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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9’ black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. McGill University Information Networks and Systems Laboratory M odeling and performance of microwave radio links in rain Joe Nader Department of Electrical Engineering, McGill University Montreal, Canada August, 1998 A thesis submitted to the Faculty o f Graduate Studies and Research in partial ful fillment of the requirements of the degree of Master o f Engineering. Copyright © Joe Nader, 1998 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1*1 National Library of Canada Bibliotheque nationale du Canada Acquisitions and Bibliographic Services Acquisitions et services bibliographiques 395 Wellington Street Ottawa ON K1A0N4 Canada 395. rue Wellington Ottawa ON K1A0N4 Canada Your 6tm Votrm rmtmrmnc m Our Him Notrm rmttrmncm The author has granted a non exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies o f this thesis in microform, paper or electronic formats. L’auteur a accorde une licence non exclusive permettant a la Bibliotheque nationale du Canada de reproduire, preter, distribuer ou vendre des copies de cette these sous la forme de microfiche/film, de reproduction sur papier ou sur format electronique. The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author’s permission. L’auteur conserve la propriete du droit d’auteur qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. 0-612-50644-4 Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Microwave radio links, operating in the millimeter wave region, must account for the effects o f rain when considering transmission loss. In this work, a theoretical model is used to generate the specific attenuations based on perturbation theory with spheroidal or Pruppacher-Pitter raindrop shapes, and Marshal 1-Palmer or Weibull drop size distributions. The specific attenuation is fitted to the power law relation with rain rate and the parameters are used in a two-component rain rate model in order to estimate the attenuation along the path. The theoretical model is simulated and compared to the ITU and Crane prediction methods. Both moderate and tropical climates are considered. A simple line-ofsight radio system is then simulated and evaluated by incorporating the rain attenu ation in the channel. Finally, three basic network blocks are discussed and ana lyzed for links affected by rain. • • • ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SOMMAIKE Les liaisons radio, fonctionnant dans les longueurs d’ondes millimetriques, doivent inclure les effets de la pluie en considerant les pertes de signal dues a la transmis sion. Base sur la methode de perturbation, un modele theorique est employe pour produire 1’attenuation specifique, utilisant des gouttes de pluie de forme spheroi'dale ou Pruppacher-Pitter et une distribution de grandeur de goutte selon Marshall-Palmer ou Weibull. Afin d ’estimer 1’attenuation sur le parcours, 1’attenu ation specifique est repesentee avec une relation exponentielle incluant Ie taux de pluie, et sont, par ia suite, utilises dans un modele de taux de pluie a deux composantes. Le modele theorique est simule et compare aux methodes de prediction de rUTT et de Crane. Des climats moderns et tropicaux sont consideres. En incorporant 1’attenuation due a la pluie dans la transmission, un systeme de communication terrestre simple est alors simule et evalue. Enfin, trois blocs elementaires de reseau sont discutes et analyses pour les liaisons radio affectees par la pluie. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to express my deepest gratitude to my supervisor Dr. Salvatore D. Morgera for his guidance throughout my graduate studies at M cGill University. I would like to thank Harris Corporation, Farinon division who financially sup ported the research. I would like to thank Carolina Dieguez for her contribution to the world rain region figures and her patience. Finally, I would like to thank my parents for their continuous support and encour agement. • • V ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONTENTS 1 ABSTRACT iii SOMMAIRE v ACKNOWLEDGMENTS vii CONTENTS ix LIST OF FIGURES xiii LIST OF TABLES xvii INTRODUCTION 1 1.1 1.2 1.3 2 Overview of Attenuation Prediction Motivation o f our Research Organization of Thesis PREDICTION OF RELIABILITY 2.1 2.2 2.3 Rain-Rate Climate Models 2.1.1 ITU-R837 Rainfall Rate Statistics 2.1.2 Crane Rain Rate Climate Zone Specific Attenuation 2.2.1 ITU-R838 Specific Attenuation Model for Rain Path Attenuation Formulas 2 2 4 5 7 7 9 11 11 13 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X CONTENTS 2.4 2.3.1 ITU-R530 Prediction Model 2.3.2 Crane Prediction Model System Performance 3 MODELING ATTENUATION AT A POINT 3.1 19 21 24 24 26 28 29 31 34 35 35 36 36 37 41 42 42 43 MODELING POINT-TO-PATH ATTENUATION 45 3.3 4.1 4.2 4.3 5 17 Derivation of the Total Cross-Section 3.1.1 Vector-eigenfiinction expansion 3.1.2 Perturbation method solution 3.1.2.1 The Zeroth-Order Approximation 3.1.2.2 The First-Order Approximation 3.1.3 Expansion Coefficients 3.1.3.1 Axisymmetry contribution 3.1.3.2 Nonaxisymmetry contribution 3.1.4 Total Cross Section Raindrop Characteristics 3.2.1 Raindrop shape 3.2.1.1 Spherical 3.2.1.2 Spheroidal 3.2.1.3 Pruppacher-and-Pitter 3.2.2 Drop Size Distribution 3.2.2.1 M arshall-Palmer dsd 3.2.2.2 Weibull dsd Theoretical Derivation 3.2 4 13 14 15 Spatial Correlation of Rain Two-Component Model 4.2.1 Model for Volume Cell Component 4.2.2 Model for Debris Component Application to Networks 4.3.1 Parallel Diversity 4.3.1.1 Model for Volume Cell Component 4.3.1.2 Model for Debris Component 4.3.2 Node Diversity 4.3.2.1Model for Volume Cell Component 4.3.2.2 Model for Debris Component 4.3.3 Relay PERFORMANCE ANALYSIS 5.1 Specific Attenuation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 48 50 52 55 55 56 57 59 59 61 62 65 66 CONTENTS 5.2 5.3 5.4 Path Links 5.2.1 Temperate Continental (Montreal) 5.2.2 Tropical Climate (Singapore) System Analysis 5.3.1 Temperate Continental (Montreal) 5.3.2 Tropical Climate (Singapore) Network Links 5.4.1 Parallel Diversity 5.4.2 Node diversity 5.4.3 Relay 6 FINAL REMARKS AND FUTURE WORK 6.1 6.2 Summary of our work Future Considerations id 74 75 80 85 85 88 91 91 93 93 95 95 98 REFERENCES 101 BIBLIOGRAPHY 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • LIST OF FIGURES Figure 2 .1 Rain attenuation prediction procedure. 6 Figure 2.2 ITU world rain regions (source [ITU837]). 7 Figure 2.3 Crane world rain regions (source [Crane80]). 10 Figure 2.4 Crane U.S. rain regions (source [Crane80]). 10 Figure 2.5 Block diagram of a communication system. 15 Figure 3.1 Geometry of plane waves scattered by a distorted raindrop 19 Figure 3.2 The raindrop models with cquivoiumetric drop radius 3.25mm. 35 Figure 3.3 The two spherical coordinates system superimposed. 39 Figure 4.1 One-dimensional idealized spatial spectrum for ln(rain rate) 47 Figure 4.2 Spatial correlation function for ln(rain rate) and for rain rate (Climate D2)48 Figure 4.3 Geometry for a parallel topology 55 Figure 4.4 Geometry for cell contribution calculations for a parallel diversity 56 Figure 4.5 Geometry for debris contribution calculations for parallel diversity 57 Figure 4.6 Geometry for a node topology. 59 Figure 4.7 Geometry for cell contribution calculations for a node topology. 60 Figure 4.8 Geometry for debris contribution calculations for node topology. 61 Figure 4.9 Geometry for relay topology. 62 Figure 5.1 Total cross section comparing spheroidal and P-P raindrop model for 15(top) and 38(bottom) GHz. 67 Figure 5.2 M-P and Weibull drop size distribution for different rain rates. 68 • • • m i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x iv LIST OF FIGURES Figure S.3 Specific attenuation varying with frequency for a 50mm/h rain rate. (Spheroidal raindrops) 69 Figure 5.4 Specific attenuation comparing drop size distribution for spheroidal raindrops and frequencies 15(top) and 38(bottom) GHz. 70 Figure 5.5 Specific attenuation comparing drop size distribution for P-P raindrops and frequencies 15(top) and 38(bottom) GHz. 71 Figure 5.6 Specific attenuation with different frequencies Weibull(bottom) dsd, assuming P-P raindrop model. Figure 5.7 Specific attenuation comparing theoretical spheroidal and P-P raindrop models with the ITU-R838 at 38GHz, for M-P(top) and Weibull(bottom) dsd. 73 Figure 5.8 Path attenuation in Montreal using ITU-R530 prediction method for a 38GHz system and fTU-R838 specific attenuation model. 76 Figure 5.9 Path attenuation in Montreal using Crane prediction method for a 38GHz system and ITU-R838 specific attenuation model. 76 Figure 5.10 Path attenuation in Montreal using the two-component model with spheroidal based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. 77 Figure 5.11 Path attenuation in Montreal using the two-component model with P-P based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. 78 Figure 5.12 Path attenuation comparison in Montreal for different prediction methods at 38 GHz and unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using M-P dsd. 79 Figure 5.13 Path attenuation comparison in Montreal for different prediction methods and unavailabilities o f 0.01 (top) and 0.001 %(bottom), theoretical derivation using Weibull dsd. 79 Figure 5.14 Path attenuation in Singapore using ITU-R530 prediction method for a 38GHz system and ITU-R838 specific attenuation model. 81 Figure 5.15 Path attenuation in Singapore using Crane prediction method for a 38GHz system and ITU-R838 specific attenuation model. 81 Figure 5.16 Path attenuation in Singapore using the two-component model with spheroidal based specific attenuation at 38GHz with M-P(top) and 82 Weibull(bottom) dsd. Figure 5.17 Path attenuation in Singapore using the two-component model with P-P based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. 83 Figure 5.18 Path attenuation comparison in Singapore for different prediction methods at 38 GHz and unavailabilities of 0.01(top) and 0.001 %(bottom), theoretical derivation using M-P dsd. 84 Figure 5.19 Path attenuation comparison in Singapore for different prediction methods and unavailabilities of 0.01(top) and 0.001%(bottom), theoretical derivation using Weibull dsd. 84 Figure 5.20 Fade margin comparison in Montreal for different prediction methods at 38 GHz for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using M-P dsd. 86 for M-P(top) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 72 LIST OF FIGURES XV Figure 5.21 Fade margin comparison in Montreal for different prediction methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd. 86 Figure S.22 Maximum path length comparison in Montreal for different prediction methods at 38 GHz for unavailabilities of 0.01 (top) and 0.001 %(bottom), theoretical derivation using M-P dsd. 87 Figure 5.23 Maximum path length comparison in Montreal for different prediction methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd. 87 Figure 5.24 Fade margin comparison in Singapore for different prediction methods at 38 GHz for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using M-P dsd. 89 Figure 5.25 Fade margin comparison in Singapore for different prediction methods for unavailabilities of 0.01 (top) and 0.001 %(bottom), theoretical derivation using Weibull dsd. 89 Figure 5.26 Maximum path length comparison in Singapore for different prediction methods at 38 GHz for unavailabilities of 0.01(top) and 0.001 %(bottom), theoretical derivation using M-P dsd. 90 Figure 5.27 Maximum path length comparison in Singapore for different prediction methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd. 90 Figure 5.28 Diversity improvement factor and ratio of the cell to debris component as a function of baseline length for jointly exceeding an attenuation of 3dB. Calculations are for a pair of terrestrial paths of 10km length with an orientation angle of 45° between the baseline and path direction. 92 Figure 5.29 Diversity improvement factor and ratio of the cell to debris component as a function of orientation angle for jointly exceeding an attenuation of 3dB. Calculations are for a pair of terrestrial paths of 10km length with a 3km baseline. 92 Figure 5.30 Diversity improvement factor and ratio of the cell to debris component as a function of separation angle for jointly exceeding an attenuation of 3dB. Calculations are for a node with paths of 10km length. 93 Figure 5 .31 Diversity improvement factor and ratio of the cell to debris component as a function of separation angle for exceeding an attenuation of 3dB. Calculations are for a relay with paths of 10km length. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES TABLE 1.1 Rain attenuation assumptions 4 TABLE 2.1 ITU-R837 Rainfall intensity exceeded (mm/h) (source [ITU837]) 8 TABLE 2.2 Crane Rain-Rate Distributions (mm/h) (source [Crane80]) 9 TABLE 2.3 Regression coefficients for estimating specific attenuation in (EQ. 2.1) 12 TABLE 2.4 Link availability 16 TABLE 3.1 Calculated Coefficients for different raindrop sizes (Spheroidal raindrops)37 TABLE 3.2 Computed deformation coefficients of raindrops (source [Oguchi77]) 38 TABLE 3.3 Calculated coefficients for different raindrop sizes (P-P raindrops) 40 TABLE 4.1 Parameters for the Two-Component model by region (source [1]). 50 TABLE 5.1 Parameters for the empirical specific attenuation evaluated at 38GHz. 74 TABLE 5.2 Rainfall Rate Statistics (mm/h) for Montreal 75 TABLE 5.3 Rainfall Rate Statistics (mm/h) for Singapore 80 TABLE 5.4 Parameters for the microwave system 85 x v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 INTRODUCTION Millimeter wave links offer large bandwidth and high speed communication for integrated multimedia services. Such links are also quick to deploy, as compared to cable and fibre optic wireline connections. The issue of greatest importance in the study of the performance o f millimeter links is rainfall. In the design o f such sys tems, the attenuation due to rain must be accurately accounted for in order to ensure system reliability and availability. Systems that are poorly designed lead to an increase in transmission errors, or worst, to an outage in the received signal. Unfortunately, the designer has to rely on rain process statistics which could con tain an insufficient amount o f data for long-term predictions. This is particularly true for tropical climates which are vaguely understood, as compared to climate regions in North-America or Europe where the largest number of observing sta tions is located [1]. The primary goal of a rain attenuation prediction method is to achieve acceptable estimates of the attenuation incurred on the signal due to rain, given the system requirements such as frequency, path length, polarization, path geometry, rain rate distributions, and availability. The International Telecommunication Union, or ITU prediction model [ITU530] is a first point o f reference for the engineer. Improve ments to the models come from measurement programs which provide further insight into the physics o f precipitation systems and also data against which their accuracy can be evaluated. These can be obtained from experimental links, which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 INTRODUCTION provide accurate data, but, generally for only one specific frequency and path geometry. 1.1 OVERVIEW OF ATTENUATION PREDICTION A power law empirical relation between the specific attenuation and the rain rate has been found to be a good approximation [01sen78]. The derivation of the spe cific attenuation is a known result from scattering theory [4], but the computation is complicated using the perturbation method [Oguchi60], because it involves the summation of spherical Bessel functions and associated Legendre functions. Therefore, the empirical relation for evaluating the specific attenuation is extremely practical, since the parameters are given in a tabulated form and calcula tions can be carried out in seconds. The path attenuation was first calculated [Lin77] by directly applying the line rain rate in the empirical relation previously discussed. Later, Crane [Crane80] found a power law relation between the line rain rate and point rain rate from which he evaluated the instantaneous rain profile. He then utilized the empirical relation between the attenuation and rain rate to compute an exponentially fitted effective path length. The attenuation along the path is then obtained by multiplying the spe cific attenuation by the computed effective path length. By making the rain rate the main parameter of the attenuation prediction model, there is no need to evaluate attenuation at different frequencies and path geome tries. 1.2 MOTIVATION OF OUR RESEARCH The evolution of computers and the advancement o f programming tools has made it possible to simulate complex models in a reasonable amount of time. The pur pose of this thesis is to model the attenuation due to rain and to analyze microwave link performance when degraded by rain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 MOTIVATION OF OUR RESEARCH 3 The modeling o f the attenuation will be divided into two parts. First, we carry out the evaluation o f the specific attenuation according to the solution o f the total cross section provided by the perturbation method used by Oguchi for spheroidal rain drops [Oguchi60]. The theory will be applied to the Pruppacher-and-Pitter shaped raindrops [Pruppacher71], which was made available through the simplification of the raindrop shape expression [Li94]. The general expression for the P-P shaped raindrops will be used to represent the spherical and spheroidal shape as well. The new expression obtained resulted in better approximation of the spheroidal shape when compared with the one used by Oguchi [Oguchi60] or M orrison and Cross [Morrison74]. This could explain why differences in the total cross section were noted between the Morrison and Cross least squares technique and the Oguchi per turbation method. Finally, the total cross section is integrated with the drop size distribution in order to obtain the specific attenuation due to rain. In the second part, the modeling of the rain rate process as a combination of two components [Crane89] is carried out, one with an exponential distribution attrib uted to volume cells and the second with a lognormal probability density function for the debris component. The model estimates the probability of exceeding a cer tain attenuation threshold. A trial and error approach results in the path attenuation for a specified time percentage. The two-component model for rain rate assumes an empirical relation for the specific attenuation. Theoretically derived specific attenuations can therefore be fitted by the least squares method with the parameters then fed into the two-component model. The choice of the two-component model was encouraged because o f its applicability to network analysis. Once the theoreti cal model was ready, the rain effect could be included in the channel o f a micro wave system and then analyzed. Two prediction methods are chosen to validate the simulated rain attenuation. The first one is the ITU Recommendation 530 [ITU530], and the second one is the Crane model as described in [Crane80]. We will limit ourselves to the study of terrestrial line-of-sight microwave links. The Earth-space links differ only by an adjustment of the path length that is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 INTRODUCTION affected by rain. Table 1.1 summarizes the different assumptions for modeling the rain attenuation. TABLE 1.1 Rain attenuation assumptions Specific Attenuation Rain Rate Distribution Raindrop Shapes Spheroidal or P-P Raindrop Size Distributions M-P or Weibull Cell Component Exponential pdf Debris Component Lognormal pdf 1.3 ORGANIZATION OF THESIS The intent of this thesis is to study the effect of different assumptions in the com putation o f microwave radio system rain attenuation and to compare the resulting approaches to the standards. Chapter 2 describes the ITU and Crane prediction methods along with their climate models. The empirical specific attenuation pro vided by ITU-R838 will be used for the calculation of the attenuation. A simple system is presented, which will be simulated with the rain attenuation modeled in the channel to complete the analysis of a microwave single link affected by rain. Chapter 3 models the specific attenuation, by computing the total cross section for a generalized expression of the raindrop shape using the perturbation method. The spherical, spheroidal, and Pruppacher-and-Pitter raindrop shapes are then fitted to this general expression. A knowledge o f the drop size distribution is needed to finalize the calculation of the specific attenuation. Chapter 4 describes the twocomponent model for modeling the rain rate distributions. The rain correlation function, which is needed for the model, is then derived from its power spectrum. Finally, the two-component model is applied on three simple network blocks. Chapter 5 describes the implementation o f the above mentioned models and includes simulation results and performance evaluations. Chapter 6 concludes the thesis with a summary of our work and suggestions for future investigation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 PREDICTION OF RELIABILITY The effects that hydrorneteor, such as cloud, raindrops, or snowflakes have on communication systems are dependent both on the system frequency and the type of particle present. The rain attenuation for centimetric (3-30GHz) or submillimetric (300-3000GHz) and, especially, millimetric (30-300GHz) microwave links becomes dominant and must be accounted for in a system design. The latter region of the spectrum is now being developed as new technology becomes available. Unfortunately, there are not enough data to completely describe the rain process as it affects the propagation of electromagnetic waves. Models are prepared to pro vide the best possible estimates given the information currently available. By tak ing the available statistical information into account, statistical predictions are made for the occurrences of events that affect the availability of communication systems. Due to the observed relation between specific attenuation and the precipitation rate, o f the form kR“ [01sen78], prediction methods are constructed with the rain rate as the primary parameter. Starting with a specification for a microwave sys tem, as can be seen in Figure 2.1, the path attenuation is calculated in three stages. The first stage, which estimates the rain rate at availability (1-p), requires the knowledge of rain rate distributions which characterize the geographical location of the link. Rain rate climate models with sufficient long-term data are readily Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 PREDICTION OF RELIABILITY available for an estimate o f these rain rate distributions. In the second stage, the attenuation at a point, expressed in decibels per kilometer, is computed using the above mentioned formula. The parameters, fitted to theoretical derivations, are pre sented in tabular form. At the last stage, an effective path length is estimated to account for the rain inhomogeneous characteristic in the horizontal. Finally, the path attenuation is derived. The effective path attenuation depends on the rain rate due to the complexity o f measuring the path attenuation for different frequencies and distances. Specification Stage i ---------------- 1 I------------ 1 r Stage 2 Stage 3 ~\ r d-p) Frequency Effective Path Length Path Attenuation Path Length F ig u re 2.1 Rain attenuation prediction procedure. Once the rain attenuation is known for some design specification, the performance of that system can be analyzed by incorporating the rain attenuation in the trans mission loss of the medium. A backward approach is also useful when the path length between the antennas is to be determined. Two prediction procedures will be considered because of their acceptability and wide use in microwave system design. The first one is the ITU-Recommendation 530 [ITU530], which is considered the standard in predicting the path attenuation based on the ITU-R837 [ITU837] climate regions and the ITU-R838 (ITU838] parameters for the specific attenuation. The second one is the Crane attenuation prediction model [Crane80], based on the Crane climate model [Crane80]. This model has an important interest in North-America. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 2.1 RAIN-RATE CLIMATE MODELS 2.1 RAIN-RATE CLIMATE MODELS The starting point for estimating rain attenuation is a knowledge o f the statistics of rainfall rate. Direct measurements o f rain attenuation are not very practical because they can only be obtained for a given frequency and path geometry and need to span several years to be statistically accurate, therefore smoothing the year-to-year variability. The preferred option is to use locally derived statistics of rainfall rate with an integration time o f 1 min. However, this data might not be available, in which case global distributions o f rainfall rate are provided for various percentages according to specified zones. There are two important climate regions that need to be considered. First, the ITUR837 “Rainfall Rate Statistics” (1TU837] is utilized in the ITU-R530 prediction model [ITU530]. Second, the Crane “Rain-Rate Climate M odel” [Crane80] is uti lized in the Crane Attenuation Model [Crane80]. 2.1.1 ITU-R837 Rainfall Rate Statistics 180 180 Figure 2.2 ISO 120 tO 00 ISO 120 90 80 30 30 60 90 120 ISO 180 120 ISO 180 ITU world rain regions (source [ITU837]). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 PREDICTION OF RELIABILITY The climate regions used by ITU are shown in Figure 2.2, and the rain rate distri butions are given in Table 2.1. The distribution gives the rain rate that is exceeded at a certain percentage o f time or year. The climate regions are labeled with letters starting with least precipitations in region A, and finishing with the letter P for higher rain rates. TABLE 2.1 ITU-R837 Rainfall intensity exceeded (nun/h) (source [ITU837]) Percentage of time (%) 1.0 0.3 0.1 0.03 0.01 0.003 A <0.1 0.8 2 5 8 14 22 B 0.5 2 3 6 12 21 32 C 0.7 2.8 5 9 15 26 42 D 2.1 4.5 8 13 19 29 42 E 0.6 2.4 6 12 22 41 70 F 1.7 4.5 8 15 28 54 78 G 3 7 12 20 30 45 65 H 2 4 10 18 32 55 83 J 8 13 20 28 35 45 55 23 42 70 100 105 150 K 1.5 4.2 12 0.001 L 2 7 15 33 60 M 4 11 22 40 63 95 120 N 5 15 35 65 95 140 180 P 12 34 65 105 145 200 250 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 2.1 RAIN-RATE CLIMATE MODELS 2.1.2 Crane Rain Rate Climate Zone The Crane climate regions have different boundaries than the ITU regions as can be seen from Figure 2.3 for the world regions and Figure 2.4 for the U.S. details. The corresponding rain rate distributions, in Table 2.2, are, however, similar in general. TABLE 2 .2 Crane Rain-Rate Distributions (nun/h) (source [Crane80]) Percentage of time (%) 0.01 0.005 0.001 1.0 0.5 0.1 0.05 A 1.7 2.5 5.5 8.0 15.0 19.0 28.0 B 1.8 2.7 6.8 9.5 19.0 26.0 54.0 C 1.9 2.8 7.2 11.0 28.0 41.0 80.0 DI 2.2 4.0 11.0 16.0 37.0 50.0 90.0 D2 3.0 5.2 15.0 22.0 49.0 64.0 102.0 D3 4.0 7.0 22.0 31.0 63.0 81.0 127.0 E 4.0 8.5 35.0 52.0 98.0 117.0 164.0 F 0.8 1.2 5.5 8.0 23.0 34.0 66.0 G 3.7 7.0 22.0 33.0 67.0 85.0 129.0 H 6.4 13.0 51.0 77.0 147.0 178.0 251.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 PREDICTION OF RELIABILITY 100 150 <0 120 00 30 120 190 79 00 00 30 30 45 ISO 150 120 90 00 30 0 30 Figure 2.3 Crane world rain regions (source [Crane801). Figure 2.4 Crane U.S. rain regions (source [Crane80]). 90 90 ISO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 2.2 SPECIFIC ATTENUATION 11 2.2 SPECIFIC ATTENUATION The fundamental quantity in the calculation o f rain attenuation statistics for terres trial paths is the specific attenuation y, representing the rain attenuation per unit distance. R. L. Olsen [01sen78] extensively analyzed the relation y = aRb between specific attenuation and rain rate from both theoretical and numerical viewpoints and concluded that this relation is an approximation of a more general series rela tion in frequency and rain rate, which reduces to the simpler form at the frequen cies and rain rates o f practical interest. In order to obtain the parameters for the formula, specific attenuations at different rain rates, computed from theoretical models, were fitted by the least squares method for a number o f frequency values. Then, the formula was tested against available data in order to verify its correctness. The ITU-Recommendation 838 [ITU838] has become the standard for the specific attenuation estimation. 2.2.1 ITU-R838 Specific Attenuation Model for Rain The specific attenuation is calculated using the following power-law relationship: YR = k R a (EQ 2.1) where yR is the specific attenuation in dB/km and R the rain rate in mm/h. The frequency-dependent coefficients k and a are given in Table 2.3 for vertical and horizontal linear polarizations, and terrestrial paths. For all other path geometries and polarization, the coefficients in (EQ. 2.1) can be calculated from the values in Table 2.3 using the following equations: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 PREDICTION OF RELIABILITY , _ + + ( * / / - ly)COS20COs2T] 2 (EQ 2J2) [ k Ha H + k v a v +• ( k Ha H - k v a v ) c o s 20 c o s 2 t ] — a where 9 is the path elevation angle and x is the polarization tilt angle relative to the horizontal, e.g., for circular polarization x = 45°. TABLE 2.3 Regression coefficients for estimating specific attenuation in (EQ. 2.1) r (GHz) 1 kv 0.0000352 2 kH «H 0.0000387 «v 0.880 0.912 0.000138 0.000154 0.923 0.963 4 0.000591 0.000650 1.075 1.121 6 0.00155 0.00175 1.265 1.308 7 1.332 0.00265 0.00301 1.312 8 0.00395 0.00454 1.310 1.327 10 0.00887 0.0101 1.264 1.276 12 0.0168 0.0188 1.200 1.217 15 0.0335 0.0367 1.128 1.154 20 0.0691 0.0751 1.065 1.099 25 0.113 0.124 1.030 1.061 30 0.167 0.187 1.000 1.021 35 0.233 0.263 0.963 0.979 40 0.310 0.350 0.929 0.939 45 0.393 0.442 0.897 0.903 50 0.479 0.536 0.868 0.873 60 0.642 0.707 0.824 0.826 70 0.784 0.851 0.793 0.793 80 0.906 0.975 0.769 0.769 90 0.999 1.06 0.754 0.753 100 1.06 1.12 0.744 0.743 120 1.13 1.18 0.732 0.731 150 1.27 1.31 0.711 0.710 200 1.42 1.45 0.690 0.689 300 1.35 1.36 0.689 0.688 400 1.31 1.32 0.684 0.683 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 PATH ATTENUATION FORMULAS 13 2.3 PATH ATTENUATION FORMULAS The general prediction procedure proceeds by choosing an availability requirement for the system. The availability is expressed in percentage of time per year that the signal should be available. In this section, a description of the two methods employed to model the rain atten uation is given for the ITU-R Recommendation 530 [ITU530], which will be con sidered the reference, and the Crane model [Crane80], which is largely used in the industry. 2.3.1 ITU-R530 Prediction Model The present ITU-R530 [ITU530] method predicts the effects of rain as an attenua tion on the signal and is considered to be valid in all parts o f the world for frequen cies up to at least 40GHz and path lengths up to 60km. The method is as follows: 1. Obtain the rain rate Ro.oi exceeded for 0.01% of the time. If this information is not available from local sources of long-term measurements, an estimate can be obtained from the information given in Recommendation ITU-R837 IITU837]. 2. Compute the specific attenuation Yr (dB/km) for the frequency, polarization and rain rate of interest using Recommendation ITU-R838 [ITU838]. 3. Compute the effective path length deff from the path length d as follows: d0 = 35e"°'°15*001 d '!f ~ (EQ 2.3) For Ro.oi > lOOmm/h, use the value of 100 mm/h in place of Ro.oi4. An estimate of the path attenuation exceeded for 0.01 % of the time is given by: *o.oi = Ygd'/f (EQ 2.4) 5. Attenuation exceeded for other percentages of time p in the range 0.001% to 1% may be deduced from the following power law: p = _ 0.12p~<0546+0043log|0,’) ‘o.oi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 2.5) 14 PREDICTION OF RELIABILITY 2.3.2 Crane Prediction Model Crane has extensively studied the effect of rain attenuation in the millimeter range. His step-by-step application of the attenuation prediction model [Crane80] is as fo llo w s: 1. Determine the rain rate distribution Rp. Locate path endpoints on Crane’s map and determine the rain climate region. From the rain climate region, obtain the rain rate distribution also supplied by the Crane model. 2. Determine the specific attenuation parameters k and a for the frequency of inter est. 3. Calculate the attenuation value Ap from R and the distance d as \ = (EQ2.6) yR<*'ff i,v5 I____ _.cS - ------ - + - ----— e«*, y z yd 8 < d < 22.5 I (EQ 2.7) 0<d<5 y with the remaining coefficients computed in the following manner B = 0 .8 3 -0 .1 7 1 ^/?,,) c = 0.026 - 0.03 In (Rp) 6 = 3.8 —0.61n(Rp) B u - —+ c o y = CLu Z = 0LC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 2.8) 15 2.4 SYSTEM PERFORMANCE 2.4 SYSTEM PERFORMANCE In the evaluation of the system performance, the attenuation due to rain is included in the channel model as illustrated in Figure 2.5. Channel J Transmitting antenna sain Ji Signal in 4 ~ t Rr, - g at ~ Free space loss L FS Rain loss - »» 1 b— ( Signal \ 1------------------------------------------------------------------------- Transmitter F igure 2.5 Receiving antenna gain ► * Receiver Block diagram of a communication system. The available signal power at the receiver may be determined as follows: = P tx + g a t ~ L f s - A p + G ar (EQ 2.9) where, PRx = receive power level, in dBm P fx = transmit power, in dBm G at = gain of transmitter antenna, in dB G ar - 8a' n ° f receiver antenna, in dB Lpg = free-space path loss, in dB Ap = rain attenuation exceeded for p% of the time on a path length d, in dB The transmitting and receiving antenna gains can be computed, assuming a para bolic shape, as follows: g at = g ar = 17.88 + 201og/ + 20 log Z) (EQ2.10) The free-space loss is given by: L fs = 92.44 + 2 0 lo g / + 20Iog</ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ2.11) 16 PREDICTION O F RELIABILITY f = frequency, in GHz D = antenna diameter, in m d = distance between antennas, in km Finally, the fade margin M, which must be positive in order to meet the specifica tion o f the design, is given by: M = P Rx - T Rx (EQ 2.12) where, When M = fade margin, in dBm T rx = receive threshold for a specified bit error rate, in dBm the fade margin is negative, the signal is received under the specified bit error rate, or worst, an outage could occur. When the availability o f the signalis not met, widely due to rain loss, the following adjustment can be made to increase the fade margin: 1. 2. 3. 4. Decrease the frequency Decrease the path length between the transmitting and receiving antennas Increase the antenna size Increase the transmit power The following table summarizes the outage time per year allowed for given avail ability percentages: TABLE 2 .4 Link availability Availability (1-p) Unavailability (p) Outage per year 99% 1% 3.7 days 99.9% 0.1% 8.8 hours 99.99% 0.01% S2.6 min 99.999% 0.001% S.3 min Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 MODELING ATTENUATION AT A POINT In order to evaluate the specific attenuation of electromagnetic waves due to rain, the total cross-section (sum of the absorption cross-section and the scattering cross-section) of raindrops must be evaluated. The scattering of a plane electro magnetic wave by spherical particles of any material was treated by Mie and is known as Mie scattering theory. Later, Stratton [4] reformulated M ie’s work using a method o f expanding the scattered fields in a series o f spherical vector wave functions. When the raindrop was found to be deformed, it was represented by a spheroid, which was considered as a perturbation of a sphere. Mushiake [Mushiake56] developed the theory of scattering of a plane electromagnetic wave by perfectly conducting spheroids. Oguchi [Oguchi60], [Oguchi64] and [Oguchi73] extended the work of Mushiake to the case o f spheroids of any mate rial. Morrison and Cross [Morrison74] modified the perturbation calculation by using an equivolumic spherical drop with an appropriate perturbation parameter, rather than the perturbation about an inscribed spherical drop. This modification improved the results considerably and gave results closer to the least squares fitting method that they were analyzing. At present, the most realistic and best-accepted distorted raindrop model is the Pruppacher-and-Pitter raindrop model [Pruppacher71], which is expressed as a cosine series. Li et al. [Li94], realizing the difficulty with a cosine series in the cal culations of the scattered fields, introduced a simplified expression for the Prup17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 MODELING ATTENUATION AT A POINT pacher-and-Pitter raindrop model. Using the simplified expression, they were able to derive the total cross-section [Li95a], [Li95b] for the Pruppacher-and-Pitter raindrop model using the perturbation method theory previously used for spheroi dal raindrops. In the following, a general expression for raindrops will be used to fit the three raindrop models: spherical, spheroidal, and PP; thus, the same equa tions can be reused in calculating the total cross-section for the three raindrops by using the proper param eter values. Oguchi studied the Pruppacher-and-Pitter raindrops using the point-matching tech nique and least squares fitting that solve the boundary-value problem [Oguchi77]. The cosine series approximation to raindrops was used in his study. In other work, Oguchi presented a review [Oguchi81] of different techniques used in the determ i nation of the scattering properties and references to papers written concerning the methods. He has also contributed [Oguchi83] to the propagation and scattering of electromagnetic wave in different hydrometeors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 3.1 DERIVATION OF THE TOTAL CROSS-SECTION Consider an electromagnetic wave passing through a raindrop as shown in Figure 3.1. Assume the raindrop shape is represented by the general simplified expression considered by Li et al. [Li94]: r = a ( l - v 1) [ / o( 0 ) + T^ - / I( 0 ) ] O < 0 £ jc (EQ3.1) 0 < $ < 2 jc where /o<e, = 1 j) (EQ 3.2) /l( e ) = [, + ^ w H (Q ) ( e - ! ] si„*e represents the step function, 6 the incident angle, a the horizontal radius of the raindrop, and Vj and v2 the upper and lower vertical deformation, respectively. HI ►y X ►y X Case I F igure 3.1 Case II Geometry of plane waves scattered by a distorted raindrop The electromagnetic fields E and H inside and outside the raindrop are governed by M axwell’s equations expressed as follows: V x V x E ( r ) - k 2E ( r ) = 0 (EQ 3.3) V x V x / f ( r ) - * 2t f ( r ) = 0 (EQ 3.4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 where MODELING ATTENUATION AT A POINT k2 = to|i0(a )e -ia ). (Iq is the free space permeability, e and a are the perm it tivity and the conductivity o f the raindrop, respectively. The permittivity and con ductivity for water were derived from Ray [Ray72]. A time dependence is assumed throughout the derivations. e iwl The boundary conditions associated with the raindrop surface can be written as: n x E in = n x E oul (EQ 3.5) h x H in = n x H ou, (EQ 3.6) where and H in ou, represent the EM fields inside and outside the raindrop scatterers, respectively, and n denotes an outward unit vector normal to the rain drop surface. The vertical and horizontal polarization is denoted by the subscripts I and II, respectively. The two pairs o f incident EM waves are expressed by: ^ ^ _ (rfcoCrsinO + zcosO)] E' i = E ^ c o s B x — s m 0 z ) e Hi = * < ,£ /- ojpo ye [<iO(-tS in 0 + 'CO50)l (EQ 3.7) (EQ 3.8) and E n = Eu ye ... H i, = (EQ 3.9) k0E „ , copo . (c o s 0 x -s in 6 z )e | i t 0(xsin8 + ;c o s e ) | _ (EQ3.10) The scattered fields due to the raindrop scatterers illuminated by the plane waves shown in (EQ. 3.7-3.10) can be obtained by solving the Maxwell equations in (EQ. 3.3-3.4). The total cross section is then calculated from the scattered EM fields due to the raindrop. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 3.1.1 Vector-eigenfunction expansion Using a technique similar to that employed by Oguchi [Oguchi60] and Stratton [4], the incident, scattered, and transmitted EM fields are expanded in term s of the fol lowing vector wave eigenfunctions: M^ k) = •nzA kr) e ) |s,n( ^ ) l e * ^ k r p z (cos [cos J (EQ3.11) 3P*(cos0) fcos - z „ ( k r ) — n-K . ou n(n = -i + [sm ]- J l)z.(k r) ^ _ Jp -(c o s 0)jsin*m<*>)}^ 3['-2B(Arr)]a/,J,(cos0)fcos/ (m<t>ne 30 [sin j sin (EQ 3.12) +- k ? T r m where zn(k r ) fcos d [rz n(kr)] I- represents the spherical Bessel functions of n-order, and />*(cos0) is the associated Legendre function. Considering the two polarizations of the incident waves illustrated in Figure 3.1, the incident electromagnetic fields are expanded in the following forms: (EQ 3.13) 'k Hl, = i|r0I I n “t ^ ^ (EQ 3.14) + b ^ A f^ A k o ) n = 1m = 0 where the spherical Bessel functions of the first kind, i.e., z n{kQr ) used in the above vector wave functions, the orthogonal properties o f Wf,mn (*0) and = j„ (k Qr ) , M r mn (*0) are and are considered, and the coefficients o f the expanded incident EM fields, b' are given by: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 MODELING ATTENUATION AT A POINT m P” (cosQ)^ _ ( 2 - 5 w)(2n + l ) ( / , - w )! .. sin 6 n(n + 1 ) (n + m)!* (EQ 3.15) dP” ( cos 9) '// ae d P ” (cosQ )i = ae (2 - 8° )(2„ n(/i + 1) (n + /n)! m P*(cos0) sin0 (EQ 3.16) -// where 5" is the Kronecker symbol: m =n (EQ 3.17) m *n The method of vector wave eigenfunction expansion is also applied to obtain the scattered EM fields and E\ h and the transmitted EM fields H s. U E\ h and H ‘. . ti Because the EM fields outside the raindrop must satisfy the Sommerfeld radiation conditions, the scattered EM fields can be expanded as follows, in terms of the spherical Hankel functions of the second kind //*2> : a : mnM £ \ ( k 0) + b'CmN g A k Q) = £ o .y y a:,mnN $ A k 0) + b\u m M $ n ( k Q) (EQ 3.18) (EQ 3.19) n= 1m=0 while the transmitted field inside the raindrop can be expanded into the following forms: a'smnM timA k ) + b ^ nN rinn( k ) (EQ 3.20) n = Im = 0 Hj, =— oiUo y y n =IfflsO (k) + b'SmnM > n (k) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ3.21) 3.1 DERIVATION OF THE TOTAL CROSS-SECTION where a * n, , a' , and boundary conditions, and k b‘ 23 are the coefficients to be determined from the denotes the propagation constant determined by the raindrop medium. The boundary conditions satisfied by the EM fields indicate the continuity o f the tangential components o f the total EM fields across the surface of the raindrop. The total EM fields outside the raindrop can be considered as the sum of the inci dent EM fields and the scattered fields: E out — E Hn E* (EQ 3.22) = W + Hs while those inside the raindrop are the transmitted fields: E in = E ‘ (EQ 3.23) = H‘ Substituting the expanded electromagnetic fields into Eq. 3.5-3.6, the expansion coefficients are solved by matching the boundary conditions. The boundary condi tions can therefore be rewritten as: E\ + E*. = Ef. //'+ //; = h\ h* //♦ //♦ (EQ 3.24) h* //♦ //♦ 1 d f ,( 0 ) 1( 6 ) = e! dB 7/« +Z (6 ) dB (EQ3.25) H' iie + H S,. if8 I d f t (Q) / KB) dB H‘ + H S, fir hr 1 d f t(B) = h! 7/9 + J 1(9, dB li> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 MODELING ATTENUATION AT A POINT 3.1.2 Perturbation method solution When the deformation of the raindrop from a sphere is small, variation of the fields from those for a sphere can be evaluated by a first-order perturbation theory. Sup posing that the parameter is small, the scattering and transmission coeffi cients of the EM fields can be considered approximately the sum of the first two terms of the expanded series with respect to the parameter ~Q%mn S "<j j, 0 “ f/tlfl a ?rm n g mn ■<35. 1 ■ (BIB g mn + where a p -« g mn b f*r/n n Bi-° 5 mn b *'emn B'.o and V' 1 - v , B*'1 f,mn ()/nn (p = (EQ 3.26) fl'1 S mn Bp i : s or t, q = 0 or 1) are the expansion coefficients to be determined. The required coefficients can be derived directly, but approximately, from this series of equations under the zeroth-order and the first-order approxima tions. 3.1.2.1 The Zeroth-Order Approximation Under the zeroth-order approximation, the solution of the unknown coefficients of the scattered and transmitted EM fields can be obtained. The four coefficient equa tions are obtained [Li95a] and given in the matrix forms as follows: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 25 '< * > P n l p> - Vn\ n = I + A £2,A(«(p) © J3 A= 1----------- 1 ©£ =o I * U 3 '«<p >- p) + P'(P) (EQ 3.27) A= 1 =0 ^ P n ] -i *,° „J *U*oA,(P> - *&tf.(CP> * B & * o * S ° ( P ) n = 1 where the parameters 3„(x) and p„ (x) are defined as follows: 3 .< * » - ^ h ? \ k ’r) * ' « r) 3' ° P = ^FT- (EQ 3.28) - 3 [(^ )y .(y r)] j ~ itva(ifcV) = a[(rr)AW(yr)] } k 'r d ( k 'r ) ( k ' r) "( with the symbol k’ denoting k and k0 . The inter-parameter operators <t>,°n and VF,° satisfy: <*>°,/r(0) = j'r(e)/»(e» [ S X ^ 5 ) +( ^ I ^ :]lsi"e‘i9 ° n r m (EQ 3.29) - 0 for / 0(6) and /,(6 ) defined in (EQ. 3.2); and other parameters p, £ are defined as P = *oa O ~ vi) 5= F Ko = M l -v ,) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 3.30) 26 MODELING ATTENUATION AT A POINT 3.1.2.2 The First-Order Approximation Using a method similar to the zeroth-order approximation, four equations are obtained [Li95a] for first-order approximate solutions. The results are given as fol lows in matrix form: W n. m L^'n. a L „ p [y „ ( P ) ]'-^ '0 Cp[;„(?p)l' + A z ° p [ h ? K p ) V * U p 3 ' - (p )- a£ i 5 p 3 v c p ) + « y i 1p p '( p ) nr/„n = I A'n. * U 3 - (P ) " BSmn3 '>(^ P) + *£«« * « (P) = 0 m' fl£--3,-<?P )- fiS-»p,(P) (EQ3.31) ro f a?n..P23'- ( P ) - Ag«,(?P)23,r-(?P) + ^ - « P 2^ '( P ) rn - ^ U p2[^ (P)1' - fls l ^ P)2^ ^ P)]' + flSm °np2^ 2)(P)l' ro f n = I - A£ j« & P ) + ^ ^ ° „ ^ 2)(P) =0 . f In . .^PnJ r e A y.[?PA (5P)]'-A *|.[pJi«)(p)]' B ^ k j n^ p ) - B ^ k 0h ^ K p ) *a. where the parameters 3„(p) and p ”(p) are defined as 3 „ (P ) Ipy'n(p)]] ■f p J (2). :«p . - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 3.32) 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 27 and the parameter operators are given by: = j F (e,/ l ( e ) [ g ] ( ^ r ) + ( ^ : ) ( ^ : ] | Sin9d8 0 0 r,'„F(9) = J f < 9 F ^ [ / « - (EO 3.33) l < g |) ] s i „ 9 , » 0 a{BF(9) = j F ( 0 ) ^ g ^ [ / ( / + l ) / > T ^ ] s m 0 d B o The associated Legendre functions have the orthogonality property in the variable range [-1,1] or [0,tc]. The solution o f the first-order scattered EM fields by the axisymmetric raindrops, i.e., spherical or spheroidal, can be easily derived. In nonaxisymmetric raindrops, i.e., Pruppacher-and-Pitter, the upper part, corresponding to 0 < 0 < ^ , and the lower part, corresponding to ^ < 0 < n , are no longer the same. Thus, the integration range changes to [-1,0] instead of [-1,1], and the associated Legendre functions are no longer orthogonal. Therefore, the solutions of the inte grals in (EQ. 3.33) are no longer available in the closed forms, so that the coeffi cients of the first-order expanded EM fields must be solved for numerically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 MODELING ATTENUATION AT A POINT 3.1.3 Expansion Coefficients To solve for the first-oder approximate coefficients of the EM fields, the zerothorder and the first-order coefficients of the expanded EM fields are expressed [Li95a] in terms o f the axisymmetric and nonaxisymmetric contribution of the raindrop as follows: A s- ° $mn A * '° ° jfmn 4 5 .0 1 f/mn A '- 0 Jfm/i A 1- 00 %mn A t . 01 jtmn B l-° £mn B s. 00 £mn B'.o B ‘.o° S'"" _ - fU T- V', ]J f,mn B'.oi £ mn _ - A s , 10 A °m n A?; 1 ffmn A t , 10 A »mn A !%mn ;1 Vl B £m s-1n 1“ B >. 1 Vl B l -10 omn B t.\o Smn £mn A?" + fV2-VO U -v , A %mn £" J B l -11 c„ mn B t.ll £m n _ . Assuming that (EQ 3.34) B*. ° i and y are very small, the high-order terms can be neglected. Only the scattered EM fields will be used in the calculation of the total cross sec tion, so that the first-order coefficients of the transmitted EM fields would not be needed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 3.1.3.1 29 Axlsymmetry contribution By substituting (EQ. 3.34) into (EQ. 3.27), the expansion coefficients o f the EM fields under the zeroth-order approximation can be derived as follows: „ tnn y„(^p)[py„(p)i, - y „ ( p ) [^ p y „ (? p ) r ^ = --------------------- rr.----------------— ----------------------------- a ' j n ( & n p h ? \ p ) r - h n \ p ) i & j aw ; (EQ 3.36) = SZ ■a ‘„ gmn n fm n r , ( 2), ,v yn( p ) [ p ^ ( P ) i>- ^ ( P ) [ p ; n(P )r o o - a , yn( C p ) [ p ^ 2)( p ) r - A l 2,(p)[Cpy„ccp)i' B s.o o _ - ° ^ ( p ) ^ p y "( *>p ) * - U ( ^ P ) tP l.|P ) l ^ )(p)[;py„(^p)], - ? y „ (;p ) tp ^ 2)(p )r _ g ^ 2>(p)fpy«cp>r- cy,cp)tp*l2>( p )r Aj,2,(p )tc p y « (5p ) i' - ? 2y«(5p)[p*«2>c p ) r = r gmn " . b, bl . a ,. ?mfl _ 5 * . b'e (EQ 3.38) = r * .fcl- Smn n (EQ 3.39) 5 The above expansion coefficients of the EM fields represent those of the Mie scat tered EM fields corresponding to the spherical raindrop scatterers. Applying a method similar to that of Oguchi [Oguchi60], the first term of (EQ. 3.35) is expressed as follows: AS. rt + 2 1 10 _ y„(£p) A >’mn ~ + W „ A K m/ l = m ax{m , n —2, I ) (EQ 3.40) n +2 -[?py«c?p)i' In A / = mux{m, n —2. !) + x f V nA)b Sm 1 n+2 B l - 10 = J,mn «(« + s LCpy„(^p>r ?p I — max(m, n - 2, i ) (EQ 3.41) -?pyn(CP) X (X/«W„a + X } n U ) b ‘m l ± X } ' V l Aai1m l / = m ax (m . n - 2. I ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 MODELING ATTENUATION AT A POINT where the inter-parameters r \ } - \ x } - 2, X/-2, and A,J-2 are defined as: = p2^ L ^ _ 7-|.(^p)2^ £ > i + 5f p2‘/[/,'2)(P>1 dp dp P) zrt^pyKCp)] i zrtpy<(p)]i fpy'i(p)] j Z;1 = Pp -T ft (EQ 3.42) (EQ 3.43) '[p */2,(p )]‘ p f Cp /KSp H i + S,*p Sp J (EQ 3.44) (EQ 3.45) (EO 3.46) d[y,(p)] = Ai = p dp d (£ p ) ' K (EQ 3.47) dp j n{ Z , P ) [ p h ? \ p ) Y - h * \ p ) & p j n& p ) Y (EQ 3.48) A2 = A^2)(p)[Cpy„(CP)]' - C2y„(?p)[p*^2)(p)]' (EQ 3.49) a n d th e parameters <PfnA, vF 1nA, T,ln<t, an d , ft|nA are represented by: fm 2+ /i~(/i — m + 1)(/i + m + 1) -----------------— ------------------------- — -— L (2 n + l)(2n + 3) . ( n + l ) 2(/i - m ) ((njf n + m) m)~I . (2n — 1)(2rt *U = + 1) J (EQ 3.50) n(n + 3 ) (n + m + l)(n + m + 2 ) , (2n + 3)(2/i + 5) B’ (/i —2)(/i + 1)(/t - m)(n - m - 1). (2/i - 3)(2/« - 1) 0, vui ‘ InA _ - I = n 2m ( n - m ) 2n-l 2m(n + m + 1) 2/t + 3 I = n+2 / = n —2 otherwise I = n- 1 / = n+ 1 0, otherwise Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 3.51) 3.1 DERIVATION OF THE TOTAL CROSS-SECTION r* 1 In* ~ 2 m ( n - 1)n(n —m) 2/i - I 2m(n + 1)(n + 2)(/i + m + 1) 2n + 3 31 I = n-l (EQ 3.52) I - n+ 1 0. otherwise 2n(n + 1)rn(n —m + 1)(« + m + 1) 2/i + I L 2 /j +3 I = n (n + l)(n —m)(n + m)~ 2n —1 (EQ 3.53) 2n(n + 2)(« + 3)(/i + m + l)(n + m + 2) (2/i + 3)(2/i + 5) I = n +2 2(/»-2)(/i2- I )(/i - m - I ) ( / i - m ) , /■ •>„ - 3)(2/i - 1) n’ (2n 0, I = n-2 otherwise with 2n+l (n + m)! (n —m)! (EQ 3.54) and the subscript A denoting axisymmetry of the raindrops. 3.1.3.2 N onaxisym m etry contribution When the raindrops are deformed axisymmetrically, the zeroth-order approximate coefficients correspond exactly to the Mie-scattering coefficients. There is an addi tional contribution, however, due to such a nonaxisymmetrical distortion. Under the sphere-based approximation, the upper and lower parts o f the sphere have dif ferent radii, a (l-v ,) and a(l-v 2), respectively. Therefore, the function U(p) such as j n(p), /*„2)(p), 3'„(p), p '„(p ), etc. can be expanded due to the nonaxisymmetry into , where H(0) denotes the step function. Substituting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 MODELING ATTENUATION AT A POINT (EQ. 3.34) into (EQ. 3.27) and applying the above expansion, the additional scat tering coefficients can be obtained [Li95a] and written as follows: - ^.v.OI l __ n(n + 1)/„A,j /«(£P) ISp/.CSp)]' / N X / = maxim. 1) \ / + n se odd / \ t f ® l n N a omi X ) / = maxim, 1) \ 1+ n = odd ( • \ / = maxim , I ) ^ I + n = even J (EQ 3.55) 04 h + 5*5-00 2 ?mn 2 J / = 1) / ♦ n s even 40 B s.oi f. mn [?py-(? p)i' * n(n + l)/„A2 ?p / JL X ♦/ s m u x (m . 1) /♦/ i = even 40 1+ M 40 / = m a xim . I) ^ + X J / = m axim . 1) / + /t = even n/4>?n/v^ m/ 2 = m a x (m , I ) / + n s odd , . Z /1<t , 2.Ar6 c m / ^ ' 2 Smn l + n s odd (EQ 3.56) The parameters <!>{{,* and Tp,* are defined by: n(n + !)(/> + m)l In + 1 (n —m)l' * ? aN = l(l+ D P f ( 0) n(n + I ) -1(1+ 1)' 1 n = I dP”( 0) dx (EQ 3.57) dP?(0) n(n+I) P"( 0) rn(n+ ! ) - / ( / + 1) dx •Fg,* = -mP?( 0)Pr(0) (EQ 3.58) In the above derivation, the following inter-parameters have been used: 2 / + I (l-m )l =0 (EQ 3.59) (EQ 3.60) The coefficients of the first-order scattered electromagnetic fields due to the nonax isymmetry of the raindrop scatterers are expressed as [Li95a]: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 DERIVATION OF THE TOTAL CROSS-SECTION 41. II °m n = 33 I AS, 1 0 _____________! ___________ 2 “mn n(n + I)/„A j OB y„(Cp) X sm l ma.r(m, 1) / + n = odd (EQ 3.61) I = max(m, I ) I s s even -Kpyn(Cp)]' / = / = m a i(m , I ) ^ 1) / + n = even / ♦ n = odd U = ig.t. 10______^___ . I,mn 2 £»»» n(« + I )/„A ; [Cpy,(?p)]' * / = /nax(m . 1) I + n = odd X o i/^ iU + w u / = mcix(m, l)L l + n - even (x/ &lnN + X l ^ l a N ^ e mi -?py„(?p) I = m ax(m , 1)1u l + n = odd gm / /s I) I + n = e v en (EQ 3.62) The symbols <t»/nA/ and T,1^ (the subscript N denotes nonaxisymmetry) are repre sented by: 1 **»]IniV = [ n l ( n - m + 1 ) 0 - m + 1)/,+ (2 n + I )( 2 /+ 1) -/(/i + l)(n + m ) 0 ~ m + I )//+ i.„_ i m Il.n + ,.„ + 1 (EQ 3.63) - « ( / + l)(/ + m ) ( n - m + l)/ ;_i. n+i +('« + 1 ) ( / + l)(/i + / n ) ( / + / n ) / , T,n/V “ L 2/ + I //+ ,-n , « ( n - m + 1), + 2n + 1 T,U = ^ £ r T T 1 Un - m+ ( / + l ) ( I + m) , 27T1 (EQ 3.64) ( n + l ) (/ i + m ) , "| 2n + 1 ''•"-•J (EQ 3.65) I “ '■» ' ^ T w 7 T T ) ["<" - m + 1)(' - m +1)/' - 1” ' (EQ 3.66) - ( n + 1)(n + m ) ( / - m + I ) / , + i . n - i + n ( /i- m + l)(/ + m )/,_, n + , - ( n + I)(/i+ /n ) ( /+ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ] 34 MODELING ATTENUATION AT A POINT and the parameter , solved by Li et al. [Li95a] for the range [-1,0] is given by: p? (x)p nx)dx 0 I* n 1 (n + m)\ 2/i + 1 (/i —m)\ _______I n +l even I —n dPp( 0) r dP?(0 )-| (EQ 3.67) n +l odd (*-/)(* 3.1.4 Total Cross Section The total cross section is defined [Oguchi60] as the ratio o f the sum of absorbed and scattered energies to the mean energy flow of the incident waves. Using a / method similar to that o f Oguchi’s [Oguchi60], the total cross section " q t under the first-order perturbation approximation is as follows: »Q r . (EQ 3.68) where, (EQ 3.69) n \ rMi-01A ” Q0 t' 471™. !,Q't ° i 11£ nl Tl 1/ " Ko m P ”( cos0) A s. 10 sin6 Qmn n dP”( cos 6) n = Im=0 4 5. 11 de %mn \ J rBs-0l\ f,mn B Z,mn i 10 B sU hmn V / ' 9P*(cos9) de mP”( cos0) sin0 ► 1 n —1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 3.70) 3.2 RAINDROP CHARACTERISTICS 35 3.2 RAINDROP CHARACTERISTICS Assumptions made on the raindrop have a great impact on the resulting specific attenuation. First, the shape of the raindrop affects the calculations of the total cross-section depending on the parameters given to the raindrop shape equation given in (EQ. 3.1) and (EQ. 3.2). The parameters a, v t, and v2 can be fitted for a spherical, spheroidal, or Pruppacher-and-Pitter raindrop shape. Second, the rain drop size distribution will affect the specific attenuation when it is integrated with the total cross-section. 3.2.1 Raindrop shape There are three different raindrop shapes that will be considered. The spherical and spheroidal [Oguchi77] shapes are axisymmetrical about the horizontal axis, while the Pruppacher-and-Pitter [Oguchi77] shape is distorted. The Pruppacher-and-Pit ter model is more realistic than the first two mentioned. The raindrop starts almost spherical, then becomes spheroidal and axisymmetric for midsize raindrops, and finally becomes distorted and nonaxisymmetric. This leads to the use of this model in heavy rain climates due to the increased presence of larger raindrops. Figure 3.2 shows the raindrop as seen by the three models. The raindrop effective radius is taken to be 3.25mm, which is considered a large drop. The spheroidal and the Pruppacher-and-Pitter consider an equivolumic raindrop, i.e., they have the same volume as the spherical case. This turned out to be an important issue when deriv ing the total cross sections. Spherical Figure 3.2 Spheroidal The raindrop models with equivolumetric drop radius 3.25mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 3.2.1.1 MODELING ATTENUATION AT A POINT Spherical For the spherical shape, the parameters to (EQ. 3.1) are straightforward: a = aQ (EQ 3.71) v, = v2 = 0 where ag is the equivolumetric drop radius. W hen these values are substituted into (EQ. 3.68), the resulting total cross-section is identical to the Mie scattering. 3.2.1.2 Spheroidal The spheroidal shape has been extensively used in the derivation of total cross-sec tion, notably [Oguchi60] and [Morrison74]. It is approximated with a relation between axial ratio dependent on the equivolumetric drop radius ag and is approxi mated by l —0. l a0, when ag is expressed in mm. This value was used by [Morrison74] and later by [Oguchi83]. Equating both volumes of sphere and spheroid for the above constraint, the parameters in (EQ. 3.68) are derived: mm a (EQ 3.72) ( I —0 . 1 a 0 ) ,/3 v, = v2 = O .Ia0 The axisymmetry is maintained by equating v t and v2, the upper and lower defor mation coefficients, respectively. Substituting the parameters into (EQ. 3.1) results in a total cross-section with only and Q l° components, which corresponds to the axisymmetric case. The raindrop shape (EQ. 3.1) along with the parameters in (EQ. 3.72) results in less error when compared to the spheroidal model used by Oguchi [Oguchi60] and Morrison [Morrison74]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 RAINDROP CHARACTERISTICS 37 The following table summarizes the values for different equivolumetric drop radius. TABLE 3.1 Calculated Coefficients for different raindrop sizes (Spheroidal raindrops) 3.2.1.3 r« > II > ag (m m ) a (mm) 0.25 0.252119 0.025 0.50 0.508622 0.050 0.75 0.769746 0.075 1.00 1.035744 0.100 1.25 1.306895 0.125 1.50 1.583501 0.150 1.75 1.865893 0.175 2.00 2.154435 0.200 2.25 2.449525 0.225 2.50 2.751606 0.250 2.75 3.061165 0.275 3.00 3.378744 0.300 3.25 3.704948 0.325 Pruppacher-and-Pitter Pruppacher and Pitter [Pruppacher71] established an equation to describe the shape of w ater drops falling at their terminal velocity in terms of the balance o f the internal and external pressure at the surfaces o f the drops. To simplify the expres sion, they introduced a cosine series, , r'(G') = a, «o 1+ cncos(«0') (EQ 3.73) fl =0 to replace the nonlinear equation and evaluated the first ten coefficients cn (n = 0, 1,2, ..., 9) o f the series for different sphere-based radii ao- This model is now well accepted and used by researchers for the calculation of microwave attenuation by rain [Oguchi77], [Oguchi81]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 MODELING ATTENUATION AT A POINT TABLE 3.2 Computed deformation coefficients o f raindrops (source [Oguchi77]> *0 (mm) co c2 Cj «4 «5 0.25 -0.386680xl0"3 -0.115996x10*2 -0.151420x10"3 0.629391x10*® 0.406408x1O'5 0.50 -0.265873xI0'2 -0.795387x1 O'2 -0.162839xl0*2 -0.104182x1O'3 0.844674x10*4 0.75 -0.693210x1O'2 -0.207063x10-' -0.474328xl0'2 -0.428598x10"3 0.267023xl0*3 1.00 -0.115073x10-' -0.342597x10-' -0.885078x10"2 -0.138068x10"2 0.593452x1O'3 1.25 -0.187815x10-' -0.559112x10*' -0.146408x10* -0.228007x10"2 0.964343x1 O'3 1.50 -0.267192x10'* -0.795319x10"' -0.211670x10-' -0.329143xI0*2 0.136429xI0*2 1.75 -0.350003x10-' -0.104166 -0.282497x10* -0.438808xl0*2 0.177493x1O'2 2.00 -0.429905x10-' -0.127923 -0.354365x10-' -0.550270x10*2 0.216212xl0'2 2.25 -0.504202x10-' -0.15000 -0.425377x10-' -0.661037x1O'2 0.251072xl0’2 2.50 -0.573820x10-' -0.170669 -0.496486x10*' -0.773188x1O'2 0.282382xl0'2 2.75 -0.639783x10-' -0.190235 -0.568738x10"' -0.889090xl0*2 0.3l0464xl0'2 3.00 -0.703284x10*' -0.209049 -0.643374x10*' -0.101157x10*' 0.335670x10"2 3.25 -0.766642x10-' -0.227796 -0.722772x10*' -0.114546x10-' 0.358806xl0'2 *0 (mm) C6 c7 c* 0.25 -0.852069x1 O'6 0.125856x10"^ -0.853347x 10‘7 0.36571 lxlO '7 0.50 -0.152054x1 O'4 0.353713x1 O'5 -0.412700x10*5 0.249148x1 O'5 0.75 -0.398766x10"4 0.107211x10“* -0.167668x10“* 0.877456x1 O*5 1.00 0.221543x1 O'3 0.545847x10“* -0.107608x1O’3 -0.573183x10“* 1.25 0.362000x1 O'3 0.891036x10“* -O.I75351xlO'3 -0.934761x10“* 1.50 0.515668x1 O'3 0.126781xl0"3 -0.248960x 10"3 -0.132844x10"3 1.75 0.676448x1 O'3 0.166094x1 O'3 -0.325320x10‘3 -0.173789x1O*3 2.00 0.832096x1O'3 0.204025x10"3 -0.398414xl0"3 -0.213118xlO"3 2.25 0.977315x10° 0.239278x10"3 -0.465649x10 3 -0.249457x1O'3 2.50 0.111374x10*2 0.272273x1 O'3 -0.527797x10"3 -0.283225x10"3 2.75 0.124315x1 O'2 0.303469x10*3 -0.585708x10‘3 -0.314883xl0"3 3.00 0.136761x1 O'2 0.333400xl0-3 -0.640364x10"3 -0.344966x1O*3 3.25 0.149126x10"2 0.363121xl0'3 -0.693735x1O*3 -0.374542x1O'3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 RAINDROP CHARACTERISTICS 39 In order to analytically derive the scattered held by the rain, a simple mathematical formula given by (EQ. 3.1) is assumed. Assuming 0'max is the angle at which the horizontal distance r'(Q 'max) s in 6 'mux in (EQ. 3.73) reaches the maximum, the translated vertical distance is equal to: (EQ 3.74) Figure 3.3 shows the two coordinate systems. Under the new coordinate, the PP raindrop model is fitted to the simplified model in (EQ. 3.1). The maximum and minimum vertical distances from the horizontal plane, on which the points with maximum horizontal distances are located, should remain unchanged in the two coordinate systems. 8=K 8= 0 8=0 8 = x F ig u re 3.3 The two spherical coordinates system superimposed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 MODELING ATTENUATION AT A POINT The parameters to (EQ. 3.1) are finally given by: (EQ 3.75) n ss 0 V| 1 sin0_ - C O S0. (EQ 3.76) cos0. (EQ 3.77) 1 + ^ c ncos(nemux) n=0 V, = 1 - -7 sin0_ 1 + X CnCOs(/,0'"“ ) n=0 Numerically, the coefficients for different raindrop sizes are com puted and shown in Table 3.3. TABLE 3.3 Calculated coefficients for different raindrop sizes (P-P raindrops) ao (mm) a (nun) 0.25 0.250194 vt 0.001649 v2 0.002991 0.50 0.502608 0.009103 0.022635 0.75 0.760110 0.022384 0.059717 1.00 1.021407 0.033650 0.100362 1.25 1.294050 0.055970 0.160456 1.50 1.575835 0.080877 0.223404 1.75 1.866817 0.106975 0.286660 2.00 2.165145 0.132021 0.345647 2.25 2.469269 0.154547 0.399411 2.50 2.778873 0.174757 0.449140 2.75 3.093901 0.193242 0.495616 3.00 3.414610 0.210211 0.540041 3.25 3.742279 0.225862 0.584559 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 RAINDROP CHARACTERISTICS 41 3.2.2 Drop Size Distribution The drop size distribution (dsd) reflects the distribution of the drop sizes as a func tion o f the rain rate. Many raindrop size distributions have been reported. The ear liest paper on the size o f raindrops was by Laws and Parsons [Laws43], where the distribution is tabulated. This distribution was used by Oguchi [Oguchi60], [Oguchi64], as well as Olsen [OIsen78] and many others. Later, the exponential drop size distribution was empirically proposed by Marshall and Palmer [Marshall48]. The latter distribution is well accepted in the meteorological domain and in radar analysis. It is also used in the derivation o f attenuation such as found in Olsen [01sen78] or Li, et al. [Li95b]. Joss and Waldvogel [Joss67] proposed another model, another exponential form, by dividing the rain into three types: drizzle (J-D), widespread (J-W), and thunderstorm (J-T). Recently, Ulbrich and Atlas [Atlas74] studied the gamma distribution as a raindrop size distribution. Finally, the Weibull drop size distribution was proposed by Sekine and Lind [Sekine82] and has been o f considerable interest in its application to rain attenua tion. Only the M-P and Weibull dsd’s will be considered later in the derivation o f spe cific attenuation. The M -P dsd is chosen because it is widely accepted, although is found to overestimate the number of small drops [01sen78]. The Weibull dsd is chosen because it was found to yield the least rms error with experimental data as opposed to the other mentioned distributions [Jiang97]. There have been numerous other interesting papers written about the Weibull dsd and its application to radio wave propagation, such as [Sekine88], [Jiang96a], and [Jiang96b]. The choice of the drop size distribution is crucial and affects the resultant specific attenuation. Limiting the distributions to these two is mostly for simplicity because they will be combined with the total cross section for both spheroidal and P-P rain drop shapes. At any specified frequency, this results in four different combinations and sets o f values for the specific attenuation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 MODELING ATTENUATION AT A POINT 3.2.2.1 Marshall-Palmer dsd The Marshall-Palmer drop size distribution [Marshall48] is expressed as: N ( a 0) = N 0e~2Aa° where a0 m ^ m r1 (EQ3.78) is the equivolumetric drop radius expressed in mm and the other parame ters are given by: N = 8000 r W A = 4.1 R~°2' mm'1 (EQ 3.79) The precipitation rate is R and is in mm/h. The M-P radar reflectivity Z-R relation is given by [2]: Z = 200/? 16 3.2.2.2 (EQ 3.80) Weibull dsd The Weibull drop size distribution [Sekine82] is expressed as: (EQ 3.81) W<«o> = where the precipitation rate R is in mm/h and N 0 = 1000 m-3 r) = 0.95/?° 14 (EQ 3.82) a = 0.26/?°42 mm The Weibull reflectivity Z-R relation is given by [Jiang96b]: Z = 285 /?148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 3.83) 43 3.3 THEORETICAL DERIVATION 3 .3T H E O R E T IC A LD E R IV A T IO N When the total cross section and the drop size distribution are determined as a function o f the drop size, the specific attenuation can be computed. First, the atten uation in decibels per meter when a single drop per cubic m eter exists is given by [Oguchi60]: dB/m drop/m3 (EQ 3.84) considering a drop of radius a Q. The specific attenuation, therefore, can be computed by integrating (EQ. 3.84) with the drop size distribution over all drop sizes. By choosing the appropriate total cross-section, i.e., case I or case II, vertical and horizontal specific attenua tions are obtained. The rain attenuation in dB per kilometer is given by [Oguchi60]: QT(a0)N(a0)da0 y = 8.6859 x dB/km (EQ 3.85) o where Q T(a 0 ) represents the total cross-section of the raindrop, expressed in m2; N( a0) , the rain drop-size distribution, expressed in m ' W 1; and a0 in mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 MODELING POINT-TOPATH ATTENUATION Given the rain attenuation o f a microwave link at a point, the calculation of the path attenuation is not straightforward due to the inhomogeneous characteristics of rain in the horizontal. An effective path length, which is affected by the rain pres ence, should be considered in that case. Due to the relation between the attenuation and the rain rate, the effective path length is simpler to estimate with the precipita tion rate as a parameter. The experimental data for path attenuations depends on the rain rate and the operating frequency, which limits its use in estimating the effective path length. Lin [Lin77] has shown that satisfactory agreement was obtained between measured attenuation values and those calculated from measured line rates, by using the relationship A = k R£ . The line rate, RL , is obtained from near-instantaneous rl = rain rates along the path L of the radio link, i.e., ( I / L ) j R d l . Harden et al. [Harden77] suggested the use of rain distributions with integration time of 10s for the near-instantaneous values and gave ratios to transfer other integration time values to the suggested ones. In their following paper [Harden78], they used readily available rainfall data and compared experi mental observations of path attenuations from 10-100GHz with the estimated val ues using L in’s method [Lin77], and concluded that the method is a practical procedure. Crane [Crane80], from the study of numerous data of rain gauge net works, found a power law relationship between the line rain rate and the point rain rate. Because the specific attenuation is a nonlinear function of point rain rate, the 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 MODELING POINT-TO-PATH ATTENUATION empirical power law relationship was numerically differentiated to obtain the instantaneous rain profile model, which is then expressed with piecewise exponen tial functions. Finally, the path attenuation is calculated from A = J k Ra{l)dl, and the result is shown in (EQ. 2.6). Crane [Crane82] presented a new model for the prediction of single path attenua tion statistics which calculates the occurrence probabilities of convective cells or widespread debris regions of rain along a propagation path. The path attenuation is then estimated by trial and error until the required occurrence probability is reached. The two-component model models the rain as a joint event with volume cell and debris contributions. A revised model was then suggested [Crane89] and found its utility in the study of site diversity performance. The latter version of the two-component model will be used in the study of pointto-path attenuations because of its ability to analyze performance o f different path geometries, especially those of a network. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 SPATIAL CORRELATION OF RAIN 47 4.1 SPATIAL CORRELATION OF RAIN The spatial correlation function for rain is needed in the two-component model theory. The spatial spectra obtained from the radar observations and from the rain gauge measurements provide a good starting point for the estimation o f a spatial correlation function. Available observations all have the same shape and spectra and usually show a power law increase in spectral density with decreasing wavenumber. By assuming an outer scale with a constant spectral density for scales larger than that scale, the integral may be performed, the process is stationary, and a correla tion function may be calculated. The minimum outer scale must be greater than 256 km. This value of the outer scale was used together with an idealized, seg mented power spectrum to create the correlation function estimate. The idealized spectrum had an energy input scale of 12 km and a rain input scale o f 4 km. The spectrum had a k '5/3, k"3, and k*1 power law region as provided by Crane [Crane90]. 10* ' 12km gjo- 10“* ll>*: Figure 4.1 One-dimensional idealized spatial spectrum for ln(rain rate) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 MODELING POENT-TO-PATH ATTENUATION The spatial correlation function for the logarithm o f rain rate is then obtained from the inverse Fourier transform o f the spatial power spectrum for variations in the logarithm o f the rain rate. The correlation function for rain rate can be obtained from the correlation function for the logarithm of rain rate by: ^2)I"** ------ Pr U i. I i ) = (EO«.1) e b- \ The correlation function pR(li,l2) depends on the rain climate zone through the parameter SD, the standard deviation of the natural logarithm o f rain rate. ln(Rain Rale) Rain Rale o.x 06 c % 0.4 C - 0.2 40 60 HO too 120 Distance, kxn F igure 4.2 Spatial correlation function for Infrain rate) and for rain rate (Climate D2) 4.2 TWO-COMPONENT MODEL The revised two-component path attenuation prediction model [Crane89] is an extension and refinement of an earlier work of Crane [Crane82]. The two-compo nent model separately addresses the contributions o f volume cells and of larger debris regions of lighter rain intensity surrounding the cells. Furthermore, it con siders the cell and debris contributions as statistically independent and sums the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 4.2 TWO-COMPONENT MODEL probabilities o f occurrence of the attenuations due to both components to estimate the total. Assuming independence, the joint probability of occurrence o f rain in a cell and in debris is small and may be neglected. The model for the volume cell sizes and intensities and lifetimes is exponential. The statistical model for the debris is lognormal. The two-component model for the empirical rain rate distribution function is: P ( r > R ) = P c ( r > R) + PD(r >R) —PCD(r > R) Pc( r > R ) = Pce~*/Rc A n R - l n R D\ PD( r > R ) = P dN [ - ^ — 2 ) (E0 4.2) PCD( r > R ) = Pc (r > R)PD(r > R ) = 0 where P(r>R) = probability that observed rain rate r exceeds specified rain rate R Pc( r > R ) = cumulative distribution function for volume cells PD(r >R) = cumulative distribution function for debris P cd(r ^ R ) = j° 'nt cumulative distribution function for cells and debris r, R = rain rate Pc = probability of cell Rc = average rain rate in cell N = normal distribution function PD = probability of debris Rd = median rain rate in debris (calculated from average of natural log arithm of rain rate) SD = standard deviation of natural logarithm of rain rate The parameters are fitted to the rain rate distributions found in [1], first by estimat ing the lognormal curve for low rain rates or high probability occurrences, and sec ond by fitting the exponential curve to the difference between the median rain rates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 MODELING POINT-TO-PATH ATTENUATION and the lognormal values at higher rain rates or lower probability occurrences. The results are shown in Table 4.1. TABLE 4.1 Parameters for the Two-Component model by region (source [1]). A B 0.023 14.3 10.8 0.178 1.44 C 0.033 19.8 12 0.293 1.31 D1 0.026 23.2 8.19 0.463 1.34 D2 0.031 14.3 9.27 0.475 1.48 D3 0.048 17.0 4 1.97 1.21 E 0.22 23.2 5.25 2.02 1.25 1.81 3.45 Pd 2.27 sD Pc 0.000088 Rd 0.205 1.49 F 0.0048 6.95 0.0994 G 0.028 50.5 9.82 1.82 1.20 H 0.048 35.4 7 2.47 1.49 8.26 4.2.1 Model for Volume Cell Component Rain cells often produce intense rainfall and cause severe attenuation to propagat ing signals over short time intervals. The average horizontal cell dimension Wc that spans the region with marginal reflectivity ZM (a factor of 2, i.e., 3 dB, below the peak value in the cell, Zy) is: W c = JS~C = J 3 3 km (EQ 4.3) When the Laws and Parsons drop size distribution is chosen, the reflectivity can be approximately related to rain rate by: Z = 400/?14 (EQ 4.4) The marginal rain rate RM at the cell edge is then: 1 _ zl 2 ZM = *vY-4 {R j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 4.5) 4.2 TWO-COMPONENT MODEL 51 and (EQ 4.6) Other relations between the marginal and the peak rain rate in the cell can be obtained when using different Z-R relations for M -P and Weibull, i.e., (EQ. 3.60) and (EQ. 3.63), respectively. Assuming the spatial rain-rate profile along a horizontal line through a rain cell is Gaussian, (EQ. 4.6) is used to establish the rain profile as a function o f the distance among the line. For a line through the cell center with distance x measured along the line from the cell center, the rain-rate profile R(x) becomes: R ( x ) = R ve ( x / W c r i n i K M/ R Y)/(. 1 /2 )2 = Rve - 1 .9S(.x/W c f (EQ4.7) and the profile for the specific attenuation is given by: y(x) = K[R(x)Ja = KRye~,9Sa(x/Wc)2 (EQ 4.8) y(x) = KRye~l/2(x/Sv) where Sv = Wc/-/3 .9 6 a. The horizontal path attenuation A j- along the x axis between terminals at -L/2 and +L/2 due to a volume cell with its center at (x,y) is expressed by: -1 /2 (1 / S v y e e - 1 /2 ((x -x V S » ) - 1 / 2 <v/S„)* - 1 / 2 ( x / S y ) e . j dx , , dx (EQ 4.9) - L/2 - x where. l/2(«) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ 4.10) 52 MODELING POINT-TO-PATH ATTENUATION and L is the horizontal length of the propagation path. The peak rain rate in the cell, Rv, for a specified Ap on the path is given by: / f i y ( x , y> a t ) — - l / 2 ( y / 5 v )J ^l/a j kS v J 2 k { F n [ { - L / 2 - x ) / S v] - F n [ ( L / 2 - x ) / S v ]}) (EQ4.11) Any peak rain rate larger than Ry occurring in the volume cell will result in a path attenuation that exceeds Ap when all other conditions are not changed. Therefore, the probability o f exceeding Ap is given by: Pc( a > A T) = Pc ( r 2 * v ,) | = Pce~*v/*C\X y = P c e~Kv(x-y-A^ / R c (EQ4.12) The result shown in (EQ. 4.12) applies for a volume cell at (x,y). To obtain the probability P{a > A T) , an integration of the conditional probability times the occur rence probability density for a volume cell at (x,y) is required over all space sur rounding the path. Assuming the occurrence probability density for a cell is uniform, and a normalization area o f A = 1.0km2 (source [1]), the integral is then given by: I L L Pc( a > A T) ^ d x d y =j f f Pce~Rv(x' y Ar)/Rcdxdy (EQ 4.13) ■j f3Sv/ 2 J L + 3 S v ) / 2 Pc ( a > A T) = =-\ AJ o f Pc e~Rv(x' '• Ar)/Rcdxdy J ( - L - 3 S v) / 2 This integral may be numerically solved using Legendre polynomials. 4.2.2 Model for Debris Component The rain debris region is associated with light to heavy rain intensities. The proba bility density function (pdf) for the debris component o f the mixed rain-rate pro cess is assumed to be jointly lognormal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 TWO-COMPONENT MODEL The lognormal probability density function for the rain debris is 53 f u , { r ) . Assuming that the path attenuation is approximately lognormally distributed, the approach is to estimate the parameters of the logarithm of the path attenuation pdf from the parameters of the point rain-rate pdf. The mean and variance of the logarithm of specific attenuation are obtained from the parameters of the rain-rate pdf by a linear relationship. Using the notation for the mean and standard deviations for linear processes and M, S \l , a for logarithmic processes, My = ln(K) + a I n ( f l D) = ln(ic) + a A /D Sy ~ ttSg and UY = E[ Yl = £ [* /? “ ] = kj ^ f u t W d r = KeaM° * i a *° (EQ 4.14) 0 a? = Var[ y] = - ^ f ^ d r = K*e2aM° * 2a^ - p* (EQ 4.15) o Parameters for y are needed instead of ln(y) to find the mean and variance of the path attenuation by integration along the path. The parameters for path attenuation (linear in A) are given by: \iA = E[A] = E ^ y ( x ) d x ] = ^ E [ i ( x ) ] d x = \lyL a I = E[{A-ViAn (EQ 4.16) = E[(^j^(icA?V)-UY)(K:/?V,)-U y)^ -^ "J (EQ 4.17) = t f f f <K* V ) - U tX k K V ') - H*) • f j NW ) , r'\x"))dr>dd'dx'dx' J n0 JJ0 n JJ n0 JJn0 where f JN(r\ r") is a jo in t probability density function to describe the relations between rain rates (linear in rain rate) at different spatial positions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 MODELING POENT-TO-PATH ATTENUATION If the spatial correlation function for specific attenuation is defined by: then f • f j Nw f PT(JC - X ) = ) , n n w d ? ' — and (EQ. 4.17) can be expressed as: <52A = o l t t PT(*' - X")dx'dx'' (EQ 4.18) ' J qJ q The spatial correlation function for the logarithm o f rain rate, Plnr , was obtained from a Fourier transform of the spatial spectrum, see Section 4.1. The spatial cor relation function for rain rate, pR, may be obtained from P lnr by [I]: - x") _ = -— 35— ;— 1 e*b - 1 1 (EQ4.19) a-SfiPl h i I x' - i -) _ . = — ^ 7 1 — The parameters for the lognormal distribution for attenuation - mean ance SA MA and vari - are obtained from p A and a A: m a = ln(pA)~5ln S I = In & 1 + -4 (EQ 4.20) Pa 1+-7 (EQ4.21) Ha ) To obtain the desired attenuation statistics, we need the pdf for the logarithm of path attenuation with the mean and variance values given by (EQ. 4.20) and (EQ. 4.21): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 APPLICATION TO NETWORKS 'l/2 ( ( ln C it) - M A) / 5 Ar fut{A) 55 (EQ 4.22) The cdf for a single path is then obtained from: P D( a > A T) = p X JA r f w W d a = PDf Jin \to(.AT) - M A\ / S A where f N(u) is the unit-normal pdf and at f N(u)du (EQ 4.23) is the attenuation level to be exceeded. Numerical integration was used to evaluate the integrals in (EQ. 4.18) and (EQ. 4.23). The final probability o f exceeding a specified threshold attenuation AT is the sum of the probabilities for the volume cell and debris components: P ( a > A t ) = P c ( a > A t ) + P D( a > A T) (EQ 4.24) 4.3 APPLICATION TO NETWORKS 4.3.1 Parallel Diversity The two-component model employs the volume cells plus the inhomogeneous debris region description to characterize the variability in rain rate and specific attenuation needed to calculate the joint occurrences of attenuation on parallel propagation paths. The two parallel path lengths are L with a base length B and angle (3 between path 1 and the base. L ✓ ✓ Path 1 # Figure 4.3 Geometry for a parallel topology Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 MODELING POINT-TO-PATH ATTENUATION 4.3.1.1 M odel for Volume Cell Component x’* \J1 uz C ell F igure 4.4 Geometry for cell contribution calculations for a parallel diversity Due to symmetry, a boundary line can be set up at y=0 midway between the paths. Any rain cell that occurs above this line will contribute more attenuation to the upper path than to the lower one. The joint probability o f exceeding Ap on both paths 1 and 2 with attenuation A t and A2, respectively, and with the higher attenu ation on the upper path (A2) is then equal to the single path probability given in (EQ. 4.12), but with the constraint that the rain cell must occur above the boundary line, y=0, and its peak value calculated with respect to path 1: Pc( a l > a 2 n a 2 > A T) = J ” e~Rv{x'-^-A^ /R^dxdy p J , + 3 S v/ 2 J J , + (L + i S v ) / 2 A 3q -t' — X X" y"_ X (EQ 4.25) 3(- l -- 3 S v) / 2 (EQ 4^6) + y. = X y. (EQ 4.27) + -fl. and, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 APPLICATION TO NETWORKS b] _ B COsP 2 sin P By 57 (EQ 4.28) For a volume cell below the middle line, similar probabilities are obtained. By add ing the occurrence probability for both cases, the joint probability o f exceeding Ap on both paths becomes: PC(a x > At r\a-,> A t ) ~ 2 /> A 4.3.1.2 f B, * 3SV/ Z -« , + <£. + 3Sy)/2 F I JQ J ( - L - 3 S v) / 2 e - B y ^ * B c y * B r A T) / R c d x d y (EQ 4 .2 9 ) Model for Debris Component The geometry o f a diversity system for use in modeling debris is illustrated in Figure 4.5. Bam p u*\y") Bun p <*\y’) F ig u re 4.5 Geometry for debris contribution calculations for parallel diversity The joint pdf of path attenuations Ai and A2 in a site diversity system can be assumed to be a joint lognormal pdf as follows: (■ 2kS \ a xA 2*I\ - c' (EQ 4.30) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 MODELING POINT-TO-PATH ATTENUATION and c denotes the joint correlation coefficient related to the correlation function of path attenuation p^A ,. a2) by: In (EQ 4.31) C and p^fA,, a 2) is approximated by: PaMi’ A 2) - Cov[A,,A2] Var[Al t f ’*BC‘**p1( J ( B s i n $ ) 2 + ( x ' - x " ) 2)dx'dx' Pa(^1’ A 2) — (EQ 4.32) ^0 Jficosfl f ( ' p 1(x\x")dx'dx" This integral is for parallel paths o f equal length L. (EQ. 4.32) was evaluated numerically. M ore complex geometries can be accommodated by recognizing that the argument o f Py is the magnitude o f the vector distance between point 1 and x" x’ on path on path 2. The joint cumulative distribution function is evaluated by: P 1 — ^ a 2 —A t ) = P d f* f ” f [ln(Ar ) jln ^ u ' u )^u du ' —M a \ / S a (EQ 4.33) Finally, the joint cumulative distribution for a balanced diversity system with the attenuation thresholds set at A r for path 1 and 2 is given by: P(a, > / t r n a 2 ^ A T) = Pc ( a x > A T r~>a2 ^ A T) + P d(.°\ - A T (~^a2 > AT) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (EQ4.34) 4.3 APPLICATION TO NETWORKS 59 4.3.2 Node Diversity In a network, a node arises when a signal has to travel through path 1 or path 2 as viewed in Figure 4.6. Both path lengths are considered to be equal to L, and a sep aration angle (3 is considered. F ig u re 4.6 Geometry for a node topology. 4.3.2.1 Model for Volume Cell Component Considering the symmetry in the geometry of Figure 4.7, a line is passed through the intersection of both paths and in an angle of p/2. When P is zero, both paths coincide and the middle point is considered for the y axis. When a volume cell is above the boundary line y=0, more attenuation is observed on the upper path than on the lower one. The jo in t probability of exceeding Ap on both paths, I and 2, and with the higher attenuation on the upper path is then equal to: P c(a ! —^2 A-j-) —t J o d xdy (EQ 4.35) with (EQ 4.36) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 MODELING POINT-TO-PATH ATTENUATION cos 5P X” jr. sinxP 2 2 + -L , - 3 cosxP y. r h . —sinx 2 2 X (EQ 4.37) and, i P' 1 - COSx 2 (EQ 4.38) - P sinx 2 . />• F ig u re 4.7 Geometry for cell contribution calculations for a node topology. Integrating similarly on Box 2 and adding results together, yields the following o t ^ A ^ ^ a \ P C(° f£(cos|J — 0.5) ♦ 3 S y/ 2 «- R v(iT. A T) / R C . . dxdy 2 Pc +t J o (EQ * * 3S*/2 rucosfi -0.5) +■3Sv / 2 - R y « . v\ A T) / R C . . ] ( - L - 3 s y)/2 e <**& Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 APPLICATION TO NETWORKS 4.3.2.2 61 Model for Debris Component Refer to Figure 4.8 for the node probability calculations due to debris. The approach is very similar to that o f Section 4.3.1.2, except for the correlation func tion of path attenuation p ^ A ,, A2) that is needed in the joint correlation coefficient c. U’.y’) F igure 4.8 Geometry for debris contribution calculations for node topology. Therefore, (EQ 4.40) ,’ 4- <- B <— P 4 r t PT(*\ x")djtdx" JoJo f ^ 2) - Ja (EQ 4.41) t Py( J ( x ’- x " ) 2 + (x"lan$)2)dx'dx" Jo , otherwise x")dx'dx" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 MODELING POINT-TO-PATH ATTENUATION Finally, the joint probability contributed from debris is evaluated by: P o (a i A T r\ A T) — P p ( “ f u")du’du" J a tJ a t = P D f [ l n ( A r ) - M A / S / # ( « M) f J 1 1 r) I----J aaimAT) i n ( A T) -- M M A\ A]/Sa-cu-)/Jl /S„-cu-)/Jl-c -c‘ (EQ4.42) Finally, the joint cumulative distribution for a node system with the attenuation thresholds set at AT for path 1 then 2 is given by: P ( a l > A T C i az > A t ) = P c ( a l > A T r \ a 1 > A T) + P D( a l > A T r \ a 2 > A T) (EQ4.43) 4.3.3 Relay The relay term is used for a geometry where the signal goes through two paths simultaneously, path 1 and 2 in the case of Figure 4.9. F ig u re 4.9 Geometry for relay topology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 APPLICATION TO NETWORKS 63 Unlike the previous geometries where the joint probability of excedance was of importance, in this case the union probability o f exceeding at on each paths, 1 and 2, needs to be evaluated as follows: I ^ A f U ^ 2 — ^7*1 = — A y ] + P [ ( l 2 — A f ] “ P \.Q \ — A f C \ ^ A t] = 2P[ax > A t ] - P{ a x > A T n a 2 Z A T] (EQ 4.44) The probability is computed from the probability o f exceeding A T on a single path, given by (EQ. 4.24), and from the joint probability of exceeding A r on both paths, given by (EQ. 4.43). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 PERFORMANCE ANALYSIS This chapter will present the implementation of the models discussed in previous chapters and includes simulation results and performance evaluations. All the models were implemented using Matlab version 5.1. This software was chosen as it is very powerful for matrix manipulations and the code is optimized in the sense that computations are vectorized. Matlab also has a multitude o f readily available functions such as the special Bessel functions and the Legendre polynomials which were needed in Chapter 3. The calculations of the derivative o f the spherical Bessel functions and the Legendre polynomials were carried out by the recurrence formulae found in [3]. The values were tested for exactness using mathematical tables for real arguments. Other utilities found in Matlab were the statistical func tions which where needed for modeling the two-component rain rate distribution model in Chapter 4. Gaussian quadrature with Legendre orthogonal polynomials was used in the evaluation of numerical integrals because a smaller number of function evaluations were needed and high accuracy was achieved when the inte grand was well-approximated by a polynomial. In all derivations o f specific atten uation, the temperature of rain is considered to be 0°C, and the incident angle 0, defined in Figure 3.1, is chosen for terrestrial line-of-sight links, i.e. 9 = rc/2. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 PERFORMANCE ANALYSIS 5 .1S P E C IF ICA T T E N U A T IO N In this section, the specific attenuation based on the theory in Chapter 3 is com puted and compared them to the ITU Recommendation 838 (TTU838] standard. For a specific frequency, there are two combination sets: 1. Raindrop shape: Spheroidal vs. Pruppacher-and-Pitter, or P-P 2. Drop size distribution: Marshall-Palmer, or M-P vs. WeibuII The spheroidal raindrop shape and the M -P drop size distribution is a well estab lished combination, compared to the P-P with Weibull combination. To obtain the theoretical specific attenuation, the total cross section and the drop size distribution are integrated together. The total cross section is plotted in Figure 5.1 for both frequencies 15 and 38GHz. The vertical polarity gives rise to lower total cross section, or tcs, than the horizontal polarity. For the same drop size distribution, or dsd, and frequency, the integration will result in higher horizontal attenuations. Using similar reasoning, the P-P raindrop model results in specific attenuations higher than the spheroidal one with constant frequency and dsd. The P-P tcs is generally higher than the spheroidal case, given the added contribution from the nonaxisymmetry of the drop shape. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 SPECIFIC ATTENUATION Vcroca! Pwlanty (15GHz) 67 Horuuntal Polarity (15GHz) *= 0.5 0.5 0 OS IS 2 IS 0 3 05 Raindrop Radius, mm 2 3 Raindrop Radiui. mm Vertical Polarity DUGH r) Horizontal Polarity (3*GHz) 1.4 1.4 Z OK 0.6 0.6 0.4 0.4 0.2 0 1 2 Raindrop Radius, mm 3 0 2^ 05 3 Raindrop Radius, mra F ig u re 5.1 Total cross section comparing spheroidal and P-P raindrop model for 15(top) and 38(bottom) GHz. The variation of the dsd with rain rate is shown in Figure 5.2 for both M -P and Weibull. The choice o f the dsd greatly affects the specific attenuation. With the MP dsd having an exponential behavior, small raindrops, less than 0.5mm in radius, occur in large number. The Weibull dsd attempts to correct that overestimation as can be seen in the graph. For larger raindrop sizes, the Weibull dsd is very close to the M-P one. For both drop size distributions, the higher the rain rate is, the larger the raindrops present. This increase results in an increase in specific attenuation with rain rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 PERFORMANCE ANALYSIS Rain Rate 5 mm/h Rain Rate 15 mm/h 1000 1500 M -P W eibull mo 600 3 e 7 500 200 0 I 1.5 ■> 2.5 3 Rain Rale 50 mm/h 0 3.5 0.5 3 3.5 R ainm e IQDmm/h 1000 mo MX) E E E e 400 WO 0 Figure 5.2 I 2_5 1.5 Raindrop Radius* mm 3 0 1 2J 3 Raindrop Radius, mm M-P and Weibull drop size distribution for different rain rates. Using a spheroidal raindrop, the specific attenuation is computed for a rain rate of 50mm/h and plotted for frequencies ranging from 3GHz to 3000GHz in Figure 5.3. The ITU specific attenuation should not be considered for frequencies higher than 400GHz due to the fact that the coefficients for the formula are not available beyond that frequency - a logarithmic interpolation yields the values beyond that frequency. Moreover, the values given by the ITU-R838 have been tested and found reliable for frequencies only up to about 40GHz. The generated specific attenuation agrees closely in that range with the ITU, both for M -P and Weibull distributions. The Weibull specific attenuation, unlike its M-P counterpart, follows the ITU one up to 100GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 SPECIFIC ATTENUATION 69 The total cross section shown in Figure 5.1 shows how the tcs increases for small drops when increasing the frequency from 15 to 38GHz. For example, the 15GHz tcs starts off at raindrops with radius 1mm compared to the 38GHz case, at rain drops with radius 0.5mm. For the first case, considering raindrops larger than 1mm, the Weibull dsd equals the M-P dsd for small rain rates, and then becomes larger for higher rain rates. The second case presents a Weibull dsd smaller than the M-P dsd for smaller rain rates, and higher at larger rain rates, considering rain drops with radius larger than 0.5mm. Horizontal Polarity (5Uram/fe) Vertical Pulanty (5Umm/h) 30 -F -O - ITU-R838 M -P Weibull 25 20 15 10 5 1 1(1 100 Frequency, GHz F ig u re 5.3 raindrops) 1000 0 Frequency. GHz Specific attenuation varying with frequency for a 50mm/h rain rate. (Spheroidal W hen the frequency increases, the total cross section changes shape to give more weight to smaller drops and less weight to larger drops, but the overall magnitude o f the tcs decreases. In the drop size distribution case, the inverse process occurs relative to the presence o f drops with respect to the rain rate. When the tcs of dif ferent frequencies are combined with the same dsd at constant rain rate, the result ing specific attenuation increases with frequency. This continues until a maximum is reached, around 200GHz where the specific attenuation starts to fall down with an increasing frequency. The wavelength at 200GHz is 1.5mm, which is in the range of the raindrop size. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 PERFORMANCE ANALYSIS Once the frequency of the system is known, the most important factor when deal ing with specific attenuation is the rain rate. To see how they are related together, the total cross section for the raindrop shape (spheroidal or P-P) is integrated with the drop size distribution (M-P or Weibull) to give the specific attenuations at 15 and 38GHz. Finally, the standard ITU-R838 is compared with the theoretical spe cific attenuations and presented in Figure 5.4 and 5.5 for the spheroidal and P-P cases, respectively. As expected, the specific attenuation is higher for horizontal than vertical polarity, and for P-P as compared to spheroidal raindrops. Vertical Polamy (!5GHz) Horizontal Polarity (15GHz) 15 15 £ e £« S •3 e -O - ITU-R838 M -P Weibull ,cr. 10 10 3 s < 5 5 C/J 00 E 40 < I(I) 50 00 50 100 Rain Rate, mm/h Rain Rate, mm/h Vertical Pulaiity (38GHz) Horizontal Pulanty (38GHz) ITU-R838 M -P Weibull 150 44) 20 2t) KM) Rain Rate, mm/h 0 50 KM) 150 Rain Rate, mm/h F ig u re 5.4 Specific attenuation comparing drop size distribution for spheroidal raindrops and frequencies 15(top) and 38(bottom) GHz. The Weibull dsd has lower presence of small drops when compared to the M -P dsd. The increased presence o f small drops found in the M -P dsd has an impact on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 SPECIFIC ATTENUATION 71 the attenuation as the frequency increases, through the total cross section, and leads to a higher rate increase in attenuation. A cross-over frequency is reached beyond 40GHz, when the M-P attenuations become larger than the Weibull case. On the other hand, the Weibull dsd has more large drops than the M-P dsd as the rain rate increases and results in Weibull attenuations higher and more apart as the rain rate increases. Beyond the cross-over frequency, the M-P attenuations are always dominant because at that point the small raindrops, less than 0.5mm, have an increased weight in the total cross section contribution. In summary, the M -P specific attenuation starts at a lower level than the Weibull case for lower frequencies and ultimately crosses it and becomes larger for higher frequencies with the difference between the two becoming more noticeable as the rain rate increases. Vertical Polarity ( 15GHr) Horizontal Polamy (ISGHz) 3(1 20 —I— -O - [T U -R 8 3 8 M -P W eibull (5 e 10 < 5 0 100 50 150 00 50 150 100 Rain Rate, rom/h Rain Rate, mm/h Vertical Pularity (38GHz) Horizontal Polarity (38GHz) SO — -O g 40 IT U -R 8 3 8 M- P W eibull 40 < 20 ■¥ 20 o 100 Rain Rate, mm/h 150 0 50 ISO Rain Rate, mm/h F ig u re 5.5 Specific attenuation comparing drop size distribution for P-P raindrops and frequencies 15(top) and 38(bottom) GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 PERFORMANCE ANALYSIS Looking at Figure S.4 for spheroidal specific attenuations, the theoretical values agree very well with the standard values, for both frequencies. In the IS GHz case, the standard falls between the M-P and Weibull values. In the other case, the stan dard follows the M-P attenuation. In the P-P case, Figure 5.5, the attenuation val ues are higher, as expected, and the ITU-R838 attenuations follow the M-P ones very closely at both frequencies. Vertical Polarity (M-P) -a g -k ~ M -O - Horizontal Pularity (M -P) 15G H z 38G H z 300G H z 40 K Ra fit < 20 0 50 0 150 50 Rain Rate, mm/h 150 100 Rain Rate, mm/h Vertical Pularity (Weibull) Hirtzontal Polarity (Weibull) 50 -a 6 40 -O - 15G H z 38G H z 300G H z 40 tc R ° fit I 30 30 < 20 20 1/510 10 0 HI) 150 00 50 150 100 Figure 5.6 Specific attenuation with different frequencies for M-P(top) and Weibull(bottom) dsd, assuming P-P raindrop model. In Figure 5.6, the frequency is compared together for M-P and Weibull dsd’s, con sidering only the P-P raindrop model. A fit to the empirical formula KRa is also plotted for the three frequencies 15, 38, and 300GHz. The theoretical values are very well approximated with the formula. Later, these values will be used for com puting the theoretical path attenuation. In the Weibull case, the 300GHz frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 5.1 SPECIFIC ATTENUATION crosses the 38GHz attenuation level due to the fact that at this high frequency, the attenuation has started to decrease passing through a maximum which depends on the rain rate. In the M-P case, the maximum has not been reached yet for any rain rate. For a better view, Figure 5.3 shows a general behavior o f the specific attenua tion with varying frequency. Taking only the 38GHz frequency, the specific attenuations are plotted in Figure 5.7 comparing the standard ones to the theoretically derived ones with spheroidal and P-P raindrop models, both for M -P and Weibull dsd’s. In the fol lowing sections, the 38GHz frequency will only be considered given its impor tance in radio communications and its commercial use. The following figure will be needed for reference later on. The P-P raindrop derived specific attenuations start very close to the spheroidal ones, and then become higher with higher rain rates. Vertical Pnianty (38GHz. M -P) Horizonul Polarity (38GHz. M-P) 50 50 —i— -O - IT U -R 8 3 8 Sp h ero id al shape P - P shape 40 30 30 < 20 5/5 10 0 50 100 Rain Rate, mm/h Vertical Polarity (38GHz. Weibull) Horizontal Pularity (38GHz. Weibull) 50 IT U -R 8 3 8 —f — Sp h ero id al shape - O - P - P shape g 40 SO 0 150 Rain Rale, mm/h 150 40 30 < 20 20 ** It) 10 0 50 101) 150 0 Rain Rale, mm/h F ig u re 5.7 50 100 150 Rain Rate, mm/h Specific attenuation comparing theoretical spheroidal and P-P raindrop models with the IT U -R 838 at 38GHz, for M-P(top) and Weibuil(bottom) dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 PERFORMANCE ANALYSIS 5.2 PATH LINKS Given the specific attenuation in decibel per kilometer, the path attenuation is esti mated. For the empirical models, an effective path length, which takes into account the statistical presence o f rain along the path, is multiplied by the specific attenua tion to give the required attenuation for a particular unavailability requirement, p. In the following, the standard derivation o f the path attenuation, ITU-R530 and Crane, are based on the standard specific attenuation given by ITU-R838. For the theoretical derivations, the two-component rain model gives the probability of exceeding a certain attenuation level. A trial and error method in finding the required unavailability requirement is employed to get the path attenuation along the link. The theoretically derived specific attenuations are fitted to the 1c Ra empir ical formula, and the parameters k and a are employed in the two-component model. Two different climates will be considered, the first one is the Montreal area which is a temperate continental climate, and the second one is Singapore which is a trop ical climate. The choice o f these two locations was encouraged by the fact that the standard prediction methods seem to work effectively in moderate climates, but fail in tropical regions. Hereafter, the frequency of 38GHz will be chosen for the simulations. The com puted parameters at that frequency for the empirical specific attenuation formula are given in Table 5.1, both for the standard ITU-R838 as well as the theoretical derivations using different raindrop models and drop size distri butions. The rain temperature is considered to be 0°C. TABLE 5.1 Parameters for the empirical specific attenuation evaluated at 38GHz. Ky 0.278 0.942 «H 0.954 M-P 0.426 0.484 0.866 0.875 Weibull 0.236 0.268 1.018 1.029 M-P 0.471 0.503 0.864 0.879 Weibull 0.266 0.278 1.010 1.034 ITU-838 Spheroidal P-P <*v 0.314 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 PATH LINKS 75 5.2.1 Temperate Continental (Montreal) The ITU climate region corresponds to K, and the Crane climate region to D l. The latitude o f 45° N and longitude o f 73° W has been used to determine the regions. A summary o f rain rate statistics is shown in Table S.2., focusing on unavailabilities of 1,0.1, 0.01, and 0.001% of the time. TABLE 5.2 Rainfall Rate Statistics (mm/h) for Montreal 1% 0.1% 0.01% 0.001% ITU - Climate K 1.5 12 42 100 Crane - Climate D l 2.2 11 37 90 T-C* - Climate Dl 2.2 10.3 36.3 86.7 (*) From (EQ. 4.2) using a trial and error technique. The ITU and Crane models have different climate regions, but the rain statistics are very similar. The two-component parameters yield values closer to the Crane rain rate distribution. Figure 5.8 and 5.9 are the path attenuations for the ITU and Crane methods, respectively. Starting with the same specific attenuation evaluated at 38GHz, dif ferent path attenuations are obtained through the calculations of the effective path length at a specified time percentage. Decreasing the unavailability of the link increases the attenuation. Consider the specific attenuation at 0.001% unavailabil ity, which is around 20dB, corresponding to a rain rate around 40mm/h. For a path length o f 8km, the maximum achievable rain attenuation if the rain occurs along the whole path is 20dB /km x 8km = 160dB. But in both standard prediction meth ods, the attenuation is around 1 lOdB. For higher unavailabilities, the difference becomes gradually less obvious, due to the fact that average rainceli diameters are smaller for the more intense convective rain than for more widespread weak rain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 PERFORMANCE ANALYSIS Vertical Polarity 08GHz) 120 KJO Horizontal Polarity (38GHz) - O - 0 .0 0 1 % — — 0 .0 1 % —*— 0 .1 % -B 1% c M) | < 60 40 20 00 □□□o o goo 2 4 Path Length, km 6 4 8 6 Path Length, km F igure 5.8 Path attenuation in Montreal using ITU-R530 prediction method for a 38GHz system and ITU-R838 specific attenuation model. Vertical Polarity (3MGHz) Horizontal Polanty (38GHz) 140 120 100 0.001% 0.01% 0.1% -e- !% -O' ——h— 120 100 a "3 e xo n ue < 60 40 0 20 0 ^ ' f & tr iT fr n n JJ u ,a n n n r w 4 Path Length, km 6 Path Length, km F igure 5.9 Path attenuation in Montreal using Crane prediction method for a 38GHz system and ITU-R838 specific attenuation model. The theoretical path attenuations presented in Figures 5.10 and 5.11 for spheroidal and P-P derived specific attenuations, respectively, show a similar behavior as compared to the standard ones. Finally, ail path attenuations are plotted together in Figures 5.12 and 5.13 for theo retical derivations using the M-P and Weibull dsd’s, respectively. Only the two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 PATH LINKS 77 smallest unavailabilities are considered because of their wider use in practice. For both raindrop models, the theoretical path attenuations agree very well with the ITU and Crane prediction methods. The standard values lie slightly below the two derived ones, or in between them for both cases o f raindrop models. Yet a more noticeable agreement is achieved when using the Weibull dsd, as shown in Figure 5.13. V e n ia l Polarity (MGHz. M -P) 140 Horizontal Polarity (33GHz. M -P) 140 -o - 120 100 T3 C HO 0.001% . —— 0.01% 1- .-e- 120 0.1% 1% 100 a 30 ** c u < 60 60 40 40 20 20 o n a a- 0 □ □ □ o -e 0 o :/r r s> o g u o n n n a o o Path Length, km Path Length, km Vertical (Hilarity (33GHz. Weihull) Horizooul Polarity (33GHz. Weibttfl) 140 120 MX) o - 0.001% . -— — 0.01% —1— . -a- 0.1% 1% a c HO e 60 JX& -- < 40 20 O □ OO 0 Path Length, km 4 Path Length, km 6 F igure 5.10 Path attenuation in Montreal using the two-component model with spheroidal based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 PERFORMANCE ANALYSIS Horizontal Polarity (38GHz. M -P) Vertical Polarity (38GHz. M -P) 140 -o- 0.001% 120 —— 0.01% —f— 0.1% !<» -s_ 1% 120 & 100 80 60 40 ex | r i tTiV n n n v ' u " n n n f > ^ 4 Path Length, km 6 20 01 p r. 0 -t-+ p -a & ■u □ □ o n -n '4 i tj-s=a s 2 * 6 Path Length, km Horizontal Polarity (38GHz. Weibull) Vertical Pulanty (38GHz. Weibull) 150 -O- 0.001% --- 0.01% - t — 0.1% -e- i% ex <3 " P cr - 0. 0 ^ nnriaOOOOOlHl 4 Path Length, km 6 ii iilllinnnnfHi 4 Path Length, km 6 Figure 5.11 Path attenuation in Montreal using the two-component model with P-P based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 5.2 PATH LINKS Vertical Pularity (38GHz. 0.01%) Hunzuntal Pulanty (38GHz. 0.01%) 80 70 60 ------ ITU ------C ran e — T - C (S p h ero id al) -°- T3 50 e % 40 e < 30 ^ © ...rrfJe- T- C (P - P> 60 2(1 9 0 ^ 10 0 0 ( Vertical Pulanty (38GHz. 0.001%) 4 6 S Horizontal Polarity (38GHz. 0.001%) 70 70 60 60 3 50 40 40 = 30 20 20 0 I 3 0 4 Path Length, km 2 3 4 Path Length, km Figure 5.12 Path attenuation comparison in Montreal for different prediction methods at 38 GHz and unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using M-P dsd. Vertical Pulanty (38GHz. 0.01%) Horizucual (Hilarity (38GHz. 0.01%) 80 ITU C ran e T - C (S p h ero id al) T -C (P -P ) 70 70 60 40 = 30 20 0 4 6 o Vertical Pulanty (38GHz. 0.00!%) < 40 8 1( 1 ) xo 60 6 Horizontal Pulanty (38GHz. 0.001%) 1CX) c 4 80 40 20 o 3 Path Length, km 4 0 2 3 4 Path Length, km Figure 5.13 Path attenuation comparison in Montreal for different prediction methods and unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 PERFORMANCE ANALYSIS 5.2.2 Tropical Climate (Singapore) The ITU climate region corresponds to P, and the Crane climate region to H. The latitude o f 1° N and longitude o f 103° E has been used to determine the regions. A summary of rain rate statistics is shown in Table 5.3. TABLE 5.3 Rainfall Rate Statistics (nun/h) for Singapore 1.0% ITU - Climate P 12 Crane - Climate H 6.4 T-C* - Climate H 12.5 0.1% 0.01% 0.001% 65 145 250 51 147 251 66.9 209.7 544.4 (*) From (EQ. 4.2) using a trial and error technique. The Tropical climate will follow the same procedure as the previous one concern ing the output results. The ITU and Crane path attenuations, shown in Figure 5.14 and Figure 5.15, respectively, have much higher values than the temperate conti nental case. The major contributor to this increase is due to higher rain rate statis tics experienced in this particular climate region; therefore, specific attenuations are higher. The effective path length is still less than the actual path length, as it is seen through the inclination of the path attenuation with increased length. When the effective path length is calculated in the ITU method, the rain rate at 0.01% for a maximum of lOOmm/h is included in the formula in order to adjust for the climate region. For the other time percentages, the effective path length at 0.01% o f the time is multiplied by an appropriate factor. Even at Singapore’s high rain rates, the effective path length becomes proportionally smaller than th e path length as it is increased. On the other hand, Crane effective path length depends directly on the rain rate, and its relation to the path length is almost linear. Both prediction methods have similar rain attenuation for the different tim e per centages; ITU estimates slightly higher values than Crane for 0.001%, and lower for the other time percentages. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 5.2 PATH LINKS Using theoretical models, path attenuations are generated and plotted in Figures 5.16 and 5.17 for the M -P and Weibull dsd’s, respectively. The path attenuations are much higher and tend to be very linear with the path length. In Table 5.3, the rain rates at 0.01% and 0.001% estimated from the two-com ponent model are higher than ITU and Crane. The rain rate distributions from [Crane80] were used in the Crane prediction method discussed in Section 2.3.2, w hile the revised ones found in [1] were used in the two-component model in Section 4.2. The latter one uses a wider range o f observations. Considering the high attenuations in the tropi cal regions, the 0.01% unavailability seems to be more appropriate and yields a difference of 30% in the rain rate when compared to the new value. Vertical Polarity (3£GHz) Horizontal Pularily (38GHz) 250 200 -o - 0.001% —f -e - 0.1% 1% — — 0.01% ,<3T a KH) & or < a I 0 0 - 1 Path Length, km 2 Path Length, km 3 4 F ig u re 5.14 Path attenuation in Singapore using ITU-R530 prediction method for a 38GHz system and ITU-R838 specific attenuation model. Vertical Polarity (3KGHz) Huri/untal Polarity <38GHz) -o- 0.001% 0 .01 % -1- 0.1% -a- i% ISO 5s 100 c < & 5(1 1 m n a Q a a 2 3 Path Length, km o 0 -0 <lrir £ 'a n n -H- n 1 a O O O O 0 -g ~ Q 2 Path Length, k m 3 F ig u re 5.15 Path attenuation in Singapore using Crane prediction method for a 38GHz system and ITU-R838 specific attenuation model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 PERFORMANCE ANALYSIS Hurizuttal Pobnty (38GHz. M-P) V e n ia l P u b n ty (38G H z. M -P ) 300 400 a - o — 0.001% — — 0.01% ■f- 0.1% 1% 2(» Path Length, km P ith Length, km H uhzuaal P ubnty (38GHz. WabuU) V e n ia l Pubnty (38GHz. WeibuU) 7 00 500 a -3 e 400 « oe 300 ** —o o o 600 700 MU .................. ...............: a® jf 500 a 400 300 < 2(X> MX) 200 a . ,T. O ■■jcr _0 | | 0 f + jr tr tr jr iTn n n n n a ttg j 100 I Path Length, km 2 Path Length, km 3 F ig u re 5.16 Path attenuation in Singapore using the two-component model with spheroidal based specific attenuation at 38GHz with M-P(iop) and Weibull(bottom) dsd. The path attenuations are compared together in Figures 5.18 and 5.19 for the theo retical derivations using the M -P and Weibull dsd’s, respectively. Only the unavail abilities of 0.01% and 0.001% are considered as in the previous case of the tem perate continental climate. In the tropical climate, the choice of the drop size distribution model turns out to be critical as can be seen by the graphs where the Weibull attenuations in Figure 5.19 are higher, especially in the 0.001% case, when compared to the M-P values in Figure 5.18. As for the raindrop model, the PP raindrop shape attenuation in both dsd’s are higher, especially for the 0.001% unavailability, because o f the greater presence of large raindrops at higher rain rates. On the average, the difference is estimated to be around 8%. The ITU and Crane model attenuations, on the other hand, are much lower than the theoretical values for the reason discussed previously. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 5.2 PATH LINKS Vertical Pobnty DSC Hz. M -P) no 500 Horizontal Polahty (38GHz. M -P) a -o- 0.001% 0.01% 4 0.1% 1% — — . ■ — — . A tte n u atio n , d B : . X 0 ^ : ............ 200 * _ jfr n n n n g P ° o f »^9 2 Path Length, km Path Length. km 700 A tte n u a tio n , dB «X) 800 -O- 0.001% 0 .1 % ■O ■ I% 600 500 4») 400 300 300 200 21X) 0 ........... t 700 0 .01 % -I— 5«) UX1 3 Horizontal Polamy (38GHz. Westell) Vertical P obnty (38GHz. Wesbtdl) 800 1 1 1 1 n n n n n-B -o • ® 'or 0 0_ - -f-t- nI nI n-a I -*3 Path Length, km 0 t " jjr 9 0 :................. 0 ^ — MX) ........so t t I I— ■» Path Length, km F igure 5.17 Path attenuation in Singapore using the two-component model with P-P based specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 PERFORMANCE ANALYSIS Vertical Pulanty (38GHz. 0.01%) Hurizuoul M aritjr (38GHz. 0.014) 250 250 200 -O - £ ITU C rane T - C (S p h ero id al) T - C ( P -P ) 200 150 150 100 100 50 0 3 0 4 4 3 Horizontal Polarity (38GHz. 0.001%) Vertical Pulanty (38GHz. 0.001%) 250 250 200 150 100 50 o 0.5 0 1 I 1.5 Path Length, km Path Length, kin F igure 5.18 Path attenuation comparison in Singapore for different prediction methods at 38 GHz and unavailabilities of 0 .01(top) and 0.001%(bottom), theoretical derivation using M-P dsd. Horizontal (Hilarity (38GHz. 0.01%) Vcnical Pubnly (38GHz. U l l l l ) ITU Crane —I— -O - 250 T - C (S p h ero id al) T - C (P -P ) 200 150 100 50 0 2 3 4 Horizontai Pulanty (38GHz. 0.(101%) Vcnical Pulanty (38GHz. 0.001 %) 350 350 300 3(1) *3 250 250 200 = 150 150 I(X) 100 50 o 0.5 1 Path Length, km 2 0 I Path Length. 1.5 2 F igure 5.19 Path attenuation comparison in Singapore for different prediction methods and unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 SYSTEM ANALYSIS 85 5.3 SYSTEM ANALYSIS With the knowledge o f the rain attenuation along the path length, a microwave sys tem can be simulated and analyzed following the model described in Section 2.4. TABLE 5.4 Parameters for the microwave system f 38.0GHz D 30.72cm Ptx 16.0dBm T rx -82.5dBm The parameters used for the model are shown in Table 5.4. Considering the diam e ter for both antennas to be 12in, or 30.72cm, the gain of the transmitter and receiver can be calculated using (EQ. 2.10). Attenuation values, from the previous section, are included in the calculation o f the fade margin for the study o f system performance in the presence of rain. 5.3.1 Temperate Continental (Montreal) When a fixed antenna diameter is considered with the values given in Table 5.4, fade margins are calculated with varying path length between the transmitting and receiving antennas. The theoretical margins are compared with the ITU and Crane model values for the lower unavailabilities and presented in Figure 5.20 for an MP dsd and in Figure 5.21 for a Weibull dsd. Similar margins are observed for the 0.01% availability, but for the 0.001% the margins are higher in the M -P case. With higher margins, the maximum distance between the antennas is increased as can be seen when looking at the zero margin values. Overall, the theoretical mod els that describe the rain effect on the microwave link are performing well when compared to the ITU or Crane model. The difference is well under the 20% bound for the P-P raindrop model, and 15% for the spheroidal case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 PERFORMANCE ANALYSIS Vertical Pulanty OSGHz. 0 .0 1 * ) —t— -O - Horizontal Polarity (38GHz.0.01%) IT U C ra n e T - C (S p h ero id al) T - C (P -P ) 40 “ 30 0 0 4 Vcnical Pulanty (38GHz. 0 .001 * ) 4 6 8 Horizontal Pulanty (38GHz. 0.001*) 40 £ 2 40 21) 0 3 4 Path Length, km Figure 5.20 Fade margin comparison in Montreal for different prediction methods at 38 GHz for unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using M-P dsd. Vertical Pulanty (3KGHZ. 0.0 1 * ) Horizontal Polarity (38GHz. 0.01*) 60 60 —— -O ITU C ra n e T - C (S p h ero id al) T -C (P -P ) 40 0 4 8 0 Vertical Pulanty (38GHz.0.001* ) 4 6 Horizontal Polarity (38GHz. 0.001*) 50 s£ e 30 20 10 0 3 Path Length, km 4 o 1 2 Path Length, km 3 4 F igure 5.21 Fade margin comparison in Montreal for different prediction methods for unavailabilities o f 0.01(top) and 0.001%(boltom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 5.3 SYSTEM ANALYSIS Vertical Polarity (38GHz. 0.01%) 8 H uruoobl Polarity (38GHz. 001% ) 8 7 7 E 6 6 45 5 4 3 --—— CITU rane - f-o - 1 00 T - C (S pheroidal) T - C (P -P ) V e rtic al P u la n t y (3X G *Iz. 0.001 % ) 20 80 40 100 H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 0 1 % ) JO T 2J .or 0 J 0 2(1 40 60 0 IU0 20 A n te n n a D ia m e te r, cm 40 60 A nten n a D ia m e te r, cm 80 100 F igure 5.22 Maximum path length comparison in Montreal for different prediction methods at 38 GHz for unavailabilities of O.Ol(top) and 0.00l%(bottom), theoretical derivation using M-P dsd. V e rtic a l P u la n t y (3 8 G H r. 0 .0 1 % ) X 7 E € H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 1 % ) 8 7 e 6 6 Zsr. 5 5 4 £3 o0 3 — — —f— -O 20 rru C rane T - C (Spheroidal) T - C ( P -P ) 40 80 100 1 0o V e rtic al P u la n ty (3 8 G H z . 0.(101% ) 20 40 80 100 H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 0 1 % ) JO 0 20 40 60 A n te n n a D ia m e te r, cm 80 100 0 20 40 60 80 100 A n ten n a D iam e te r. F igure 5.23 Maximum path length comparison in Montreal for different prediction methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 PERFORMANCE ANALYSIS Figures 5.22 and 5.23 show the critical zero margin values with varying antenna diameter. The maximum path length is not appreciably affected after a certain antenna diam eter is reached. The construction of bigger antennas does not appear to be justified, particularly since they become more difficult to align and are quite costly. The effect o f different drop size distributions is as important as the choice of the raindrop shape, where the lowest distances are obtained with a P-P raindrop shape with a Weibull dsd, as opposed to the highest separation between antennas when the spheroidal model is chosen with an M-P dsd. 5.3.2 Tropical Climate (Singapore) The theoretical margins are compared with ITU and Crane model values, for the lower unavailabilities and presented in Figure 5.24 for an M -P dsd and Figure 5.25 for a Weibull dsd. Due to the higher rain rates in this region, the lower margins that are witnessed in these figures were to be expected. When considering the theoreti cal values, the ITU and Crane predictions are almost double for the maximum dis tances, the point where zero margin is reached. When considering the theoretical models, the choice o f the drop size distribution model has a m ore crucial impact on the margin than the choice of the raindrop model. One should consider the diffi culty in predicting the performance of a microwave link in the tropical regions due to the intensity o f rainfall as well as the added fluctuations from one year to the other. With this in mind, the maximum distances between the transmitting and receiving antennas should be set to a worst case scenario. Operating on a smaller frequency may not be possible because of restrictions from frequency allocations within radio regulations, or even because of saturation of frequency reuse. The last attempt to increase the distance between antennas could lead to looser availability requirements, by accepting the limitations imposed by nature. Finally, Figure 5.26 and 5.27 illustrate the maximum path length between antennas for varying antenna diameter size. M -P dsd model shows better improvement than the Weibull case for increasing antenna diameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 5.3 SYSTEM ANALYSIS Vertical Pulanty (38GHz. 0.014) 100 IU0 80 -O - e ITU C ran e T - C (S pheroidal) T - C (P -P ) Horizontal P ubnty (38GHz. 0 .0 1 4 ) 80 a ~3 e 60 'ta zao 1 40 0 Id) 3 0 4 Vcnical Polarity (38GHz. 0.0014) 100 80 I 2 3 4 Horizontal Pularity (38GHz. 0.0 0 1 4 ) 80 E a 60 c ea a 40 0 0J 1 o IS os Path Length. km I Path Length, km 2 F ig u re 5.24 Fade margin comparison in Singapore for different prediction methods at 38 GHz for unavailabilities of 0.01 (top) and 0.001 %(bottom), theoretical derivation using M -P dsd. 100 Vertical Polarity (3*GHc. <1.01*1 -H— -O - e Horizontal Pulanty (38GHz. 0 .0 1 4 ) ITU C ran e T - C (S pheroidal) T -C (P -P ) 80 a C 60 40 40 0 I 2 3 2 0 4 Vertical P ubnty (38GHz. 0.0014) 3 4 Horizontal Polarity (38GHz. 0 .0 0 1 4 ) 100 80 80 E a ~3 _e 40 3 u. 20 0 <L5 1 Path Length, km 2 00 0J I Path Length, km F ig u re 5.25 Fade margin comparison in Singapore for different prediction methods for unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 PERFORMANCE ANALYSIS Vertical Polarity (38GHz, 0.01%) HurizonaJ Pularity (38GHz. 0.01%) IJ : ______ rru — — C rane —f — T - C (S p h ero id al) - o - T - C (P -P ) 0.5 0 20 40 60 *o o (00 Vertical Pulanty (38GHz. 0.001%) 20 40 60 SO Horizontal Pularity (38GHz. 0.001%) E 1.5 JZ S. n —eh =s 0 20 40 Antenna Diameter, cm 80 100 0 20 40 Antenna Diameter, cm =^F= = 6 80 100 F ig u re 5.26 Maximum path length comparison in Singapore for different prediction methods at 38 GHz for unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using M-P dsd. Vertical Pulanty (38GHz. 0.01%) Horizontal Pulanty (38GHz. 0.01%) 3 2.5 1.5 — &■ — -O O 20 40 rru C rone T - C (S p h ero id al) T - C (P -P ) 60 80 0.5 KM) 0 Vertical Polamy(38GHz. 0.001%) 20 40 80 100 Horizontal Pulanty (38GHz, 0.001%) 1J E 1.5 c ^ 0.5 0 20 40 60 Antenna Diameter, cm 80 100 0 20 40 60 Antenna Diameter, cm 80 100 F ig u re 5.27 Maximum path length comparison in Singapore for different prediction methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 NETWORK LINKS 91 5.4 NETWORK LINKS Given the horizontal inhomogeneous characteristic o f rain, microwave network links can improve in performance if certain considerations are well accounted for. For example, an open network, where all nodes are connected together with increasing bandwidth until the main node is reached, is extremely unreliable because if one link is disconnected due to rain, then all the remaining links down the line are lost as well. A ring topology, on the other hand, benefits from redun dancy and provides the opportunity to take the opposite route in case a link is affected by a rain cell. The three basic network blocks will be considered and analyzed. The two-compo nent model is used, and the temperate continental clim ate D1 is assumed. The gen eral approach is to analyze one parameter while the others are set in a way so not to affect the analysis. The occurrence probabilities are compared to the single link case to give an idea o f the improvement factor. Also, the cell to debris contribu tions will be plotted in order to spot the ranges with minimum cell contribution due to their larger impact on system performance. 5.4.1 Parallel Diversity Consider the parallel diversity geometry described in Figure 4.3. The attenuation level is set at 3dB and the path lengths at 10km. First, the baseline B is varied with orientation angle P fixed at 45°. The results are shown in Figure 5.28. The refer ence single link was taken to be the same size as the two parallel path lengths. The figures shows a reasonable behavior with an improvement factor starting to be 1 for a baseline distance o f 0km, single link case, and starts increasing as the separa tion is bigger.The contribution from the volume cell remains comparable to the debris contribution until a baseline distance of 3km is reached, when it starts to decay rapidly. This corresponds to a separation between the paths of the order of the cell width. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 PERFORMANCE ANALYSIS A similar analysis was done for changes in baseline orientation relative to the direction of the parallel paths. This is illustrated in Figure 5.29. The baseline is taken to be 3km reducing the cell contribution, thus concentrating on the effect of the orientation angle. For the overlapping parallel paths, the case o f 0° angle, the contribution com es equally from cell and debris. The more perpendicular the paths are, the more improvement is realized. aac 0.1 — -O ' 0.01 Improvement Factor Ratio Cell to Debris 1 3 4 5 Baseline, km Figure 5.28 Diversity improvement factor and ratio of the ceil to debris component as a function of baseline length for jointly exceeding an attenuation of 3dB. Calculations are for a pair of terrestrial paths o f 10km length with an orientation angle of 45° between the baseline and path direction. 'r t x . : : : — -O 0 ?:€h > Improvement Factor Ratio Cell to Debris 15 30 45 Hi Orientation, deg 75 90 F igure 5.29 Diversity improvement factor and ratio of the cell to debris component as a function of orientation angle for jointly exceeding an attenuation of 3dB. Calculations are for a pair of terrestrial paths o f 10km length with a 3km baseline. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 NETWORK LINKS 93 5.4.2 Node diversity The node diversity, illustrated in Figure 4.6, is analyzed in Figure 5.30 for an atten uation level o f 3dB and path lengths of 10km with a reference taken as the single link with same path length. Increasing the separation angle between the paths shows an improvement with a fall off in the cell contribution for angles beyond 80°. A maximum improvement factor is reached at an angle o f 135°. — rr *“*». 3---------- 1 ® © - o - ^ - o - a ^ ..... 0.01 0 k : : - : 30 60 90 120 Scpenuon Angle. <lcg : : /•: 150 ■: 180 F igure 5.30 Diversity improvement factor and ratio of the cell to debris component as a function of separation angle for jointly exceeding an attenuation of 3dB. Calculations are for a node with paths of 10km length. 5.4.3 Relay Considering a network node could function as a relay, the topology illustrated in Figure 4.9 is considered. The attenuation level is set at 3dB with path lengths of 10km. The occurrence probabilities are compared to the single link that starts at the beginning of path 1 and finishes at the end of path 2, with a length of 2 L sin |. In order to justify the use of the relay, the lengths of paths 1 and 2 should be less than or equal to their base length. In geometric analysis, this is ensured with a separa tion angle larger than 60°. W hen the rain is taken into consideration, the simula- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 PERFORMANCE ANALYSIS tion, illustrated in Figure 5.31, show that a minimum separation angle o f 80° must be considered in order to have an improvement factor larger than unity. — — Improvement Factor - O - Ratio Ceil to Debris :____- " " ............. / ......... 2> ^ '0 - 0 - 4 2T / 7 ' e , ( 0 .1 1 0 l « 30 * * ■ * 60 80 90 120 Scpcrauon Angle, deg ■ * 150 * 180 F igure 5.31 Diversity improvement factor and ratio of the cell to debris component as a function of separation angle for exceeding an attenuation of 3dB. Calculations are for a relay with paths of 10km length. In summary, when designing for network links, the following should be consid ered: 1. the topology of the network should ensure alternative paths in case of outage of one individual link. 2. parallel links should be at least 3km apart. 3. adjacent links should be separated with an angle of at least 80°. The design o f network individual links can be achieved as discussed in Section 5.3, while the overall performance of the network should exceed the individual ones. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 FINAL REMARKS AND FUTURE WORK There is a continuing need for higher frequency spectrum use in microwave com munications as the lower frequencies are being saturated. Understanding the the ory in predicting attenuations due to rain is crucial in the harnessing of the higher frequencies. In this thesis, we have combined the perturbation method for estimat ing the total cross section and, ultimately, the specific attenuation, and incorpo rated it with the two-component model for the prediction of point-to-path attenuations. The procedure was successful in the moderate climates, but some val idation is required in the tropical climates, especially when modeling the rain rate distributions. We will give a summary of the work found throughout the thesis and finish with suggestions for future improvements of the models. 6.1 SUMMARY OF OUR WORK In Chapter 1, we presented a brief overview of the rain attenuation calculation pro cedure. In the prediction procedure, the rain rate is a critical parameter and its long-term statistical knowledge is crucial for system reliability. Chapter 2 provided a review of the prediction of reliability for a microwave link when degraded by precipitation. The prediction methods are based on the power 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 FINAL REMARKS AND FUTURE WORK law relation that exists between the rain rate and the specific attenuation. This rela tion has simplified the computation of the attenuation considerably by further modeling the effective path length as a function of the rain rate. The procedures used in the ITU and Crane models for predicting the rain attenuation were pre sented in detail. At the end of the chapter, a description of a simple system that incorporates the rain attenuation was illustrated for performance analysis. The modeling of specific attenuation at a point in space was introduced in Chapter 3. By using a general equation to represent the raindrop shape, perturbation theory is used to solve the boundary conditions at the drop surface in order to solve for the scattering coefficients, which were needed for the total cross section derivation. Finally, the raindrop size distribution was estimated based on experimental obser vations, and combined with the total cross section in order to calculate the specific attenuation. A new equation for the spheroidal raindrop has been suggested and was shown to be a better approximation than the one used in the literature. More over, a more realistic Pruppacher-and-Pitter raindrop shape with its nonaxisymmetry is fitted to the general drop shape equation. With one equation to represent the different raindrop shapes, it was possible to use the same equations, but different parameter values, to calculate the total cross section. Once the specific attenuation is known at a point, a point-to-path scheme is required to obtain the total attenuation along the path length. Chapter 4 describes the two-component rain rate model and its application to the single path geometry as well as to other ones that define simple network blocks. In doing so, the required spatial correlation o f rain is estimated from the power spectrum o f logarithmic variations in rain rate. Different model assumptions were combined together in Chapter 5 to analyze their impact on system performance. The spheroidal and Pruppacher-and-Pitter raindrop shapes were considered. The P-P raindrop shape attenuations were higher than the spheroidal case due to the nonaxisymmetry added contribution. As for the drop size distribution, both the Marshall-and-Palmer and the Weibull distributions were Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 SUMMARY OF OUR WORK 97 looked at. The Weibull dsd has shown better agreement with the ITU specific attenuation model, for frequencies up to 100GHz. For variations in the rain rate, the theoretical specific attenuations were in good agreement with the ITU standard for the frequencies of IS and 38GHz that were considered. The relation between rain rate and specific attenuation being a good approximation for the theoretical attenuations, the parameters of the formula were employed in the two-component model to obtain estimates of the path attenuations at different time percentages. These values were then compared with the ITU and Crane mod els for moderate and tropical climates. In the former climate, good agreement was found between the two. In the latter climate, the two-component model gave rise to higher rain attenuation values because different rain rate distribution values were considered. In order to analyze the performance of microwave radio links in rain, a simulation was carried out incorporating the rain attenuation in the channel model. For the moderate climate, the fade margins between theoretical values and ITU and crane models were found to be within the lOdB range, with lower margins for the theo retical case. As for the tropical climate, once again, the theoretical case was very conservative with differences that went as high as 30 dB. From the plot of maxi mum path length between the transmitting and receiving ends, some improvement was found when the diameter of the antennas was increased, but would definitely be outweighed by the complexity and cost of such a process. To complete the microwave performance analysis, network links were analyzed and useful recommendations were noted, toward the end of Chapter 5, for the design o f networks. The analysis was based on three simple network blocks, which were parallel, node, and relay topologies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 FINAL REMARKS AND FUTURE WORK 6.2 FUTURE CONSIDERATIONS In this thesis, we have studied the development and simulation of theoretical m od els in the analysis o f microwave link performance. With a direct contact w ith the ory, models can be modified and improved as more data becomes available for higher frequencies and different locations. The modeling of rain is still an issue to be explored, both for microwave propaga tion and meteorological studies. The study of radar data has been very useful in understanding the rain process, by providing enormous amounts of data for longperiods of time. There is still a necessity for experimental data for comparison with theoretical attenuations and for validation of models. NASA is working on a TRM M project, Tropical Rainfall Measuring Mission, which consist of a satellite for observing rainfall in the tropics and a ground validation support for calibrating the satellite measurements. While the scattering theory by a raindrop is almost perfected, the drop size distri bution needs to be validated in different parts of the world, especially in the trop ics, where the rain rates are higher than the normal. For example, Li, et al. [Li95b] have derived the drop size distribution for Singapore’s tropical environment from the rainfall attenuation data at a frequency of 21.225GHz over a short path length of about 1km, and resulted in large-scale raindrops with larger density than in the moderate climatic region, such as the one described by the M-P distribution. Another aspect to the prediction of rain attenuation is the rain rate distribution which is being updated as more data is available. This case was experienced in the thesis between the two-component model, which is based on the latest distribu tions extracted from [1], and the Crane prediction method which is based on the earliest distribution taken from [CraneSO]. On the other hand, the rain rate model described by the two-component model seems to be a good method for path atten Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 FUTURE CONSIDERATIONS 99 uation predictions given that its parameters are fitted to the appropriate rain rate distributions. As a future work, the two-component model can be modified in order to simulate more sophisticated network topologies which can include several links at the same time. The knowledge acquired from these studies can help improve the perfor mance o f these networks and minimize the effect of rain on the overall system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [ A tla s 7 4 ] D. Atlas and C. W. Ulbrich, “The physical basis for attenuation-rainfall relationships and the measurement of rain fall parameters by combined attenuation and radar methods,” J. R e c h . A tm o s., vol. 8, pp. 275-298, Jan.-Jun. 1974. [C ra n e8 0 ] R. K. Crane, “Prediction of attenuation by rain,” T ra n sa c tio n s o n C o m m u n ic a tio n s, IE E E vol. 28, n. 9, pp. 1717- 1733, Sep. 1980. I C rane82 J R. K. Crane, “A two-component rain model for the pre diction of attenuation statistics,” R a d io S c ie n c e , vol. 17, n. 6, pp. 1371-1387, Nov. 1982. [C rane89J R. K. Crane, “A two-component rain model for the pre diction of site diversity performance,” R a d io S c ie n c e , vol. 24, n. 6, pp. 641-665, Sep.-Oct. 1989. [C ra n e9 0 ] R. K. Crane, “Space-time structure of rain rate fields,” J o u r n a l o f g e o p h y s ic a l research, vol. 95, n. D3, pp. 2011- 2020, Feb. 1990. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 REFERENCES [H arden78] B. N. Harden, J. R. Norbury and W. J. K. While, “Estima tion of attenuation by rain on terrestrial radio links in the UK at frequencies from 10 to 100 GHz,” O p tic s, a n d A c o u s tic s , [H a r d e n 7 7 ] M ic r o w a v e s , vol. 2, n. 4, pp. 97-104, Jul. 1978. B. N. Harden, J. R. Norbury and W. J. K. White, “Mea surements of rainfall for studies of millimetric radio attenuation,” M ic ro w a v e s, O p tic s , a n d A c o u s tic s , vol. 1, n. 6, pp. 197-202, Nov. 1977. [ IT U 5 3 0 ] ITU-R Recommendation 530, “Propagation data and pre diction methods required for the design of terrestrial lineof-sight systems”, P N [ IT U 8 3 7 ] S e r ie s . 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Further reproduction prohibited without permission. REFERENCES [Joss67] J. Joss, J.C. Thomas, and A. Waidvogel, “The variation of raindrop size distribution at Locarno,’’ P ro c . n a tio n a l c o n fe r e n c e o n c lo u d p h y s ic s , pp. [L aw s43] o f in te r 369-373, 1967. J. O. Laws, and D. A. Parsons, “The relation of raindropsize to intensity,’’ T ra n sa c tio n s o f th e A m e r ic a n ic a l U n io n , [U 95aJ 103 G eo p h ys vol. 24, pp. 452-460, 1943. L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Microwave attenuation by realistically distorted rain drops: Part I-Theory,” I E E E P ro p a g a tio n , [Li95b] T ra n s a c tio n s o n A n te n n a s & vol. 43, n 8, pp. 811-822, Aug. 1995. L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Micro wave attenuation by realistically distorted raindrops: Part 11-Predictions,” I E E E a g a tio n , [U 94] T ra n sa c tio n s o n A n te n n a s & P ro p vol. 43, n. 8, pp. 823-828, Aug. 1995. L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Integral equation approximation to microwave specific attenua tion by distorted raindrops: The spheroidal model,” I E E E S IC O m C lE , ILin77J pp. 658-662, 1995. S. H. Lin, “A method for calculating rain attenuation dis tributions on microwave paths,” B e ll S y s t. Tech. J ., vol 54, n. 6, pp. 1051-1086, 1975. [M arshall48] J.S. Marshall and W.M.K. Palmer, “The distribution of raindrops with size,” J. M e te o ro L , vol. 5, pp. 165-166, 1948. [M orrison74] J. A. Morrison and M.-J. Cross, “Scattering of a plane electromagnetic wave by axisymmetric raindrops,” B e ll S y s te m T e c h n ic a l J o u r n a l, T he vol. 53, n. 6, pp. 955- 1019, Jul.-Aug. 1974. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 REFERENCES [Mushiake56] Y. Mushiake, “Backscattering for arbitrary angles of inci dence of a plane electromagnetic wave on a perfectly conducting spheroid with small eccentricity,” Journal o f Applied Physics, vol. 27, n. 12, pp. 1549-1556, Dec. 1956. [Oguchi83] T. Oguchi, “Electromagnetic wave propagation and scat tering in rain and other hydrometeors,” Proceedings o f the IEEE, vol. 71, n. 9, pp. 1029-1078, Sep. 1983. [Oguchi8I J T. Oguchi, “Scattering from hydrometeors: A survey,” Radio Science, vol. 16, n. 5, pp. 691-730, Sep.-Oct. 1981. [Oguchi77] T. Oguchi, “Scattering properties of Pruppacher-and-Pitter form raindrops and cross polarization due to rain: Cal culations at II, 13, 19.3, and 34.8 GHz,” Radio Science, vol. 12, n. 1, pp. 41-51, Jan.-Feb. 1977. [ Oguchi73 J T. Oguchi, “Scattering properties of oblate raindrops and cross polarization of radio waves due to rain: Calcula tions at 19.3 and 34.8 GHz,” Journal o f the Radio Research Laboratories, vol. 20, no. 102, pp. 79-118, Sep. 1973. [Oguchi64] T. Oguchi, “Attenuation of electromagnetic wave due to rain with distorted raindrops (Part II),” Journal o f the Radio Research Laboratories, vol. 11, no. 53, pp. 19-44, Jan. 1964. [Oguchi60] T. Oguchi, “Attenuation of electromagnetic wave due to rain with distorted raindrops,” Journal of the Radio Research Laboratories, vol. 7, no. 33, pp. 467-485, Sep. 1960. [Olsen78] R. L. Olsen, D. V. Rogers, and D. B. Hodge, “The aRb relation in the calculation of rain attenuation," IEEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES 105 Transactions on Antennas & Propagation, vol. 26, n. 2, pp. 318-329, Mar. 1978. [ Pruppacher71 ] H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” Journal o f Atmospheric Sciences, vol. 28, pp. 86-94, Jan. 1971. [Ray72] P. S. Ray, “Broadband complex refractive indices of ice and water,” Applied Optics, vol. 11, n. 8, pp. 1836-1844, Aug. 1972. [Sekine88J M. Sekine, T. Musha, and C.-D. Chen, “Rain attenuation from Weibuil raindrop-size distribution,” Conference Proceedings - European Microwave Conference, n. 18, pp. 423-428, 1988. [Sekine82] M. Sekine and G. Lind, “Rain attenuation of centimeter, millimeter and sub-millimeter radio waves,” Proc. o f the 12th European microwave conference, Helsinki, Finland, pp. 584-589, 1982. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY [1] R. K. Crane, “Electromagnetic wave propagation through rain,” A WileyInterscience publication, 1996. [2] R. R. Rogers, “A short course in cloud physics,” Pergamon Press, 1989. [3] M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions with formulas, graphs, and mathematical tables”, U.S. Govt. Print. Off., 1964. [4] J. A. Stratton, “Electromagnetic Theory” McGraw-Hill Book Company, 1941. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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