate, at different heights. 4. Sounding balloon stations. 5. Rain, Temperature and Humidity Gages along the propagation paths. While it is true that such an experiment would take a considerable amount of resources, it may be the only way to put the question of the melting layer and its effect on to rest. It would also be less expensive than all these fragmented experiments that raise more questions than provide answers.  11.4 Concluding Remarks Even though propagation work may seem dull and a bit bewildering at first glance, it is indeed very exciting and satisfying. A new student to the field should first investigate what propagation study is and how it relates to system design. Much more work can and should be done in this field, whether it be interference or fading and I wish good luck and happy sailing to those who decide to pursue the field.  University of British Columbia  197  Electrical Engineering  References  References  [1]  COST 205 [1985] Influence of the atmosphere on radio propagation on satellite earth paths at frequencies  above  10 GHz. Commission of the European  Communities. [2]  ITU-R Doc. 5/BL/46-E [1994] Propagation Data and Prediction Methods Required for the Design of Earth-Space Telecommunications Systems.  [3]  Ippolito, L.J. [1989] Propagation Effects Handbook for Satellite Systems Design. N A S A Reference Publication 1082(04).  [4]  ITU-R Revision Rep. 452-4 [1991] Prediction procedure for the evaluation of microwave interference between stations on the surface of the earth at frequencies above 0.7 GHz, Doc. 5/14(Add.l).  [5]  COST Project 210 [1991] Influence of the atmosphere on interference between radio communications systems at frequencies above 1 GHz. Final report of COST 210 management committee, E U R 12407, ISBN 92-826-2400-5, Commission of the European Communities, Luxemburg.  [6]  Capsoni, C , Barbaliscia, F., Martelluci, A., Ordano, L., Paraboni, A., and Tarducci, D.  [1990]  Study of interference  by rain scatter. Final  report:  ESTEC  7625/88/NL/PB (SC). Dipartimento Di Elettronica, Politecnico Di Milano, Milano, Italy. [7]  Capsoni, C , Barbaliscia, F., Martelluci, A., Ordano, L., Paraboni, A., Tarducci, D. and J.P.V. Baptista [1992] Study of interference by rain scatter. ESA Journal, Vol. 16, pp. 171-192.  198  References  [8]  Awaka, J. [1989] A three-dimensional rain cell model for the study of interference due to hydrometeor scattering. Journal ofthe Radio Research Laboratories, Vol.  36, No. 147, 13-44. [9]  Fabry, F., Bellon, A . and Zawadzki, I. [1994] The analysis of melting layer meteorology for the interference application. Final report: C R C 36001-3-3537. J.S. Marshall Radar Observatory, McGill University, Montreal, Canada.  [10] Fabry, F. and Austin, G. L. [1991] Stratiform rain estimates by radar: a vertically pointing radar perspective. 25th international  conference on radar  meteorology,  824-827, Paris, France. [11] Klassen, W. [1987] Radar reflectivity and attenuation of radio waves in the melting layer of precipitation. PB88-121553, Stichting voor de Technische Wetenschappen Utrecht, Netherland. [12] Klassen, W. [1988] Radar observations and simulation of the melting layer of precipitation, J. Atmos. Sci., Vol. 45, No. 24, 3741-3753. [13] Klassen, W. [1990] Attenuation and reflection of radio waves by a melting layer of precipitation. IEE Proceedings, Vol. 137, No. 1, 39-44. [14] Gibson, C , Kochtubajda, B., and Bergwall, F. [1991] Feasibility study for the extraction of melting layer statistics from archived Alberta S-band weather radar observations. Alberta Research Council, Edmonton, Alberta. [15] Ffulays, R.A. [1992] Precipitation scatter interference on communication links with emphasis on the melting-snow layer. M.A.Sc. Thesis, Univ. British Columbia, Vancouver, Canada. [16] Ffulays, R.A. and Kharadly, M.M.Z. [1993] Hydrometeor scatter interference on communication links with emphasis on the melting-snow layer. PIERS'93, JPL, California Institute of Technology, Pasadena, California.  199  References  [17] Kharadly, M . [1993] The analysis of melting layer geometry for interference applications. Final report: C R C 36001-2-3506/01-ST, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. [18] Kharadly, M . M . Z and Hulays, R.A. [1994] Estimation of melting-snow layer attenuation and scattering on microwave communication links. 24th European Microwave Conference, Cannes, France. [19] ITU-R Document  12-3/29 (Rev. 1) [1992] Effect of the melting layer on  hydrometeor interference and coordination distance. Canada submission. [20] Olsen, R.L., Rogers, D.V., Hulays, R.A. and Kharadly, M.M.Z. [1993] Interference due to hydrometeor scatter on satellite communication links. Proc. IEEE, Vol. 81, 914-922. [21] rTU-R Rec. 618-2 [1992] Propagation data and prediction methods required for the design of earth-space telecommunications systems, ITU, Geneva. [22] Van der Star, J. and Kharadly, M . [1981] Measurement of copolar attenuation through the bright band at 4 Ghz and 7 GHz. Annates des Telecommunications, vol. 36, #1-2. 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[1980] A Gaussian rain-cell model for prediction of rain effects on millimetre wave propagation, Report No. 1980-2 from Virginia Polytechnic (Dept. of Electrical Engineering) to NASA.  [34]  Capsoni, C , Fedi, F., Magistroni, A., Paraboni, A., and Pawlina, A. [1987] Data and theory for a new model of the horizontal structure of rain cells for propagation applications. Radio Science, Vol. 22, No. 3, 395-404.  201  References  [35] Capsoni, C , Matricciani, E. and Mauri, M. [1985] Profile statistics of rain in slant path as measured with a radar. Alta Frequenza, Vol. LIV, No. 2, 50-57. [36] Bogush, A. J. [1989] Radar and the atmosphere. Artech House, Norwood, MA, U.S.A [37] CCIR Rep. 563-4 [1990] Radiometeorological data. Annex to Vol. V, ITU, Geneva. [38] Crane, R.K. [1980] Prediction of attenuation by rain, IEEE Trans. Comm., vol. COM-28, pp. 1717-1733. [39] Segal, B. [1979] High-intensity rainfall statistics for Canada. CRC Report No. 1329-E, Communications Research Centre, Ottawa. 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[1984] Microphysical processes of melting snowflakes detected by two-wavelength radar. Part I: Principle of measurement based model calculation. J. Meteo. Soc, Vol. 62, No. 4, 650-667. [60] Matsuo, T., and Sasyo, Y. [1981] Melting of snowflakes below freezing level in the atmosphere. J. Meteor. Soc. Of Japan, Vol. 59, No. 1, 10-24. [61] Pace, J. [1980] Microphysical and thermodynamic characteristics through the melting layer. Dept. Atmospheric Sci., Univ. Wyoming, USA. [62] Best, A. [1950] Empirical formulae for the terminal velocity of drop falling through the atmosphere. Quart. J. R. Met. Soc, vol. 76, pp. 302-311. [63] Medhurst, R. [1965] Rainfall attenuation of centimeter waves: comparison of theory and measurement. IEEE Trans. Ant. Prop., Vol. AP-13, 550-564. [64] Magano, C. and Nakamura, T. [1965] Aerodynamic studies of falling snowflakes. Jour. Meteoroi. Soc. Jap., Vol. 43, No. 3, 139. [65] Laws, J. and Parsons, D. [1943] The relation of raindrop-size to intensity. Trans. Amer. 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Atmospheric Sci., vol. 36, 2093-2107. [73] Marshall, J. and Gunn, K. [1952] Measurement of snow parameters by radar. /. Meteo., vol. 9, 322-327. [74] Battan, L. [1977] Rain resulting from melting ice particles. J. Appl. Meteor., vol. 16,595-604. [75] Leito, M.J., Watson, P.A. and Brussard, G. [1984] Report No. 352, Univer. Bradford. Interm Report for ESA/ESTEC Contract. [76] Kharadly, M.M.Z., and Hulays, R.A. [1992] Melting layer attenuation on earthsatellite links. Proc. APSWIV, 105-114. [77] Hulays, R.A., and Kharadly, M.M.Z. [1993] Modeling of melting-snow particles for  scattering and attenuation calculations. ICAP'93, Heriot-Watt University,  Edinburgh, U.K, 873-876. [78] Awaka, J. [1992] Attenuation and backscatter properties of bright band computed at 13.8 Ghz. Proc. ISAP'92, Sapporo, Japan, 1057-1060.  205  References  [79] Leitao, M. and Watson, P.A. [1986] Method for prediction of attenuation on earthspace links based on radar measurements of the physical structure of rainfall. IEE Proc. F, 11, 429-440. [80] Kozu, T., Awaka, J., and Nakamura, K. [1988] Rain attenuation ratios on 30/20 and 14/12 Ghz satellite to earth paths. Radio Science, Vol. 23, 409-418. [81] Kharadly, M.M.Z., and Hulays, R.A. [1992] Prediction of melting layer attenuation. Proc. 1SAP'92, Sapporo, Japan, 1069-1072. [82] Aden, A. and Kerker, M. [1951] Scattering of electromagnetic waves from two concentric spheres. J. Appl. Phys., vol. 22, 1242-1246. [83] Maxwell-Garnet, J. [1904] Colors in metal glasses and in metallic films. Phil. Trans. Roy. Soc, 385-420. [84] Bohren, C. and Battan, L. [1982] Radar scattering by spongy ice spheres. J. Atmospheric Sci., vol. 39, 2623-2629. [85] Kharadly, M. [1989] Scattering by rain and melting-snow.  ICAP'89, Univ.  Warwick, U.K., 406-410. [86] Olsen, R., Rogers, D.V., and Hodge, D.B. [1978] The aR relation in the calculation b  of rain attenuation. IEEE Trans. Ant. Prop., Vol. AP-26, No. 2, 318-329. [87] Brent, R. P. [1978] A fortran multiple-precision arithmetic package. ACM transactions on Mathematical Software, Vol. 4, No. 1, 57-70. [88] Brent, R. P. [1978] Algorithm 524: MP, a fortran multiple-precision arithmetic package. ACM transactions on Mathematical Software, Vol. 4, No. 1,71-81. [89] Kharadly, M. [1992] Precipitation scatter interference on communication links with emphasis on the melting snow layer. Final report: CRC 36001-0-3569/01-SS, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada.  206  References  [90] CCIR Rep. 721 [1990] Attenuation by hydrometeors, in particular precipitation, and other atmospheric particles. Vol. V, ITU, Geneva. [91] CCIR Rep. 569-3 [1986] The evaluation of propagation factors in interference problems between stations on the surface of the earth at frequencies above about 0.5 GHz. Document 5/1051-E, ITU, Geneva. [92] CCIR Doc. 5C/GREY/14 [1993] Proposed revisions to recommendation 620-1: results of tests on the effect of hydrometeors above 0°C isotherm on scatter interference. Canada submission, ITU, Geneva. [93] Morgan, W.L. and Gordon, G. D. [1989] Communications satellite handbook. John Wiley & Sons, Toronto, Canada. [94] Gagliardi, R. M . [1991] Satellite Communications, 2  nd  edition. Van Nostrand  Reinhold, New York, N Y , USA. [95] Proakis, John G. [1995] Digital Communications, 3  rd  edition. McGraw-Hill Inc.,  Toronto, Canada. [96] Goddard, J.W.F. and D'Amico, M . [1993] Melting layer studies at S Band and comparisons with physical model. Eighth International Conference on Antenna and Propagation, Edinburgh, UK, 881-884. [97]  COST 205 [1985] Influence of the Atmosphere On Radiopropagation On Satellite Earth Paths At Frequencies Above 10 GHz. Cost 205 Project.  [98] Russchenberg, H . W. J. and Ligthart, L.P. [1996] Backscattering by and Propagation Through the Melting Layer of Precipitation: A New Polarimetric Model. IEEE Transaction on Geoscience and Remote Sensing, Vol. 34, No. 1, 3-14. [99] D'Amico, M . [1991] Melting Layer Modeling And Scattering by Raindrops. Internal ESTEC Working paper N. 1618. [100] Zawadzki, I [1996] private correspondence.  207  References  [101] Russchenberg,  H . W. J. and Ligthart, L.P. [1992] Backscattering By And  Propagation Through the Melting Layer of Precipitation. ESTEC Final Report PO 122859. [102] Zhang, W., Karhu, S.I and Salonen, E T . [1994] Precipitation of Radiowave Attenuation Due to a Melting Layer of Precipitation. IEEE Transactions on Antenna and Propagation , Vol. 42, No. 4, 492-500. [103] Zhang, W. and Salonen, E T . [1996] Bistatic Scattering of Radiowaves by a Melting Layer of Precipitation. IEEE Transactions on Antenna and Propagation , Vol. 44, No. 8, 1057-1062. [104] Zhang, W. [1994] Scattering of Radiowaves by a Melting Layer of Precipitation in Backward  and  Forward  Directions. IEEE Transactions on Antenna and  Propagation , Vol. 42, No. 3, 347-356. [105] Matsumoto, A . and Nishitsuji, E. [1971] SHF and E H F Propagation in Snowy Districts. Monograph Ser. of the Res. Inst. Of Appl. Electricity, Hokkaido University, Sapporo, Japan. [106]Cheah, -J.Y.C. [1993] Wet Antenna Effect on V S A T Rain Margin. IEEE Transactions on Communications, Vol. 41, No. 8, 1238-1244.  208  Appendix A : Interference Models  Appendix A: Interference Models  In this appendix, the programs and files necessary to run the interference models are presented. These are: 1. drive_a.f: This is the driver program that generates the input parameters to the interference model ddd_mod.f using the Awaka meteorological model. 2. drive_c.f: This is the driver program that generates the input parameters to the interference model ddd_mod.f using the Capsoni meteorological model. 3. drive_i.f: This is the driver program that generates the input parameters to the interference model itu_mod.f using the ITU-R meteorological model. 4. ddd_mod.f: This is the program that calculates the interference level for a single case l  for the 3D (exponential rain-cell) model. 5. itu_mod.f': This is the program that calculates the interference level for a single case for the ITU-R model. 6. paramet: This is the file containing the general parameters of the link. 7. ml_pr: This is the file containing the percentages of presence of the melting layer as a function of rain rate. 8. hfr_dh: This is the file containing the variations of the rain height around its median. 1 Some of the subroutines in the program were written by Tim Vlaar, specifically the routine to compute the intersection between a cylinder and a line.  University of British Columbia  209  Electrical Engineering  Appendix A: Interference Models  9. hydro: This is the file containing the particle-size distribution of rain. 10. rain_pr: This is the file containing the cumulative distribution function of rain, cdf.m is the output file containing the cumulative distribution functions of transmission loss. A l l the above files are explained below. The input files and the programs are also provided in electronic format on the diskettes appended to this report. For each model, a subdirectory is created where the programs reside along with their required input files.  A.1 Software and Hardware Requirements The simulation programs are guaranteed to work under the environments described below. Even though the programs should run under similar environments, the author cannot guarantee this and users should abide by the system requirements set below.  A . l . l Software requirements The program is written in Fortran-77, compiled using the SUN Fortran-77 compiler and run in a UNIX environment.  A.1.2 Hardware requirements  A.l.2.1 ITU-R program The program was used on SPARC IPX, SPARC 2 and SCPARC 5 computers with a minimum of 16 megabytes of R A M . A.1.2.2 Awaka program The program was used on SPARC 2 and SCPARC 5 computers with a minimum of 32 megabytes of R A M .  University of British Columbia  210  Electrical Engineering  Appendix A: Interference Models  A.1.2.3 Capsoni program The  program was used on SPARC 5 computer with a minimum of 64 megabytes of  RAM.  A.2 Running the Interference Programs  A.2.1 Program inputs In order for the program to run, it is necessary to provide the correct input format. The input files for all three models are the same and are described below. In the sections below, the text of the input files are in italic.  A.2.1.1 paramet 20.0 3.0 3.0 131.0 0.12 0.086 1.8 70.0 0.0 0.00314 0.0 59.0 89.0 180.0 0.0279 0.0 40.5 -17.0 1.6 5.8788 2 6 1 1 3 5 10.0  ,frequency (GHz) , m e d i a n r a i n h e i g h t (km) , c o m m o n v o l u m e h e i g h t (km) , distance b e t w e e n trans, a n d r e c e i v . (km) , h e i g h t o f t r a n s m i t t e r (km) , h e i g h t o f r e c e i v e r (km) , i n i t i a l snow density ( v a l i d b e t w e e n 0.1-0.3, 1.8: v a r i a b l e ) , t h e t a o f t r a n s m i t t e r (1 - e l e v , d e g r e e ) , p h i o f the t r a n s m i t t e r ( d e g r e e ) , h a l f b a n d w i d t h of t r a n s m i t t e r (rad) , a l p h a f o r t r a n s m i t t e r (0: H o r i z o n t a l , 1: V e r t i c a l ) , gain of transmitter , t h e t a o f r e c e i v e r (1 - e l e v , d e g r e e ) , p h i o f the r e c e i v e r ( d e g r e e ) , h a l f b a n d w i d t h o f r e c e i v e r (rad) , a l p h a f o r the r e c e i v e r (dummy variable) , g a i n o f the r e c e i v e r , K(dB) , H a l f - b e a m w i d t h f o r r e c e i v e r m a i n lobe ( d e g . ) , H a l f - b e a m w i d t h f o r r e c e i v e r secondary lobe ( d e g . ) , m o d e l select (use 2 f o r a w a k a m o d e l ) , a t t e n u a t i o n select — select 6 , s c a t t e r i n g select — select 1 , l : - 6 . 5 ; 2 : - 4 . 0 (reflectivity roll-off) , l:S=h/H;2:S=(h/H)*2;S=(n/H)*3 (melting profile) , M L thickness - select , steps o f i n t e g r a t i o n  University of British Columbia  5 (meters)  211  Electrical Engineering  Appendix A: Interference Models  A.2.1.2 ml_pr % percentage of time the melting layer is present versus % rain rate % % source: McGill University" % % Rain rate (mm/h.) % of time % 10 , # of data points 0.5 88.5 , 1.25 87.0 '. 2.5 80.8 5.0 72.0 10.0 ' 5 5 . 7 15.0 44.0 20.0 32.0 30.0 15.5 37.0 1.5 40.0 0.0  '  ,  A.2.1.3 hfr_dh % Cumulative distribution ofthe rain height relative to % its median value % % source: COST210 report 3.28 %  % Rain height difference(km) Exceedence probability % 17 ', # of data points -1.625 100.0 -1.375 99.1 -1.125 96.9 -0.875 91.0 -0.625 80.0 -0.375 68.5 -0.125 56.5 0.125 44.2 0.375 33.5 0.625 24.0 0.875 16.3 1.125 10.2 1.375 6.1 1.625 3.4 1.875 1-8 2.125 0.9 2.375 0.0  University of British Columbia  212  Electrical Engineering  Appendix A : Interference Models  A.2.1.4 hydro The table in the file below is the Law and Parson particle-size distribution. The cells (2,1)-(15-1) represent the sizes of the rain particles and (2,2)-(15,2) are their corresponding velocities. The cells (1,3)-(1,11) are the rain rates and cells (2,3)-(15,11) the percentages of rainfall that arrive from particles of a certain size for a given rain rate. The user may change the type of distribution at will to fit his/her needs.  0.0000 0.0000 0.25DO 1.25DO 2.50D0 5.00D0 12.5DO 25.0D0 50.0D0 100.DO 150.DO 0.05D0 2.06D0 28.0D0 10.9D0 7.30D0 4.70D0 2.60D0 1.70D0 1.20D0 1.00D0 1.00D0 0.10D0 4.03DO 50.1 DO 37.1 DO 27.8D0 20.3DO 11.5DO 7.60D0 5.40D0 4.60D0 4.10D0 0.15D0 5.40D0 18.2D0 31.3D0 32.8D0 31.0D0 24.5D0 18.4D0 12.5D0 8.80D0 7.60D0 0.20D0 6.49D0 03.0D0 13.5D0 19.0D0 22.2D0 25.4D0 23.9D0 19.9D0 13.9D0 11.7D0 0.25D0 7.41D0 00.7D0 04.9D0 7.90D0 11.8D0 17.3D0 19.9D0 20.9D0 17.1 DO 13..9D0 0.30D0 8.06D0 OO.ODO 01.5DO 3.30D0 5.70D0 10.1 DO 12.8D0 15.6D0 18.4D0 17.7DO 0.35D0 8.53D0 OO.ODO 00.6D0 1.10D0 2.50D0 4.30D0 8.20D0 10.9D0 15.0D0 16.1 DO 0.40D0 8.83D0 OO.ODO 00.2D0 0.60D0 1.00D0 2.30D0 3.50D0 6.70D0 9.00D0 11.9D0 0.45D0 9.00D0 OO.ODO OO.ODO 0.20D0 0.50D0 1.20D0 2.10D0 3.30D0 5.80D0 7.70D0 0.50D0 9.09D0 OO.ODO OO.ODO OO.ODO 0.30D0 0.60D0 1.10D0 1.80D0 3.00D0 3.60D0 0.55D0 9.13D0 OO.ODO OO.ODO OO.ODO OO.ODO 0.20D0 0.50D0 1.10D0 1.70D0 2.20D0 0.60D0 9.I4D0 OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO 0.30D0 0.50D0 1.00D0 1.20D0 0.65D0 9.14D0 OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO 00.0D0.0.20D0 0.70D0 1.00D0 0.70D0 9.14D0 OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO OO.ODO 0.30D0  A.2.1.5 rain_pr % Concurrent rainfall statistics % (Chilbolton-Baldock) - 2 years %  % source: COST210 report 5.44 %  -  % Rain rate (mm/h.)  '  % of time  %  7  , # of data points 1.9 4.3 8.3 15.0 26.3 42.0 62.0  University of British Columbia  1.0 0.3 0.1 0.03 0.01 0.003. 0.001  213  Electrical Engineering  Appendix A: Interference Models  A.2.2  Program outputs  A.2.2.1 interm Ignore this file.  A.2.2.2 CDF.m The CDF.m file is in matlab format as follow:  RSUM=[  % Original  model exceedence percentages (Rain only)  1: L_R=[  % Original model interference levels (Rain only)  1: MSUM=[  ' % Modified  model exceedence percentages (ML, 100 %)  /; L_M=[  '. % Modified model interference levels (ML, 100 %)  I:  RMSUM=[  % Modified model exceedence percentages (ML, x % - ignore for Awaka and Capsoni  models)  University of British Columbia  214  Electrical Engineering  Appendix A : Interference Models  1; L_RM=[  % Modified model interference levels (ML, x % - ignore for Awaka and Capsoni models)  The results for method two are obtained by a melting-layer presence probability-weighed interpolation between L_R and L_M. A.2.3 Compiling and running the models When running the simulation models, please follow the steps outlined below: 1. Compile and link the driver programs with the model-programs: f77 -o drive_a.exe drive_a.f ddd_mod.f f77 -o drive_c.exe drive_c.f ddd_mod.f f77 -o drive_i.exe drive_i.f itu_mod.f 2. Make sure that the following files are in the same directory where you intend to run the program: paramet ml_pr hfr_dh hydro rain_pr 3. Run the program in low priority: priority invoke.exe >&! out &  University of British Columbia  215  Electrical Engineering  Appendix A : Interference Models  4. Remove "temp4", "interm", "out" and "GINP" files. 5. The relevant output is in "CDF.m". Use matlab to plot data.  A.3 Program Listings  A.3.1 3D (Awaka, Capsoni) model (ddd_mod.f) This program is available in the appended disks in the awaka and capsoni subdirectories, along with the other programs and input files necessary to run the simulation model.. A.3.2 ITU-R model (itu_mod.f) This program is available in the appended disks in the itu_r subdirectory, along with the other programs and input files necessary to run the simulation model.  A.3.3 Awaka driver program (drive_a.f) This program is available in the appended disks in the awaka subdirectory, along with the other programs and input files necessary to run the simulation model.  A.3.4 Capsoni driver program (drive_c.f) This program is available in the appended disks in the capsoni subdirectory, along with the other programs and input files necessary to run the simulation model.  A.3.5 ITU-R driver program (drive_i.f) This program is available in the appended disks in the itu_r subdirectory, along with the other programs and input files necessary to run the simulation model. .  University of British Columbia  216  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  Appendix B: Scattering and Attenuation Models and Programs  M i e scattering method for spheres and concentric sphere  B.l  Scattering theory  B.l.l  A melting particle is either represented by a sphere with average permittivity e„, or a two concentric spheres particle, with a layer of water of permittivity e surrounding a dry w  snow core of permittivity e . The scattering of a plane electromagnetic wave due to two s  concentric spheres, with different refractive indices, has been treated by Aden and Kerker [27]. The total cross section Q , the total scattering cross section Q , and the backscattering t  s  cross section Gb of a sphere of radius a with a concentric spherical shell of thickness a,„s  a  s  (figure 6-1) and whose refractive indices are N (N,=y[e^) and  N (N - f£^),  t  2  i y  respectively, are given by: (B.l) K  • n=l  a^I(2^/)! K| M j v  •• . (B.2)  +  University of British Columbia  217  Electrical Engineering  Appendix B: Scattering and Attenuation Programs  K  ±(-!V(2n  (B.3)  l)(a:-b:  +  where the propagation constant for free space k = 2n/X, with X as the wavelength. The values of a and b[ are calculated by applying the boundary conditions to the wave n  equations in the three distinct media (inner sphere, outer sphere and free space). Then, i { 2) 3 n {a )A a  A  +  1  n  n  h[ \a )  2  4  (B.4)  n {a )A  2  h  2  +  n  2  4  Vn{ ) l-Jn{ 2) 2 a  A  a  A  (B.5)  2  ^ {a )A h' {a )A 2>  n  2  l+  n  2  2  where Ay, A , A3 and A are given by: 2  4  A =.N ^{N a ){j^N a )hl \N a )-j {N aX {N a )} 2  l  2  2  l  l  2  >  2  2  l  n  2  2  2  (B.6a)  +N^2Jn{N a,){^{N aX \^2^)-^{N a )j {N a )} 2  l  2  2  A =  !  n  2  2  N^iiN^^n^N^j^N.a^-n^N^yj'iN.a^}  2  .  (B.6b)  +^ Jn{^ ^)lni{N a )v {N a )-r ^N a,)r {N a )} h  I  1  2  2  h  n  2  l  1  2  1 n  2  2  (B.6c)  (B.6d)  + ^^(^/)U(^2K ^2«/)-;n(^/K (^ )} 2  2;  2  where a/ and a are the size parameters of the inner and outer spheres and are given by 2  ka and ka , respectively. Here,;„ is the spherical Bessel function of the first kind and h[  2)  s  m  is the spherical Hankel function of the second kind and is also known as the spherical Bessel function of the fourth kind. The values of n and r\ are given by: J  h  n  1 d{Zj (Z)}  (B.7a)  n  Z  n  dZ  University of British Columbia  218  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  d{zh[ \z)} 2  (B.7b).  dZ  In order to obtain the solution for the single sphere representation, we can either take the inner radius (a ) to zero or equate it to the outer radius (a ) . Equating the inner radius to s  m  zero, A/, A 2, A3, and A4 reduce to: A = N Jn{N,a )  (B.8a)  ^=-rii[N,a )  (B.8b)  l  l  2  2  ^i=N,ni{N,a )  .  2  (B.8C)  A =-h[N,a ) 4  (B.8d)  2  The attenuation, in dB, for one particle of size /, is then given by:  •  a =-4.343x10 Q s  i  ti  (B.9)  and that of a whole distribution of m particle-sizes of number density N-, is: (B.10)  « = £ a , =-4.343xl0 X62,,/V, 5  1=1  The scattering for the whole distribution of particles is obtained in a similar fashion:  (B.l 1)  1=1  m 1=1  The equivalent reflectivity is obtained from the scattering values using: m +2 x!0 5 m -\ 2  Z =lOlog 10 e  scat K  (B.12)  1:  l  University of British Columbia  219  Electrical Engineering  Appendix B : Scattering and Attenuation  Programs  B.1.2 Simulation S e r i o u s problems  were encountered in p r o g r a m m i n g  equations B . 1 - B . 6 using single,  double or extended double p r e c i s i o n floating point arithmetic. T h i s is e s p e c i a l l y true when operating  close to the resonance frequency  o f the particle. S i m p l y ,  single, double or extended double precision floating point arithmetic  traditional  cannot  provide  e n o u g h accuracy to m o d e l the above equations in a straight forward manner. F o r this purpose, a m u l t i p l e - p r e c i s i o n floating-point [88])  is used.  A r-digit floating-point  arithmetic p r o g r a m " m p . f  ([87]  number is represented as an integer array of  d i m e n s i o n at least t+2. T h e first w o r d is used to identify the sign, the second w o r d for the exponent and the rest for the n o r m a l i z e d fraction. T h e user can specify the size o f the integer array, where the m u l t i p l e p r e c i s i o n n u m b e r is stored, and the size o f the w o r k i n g f i e l d , where the computations  takes place. T h e larger the size of the fields, the more  p r e c i s i o n is available. H o w e v e r , the computation time increases dramatically w i t h higher accuracy. T h e p r o g r a m " m p . f consists o f subroutines w h i c h include s i m p l e arithmetic operations (addition, subtraction, m u l t i p l i c a t i o n , d i v i s i o n ) , elementary functions (log, exp., s i n , tan, arcsin, etc ...) and c o n v e r s i o n tools between integer, real, and d o u b l e - p r e c i s i o n numbers to m u l t i p l e p r e c i s i o n format and vice-versa. S i n c e the real, and d o u b l e - p r e c i s i o n numbers conversion  to m u l t i p l e p r e c i s i o n  format  does not  work  satisfactorily,  only  integer  c o n v e r s i o n s to m u l t i p l e - p r e c i s i o n format is used. T h i s is done by m u l t i p l y i n g the real n u m b e r by 10" to get an integer value, convert it to m u l t i p l e - p r e c i s i o n format and then d i v i d e by 10" u s i n g " m p . f d i v i s i o n . N o problems are encountered w h e n c o n v e r t i n g f r o m m u l t i p l e - p r e c i s i o n format to integer, real, and double p r e c i s i o n numbers.  University of British Columbia  220  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  The program "mp.f deals with real numbers and complex representations are not available. For this purpose, we set Z = (x,y), where Z is the complex number of real part x and imaginary part y. Subroutines are developed to perform complex numbers arithmetic operations (addition, subtraction, multiplication, division), elementary functions (sin, cos, tan, cot, abs, arctan, sqrt). Also the spherical Bessel function of the first kind and the spherical Hankel function of the second kind with their derivatives, as in equation B.7, are implemented. The spherical Bessel and Hankel functions are implemented using the following expansion: y (Z)=^;„_,(Z)-;„_ (Z)  (B.l 3a)  h?\Z)=^h?_\(Z)-h?MZ)  (B.13b)  f l  2  Similarly the derivatives were expressed as:  n (Z) = ~j (Z) J  n  n  + j _ {Z) n  (B.14a)  l  B.1.3 Software In this section, the programs necessary to run the Mie scattering and attenuation programs are presented. These are: 1. mie_co.f: This is the Mie scattering program that generates the attenuation and scattering properties for the melting-snow particles, assuming a concentric-sphere representation (Model A). 2. mie_cp.f: This is the Mie scattering program that generates the attenuation and, scattering properties for the melting-snow particles for models B1 and B2.  University of British Columbia  221  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  3. mie_sp.f: This is the Mie scattering program that generates the attenuation and scattering properties for the melting-snow particles, assuming a spongy-sphere representation (Models C l and C2).  *  .  4. mp.f: This is the multiple precision program used by the above models to perform multiple precision computations. 5. comp3.f: This is the program developed to allow multiple precision calculations using complex numbers. 6. hydro: This is the file containing the particle-size distribution of rain, the initial-snow density and the different melting rates. 7. frequ: This is the file containing the frequency information. "atten" and "scatter" are the output files of the above three Mie scattering programs. These files contain the attenuation and scattering values for the individual particles. These files are then renamed and used as inputs to generate the precipitation attenuation and scattering properties. The files necessary to run this program are: 1. atten.f: This is the program that uses the "atten" and "scatter files generated by the Mie scattering programs to generate the precipitation's specific scattering and attenuation. The program can also generate total and excess attenuation and reflectivity in the melting layer among other tasks. These tasks will not be covered here, however. 2. atten: This is the file that contains the attenuation values for the individual particles comprising the precipitation medium. 3. scatter: This is the file that contains the scattering values for the individual particles comprising the precipitation medium.  University of British Columbia  222  Electrical Engineering  Appendix B: Scattering and Attenuation Programs  The inputs and outputs of the programs need to be discussed later in the appendix. The input files and the programs are also provided in electronic format on the diskettes appended to this report.  B.l.3.1 Software and Hardware Requirements The Mie scattering programs are guaranteed to work under the environments described below. Even though the program should run under similar environments, the author cannot guarantee this and users should abide by the system requirements set below.  B. 1.3.1.1 Software requirements The programs is written in Fortran-77, compiled using the SUN Fortran-77 compiler and run in a UNIX environment.  B.1.3.1.2 Hardware requirements The Mie scattering programs were used on SPARC IPX, SPARC 2 and S C P A R C 5 computers with a minimum of 16 megabytes of R A M . The "atten.f' could run on a lower end machine.  B.l.3.2 Running the Mie Scattering Programs  B.1.3.2.1 Program inputs In order for the, program to run, it is necessary to provide the correct input format. The input files for all three models are the same and are described below. In the sections below, the text of the input files are in italic. B.1.3.2.1.1 frequ  University of British Columbia  223  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  / , number offrequencies (Recommended you stick to 1) 9.344 , frequency in GHz 20 , number of iteration in equations 4.1-4.3 B.1.3.2.1.2  hydro  The table in the file below is the Law and Parson particle-size distribution. The cells (2,1)-(15-1) represent the sizes of the rain particles and (2,2)-( 15,2) are their corresponding velocities. The cells (1,3)-(1,11) are the rain rates and cells (2,3)-( 15,11) the percentages of rainfall that arrive from particles of a certain size for a given rain rate. The user may change the type of distribution at will to fit his/her needs.  0.0000,0.0000,0.25D0J. 25D0.2.50D0,5.00 DO, 12.5D0,25~.0D0,50.0D0,100.D0,150.D0 0:05DO,2.06DO,28.0DO,10.9DO,7.30DO,4.70DO,2.60DO,1.70DO,1.20DO,1.00DO,1.00DO 0.10D0,4.03D0,50.1D0,37.1D0,27.8D0,20.3D0,11.5D0;7.60 DO, 5.40D0,4.60D0,4.10D0 0.15D0,5.40D0,18.2D0,3J.3D0,32.8D0,31.0D0,24.5D0,18.4D0;i2.5D0,8.80D0,7.60D0 0.20D0,6.49D0,03.0D0,13.5D0,J9.0D0,22.2D0,25.4D0,23.9D0,19.9D0,13.9D0,11.7DO 0.2 5 DO, 7.41 DO, 00.7 DO, 04.9D0,7.90D0,11. 8D0,17.3 DO, 19.9D0.20.9D0,17.1 DO, 13.9D0 0.30D0,8.06D0,00.0D0,01.5D0.3.30D0,5.70D0,10.1 DO, 12.8D0,15.6 DO, 18.4 DO, 17.7 DO 0.35D0,8.53D0,00.0D0,00.6D0,1.10D0,2.50D0,4.30D0,8.20D0,10.9D0.15.0D0,16./DO 0.40D0,8.83 DO, 00.0D0,00.2 DO, 0.60D0.1.00D0,2.30D0.3.50D0,6.70 DO, 9.00 DO, 11.9 DO 0.45 DO, 9.00D0,00.0D0,00.0D0,0.20D0.0.50D0,1.20D0.2.10D0,3.30 DO, 5.80 DO, 7.70D0 0.50D0,9.09D0,00.0D0,00.0D0,00.0D0,0.30D0,0.60D0,1.10D0J. 80D0,3.00D0,3.60D0 0.5 5 DO, 9.13 DO, 00.0D0,00.0D0,00.0D0,00.0D0,0.20D0,0.50D0,1.10D0,1.70 DO, 2.20 DO 0.60D0,9.14D0,00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,0.30D0,0.50 DO, 1.00D0,1.20D0 0.65 DO, 9.14 DO, 00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,0.2 0D0,0.70 DO, 1.00D0 0. 70 DO, 9.14 DO, 00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,00.0D0,0.30 DO 1.5D0 , dummy variable 1. DO , dummy variable 0.9D0 , 1 -initial snow density 60 , number of melting degrees S for which calculations should be done 0.0 , this and all the following lines are the different Ss used in the calculations 0.02 0.05 0.06 0.1 0.2 0.3  0.4 0.5 0.6 0.7 0.8 0.9 1.0  University of British Columbia  224  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  B.1.3.2.2 Program output Two output files are generated by the Mie scattering, programs. These are "atten" and • "scatter" files; both have the same structure. The rows represent the scattering or attenuation values for the different melting degrees and the columns for the different resultant rain sizes.  B. 1.3.2.3 Compiling and running the models When running the Mie scattering programs, please follow the steps outlined below: 1. Compile and link the Mie scattering programs, comp.f and mp.f: 177 -o mie_co.exe mie_co.f comp.f mp.f f77 -o mie_cp.exe mie_cp.f comp.f mp.f f77 -o mie_sp.exe mie_sp.f comp.f mp.f 2. Make sure that the "hydro" and "frequ" files are in the same directory where you intend to run the program. 3. Make sure that you deleted "atten" and "scatter" files. 4. Run the program in low priority. 5. The relevant outputs are in "atten" and "scatter" files. B.l.3.3 Generating the Precipitation Attenuation and Scattering Characteristics The reason for using a distinct program to sum up the contributions of the different particles that make up the precipitation medium is flexibility. The Mie scattering programs are computer intensive (i.e., they simply take a long time to run) and it may be desirable to re-use the attenuation and scattering properties of particles rather than regenerating them all over. This program can perform several functions. These are:  University of British Columbia  225  Electrical Engineering  Appendix B : Scattering and Attenuation  l.  Programs  Calculates specific scattering and attenuation in the melting layer.  '2. Calculates the total, excess attenuation and reflectivity in the melting layer; 3. Calculates the ^^relationship for rain and snow attenuation and scattering. 4. Calculates attenuation-corrected peak reflectivities. This was used to derived the initial-snow density relationship. 5. Calculates attenuation-corrected peak reflectivities in the melting layer and their relative positions. In this work, we will only consider task.l. B.1.3.3.1 Inputs In order to run this program, the "atten" file should be appropriately renamed. The appropriate names are given in the "FILE_IN" subroutine in the atten.f program. When the program invokes the user to which task is desired, enter "1". In choosing the models, "1" is for the concentric sphere model while "2" is for the spongy-sphere model.  B. 1.3.3.2 Outputs The output of the program is quite simple. Three columns are generated. The first column is the degree of melting. The second is the attenuation and the third is the scattering cross section.  B.1.3.3.3 Compiling and running the program When running the simulation program, please follow the steps outlined below:  University of British Columbia  226  Electrical Engineering  Appendix B : Scattering and Attenuation  Programs  1. Make sure that the input attenuation and scattering files are named appropriately. If they are not, either rename them or edit the "FILE_IN" subroutine in the "atten. f" program. 2. Compile and run the program B.1.3.4 Program Listings  B.1.3.4.1 mie_co.f This program is available in the appended disks in the Mie subdirectory, along with the other programs and input files necessary to run the simulation model.  B.1.3.4.2 mie_cp.f This program is available in the appended disks in the Mie subdirectory, along with the other programs and input files necessary to run the simulation model.  B.1.3.4.3 mie_sp.f This program is available in the appended disks in the Mie subdirectory, along with the other programs and input files necessary to run the simulation model.  B.1.3.4.4 comp.f This program is available in the appended disks in the Mie subdirectory.  B.1.3.4.5 mp.f This program is available in the appended disks in the Mie subdirectory.  B.1.3.4.6 atten.f This program is available in the appended disks in the Mie subdirectory. University of British Columbia  227  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  B.1.3.5 Input file Listings Some of the attenuation and scattering input files for the atten.f program are given in mie/conc and mie/spongy directories for the concentric and spongy-sphere representation, respectively.  B.2 Artificial dielectric model for attenuation [26] Since Mie computations.of attenuation and scattering are computer intensive, simplified models are used to simulate scattering and attenuation in rain and melting-snow. The artificial dielectric model treats rain as an artificial dielectric medium composed of onesize particles of equivalent radius a and number density TV. The attenuation a (dB/km), at a frequency/ (GHz), is then given by: a =9.1xl0 g'Nf  (B.15)  •  4  where the effective value of polarizability at high frequencies g -g' - jg" is given by: e  TT \r  e  e  (B.16) J  J  where g is the low-frequency value of polarizability, f is the characteristic frequency of a r  rain drop ( f = 4 c / 2 n a , where c is the speed of light in air and £ = 0.866)"and n varies r  between 1 and 2 according to: n  j  +  i o o ( f  /  f  r  f  (  B  1  7  )  i+ioi{f/f )  2  r  The low frequency polarizability for a spherical water droplet (For particles which are not spherical in shape, refer to [26]) is given by:  University of British Columbia  228  Electrical Engineering  Appendix B: Scattering and Attenuation Programs  g = 4mf  "}  (B.18)  \ - o) £  2£  w  where e and £b are the permittivities of water and air, respectively. w  The representative rain particle radius (a=a  for S = 1, rain) is calculated from equation  R  B.29. The velocity (VR) can be obtained from either table 5-5 or equation B.17. The number of particles per unit volume (N = N ) is obtained using equation 5.30. For a more R  thorough treatment, refer to [26]. B.2.1 Artificial dielectric model for melting layer attenuation ( A D M 1) The above model was applied to the melting-snow layer [26]. The model assumed the melting particles to be in the form of two concentric spheres with a snow core of radius a  s  and permittivity e surrounded by a water shell of thickness (a - aj . The low frequency m  s  polarizability of the particle becomes:  (£ -£ )(2£ +£ )w  0  w  s  8 = 4<  ^  (£ -£ ){2£ +£ ) w  s  w  0  ^ T y  (B-19)  (£ +2£ )(2£ +£ )-2\^-) w  0  w  ( -£ )(£„-£ )  s  £w  s  0  \ mJ a  The representative melting-snow particle radius (a  m  ) and the corresponding number  density for a certain melting rate S (between 0, for dry snow, and 1, for rain), associated with a certain position in the melting layer, are calculated using equations 5.29 and 5.30, respectively. In [26], the representative melting particle, a , for a certain melting rate S is obtained by m  applying equation 5.1 on the representative particle derived for rain, a , using equation R  University of British Columbia  229  Electrical Engineering  Appendix B: Scattering and Attenuation Programs  5.29. The melting-snow particle velocity and number- are derived using equations 5:25 and 5.30, respectively. Because of the deviation of the results from Mie-derived values for the melting-snow layer, a multiplication correction factor is introduced [81] that brought the results of the model closely to the attenuation results calculated using Mie scattering: f  {2 + S) — +l fr  CF1 =  — + 2-S  .  l-S  2-S  (B.20)  2+S  fr  where/is the frequency in GHz. B.2.2 Artificial dielectric model for melting layer attenuation ( A D M 2) The above approach to calculating the representative melting particle was considered unsatisfactory and a new procedure was formulated [81]. The representative particle sizes in both the melting layer for a certain melting ratio S and the rain medium ( 5 = 1 ) are found. Using Equation 5.25, the velocity of the melting particle is obtained and subsequently the number density. Because of the deviation of the results from that of the exact values for the melting-snow layer, a multiplication correction factor is introduced [81] that brought the results of the model closely to those calculated using Mie scattering:  n a (\-S) 2  CF2 = {n + S(\-n)} 1 +  n  28  (1-55)/ 1+ 5/  (B.21)  r  where a is the radius of the representative particle, <5is the skin depth of water given by: m  8 = 100/  (B.22)  Re{20.958f^£^)  University of British Columbia  230  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  For rain (S = /), the correction factors (CF1 and CF2) tends to 1. It should be noted that the velocity of the representative melting particle is obtained using equation 5.25. which is a function of the velocity of the representative particle in rain which is not necessarily equivalent to the representative melting-snow particle (i.e., the representative melting-snow particle can generate a different rain drop size if allowed to melt). However, since the attenuation models (with the correction factors) give comparable results to those obtained by Mie scattering techniques, their use can readily be justified.  B.3  Artificial dielectric model for scattering [77]  The model treat rain as an artificial dielectric medium composed of one-size particles of equivalent radius a  .The equivalent, or representative, radius is the same one in the  m  attenuation model in section B.3.2. Under the effect of an external electric field, the drop may be represented by an equivalent dipole of moment. Due to the complex nature of scattering, several correction factors were needed to bring the artificial dielectric scattering results closer to those obtained using Mie scattering. The scattering due to a rain medium of a rain rate R, equivalent radius a, and a frequency /, is given by: oie ,<j>) = oiaU F{M")x  F{n)x  2  -2  2n  oia) = 4{k a) 0  (B.23)  N  o  (B.24)  F{e,<p)  1"  F{e,<p)= s m d  f/fr  + 1+  £ {sin 2  9 cos<p+0.5cos  4  d)  (B.25)  f/fr  University of British Columbia  231  Electrical Engineering  Appendix B: Scattering and Attenuation  F{M')  = 1 +  AT 2.6  F(n)  2.6  \  l +R/300  r  2n{2.5n-l)  (B.26)  sin 6 cos <p  600  V  •f/f  U + n)  =  R  ]+•  U +  Programs  (///,.). •+——-t—ril-R/150)  (B.27)  n-f/f j r  where, k =2n / X, where Ac = free space wavelength of the incident wave 0  2+  200[f/f )  l+  20l(f/f f  3  r  r  f = c/Ar, where c is the velocity of light in free space r  \=2TW/E,  % = 0.866(l  where / i s in GHz  + I.5xlO~ /), 4  £o = permittivity of free space £ = permittivity of water w  M = M'-iM"\%  the refractive index of water (=£ ) l/2  9, <p = polar and azimuth angles, as in figure 6-8. N = number of melting-snow particles per unit volume The model is extended into the melting-snow layer with the introduction of a multiplication factor F(S) given by [50]: is  \(i-s)  4  FiS)-.  150+R  1  +  J  {f/f ) r  f  R + 100 100  '  25S  150  1-0.5—  V-  \100j 1+  s i n <p  R  5  (B.28) if'fr)  2  fr  2n  2  cos <p sin 6  2  This was subsequently changed to [13]:  University of British Columbia  232  Electrical Engineering  Appendix B : Scattering and Attenuation Programs  lis  ti-sY 150 150  +  25{S + 0.002) R  ( )2  1  +  f/fr  F(S): R + 100 100  1  -0.5—sin  2  fr  University of British Columbia  S  ^R  v  2^  5  +  100 j  >*[f'fr)  2n  (B.29) • cos (j> sin 9  2  233  Electrical Engineering  Appendix C : ( C O S T 2 1 0 + Modifications) Links Simulations  Appendix C: COST 210 + Modifications Links Simulations  C.1 COST 210 Experimental Links Simulations In order to ascertain the validity of the interference models, they are compared to several experimental links, part of the COST 210 project in Europe to study interference [5]. Both the original and modified rTU-R and Awaka models are used in the comparison, with 6.5 dB reflectivity roll-off in the ice/snow region above the melting layer. The parameters of the links are listed in table C - l , their measured transmission loss in table C2,. and rain rate statistics in table C-3 and they consist of short and long paths, different frequencies and gain, different common volume heights, etc, ... Some are measured interference statistics (all paths with the exceptions of S1-S5) while others are radar simulations (S1-S5). Therefore, the results in these simulations (S1-S5) may not provide a good indication of the effect of the inclusion of the melting layer. The transmission loss cumulative distribution functions for the different sites are given in figures C - l to C-24 for the ITU-R and the Awaka with both the original and modified models to account for the effect of the melting layer.  University of British Columbia  234  Electrical Engineering  Appendix C : ( C O S T 210 + Modifications) Links Simulations  C.2 Other Simulations In this section we examine the effect of the melting layer on the Bf link for different frequencies. In this section, we assume the rain height to be 3 km placing it in the middle of the common volume. The interference levels generated for/= 4, 8, 11.2, 20,'30 and 40 GHz are shown in figures C-25 to C-30, respectively.  University of British Columbia  -235  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  vO  fi  1-8  o  I ON I CS  "3 a 1  Si  CS'  CN  •g  "3  in  o  Tf  ON  s  a  Tf  'I  GO  Tf'  fi, a I* •c  ©'  ©  •c  I CD I  p  CN  3 •fi  p ©  TI-  00  8  CS  ON  vo vo  cs  5  si p  ©  © cs © ©  o  o  oo  cs  00  E VO  8 2 >/">  CN  "S s o  a  OS  VO  CN  I*  ^  Tf'  o _  © cs  Ov o\  o  fi •c  ©  oo  o cs  © ov r -  Ov  CS  s  "8 a  vo ©  © Ov  12 5 is r»'  •c  W)  25  cs  © Ov f l  CS  — cs  VO  I voj Ov  © °°.  -< vo  CS  I. I  8  3  00  00  I: •a CO  601  -a  >  University of British Columbia  50.  ,3 3 236  oa  9  00 O  f  Electrical Engineering  Appendix C : ( C O S T 210 + Modifications)'Links Simulations  % Time Ll L2 L3 L4 L5 L6 L7 L8 Bf Bb Ff Fb Gz Dl D2 D3 D4 D5 D6 SI S2 S3 S4 S5  1.0 135.7 136.3 137.4 139.0' 141.5 143.7 . 146.3 148.3 149.2 154.0  0.3 131.6 132.0 132.8 134.3 136.8 139.6 142.6 145.6 143.7 148.6  0.1 128.2 128.6 129.8 131.1 133.0 135.7 139.5 142.4 139.6 145.4  -  -  -  -  -  125.3 136.7 141.6 143.2 146.0 140.6 146.0 138.2 141.1 145.3 151.1 160.0  120.0 , 133.1 137.5 138.4 141.3 135.8 145.6 133.8 136.8 140.6 146.0 154.5  0.03 124.6 .125.1 126.7 128.0 129.6 131.7 136.5 139.2 136.6 142.7 147.8  -  -  116.9 130.4 134.6 135.1 137.3 132.3 142.3 130.8 .134.1 137.5 142.5 .150.8  114.5 127.7 132.0 132.3 134.0 129.4 139.2 128.0 131.6 134.8 139.3 147.7  0.01 121.4 122.0 123.8 125.0 126.9 128.0 133.8 136.4 134.6 140.9 146.8 148.5 112.3 125.6' 129.6 130.1 131.1 127.1 135.6 125.8 129.8 132.2 136.4 145.2  0.003 118.7 119.4 120.7 121.9 122.8 124.8 129.5 133.5 133.4. 139.6 145.8 147.4 110.3 123.7 126.7 126.8 ' 127.4 125.0 130.4 123.8 128.0 129.4 133.1 142.3  0.001 1 17.2 1 17.6 118.4 119.6 120.6 122.0 125.1 130.5 . .132.5. 138.5 145.1 146.9 109.4 122.1 123.9 123.3 124.7 123.1 126.6 122.4' 126.2 127.1 129.6 139.4  Period 30 months 30 months 30 months 30 months 30 months 30 months . 30 months 30 months 2 years 2 years 8 months 8 months • 2 years 8 months 8 months 8 months 8 months 8 months 8 months 2 years 2 years 2 years 2 years 2 years  Table C-2: Measured transmission loss statistics for the different experimental links  C h i l b o l t o n - B a l d o c k 2 years (Bf, B b ) 0.01 0.1 0.03 1.0 0.3 Time % 26.3 15.0 8.3 1.9 4.3 Rain rate L e i d s c h e n d a m - 30 months (L1-L8) 0.071 0.84 0.32 0.17 2.6 Time % 10.0 6.67 3.33 5.0 Rain rate 1.67 N o r d h e i m - D a r m s t a d t - 2 years ( A p p l i c a b l e to 0.31 1.24 0.81 2.93 2.77 Time % 4.0 1.6 2.5 1.0 0.63 Rain rate  0.003 42.0  0.001 62.0  0.0088 0.0025 0.0006 0.031 63.33 40.0 25.0 15.0 f o r w a r d scatter geometries) (D1-D4) 0.0183 0.0097 0.062 0.033 0.128 40.0 '25.0 16.0 10.0 6.3 to b a c k w a r d scatter geometries) ( D 5-D6) (Applicable stadt - 19 months 0.01 0.0196 0.035 0.067 0.137 0.32 1.22 0.81 2.48 40.0 25.0 16.0 10.0 6.3 4.0 1.6 2.5 1.0 t - 8 months (Ff. F b ) 0.017 0.031 0.049 0.09 0.173 0.37 1.24 0.88 2.27 40.0 25.0 16.0 10.0 6.3 4.0 2.5 1.6 1.0  Nordheim - Darm 2.67 Time % 0.63 Rain rate F u l d a - D armstad 2.36 Time % 0.63 Rain rate ars ( G z) G r a z - 2 ye 1.92 1.27 0.84 Time % 1.6 2.5 1.0 Rain rate C h i l b o l t o n R a d a r Simuk i t i o n s 1.18 1.66 2.05 Time% 1.6 0.63 1.0 Rain rate  0.49 4.0 2 Years 0.707 2.5  0.113 10.0  0.056 16.0  0.032 25.0  0.0203 40.0  0.0094 63.0  (S1-S5) 0.403 0.21 6.3 . 4.0  0.115 10.0  0.061 16.0  0.027 25.0  0.009 40.0  0.25 6.3  0.0035 63.0  0.0004 100.0  0.0033 63.0.  0.0003 100.0  0.0063 63.0  0.0006 100.0  0.0033 100.0  0.0007 158.5  ' 0.002 63.0  0.0003 100.0  Table C-3: Rain rate statistics for the different experimental sites described in table C-l.  University of British Columbia  237  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -100  ,-3 _  io "  .i<r*  io"'  10  % Time  Figure C-l: Transmission loss characteristics for the Gz link. The Awaka model (original; , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).'  -110  -120 CQ "O w  8-130 c o 'w CO  '1-140 c  0 -150  -160 10" L  10"  -1  % Time  10  10"  Figure C-2: Transmission loss characteristics for the Ll link. The Awaka model (original , modified ) and the ITU-R model (original , modified - . - . - . - . j are compared to the experimental results (+):  University of British Columbia  238  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  % Time Figure C-3: Transmission loss characteristics for the L2 link. The Awaka model (original , modified ) and the ITU-R model (original -, modified -.-.-.-.) are compared to the experimental results (+).  -110  -120F  m T3 CO CO  .  o -130 c o  i-140 c CO  -150  -160 10" L  10"  % Time  10-1  10"  Figure C-4: Transmission loss characteristics for the L3 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  239  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110  % Time Figure C-5: Transmission loss characteristics for the L4 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  -110 -1204  Figure C-6: Transmission loss characteristics for the L5 link. The Awaka model (original , modified ) and the CCIR model (original , modified -.'-.-.-.) are compared to the experimental results (+).  University of British Columbia  240  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110  -120 h  Figure C-7: Transmission loss characteristics for the L6 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  -110  -120h  % Time Figure C-8: Transmission loss characteristics for the L7 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  241  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110r  -120 CO  T3  % Time  Figure C-9: Transmission loss characteristics for the L8 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  .  -110r—  •  ,  r  _  _  r  _  ^  ,  % Time Figure C-10: Transmission loss characteristics for the D l link. The Awaka model (original , modified , ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  242  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110  % Time Figure C - l l : Transmission loss characteristics for the D2 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  -110  % Time Figure C-12: Transmission loss characteristics for the D3 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  243  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110  % Time Figure C-13: Transmission loss characteristics for the D4 link. The Awaka model (original , modified ) and the ITU-R model (original ——, modified -.-.-.-.) are compared to the experimental results (+).  -110  Figure C-14: Transmission loss characteristics for the D5 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  244  Electrical Engineerin  Appendix C: (COST 210 + Modifications) Links Simulations  -110  -120h  % Time Figure C-15: Transmission loss characteristics for the D6 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  —1101  •  •  •  •  ,  -120h  % Time Figure C-16: Transmission'loss characteristics for the SI link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  245  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -110  % Time Figure C-17: Transmission loss characteristics for the S2 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  -110  Figure C-18: Transmission loss characteristics for the S3 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  246  Electrical Engineering  Appendix C: (COST.210 + Modifications) Links Simulations  -11.0'r  -120-  m T3  % Time Figure C-19: Transmission loss characteristics for the S4 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results \ ). T  _120i  •  • ••  •  •  .  -130CQ T3  % Time Figure C-20: Transmission loss characteristics for the S5 link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-:-.-.) are compared to the experimental results (+).  University of British Columbia  247  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -120  -130r CQ TJ  w co o  -140  c• o 'co  CO  E -150 CO  c CO I—  1- -160r  -170  % Time Figure C:21: Transmission loss characteristics for the Ff link. The Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  -120  % Time Figure C-22: Transmission loss characteristics for the Fb link. The Awaka model (original _ _ , modified ) and the ITU-R model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  248  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -120r  -130 -  % Time Figure C-23: Transmission loss characteristics for the Bf link. The Awaka model (original _, modified ) and the ITU-R model (original., modified -.-.-.-.) are compared to the experimental results (+,).  -120  -170 -3 10  10"  % Time  10"  10"  Figure C-24: Transmission loss characteristics for the Bb link. The Awaka model (original , modified ) and the CCIR model (original , modified -.-.-.-.) are compared to the experimental results (+).  University of British Columbia  249  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -120 m -130 TJ  % Time Figure C-25: Transmission loss characteristics for the Bf link (h = 3.0 km,/= 4.0 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  -120  -130 h  2 -160-  -170 10"  •  1  3  • 10"  —  —  i  .  10"  2  1  .  1 •  10°  % Time Figure C-26: Transmission loss characteristics for the Bf link (h = 3.0 km,/= 8.0 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  University of British Columbia  250  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -120  % Time Figure C-27: Transmission loss characteristics for the Bf link (h = 3.0 k m , / = 11.2 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  -120  -130 -  m  % Time Figure C-28: Transmission loss characteristics for the Bf link (h = 3.0 k m , / = 20.0 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  University of British Columbia  251  Electrical Engineering  Appendix C: (COST 210 + Modifications) Links Simulations  -120  -130+  % Time Figure C-29: Transmission loss characteristics for the B f link (h = 3.0 k m , / = 30.0 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  -120r  -130-  m  % Time Figure C-30: Transmission loss characteristics for the Bf link (h = 3.0 k m , / = 40.0 GHz) using the Awaka model (original , modified ) and the ITU-R model (original , modified -.-.-.-.) FRm  University of British Columbia  252  Electrical Engineering  Appendix D:. Attenuation M o d e l  Appendix D: Attenuation Model  In this appendix, the files necessary to run the attenuation model are presented. These are: • 1. driver.f: This is the driver program that generates the input parameters to the attenuation model.  '  2. fade.f: This is the program that calculates the attenuation that a signal suffers. 3. paramet: This is the file containing the general parameters of the link. 4. ml_pr: This is the file containing the percentage of presence of the melting layer as a function of rain rate. 5. hfr_dh: This is the file containing the variations of the rain height around its median. 6. hydro: This is the file containing the particle-size distribution. 7. rain_pr: This is the file containing the cumulative distribution function of rain, cdf.m is the output file containing the cumulative distribution functions of transmission loss. A l l the above files are explained below. The input files and the programs are also provided in electronic format on the diskette appended to this report.  D.1 Software and Hardware Requirements The simulation program is guaranteed to work under the environments described below. Even though the program should run under similar environments, the author cannot guarantee this and users should abide by the system requirements set below.  University of British Columbia  253  Electrical Engineering  Appendix D: Attenuation Model  D.l.l  Software requirements  The program is written in Fortran-77, compiled using the SUN Fortran-77 compiler and run in a UNIX environment.  D.1.2 Hardware requirements The program was.used on SPARC 2 and SPARC 5 computers with a minimum of 32 megabytes of R A M .  D.2 Running the Attenuation Program  D.2.1  Program inputs  In order for the program to run, it is necessary to provide the correct input format. The input files to this model are the same as those for the interference models. In the paramet file, however, there are several variables that do not enter into the calculation and they are in the file as dummy variables.  D.2.1.1 General parameters file 20.0 3.0 3.0 131.0 0.12 0.086 1.8 70.0 0.0 0.00314 0.0 59.0 89.0 180.0 0.0279 0.0 40.5 -17.0  ,frequency (GHz) , m e d i a n r a i n h e i g h t (km) , c o m m o n v o l u m e h e i g h t (km) — dummy variable , d i s t a n c e b e t w e e n trans, a n d r e c e i v . (km) , h e i g h t o f t r a n s m i t t e r (km) , h e i g h t o f r e c e i v e r (km) , i n i t i a l snow density ( v a l i d b e t w e e n 0.1-0.3, 1.8: v a r i a b l e ) , theta o ftransmitter (1-elev, degree) , p h i o f the t r a n s m i t t e r ( d e g r e e ) - dummy variable , h a l f b a n d w i d t h o f t r a n s m i t t e r ( r a d ) - dummy variable , a l p h a f o r t r a n s m i t t e r (0: H o r i z o n t a l , 1: V e r t i c a l ) - dummy variable , g a i n o f t r a n s m i t t e r - dummy variable , t h e t a o f r e c e i v e r ( 1 - e l e v , d e g r e e ) - dummy variable , p h i o f the r e c e i v e r ( d e g r e e ) - dummy variable , h a l f b a n d w i d t h o f r e c e i v e r ( r a d ) - dummy variable , a l p h a f o r the r e c e i v e r (dummy v a r i a b l e ) - dummy variable , g a i n o f the r e c e i v e r - dummy variable , K ( d B ) - dummy variable  University of British Columbia  254  Electrical Engineering  Appendix D:-Attenuation M o d e l  1.6 5.8788 2 6 1 1 3 5 10.0  . Half-beam width for receiver main lobe (deg.) - dummy variable ', Half-beam width for receiver secondary lobe (deg.) - dummy variable , model select (use 1 for the concentric and 2 for spongy) , attenuation select (use 1 for the concentric and 2 for spongy) , scattering select - dummy variable , 1 :-6.5;2:-4.0 (reflectivity roll-off) - dummy variable , l:S=h/H;2:S=(h/H) 2;S=(h/H) 3 (melting profde) . , ML thickness — select 5 , steps of integration (meters) A  A  D.2.1.2 Meteorological files The general meteorological files ml_pr, hfr_dr, hydro, and rain_pr, have the same structure and properties as those used for the interference models. For a detailed description of these files, please refer to appendix A .  D.2.1.3 Mie scattering and attenuation files In order to run the above model, it is necessary to provide the Mie-generated attenuation files. These files are the same as the input files for the "atten.f' program in appendix B.  D.2.2 Program outputs  D.2.2.1 interm Ignore this file.  D.2.2.2 C D F . m The CDF.m file is in matlab format as follow:  RSUM-[  % Exceedence percentages (Rain only)  I: L_R=[  % Interference levels (Rain only)  University of British Columbia  255  Electrical Engineering  Appendix D: Attenuation M o d e l  7/ MSUM=[  % Exceedence percentages (ML, 100 %)  L_M=[  % Interference levels (ML, 100%)  RMSUM=[  % Exceedence percentages (Ignore this)  7/ L_RM=[%  Interference levels (Ignore this)  I; N_RMSUM=[  % Exceedence percentages (Method I)  7;  N_L_RM=[  % Interference levels (Method I)  I: The results for method two are obtained by a melting-layer presence probability-weighed interpolation between L_R and L _ M . D.2.3 Compiling and running the models When running the simulation model, please follow the steps outlined below: 1. Compile and link driver.f and fade.f programs:  University of British Columbia  256  Electrical Engineering  Appendix D: Attenuation Model  f77 -o fade.exe driver.f fade.f 2. Make sure that the following files are in the same directory where you intend to run the program: the correct "atten" file paramet ml_pr • hfr_dh  ,  hydro rain_pr 3. Run the program in low priority:  '  priority fade.exe >&! out & 4. Remove "temp4", "interm", "out" and "GINP" files. 5. The relevant output is in "CDF.m". Use matlab to plot data.  D.3 Program Listings  D.3.1 driver.f This program is available in the appended disks in the fade subdirectory, along with the other programs and input files necessary to run the simulation model.  D.3.2 fade.f This program is available in the appended disks in the fade subdirectory, along with the other  programs  and  University of British Columbia  input  files  necessary  257  to  run  the  simulation  model.  Electrical Engineering  

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