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Precipitation scatter interference on communication links with emphasis on the melting-snow layer Hulays, Rafeh Ahmad 1992

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PRECIPITATION SCATTER INTERFERENCE ON COMMUNICATION LINKS WITH EMPHASIS ON THE MELTING-SNOW LAYER by RAFEH AHMAD HULAYS B.Sc., Monmouth College, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April, 1992 © Rafeh Ahmad Hulays, 19 2  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  Electrical Engineering  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  April 13/1992  Abstract  Abstract A geometrical model has been developed to calculate hydrometeor interference between different microwave systems sharing the same frequency. The model is capable of calculating the interference for any combination of transmitter-receiver geometry and the program is flexible enough to allow for many assumptions related to the spatial and vertical structure of the rain cell. Furthermore, it can easily accommodate different attenuation and scattering models. The study also focuses on the melting-snow layer and it is found that this layer plays a significant role in the interference calculations. The melting layer significantly increases the interference in the 1-8 GHz range, and moderately in the 8-12 GHz. On the other hand, the melting layer results in a significant decrease in the interference level at higher frequencies, especially in the 30-40 GHz range. The study also examines the effect of the ice/snow region above the melting layer and it is concluded that this region plays an important role in the interference calculations, especially at higher frequencies. Three examples of interference geometries are examined in Chapter 4. The first deals with the interference from an up-link to terrestrial links in the near-forward direction, the second deals with the interference from an up-link to terrestrial links in the near-backward direction and the third deals with the interference from an up-link to a satellite in the forward direction. A comparison is made between two rain-cell models in Chapter 5. The COST 210 rain-cell model, which is adopted by the CCIR (International Radio Consultative Committee), is compared with the more physical Capsoni rain-cell model. ii  Abstract  A new empirical attenuation formula for rain and melting-snow has been developed, which, unlike previous formulae, has the frequency as a separate parameter. For detailed analysis, refer to Appendix D.  iii  Table of Contents  Table of Contents  page  ii  Abstract ^ List of Figures ^  viii  List of Tables ^  xiii  List of Symbols ^  xiv  Acknowledgment ^  xvii  Chapter 1^Introduction ^  1  Chapter 2^Hydrometeors: Structure and Characteristics ^ 3 2.1 Hydrometeor Structure ^ 2.1.1 Rain cells ^  3 3  2.1.2 Spatial structure of rain ^ 5 2.1.2.1 The Melting-snow layer (Bright Band) ^ 5 2.2 Electromagnetic wave propagation in Hydrometeors ^ 7 2.2.1 Hydrometeor scatter ^  7  2.2.2 Hydrometeor attenuation ^ 8 2.2.2.1 Kharadly 1st model for attenuation [10] 8 2.2.2.2 Kharadly 2nd model for attenuation [12] 9 2.2.2.3 Kharadly 3rd model for attenuation [12] 9 2.2.2.4 Kharadly 4th model for attenuation . . . 9 2.2.2.5 Empirical model 10 iv  Table of Contents  Chapter 3^The Interference Model ^  16  3.1 Approximate Radar equation ^ 16 3.2 Antenna gain pattern [7] ^  19  3.3 The "Universal Model" ^  21  Chapter 4^Interference Calculations ^ 4.1 Introduction ^  23 23  4.1.1 Scattering in the ice/snow region ^ 23 4.2 Interference from up-link to terrestrial links in the near-forward direction ^  24  4.2.1 Description ^  24  4.2.2 Computed results ^  27  4.3 Interference from up-link to terrestrial links in the near-backward direction ^  38  4.3.1 Description ^  38  4.3.2 Computed results ^  38  4.4 Interference from up-link to satellite in the forward direction ^  49  4.4.1 Description ^  49  4.4.2 Computation results ^  51  4.5 Summary ^ V  52  Table of Contents  Chapter 5^COST 210 rain-cell model ^ 5.1 Introduction ^  61 61  5.2 COST 210 rain cell model [7] ^ 61 5.3 Results for sample interference geometries ^ 64 5.4 Conclusion ^ Chapter 6^Discussion and Conclusions ^  65 77  6.1 Effect of melting-snow layer ^  77  6.2 Effect of rain height, Hm ^  78  6.3 Effect of rain rate ^  79  6.4 Effect of frequency ^  79  6.5 Reminder ^  79  Chapter 7^Suggestions for future research ^  80  Appendix A^Precipitation modeling ^  82  A.1 Rain medium ^  82  A.2 Melting-snow medium ^  83  Appendix B^Kharadly attenuation models ^  85  B.1 Artificial dielectric model ^  85  B.2 Corrected attenuation models ^ 86 B.2.1 Kharadly 3rd model for attenuation ^ 86 B.2.2 Kharadly 4th model for attenuation ^ 87 Appendix C^Kharadly scattering model [11] ^  88  Appendix D^Empirical formula for attenuation ^  92  vi  Table of Contents  Appendix E^The program for the "Universal Model" ^ 101 Appendix F^CCIR Document 12-3/29 (Rev. 1) and supplement . . ^ 138 References ^  153  vii  List of Figures  List of Figures  page Figure 2.1^Rain rate distribution in a 20 km radius rain-cell for^4 different ro Figure 2.2^Two views of the radar bright band: at the left a vertical ^6 profile of reflectivity and Doppler velocity as measured with vertically pointing Doppler Radar; at right a PPI map at 8° elevation on which the melting layer appears as a bright ring at about 12 miles. Figure 2.3(a)^A comparison of the attenuation profile of the melting layer ^11 between Kharadly 3rd and 4th attenuation models, and the Exact calculations for f (frequency) = 1.0 and 10.0 GHz and =  0.1, 0.2, 0.3  Figure 2.3(b)^A comparison of the attenuation profile of the melting layer ^12 between Kharadly 3rd and 4th attenuation models, and the Exact calculations for f (frequency) = 20.0 and 30.0 GHz and p s = 0.1, 0.2, 0.3 Figure 2.3(a)^A comparison of the attenuation profile of the melting layer ^14 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 1.0 and 10.0 GHz (p s = 0.1). viii  List of Figures  Figure 2.3(b)  A comparison of the attenuation profile of the melting layer ^15 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 20.0 and 40.0 GHz (p s = 0.1).  Figure 3.1  General Interference geometry ^  Figure 3.2  Simulation of the gain of an antenna with K=-15, ^20  17  al = 0.6, (12 = 5.5.  Figure 4.2.1  Interference from up-link to terrestrial link geometry in the ^25 near forward direction.  Figure 4.2.2(a) Interference versus rain rate for various frequencies (f in^30 GHz), ps (m = 1—p,), and for Hm = 2.0 km. Figure 4.2.2(b) Same as Figure 4.2.2(a), with Hm = 2.5 km.^31 Figure 4.2.2(c) Same as Figure 4.2.2(a), with Hm = 3.0 km.^32 Figure 4.2.2(d) Same as Figure 4.2.2(a), with Hm = 3.5 km.^33 Figure 4.2.2(e) Same as Figure 4.2.2(a), with Hm = 4.0 km.^34 Figure 4.2.3(a) Interference versus rain rate for various rain heights (Hm), ^35 and for f (frequency) = 1, 5, 10, 20 GHz. Figure 4.2.3(b) Same as Figure 4.2.3(a), with f = 30 GHz.^36 Figure 4.2.3(c) Same as Figure 4.2.3(a), with f = 40 GHz.^37 Figure 4.3.1  Interference from up-link to terrestrial link geometry in the ^39 near backward direction.  Figure 4.3.2(a) Interference versus rain rate for various frequencies (f in ^41 GHz),p, (m = 1—p,), and for Hm = 2.0 km. Figure 4.3.2(b) Same as Figure 4.3.2(a), with Hm = 2.5 km.^42 ix  List of Figures  Figure 4.3.2(c) Same as Figure 4.3.2(a), with Hm = 3.0 km ^43 Figure 4.3.2(d) Same as Figure 4.3.2(a), with Hm = 3.5 km. ^44 Figure 4.3.2(e) Same as Figure 4.3.2(a), with Hm = 4.0 km.^45 Figure 4.3.3(a) Interference versus rain rate for various rain heights (Hm), ^46 and for f (frequency) = 1, 5, 10, 20 GHz. Figure 4.3.3(b) Same as Figure 4.3.3(a), with f = 30 GHz.^47 Figure 4.3.3(c) Same as Figure 4.3.3(a), with f = 40 GHz.^48 Figure 4.4.1  Interference from up-link to satellite geometry in the ^50 forward direction.  Figure 4.4.2(a) Interference versus rain rate for various frequencies (f in  ^  53  GHz), p s (m = 1—p s ), and for Hm = 2.0 km. Figure 4.4.2(b) Same as Figure 4.4.2(a), with Hm = 2.5 km. ^54 Figure 4.4.2(c) Same as Figure 4.4.2(a), with Hm = 3.0 km. ^55 Figure 4.4.2(d) Same as Figure 4.4.2(a), with Hm = 3.5 km. ^56 Figure 4.4.2(e) Same as Figure 4.4.2(a), with Hm = 4.0 km. ^57 Figure 4.4.3(a) Interference versus rain rate for various rain heights (Hm),^58 and for f (frequency) = 1, 5, 10, 20 GHz. Figure 4.4.3(b) Same as Figure 4.4.3(a), with f = 30 GHz. ^59 Figure 4.4.3(c) Same as Figure 4.4.3(a), with f = 40 GHz. ^60 Figure 5.1  COST 210 rain-cell model  ^  Figure 5.2(a,b) A comparison between Capsoni and COST 210 rain cell  62 ^  models with and without a melting snow layer for Hm = 2.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  66  List of Figures  Figure 5.2(c,d) A comparison between Capsoni and COST 210 rain cell ^67 models with and without a melting snow layer for Hm = 2.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz. Figure 5.3(a,b) A comparison between Capsoni and COST 210 rain cell ^68 models with and without a melting snow layer for Hm = 2.5 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz. Figure 5.3(c,d) A comparison between Capsoni and COST 210 rain cell^69 models with and without a melting snow layer for Hm = 2.5 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz. Figure 5.4(a,b) A comparison between Capsoni and COST 210 rain cell ^70 models with and without a melting snow layer for Hm = 3.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz. Figure 5.4(c,d) A comparison between Capsoni and COST 210 rain cell^71 models with and without a melting snow layer for Hm = 3.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz. Figure 5.5(a,b) A comparison between Capsoni and COST 210 rain cell ^72 models with and without a melting snow layer for Hm = 3.5 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz. Figure 5.5(c,d) A comparison between Capsoni and COST 210 rain cell^73 models with and without a melting snow layer for Hm = 3.5 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz. Figure 5.6(a,b) A comparison between Capsoni and COST 210 rain cell^74 models with and without a melting snow layer for Hm = 4.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  xi  List of Figures  Figure 5.6(c,d) A comparison between Capsoni and COST 210 rain cell ^75 models with and without a melting snow layer for Hm = 4.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz. Figure C.1^Scattering geometry of a rain particle due to an incident ^89 electromagnetic wave. Figure D.1^A comparison of the attenuation profile of the melting layer ^97 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 1.0 and 5.0 GHz (p 3 = 0.1). Figure D.2^A comparison of the attenuation profile of the melting layer ^98 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 10.0 and 20.0 GHz (p s = 0.1). Figure D.3^A comparison of the attenuation profile of the melting layer ^99 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 30.0 and 40.0 GHz (p s = 0.1). Figure D.4^A comparison of the attenuation profile of the melting layer ^100 between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 70.0 and 100.0 GHz (p s = 0.1).  xi i  List of Tables  List of Tables  page Table 4.1  Chilbolton-Baldock path parameters [7]  24  Table 4.2  Measured transmission loss for the Chilbolton-Baldock  26  path at 11.2 GHz as a function of rain rate and percentage of time Table 4.3  The converted input parameters for the model-program  27  Table 4.4  Parameters used for the interference calculations from  49  up-link to satellite Table A.1  Drop size distribution and their velocities for various precipitation rates [14]  82  ^ ^  • • ▪ •  List of Symbols  List of Symbols  representative rain drops radius.  a  ^a nt i^melting-snow ^aRi^rain  particles radius.  drops radius.  ^dc^diameter of the COST 210 rain-cell. frequency. fr  the resonant frequency of the melting-snow particle. low-frequency polarizability.  •  ^ge^high-frequency polarizability.  gr (0)^normalized radiation pattern of the receiving antenna at an angle Co from the main lobe axis. ^n(d)^the Pi  raindrop-size distribution.  fraction of the volume of rain VR composed of the rain drops of radius aRt•  r,^the  distance at which the rainfall rate decrease by a factor of lie (Capsoni rain cell).  r e f f effective Earth radius. ^f  m^the  truncated Capsoni rain cell radius.  rrt link length between transmitter and receiver. fall velocity of the melting-snow spheres of radius a rm. vm i vRi fall velocity of the rain drop of radius aRi. ^(xc,  Yc, zc)^the  rectangular coordinates of the bottom of the rain cell.  (x r , yr, zr)^the rectangular coordinates of the receiver. ^(xt,  yt, zt)^the rectangular coordinates of the transmitter. sum of the sixth powers of the diameters of all hydrometeors • per unit volume (reflectivity) Attenuation coefficient due to atmospheric gases.  Aa  •  the normalized melting-snow attenuation. xiv  • • • •  List of Symbols  Ap  Attenuation coefficient due to precipitation. receiving antenna gain. Gr Gt transmitting antenna gain. He the rain cell height. Hm rain height; also the height of the top of the melting layer. a parameter connected with the receiver normal radiations pattern. It determines the gain level of the side lobe of the antenna. • transmission loss. number of representative rain drops of radius a per unit volume. Nm number of melting-snow particle per unit volume (rain drop density). Pr average interference power received. Pt transmitted power. R rain rate. distance from the common volume to receiver. Rr the rain rate where the Capsoni rain cell is truncated. Rm i n ^Rt^distance from the transmitter to the common volume. RM the peak rain rate at the centre of the rain cell (Capsoni rain cell). • ratio of the melted to the total volume in the melting-snow particle. S. surface area of the narrow beam antenna perpendicular to the main beam axis (also known as the 3dB foot-print). • melting-snow layer thickness. VR volume of rain. a2) parameters connected with the receiver normal radiations pattern. • phase coefficient due to precipitation. ^7(Rt, Rr)^the propagation loss due to precipitation and atmospheric gases along the path Rt + Rr •  xv  • •  List of Symbols  skin depth of water. = complex permitivity of water. ^CO^complex permitivity of free space. receiver loss factor. ^Tit^transmitter loss factor. • double-sided half power beamwidth of the narrow beam antenna. (Or, Or) the rectangular coordinates of the receiver main lobe axis relative to the receiver coordinate axes. (et, 0i) the spherical coordinates of the transmitter main lobe axis relative to the transmitter coordinate axes. A wavelength of the transmitter electromagnetic wave. ▪ the rain cell radius. P • the density of snow in the melting-snow particle. bistatic scattering cross section in the direction of O and (Figure C.1). • the bistatic cross section of a drop of equi-volume radius a in obi^cb, a ) the direction of (O, cb) . Rayleigh scattering. Qs FR attenuation outside the COST 210 rain-cell. -  xvi  Acknowledgment  Acknowledgment I would like to express my appreciation to Dr. M.M.Z. Kharadly who has provided me with much needed support, supervision and suggestions throughout the course of my studies at UBC. I also would like to thank Dr. R.L. Olsen and Dr. D.V. Rogers of the Communications Research Centre for their assistance and helpful advice. I would also like to thank Mr. Tim Vlaar and Mr. Smrz Honza for their help during the summer of 1991. The help of Mr. Vlaar in developing the empirical model is greatly appreciated. I would like to thank all my friends and teachers since everyone of them left their mark in my life and character. This research was supported by the Communications Research Centre, Department of Communication, under contract number 36001-0-3596/01—SS Finally, I would like to thank my mother and father, whom I long to see soon and whose advice has always been with me.  xvii  Chapter 1—Introduction  Chapter 1 Introduction The ever increasing demand on a limited radio-frequency spectrum has necessitated the sharing of frequencies by a number of services. This frequency sharing increases the possibility of interference. In system planning, an engineer has to be able to establish a reliable system which can distinguish between the incoming signal and interference caused by other systems using the same frequency. To do this, it is necessary to estimate the mutual interference between the different radio systems. This is by no means an easy task (and is getting harder with the increasing congestion of the radio-frequency spectrum). There are many mechanisms that can cause interference [6]: — line of sight; — diffraction over isolated obstacles; — diffraction over irregular terrain; — tropospheric forward scatter; superrefraction, with or without reflection; ducting; scatter from hydrometeors; reflections from aircraft. Microwaves are scattered by hydrometeors such as rain, snow, melting-snow, and ice particles. This scattering is one of the possible causes of interference between communication links operating at the same frequency. It becomes then necessary to quantify this interference in order to be able to design more reliable communication links. 1  Chapter 1—Introduction  Recently, considerable work has been done on interference caused by hydrometeor [1,2,7]. So far, however, the issue of the melting-snow layer has not been considered, even though the presence of the melting layer tended to be a significant source of error in radar measurement of rain rate [17] The purposes of this study are: 1. to develop a general geometrical model which can be used in conjunction with any attenuation and scattering model; 2. to develop a "universal" program to calculate the interference for a wide range of geometries and variables; 3. to study the effect of those variables and geometries on interference due to rain, and 4. to study the effect of the melting-snow layer on the interference problem. The possible geometries involved in the interference calculations are numerous. To overcome this, a "universal" model-program has been developed. While this program is capable of calculating the interference for all geometries and variables, only the most likely scenarios (which are still very numerous) will be considered (e.g., geometries involving the interference from up-link to satellite, interference from up-link to terrestrial link and variables such as antenna gain, rain rate and rain-cell structure).  2  Chapter 2—Hydrometeors: Structure and Characteristics  Chapter 2 Hydrometeors:  Structure and Characteristics 2.1 Hydrometeor Structure Hydrometeor scattering is observed when a rain cell overlaps with the common volume of transmitting and receiving antennas. This scattering and the subsequent interference depends on the rain cell, its rain intensity, height, radius, etc. It thus becomes very important to model the rain cell as accurately as possible for interference calculations. Many models have been developed to describe rain processes [eg. 3,6,7]; the most realistic is that of Capsoni [3]. The Capsoni model will be the basis for the spatial distribution of rain in the present work.  2.1.1 Rain cells The horizontal pattern of the rain cell has been represented by an analytical expression with an exponential shape having rotational symmetry [2]:  R( z , y ) = Rme ro  ^  (2.1)  where r is the distance of the point (x, y) from the rain cell centre, ro is the radius at which rainfall rate decreases by a factor of 1/e, and RM is the peak rain rate at the centre of the cell (Figure 2.1) The rain cell is truncated at a certain distance f,,, where the rain rate is R m i n [2]: Rmin  = Rme —i. "1 / r °^ 3  (2.2)  Chapter 2--Ifydrometeors: Structure and Characteristics  0.9 0.8  3 c4  0.7 0.6 0.5 0.4 0.3 0.2 0.1  -15  ^  -10^-5^0^5^10  ^ ^ 20 15  distance (r) from the centre of the rain cell in km  Figure 2.1 Rain rate distribution in a 20 km radius rain-cell for different I-.  Chapter 2--Ilydrometeors: Structure and Characteristics  Beyond this point, we assume that the effect of rain is negligible. Equation 2.2 can now be rewritten as: r ° =^In(RA I I Rmsn) ^  (2.3)  The radius of the rain cell is given by:  = 10 — 1.5 x log io Rm^(2.4) where^is in km and RM is in mm/h.  2.1.2 Spatial structure of rain Two rain-cell structures are considered. The first is a rain-only medium. The other is when snow forms and then melts introducing a melting-snow layer. 2.1.2.1 The Melting-snow layer (Bright Band)  When it is warm enough for the snow to melt before reaching the ground, there is often observed a layer of high reflectivity just below the 0°C isotherm. This phenomenon was observed as far back as the forties and it became known as the radar "bright band." The melting-snow layer is the region in which the precipitation changes from snow to rain (Figure 2.2). As snowflakes descend into the melting-snow layer, they become highly "reflective." The most important reason for the increase in reflectivity is that the dielectric constant of water is four times higher than that of ice [19]. Another reason for the high reflectivity is the large size and low velocity of the melting-snow particles relative to those of rain drops. Continuing to melt while descending, the snowflakes become smaller in size and 5  Chapter 2--Hydrometeors: Structure and Characteristics  Refiechinfy foe* dal  Figure 2.2 Two views of the radar bright band: at the left a vertical profile of reflectivity and Doppler velocity as measured with vertically pointing Doppler Radar; at right a PPI map at 8° elevation on which the melting layer appears as a bright ring at about 12 miles. [Rogers]  Chapter 2—Hydrometeors: Structure and Characteristics  faster with less concentration. This will cause a decrease in the reflectivity as the rain medium is approached. A formula for the melting layer thickness (in meters) has been suggested by Klassen [13]: T = 100z 0 ' 17^(2.5)  where z is the reflectivity as given in [6]: z = 400// 1.4^(2.6) where R is the rain rate in mm/h. The melting layer disappears at high rain rates; it is usually assumed that it disappears above a rain rate of 30 mm/h.  2.2 Electromagnetic wave propagation in Hydrometeors The most accurate method to model melting layer scattering and attenuation is through the use of Mie scattering techniques for spherical rain droplets [11]. The advantage of using this technique is obvious — its accuracy. This method is quite complicated and thus undesirable in computer models. Another technique is to have the data for scattering and attenuation stored in files. Unfortunately, such files occupy a very large chunk of memory and they slow the system considerably. New models, that are simpler, but less accurate, have been developed in order to model hydrometeor scattering and attenuation [6, 10, 11, 12].  2.2.1 Hydrometeor scatter A Hydrometeor scattering model has been developed by Kharadly [11]. This model is both simple and relatively accurate in the 1-40 GHz range. A thorough description of 7  Chapter 2—Hydrometeors: Structure and Characteristics  the model is given in Appendix C. The model has "good" agreement with the "exact" results calculated by Kishk using Mie scattering [11]. At the top of the melting layer (S=O, where S is the ratio of the melted to the total volume in the melting-snow particle), the reflectivity is assumed to be the same as that of rain (S=1). The reflectivity then decreases by —6.5 dB per kilometer.  2.2.2 Hydrometeor attenuation  )  Attenuation plays an important role in the interference problem. On one hand, an increase in attenuation may decrease the interference. On the other hand, it may force the transmitting station to increase its transmitting power thus further aggravating the interference. Since attenuation affects the incident and scattered signals and since this attenuation varies as a function of rain rate and melting ratio (S), accounting for it occupies much of the computer time in the interference calculations. It is then desirable to use models that are reasonably accurate and simple. The models developed by Kharadly [10, 12] are simple and flexible; they can readily accommodate changes in physical assumptions relating to drop-size distributions, rain drop shapes, the density of the snow in the melting-snow particle, etc. An empirical formula has also been developed (Appendix D). While this formula is simpler, it is not flexible.  2.2.1 Kharadly 1st model for attenuation [10]  The melting-snow particles are considered to be spherical, of the same number and (relative) size distribution as the resulting rain drops. The radius of the representative particle is calculated using equation A.4. 1  ^  For a thorough analysis refer to the Appendix A and B (Kharadly's models [10, 12]) and Appendix D for empirical model.  8  Chapter 2—Hydrometeors: Structure and Characteristics  2.2.2 Kharadly 2nd model for attenuation [12]  The above model does not satisfy the conservation of mass criterion since it does not take into account the effect of the changing velocities of the melting-snow particles on the number density and hence the drop-size distribution. This model has been amended to include the effect of the velocity. 2.2.3 Kharadly 3rd model for attenuation [12]  Because of the deviation of the results of the 1st model from that of the exact values for the melting-snow layer, Kharadly introduced a correction factor that brought the results of the model closely to the exact attenuations calculated using Mie scattering. The correction factor is given by [12]: Factorl =  ni(2^+ 1]  { 2 — 51 (1-s) ^ x -1-- + 2 — S^2 + S  (2.7)  where f is the frequency in GHz, fr is the resonant frequency of the melting-snow particle, S is the melting degree , defined as the melted to the total volume in the representative melting-snow particle, and n is defined in Appendix B. 2.2.4 Kharadly 4th model for attenuation  Because of the deviation of the results of the 2nd model from that of the exact values for the melting snow layer, Kharadly introduced a correction factor that brought the results of the model closely to the exact attenuations calculated using Mie scattering. The factor is given by:  n2e/(1 S)  ^[^1110  Factor2 = [fn S(1 — nil {1 +^26^1 + —  9  (1 — 58)1  (2.8)  Chapter 2 Hydrometeors: Structure and Characteristics —  where a is the radius of the representative particle, 6 is the skin depth of water = .  ^ with C1 = 20.958, f is the frequency in GHz and e is the complex Real[f permittivity of water as given in Appendix B. The range of applicability of Kharadly formulas is between 1-40 GHz Although the above formulas were developed with the assumption that the density of the snow core in the melting layer (AO is 0.1, they still apply with a reasonable degree of accuracy for a wide range of A, (typically between 0.1 and 0.3). Figure 2.3 shows how close the 3rd and the 4th attenuation models are to the "exact" results for this range. We also note that Kharadly 3rd model yields the best results.  2.2.5 Empirical model Since the frequency stays constant during the interference calculations, it would be useful to have an equation where the frequency variable is separable from all other variables, which is not the case in any of the above models. This has been achieved through the development of an empirical formula for attenuation based upon the exact values [12] and is given by:  Ap A n (R,^) x ale  where, An(R , S) m  (2.9)  i sal —1 e —biSal m2 sa2-1 e —b 2 S° 2 m3e —b3S 1  (2.10)  with S < 1 with, Ali(R)  + 2Ro.0002 = Co + CiRm°3c  2Ro.0002 M2(R) = Do + Di R°.°°3 ^ 10  (2.11)  •  Chapter 2-Hydrometeors: Structure and Characteristics  f = 1.0 GHz  100.00 ,--.^90.00  -  I  ,1  ^  Kharadly 3rd attenuation model ^ Kharadly 4th attenuation model Exact calculations, Ps = 0.1 x Exact calculations, Ps = 0.2 + Exact calculations, Ps = 0.3  so.00 A 70.00  m 60.00 C  30.00 40.00  v 30.00 20.00 10.00 0.00 0.00  ^  0.20  ^  0.40^0.60  ^  0.80  ^  1.00  Degree of melting (S)  f = 10.0 GHz  P ;, 7.00  - ? t. i ,■ Kharadly 3rd attenuation model 6.00 _^,,t Kharadly 4th attenuation model .,t i:^'A s = 0.1 E .t^ Exact calculations, P 3.00 ;^'t^x Exact calculations, Ps = 0.2 i^'t x.^\^ + Exact calculations, Ps = 0.3 *•-• 4.00 i ,...--^k.t. 0 I ,:^IC,^ 't. If .'^\^'.;, ro= 3.00 -  C < 2.00  I): rr. ,^s•^..% i‘ /,' ^- -4.  li,:^.%,` i•. ...  1.00  0.00 0.00  0.20  0.40  0.60  0.80^1.00  Degree of melting (S)  Figure 2.3(a) A comparison of the attenuation profile of the melting layer between Kharadly 3rd and 4th attenuation models, and the Exact calculations for R (rain rate) = 12.5 mm/h, f (frequency) = 1.0 and 10.0 GHz and p a  11  =  0.1, 0.2, 03  -^  Chapter 2-Hydrometeors: Structure and Characteristics  18 .00  ^ Kharadly 3rd attenuation model  16.00  ^ Kharadly 4th attenuation model Exact calculations, Ps = 0.1 x Exact calculations, P = 0.2 + Exact calculations, s = 0.3  14.00  •  I  f= 20.0 GHz  12.00  10.0o  x  ,,t  0 8.00  -  '  '^• '  6.00  4.00  4  2.00  0.00 0.00  0.20  0.40^0.60  0.80  1.00  Degree of melting (S) 25.00  f = 40.0 GHz Kharadly 3rd attenuation model Kharadly 4th attenuation model Exact calculations, Ps = 0.1 x Exact calculations, PS = 0.2 + Exact calculations, Ps = 0.3  rY  20.00  E T7 .0  x  15.00  C 0  /,'  10.00  / ,'  7  It,' 5.00  -  0.00 0.00  0.20  0.60  0.40  0.80  1.00  Degree of melting (S)  Figure 2.3(b) A comparison of the attenuation profile of the melting layer between Kharadly 3rd and 4th attenuation models, and the Exact calculations for R (rain rate) = 12.5 mm/h, f (frequency) = 20.0 and 40.0 GHz and p, = 0.1, 0.2, 0.3  12  Chapter 2—Hydrometeors: Structure and Characteristics  Co, C1, C2, Do, D1, D2, al, a, # are frequency dependent constants. A n (R , ^)  approaches unity when S = 1.  The advantage of this formula is its simplicity and ease of use. Its disadvantage is that the formula does not, so far, take into account the average density of the snow in the melting-snow particle. The formula does represent the attenuation "quite well" from 1-100 GHz, however (Figure 2.4(a,b)). For a complete description, refer to Appendix D.  13  Attatuation in dB/km x le 210.00 200.00  Attenuadon in dB/km 16.00 15.00  190.00 180.00  14.00  170.00  13.00  160.00  12.00  150.00  11.00  140.00 130.00  10.00  120.00  9.00  110.00  8.00  100.00 90.00  7.00  i•-• 80.00 4:•  6.00  70.00  5.00  60.00 50.00  4.00  40.00  3.00  30.00  2.00  20.00  1.00  10.00  0.00  0.00 -10.00  0.00  ^ 1.00 0.50 Degree of melting (S)  0.00  ^  0.50  ^  1.00  Degree of melting (S)  Figure 2.3.(a) A comparison of the attenuation profile of the melting layer between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 1.0 and 10.0 GHz (p. = 0.1).  Attenuation in dB/km  Attenuation in dB/l® .xact calculations, in•0.9 Empirical formula Kharadly 3rd attenuation model  36.00 45.00  34.00 32.00  40.00  30.00 28.00  35.00  26.00 24.00  30.00  22.00 20.00  25.00  18.00 16.00  20.00  14.00 12.00  15.00  10.00 8.00  10.00  6.00 4.00  5.00  2.00 0.00  0.00 0.00  ^  0.50  ^  Degree of melting (S)  1.00  0.00  0.50  ^  1.00  Degree of melting (S)  Figure 2.3.(b) A comparison of the attenuation profile of the melting layer between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 20.0 and 30.0 GHz (p, = 0.1).  Chapter 3—The Interference Model  Chapter 3 The Interference Model 3.1 Approximate Radar equation The interference power received by an antenna due to the scattering of the electromagnetic wave by precipitation is given by [1] [8]:  C ')  1_ Pr _ A 2 GtGoor [ gi(OfirM a e RO.dV^(3.1) L 71);^(4703^i^Rpt?^-Y(Rt, , Vol  Where  gt  L  =  transmission loss  Pr  =  average interference power received  Pt  =  transmitted power  G1  =  transmitting antenna gain  Gr  =  receiving antenna gain  (0  g r (0) 7t  7  = normalized radiation pattern of the transmitting antenna at an angle 14 from the main lobe axis = normalized radiation pattern of the receiving antenna at an angle 3 from the main lobe axis = transmitter loss (loss factor < 1) - for simplicity assume 1 (no loss)  71r^=  receiver loss (loss factor < 1) - for simplicity assume 1 (no loss)  lit  =  distance from the transmitter to dV  Rr  = =  distance from dV to receiver  A  wavelength of the transmitter electromagnetic wave 16  Chapter 3—Intetference Model and System parameters  Figure 3.1 General Interference geometry  17  ^7  • •  Chapter 3—The Interference Model  7(11t, Rr) = the propagation loss due to precipitation and atmospheric gases along the path R i + R, •  bistatic scattering cross section in the direction of B and (Figure C.1)  The propagation loss y is given by [1]: lt+1, —0.1 A p dr-0.1 f A s dr  =  10  f  ^  (3.2)  where,  Ap  Attenuation coefficient due to precipitation  Aa  Attenuation coefficient due to atmospheric gases. For simplicity we will assume that the attenuation due to gases is negligible (A a = 0)  it, lr  the distances that the electromagnetic wave traverses the rain cell along the transmitter and receiver directions, respectively  The bistatic cross section a (O, c'b) is given by .  amax  a 0,^=^4,,^= / n(a)cy bi (0,i4,a)da  ^  (3.3)  0 where n(d) is the raindrop-size distribution, a is the radius of the raindrop, and Obi (a, a) is the bistatic cross section of a drop of equi-volume radius a in the direction of (9, (¢). In order to simplify the Radar equation, the narrow beam approximation to either one of the antennas may safely be introduced. Since the transmitter and receiver parameters 18  Chapter 3—The Interference Model  are interchangeable, we will assume that the antenna with the narrow-beam approximation has the subscript '1' and the other antenna has the subscript '2' as shown below: r A 2 GtG r f = P ^L^Pt^(4703 Vol  1  ) 92(0)o. ^ (0g 0 ( (e, (3.4) 134113^^y.dV  91  2  (  )  Using the narrow-beam approximation, dV = Sa dr(where S. is the 3dB circular surface area perpendicular to the main beam axis), equation 3.4 becomes 1 1+ 1 2  r e^tA  (  —0.1 f A r dr  Ss9)cy c) 0 — = _^ dr g ^ x 10^ (3.5)  ^2P P^ r A 2 GiG,0 2 h f  ,  L^Pt^2567r2^113 To  3.2 Antenna gain pattern [7] The standard method for representing the main lobe of an antenna is through the Gaussian-shaped pattern [7]: G1(B) =  e  —41n(  ^  where G i (B) is the gain at angle B from the main axis and  al  (3.6)  is the double-sided  half-power bandwidth. In order to represent the secondary lobe, we also assume a Gaussian-shaped pattern, but with a larger half-power bandwidth and a gain of K dB below the main lobe (Figure 3.2) : G 20) = 10 0.1K x e —iin2(*) 19  ^  (3.7)  Chapter 3—Interference Model and System parameters  0. 1  .1^4is^i^(is^2 ,,  angle in degrees  Figure 3.2 Simulation of the gain of an antenna with K=-15. al = 0.6, a3 = 5.5.  Chapter 3—The Interference Model  The total radiation pattern thus becomes: G^Gi(e)  + G2(e)^ (9) = 1 + 100.1K  (3.8)  3.3 The "Universal Model" A program have been developed to implement the above equations. This program can calculate the interference for any configuration of transmitter-receiver geometry. It accepts the following input variables: 1. the rectangular coordinates (xi, yi, z2) of the transmitter. Initially, we will consider that the transmitter is located at the centre of the main coordinate system, and hence the coordinates of the transmitter are (0,0,0) 2. the spherical coordinates (Ot, (ki) of the transmitter main lobe axis relative to the transmitter coordinate axes 3. the rectangular coordinates (x r , y r , z r ) of the receiver 4. the spherical coordinates (O r , O r ) of the receiver main lobe axis relative to the receiver coordinate axes 5. the polarization of the transmitter 2 6. the transmitter and receiver gains 7. the transmitter double-sided half-power bandwidth 8. the parameters connected with the receiver normal radiations pattern (a l , a 2 , K) 9. the rectangular coordinates (x e , y c , z,) of the bottom of the rain cell 10. the rain cell radius (p) and Height (Hc) 2  ^  We assume that the receiver accepts input electromagnetic wave regardless of polarization  21  Chapter 3—The Interference Model  11. the height (Hm — T) and the thickness (T) of the melting snow layer. Thickness will be zero in the absence of a melting snow layer 12. the rain rate at the centre of the rain cell RM and the distance (r o ) at which rainfall rate decrease by a factor of lie 13. the integration steps which largely determine the accuracy of the program 14. the frequency used 15. the density of the snow in the melting snow particle A, 16. the scattering and attenuation model to be used in the calculations. For scattering, we are limited to Kharadly's model. For attenuation, we can choose from Kharadly's 1st, 2nd, 3rd, 4th models and the empirical formula. Also note that: 1. The z-axis for all the above mentioned coordinate systems is in the direction of the vertical edge of the rain cell, in the direction opposite to the rain fall. 2. In order for the program to work correctly, at least one of the antennas has to satisfy the narrow-beam approximation. The program can also compute the interference for different melting layer profiles. However, a different subroutine is needed for each profile. Another method, which has not yet been implemented, is to have an external data file that contains the shape of the melting layer. The advantage is to avoid changing the program and recompiling it every time we introduce a different profile. The disadvantage of this procedure is that using an external file will add to the computation time.  22  Chapter 4--interference Calculations  Chapter 4 Interference Calculations 4.1 Introduction 4.1.1 Scattering in the ice/snow region Rayleigh scattering is generally assumed for the ice/snow region above the meltingsnow layer in the rain cell. The scattering cross section per unit volume at the top of the melting layer is given by [7]: c—1 2 Cr s = A4 + 2 Z x 10 —18 7r 4  m 2 /m 3^(4.1)  where c = complex relative permittivity of water, A is the wavelength, and z is the sum of the sixth powers of the diameters of all hydrometeors per unit volume. The magnitude of z is also given by the following empirical formula [6]: z = 400R 1 ' 4  772  6  77"/  —3  (4.2)  where R is the rain rate. The scattering decreases by —6.5 dB/km as we move higher into the ice/snow region. On the other hand, the attenuation in the ice/snow region is negligible and is assumed to be zero. Several transmitter-receiver systems will be considered below, in our study of interference caused by the rain and melting-snow. It will seem that the interference will vary depending on several factors, which will include frequency, rain rate, the height of the melting snow layer, its thickness, and the density of the snow in the melting-snow particle. 23  Chapter 4—Interference Calculations  4.2 Interference from up-link to terrestrial links in the near-forward direction 4.2.1 Description The calculations are performed for an experimental link which is part of the European COST 210 project [7] dealing with the influence of the atmosphere on interference between radio communication systems. The Chilbolton-Baldock (England) path [7] has been chosen because of the availability of the measured transmission loss. The geometry of interference is shown in Figure 4.2.1. A radio wave transmitted by an up-link toward a satellite is scattered by rainfall. The scattered electromagnetic wave interferes with a terrestrial receiving station operating at the same frequency and sharing a common volume. In this case the main lobe axes of the two antennas intersect. In order to maximize the interference, the centre of the rain cell is positioned at the intersection of antenna beam axes. The parameters used in the calculations are listed below, in Table 4.1. The measured (experimental) transmission loss of the path for 11.2 GHz frequency Chilbolton-Baldock path Station separation in km (rrt)  131 km  Scatter geometry  Vertical plane  Transmitting antenna gain in dB  59.0 dB  Receiving antenna gain in dB  40.5 dB  Transmitting antenna 3dB Beamwidth in degrees  0.18 degrees  Receiving antenna 3dB Beamwidth in degrees  1.6 degrees  Transmitting antenna elevation angle in degrees  20.0 degrees  Receiving antenna elevation angle in degrees(f r )  1.0 degrees  Transmitting antenna height from sea level in km (h e )  0.12 km  Receiving antenna height from sea level in km (h r  0.086 km  )  Table 4.1 Chilbolton-Baldock path parameters [7]  24  Chapter 4—Interference Calculations  (Graph not to scale)  rain cell Terrestrial receiving station  Transmitting antenna  Figure 4.2.1 Interference from up-link to terrestrial link geometry in the near forward direction.  25  Chapter 4--Interference Calculations  % of time 1.0 0.3 0.1 0.03 0.01 0.003 0.001  Rain Rate in mm/h 1.9 4.3 8.3 15.0 26.3 42.0 62.0  Transmission loss in dB 149.2 143.7 139.7 136.6 134.6 133.4 132.5  Table 4.2 Measured transmission loss for the Chilbolton-Baldock path at 11.2 GHz as a function of rain rate and percentage of time  and 2.1 km average rain height (Hm) is given in Table 4.2 as a function of rain rate and percentage of time. The geometrical parameters in Table 4.1 were converted to the common Cartesian system used in the model-program. The transmitting antenna is chosen as the origin of the system, with the horizontal plane as the x-y plane, the x axis pointing in the direction of the receiver, and the z axis pointing vertically upward. We now define an angle b, subtended at the Earth's centre by the link length, rrt, assuming an effective Earth radius of r e ff = 8500 km: ^  = rrri e ff  4.3  rad^  The Cartesian receiver angle is then calculated by: f r 1 = 90 — B r  = arcsin(cose r sinb sine r cos(5)  ^  4.4  and the coordinates of the receiving antenna becomes ^ (xr, yr, zr) = (rri, 0,  hr  —  ht  —  rrt 2 )  The converted input parameters for the model-program are given in Table 4.3: 26  4.5  Chapter 4--Interference Calculations  4.2.2 Computed results The results of the computations for this case are plotted in Figures 4.2.2(a,b,c,d,e) and 4.2.3(a,b,c). Figures 4.2.2(a,b,c,d,e) show the transmission loss versus rain rate for Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km, respectively. The transmission loss calculated for the following frequencies, f = 1.0, 5.0, 10.0, 20.0, 30.0 and 40.0 GHz is shown in each one of these figures. Also, we calculated the interference caused by a rain-only cell, and a rain cell with a melting-snow layer with p s = 0.1, 0.2, 0.3 at each frequency.  Scattering Attenuation Profile of the melting layer (xt, Yt, zt) of the transmitter (9 i , fit of the transmitter Polarization of the transmitter 0 1/2 of the transmitter (x r , y r , z r ) of the receiver (Or, Or of the receiver (ai,a2,K) of the receiver rain cell height (He ) Gain of the transmitter Gain of the receiver (x c , yc, z c ) of the rain cell Frequency Rain rate Hm )  )  p.,  Kharadly scattering model Kharadly 3rd attenuation model S = h/H (linear) (0, 0 , 0) in km (70, 0) degrees 0 degrees (vertical) 0.00314 rad (131, 0, -1.0435) km (88.117, 180) degrees (1.6, 4.5, -15) 10.0 km 794328 11220 (7.912,0,0) km 1, 5, 10, 20, 30, 40 GHz 0.5-150 mm/h. 2.0, 2.5, 3.0, 3.5, 4.0 km 0.1, 0.2, 0.3  Table 4.3 The converted input parameters for the model-program 27  Chapter 4—intelference Calculations  The centre of the common volume in this geometry is at 3 km from the ground level (Hm = 3.0 km). We notice that it is when the rain height is in the common volume that we get the maximum effect of the melting-snow layer. The layer increases the interference at the lower frequencies and decreases it for the higher frequencies. At optimum rain height (Hm = 3 km) and at a frequency of f=1.0 GHz (Figure 4.2.2(c)), the interference enhancement caused by the melting layer for p s = 0.1 is 2.0, 4.2, 7.0, 10.5 dB for rain rate of 1.0, 3.0, 10.0, 30.0 mm/h, respectively. The interference level decreases for A, = 0.2 and p s = 0.3. Nonetheless the enhancement remains significant at 1.5, 3.0, 5.0, 8.0 dB for p s = 0.2 and 1.0, 2.0, 4.0, 6.5 dB for A, = 0.3. As the frequency is increased, we note that the effect of the melting-snow layer on the transmission loss decreases. The melting-snow layer enhancement decreases to 2.0, 4.0, 6.1, 7.5 dB, and 1.8, 3.0, 3.0, 0.0 dB for f = 5.0 and 10 GHz respectively. An interesting observation occurs when the frequency is increased further. The melting layer starts to degrade, instead of enhance, the interfering signal. For f = 20 GHz, the melting-snow layer causes a drop in the interfering signal by —2.0 and —7.5 dB for rain rate of 10.0 and 30.0 mm/h, respectively. This degradation becomes more pronounced at still higher frequencies. For A, = 0.1, the degradation becomes 0.0, —0.8, —4.0, —12.0 dB for f = 30.0 GHz and —0.5, —1.9, —6.0, —18.0 dB for f = 40.0 GHz. Higher values of p s tends decrease this gap but not by much. For A, = 0.2, the gap becomes —0.5, —1.2, —4.8, —16.0 dB and for A, = 0.3, the gap reduces to —0.5, —1.1, —4.0, —12.0 dB. As the rain height moves out of the centre of the common volume, the enhancement due to the melting-snow layer declines considerably. At Hm = 2.5 or 3.5 km (Figure 4.2.2(b) and Figure 4.2.2(d), respectively), the effect of the melting layer is still considerable at 1.5, 2.75, 5.0, 7.5 dB for f = 1 and 1.2, 2.5, 4.5, 6.0 dB for f = 5 GHz and Hm = 28  Chapter 4—Interference Calculations  3.5 km. The enhancement is slightly greater for Hm = 2.5 km. For Hm = 2.0 and 4.0 km (Figure 4.2.2(a) and Figure 4.2.2(e), respectively), the enhancement becomes very small. For Hm = 4.0 km, the enhancement becomes 0.8, 1.2, 2.2, 2.5 for f = 1.0 GHz and 0.5, 1.0, 2.0, 2.5 for f = 5 GHz. For Hm = 2.5 km, the enhancement is slightly greater. Figures 4.2.3(a,b,c) shows the transmission loss versus rain rate for f = 1.0, 5.0, 10.0, 20.0, 30.0, 40.0 GHz and Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km for a rain-only cell. We observe that for f = 1.0, 5.0, 10.0 GHz, the interference increases with Hm. This is due to the higher reflectivity of rain compared to that in the ice/snow region. For f = 20 GHz, the interference starts to decrease as Hm is increased, in the high rain rate region. For higher frequencies this phenomenon becomes more severe and the interference at Hm = 2.0 is 68.5 dB higher than the interference at Hm = 4.0 for rain rate of 100 mm/h and f = 40.0 GHz (Figure 4.2.3(c)). This leads us to conclude that at higher frequencies and rain rates, the effect of the ice/snow region in the rain cell is more significant than that of rain despite its lower reflectivity. Even though an exact comparison between the measured transmission loss (Table 4.2) and our calculations is not possible because of frequency difference and because, in reality, Hm acts as a random variable rather than the deterministic values we assume, we observe that the experimental transmission loss adjusted for atmospheric attenuation (discrete data in Figure 4.2.2(a)) agrees well with our calculations for f = 10.0 GHz (Figure 4.2.2(a)).  29  Chapter 4--interference Calculations  Transmission loss in dB -120.00  le+00  Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  3  le+01  3  le+02  Figure 4.2.2(a) Interference versus rain rate for various frequencies (f in GHz), p, (m = 1—p,), and for Hm = 2.0 km.  30  Rain Rate in mm/h.  Chapter 4—Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  le+00  ^ ^ ^ ^ 3 le+01 3 Ie+02  Figure 4.2.2(b) Same as Figure 4.2.2(a), with Hm = 2.5 km.  31  Rain Rate in mm/h.  Chapter 4 Interference Calculations —  Transmission loss in dB I  -128.00 -130.00 -  Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -132.00 -134.00 -136.00 -138.00 -140.00 -142.00 -144.00 -146.00 -148.00 -150.00 -152.00 — 4* -154.00 -156.00 -158.00  '....  -160.00 -162.00 -164.00 -166.00 -168.00 -170.00 — •a -172.00 -174.00  I le+00  I  1^1^1  3  Ie+01  3  Ie+02  Figure 4.2.2(c) Same as Figure 4.2.2(a), with Hm = 3.0 km.  32  Rain Rate in mm/h.  Chapter 4--Interference Calculations  - Transmission loss in dB  Rain M_L,m=0.9  -130.00 -135.00  M_L,m4.7  -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00 -180.00 -185.00 -190.00 -195.00 -200.00 -205.00 -210.00 le+00  3  le+01  3  le+02  Figure 4.2.2(d) Same as Figure 4.2.2(a), with Hm = 3.5 km.  33  Rain Rate in nun/h.  Chapter 4—Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m4.7  -130.00 -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00 -180.00 -185.00 -190.00 -195.00 -200.00 -205.00 -210.00 -215.00 le+00  ^Rain Rate in mm/h. ^ ^ ^ ^ 3 le+01 3 le+02  Figure 4.2.2(e) Same as Figure 4.2.2(a), with Hm = 4.0 km.  34  Chapter 4—Interference Calculations  Transmission loss in dB Hm=2.0 km Hm=2.5 km  -125.00  Hm=3.0 km Hm=3.5 km  -130.00  Hm=4.0 km  -135.00  -140.00  -145.00  -150.00  -155.00  -160.00  -165.00  -170.00  -175.00 le+00  ^ ^ ^ ^ le+02 3 le+01 3  Figure 4.2.3(a) Interference versus rain rate for various rain heights (Hm), and for f (frequency) = 1, 5, 10, 20GHz.  Rain Rate in nun/h.  Chapter 4-- Interference Calculations ,  Transmission loss in dB -120.00 OIM  Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km  -125.00 -130.00  Fin3=1.6k1  -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00 -180.00 -185.00 le+00  ^ I^I^I ^ ^ 3 le+01 le+02  Figure 4.2.3(b) Same as Figure 4.2.3(a), with f = 30 GHz.  36  Rain Rate in mm/h.  Chapter 4—Interference Calculations  Transmission loss in dB -120.00 —  1  ^^ 1  1  Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km  -125.00 -130.00 -135.00 —  . .........  -14o.00 ..........  . ..^.. ...  ani.1.6k1  ..... ..=, ■ .^ . ... -  ..  ... `  Ik  ...... " C.^•.. ♦^5,..  e.^  ......—...^  -145.00 —  ...  "...  .. •■ ^ %  ••■•  -150.00 -  ♦ ♦^• • s  %`%  -155.00 -160.00 -  .  1==40.0 GHz  -165.00 -170.00 -175.00 -180.00 — -185.00 -190.00 --195.00 — -200.00 -205.00 -210.00 —  -215.00 — 1^1^I^1^  le+00^3^le+01^3^  1  1  le+02  Figure 4.2.3(c) Same as Figure 4.2.3(a), with f = 40 GHz.  37  Rain Rate in mm/h.  Chapter 4—interference Calculations  4.3 Interference from up-link to terrestrial links in the near-backward direction  4.3.1 Description The geometry of the interference is close to that of the near-forward scattering (Figure 4.3.1). The only difference is that the receiving antenna is 180 degrees from the previous case but maintaining the same distance to the common volume. The common volume remains 3km high.  4.3.2 Computed results The results of the computations in this example are given in figures 4.4.2(a,b,c,d,e) and 4.4.3(a,b,c). Figures 4.4.2(a,b,c,d,e) show the transmission loss versus rain rate for Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km, respectively. The transmission loss calculated for the following frequencies, f = 1.0, 5.0, 10.0, 20.0, 30.0 and 40.0 GHz is shown in each of these figures. Also, we calculated the interference caused by a rain-only cell, and a rain cell with a melting-snow layer with p s  =  0.1, 0.2, 0.3 at each frequency.  The maximum effect of the melting snow layer occurs when Hm = 3.0 km (Figure 4.3.2(c)). For f = 1.0 GHz, the interference enhancement caused by the melting snow layer for p s = 0.1 is 1.8, 3.0, 6.0, 9.0 dB for rain rate of 1.0, 3.0, 10.0, 30.0 mm/h, respectively. This interference decreases for p s  =  0.2 and p, = 0.3. Nonetheless the enhancement  remains considerable. The values of the interference enhancement calculated in this case are slightly lower than those calculated in the near-forward direction. As we increase the frequency, the melting layer enhancement decreases to 1.5, 2.5, 5.25, 6.0 dB and 1.1, 2.1, 2.5, 0.8 dB for f = 5.0 and f = 10 GHz respectively. At higher frequencies (f = 20.0, 30.0, 40.0 GHz) the melting snow layer degrades the interference signal. For A, = 0.1, the degradation becomes —0.4, —1.0, —2.5, —4.0 for f = 30 GHz 38  Chapter 4—Interference Calculations  (Graph not to scale)  rain cell  Transmitting antenna  Figure 4.3.1 Interference from up-link to terrestrial link geometry in the near backward direction.  39  Chapter 4—Interference Calculations  and-0.5, —1.5, —2.5, —1.75 for f = 40.0 GHz. Higher p s values help reduce the gap. This reduction is significantly lower than the one experienced in the previous example. As Hm moves away from the centre of the common volume, the effect of the meltingsnow layer decreases. At Hm = 3.5 km (Figure 4.3.2(d)), the effect of the melting layer is still considerable at 1.2, 2.5, 4.0, 6.25 dB for f = 1.0 and 1.0, 2.0, 3.0, 3.7 dB for f = 5 GHz. The enhancement is slightly greater for Hm = 2.5 km (Figure 4.3.2(b)). This greater enhancement is due to the closer proximity of the melting snow layer to the centre of the common volume. Again the interference enhancement is slightly less than the previous example. For Hm = 2.0, and Hm = 4.0 the interference enhancement decreases significantly. For Hm = 4.0 km, the enhancement becomes 0.8, 1.4, 2.0, 2.5 dB for f = 1.0 GHz and 0.5, 1.0, 1.3, 1.5 dB for f = 5.0 GHz. For Hm = 2.0 km, the enhancement is slightly greater. Figures 4.3.3(a,b,c) show the transmission loss versus rain rate for f = 1.0, 5.0, 10.0, 30.0, 40.0 GHz and Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km for a rain-only cell. We observe that for f = 1.0, 5.0, 10.0 GHz, the interference increases with Hm at the lower frequencies. This is also true for the higher frequencies coupled with low rain rate. However, the interference drops considerably with high rain rates as Hm increases. For rain rate of 100 mm/h and f = 40 GHz (Figure 4.3.3(c)), the interference for Hm = 2 km is 14.5 dB higher than that for Hm = 4.0 km. We also observe that the reduction of the interference signal is much less than in the previous example. Comparing the current results with those in the near-forward case, we observe that the two are comparable for lower frequencies and rain rates. For a combination of higher frequency and high rain rate, the difference between the two is very large to be accounted for by scattering properties alone. This difference can only be due to different attenuation paths for the transmitted and scattered waves between both cases.  40  Chapter 4---Interference Calculations  Transmission loss in dB -120.00  Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -125.00 -130.00 -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00 le+00  ^ ^ ^ ^ le+02 le+01 3 3  Figure 4.3.2(a) Interference versus rain rate for various frequencies (f in GHz), p, (m = 1—p,), and for Hm = 2.0 km.  41  Rain Rate in nun/h.  Chapter 4--Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -125.00 -130.00 -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00 le+00  ^^ ^^ le+02 le+01 3 3  Figure 4.3.2(b) Same as Figure 4.3.2(a). with Hm = 2.5 km.  42  Rain Rate in mm/h.  Chapter 4--Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m41.8 M_L,m=0.7  -125.00 -130.00 -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00  le+00  ^^ ^^ le+02 le+01 3 3  Figure 4.3.2(c) Same as Figure 43.2(a), with Hm = 3.0 km.  43  Rain Rate in mm/h.  Chapter 4--Inserference Calculations  Transmission loss in dB -125.00 —  Rain  -130.00 —  M_L,m=0.8 M_L,m=0.7  -135.00 —  -140.00 —  -145.00 —  -150.00 —  -155.00 —  -160.00 —  -165.00  -170.00  le+00  3  le+01  3  Rain Rate in mm/h. le+02  Figure 4.3.2(d) Same as Figure 4.3.2(a), with Hm = 3.5 km.  44  Chapter 4--Interference Calculations ..•  Transmission loss in dB -125.00  Rain M_L,m=0.9 M_L,m.8 M_L,m.7  -130.00  -135.00  -140.00  -145.00  -150.00  -155.00  -160.00  -165.00  -170.00  le+00  ^^ ^^ 3 le+02 3 Ie+01  Figure 4.3.2(e) Same as Figure 4.3.2(a). with Hm = 4.0 km.  45  Rain Rate in mm/h.  Chapter 4 Interference Calculations —  Transmission loss in dB Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km Hm=4.0 km  -125.00 -130.00 -135.00 -140.00 -145.00 -150.00 -155.00 -160.00 -165.00 -170.00 -175.00  1  1  le+00  3  3  le+02  Figure 4.3.3(a) Interference versus rain rate for various rain heights (Hm), and for f (frequency) = 1, 5, 10, 20GHz.  Rain Rate in mm/h.  Chapter 4--Interference Calculations  Transmission loss in dB Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km Hm=4.0 km  -122.00 -123.00 -124.00 -125.00 -126.00 -127.00 -128.00 -129.00 -130.00 -131.00 -132.00 -133.00 -134.00 -135.00 -136.00 -137.00 -138.00 -139.00 -140.00 -141.00 -142.00 -143.00 -144.00 -145.00 Ie+00  ^ ^ ^ ^ 1e+02 1e+01 3 3  Figure 4.3.3(b) Same as Figure 4.3.3(a), with f = 30 GHz.  47  Rain Rate in mm/h.  Chapter 4--Interference Calculations  Transmission loss in dB Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km Hm=4.0 km  -122.00 -123.00 -124.00 -125.00 -126.00 -127.00 -128.00 -129.00 -130.00 -131.00 -132.00 — -133.00 -134.00 -135.00 -136.00 -137.00 -138.00 -139.00 -140.00 -  1=40.0 GHz  -141.00 — " -142.00 -143.00 -144.00 L le+00^3^le+01^3^le+02 Figure 4.3.3(c) Same as Figure 4.3.3(a), with f = 40 GHz.  48  Rain Rate in nun/h.  Chapter 4—Intelference Calculations  4.4 Interference from up-link to satellite in the forward direction 4.4.1 Description The geometry of the interference is shown in Figure 4.4.1. A radio-wave transmitted by an up-link toward a satellite is scattered by rainfall. The scattered electromagnetic wave interferes with another nearby satellite operating at the same frequency. The parameter used in these calculations are listed in Table 4.4. The geometrical parameters are given in Cartesian coordinates . Scattering Attenuation Profile of the melting layer (xs, yt, zt) of the transmitter (Os , Os ) of the transmitter Polarization of the transmitter 0 1 / 2 of the transmitter (x i., yr, zr) of the receiver (O r , (/),.) of the receiver (a l , a2, K) of the receiver rain cell height (11s ) Gain of the transmitter Gain of the receiver (x c , Yc, zc) of the rain cell Frequency Rain rate Hm m (m = 1 — AO  Kharadly's scattering model Kharadly 3rd attenuation model S = h/H (linear) (0, 0 , 0) in km (70, 0) degrees 0 degrees (vertical) 0.00314 rad (3291, 0, 1900) km (110, 180) degrees (3.0, 7.5, -10) 10.0 km 794328 5011.87 (7.912,0,0) km 1, 5, 10, 20, 30, 40 GHz 0.5-150 mm/h. 2.0, 2.5, 3.0, 3.5, 4.0 km 0.9, 0.8, 0.7  Table 4.4 Parameters used for the interference calculations from up-link to satellite 49  Chapter 4—Interference Calculations  (Graph not to scale)  receiving satellite  rain cell  Transmitting antenna  Figure 4.4.1 Interference from up-link to satellite geometry in the forward direction.  50  Chapter 4--Inuetference Calculations  4.4.2 Computation results The results of the computations in this example are given in figures 4.4.2(a,b,c,d,e) and 4.4.3(a,b,c). Figures 4.4.2(a,b,c,d,e) show the transmission loss versus rain rate for Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km respectively. The transmission loss calculated for the following frequencies, f = 1.0, 5.0, 10.0, 20.0, 30.0 and 40.0 GHz is shown in each one of these figures. Also, we calculated the interference caused by a rain-only cell, and a rain cell with a melting snow layer with  ps =  0.1, 0.2, 0.3 at each frequency.  In this case, it is noticed that for f = 1, 5, 10 GHz, the enhancement caused by the melting layer is not only significant but it remains strong for a wide range of Hm. With a rain rate of 30 mm/h and frequency of 5.0 GHz the enhancement is 4.0, 6.0, 8.8, 8.0, 3.75 dB for Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km, respectively (Figures 4.4.2(a), (b), (c), (d) and (e), respectively). The interference level is also significant at 10.0 GHz where, for 30 mm/h, the enhancement becomes 3.0 ,4.0 ,5.0 ,3.7, 1.2 dB. At higher frequencies, we observe that the melting layer tends to reduce the interference signal for higher rain rates. This reduction increases with frequency, rain rate and Hm. For a rain rate of 30.0 mm/h and a frequency of 40.0 GHz, the interference signal is reduced by 4.0, 6.0, 6.2, 7.5, 8.0 dB for Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km, respectively. We again observe that  !I s  plays an  important role in enhancing or reducing the interference level. Figures 4.4.3(a,b,c) show the transmission loss versus rain rate for f = 1.0, 5.0, 10.0, 20.0, 30.0, 40.0 GHz and Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km for a rain-only cell. It is observed that at the lower frequencies (f = 1, 5, 10 GHz), the interference increases with higher Hm. At the higher frequencies, the same is true for low rain rates, but for high rain rates, the interference decreases sharply for higher values of Hm. For a rain rate of 100 mm/h and a frequency of 40.0 GHz, the interference for Hm = 2.0 km is 36 dB 51  Chapter 4—Interference Calculations  higher than that for Hm = 4.0 km (Figure 4.4.3(c)).  4.5 Summary The above three examples show that the melting-snow layer significantly affects the transmission loss. The maximum effect occurs when the melting-snow layer exists in the common volume. But even outside the common volume, we noticed that the melting layer did exert considerable influence. We observed also that the melting layer tends to increase the interference level at the lower frequencies and decrease it for higher frequencies. We also observed that the ice/snow region significantly contributes to the interference level at the higher frequencies. The attenuation by rain and melting-snow at high frequencies degrades the scattered signal, thus considerably reducing the interference level from rain and melting-snow. The scattered wave from the ice/snow region does not suffer from attenuation (except if the scattered wave intersects the melting layer or rain. This is limited to the lower parts of the ice/snow region) and thus contributes significantly to the interference level.  52  Chapter 4--Interference Calculations  Transmission loss in dB Rain  -195.00  M_L,m=0.9 M_L,m=0.8  -200.00  M_L,m4.7  -205.00  -210.00  -215.00  -220.00  -225.00  -230.00  -235.00  -240.00  -245.00  le+00  ^Rain Rate in mm/h. ^ ^ ^ ^ 3 1e+01 3 le+02  Figure 4.4.2(a) Interference versus rain rate for various frequencies (f in GHz), p, (m = 1—p,), and for Hm = 2.0 km.  53  Chapter 4--Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -200.00 -205.00 -210.00 -215.00 -220.00 -225.00 -230.00 -235.00 -240.00 -245.00 le +00  1  3  3  le+02  Figure 4.4.2(b) Same as Figure 4.4.2(a). with Hm = 2.5 km.  54  Rain Rate in mm/h.  Chapter 4—Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8  -200.00 -202.00 -204.00 -206.00 -208.00 -210.00  0 Go  -212.00 -214.00 -216.00 -218.00 -220.00 -222.00 1..  -224.00 -226.00 -228.00 -230.00 -232.00 -234.00 -236.00 -238.00 -240.00 -242.00 -244.00 -246.00 H^I le+00  I ^—  le+01  3  3  1e+02  Figure 4.4.2(c) Same as Figure 4.4.2(a). with Hrn = 3.0 km.  55  Rain Rate in mm/h.  Chapter 4--Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -202.00 -204.00 -206.00 -208.00 -210.00 -212.00 -214.00 -216.00 -218.00 -220.00 -222.00 -224.00 -226.00 -228.00 -230.00 -232.00 -234.00 -236.00 -238.00 -240.00 -242.00 -244.00 le+00  ^Rain Rate in mm/h. ^ ^ ^^ 3 le+01 3 le+02  Figure 4.4.2(d) Same as Figure 4.4.2(a). with Hm = 3.5 km.  56  Chapter 4--Interference Calculations  Transmission loss in dB Rain M_L,m=0.9 M_L,m=0.8 M_L,m=0.7  -202.00 -204.00 -206.00 -208.00 -210.00 -212.00 -214.00  /Sr .. -216.00 —^ •■ %, 1.. ../: f^...% •■ .., . -^ 218.00 — ...% .e.s^ W /'^ -220.00 —^ ...% .:/i"^•^% -222.00 —ive / .% -224.00 —^ '''% -226.00  -  —  -228.00 — -230.00 —\\ t /;•7 -232.00 I—^ 0 CI /44'^ 1..^ -234.00 —^  _  -236.00 -238.00 -240.00 -242.00 -244.00 — le+00  ^ ^ ^^ le+02 3 le+01 3  Figure 4.4.2(e) Same as Figure 4.4.2(a), with Hm = 4.0 km.  57  Rain Rate in mm/h.  Chapter 4--Interference Calculations  Transmission loss in dB 1  1  1  Hm=2.0 km Hm=2.5 km Hm=3.0 km Hm=3.5 km Hm=-4.0 km  -200.00 -205.00 -210.00  f',•  10 .  0 ..  •^.  .4. •^.„^ ,  -=-:...-•••^  -215.00  „.."" ,...  #  , 1  -220.00 -225.00 -230.00  GP: f:-;:-.*** *  itf.„-Z ^  ,  0P--?-':"--e^ s:,-.',-..: • „'-'-' ..,-..••  ii,/  '-,--  -....„--  .:---.:,•-•  ,  „  -235.00  -,  ,  1 .0 ol'  t  -240.00 -245.00 1^1^1^1^1  le+00^3^1e+01^3^le+02 Figure 4.4.3(a) Interference versus rain rate for various rain heights (Hm), and for f (frequency) = 1. 5, 10, 20GHz.  Rain Rate in mm/h.  Chapter 4  —  lnierference Calculations  Transmission loss in dB 196.00 —^I^I^I^I^I^— Hm=2.0 km Hm=2.5 km -198.00 Hm=3.0 km -200.00 Hm=3.5 km Hm=4.0 km -202.00 -204.00 -206.00 —  .1..".-^..,, ss . -208.00 —^ 5^•^ — 5^s N.^5^•s N.^5 ' -210.00 — ^ , -..:''......^ N.^5 .^ -212.00 —  .^. -^ A"::::.." \ N^5 ^="::"^ -214.00 — ...* ^  ^,^  ....^.  r  -216.00 —  ^f=30.0 -218.00 -220.00  -  ■  GHz^  7^^  -222.00 —^  —  .  . \ %^. ,  . ^ •  ■  -224.00 —^  ^^—  -226.00 —^  A^A  t^— t % -  -228.00^ -230.00 L^  %  -232.00 7^  ^I  ^1^I^I^1 le+00^3^le+01^3^le+02  Figure 4.4.3(b) Same as Figure 4.4.3(a), with f = 30 GHz.  59  Rain Rate in mm/h.  Chapter 4 Interference Calculations —  Transmission loss in dB -194.00 Hm=2.0 km Hm=2.5 km  -196.00 -198.00 -200.00 -202.00 .......... ............ ....... .............. -204.00 —^ -206.00 —  Hm=3.0 km Hm=3.5 km Hm=4.0 km  ..•-0' ........  -208.00 —  .‘.. .. -210.00 —^0:....^ .^ .. .., .^ ---::::-..--'^ .^. . -212.00 — :::.--^ .^s.., -214.00 — -216.00 — •  -218.00 —^ -220.00 -  222.00  —^  f 40.0 GHz  S.  -  -224.00 — -226.00 — -228.00 r-230.00 -232.00 -234.00 — -236.00 -238.00 1— -240.00 -Ie+00^3^le+01^3^le+02 Figure 4.4.3(c) Same as Figure 4.4.3(a), with f = 40 GHz.  60  Rain Rate in mm/h.  Chapter 5—COST 210 rain-cell model  Chapter 5 COST 210 rain-cell model 5.1 Introduction The CCIR working party 5C has recently adopted the COST 210 model [7] as a basis for predicting transmission loss [5]. An accompanying document was presented by Canada [4] which showed that, using the COST 210 rain cell model, the introduction of a melting snow layer in an optimum position significantly affects the transmission loss up to 11 GHz. It also concluded that the melting layer should be taken into consideration while calculating the interference level, since the interference level introduced by the presence of the melting layer is larger than that introduced in changing from one composite climate to another. This is an attempt to expand on the original study and to compare the COST 210 rain model with Capsoni's model and to see if the COST 210 rain cell is able to model the effect of the melting snow layer for a wide range of rain heights and frequencies. Readers should be reminded that it is only the COST 210 rain cell geometry that is implemented and not their interference calculation methodology. To calculate the interference, the method outlined in Chapter 3 was applied with provisions to account for the attenuation outside the rain cell (refer below). This method yielded results similar to those calculated by the COST 210 program for the Chilbolton-Baldock path [7]. 5.2 COST 210 rain cell model [7] The rain cell centre is assumed to be at the intersection of the main beam antenna axes (i.e. centre of the common volume). Scattering is assumed to occur within one fixed, cylindrical rain cell of circular cross-section. The diameter of the cell depends on 61  Chapter 5—COST 210 rain-cell model  -cast II ■  aXilliFf.110:4110 :412ILE LTACML TICBCt "1.2"Zi. !.:116.15t tMaSr illICt; Z152.11.'  ,,,,  .  ;;;;Iirr exti:: ,  Figure 5.1 COST 210 rain-cell model  62  Chapter 5—COST 210 rain-cell model  the rainfall rate as: dc = 3.3R 0.08  ^  -  5.1  On the other hand, attenuation occurs inside and outside the rain cell. Inside the rain cell, the empirical formula for attenuation is used. Outside the rain cell, the attenuation F R , between the edge of the rain cell and a point at distance d is given by the following  exponential function: 1 — e-d/r,n  r R = Ap r m  dB/km^ 5.2  where r m , the scale length for rain attenuation, is given by: (R+1)° 19 r m = 600R 0.510 -  -  =  km^  5.3  elevation angle, and A p is the specific attenuation for rain, calculated from the  attenuation empirical formula, in dB/km. Equation 5.2 is valid if the whole path is below the rain height Hm. If only part of the path — let us say between distances di and d2 from the edge of the rain cell — is below the rain height: u/rm^e – d 2 /r ni )  FR =  (e–d  dB/km^5.4  COSE  For those portions of the propagation path that are above Hm, zero attenuation is assumed. The diameter of the melting layer cell is assumed to be the same as that of the rain below it. The melting layer attenuation is assumed to reduce at the same exponential rate as the rain attenuation outside the core cell. Since the specific attenuation varies with the height within the melting-snow layer, a numerical integration is carried out in the vertical direction: F R = rm COSE  . i=1  ^Api (e—d,/rm e—ds+iirni)  63  5.5  Chapter 5—COST 210 rain-cell model  where A p t is the average specific attenuation in the region between di and di  +i  in the  melting-snow layer. n is the number of integration steps. Despite this addition to the rain cell, we will continue to refer to it as the COST 210 rain cell.  5.3 Results for sample interference geometries The calculations are done for the geometry described in section 4.2 of Chapter 4. Figures 5.2(a,b,c,d)-5.6(a,b,c,d) show the transmission loss at f = 1.0, 5.0, 10.0, 20.0, 30.0, 40.0 GHz and Hm = 2.0, 2.5, 3.0, 3.5, 4.0 km for both the Capsoni rain cell model and the COST 210 rain cell model. The scattering model used in both cells is that of Dr. Kharadly. The Empirical model is used for attenuation. This is somewhat different from the COST 210 model where they use modified Rayleigh scattering for rain and a different attenuation model. Neither COST 210 attenuation nor scattering models account for the melting-snow region. For the rain-only cells, we observe that both models predict similar transmission losses for all Hm and rain rates at lower frequencies (f = 1.0, 5.0, 10.0 GHz). However, the two models' results differ considerably at higher frequencies and especially for the lower rain rates. For Hm = 3.0 km and f = 40 GHz (Figure 5.4 (d)), the Capsoni model interference level is 15.0, 16.0, 13.5, 4.2, 1.5 dB higher than the interference level calculated using COST 210 rain cell for R = 1.0, 3.0, 10.0, 30.0 mm/h, respectively. As Hm increases, the two models' interference curves seems to produce better agreement. For Hm = 3.0 km (Figures 5.4 (a),(b),(c),(d)), which is the optimum position of the melting layer in the common volume, the COST 210 rain cell models the effect of the melting snow layer very nicely. For Hm = 3.5 km (Figures 5.5 (a),(b),(c),(d)), the COST 210 cell tends to overestimate the enhancement caused by the melting snow 64  Chapter 5—COST 210 rain-cell model  layer. This is in contrast to Hm = 2.5 km (Figures 5.3 (a),(b),(c),(d)), where the COST 210 model underestimates its effect considerably. For Hm = 2.0, and 4.0 km (Figures 5.2 (a),(b),(c),(d) and Figures 5.5 (a),(b),(c),(d), respectively), the melting layer does not enter into consideration since it is out of the path of the transmitting beam. We observe from the Capsoni model that for Hm = 2.0 and 4.0 km, the melting layer does play a role (albeit reduced) in the interference problem. The reason that the COST 210 cell does not account for the melting layer is the small radius of the COST 210 cell. 5.4 Conclusion The computation time for the COST 210 model is much less than that of the Capsoni rain model. This is directly related to the radius of the core rain cell of COST 210. For rain rates of 0.5, 2.5, 5.0, 12.5, 25.0, 50.0, 100.0, 150.0 mm/h, the radius of the Capsoni rain cell is 10.45, 9.4.0, 8.95. 8.35, 7.90, 7.45, 7.00, 6.74 km respectively. On the other hand the radius of the COST 210 rain cell is 1.74, 1.53, 1.45, 1.35, 1.28, 1.21, 1.14, 1.11 km. We can see readily that the COST 210 rain cell, which is about 6 times smaller than the Capsoni model, will save a considerable amount of computer time. For lower frequencies, we notice that the Capsoni and COST 210 rain models yield similar results for all Hm values. For higher frequencies, we see that there is a large difference between the two models. The COST 210 model interference level is much lower than that of the Capsoni cell for lower rain rate. For a high rain rate and lower frequencies, the COST 210 model yields the higher interference level. Since the Capsoni rain cell is the more realistic, it is safe to assume that the COST 210 model will underestimate the interference at higher frequencies (and large station separation) and rain rate. 65  Transmission loss in dB psomun  NPsOnTAW§  .--  -130.00  Cost,ram  -126.00 -128.00  -135.00  -130.00 -140.00  -132.00 -134.00  -145.00  -136.00 -150.00  -138.00 -140.00  -155.00  142.00 -144.00  -160.00  -146.00 -165.00  -148.00 -150.00  -170.00  -152.00  -175.00  -154.00 -156.00  -180.00 ^ ^ ^ ^ le+02 3 1e+01 3 1e+00 Rain Rate in tnin/h.  ^  -158.00 le+00  ^ ^ ^ ^ Ie+02 -3 1e401 3 Rain Rate in mm/h.  Figure 5.2(a,b) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 2.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  Transmission loss in dB  ^  Transmission loss in dB Capsoni,rain Gpsamm0.9 Cost,rain  -120.00 -122.00  -122.00  -124.00  -124.00  -126.00  -126.00  -128.00  -128.00  -130.00  -130.00  -132.00  -132.00  -134.00  -134.00  -136.00  -136.00  -138.00  -138.00 -140.00  -140.00  -142.00  -142.00  -144.00  -144.00  -146.00  -146.00  -148.00  -148.00  -150.00  -150.00  -152.00  -152.00  -154.00  -154.00  -156.00  -156.00  -158.00  -158.00 le+00  -16000 ^ ^ ^ ^ ^ le+02 3 le+01 3 Rain Rate in nun/h.  ^  le+00  ^ ^ ^ ^ 3 le+01 le+02 3 Rain Rate in nun/h.  Figure 5.2(c,d) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 2.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz.  Transmission loss in dB  Transmission loss in dB  n—riain psoru, -120.00  -125.00  Cost.rain  -122.00  Nsi,r7);0-.0-  -124.00  -130.00  -126.00 -135.00  -128.00 -130.00  -140.00  -132.00 -145.00  -134.00 -136.00  -150.00  -138.00 -155.00  -140.00 -142.00  O. -16o.00 00  -144.00  -165.00  -146.00 -148.00  -170.00  -150.00 -152.00  -175.00  -154.00 le+00  3  le+01 Rain Rate m mm/h.  3  1e+02  ^  1e400  ^ ^ ^ ^ 1e+02 3 1e401 3 Rain Rate in MITI&  Figure 5.3(a,b) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 23 km, an& (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  Transmission loss in dB  Transmission loss in dB -118.00  -116.00  Capsoni,rain  -118.00  tapsommii0.§-  -120.00  -120.00  -122.00  -122.00  -124.00  -124.00  -126.00  -126.00  Cost,ram tost.nn.0.9  -128.00  -128.00  -130.00  -130.00  -132.00  -132.00  -134.00  -134.00  -136.00  -136.00  -138.00  -138.00  -140.00 -142.00  -140.00  -144.00  -142.00  -146.00  -144.00  -148.00  -146.00  -150.00  -148.00  -152.00  -150.00  -154.00  -152.00  -156.00 -158.00  -154.00  -160.00  -156.00 1e400  ^  3^le+01  ^ ^ le402 3  Rain Rate in mm/h.  le+00  ^  3^1e401^3  ^  Rain Rate in trviiih.  Figure 5.3(c,d) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 2.5 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz.  1e402  Transmission loss in dB  Transmission loss in dB  -133.00  Capsom,rain  titiTontiiRiV  -134.00  -130.00  Cost,nun  -135.00 -136.00  -135.00  -137.00 -138.00  -140.00  -139.00 -140.00  -145.00  -141.00 -150.00  -142.00 -143.00  -155.00  -144.00 -145.00  O  -160.00  -146.00 -147.00  -165.00  -148.00 -149.00  -170.00  -150.00 -151.00  -175.00 le+00  3  ^  le+01  ^ ^ le+02 3  Rain Rate in mm/h.  ^  -152.00 le+00  3  ^  le+01  ^  3  Rain Rate in nvn/h.  Figure 5.4(a,b) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 3.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  le+02  -  Transmission loss in dB  ^  Transmission loss in dB  -136.00  -138.00  -138.00  -140.00 -142.00  -140.00  -144.00  -142.00  -146.00  -144.00  -148.00 -150.00  -146.00  -152.00  -148.00  -154.00 -156.00  -150.00  -158.00  -152.00  -160.00  -154.00  -162.00 -164.00  -156.00  • -166.00  -158.00  -168.00  -160.00  -170.00  -162.00  -172.00 -174.00  -164.00  -176.bo  -166.00  -178.00 1e400  ^  3^1e+01^3 Rain Rate in mm/h.  ^  le+02  le+00  ^ ^ ^ 3 le+01^3 le+02 Rain Rate in mm/h.  Figure 5.4(c,d) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 3.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz.  Transmission loss in dB  Transmission loss in dB  -130.00  -133.00 -134.00 -135.00  -135.00  -136.00 -137.00  Capsommun  -138.00 -139.00  -140.00  -140.00 -141.00 -142.00  -145.00  -143.00 -150.00  -144.00 -145.00  -155.00  -146.00 -147.00 -148.00  ts..)^  -149.00 -150.00  -160.00  -151.00  -165.00  -152.00 -153.00  -170.00  -154.00 -155.00 -156.00  -175.00 le+00  3  ^  1e+01  ^ ^ le+02 3  Rain Rate in mint. ^  le+00  3  Ie+01  3  • Rain Rate in mm/h.  Figure 5.5(a,b) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 33 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  le+02  Transmission loss in dB -135.00 -140.00 -140.00  Cost,rain  -145.00 -150.00  -145.00  -155.00 -150.00  -160.00 -165.00  -155.00  -170.00 -175.00  -160.00  -180.00 -165.00  -185.00 -190.00  -170.00  -195.00 -175.00  -200.00 -205.00  -180.00  -210.00 -185.00  -215.00 -220.00  -190.00 le+00  ^  3^Ie+01^3  ^  le+02  Rain Rate in =Va. ^  ^  le+00  ^  3^15+01  ^  3  ^  Rain Rate in nun/h.  Figure 5.3(c,d) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 3.5 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz.  15402  Transmission loss in dB  ^  Transmission loss in dB Capsoni.rain -134.00  Cos t.rai n —  -136.00 -138.00 -140.00 -142.00 -144.00 -146.00 -148.00  -13o.00 -152.00 -154.00 -156.00 le+00 Rain Rate in nun/h.  ^ ^ ^ ^ le+02 3 1e401 3 Rain Rate in nun /h.  Figure 5.6(a,b) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 4.0 km, and: (a) f (frequency) = 1, 5, 10 GHz, (b) f = 20 GHz.  Transmission loss in dB  Transmission loss in dB  -135.00  -135.00  Npsoni,m=0.9 Cost,rain tost.rn.0.9  -140.00 -140.00  -145.00 -150.00  -145.00  -155.00 -150.00  -160.00 -165.00  -155.00  -170.00 -175.00  -160.00  -180.00 -165.00  -185.00 -190.00  -170.00  -195.00 -175.00  -200.00 -205.00  -180.00  -210.00 -185.00  -215.00  -190.00 le4.00  ^  3^1e401^3 Rain Rate in mm/h.  ^  -220.00 le+02  le+00  ^  3^1e401^3  ^  Rain Rate in mm/h.  Figure 5.6(c,d) A comparison between Capsoni and COST 210 rain cell models with and without a melting snow layer for Hm = 4.0 km, and: (c) f (frequency) = 30 GHz, (d) f = 40 GHz.  le+02  Chapter 5—COST 210 rain-cell model  Also, there are inherent weaknesses in the COST 210 rain model. The model is not a physical one where it accurately describes an actual rain cell.As we stated before, the major advantage of the COST 210 rain cell is its small radius. This advantage turns into a disadvantage when modeling the melting snow layer. As we observed, the COST 210 rain cell models the effect of the melting layer quite nicely when that region is near the center of the common volume. On the other hand, if the melting layer height (which is a random variable) moves upward or downward, the model will not be able to account for its effect beyond a relatively short distance. In general, the COST 210 model seems to be acceptable for modeling the interference for terrestrial stations. It would be quite interesting to extend the model to find out if it can reasonably estimate interference on receiving satellites. To better judge the COST 210 rain cell, a comparison of the statistical transmission loss is in order. This is currently beyond the scope of this thesis, but should be dealt with at a future date.  76  Chapter 6—Discussion and Conclusions  Chapter 6 Discussion and Conclusions 6.1 Effect of melting-snow layer The results introduced in Chapter 4 show that the melting-snow layer plays an important role in the transmission loss in the 1-40 GHz frequency spectrum. This role is by no means uniform. At lower frequencies (f = 1-10 GHz), the melting-snow layer plays a significant role in increasing the interference level. This enhancement is much higher for the f = 1 and 5 GHz than for f = 10 GHz. This is the region where most of today's radio communications is handled. The congestion of the frequency spectrum is pushing for the use of the higher frequencies. Because of the high attenuation associated with these frequencies in the melting-snow layer, outages will become more frequent. These outages will become more important to system engineers than signal interference. On the other hand, stations might increase their transmission power to avoid outages. This will generate a stronger scattered signal and thus a higher interference potential. Because of increased attenuation, the melting-snow layer decreases the effect of the ice/snow region. This is especially true for higher frequencies. However, the effect of this attenuation is limited to the lower part of the ice/snow region because of the small elevation angle of the scattered wave. Because of the importance of the ice/snow region at higher frequencies, more research is needed to model the scattering more accurately. At low frequencies, the attenuation is small enough that it will not offset the increase in reflectivity of the melting-snow layer. At high frequencies, the attenuation by the melting-snow layer becomes great enough to smother any increase in the scattering of the melting-snow layer. 77  Chapter 6—Discussion and Conclusions  It is also observed that for a high directivity antenna, the melting-snow layer interference can increase or decrease depending on the position of the main lobe axis of the receiver relative to the melting-snow layer. If the main lobe axis of the high directivity antenna intersects or is near the melting-snow layer, its effect will be greater. Also we notice that the density of the core of the melting-snow particles plays an effect — albeit not great — in the calculations. We see that the higher the density is, the smaller is the effect of the melting layer. This can be attributed to the reduction of the particle sizes resulting from higher average density. The melting layer has been assumed to have a linear melting ratio (S=h/H; where h is the distance down from the top of the melting layer, H is the thickness of the melting layer, S is the ratio of the melted volume to the total volume). This profile provides for a narrow region in the melting layer where attenuation and scattering peaks (around S = 0.1). Kharadly [1992] suggests that this profile underestimates the effect of the melting layer and that different profiles might have to be used.  6.2 Effect of rain height, Hm The effect of increasing rain height in the rain cell is quite interesting. We can see that for lower frequencies, the higher Hm causes a higher interference level meanwhile at higher frequencies, the interference decreases with a higher Hm. The reason for this phenomenon is the ice/snow region above the rain region. In this region we have scattering but no attenuation and since scattering increases but with no attenuation to offset it, the interference increases. This leads us to the conclusion that considerable interference for high frequency can be achieved if a high directivity antenna intersects the ice/snow region. 78  Chapter 6—Discussion and Conclusions  6.3 Effect of rain rate It is very hard to talk about the effect of rain rate without mentioning frequency. For lower frequencies, interference increases with rain rate, since higher rain rates translates into higher scattering cross section. For higher frequencies, higher rain rates translate into very high attenuation levels and thus lower interference.  6.4 Effect of frequency For lower frequencies, rain and melting-snow attenuation is negligible and interference can present problems for radio systems operating at the same frequencies. For higher frequencies, the interference problem seems to disappear since the high attenuation will smother any potential interference wave. Outages, due to the high attenuation level, seems to be a much more serious problem for higher frequency systems. Yet because of this high attenuation level, systems will be forced to increase their transmitting power during these periods and thus increasing the scattered power and thus the interfer6nce.  6.5 Reminder A model-program has been developed to calculate the interference caused by hydrometeors. This model-program is — unlike the COST 210 model — capable of calculating the interference for any given geometry. Also a study was conducted about the effect of the melting layer on interference. It was found that the melting layer significantly enhances the interference for lower frequencies and should not be ignored. This is especially true in the case of satellite interference. 79  Chapter 7—Suggestions for future research  Chapter 7 Suggestions for future research There is much work — both theoretical and experimental — that needs to be done on the subject in the future. Some suggestions for future work are: 1. There should be further work on the program to make it more efficient. This can be done by utilizing more efficient routines or by fitting some of the parameters used in Kharadly models into equations. These parameters include the radius of the representative particle and the number of particles per unit volume. The equations will be a function of rain rate and the melting ratio (S). This will make it unnecessary to interpolate them from Table A.1 every time we need to calculate the attenuation. 2. More work needs to be done on the program to make it user-friendly. 3. Introduce the statistics of rain to the model. To do that, more study is required on the statistics of the melting layer, its thickness and height and if any relation exists between them. Currently, there is considerable work being done in this field by the Alberta Research Council [9] 4. Include real earth and space coordinates into program (longitude, latitude). 5. Use the program to study more diverse geometries and a wider range of parameters. 6. Include gaseous attenuation into the calculations since for f = 50 GHz the attenuation is 15 dB/km. Fog attenuation might also be significant over long distances. Fog attenuation is 0.1 dB/km for f = 50.0 GHz. This will translate into 10.0 dB for a 100.0 km path length. 7. Further research is needed to see if the COST 210 model can be extended to interference on satellites. 80  Chapter 7—Suggestions for future research  8. More work is needed on the interference caused by hydrometeors on low gain systems. This can be useful in cellular communications. 9. Since the Kharadly scattering model does not extend to the snow region, more research should be done to extend it. Also more work should be done to make it more accurate. 10. Develop formulas for scattering and attenuation for higher frequencies. As a first step, it will be useful to extend the scattering formulas to 100.0 GHz since the empirical model for attenuation is valid over that range. 11. More research is needed concerning the structure of the melting-snow region including its profile. The Alberta research [9] can provide valuable information on the subject. 12. More research is needed in the area above the melting snow layer. Again, the Alberta research [9] is bound to shed light on this subject.  81  Appendix A- Precipitation modeling  Appendix A Precipitation modeling A.1 Rain medium The rain medium consists of drops of water drops of different sizes falling at a velocity depending on the size of the drop. For most applications, it is reasonable to assume that these particles are spheres. The size distribution of the spheres and their velocities v are given in Table A.1 [14]. Precipitation  0.25  1.25  2.5  5  12.5  25 •  50  100  150  rate (mm/h) Drop size (cm)  v  Percent of the total volume  (m/s)  0.05  28.0  10.9  7.3  4.7  2.6  1.7  1.2  1.0  1.0  2.06  0.1^•  50.1  37.1  27.8  20.3  11.5  7.6  5.4  4.6  4.1  4.03  0.15  18.2  31.3  32.8  31.0  24.5  18.4  12.5  8.8  7.6  5.40  0.2  3.0  13.5  19.0  22.2  25.4  23.9  19.9  13.9  11.7  6.49  0.25  0.7  4.9  7.9  11.8  17.3  19.9  20.9  17.1  13.9  7.41  0.3  1.5  3.3  5.7  10.1  12.8  15.6  18.4  17.7  8.06  0.35  0.6  1.1  2.5  4.3  8.2  10.9  15.0  16.1  8.53  0.4  0.2  0.6  1.0  2.3  3.5  6.7  9.0  11.9  8.83  0.2  0.5  1.2  2.1  3.3  5.8  7.7  9.00  0.3  0.6  1.1  1.8  3.0  3.6  9.09  0.2  0.5  1.1  1.7  2.2  9.13  0.3  0.5  1.0  1.2  9.14  0.2  0.7  1.0  9.14  0.3  9.14  0.45 0.5 0.55 0.6 0.65 0.7  Table A.1 Drop size distribution and their velocities for various precipitation rates [14]  82  Appendix A  —  Precipitation modeling  The model can further be simplified by considering the rain medium to be composed of rain drops of a representative radius a. The rain rate (R) can then be given by: R= 48r x 10 5 d 9 vN^  where,  A. (30)  ] —1/3  a = [ 1/ E ,3Pi^ i^...„ Ri  A. (31)  pi is the fraction of the volume of rain VR composed of the rain drops of radius a R i .  The number of representative rain drops in this fictitious rain medium is given by:  N— ^ 487r x 10 5 a 3 v  A. (32)  where R is measured in mm/h, a in cm, v in m/s, and N in cm -3 . The velocity can be found using Table A.1.  A.2 Melting-snow medium Unlike the rain medium, the melting snow layer is "essentially inhomogeneous". It is the region at the zero isothermal where snow melts into rain. Because of the difference in density between the snow and water, the snow particle starts to decrease in size as it melts with the water forming a layer on the outside of the snow particle. The region between where the melting starts and where the melting ends is called the "melting-snow layer". For our melting-snow layer, we assume the following: 1. The melting layer has a steady thermal structure. 2. A steady supply of snowflakes of prescribed size is maintained at the 0°C level, at the top of the melting region. The relative distribution of those particles is the same as that in Table A.1. 83  ^  •^  Appendix A— Precipitation modeling  3. There is no aggregation or breakup of snowflakes in the melting region. 4. Snowflakes have spherical shapes. 5. The melted water forms a coat around the snowflake. 6. Growth by collision and coalescence with cloud drops and by condensation of water vapor is ignored. 7. The melting particle increases in size as we move from the rain medium (S = 1) in the bottom of the melting layer to the top of the melting layer (S = 0). The radius of the representative melting snow layer is given by:  a= ( Ps + (1 — p.,)S)Y a'Rt.v  I —1/3  A. (33)  PivRi  [  M  •  Where ^Ps^density of the snow core of the particle  vRi^fall velocity of the rain drop of radius aR, •  fall velocity of the corresponding melting-snow spheres with a degree of melting S  The velocity of the melting snow particle is given by: v rn i = 1.5 + (vRi —  A. (34) 1.5) sin77)^ (-  S is the ratio of the melted volume of water to the total volume of the melting snow particle. Two models have been developed by Kharadly [10, 12] for the melting-snow layer. In [10], the effect of the fall velocity on number density is ignored which led to the violation of the conservation of mass criterion. In [12], the effect of fall velocity is taken into consideration. 84  Appendix B- Kharadly attenuation models  Appendix B Kharadly attenuation models B.1 Artificial dielectric model The specific attenuation A p and the phase # are given from the general expression for the propagation characteristic [10]: -y A p + j#=-- 21- f (—itoe) 1 / 2^B.1 From the above expression the attenuation and phase can be easily approximated by:  A p 9.1g, N f x 10 4  B.2  A# 6g: N f x 10 5  e  where the effective value of the polarizability at high frequencies g e = g — jg en is given by: ge =  g  B.3  1+  with,  n=  2+  Y(*)m  B.4  1 + (Y + 1) (-kr  and, fr -  27ra  where m = 2, Y = 100, and ( = 0.81. The low-frequency value of the polarizability for a two-concentric sphere is given by: 3 1) (e2^1 )( 2 €2 + el) — ( IL a2 ) (e2 c1)(2€2 + g =^ 3 (e2 2)(2f2 + El) 2( 102' ) (e2 el)(E2 1) —  85  —  B.5  Appendix B—Kharadly attenuation models  where ai, a2, El, E2 are the radius of the inner sphere, the radius of the outer sphere, the permittivity of the inner sphere, and the permittivity of the outer sphere, respectively. The permittivity of water, 6, is given in [18]: III ^  B.6  E_ 1 -11  I = c oo +  E  rt  (E s — 1 00 ) [1 + (A s /A) 1 ' sin (wt. /2)] 1 + 2(A s /A) 1 ' sin (air/2) + (A s /A) 2(1—a)  ^B.7  . ^(Es — c„,,)(A s /A) l—a cos (air/2)^TA 1 + 2(A s /A) 1 ' sin (air/2) + (A s /A) 2(1—a) + 18.8496 x 10 10  B.8  where E s = 78.54 x [1.0 — 4.597 x 10 -3 (t — 25.0) + 1.19 x 10 -5 (t — 25.0) 2 2.8 x 10 -8 (t — 25.0) 3 ] T =  12.5664 x 10 8  c c, = 5.27137 + 0.0216474t — 0.00131198t 2 a =^—16.8129 t+273 + 0.0609265 A s = 0.00033836e 2513.98 /( t + 273 )  B.2 Corrected attenuation models B.2.1 Kharadly 3rd model for attenuation Because of the deviation of the results of the 1st model from that of the exact values for the melting-snow layer, Kharadly introduced a correction factor that brought the results of the model closely to the exact attenuations calculated using Mie scattering. The correction factor is given by [12]: 1 n {(2 + S)1 + 1] 1 1 2 — S1( Factor 1 = f + 2 — S^x t 2 -1-Sf ;  86  1—s)  (2.7)  Appendix B—Kharadly attenuation models  where  f is the frequency in GHz, f is the resonant frequency of the melting-snow r  particle, S is the melting degree , defined as the melted to the total volume in the representative melting-snow particle.  B.2.2 Kharadly 4th model for attenuation Because of the deviation of the results of the 2nd model from that of the exact values for the melting snow layer, Kharadly introduced a correction factor that brought the results of the model closely to the exact attenuations calculated using Mie scattering. The factor is given by:  n2d(1 S Factor2 = [{n + S(1— n)}{1+ ^26  )[^(1  5ill S)f  - (1 S)  1+^ (2.8) 5 fr  —^  where a is the radius of the representative particle,  —  b is the skin depth of water =  ^ with C1 = 20.958, f is the frequency in GHz and E is the complex Real[f permittivity of water. The range of applicability of Kharadly's formulas is between 1-40 GHz  87  Appendix C—Kharadly scattering model  Appendix C Kharadly scattering model [11] When an electromagnetic wave is incident on a dielectric sphere, it scatters (Figure C.1) . This scattering can be calculated using Mie scattering. This technique is computer intensive, and thus, not efficient to use. Other approaches have been developed by researchers in the field One of the simpler techniques to model the scattering from rain has been developed by Kharadly [11]. Kharadly has based his model on two assumptions 1. A rain drop or a melting-snow particle, under the effect of an incident electric field, behaves as a point dipole. 2. The rain medium which has particles of different sizes can be represented with a fictitious medium of particles of the same geometry, but with the same particle size. After introducing correction factors to deal with some of the inaccuracy introduced as a result of the simplification of the model, Kharadly concluded the following formula for hydrometeor scattering [11]: Q(0, (19) = cr(d) x F(M  where, er (a)  =  (k o d) 2 n  F(0, c/)) =[ sin! 0 ± 1  "  ) x F(n) x F(S) x N m^C.1  7ra-2 c — c o 2^ E  ±  L f' + i-  2  2E0  (f)  2n F ( 1)  4)  )  1 cos 4 0) (^4) sin s 0 cos -1--  C.1(a)  C.1(b)  C.1(c) M" [^M"^R )} sin 0 cos 4) "")= 1 2.6 1 — 2.6 ( 1 600^ 88  Appendix C—Kharadly Scattering model  Figure C.1 Scattering geometry of a rain particle due to an incident electromagnetic wave.  Appendix C—Kharadly scattering model  F(n) =  ln  ( /^n) 2 [ n 2 ^ i+ 300 R^(f)2 ± 2n(2.5n — 1) 1 ± n 2 L^2^1 ±  2  [  F(S) = {n  1 (1—.9  4  25S^( R ) 6'.5 X .100 ) /50 ± R X^ 1 + (^ 150  02  (  [^ R + 100 1 0.5  f sin 2 gaJ_ ) ±  1±  100^fr  2  (f) cos 0 sin 0 } 1—g  202  where,  o  k = -24 Ao = free-space wavelength n=  2+200(17) 3  1+201(f) 3 f = frequency  r  f = t, where c is the velocity of light in free space  Ar  -  27iti  outer radius of the representative melting-snow sphere = 0.866(1 + 1.5 x 10 -4 f), where f is in GHz co = permittivity of free space = permittivity of water M = M' - N H ,  is the refractive index of water (= e 1/2 r )  0,0 = polar and azimuthal angles, as in FigA.1 N. = number of melting-snow particle per unit volume (rain drop density) of melted snow S = degree of melting — volume total volume The radius of the representative melting snow sphere is given by: 90  C.1(e)  Appendix C—Kharadly scattering model  =[(0 .1 + 0 .9S)V Ps VRi —113 a Rs. v nu where pi = fraction of rain drop of radius aRi vRi = fall velocity of rain drop of radius aRi v m i = fall velocity of corresponding melting-snow spheres with a degree of melting S  = 1.5 + ( vR, — 1.5) sin  V  91  Appendix D—Empirical formula for attenuation  Appendix D Empirical formula for attenuation The most convenient method for modelling the attenuation of rain is by putting it in the following form [16]:  Ap = a( f )R 1  ^D.1  where a and /3 have been found using a program which implements the least square data fitting technique. Given a value of 13, the program will find the a that corresponds to the most accurate fit with the original (given) data. a, and j3 have been calculated for 27 frequencies ranging from 1-100 GHz. The original values of attenuations are the Mie calculations done by Kishk [12]. a, and /3 have then been fitted in two equations in function of frequency. These two equations are:  /3(f) = [ 1 *  a(f)  = [f2.1  f2  f6  f3  f2.5  f4  f4  f2 e  -f }]  +1.16933000000 -0.25154000000 -2.45000 x 10 -4 +3.50920 x 10 -6 -1.46357 x 10 -8 -0.44110000000 _ +0.14282500000 _  D.1(a)  f8  +3.4777654 x 10 -88+3.8866095 x 10 -12 f71 +2.9044983 x 10 -85 -3.8528788 x 10 -08 +2.3261638 x 10 -18 -4.8164211 x 10 -14  D. 1(b)  92  Appendix D—Empirical formula for attenuation  The next step was to extend the formula to the melting snow layer. We did that by dividing the values of the attenuations for the different melting degrees (S) by the attenuation for rain. We call the result value of division as the melting snow layer normalized attenuation. A formula is found that gives a good fit for the normalized attenuation:  An(R,f =  ^sai-1 e —bi^+ m2 sa2 —1 e — b2 S a2 + m3 e —b3 S + 1^D.2  Then,  Ap = A n x a( f ) len^  D.3  In order to simplify equation D.2, the following assumptions can be safely made: = 18 b2 = 6.3 a2= 1.7 M3 = —1 b3 =  230  Equation D.2 then becomes:  A n(R, f ,S) =^sai —1 e -18Sal + m2  A formula to fit al is found: 93  SO.7 e -6.3S 1^e -230S  + 1^D.4  Appendix D—Empirical formula for attenuation  491(f) =[ 1  if_  711  f2.1 e— f ]  +1.6806 —2.791 x 10 -3 —3.41706 +5.0493 +0.4707755 —2.26293  D4(a)  M1 and M2 are found to satisfy the following equations: ^ Ml(R,f) = C0(f) Cicoli c1 C2(f) R e2  D.5  M2(R,f) = Do(f) DicoR di D2(f)R d2  where Cl = d1 = 0.003 c2 = d2 = 0.0002 Equation D.5 then becomes: Ml(R,f)^Co(f)  C1(f) R0.003^y 2(f ) R0.0002^  D.6  d2 M2(R,f) = Do(f) Di(f)R dl + D2( f)R  Where Co(f), CO), C2(f) are given by:  f2 e — f f.2^-.01  f. 7 e — f sin (-0 ] [Xin] for 1.0 < f < 12.0 in Hz ton ef-100 ^ In (f) cosh( f1 sin (43 ) ] [Y1,2 ] —40) cosh(f —20)^ for 12.0 < f < 100.0 in GHz  D.6(a)  94  Appendix D- Empirical formula for attenuation  X1 n =  Yin =  n=0 - 4.46871 x 10 10 +1.40323 x 10 08 -6.96003 x 10 08 +4.50656 x 10 10 - 6.42129 x 10 07 +8.73118 x 10 08 +1.40150 x 10 06  n=1 - 3.17499 x 10 °9 +1.00081 x 10 07 - 4.93871 x 10 °7 +3.20179 x 10 °9 - 4.56513 x 10 06 +6.21166 x 10 °7 +9.94801 x 10 °4  n=2 +4.78623 x 10 10 - 1.50331 x 10 08 +7.45011 x 10 08 - 4.82676 x 10 10 +6.87778 x 10 07 - 9.35236 x 10 °8 - 1.50098 x 10 06  n=0  n=1  n=2  -6.29528 x 10 10 +6.29636 x 10 10 +5.96653 x 10 05 - 3.88357 x 10 °5 - 1.92394 x 10 08 -1.82230 x 10 05 +8.65689 x 10 °4  -4.08460 x 10 09 +4.08533 x 10 09 +4.25990 x 10 °4 - 2.78377 x 10 04 -1.24892 x 10 07 -1.28176 x 10 04 +6.23398 x 10 03  +6.70282 x 10 10 -6.70397 x 10 1° - 6.39256 x 10 05 +4.16678 x 10 05 +2.04855 x 10 08 +1.95046 x 10 05 - 9.28018 x 10 04  Also, Do(i), Di(f), D2(f) are given by:  1[1 f f 2 f 3 fe - f f 2 e - f sin (0.6(f - 2.8)) in (1)][X2n] for 1.0 < f < 20.0 in GHz Dn(f) =^[ 1 f f2 f3 f e -f f2e-f sin (0.6(f - 2.8)) In (f)1[Y2n] for 20.0 < f < 100.0 in GHz  D.6(b)  n=1  n=2  -1.49368 x 10 06 +9.95647 x 10 05 -6.31410 x 10 °4 +1.41351 x 10 °3 +2.97214 x 10 °5 +1.13469 x 10 06 +5.31790 x 10 04 - 1.48261 x 10 06  +2.24852 x 10 07 -1.49959 x 10 07 +9.51422 x 10 °5 - 2.13062 x 10 04 - 4.47162 x 10 06 - 1.70713 x 10 07 - 8.03303 x 10 05 +2.23186 x 1007  n=0  X2n =  - 2.09914 x 10 °7 +1.40003 x 10 07 -8.88279 x 10 °5 +1.98926 x 10 °4 +4.17453 x 10 06 +1.59365 x 10 °7 +7.50124 x 10 05 _ -2.08356 x 10 07  95  ^  Appendix D--Empirical formula for attenuation  n=0  n = 1^n = 2^-  ^—3.15028 —4.43961 x 10 07 x 10 06^+4.75465 x 10 °7 ^—6.33473 —8.94385 x 10 05 x 10 °4^+9.57732 x 10 °5 +6.29234 x 10 03 +4.45502 x 10 02^—6.73784 x 10 03 —1.90282 x 10 01 —1.34692 x 10 0°^+2.03751 x 10 01 1/2n = —2.52272 x 10 15 —1.80339 x 10 14^+2.70306 x 10 15 +1.27660 x 10 14 +9.12578 x 10 12^—1.36786 x 10 14 —2.29312 x 10 04 1.62315 x 10 °3^+2.45541 x 10 °4 +1.95225 x 10 07 +1.38423 x 10 06^—2.09068 x 10 °7 Figures D.1-4 show that the empirical model agrees with the exact calculations. ^-  Although the formulas seems to be huge, their computer running time is quite short. Also, since the frequency remains constant during the integration, we can calculate the variables which are function of frequency at the beginning of the program and then we will be left with a simple formula for attenuation. Unfortunately, this formula assumes that the density of the core in the melting snow particle to be 0.1. However it should not be very difficult to incorporate the density of the core into the equation without increasing the computer-running time of the formula considerably.  96  Attenuation in dB/km x 10 -3 210.00  Attenuation in dB/kut  200.00 4.50  190.00 180.00 170.00  4.00  160.00 150.00  3.50  140.00 130.00  3.00  120.00 110.00  2.50  100.00 90.00  2.00  80.00 70.00  1.50  60.00 50.00  1.00  40.00 30.00  0.50  20.00 10.00 0.00 -10.00  0.00 0.00  ^  0 in  ^  Degree of melting (S)  1.00  ^  ^  0.00  ^  0.50  ^  1.00  Degree of melting (S)  Figure D.1 A comparison of the attenuation profile of the melting layer for Kharaclly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 1.0 and 5.0 GHz (p, = 0.1).  Attenuation in dB/km  Attenuation in dB/km 36.00  15.00  Exact calculations, m-0.9 Empirical formula  34.00  KhamcUy 1rd attenuation model  14.00  32.00  13.00  30.00  12.00  28.00  16.00  26.00  11.00  24.00  10.00  22.00  9.00  20.00  8.00  18.00  7.00  16.00  coo 6.00  14.00  5.00  12.00  t)  ,  10.00  4.00  8.00  3.00  6.00  2.00  4.00  1.00  2.00  0.00  0.00 0.00  ^  0.50  ^  Degree of melting (S)  1.00  ^  ^  0.00  ^  0.50  ^  1.00  Degree of melting (S)  Figure D.2 A comparison of the attenuation profile of the melting layer between Kharadly 3n1 attenuation model, Empirical model, and the Exact calculations for f (frequency) = 10.0 and 20.0 GHz (p, = 0.1).  Attenuation in dB/lm 44.00 42.00 40.00 38.00 36.00 34.00 32.00 30.00 28.00 26.00 24.00 22.00 20.00 18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 -2.00  Attenuation in dBAmt  45.00 40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.00  0.50 Degree of melting (S)  1.00  ^  0.00  0.50  1.00  Degree of melting (S)  Figure D3 A comparison of the attenuation profile of the melting layer between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 30.0 and 40.0 GHz (p, = 0.1).  Attenuation in d13/1cm  Attenuation in dB/lrm  tract calculations, m=0.9 iircal fornat kharadly 3rd attenuation mode!  65.00  55.00  60.00  50.00  55.00  45.00  50.00 40.00  45.00  35.00  40.00  30.00  35.00  25.00  30.00 25.00  20.00  20.00 15.00 15.00 10.00  10.00  5.00  5.00  0.00  0.00  0.00  0.50 Degree of melting (S)  1.00 ^  0.00  0.50  ^  1.00  Degree of melting (S)  Figure D.4 A comparison of the attenuation profile of the melting layer between Kharadly 3rd attenuation model, Empirical model, and the Exact calculations for f (frequency) = 70.0 and 100.0 GHz (p. = 0.1).  The program for the "Universal Model"  Appendix E The program for the "Universal Model" The following is a listing of the program used in the calculations of the interference for the modified Capsoni rain model:  *  DOUBLE PRECISION FREQ,T,QUANTITY,THETA,PHI, • XH,YH,ZH,XTHE_T,XPHIT, • XR,YR,ZR,XXXXXX,YYYYYY, • TR_ALPHA,TR_HALF_THETA,RE_ALPHA,RE_HALF_THETA, • H_MELT,H_THICKNESS,H_RAIN,RAD,RMAX,FREQ,QUANTITY,T, • D_RHO,G1,G2,HBW1,HBW2,K,MINT INTEGER TASKNUMBER1,TASKNUMBER2,SENTINAL CHARACTER*2 DECISION CALL AT_SC_MENU(TASKNUMBER1,TASKNUMBER2) CALL POSITIONINPUT(XH,YH,ZH,XTHE_T,XPHI_T, • XR,YR,ZR,XXXXXX,YYYYYY) CALL GEO_INPUT(TR_ALPHA,TR_HALFTHETA,RE_ALPHA, • RE_HALF_THETA,DECISION,H_MELT,H_THICKNESS,H_RAIN,RAD, • RO,RMAX,FREQ,QUANTITY,T,DRHO,G1,G2,HBW1,HBW2,K,MINT)  *  CALL • • • •  GEOMETRIC_MODEL(XH,YH,ZH,XTHE_T,XPHI_T, XR,YR,ZR,XXXXXX,YYYYYY,TR_ALPHA,T,QUANTITY, TR_HALF_THETA,RE_ALPHA,RE_HALF_THETA,DECISION, H_MELT,H_THICKNESS,H_RAIN,RAD,RO,RMAX,FREQ,D_RHO,G1,G2, THETA,PHI,TASKNUMBER1,TASKNUMBER2,HBW1,HBW2,K,MINT)  STOP END ******************************************************************************* *************** AT SC MENU **************************************************** ******************** IT********************************************************* -  *  SUBROUTINE AT_SC_MENU(TASKNUMBER1,TASKNUMBER2) *  INTEGER TASKNUMBER1, TASKNUMBER2 READ*,TASKNUMBER1 READ*,TASKNUMBER2  RETURN END ******************************************************************************* **************** SUBROUTINE GE() INPUT ***************************************** *******************************7*********************************************** * SUBROUTINE GEO_INPUT(TR_ALPHA,TR_HALF_THETA,RE_ALPHA, • RE HALF THETA,DECISION,H MELT,H THICKNESS,H RAIN,RAD,RO, • RMÄX,FRf Q,QUA NTITY,T,DRT10,G1,G,HBW1,HBW2,R,MINT ) DOUBLE PRECISION TR_ALPHA,TR_HALF_THETA,RE_ALPHA, • RE_HALF_THETA,H_MELT,H_THICKNESS,H_RAIN,RAD,RO,RMAX,  101  The program for the "Universal Model"  *  FREQ,QUANTITY,T,DRHO,G1,G2,HBW1,HBW2,K,MINT CHARACTER*2, DECISION READ*,TR_ALPHA READ*,TR_HALF_THETA READ*,RE_ALPHA READ*,REHALF_THETA READ*,DECISION IF (DECISION.eq.'Y')THEN READ*,H MELT READ*,H_THICKNESS ENDIF READ*,H RAIN READ*,RAD READ*,R0 READ*,RMAX READ*,FREQ READ*,T READ*,QUANTITY READ*,D_RHO READ*,G1 READ*,G2 READ*,HBW1 READ*,HBW2 READ*,K READ*,MINT  RETURN END ******************************************************************************* *************** SUBROUTINE GEOMETRIC MODEL ************************************ ************************************T****************************************** XH,YH,ZH^XTHE_T,XPHI_T XR,YR,ZR^XXXXXX,YYYYYY TR_ALPHA^TR_ HALF _ THETA RE HALF THETA DECISION^H_MELT H_THICKNESS H_RAIN RAD RO RMAX FREQ D_RHO G1 G2 T_X,T_Y,T_Z R X,R_Y,R_Z Y1,Z1 XY, X2, Y2, Z2 X, Y, Z X1P,Y1P,Z1P X2P,Y2P,Z2P X11,Y11,Z11^XPR,YPR,ZPR^XDPR,YDPR,ZDPR-  COORDINATES OF RAIN CELL (BOTTOM CENTER) - DIRECTION OF MAIN BEAM ON TRANSMITTER COORDINATES OF RECEIVER - DIRECTION OF MAIN BEAM ON RECEIVER ANGLE OF ALPHA ON TRANSMITTER THETA (HALF) FOR THE TRANSMITTER THETA (HALF) FOR THE RECEIVER 'Y' THERE IS A MELTING LAYER HEIGHT WHERE MELTING LAYER STARTS THICKNESS OF MELTING LAYER HEIGHT OF RAIN CELL RADIUS OF RAIN CELL RAIN RATE DISTRIBUTION VARIABLE MAXIMUM RAIN RATE FREQUENCY INCREMENT FOR POSITION OF MAIN BEAM GAIN OF TRANSMITTER GAIN OF RECEIVER DIRECTION OF TRANSMITTER MAIN LOBE DIRECTION OF RECEIVER MAIN LOBE 1ST INTERSECTION OF RAIN CELL AND LOBE OF TRANSMITTER 2ND INTERSECTION OF RAIN CELL AND LOBE OF TRANSMITTER INTERSECTION OF RAIN CELL AND TRANSMITTED BEAM 1ST INTERSECTION OF RAIN CELL AND LOBE OF RECEIVER 2ND INTERSECTION OF RAIN CELL AND LOBE OF RECEIVER CLOSEST INTERSECTION OF RAIN CELL AND LOBE OF RECEIVER LOCATION OF RECEIVER RELATIVE TO NEW AXIS (X',Y',Z') NEW POSITION OF RECEIVER AFTER TRANSLATION OF TRANSMITTER TO RAIN CELL DPRHO,DPTHETA, ^ SPHERICAL COORDINATES OF RECEIVER WHEN ORIGINAL DPPHI COORDINATE SYSTEM HAS BEEN ROTATED AND TRANSLATED  102  The program for the "Universal Model"  QUANTITY S  HMIN HMAX HFR HM HEIGHT RATE ATTEN,ATTE2 THETA PHI XN,YN,ZN YYYYYY AG, AH RT TASKNUMBER1 TASKNUMBER2 HBW1 HBW2  K  *  AX,AY,AZ  - TEMPERATURE - THE MASS VALUE OF THE RAIN, SNOW LAYER - THE PARAMETER A/B OF THE MELTING SNOW LAYER - BOTTOM HEIGHT OF RAIN CELL MEASURED FROM TRANSMITTER - TOTAL HEIGHT OF RAIN CELL MEASURED FROM TRANSMITTER - HEIGHT OF FREEZING LAYER MEASURED FROM TRANSMITTER - HM HEIGHT OF MELTING LAYER MEASURED FROM TRANSMITTER - HEIGHT OF BEAM IN RAIN CELL - RAIN RATE - ATTENUATION DB/KM AND FROM GAMMA RESPECTIVELY - THETA DIRECTION OF SCATTERTING - PHI DIRECTION OF SCATTERING - COORDINATES OF X,Y,Z WHEN AXIS TRANSLATED TO (XR,YR,ZR) - ANGLE BETWEEN SCATTERED BEAM BEING RECEIVED AND DIRECTION OF RECEIVING ANTENNA - ATTENUATION DUE TO GAS AND HUMIDITY RESPECTIVELY - 3X3 MATRIX TO ROTATE AXIS - ATTENUATION TASKNUMBER - SCATTEING TASKNUMBER - HALF POWER BEAM WIDTH FOR RECEIVING ANTENNA MAIN LOBE - HALF POWER BEAM WIDTH FOR RECEIVE ANTENNA SECONDARY LOBE - GAIN OF SECONDARY LOBE RELATIVE TO MAIN LOBE - COORDINATE OF ALPHA TRANSMITTER  * *********************** *********** *********************** ********** ***********  *  SUBROUTINE GEOMETRIC MODEL(XH,YH,ZH,XTHE T,XPHI_T, • XR,YR,ZR,XXXXXXTYYYYYY,TR_ALPHA,T,QUA NTITY, • TRHALFTHETA,REALPHA,RE_HALFTHETA,DECISION, • H_MELT,H_THICKNESS,H_RAIN,RAD,RO,RMAX,FREQ,D_RHO,G1,G2, • THETA,PHI,TASKNUMBER1,TASKNUMBER2,HBW1,HBW2,K,MINT) DOUBLE PRECISION XH,YH,ZH,XTHE_T,XPHI_T,S,T,QUANTITY, • XR,YR,ZR,XXXXXX,YYYYYY,TR ALPHA,RATE,ATTEN,HMIN, • TR_HALF_THETA,RE_ALPHA,RE:HALF_THETA,HMAX1,HFR,HM,HEIGHT, • H_MELT,H_THICKNESS,H_RAIN,RAD,RO,RMAX,FREQ,D_RHO,G1,G2,GT, • A_X,A_Y,AZ,RHO,TX,T_Y,T_Z,X1,Y1,Z1,X2,Y2,Z2, • R1,R2,R,RX,RY,RZ,RT(3,3),X,Y,Z,X0,Y0,ZO,H1T, • R1P,R2P,X1P,Y1P,Z1P,X2P,Y2P,Z2P,X11,Y11,Z11,XPR,YPR,ZPR, • XTRAN,YTRAN,ZTRAN,XDPR,YDPR,ZDPR,DPRHO,DPTHETA,DPPHI, • THETA,PHI,RTATT,XN,YN,ZN,YYYYYY,AG,AH,VAR,F,VARDUM, • HBW1,HBW2,K,XI1,YI1,ZI1,XYZ,RI,G,GE,ANGI,TEML,MINT, • NARR(2,3),ANUM,BNUM,A1NUM,RADARCONST INTEGER FLAG,TASKNUMBER1,TASKNUMBER2 CHARACTER*2 DECISION VAR=0.D0 T=0.D0 R=1.D0 CALL SPHERE2RECT(TX,TY,TZ,XTHET,XPHIT,R) X0=0 YO=0 Z0=0 CALL LINE CYLINDER INTERSECT(X0,YO,Z0,XH,YH,ZH,RAD,MINT, IF (FLAG.LE.0) THEN PRINT*, '-99999' RETURN ENDIF IF (TASKNUMBER1. EQ. 5) THEN CALL A_B(FREQ,ANUM,BNUM) CALL N_CALC(FREQ,NARR)  103  The program for the "Universal Model"  CALL Al_CALC(FREQ,A1NUM) ENDIF A X=0 A Y=DSIND(TR_ALPHA) A:Z=DCOSD(TRALPHA) CALL SPHERE2RECT(RX,RY,RZ,XXXXXX,YYYYYY,R) CALL ROT_PARAMETERS(T  X,T Y,T Z,A X,A Y,A  Z,RT)  RT IS A 3 X 3 ARRAY WHICH HAS X',Y',Z' IN THE FIRST, SECOND AND THIRD ROWS RESPECTIVELY. R1=DSQRT(X1**2+Y1**2+Z1**2) R2=DSQRT(X2**2+Y2**2+Z2**2) IF (R1.GT.R2) THEN RHO=R2+D_RH0/2 RI=R1 HEIGHT=Z1 ELSE RHO=R1+D_RH0/2 RI=R2 HEIGHT=Z2 ENDIF 600 CALL SPHERE2RECT(X,Y,Z,XTHE_T,XPHI_T,RHO) R2=DSQRTHXR-X)**2+(YR-Y)**2+(ZR-Z)**2) XI' = X - XR YI1 = Y - YR ZI1 = Z - ZR XYZ = DSQRT(XI1**2 + YI1**2 + ZI1**2) XII^XI1/XYZ YI1 = YI1/XYZ ZI1 = ZI1/XYZ CALL LINE_CYLINDER_INTERSECT(XR,YR,ZR,XH,YH,ZH,RAD,MINT, 6^X1P,Y1P,Z1P,X2P,Y2P,Z2P,XI1,YI1,ZI1,FLAG) R1P=DSQRT((X1P-XR)**2+(Y1P-YR)**2+(Z1P-ZR)**2) R2P=DSQRT((X2P-XR)**2+(Y2P-YR)**2+(Z2P-ZR)**2) IF (R1P.GT.R2P)THEN X11=X2P Y11=Y2P Z11=Z2P ELSE X11=X1P Y11=Y1P Z11=Z1P ENDIF FINDING (XR,YR,ZR) RELATIVE TO NEW AXIS (X',Y',Z')-->(X'R,Y'R,Z'R) XPR=RT(1,1)*XR+RT(1,2)*YR+RT(1,3)*ZR YPR=RT(2,1)*XR+RT(2,2)*YR+RT(2,3)*ZR ZPR=RT(3,1)*XR+RT(3,2)*YR+RT(3,3)*ZR CALL TRANSLATION(X0,YO,Z0,RHO,Y0,Z0,XTRAN,YTRAN,ZTRAN) XDPR=XPR-XTRAN YDPR=YPR-YTRAN ZDPR=ZPR-ZTRAN CALL RECT2SPHERE(XDPR,YDPR,ZDPR,DPTHETA,DPPHI,DPRHO) FINDING S HMAX1=H_RAIN+ZH  104  The program for the "Universal Model"  HFR=ZH+HMELT+H_THICKNESS HM=H MELT+ZH HMIN=ZH CALL GET_S(HMAX1,HFR,HM,HMIN,Z,S,DECISION) CALCULATING RAIN RATE CALL RAINRATE(XH,YH,X,Y,RMAX,RO,RATE) CALCULATE THE SPECIFIC ATTENUATION CALL ATT_TASK(FREQ,T,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) & TATT=ATTEN*DRH0/1000+TATT CALCULATE THE SCATTERING CROSS SECTION CALL SCAT_TASK(F,DPTHETA,DPPHI,S,RATE,FREQ,T,QUANTITY, & TASKNUMBER2,DECISION,H_THICKNESS) IF (DECISION.EQ.'Y') THEN H1T = HFR ELSE H1T = HMAX1 ENDIF IF (S.EQ.O.D0) THEN TEML = 10**(-0.00065*ABS(Z-H1T)) F = TEML*F ENDIF  * *^CALCULATING *  *  * * * * *  * * * *  CALL & & &  TOTAL RAIN ATTENUATION  RAIN_TOTAL_ATTENUATION(HMAX1,HFR,HM,HMIN,FREQ,T, QUANTITY,X,Y,Z,X11,Y11,Z11,RTATT,XH,YH,RMAX,RO,DECISION, TASKNUMBER1,NARR,A1NUM,ANUM,BNUM,XR,YR,ZR,H_THICKNESS, RAD,XTHET)  TRANSLATING X,Y,Z TO THE RECEIVER AXIS. SINCE THE COORDINATES (X,Y,Z) ARE MEASURED FROM AN AXIS AT (0,0,0) ALL I HAVE TO DO IS SUBTRACT THE POSITION OF THE RECEIVER FROM THE POINT AND THUS GET THE TRANSLATED POINT. XN=X-XR YN=Y-YR ZN=Z-ZR CALCULATING ANGLE BETWEEN UNIT VECTOR AND RECEIVER MAIN LOBE CALL ANGLE(XN,YN,ZN,R_X,R_Y,R_Z,ANGI) CALL ANTENNA_GAIN(ANGI,HBW1,HBW2,K,G) CALL RECEIVING_GAIN(G2,ANGI,GE) CALL GAS_HUMIDITY_ATTENUATION(AG,AH) VARDUM=10**(-0.1D0*(TATT+RTATT+AG+AH)) VAR=VAR+((F*G*D_RHO/(DPRHO**2))*VARDUM) RHO=RHO+D RHO IF (RHO.LT .RI) GOTO 600 CALL CONST_RADAR(FREQ,TR_HALF_THETA,G1,G2,GT,RADAR_CONST) VAR=VAR*RADAR_CONST VAR = 10.*DLOG10(VAR) PRINT*, VAR RETURN  105  The program for the "Universal Model"  END ******************************************************************************* ***************SUBROUTINE ROT PARAMETERS ************************************* *****************************T************************************************* THIS SUBROUTINE FINDS THE PARAMETERS IN ORDER TO ROTATE THE X-AXIS *^IN THE DIRECTION OF PROPAGATION. ******************************************************************************* * SUBROUTINE ROTPARAMETERS(T X,T Y,T Z,A X,A Y,A Z,T)  *  DOUBLE PRECISION T X,T_Y,T_Z,A X,A Y,A_Z, DUMMY,X,Y,Z,NIX,N_Y,N_Z,TT3,3 T, & Z X,Z Y,Z Z,Y X,Y Y,Y Z &  *  *  N_X=0 IF ((T_X.EQ.1).AND.(T_Y.EQ.0).AND.(T_Z.EQ.0))THEN Z_X=A_X Z_ Y=A Y Z Z=A_Z ELSE N_Y=-TZ/DSQRT(T_Z**2+T_Y**2) N_Z=T_Y/DSQRT(T_Z**2+T_Y**2) DUMMY=(A_Y*N_Y+N_Z*A_Z)*(1-T_X) Z_X=T_X*A_X Z_Y=N_Y*DUMMY+T_X*A_Y Z Z=N Z*DUMMY+T_X*A_Z CALL tROSS_PRODUCT(A X,A Y,A Z,N X,N Y,N Z,X,Y,Z) Z_X=Z_X-X*DSQRT(1-T_X**2) Z_Y=Z_Y-Y*DSQRT(1-T_X**2) Z_Z=Z_Z-Z*DSQRT(1-T_X**2) ENDIF CALL CROSS_PRODUCT(Z X,Z Y,Z Z,T X,T Y,T Z,Y X,Y Y,Y Z) T(1,1)=T_X T(1,2)=T_Y T(1,3)=T_Z T(2,1)=Y_X T(2,2)=YY. T(2,3)=Y_Z T(3,1)=Z_X T(3,2)=Z_Y T(3,3)=Z_Z  RETURN END ******************************************************************************* ********** SUBROUTINE ANTENNA GAIN ******************************************** *****************************T************************************************* * THIS SUBROUTINE CALCULATES THE GAIN OF AN ANTENNA BY IMPLEMENTING * AN ANTENNA LOB THROUGH THE USE OF TWO GAUSSIAN APPROXIMATIONS. * G1(theta)=exp(-41n2(theta/hbw1)-2) G2(theta) = 10"(K/10)*exp(-41n2(theta/hbw2) 2) * * * G(theta) = G1 (theta) + G2(theta) * G(theta) = G(theta)/G(0) * THETA^- ANGLE OF RECEPTION * * HBW1,HBW2 - HALF BEAM WIDTHS OF Gl,G2 RESPECTIVELY * K^- INPUT PARAMETER TO HELP DETERMINE THE SHAPE OF THE ANTENNA * GAIN PARAMETERS. -  * ********* ***** **************************** ***** **************************** **** SUBROUTINE ANTENNAGAIN(THETA,HBW1,HBW2,K,G)  106  The program for the "Universal Model"  DOUBLE PRECISION THETA, HBW1,HBW2,K,G,G1,G2 CALCULATING G1(THETA) AND G2(THETA). THETA = ABS(THETA) G1=DEXP(-4.0DO*DLOG(2.0D0)*((THETA/HBW1)**2)) G2=10**(K/10)*DEXP(-4.0DO*DLOG(2.0D0)*((THETA/HBW2)**2)) CALCULATING FINAL GAIN - G. NOTE: G(0) HAS BEEN ALREADY SIMPLIFIED. G=(G1+G2)/(1+10**(K/10)) RETURN END ******************************************************************************* *************** SUBROUTINE GAS HUMIDITY ATTENUATION *************************** ******************************T********7.*************************************** * SUBROUTINE GAS_HUMIDITY_ATTENUATION(AG,AH) * ^DOUBLE PRECISION AG,AH *  AG = O.DO AH = O.DO RETURN END *************** SUBROUTINE GET S ********************************************** ******************************,T************************************************ * THIS SUBROUTINE CALCULATES THE VALUE OF S IN THE TRANSITION STAGE * FROM SNOW TO RAIN AT ANY HEIGHT OF THE RAIN CELL. ******************************************************************************* * SUBROUTINE GET_S(HMAX1,HFR,HM,HMIN,HEIGHT,S,DECISION) DOUBLE PRECISION HMAX1,HFR,HM,HMIN,HEIGHT,S CHARACTER*2 DECISION  *  IF (DECISION.EQ.'Y') THEN IF ((HEIGHT.GE.HMIN).AND.(HEIGHT.LE.HM ))THEN S=1.D0 ELSEIF ((HEIGHT.GE.HM ).AND.(HEIGHT.LE.HFR)) THEN S=(HFR-HEIGHT)/(HFR-HM) ELSEIF (HEIGHT.GT .HFR) THEN S=0.D0 ENDIF ELSE IF (HEIGHT.LE.HMAX1) THEN S = 1.0D0 ELSE S = O.DO ENDIF ENDIF  RETURN END ******************************************************************************* *************** SUBROUTINE LINE CYLINDER INTERSECT **************************** *******************************7********T************************************** * THIS SUBROUTINE COMPUTES THE INTERSECTION BETWEEN A LINE IN * THREE SPACE AND A CYLINDER. THE CYLINDER IS IN THE Z-DIRECTION * XO,YO,ZO ARE THE POSITION WHERE THE LINE BEGINS, A,B ARE THE CENTER * OF THE CYLINDER AT X,Y RESPECTIVELY, H IS THE HEIGHT OF THE CYLINDER, * AND X1,Y1,Z1 AND X2,Y2,Z2 ARE THE INTERSECTION POINTS. FLAG IS  107  The program for the "Universal Model"  A VARIABLE RETURNED TO TELL US IF THERE IS ONE INTERSECTION, TWO INTERSECTIONS OR NO INTERSECTIONS FOR FLAG =0,1,-1 RESPECIVELY. IN ADDITION, XD,YD,ZD GIVES THE DIRECTION OF THE LINE, R IS THE RADIUS OF THE CYLINDER AND H IS THE HEIGHT OF THE CYLINDER. T1 AND T2 ARE THE VARIABLES USED FOR THE PARAMETRIC EQUATIONS * * * *  NOTE*** IF THE LINE GOES STRAIGHT THROUGHT THE MIDDLE OF THE CYLINDER IT WILL NOT SHOW INTERSECTION, SO IT IS EASY TO PUT THE LINE ON A SMALL ANGLE. THE RESULTS SHOULD BE VERY CLOSE.  ** ******* ******************* ******* ******************* ******* ****************** *  SUBROUTINE LINE_CYLINDER_INTERSECT(X0,YO,Z0,A,B,C,R,HH,X1,Y1,Z1, & X2,Y2,Z2,XD,YD,ZD,FLAG) DOUBLE PRECISION XO,YO,ZO,A,B,C,R,H,X1,Y1,Z1,X2,Y2,Z2, DISCRIMINANT,AA,BB,CC,T1,T2,XD,YD,ZD,HH,A1X,A1Y, & & AlZ,A2X,A2Y,A2Z,A11,A22,R1B,R2B,PHILPHI2,B11 INTEGER FLAG H=HH+C AA=XD**2.000+YD**2.000 BB=2.0D0*(X0*XD-A*XD+YO*YD-B*YD) CC=X0**2.0DO+YO**2.0D0+A**2.0D0+B**2.0D0-2*A*X0-2*B*Y0 CC=CC-R**2.0D0 DISCRIMINANT=BB**2.0D0-4*AA*CC IF (DISCRIMINANT .LT. 0) THEN FLAG=-1 ELSEIF (DISCRIMINANT .EQ.0) THEN FLAG=0 T1=-BB/(2.0DO*AA) Z1=XD*T+ZO IF ((Z1.LT.C) .OR. (21 .GT. H)) THEN FLAG=-1 ELSE X1=XD*T1+X0 Y1=XD*T1+Y0 X2=X1 Y2=Y1 Z2=Z1 ENDIF ELSEIF (DISCRIMINANT .GT. 0) THEN FLAG=1 T1=(-BB-DSQRT(DISCRIMINANT))/(2*AA) T2=(-BB+DSQRT(DISCRIMINANT))/(2*AA) Z1=ZD*T1+Z0 Z2=ZD*T2+Z0 IF (((Z1.LT.C).AND.(Z2.LT.C)).OR.((Z1.GT.H).AND.(Z2.GT.H))) &^THEN FLAG=-1 ELSEIF ((Z1.EQ.H).AND.(Z2.GT.H)) THEN FLAG=0 X1=XD*T1+X0 Y1=YD*T1+Y0 X2=X1 Y2=Y1 Z2=Z1 ELSEIF ((Z1.EQ.C).AND.(Z2.LT.C)) THEN FLAG=0 X1=XD*T1+X0 Y1=YD*T1+Y0  108  The program for the "Universal Model"  X2=X1 Y2=Y1 Z2=Z1 ELSEIF ((Z1.GT.H).AND.(Z2.EQ.H)) THEN FLAG=O X2=XD*T2+XO Y2=YD*T2+YO X1=X2 Y1=Y2 Z1=Z2 ELSEIF ((Z1.LT.C).AND.(Z2.EQ.C)) THEN FLAG=0 X2=XD*T2+X0 Y2=YD*T2+Y0 X1=X2 Y1=Y2 Z1=Z2 ELSEIF ((Z1.GT.H).AND.((Z2.LE.H).AND.(Z2.GE.C))) THEN X2=XD*T2+X0 Y2=YD*T2+Y0 Z1=H T1=(Z1-Z0)/ZD X1=XD*T1+X0 Y1=YD*T1+Y0 ELSEIF ((Z1.LT.C).AND.((Z2.LE.H).AND.(Z2.GE.C))) THEN X2=XD*T2+X0 Y2=YD*T2+Y0 Z1=C T1=(Z1-Z0)/ZD X1=XD*T1+XO Y1=YD*T1+Y0 ELSEIF ((Z1.LT.C).AND.(Z2.GT.H)) THEN Z1=C Z2=H T1=(Z1-Z0)/ZD T2=(Z2-Z0)/ZD X1=XD*T1+X0 Y1=YD*T1+Y0 X2=XD*T2+X0 Y2=YD*T2+Y0 ELSEIF ((Z2.LT.C).AND.(Z1.GT.H)) THEN Z2=C Z1=H T1=(Z1-Z0)/ZD T2=(Z2-Z0)/ZD X1=XD*T1+X0 Y1=YD*T1+Y0 X2=XD*T2+X0 Y2=YD*T2+Y0 ELSEIF(((Z1.LE.H).AND.(Z1.GE.C)).AND.(Z2.GT.H)) THEN Z2=H T2=(Z2-Z0)/ZD X1=XD*T1+X0 Y1=YD*T1+Y0 X2=XD*T2+X0 Y2=YD*T2+Y0 ELSEIF(((Z1.GE.C).AND.(Z1.LE.H)).AND.(Z2.LT.C)) THEN Z2=C T2=(22-Z0)/ZD X1=XD*T1+X0 Y1=YD*T1+Y0 X2=XD*T2+X0 Y2=YD*T2+Y0 ELSE X1=XD*T1+XO  109  The program for the "Universal Model"  Y1=YD*T1+YO X2=XD*T2+X0 Y2=YD*T2+YO ENDIF ENDIF IF (ABS(ZD).EQ.(1.0)) THEN FLAG = 1 X1 = XD Yl = YD Zl = C X2 = XD Y2 = YD Z2 = C + HH ENDIF IF (FLAG.EQ.1) THEN AiX = X1 - XO A1Y = Yl - YO A1Z = Zl - ZO A2X = X2 - XO A2Y = Y2 - YO A2Z = Z2 - ZO All = DSQRT(A1X**2 + A1Y**2 + A1Z**2) A22 = DSQRT(A2X**2 + A2Y**2 + A2Z**2) Bll = DSQRT(XD**2 + YD**2 + ZD**2) R1B = A1X*XD + A1Y*YD + A1Z*ZD R2B = A2X*XD + A2Y*YD + A2Z*ZD PHI1 = R1B/(All*B11) PHI2 = R2B/(A22*B11) IF (All.EQ.O.DO) THEN PHI1 = R1B/B11 ENDIF IF (A22.EQ.O.DO) THEN PHI2 = R2B/B11 ENDIF IF ((PHIl.LT.O.D0).AND.(PHI2.GE.O.D0)) THEN X1 = XO Yl = YO Zl = ZO ELSEIF ((PHI2.LT.O.D0).AND.(PHIl.GE.O.D0)) THEN X2 = X1 Y2 = Yl Z2 = Zl X1 = XO Yl = YO Zl = ZO ELSEIF ((PHIl.LT.O.D0).AND.(PHI2.LT.O.D0)) THEN FLAG = -1 ENDIF ENDIF RETURN END ******************************************************************************* ******************************************************************************* *************** SUBROUTINE POSITIONINPUT ************************************** ******************************************************************************* * THIS SUBROUTINE INPUTS THE POSITION OF THE RAIN CELL (XN,YN,ZN), * POSITION OF TRANSMITTER (XT,YT,ZT), DIRECTION OF TRANSMISSION * (XTHE_T,XPHI_T), POSITION OF RECEIVER (XR,YR,ZR), AND DIRECTION OF RECEPTION (XXXXXX,YYYYYY) * *******************************************************************************  * *  SUBROUTINE POSITIONINPUT(XH,YH,ZH,XTHE_T,XPHI_T, &^XR,YR,ZR,XXXXXX,YYYYYY) DOUBLE PRECISION XH,YH,ZH,XTHET,XPHIT,XR,YR,ZR,  110  The program for the "Universal Model"  &  *  XXXXXX,YYYYYY  READ*,XH READ*,YR READ*,ZH READ*,XTHE_T READ*,XPHI_T READ*,XR READ*,YR READ*,ZR READ*,XXXXXX READ*,YYYYYY  RETURN END ***************************************************************************** ***************************************************************************** *************** SUBROUTINE ROTATION ***************************************** ****************************************************************************** * THIS ROUTINE CALCULATES THE VALUES OF THE POINT TRANSFORMED TO THE * NEW COORDINATE SYSTEM. X,Y,Z ARE THE VALUES IN THE OLD COORDINATE * SYSTEM. XB,YB,ZB ARE THE VALUES OF X,Y,Z IN THE NEW COORDINATE SYSTEM. * THE TRANSFORMATION MATRIX IS: * IT11 T12 T131 * IT21 T22 T231 * IT31 T32 T331 ******************************************************************************* * SUBROUTINE ROTATION(T11,T12,T13,T21,T22,T23,T31,T32,T33, X,Y,Z,XB,YB,ZB) & DOUBLE PRECISION T11,T12,T13,T21,T22,T23,T31,T32,T33, X,Y,Z,XB,YB,ZB & COMPUTING THE NEW X,Y,Z FROM THE TRANSFORMATION MATRIX XB=T11*X+T12*Y+T13*Z YB=T21*X+T22*Y+T23*Z ZB=T31*X+T32*Y*T33*Z RETURN END ******************************************************************************* *************** SUBROUTINE UNIT VECTOR **************************************** *******************************T*********************************************** * THIS SUBROUTINE CALCULATES THE UNIT VECTOR BETWEEN TWO POINT IN * SPACE. ******************************************************************************* SUBROUTINE UNIT VECTOR(X1,Y1,Z1,X2,Y2,Z2,X,Y,Z) DOUBLE PRECISION X1,Y1,Z1,X2,Y2,Z2,X,Y,Z,DUMMY,DISTANCE  *  DUMMY=DISTANCE(X1,Y1,Z1,X2,Y2,Z2) X=(X2-X1)/DUMMY Y=(Y2-Y1)/DUMMY Z=(Z2-Z1)/DUMMY  RETURN END ****************************************************************************** *************** FUNCTION DISTANCE ******************************************** ****************************************************************************** * THIS FUNCTION CALCULATES THE DISTANCE BETWEEN TWO POINTS IN SPACE. AND THE UNIT VECTOR BETWEEN THE TWO POINTS *  111  The program for the "Universal Model" ****************************************************************************** * * *  *  DOUBLE PRECISION FUNCTION 1DISTANCE(X1,Y1,Z1,X2,Y2,Z2) DOUBLE PRECISION X1,Y1,Z1,X2,Y2,Z2,X,Y,Z X=(X1-X2)**2.D0 Y=(Y1-Y2)**2.D0 Z=(Z1-Z2)**2.D0 DISTANCE=DSQRT(X+Y+Z)  RETURN END ****************************************************************************** *************** FUNCTION RADAR CONST ***************************************** ******************************T*********************************************** • THIS FUNCTION COMPUTES THE RADAR CONSTANT FROM THE INPUTS HALF_THETA • G1,G2,GT (GAINS) AND FREQUENCY IN GHz. ****************************************************************************** * SUBROUTINE CONST_RADAR(FREQ,HALF_THETA,G1,G2,GT,RADAR_CONST) * DOUBLE PRECISION FREQ,HALFTHETA,G1,G2,GT,RADAR_CONST ^PARAMETER(P1=3.14159265359D0,C=2.9979244574D8) * GT=1.D0 RADAR_CONST=(1/(256*PI**2.D0))*((C/(FREQ*1.D9))**2.D0) RADARCONST=RADARCONST*(HALFTHETA**2.D0)*Gl*G2*GT RETURN END ****************************************************************************** ****************************************************************************** *************** SUBROUTINE TRANSLATION *************************************** ****************************************************************************** • THIS SUBROUTINE GIVES THE NEW COORDINATES XTRAN,YTRAN,ZTRAN OF THE POINT • X,Y,Z IN RELATION A NEW SET OF AXES XO,YO,ZO. ****************i************************************************************* * SUBROUTINE TRANSLATION(XO,YO,ZO,X,Y,Z,XTRAN,YTRAN,ZTRAN) * DOUBLE PRECISION XO,YO,ZO,X,Y,Z,XTRAN,YTRAN,ZTRAN * XTRAN=X-X0 YTRAN=Y-YO ZTRAN=Z-ZO RETURN END ******************************************************************************* ******************************************************************************* *************** SUBROUTINE ANGLE ********************************************** ******************************************************************************* • THIS SUBROUTINE COMPUTES THE ANGLE BETWEEN TWO 3-DIMENSIONAL • VECTORS. THE EQUATION USED IS COS(PHI)=X1*X2+Yl*Y2+Z1*Z2 * SQRT(IA111A21) • WHERE PHI IS THE ANGLE BETWEEN THE VECTORS. ******************************************************************************* * SUBROUTINE ANGLE(X1,Y1,Z1,X2,Y2,Z2,PHI) * DOUBLE PRECISION X1,Y1,Z1,X2,Y2,Z2,NUM,DEN,PI,PHI, TEMPI PARAMETER(P1=3.14159265359D0) *  112  The program for the "Universal Model"  NUM=X1*X2+Yl*Y2+Z1*Z2 DEN=DSQRT((X1*Xl+Yl*Y1+Z1*Z1)*(X2*X2+Y2*Y2+Z2*Z2)) TEMPI = NUM/DEN IF (TEMPl.GT.1.D0) TEMPI = 1.D0 PHI=DACOSD(TEMP1) RETURN END ******************************************************************************* *************** SUBROUTINE CROSS PRODUCT ************************************** ****************************************************************************** THIS ROUTINE COMPUTES THE CROSS PRODUCT OF TWO VECTORS. FOR EXAMPLE A=X1+Y1+Z1, B=X2+Y2+Z2 • THE CROSS PRODUCT OF A X B IS COMPUTED. • ******************************************************************************* SUBROUTINE CROSS PRODUCT(X1,Y1,Z1,X2,Y2,Z2,X,Y,Z) DOUBLE PRECISION X1,Y1,Z1,X2,Y2,Z2,X,Y,Z X=Y1*Z2-Y2*Z1 Y=X2*Z1-Xl*Z2 Z=X1*Y2-X2*Y1 RETURN END ******************************************************************************* ******************************************************************************* *************** SUBROUTINE SPHERE2RECT **************************************** ******************************************************************************* THIS ROUTINE CONVERTS THE SPHERICAL COORDINATES ENTERED TO • RECTANGULAR COORDINATES. WHERE THE INPUTS ARE THETA(DEG),PHI(DEG), • • RHO AND THE OUTPUTS ARE X,Y,Z ******************************************************************************* SUBROUTINE SPHERE2RECT(X,Y,Z,THETA,PHI,RHO) DOUBLE PRECISION X,Y,Z,THETA,PHI,RHO X=RHO*DSIND(THETA)*DCOSD(PHI) Y=RHO*DSIND(THETA)*DSIND(PHI) Z=RHO*DCOSD(THETA) RETURN END ******************************************************************************* ******************************************************************************* *************** SUBROUTINE RECT2SPHERE **************************************** ******************************************************************************* THIS ROUTINE CONVERTS THE RECTANGULA4R COORDINATES ENTERED TO • SPHERICAL COORDINATES. WHERE THE INPUTS ARE X,Y,Z AND • THE OUTPUTS ARE RHO, THETA(DEG), PHI(DEG). • ******************************************************************************* SUBROUTINE RECT2SPHERE(X,Y,Z,THETA,PHI,RHO) DOUBLE PRECISION X,Y,Z,THETA,PHI,RHO THETA=DATAND(DSQRT(X*X+Y*Y)/Z) PHI=DATAND(Y/X) RHO=DSQRT(X*X+Y*Y+Z*Z) IF ((X.LE.0).AND.(Y.GE.0)) THEN PHI = 180.DO + PHI ELSEIF ((X.LE.0).AND.(X.LE.0)) THEN  113  The program for the "Universal Model"  PHI = PHI + 180.DO ELSEIF ((X.GE.0).AND.(Y.LE.0)) THEN PHI = PHI + 360.DO ENDIF IF ((ABS(Z).GT.O).AND.(X.EQ.0).AND.(Y.EQ.0))THEN PHI=O ENDIF IF (Z.LT.0) THEN THETA = 180.DO + THETA ENDIF RETURN END ******************************************************************************* *************** SUBROUTINE SCAT TASK ***************************************** *******************************T***********************************************  * *  *  *  SUBROUTINE SCATTASK(F,THETA,PHI,S,RATE,FREQ,T,QUANTITY, • TASKNUMBER2,DECISION,H_THICKNESS) DOUBLE PRECISION F,THETA,PHI,S,RATE,FREQ,T,QUANTITY, H_THICKNESS INTEGER TASKNUMBER2 CHARACTER*2 DECISION IF (TASKNUMBER2.EQ.1) THEN CALL SCATTERING(F,THETA,PHI,S,RATE,FREQ,T,QUANTITY,DECISION, H_THICKNESS) ENDIF  RETURN END ******************************************************************************* **************** SUBROUTINE SCATTERING **************************************** ******************************************************************************* THIS SUBROUTINE COMPUTES FO, F(D)  * ******************************************************************************* * SUBROUTINE SCATTERING(F,THETA,PHI,S,RATE,FREQ,T,QUANTITY,DECISION, H_THICKNESS) DOUBLE PRECISION F,THETA,PHI,S,RATE,FO,F THETA PHI,QUANTITY, • FMDP,FD,FS,FD1,FD2,FD3,PI,T,FREQ,AiiEP,FRfQU ENCY,FREQR, • N,NUM,F01,F02,F03,K,C,XI,STEMP,H_THICKNESS,RTEMP,REF COMPLEX*16 E,M,MDP CHARACTER*2 DECISION PARAMETER(P1=3.14159265359D0,C=2.9979244574D10) STEMP=1.D0 IF (THETA.GT.90.D0) THEN THETA=180.DO-THETA ENDIF IF (PHI.GT.180.D0) THEN PHI=360.DO-PHI ENDIF IF (S.LT.0.008D0) THEN STEMP=S S=1.D0 ENDIF IF (RATE .LT. 0.25D0) THEN RTEMP = RATE RATE = 0.25D0 ENDIF CALL A(S,RATE,AREP,QUANTITY)  114  The program for the "Universal Model"  * *  *  * *  *  XI=0.866D0*(1.DO+FREQ*1.5D-4) FREQR=C*XI/(2.D0*PI*A_REP) FREQUENCY=FREQ*1.D9 K=FREQUENCY/FREQR N=(2.D0+200.D0*(K)**3.D0)/(1.D0+201.D0*(K)**3.D0) CALL NUMBER(RATE,NUM,S,QUANTITY) CALL PERMATIVITY(T,FREQUENCY,E,M) F01=400.DO*NUM*(XI**(2.DO*N))*PI*A_REP**2.D0 F02=(ABSNE-1.D0)/(E+2.D0)))**2.D0 F03=((K)**(2.DO*N))/(1.D0+(K)**(2.DO*N)) FO=F01*F02*F03 FD1=((l.DO+N)**2.D0)/(2.DO*N*(2.5D0*N-1.D0)) FD2=((N**2.DO+K)/(1.D0+(N**2.D0)*K))**N FD3=(1.DO+RATE/300.D0)/2.D0 FD3=FD3+((K**2.D0)/(1.DO+K**2.D0))*(1.DO-RATE/150.D0) FD=FD1*FD2*FD3 MDP=-DIMAG(M) CALL FTP_FM_FS(F_THETA_PHI,FMDP,FS,XI,K,N,RATE,S,THETA,PHI,MDP) F =FO*F_THETAPHI*FD*FMDP*FS IF (STEMP.LT.0.008D0) THEN S=STEMP F=F/1.D0 IF (DECISION.NE .'Y') THEN F = F*1.D0 ENDIF IF ((DECISION.EQ.'Y').AND.(HTHICKNESS.EQ.O.D0)) THEN F = F*1.D0 ENDIF ENDIF IF (RATE .LT. 0.25D0) THEN F = F*RTEMP/0.25D0 RATE = RTEMP ENDIF  IF (S.EQ.O.DO) THEN REF = 400.DO*RATE**1.4 F = F02*REF*(PI**5)*(1.D-18)*(FREQUENCY/(C*0.01D0))**4 ENDIF RETURN END ******************************************************************************* ******************************************************************************* ********* SUBROUTINE FTP FM FS ************************************************ ************************T**T*************************************************** * SUBROUTINE FTP_FM_FS(F_THETA_PHI,FMDP,FS,XI,K,N,RATE,S,THETA, PHI,MDP) DOUBLE PRECISION F THETA PHI,FMDP,FS,XI,K,N,RATE,S,THETA, PHI,MDP,F_TH1TF_ T H2TF_ T H3,FMDP1,FMDP2,FS1,FS2,FS3 F TH1=(DSIND(THETA))**(2.DO/N) FITH2=(K/(1.DO+K))*(XI**2.D0) F TH3=(((DSIND(THETA))**3.D0)*DCOSD(PHI)) F TH3=F TH3+.5D0*(DCOSD(THETA))**4.D0 FITHETAIPHI=(FTH1+FTH2*FTH3)**N FMDP1=1.D0-(MDP/2.6D0)*(1.DO+RATE/600.D0) FMDP2=(MDP/2.6D0)*DSIND(THETA)*DCOSD(PHI) FMDP=1.DO+FMDP1*FMDP2  115  The program for the "Universal Model"  *  FS1=((150.D0/(150.DO+RATE))*25.D0*S/(1.DO+K**2.D0)) FS1=(N**2.D0)*FS1**((1.D0-S)**4.D0) FS1=FS1*DSQRT(RATE/100.D0) FS2=((RATE+100.D0)/100.D0)*(1.D0-.5DO*K*(DSIND(PHI))**2.D0) FS3=((1.DO+K**2.D0)/(2.D0*N**2.D0))*DCOSD(PHI)*DSIND(THETA) FS=(FS1*(FS2+FS3))**(1.D0-S) RETURN END ******************************************************************************* ********** SUBROUTINE GET ATTEN *********************************************** *************************T***************************************************** • THIS SUBROUTINE INTERPOLATES THE ATTENUATION TABLE TO FIND THE • ATTENUATION FOR ANY RAIN RATE BETWEEN 1.25 AND 150 mm/hr. AND • ANY S BETWEEN 0.0 AND 1.0. • NOTE: IF THE RAIN RATE IS GREATER THAN 50 mm/hr. AND S IS NOT • EQUAL TO 1.0 THEN THE RESULT WILL BE INCORRECT. WE DO NOT HAVE VALUES • FOR THESE RAIN RATES AND S. • AT_TABLE IS THE RETURNED ATTENUATION VALUE FROM THE TABLE. *  • • * • • •  *  AT^RATE^S^AT_TABLE -  ARRAY OF ATTENUATION FOR DIFFERENT RAIN RATES AND S THE ENTERED RAIN RATE THE ENTERED S THE RETRIEVED ATTENUATION FROM THE ARRAY AT (INCLUDING INTERPOLATION) R,SS^- ARRAYS OF RAIN RATE AND S TO FIND POSITION IN ARRAY DUM1,DUM2- DUMMY VARIABLE TO HELP CALCULATE THE INTERPOLATED RESULT.  ******************************************************************************* *  SUBROUTINE GETATTEN(AT,RATE,S,ATTABLE) DOUBLE PRECISION AT(19,9),RATE,S,AT_TABLE,R(9),SS(19), DUM1,DUM2 INTEGER I,P1,R2,S1,S2 RAIN RATE AND S ARRAYS RESPECTIVELY TO FIND POSITION IN ARRAY DATA R/0,1.25,2.5,5,12.5,25,50,100,150/ DATA SS/0,.02,.04,.06,.08,0.1,.12,.14,.16,.18,.2,.3,.4,.5,.6, .7,.8,.9,1.0/ AT(0,0)=0 FINDING POSITION OF ELEMENT.  *  DO 10 1=1,8 IF ((RATE.GT .R(I)).AND.(RATE.LE.R(I+1))) THEN R1=1 R2=I+1 ENDIF 10^CONTINUE DO 20 1=1,18 IF ((S.GT.SS(I)).AND.(S.LE.SS(I+1))) THEN S1=1 S2=I+1 ENDIF 20 CONTINUE IF ((RATE.EQ.0).OR.(S.EQ.0))THEN AT_TABLE=0 ELSE DUM1=(AT(S1,R2)-AT(S1,R1))*(RATE-R(R1))/(R(R2)-R(R1)) +AT(S1,R1) DUM2=(AT(S2,R2)-AT(S2,R1))*(RATE-R(R1))/(R(R2)-R(R1))  116  The program for the "Universal Model"  *  +AT(S2,R1) AT_TABLE=(DUM2-DUM1)*(S-SS(S1))/(SS(S2)-SS(S1))+DUM1 ENDIF  RETURN END ******************************************************************************* ********* SUBROUTINE GET ARRAY ************************************************ ************************T****************************************************** • THIS SUBROUTINE READS FROM THE FILE 'DATA1' ALL THE S AND RAIN RATE • VALUES FOR ONE FREQUENCY. IT THEN STORES THESE VALUES IN AN ARRAY • CALLED AT(S,RATE). ******************************************************************************* * SUBROUTINE GETARRAY(FREQ,AT) DOUBLE PRECISION FREQ,AT(19,9),F(28),FR,R,SS INTEGER I,J,K DATA F/1,1.5,2,2.5,3,3.5,4,5,6,7,8,9,9.6,10,11,12,15,20,25, 30,35,40,50,60,70,80,90,100/ OPEN(UNIT=10,FILE='datal',STATUS='OLD')  * *  DO 20 J=1,28 IF (F(J).EQ.FREQ) THEN K=J J=28 ENDIF 20^CONTINUE J=153*(K-1) DO 10 I=1,J READ(10,*) 10^CONTINUE READ(10,*),FR DO 30 1=2,9 READ(10,*),R DO 40 J=2,19 READ(10,*),SS,AT(J,I) 40^CONTINUE 30^CONTINUE DO 50 1=1,9 AT(1,I)=0 50 CONTINUE DO 60 1=1,19 AT(I,1)=0 60 CONTINUE CLOSE(UNIT=10,STATUS='KEEP')  RETURN END ******************************************************************************* ********** SUBROUTINE GET AFB ************************************************* *************************T***************************************************** • THIS SUBROUTINE INTERPOLATES THE ATTENUATION, FORWARD, AND BACKWARD • SCATTERING FOR ANY RAIN RATE BETWEEN 1.25 AND 150 mm/hr. AND • ANY S BETWEEN 0.0 AND 1.0. • NOTE: IF THE RAIN RATE IS GREATER THAN 50 mm/hr. AND S IS NOT • EQUAL TO 1.0 THEN THE RESULT WILL BE INCORRECT. WE DO NOT HAVE VALUES • FOR THESE RAIN RATES AND S. AFB_TABLE IS THE RETURNED ARRAY FORM 'datal' ARRY^- THE ARRAY IN WHICH THE VALUES OF ATTENUATION, 100 * FORWARD, AND 100* BACKWARD SCATTERING. RATE^- THE ENTERED RAIN RATE S^- THE ENTERED VALUE OF S  117  The program for the "Universal Model"  • • • • • *  AFB_TABLE- THE VALUE OBTAINED FROM THE ARRAY AFTER INTERPOLATION, (1) ATTENUATION, (2) FORWARD SCATTERING, (3) BACKWARD SCATTERING R,SS^- THE VALUES TO WHICH THE ENTERED RAIN RATE AND S ARE COMPARED. DUM1,DUM2- DUMMY VARIABLES R1,R2^- INDEX VALUES TO GIVE POSITION OF ENTERED RAIN RATE IN ARRAY R S1,S2^- INDEX VALUES TO GIVE POSITION OF ENTERED S IN ARRAY SS  ******************************************************************************* *  SUBROUTINE GET_AFB(ARRY,RATE,S,AFB_TABLE)  * • *  DOUBLE PRECISION ARRY(19,9,3),RATE,S,AFB_TABLE(3),R(9),SS(19), • DUM1(3),DUM2(3) INTEGER I,R1,R2,S1,S2 DATA TO WHICH ENTERED RATE AND S WILL BE COMPARED TO FIND POSITION DATA R/0,1.25,2.5,5,12.5,25,50,100,150/ DATA SS/0,.02,.04,.06,.08,0.1,.12,.14,.16,.18,.2,.3,.4,.5,.6, .7,.8,.9,1.0/ & IF THE RAIN RATE OR S = 0 THEN RETURN ZERO FOR ATTENUATION, FORWARD SCATTERING, AND BACKWARD SCATTERING IF ((RATE.EQ.0).OR.(S.EQ.0)) THEN AFB_TABLE(1)=0 AFB TABLE(2)=0 AFB_TABLE(3)=0 ELSE FINDING POSITION OF RAIN RATE ELEMENT.  DO 10 1=1,8 IF ((RATE.GT .R(I)).AND.(RATE.LE.R(I+1))) THEN R1=I R2=I+1 ENDIF 10 CONTINUE * FINDING POSITION OF S ELEMENT. DO 20 1=1,18 IF ((S.GT.SS(I)).AND.(S.LE.SS(I+1))) THEN S1=I S2=I+1 ENDIF 20 CONTINUE  * • • *  LINEARLY INTERPOLATING TO FIND VALUE OF ATTENUATION, FORWARD SCATTERING, AND BACKWARD SCATTERING. DO 30 1=1,3 DUM1(I)=(ARRY(S1,R2,I)-ARRY(S1,R1,I))*(RATE-R(R1)) /(R(R2)-R(R1))+ARRY(S1,R1,I) DUM2(I)=(ARRY(S2,R2,I)-ARRY(S2,R1,I))*(RATE-R(R1)) /(R(R2)-R(R1))+ARRY(S2,R1,I) CALCULATING ACTUAL RESULT.  *  AFB_TABLE(I)=(DUM2(I)-DUM1(I))*(S-SS(S1))/(SS(S2)-SS(S1)) +DUM1(I) 30^CONTINUE ENDIF  118  The program for the "Universal Model"  RETURN END ******************************************************************************* ********* SUBROUTINE GET AFBARRY *** * * ****************************************** ************************ T****************************************************** , • THIS SUBROUTINE READS FROM THE FILE 'DATA1' ALL THE S AND RAIN RATE • VALUES FOR ONE FREQUENCY. IT THEN STORES THESE VALUES IN AN ARRAY • CALLED ARRY(S,RATE). * * F^- POSSIBLE FREQUENCIES FREQ^- CORRECT FREQUENCY USED • • ARRY^- ARRAY CONTAINING VALUES FOR ATTENUATION, FORWARD SCATTERING, AND BACK SCATTERING • FR^- FREQUENCY IN 'datal' FILE (NOT USED) * ******************************************************************************* * SUBROUTINE GET_AFBARRY(FREQ,ARRY) * DOUBLE PRECISION FREQ,ARRY(19,9,3),F(28),FR,R,SS INTEGER I,J,K  *  F - THE FREQUENCIES USED DATA F/1,1.5,2,2.5,3,3.5,4,5,6,7,8,9,9.6,10,11,12,15,20,25, 30,35,40,50,60,70,80,90,100/ OPENING FILE 'datal' TO READ ATTENUATION, FORWARD SCATTERING, AND BACK SCATTERING. OPEN(UNIT=10,FILE='datal',STATUS='OLD') FINDING POSITION OF FREQUENCY DO 20 J=1,28 IF (F(J).EQ.FREQ) THEN K=J J=28 ENDIF CONTINUE SKIPPING THROUGH FILE 'datal' TO CORRECT FREQUENCY. J=153*(K-1) DO 10 I=1,J READ(10,*) CONTINUE READING FREQUENCY READ(10,*),FR READING IN ATTENUATION, FORWARD SCATTERING, AND BACK SCATTERING FOR DIFFERENT RAIN RATES.  *  DO 30 1=2,9 READ(10,*),R DO 40 J=2,19 READ(10,*),SS,ARRY(J,I,1),ARRY(J,I,2),ARRY(J,I,3) ARRY(J,I,2)=ARRY(J,I,2)*100 ARRY(J,I,3)=ARRY(J,I,3)*100 40^CONTINUE 30 CONTINUE SETTING ATTENUATION, FORWARD SCATTERING, AND BACK SCATTERING TO ZERO  119  The program for the "Universal Model"  • *  * *  WHEN S=0 AND/OR RAIN RATE =0'  DO 50 1=1,9 ARRY(1,I,1)=0 ARRY(1,I,2)=0 ARRY(1,I,3)=0 50 CONTINUE DO 60 1=1,19 ARRY(I,1,1)=0 ARRY(I,1,2)=0 ARRY(I,1,3)=0 60 CONTINUE CLOSE(UNIT=10,STATUS='KEEP')  RETURN END ******************************************************************************* **************** SUBROUTINE RAIN TOTAL ATTENUATION **************************** ********************************T*****V**************************************** * • THIS SUBROUTINE CALCULATES THE TOTAL RAIN ATTENUATION ALONG THE • LINE CONNECTING THE RECEIVER AND THE POINT TO WHICH THE TRANSMITTER • TRANSMITS TO. * X11,Y11,Z11 - COORDINATES OF INTERSECTION OF MAIN BEAM OF RECEIVER AND RAIN CYLINDER X, Y, Z - COORDINATES WHERE TRANSMITTER TRANSMITS TO - TOTAL RAIN ATTENUATION RTATT DT - SMALL INCREMENT OF T ALONG BEAM AXIS XD,YD,ZD - DIRECTION OF LINE CONNECTING (X11,Y11,Z11) AND (X,Y,Z) XT, YT, ZT - COORDINATES OF FIRST POINT BETWEEN OTHER COORDINATES XA, YA, ZA - POINTS ALONG LINE CONNECTING THE TWO POINTS RATE - CALCULATED RAIN RATE - CENTER OF RAIN CELL XH, YH - MAX RAIN RATE (AT CENTER OF CELL) RMAX - RAIN RATE DISTRIBUTION VARIABLE RO DECISION - IS THERE A MELTING LAYER?  ******************************************************************************* SUBROUTINE RAIN_TOTAL_ATTENUATION(HMAX1,HFR,HM,HMIN,FREQ, • TE11,QUANTITY,X,Y,Z,X11,Y11,Z11,RTATT,XH,YH,RMAX,RO,DECISION, • TASKNUMBER1,NARR,A1NUM,ANUM,BNUM,XR,YR,ZR,H_THICKNESS,RAD, • XTHET)  *  DOUBLE PRECISION X11,Y11,Z11,X,Y,Z,T,RTATT,DT,XD,YD,ZD, • XT,YT,ZT,DR,XA1,YA1,ZA1,TP1,HMAX1,HFR,HM,HMIN,S,RATE, RTATT,FREQ,TEMP,QUANTITY,XH,YH,RMAX,RO,HMAX1, • • NARR(2,3),A1NUM,ANUM,BNUM,ROM,D,XR,YR,ZR,ANGEP, • H_THICKNESS,XP1,YP1,ZP1,XM1,YM1,ZM1,MELTATTEN,XA2,YA2, • ZA2,RAD,XTHE_T,TP2,TE11,ATTEN CHARACTER*2 DECISION INTEGER I,TASKNUMBER1 XD=X11-X YD=Y11-Y ZD=Z11-Z T=(Z11-Z)/ZD RTATT=0 DT=T/20.0D0 XT=XD*DT+X YT=YD*DT+Y ZT=YD*DT+Z DR=DSQRT((XT-X)**2+(YT-Y)**2+(ZT-Z)**2)  120  The program for the "Universal Model"  TP1 = -DT/2.D0 DO 10 1=1,20 TP1=DT+TP1 XA1=XD*TP1+X YA1=YD*TP1+Y ZA1=ZD*TP1+Z CALL GET_S(HMAX1,HFR,HM,HMIN,ZA1,S,DECISION) CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) RTATT=RTATT+ATTEN*DR/1000 10 CONTINUE RETURN END ******************************************************************************* **************** ATT TASK ***************************************************** ********************7**********************************************************  *  SUBROUTINE ATT_TASK(FREQ,T,RATE,S,ATTEN,QUANTITY, TASKNUMBER1,NARR,A1NUM,ANUM,BNUM)  *  * *  DOUBLE PRECISION FREQ,T,RATE,S,ATTEN,QUANTITY,ATTE2, NARR(2,3),A1NUM,ANUM,BNUM INTEGER TASKNUMBER1 IF (TASKNUMBER1.EQ.1) THEN CALL LP(FREQ,T,RATE,S,ATTEN,QUANTITY) ELSEIF (TASKNUMBER1.EQ.2) THEN CALL C_LP(FREQ,T,RATE,S,ATTEN,QUANTITY) ELSEIF (TASKNUMBER1.EQ.3) THEN CALL M_ATTENUATION(FREQ,T,RATE,S,ATTEN,ATTE2,QUANTITY) ELSEIF (TASKNUMBER1.EQ.4) THEN CALL MC ATTENUATION(FREQ,T,RATE,S,ATTEN,ATTE2,QUANTITY) ELSEIF (TASKNUMBER1.EQ.5) THEN CALL SATTCALC(S,NARR,RATE,A1NUM,ANUM,BNUM,ATTEN) ENDIF RETURN END  ****************************************************************************** **************** SUBROUTINE EMPIRICAL) ATTENUATION**************************** **************************************7***************************************  *  SUBROUTINE SATTCALC(S,N,RATE,A1,A,BNUM,ALPHA) DOUBLE PRECISION S,N(2,3),RATE,A1,M1,M2,FS, A2,M3,B3,B2,B1,A,BNUM,ALPHA PARAMETER(A2=1.7,M3=1,B3=230,B2=6,B1=20) IF (S.LT.1.0D0) THEN M1=N(1,1)+N(1,2)*RATE".003+N(1,3)*RATE".0002 M2=N(2,1)+N(2,2)*RATE".003+N(2,3)*RATE".0002 FS=Ml*DEXP(-B1*S**A1)*S**(A1-1) FS=FS+M2*DEXP(-B2*S**A2)*S**(A2-1) FS=FS-M3*DEXP(-B3*S)+1  121  The program for the "Universal Model"  ELSE FS=1.0D0 ENDIF ALPHA=FS*A*RATE**BNUM RETURN END ******************************************************************************* *************** Al CALL ******************************************************* ******************T************************************************************ • THIS SUBROUTINE CALCULATES al FOR THE SUBROUTINE SATTCALC TO USE • IN THE. EQUATION FOR F(S) (SEE SATTCALC) * ******************************************************************************* * SUBROUTINE Al_CALC(FREQUENCY,A1) * DOUBLE PRECISION FREQUENCY,F,C(6),A1 * DATA C/1.680592398562237e+00, • -2.790804733796220e-03, • -3.417061223974332e+00, 5.049293127489146e+00, 4.707754939384348e-01, • -2.262928752321020e+00/ *  F=FREQUENCY A1=C(1)+C(2)*F+C(3)*F**-1+C(4)*F**-2+C(5)*DEXP(-F)*F**2.1 A1=Al+C(6)*F**-3 RETURN END  * ******************************************************************************* ***************** N CALL ****************************************************** *******************T*********************************************************** THIS SUBROUTINE CALCULATES Ni AND N2 WHICH ARE NEEDED TO CALCULATE MI AND M2. * N[1]^([2],^[3]) 1. 1 OR 2 FROM M1 OR M2 RESPECTIVELY 2. 1 or 2 FOR LOW AND HIGH FREQUENCY RANGES. 1=1-12 GHz.,^2=12-100^GHz.^IF^[1] =^1^(M1) 1=1-20 GHz.,^2=20-100^GHz.^IF^[1] = 2^(M2) 3. CAN BE 1,2,OR 3 FOR THE VARIABLE N1,N2,OR N3. * ******************************************************************************* SUBROUTINE NCALC(FREQUENCY,N) DOUBLE PRECISION F,FREQUENCY,N1(2,3,7),N2(2,3,8),N(2,3) INTEGER I  *  DATA (N1(1,1,I),I=1,7)/-4.468710233696790e+10, 1.403226146957391e+08, -6.960027662822775e+08, 4.506558794171744e+10, -6.421292889699095e+07, 8.731179500736722e+08, 1.401495928213695e+06/ DATA (N1(2,1,1),1=1,7)/-6.295278387177499e+10,  122  The program for the "Universal Model"  6.296355557484093e+10, 5.966553195229598e+05, -3.888357198437326e+05, -1.923936894943743e+08, -1.822301639050625e+05, 8.656888811513758e+04/ DATA (N1(1,2,I),I=1,7)/-3.174987655808701e+09, 1.000814015141864e+07, -4.938711808841844e+07, 3.201787180190318e+09, -4.565132790730321e+06, 6.211655984741843e+07, 9.948011000116054e+04/ DATA (N1(2,2,1),1=1,7)/-4.084602058168248e+09, 4.085333281240790e+09, 4.259904738584688e+04, -2.783771486082181e+04, -1.248919873419444e+07, -1.281758620243883e+04, 6.233978784040986e+03/ DATA (N1(1,3,I),I=1,7)/ 4.786234527835458e+10, -1.503307914573111e+08, 7.454010806448646e+08, -4.826764045492385e+10, 6.877779221467581e+07, -9.352359718254586e+08, -1.500976973852585e+06/ DATA (N1(2,3,1),I=1,7)/6.702815890731934e+10, -6.703966103497658e+10, -6.392557379690040e+05, 4.166781931398259e+05, 2.048548402103934e+08, 1. 950457764835905e+05, -9.280180321809524e+04/ DATA (N2(1,1,1),1=1,8)/-2.099142532327721e+07, 1.400026226688738e+07, -8.882786193005802e+05, 1.989261675140147e+04, 4.174527682005403e+06, 1.593655028099594e+07, 7.501240296862618e+05, -2.083589564285219e+07/ DATA (N2(2,1,I),I=1,8)/-4.439610851013492e+07, -8.943850388853900e+05, 6.292344533547490e+03, -1.902820302670007e+01, -2.522716953072156e+15, 1.276598780066267e+14, -2.293117873033613e+04, 1.952254679706120e+07/ DATA (N2(1,2,I),I=1,8)/-1.493679774541824e+06, 9.956472852260806e+05, -6.314103317406768e+04, 1.413509106519248e+03, 2.972144450610339e+05, 1.134692648445301e+06, 5.317895603908228e+04, -1.482614399143066e+06/  123  The program for the "Universal Model"  DATA (N2(2,2,1),1=1,8)/-3.150283332144003e+06, -6.334729039970136e+04, 4.455018565359137e+02, -1.346924157214865e+00, -1.803394805985552e+14, 9.125776857644061e+12, -1.623153722957795e+03, 1.384227230004449e+06/ DATA (N2(1,3,I),1=1,8)/ 2.248517939626698e+07, -1.499594585442772e+07, 9.514215725554167e+05, -2.130616396424133e+04, -4. 471624512634573e+06, -1.707130423100917e+07, -8.033028871120176e+05, 6.^ 2.231858202049057e+07/  * *  *  * *  DATA (N2(2,3,I),I=1,8)/4.754646862535044e+07, 9.577316084775798e+05, -6.737837935928022e+03, 2.037509693681765e+01, 2.703059184218280e+15, -1.367857979044508e+14, 2.455407948296932e+04, -2.090678544294405e+07/ F=FREQUENCY IF ((F.GE.1).AND.(F.LT.12)) THEN DO 10 1=1,3 N(1,I)=N1(1,I,1)+N1(1,I,2)*F*F*DEXP(-F)+N1(1,I,3)*F**.2 N(1,I)=N(1,I)+N1(1,I,4)*F**.01+N1(1,I,5)*F**.7 N(1,I)=N(1,I)+N1(1,I,6)*DEXP(-F)+N1(1,I,7)*DSIN(F/.9) 10^CONTINUE ELSEIF ((F.GE.12).AND.(F.LE.100)) THEN DO 20 1=1,3 N(1,1)=N1(2,I,1)+N1(2,I,2)*F**.003+N1(2,I,3)*EXP(-100+F) N(1,I)=N(1,I)+N1(2,I,4)*(1/DCOSH(F-20))+N1(2,I,5)*DLOG(F) N(1,I)=N(1,I)+N1(2,1,6)*(1/DCOSH(F-40)) N(1,I)=N(1,I)+N1(2,I,7)*DSIN(F/.08) 20^CONTINUE ENDIF IF ((F.GE.1).AND.(F.LT.20)) THEN DO 30 1=1,3 N(2,I)=N2(1,I,1)+N2(1,I,2)*F+N2(1,I,3)*F**2+N2(1,I,4)*F**3 N(2,I)=N(2,I)+N2(1,1,5)*F*DEXP(-F)+N2(1,I,6)*F*F*DEXP(-F) N(2,I)=N(2,I)+N2(1,I,7)*DSIN(.6*(F-2.8))+N2(1,I,8)*DLOG(F) 30^CONTINUE ELSEIF ((F.GE.20) .AND.(F.LE.100)) THEN DO 40 1=1,3 N(2,I)=N2(2,I,1)+N2(2,I,2)*F+N2(2,I,3)*F**2+N2(2,I,4)*F**3 N(2,I)=N(2,I)+N2(2,I,5)*F*DEXP(-F)+N2(2,I,6)*F*F*DEXP(-F) N(2,I)=N(2,I)+N2(2,1,7)*DSIN(.6*(F-2.8))+N2(2,I,8)*DLOG(F) 40^CONTINUE ENDIF RETURN END  * ******************************************************************************* ***************** A g *******************************************************  124  ^  The program for the "Universal Model" ******************************************************************************* •^ THIS SUBROUTINE COMPUTES THE VALUE OF a AND B FROM THE EQUATION * b ATTENUATION = a*r * ******************************************************************************* * * r^- RAIN RATE a^- VARIABLE DEPENDENT UPON FREQUENCY. * b^ VARIABLE DEPENDENT UPON FREQUENCY. • Cl^- CONSTANTS FOR b • C2^- CONSTANTS FOR a * ******************************************************************************* * SUBROUTINE A_B(FREQUENCY,A,B) * DOUBLE PRECISION FREQUENCY,F,A,B,C1(9),C2(11) * DATA C1/1.175693749863705D+00, -5.372610249096163D-03, -1.451430379276010D-04, 2.483615756859940D-06, -1.052429358496198D-08, -4.112419378557529D-01, -1.202673982695477D-01, 3.133414878851311D-02, -2.717811349173657D-02/ -  DATA C2/3.432142878826195D-03, 2.930235190132424D-09, -9.239923589608143D-11, 9.776864619131741D-13, -3.485507759140589D-15, 6.906393023884164D-04, 5.450508196170798D-06, -1.023109914867535D-02, 1.232762310835401D-02, 1.752618764932525D-02, 1.566935461598425D-04/ F=FREQUENCY CALCULATING b B=C1(1)+C1(2)*F+C1(3)*F**2+C1(4)*F**3+C1(5)*F**4 B=B+C1(6)*F*F*DEXP(-F)+C1(7)*F**(-2)+C1(8)*DLOG(F) B=B+C1(9)*(1/(DCOSH(.2*(F-13)))) CALCULATING a A=C2(1)*F**1.4+C2(2)*F**5+C2(3)*F**6+C2(4)*F**7+C2(5)*F**8 A=A+C2(6)*DEXP(-F)*F**3+C2(7)+C2(8)*F+C2(9)*DLOG(F) A=A+C2(10)*DEXP(-F)+C2(11)*F**-9 RETURN END ******************************************************************************* **************** SUBROUTINE M ATTENUATION (M L&P) ***************************** *****************************7************************************************* * ^SUBROUTINE MATTENUATION(FREQ,T,RATE,S,ATTEN,ATTE2,QUANTITY) * DOUBLE PRECISION FREQ,T,RATE,S,A_REP,N,FREQR,ATTEN,GEDP,NUM,EA DOUBLE PRECISION FREQUENCY,QUANTITY,ATTE2,C,RTEMP  125  The program for the "Universal Model"  *  COMPLEX*16 G,GE,EPSILON,GAMMA PARAMETER(P1=3.14159265359DO,EA=1.DO,C=2.9979244574D10) IF (RATE .LT. 0.25D0) THEN RTEMP = RATE RATE = 0.25D0 ENDIF IF (S.LT..002) THEN ATTEN=0 ATTE2=0 RETURN ENDIF FREQUENCY=FREQ*1.D9 CALL A(S,RATE,A REP,QUANTITY) FREQR=0.866D0*C/(2.D0*PI*A_REP) N=(2.D0+100.D0*(FREQUENCY/FREQR)**2.D0) N=N/(1.D0+101.D0*(FREQUENCY/FREQR)**2.D0) CALL NUMBER(RATE,NUM,S,QUANTITY) CALL G_LOWCASE(S,A_REP,G,T,FREQUENCY) GE=G/DCMPLX(1.D0,(FREQUENCY/FREQR)**N) GEDP=-DIMAG(GE) ATTEN=9.1D0*GEDP*NUM*1.D4*FREQ EPSILON=EA*(1.D0+GE*NUM) GAMMA=2.D0*PI*FREQUENCY*CDSQRT(-4.D0*PI*1.D-7*EPSILON) ATTE2=DREAL(GAMMA)  * *  IF (RATE .LT. 0.25D0) THEN ATTEN = ATTEN*RTEMP/0.25D0 RATE = RTEMP ENDIF RETURN END  ******************************************************************************* **************** SUBROUTINE MC ATTENUATION (MC L&P)**************************** ******************************7************************************************  * *  SUBROUTINE MCATTENUATION(FREQ,T,RATE,S,ATTEN,ATTE2,QUANTITY) DOUBLE PRECISION FREQ,T,RATE,S,A_REP,N,FREQR,ATTEN,GEDP,NUM,EA DOUBLE PRECISION FREQUENCY,QUANTITY,ATTE2,C,FACTOR,GAM,RTEMP COMPLEX*16 G,GE,EPSILON,GAMMA,E,M PARAMETER(P1=3.14159265359DO,EA=1.D0,C=2.9979244574D10, C1=20.958228) IF (RATE .LT. 0.25D0) THEN RTEMP = RATE RATE = 0.25D0 ENDIF IF (S.LT..002) THEN ATTEN=0 ATTE2=0 RETURN ENDIF FREQUENCY=FREQ*1.D9 CALL A(S,RATE,A_REP,QUANTITY) FREQR=0.866D0*C/(2.D0*PI*A_REP) N=(2.D0+100.D0*(FREQUENCY/FREQR)**2.D0) N=N/(1.D0+101.D0*(FREQUENCY/FREQR)**2.D0) CALL NUMBER(RATE,NUM,S,QUANTITY) CALL G_LOWCASE(S,A_REP,G,T,FREQUENCY)  126  The program for the "Universal Model"  *  *  * *  GE=G/DCMPLX(1.D0,(FREQUENCY/FREQR)**N) GEDP=-DIMAG(GE) ATTEN=9.1DO*GEDP*NUM*1.D4*FREQ EPSILON=EA*(1.DO+GE*NUM) GAMMA=2.D0*PI*FREQUENCY*CDSQRT(-4.D0*PI*1.D-7*EPSILON) ATTE2=DREAL(GAMMA) CALL PERMATIVITY(T,FREQUENCY,E,M) GAM=100/REAL(FREQ*C1*CDSQRT(-E)) FACTOR=1+((1-5*S)/5)*FREQUENCY/FREQR FACTOR=FACTOR*((N**2)*A_REP*(1-S)/(2*GAM))+1 FACTOR=(FACTOR*(N+S*(1-N)))**(1-S) ATTEN=ATTEN*FACTOR IF (RATE .LT. 0.25D0) THEN ATTEN = ATTEN*RTEMP/0.25D0 RATE = RTEMP ENDIF RETURN END  ******************************************************************************* **************** SUBROUTINE LP ************************************************ *******************************************************************************  * *  *  *  *  *  *  SUBROUTINE LP(FREQ,T,RATE,S,ATTEN,QUANTITY) DOUBLE PRECISION FREQ,T,RATE,S,AREP,N,FREQR,ATTEN,GEDP,NUM,EA DOUBLE PRECISION FREQUENCY, QUANTITY, C, RTEMP COMPLEX*16 G,GE PARAMETER(PI=3.14159265359D0,EA=1.DO,C=2.9979244574D10) IF (RATE .LT. 0.25D0) THEN RTEMP = RATE RATE = 0.25D0 ENDIF IF (S.LT..002) THEN ATTEN=0 RETURN ENDIF FREQUENCY=FREQ*1.D9 CALL NUMBER2(RATE,NUM,S,A_REP,QUANTITY) FREQR=0.866D0*C/(2.D0*PI*A_REP) N=(2.D0+100.D0*(FREQUENCY/FREQR)**2.D0) N=N/(1.D0+101.D0*(FREQUENCY/FREQR)**2.D0) CALL NUMBER(RATE,NUM,S,QUANTITY) CALL G_LOWCASE(S,A_REP,G,T,FREQUENCY) GE=G/DCMPLX(1.D0,(FREQUENCY/FREQR)**N) GEDP=-DIMAG (GE) ATTEN=9.1DO*GEDP*NUM*1.D4*FREQ IF (RATE .LT. 0.25D0) THEN ATTEN = ATTEN*RTEMP/0.25D0 RATE = RTEMP ENDIF RETURN END  * ******************************************************************************* **************** SUBROUTINE C LP (CORRECTED) ********************************** *****************************T*************************************************  127  The program for the "Universal Model" *  *  *  *  SUBROUTINE CLP(FREQ,T,RATE,S,ATTEN,QUANTITY) DOUBLE PRECISION FREQ,T,RATE,S,AREP,N,FREQR,ATTEN,GEDP,NUM,EA DOUBLE PRECISION FREQUENCY,QUANTITY,C,RTEMP COMPLEX*16 G,GE PARAMETER(P1=3.14159265359D0,EA=1.DO,C=2.9979244574D10) IF (RATE .LT. 0.25D0) THEN RTEMP = RATE RATE = 0.25D0 ENDIF IF (S.LT..002) THEN ATTEN=0 RETURN ENDIF FREQUENCY=FREQ*1.D9 CALL NUMBER2(RATE,NUM,S,A_REP,QUANTITY) FREQR=0.866D0*C/(2.D0*PI*A_REP) N=(2.D0+100.D0*(FREQUENCY/FREQR)**2.D0) N=N/(1.D0+101.D0*(FREQUENCY/FREQR)**2.D0) CALL NUMBER(RATE,NUM,S,QUANTITY) CALL G_LOWCASE(S,A_REP,G,T,FREQUENCY) GE=G/DCMPLX(1.D0, (FREQUENCY/FREQR)**N) GEDP=-DIMAG (GE) ATTEN=9.1DO*GEDP*NUM*1.D4*FREQ  *  FACTOR=N*((2+S)*FREQUENCY/FREQR+1)/(FREQUENCY/FREQR+2-S) FACTOR=(FACTOR**(1-S**2))*((2-S)/(2+S))**(1-S) ATTEN=ATTEN*FACTOR IF (RATE .LT. 0.2500) THEN ATTEN = ATTEN*RTEMP/0.25D0 RATE = RTEMP ENDIF RETURN END  * ******************************************************************************* ******************************************************************************* ************** SUBROUTINE G LOWCASE ******************************************* ***************************T*************************************************** • THIS SUBROUTINE CALCULATES g FROM THE EQUATION IN TABLE III • PG. 295 OF 'IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION', • DATED FEB. 1988.^THE NAME OF THE ARTICLE IS "A SIMPLIFIED • APPROACH TO THE EVALUATION OF EMW PROPAGATION CHARACTERISTICS • IN RAIN AND MELTING SNOW. IT USES THE SUBROUTINE 'PERMATIVITY' • TO GET THE PERMATIVITY OF OUTER LAYER. ******************************************************************************* SUBROUTINE GLOWCASE(S,A_REP,G,T,FREQUENCY) COMPLEX*16 E,G,M,NUMER,DEN DOUBLE PRECISION S,AREP,PI,Z,EA,E1,T,FREQUENCY PARAMETER(P1=3.14159265359D0,E1=1.20DO,EA=1.0D0) CALL PERMATIVITY(T,FREQUENCY,E,M)  *  Z=4.DO*PI*A_REP**3.D0 NUMER=(E-EA)*(2.D0*E+E1)-(1.D0-S)*(E-E1)*(2.D0*E+EA) DEN=(E+2.D0*EA)*(2.D0*E+E1)-2.D0*(1.D0-S)*(E-E1)*(E-EA) G=Z*NUMER/DEN  128  The program for the "Universal Model"  RETURN END ******************************************************************************* ******************************************************************************* ***************** SUBROUTINE PERMATIVITY ************************************** ******************************************************************************* * THIS SUBROUTINE INPUTS THE TEMPERATURE T IN DEG CELSIUS AND THE * FREQUENCY IN Hz. ****************************************************************************** *  SUBROUTINE PERMATIVITY(T,FREQUENCY,E,M) REAL*8 LS,ALPHA,EI,TAO,ES,E1,E2,T,L,C,PI,ET REAL*8 FREQUENCY COMPLEX*16 M,E C = 2.997924574D+10 PI = 3.141592654D0 TAO = 12.5664D+08 L = C/FREQUENCY LS=0.00033836D0*DEXP((2513.98D0/(T+273.D0))) ALPHA=-16.8129D0/(T+273.D0) + 0.0609265 EI=5.27137D0+0.0216474*T-0.00131198*T**2 ES=1.0D0-4.597D-03*(T-25.D0) ES= ES+1.19D-05*(T-25.D0)**2 - 2.8D-08*(T-25.D0)**3 ES = ES*78.54D0  * *234567890123456789012345678901234567890123456789012345678901234567890 ET = 1.D0 + 2.D0*(LS/L)**(1.D0-ALPHA)*DSIN(ALPHA*PI/2.D0) ET = ET + (LS/L)**(2.D0*(1.D0-ALPHA)) El = (ES-EI)*(1.D0+(LS/L)**(1.D0-ALPHA)*DSIN(ALPHA*PI/2.D0)) El = El/ET El = El + EI * *  E2 = (ES-EI)*(LS/L)**(1-ALPHA)*DCOS(ALPHA*PI/2.D0)/ET E2 = TAO*L/18.8496D+10 + E2  E= DCMPLX(E1,-E2) M = CDSQRT(E) END ******************************************************************************* ******************************************************************************* ***************** SUBROUTINE NUMBER ******************************************* ******************************************************************************* THIS SUBROUTINE CALCULATES 'NUM', THE NUMBER OF DROPS PER UNIT VOLUME. * IT USES THE SUBROUTINES WHICH CALCULATE THE RAIN VELOCITY (VELRAIN) * AND REPRESENTATIVE RAIN RADIUS WHEN S=1 (NO SNOW). * ******************************************************************************* SUBROUTINE NUMBER(RATE,NUM,S,QUANTITY) DOUBLE PRECISION RATE,VELR,S,NUM,A_REP,V,S2 PARAMETER(P1=3.14159265359D0) S2=1.D0  *  CALL A(S2,RATE,A_REP,QUANTITY) CALL VELRAIN(A_REP,VELR) V=(1.5D0+(VELR-1.5D0)*DSIN(S*PI/2.D0)) NUM=RATE/(48.0DO*PI*1.0D5*V*(AREP**3.D0))  RETURN END *******************************************************************************  129  The program for the "Universal Model" ***************** SUBROUTINE NUMBER2 ****************************************** ******************************************************************************* • THIS SUBROUTINE CALCULATES 'NUM', THE NUMBER OF DROPS PER UNIT VOLUME. • IT USES THE SUBROUTINES WHICH CALCULATE THE RAIN VELOCITY (VELRAIN) • AND REPRESENTATIVE RAIN RADIUS WHEN S=1 (NO SNOW). ******************************************************************************* * SUBROUTINE NUMBER2(RATE,NUM,S,AREP,QUANTITY) DOUBLE PRECISION RATE,VELR,S,NUM,A_REP,V,S2,QUANTITY PARAMETER(P1=3.14159265359D0) S2=1.D0  *  CALL A(S2,RATE,A_REP,QUANTITY) CALL VELRAIN(AREP,VELR) NUM=RATE/(48.00 *PI*1.0D5*VELR*(A_REP**3)) V=(1.5D0+(VELR-1.5D0)*DSIN(S*PI/2.D0)) AREP=AREP/M1-QUANTITY)+QUANTITY*S)**.33333333333333D0) NUM=NUM*VELR/V RETURN END  * ******************************************************************************* *************** SUBROUTINE A ************************************************** ******************************************************************************* • THIS SUBROUTINE COMPUTES THE VALUE OF A REP, WHICH IS THE • REPRESENTATIVE RADIUS OF THE SNOW AND RAIN MIXTURE. IF S • IN THE FORMULA IS SET TO EQUAL '1', THIS ROUTINE WILL RETURN THE • VALUE OF A RAIN (JUST THE RAIN RADIUS). THE VARIABLE 'RATE' • IS THE RAIN RATE IN mm/hr. ******************************************************************************* * SUBROUTINE A(S,RATE,AREP,QUANTITY) * DOUBLE PRECISION PI,DROP(15,11),RATE,P(14),DUMMY,A_REP, NORMALIZE,S,QUANTITY INTEGER 1,J, PARAMETER(PI=3.1415926535900) OPEN(UNIT=10,FILE='hydro',STATUS='OLD') 3000 FORMAT(10(D6.0,X),D6.0) READ(10,3000)((DROP(I,J),J=1,11),I=1,15) CLOSE(UNIT=10,STATUS='KEEP')  *  *  DO 110 J=3,10 IF ((DROP(1,J).LE.RATE).AND.(DROP(1,J+1).GE.RATE)) THEN DUMMY=(RATE-DROP(1,J))/(DROP(1,J+1)-DROP(1,J)) DO 100 1=2,15 P(I-1)=DUMMY*(DROP(I,J+1)-DROP(I,J))+DROP(I,J) 100^CONTINUE J=10 END IF 110 CONTINUE NORMALIZE=0 DO 120 1=1,14 NORMALIZE=NORMALIZE+P(I) 120 CONTINUE DUMMY=0 DO 130 1=1,14 VS=1.5+(DROP(I+1,2)-1.5)*DSIN(PI*S/2) DUMMY=DUMMY+DROP(I+1,2)*(P(I)/NORMALIZE)/  130  The program for the "Universal Model"  * *  (((DROP(I+1,1)/2)**3)*VS) 130 CONTINUE AREP=(DUMMY*((l.DO-QUANTITY)+QUANTITY*S))**(-.3333333333D0) RETURN END  * ******************************************************************************* ************** SUBROUTINE VELRAIN ********************************************* ******************************************************************************* • RAIN VELOCITY WITH LINEAR INTERPOLATION. • THIS SUBROUTINE TAKES AS INPUT, THE DIAMETER OF THE RAINDROP (DIA), • AND ACCORDING TO THE TABLE ON PG.552 OF 'IEEE TRANSACTIONS ON • ANTENNAS AND PROPAGATION' AN ARTICLE CALLED "RAINFALL ATTENUATION OF • CENTIMETER WAVES: COMPARISON OF THEORY AND MEASUREMENT" BY • RICHARD G. MEDHURST, DATED JULY 1965, IT GIVES THE VELOCITY. • LINEAR INTERPOLATION IS USED TO GET THE VELOCITY • FOR WHICH THE DIAMETER IS NOT SPECIFIED IN THE TABLE. ******************************************************************************* * SUBROUTINE VELRAIN(AREP,VELR) * DOUBLE PRECISION VELR,AREP,AREP,DUMMY * AREP=AREP*2.D0 * DUMMY=DEXP(-DSQRT(115.D0)*(AREP+.05D0)) VELR=5.44704233688D0-6.47412769848D0*DUMMY VELR=VELR-78.0827787323D0*(AREP+.05D0)*DUMMY VELR=VELR+6.883324065D0*DSQRT(AREP)-4.278474844DO*AREP**2.D0 RETURN END  * ******************************************************************************* ************ SUBROUTINE RAINRATE ********************************************** ******************************************************************************* • THIS SUBROUTINE CALCULATES THE RAIN RATE AT ANY POINT IN A RAIN CELL. * • XO,Y0^- BOTTOM CENTER COORDINATE LOCATION OF THE RAIN CELL. • X,Y^- COORDINATE LOCATION WHERE RAIN RATE IS TO BE CALCULATED. • RMAX^- MAXIMUM RAIN RATE LOCATED AT (XO,Y0). • RATE^- RETURNED RAIN RATE • RO^- INPUT PARAMETER WHICH CONTROLS THE DISTRIBUTION OF THE RAIN RATE IN THE RAIN CELL. * ******************************************************************************* * SUBROUTINE RAINRATE(X0,Y0,X,Y,RMAX,R0,RATE) * DOUBLE PRECISION XO,YO,X,Y,RMAX,RO,RATE * • CALCULATION OF RAIN RATE AT LOCATION (X,Y) * RATE=RMAX*DEXP(-DSQRTUX-X0)**2+(Y-Y0)**2)/R0) * RETURN END  To implement the COST 210 rain cell model, replace RAIN_TOTAL_ATTENUATION 131  The program for the "Universal Model"  by:  ******************************************************************************* **************** SUBROUTINE RAIN TOTAL ATTENUATION **************************** ********************************7*****T****************************************  *  • • •  THIS SUBROUTINE CALCULATES THE TOTAL RAIN ATTENUATION ALONG THE LINE CONNECTING THE RECEIVER AND THE POINT TO WHICH THE TRANSMITTER TRANSMITS TO.  •  X11,Y11,Z11 - COORDINATES OF INTERSECTION OF MAIN BEAM OF RECEIVER AND RAIN CYLINDER X,Y,Z^- COORDINATES WHERE TRANSMITTER TRANSMITS TO RTATT^- TOTAL RAIN ATTENUATION DT^- SMALL INCREMENT OF T ALONG BEAM AXIS XD,YD,ZD^- DIRECTION OF LINE CONNECTING (X11,Y11,Z11) AND (X,Y,Z) XT,YT,ZT^- COORDINATES OF FIRST POINT BETWEEN OTHER COORDINATES XA,YA,ZA^- POINTS ALONG LINE CONNECTING THE TWO POINTS RATE^- CALCULATED RAIN RATE XH,YH^- CENTER OF RAIN CELL RMAX^- MAX RAIN RATE (AT CENTER OF CELL) RO^- RAIN RATE DISTRIBUTION VARIABLE DECISION^- IS THERE A MELTING LAYER?  *  • • • • • • • • • • •  *  *******************************************************************************  *  SUBROUTINE RAINTOTAL_ATTENUATION(HMAX1,HFR,HM,HMIN,FREQ, TE11,QUANTITY,X,Y,Z,X11,Y11,Z11,RTATT,XH,YH,RMAX,RO,DECISION, • TASKNUMBER1,NARR,A1NUM,ANUM,SNUM,XR,YR,ZR,H_THICKNESS,RAD, • • XTHE_T) DOUBLE PRECISION X11,Y11,Z11,X,Y,Z,T,RTATT,DT,XD,YD,ZD, XT,YT,ZT,DR,XA1,YA1,ZA1,TP1,HMAX1,HFR,HM,HMIN,S,RATE, • RTATT,FREQ,TEMP,QUANTITY,XH,YH,RMAX,RO,HMAX1, • NARR(2,3),A1NUM,ANUM,BNUM,ROM,D,XR,YR,ZR,ANGEP, • H_THICKNESS,XP1,YP1,ZP1,XMLYM1,ZM1,MELTATTEN,XA2,YA2, • ZA2,RAD,XTHE_T,TP2,TE11,ATTEN • CHARACTER*2 DECISION INTEGER LTASKNUMBER1 XD=X11-X YD=Y11-Y ZD=Z11-Z T=(Z11-Z)/ZD RTATT=0 DT=T/20.0D0 XT=XD*DT+X YT=YD*DT+Y ZT=YD*DT+Z DR=DSQRT((XT-X)**2+(YT-Y)**2+(ZT-Z)**2) TP1 = -DT/2.D0 DO 10 1=1,20 TP1=DT+TP1 XA1=XD*TP1+X YA1=YD*TP1+Y ZA1=ZD*TP1+Z CALL GET_S(HMAX1,HFR,HM,HMIN,ZA1,S,DECISION) CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE)  132  The program for the "Universal Model"  CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR, A1NUM, ANUM, BNUM) RTATT=RTATT+ATTEN*DR/1000 10 CONTINUE CALL GETS(HMAX1,HFR,HM,HMIN,Z11,S,DECISION) CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE) ROM = 600.D0*RATE**(-0.5)*10.D0**(-(RATE+1.D0)**0.19) ROM = ROM*1000.D0 IF (S.EQ.1.0D0) THEN D = DSQRT((X11-XR)**2+(Y11-YR)**2) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) ATTEN = ATTEN*ROM*(1.D0-DEXP(-D/ROM))/1000.D0 ANGEP = D/DSQRT((X11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN = ATTEN/ANGEP ENDIF IF (S.EQ.0.D0.AND.H_THICKNESS.EQ.0.D0) THEN ZP1 = HFR S = 1.D0 D = DSQRT((X11-XR)**2+(Y11-YR)**2) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) XD = X11-X YD^Yll-Y ZD = Z11-Z XP1 = XD*(ZP1-Z)/ZD + X YP1 YD*(ZP1-Z)/ZD + Y ZP1 = ZD*(ZP1-Z)/ZD + Z D2 - DSQRTHX11-XR)**2+(Y11-YR)**2) D1 = DSQRTUX11-XP1)**2+(Y11-YP1)**2) ATTEN = ATTEN*(DEXP(-D1/ROM) - DEXP(-D2/ROM))/1000.D0 ANGEP = D/DSQRT((X11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN = ATTEN*ROM/ANGEP ENDIF IF (S.LT.1.D0.AND.S.GT.O.D0) THEN ZM1^HM XD = X11-X YD = Y11-Y ZD = Z11-Z XM1 = XD*(ZM1-Z)/ZD + X YM1 = YD*(ZM1-Z)/ZD + Y ZM1 = ZD*(ZM1-Z)/ZD + Z T=(ZM1-Z11)/(ZM1-Z11) MELTATTEN = 0.D0 DT-T/20.0D0 TP1 = -DT/2.D0 DO 420 1=1,20 TP1=DT+TP1 XA1=(XM1-X11)*TP1+X11 YA1=(YM1-Y11)*TP1+Yll ZA1=(ZM1-Z11)*TP1+Z11 CALL GET_S(HMAX1,HFR,HM,HMIN,ZA1,S,DECISION) CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR, A1NUM, ANUM, BNUM) D2 = DSQRTHX11-XA1)**2+(Y11-YA1)**2) IF (I.EQ.1) THEN D1 = 0 ELSE TP2 = TP1 - DT XA2^(XM1-X11)*TP2 + X11 YA2 = (YM1-Y11)*TP2 + Yll  133  The program for the "Universal Model"  ZA2^(ZM1-Z11)*TP2 + Z11 D1 = DSQRTHX11-XA2)**2+(Y11-YA2)**2) ENDIF ATTEN = ATTEN*(DEXP(-D1/ROM) - DEXP(-D2/ROM))/1000.D0 D = DSQRTHX11-XR)**2+(Y11-YR)**2) ANGEP = D/DSQRT((X11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN = ATTEN*ROM/ANGEP MELTATTEN = MELTATTEN + ATTEN 420^CONTINUE S = 1.D0 CALL RAINRATE(XH,YH,X11,Y11,RMAX,R0,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) D2 = DSQRT((X11-XR)**2+(Y11-YR)**2) D1 = DSQRT((X11-XM1)**2+(Y11-YM1)**2) ATTEN^ATTEN*(DEXP(-D1/ROM) - DEXP(-D2/ROM))/1000.D0 ANGEP = D2/DSQRTHX11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN = ATTEN*ROM/ANGEP ATTEN = ATTEN + MELTATTEN ENDIF IF (S.EQ.O.DO.AND.H_THICKNESS.NE.0.D0) THEN ZP1 = HFR ZM1 = HM XD = X11-X YD = Yll-Y ZD = Z11-Z XP1 = XD*(ZP1-Z)/ZD + X YP1 = YD*(ZP1-Z)/ZD + Y ZP1 = ZD*(ZP1-Z)/ZD + Z XM1 = XD*(ZM1-Z)/ZD + X YM1 = YD*(ZM1-Z)/ZD + Y ZM1 = ZD*(ZM1-Z)/ZD + Z T=(ZM1-ZP1)/(ZM1-ZP1) MELTATTEN = 0.D0 DT=T/20.0D0 TP1 = -DT/2.D0 DO 40 1=1,20 TP1=DT+TP1 XA1 = (XM1-XP1)*TP1+XP1 YA1 = (YM1-YP1)*TP1+YP1 ZA1 = (ZM1-ZP1)*TP1+ZP1 CALL GET S(HMAX1,HFR,HM,HMIN,ZA1,S,DECISION) CALL RAIi4RATE(XH,YH,X11,Y11,RMAX,RO,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) D2 = DSQRT((X11-XA1)**2+(Y11-YA1)**2) IF (I.EQ.1) THEN D1 = DSQRTUX11-XP1)**2 + (Y11-YP1)**2) ELSE TP2 = TP1 - DT XA2 = (XM1-XP1)*TP2+XP1 YA2 = (YM1-YP1)*TP2+YP1 ZA2 = (ZM1-ZP1)*TP2+ZP1 D1 = DSQRT((X11-XA2)**2+(Y11-YA2)**2) ENDIF ATTEN = ATTEN*(DEXP(-D1/ROM) - DEXP(-D2/ROM))/1000.D0 D = DSQRTUX11-XR)**2+(Y11-YR)**2) ANGEP = D/DSQRTUX11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN = ATTEN*ROM/ANGEP MELTATTEN = MELTATTEN + ATTEN 40^CONTINUE S = 1.D0  134  The program for the "Universal Model"  CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) D2 = DSQRT((X11-XR)**2+(Y11-YR)**2) D1 = DSQRT((X11-XM1)**2+(Y11-YM1)**2) ATTEN = ATTEN*(DEXP(-D1/ROM) - DEXP(-D2/ROM))/1000.D0 ANGEP = D2/DSQRTHX11-XR)**2+(Y11-YR)**2 + (Z11-ZR)**2) ATTEN ATTEN*ROM/ANGEP ATTEN ATTEN + MELTATTEN ENDIF RTATT = RTATT + ATTEN D = SQRT(XH**2 + YH**2) ATTEN = 0.D0 IF (D.GT.RAD) THEN D = D - RAD S = 1.D0 CALL RAINRATE(XH,YH,X11,Y11,RMAX,RO,RATE) CALL ATT_TASK(FREQ,TE11,RATE,S,ATTEN,QUANTITY,TASKNUMBER1, NARR,A1NUM,ANUM,BNUM) ATTEN = ATTEN*ROM*(1.DO-DEXP(-D/ROM))/1000.D0 ANGEP = 90.DO - XTHE_T ANGEP = DCOSD(ANGEP) ATTEN = ATTEN/ANGEP ENDIF RTATT = RTATT + ATTEN RETURN END  The following is a sample input to the program:  5^, Select attenuation model 1^, Select Scattering model 0^f (X, 0^ 0^  ,  ,  60^, 0^, 200000^, 0^I 0^, 89.14063, 180^, 0^, 0.02^, 0^, 0.02^, Y^,  y,  [note 1] [note 2]  z) of the centre of the rain cell THE_T [note 3] _ PHI __T [note 3] (x, y, z) of the receiver THE_R [note 4] _ [note 4] PHI __R TR_ ALPHA [note 5] TR_^_ HALF THETA [note 6] RE _ALPHA [note 7] [note 7] RE_^_ HALF THETA Is there a melting snow layer [note 8] 135  The program for the "Universal Model"  5200^, height of melting layer (Hm - T)  800^, thickness of melting layer (T) 6000^, Hc 20000^, radius of the rain cell 10000^ro 12.5^, The maximum rain rate at the centre of the rain cell 1^, Frequency in GHz 0^, Temperature in degrees Celcius .9^m^ [note 9] 50^, Steps of integrations in meters 100000 , Transmitter gain 50000^, Receiver gain 0.4^ [note 10] 5.5^ [note 10] -18^ [note 10] 6000^Hc  note 1^An input to select the attenuation model to be used in the calculations. "1"is used for Kharadly's first attenuation model "2"is used for Kharadly's third attenuation model "3"is used for Kharadly's second attenuation model "4"is used for Kharadly's fourth attenuation model "5" is used for the empirical formula note 2^An input to select the Scattering model to be used in the calculations.  note 3  ^  Cuurentelly, we have only one scattering model (Kharadly's). THE_T is B t PHI_T is cbt  note 4^• THE_R is Or PHI_R is O r 136  The program for the "Universal Model"  note 5^Determine the polarization of the transmitting antenna "0", "180", "360" is for vertical polarization "90", "270" is for horizental polarization note 6 note 7 note 8  A number can be chosen between 0 to 360. • The double-sided half-power bandwidth of the transmitter parameter that currentelly does not enter into calculations Is there a melting snow layer: If "N"o, then skip ignore the next two lines (do not enter them) If "Y"es, then enter (Hm - T) and T  note 9  The No case need not be used since we can enter T = 0 m = 1 — ps m ranges between 0.7 to 0.9  note 10  The empirical formula will treat m to be 0.9 whatever the input is In this case:  (a l , a2, K) = (0.4, 5.5, — 18) for the receiver antenna note^The Transmitter is assumed to be at the centre of the main coordinate system. The program can be (and should be) further refined as to make it more efficient and to incorporate more features.  137  CCIR Document 12-3/29 (Rev. 1) and supplement  Appendix F CCIR Document 12-3/29 (Rev. 1) and supplement Canada's contribution to CCIR Study group (Working Party 5C), document 12-3/29 (Rev. 1) titled EFFECT OF THE MELTING LAYER ON HYDROMETEOR SCATTER INTERFERENCE AND COORDINATION DISTANCE is reproduced on p. 139-148. [4] The error statistics for COST 210 [7] paths, extracted from a paper to be submitted to the IEEE Proceedings, is reproduced on p. 149-152. *  Olsen, R.L., Kharadly, M.M.Z. and Hulays, R.A, "Effect of hydrometeors in and above the melting layer on scatter interference," IEEE Proceedings on Electromagnetic Propagation in Rain, early 1993.  138  CCIR Document 12 3129 (Rev. 1) and supplement -  1 Delayed Contribution Document 12-3/29 (Rev. 1)  Documents CCIR Study Groups Period 1990-1994  8 January 1992  Driginal: English  Received: 16 January 1992  EFFECT OF THE MELTING LAYER ON HYDROMETEOR SCATTER INTERFERENCE AND COORDINATION DISTANCE 1. Introduction The Working Party 5C Revision of Recommendation 620 (Doc. SC/TEMPS) includes a simple modification to the model for the hydrometeor scatter mechanism of Report 724-2 to take into account the 6.5 dB/km fall off in reflectivity above the rain height. Because the old method in Report 724-2 assumed a constant reflectivity up to heights above the rain height, the effect of the modification is to significantly reduce the coordination distance In many circumstances. The intent of the modification to bring the model into better agreement with physical reality in a relatively simple way Is believed to be a good one. Unfortunately, the modification may now err too much in the optimistic direction, particularly for the 4-6 GHz band, because it ignores the effect of the melting layer. A Canadian document considered by Working Party 5C at its recent meeting gave some preliminary model calculations of the effect of the melting layer on relative interference levels. The predicted enhancement of interference as a result of the melting layer was the same order as that indicated by some limited experimental results available at the time [COST 210, 1991]. This document presents some additional model calculations that illustrate the effect of both melting layer scattering and attenuation on coordination distance. Further comparison is also made with experimental results available for frequencies above 11 GHz [COST 210, 1991]. 1. Model for scattering cross-section of melting snowflakes The model for the scattering cross-section of melting snowflakes [Kharadly, 1990] is based on an extension of three main physically-based empirical approximations investigated by Kharadly and Choi [1988) for attenuation by melting snow flakes. First of all, it is assumed for the purposes of interference calculations that the melting snow flakes can be approximately represented by water-coated snow spheres (The density of 0.1 g/cm 3 of the snow core was based on measurements of Matsumoto and Nishltsujo [1971].) Secondly, a physically-based empirical extension of the Rayleigh scattering cross-section is introduced for frequencies above the Rayleigh region. Thirdly, it is assumed that a distribution of particle sizes can be replaced by a single particle of representative size. Three additional empirical correction factors were also introduced by Kharadly [1990] to give improved agreement with Mie scattering calculations. The particle-size distribution used to determine the representative particle size for the melting layer is such that it reduces to a Laws and Parsons drop-size distribution for the rain below and satisfies the conservation of mass criterion (Kharadly and Kishk, 1991]. (The Initial distribution used by Kharadly and Choi [1988] for melting snow particles violated conservation of mass.) Example comparisons between the model cross-sections (Model I) and Mie scattering cross sections for the forward scattering direction are given in Figures 1 and 2 for rain (degree of melting S.1) and a modelled melting snow medium with 10% by volume of the outer shell of the particle melted (5.0.1). (The Model II curves given are based on a different set of assumptions and are not used here [Kharadly, 1990].) These results indicate the validity of all approximations noted above except the first, the shape and composition of the particles themselves. An additional investigation [Kharadly, 1991a] has demonstrated that uniformly-randomly-oriented (and even to some extent non-uniformly139  CC1R Document 12-3129 (Rev. 1) and supplement  2 randomly-oriented) water-coated snow spheroids can be adequately approximated by water-coated snow spheres for the purposes of interference calculations. The water-coated snow sphere model employed results in a radar reflectivity peak in the melting layer of about 16 dB with respect to that of rain of equivalent rain rate. This compares with some values of about 12 dB observed from radar measurements [Klassen, 1988]. Any such measurements, however, will tend to reduce the peak value because of volumetric averaging. Even more averaging will occur in radar measurements of statistical reflectivity profiles [e.g., COST 210, 1991]. The model is sufficiently flexible, however, so as to allow the peak reflectivity to be adjusted to fit the data. An increase in the snow core density, for example, will reduce the size of the model snow scatterers and therefore the reflectivity peak. Such adjustments can also be made to obtain best fits to actual melting layer attenuation data [Kharadly and Kishk, 1991]. 2. Models for specific attenuation by melting snowflakes Three models have been developed for specific attenuation of melting snowflakes [Kharadly 1991 b], all of which employ the three main physically-based empirical approximations noted above. The differences between the models are the additional empirical correction factors which have been introduced to give improved agreement between the model calculations and Mie calculations for water coated snow spheres. Finally, a totally empirical model has been developed [Hulays, 1991] of the form A. A n(R,S) aRb^  (1)  where R is the precipitation rate and S is the degree of melting, with aRb the well-known form for rain [Olsen et al., 1978]. This model, which also gives good agreement with Mie scattering calculations, is conveniently used for the interference calculations presented in this document. 3. Macroscopic meteorological models The horizontal structure of the fixed-position rain cell employed in the rain scatter and rain attenuation calculations is that currently assumed in Revised Recommendation 452-4 (Doc. 5C/TEMP9) and used previously in Report 569-4 and also by COST 210 [1991]. The center of the cell is positioned at the intersection of the antenna beam axes. The diameter of the melting layer cell is assumed to be the same as that of the rain cell below it, and the melting layer attenuation is assumed to reduce at the same exponential rate as the rain attenuation outside the core cell. Since the specific melting layer attenuation varies with height within the melting layer, a numerical integration is carried out in the vertical direction. The depth D m of the melting layer is assumed to vary with reflectivity factor Z of the rain in the form [Klassen, 1988] Dm .100 Z 0 . 17^( Dm is related to the precipitation rate R through the relation Z • 400R 1.4  .  The height of the melting layer is assumed to be fixed, but the effect of different values of this fixed height have been calculated. As demonstrated elsewhere [COST 210, 1991], the height of the melting layer varies considerably and the most accurate calculations should assume a distribution of heights. However, within any given month such as the worst month, the range in distribution of heights will be smaller and the use of a fixed height should give a reasonably close upper bound to the effect of the melting layer. The variation in the melting profile of the melting layer (i.e., variation in S between 0 at the by and 1 at the bottom) is assumed to be linear Kharadly and Choi, 1988]. Other profiles are considered elsewhere [Kharadly and Kishk, 1991]. 140  2)  CCIR Document 12 3129 (Rev. 1) and supplement -  3 The effect of the ice and dry snow medium above the melting layer was not Included in the calculations of the earlier Canadian document considered by Working Party 5C. This approximation has now been eliminated. Rayleigh scattering by the ice and snow medium above the melting layer is assumed along with a reflectivity of Z • 400R 1 4 at the lower boundary of this medium (the so-called 'rain height") decreasing with height at the rate of -6Z dB/km as required In Revised Recommendations 620 (Doc. 5C/TEMP9) and 452-4 (Doc. 5C/TEMP9). The elimination of this approximation has turned out to be more Important than previously believed since the attenuation of an Intervening melting layer tends to reduce the contribution of this ice and snow region relative to Its contribution without a melting layer, the effect of course increasing with Increasing frequency. The overall result at the higher frequencies is that the relative effect of Including the melting layer in the model calculations is less, even when the melting layer is at the optimum height. However, since existing experimental data [COST 210, 1991] are used as a reference, the difference does not change the conclusions. .  4. Other assumptions Other assumptions made In the calculations are consistent with the 'extended CCIR model" discussed elsewhere [COST 210, 1991). These include an assumption of no polarization mismatch, a narrow-beam approximation for the earth-station antenna, and Gaussian-shaped main-lobe and side-lobe patterns (highest side-lobe gain of 15 dB down assumed) for the terrestrial antenna. Atmospheric attenuation, ignored in the calculations of the earlier Canadian document, Is now included using the model in Revised Recommendation 452-4 (Doc. 5C/TEMP9), although the contribution is small at the frequencies considered. 5. Results for sample Interference geometries Two curves of transmission loss as a function of rainrate are given in Figure 3 for the experimental parameters of the Chilbolton-Baldock link [COST 210, 1991] (e.g., frequency of 11.2 GHz and a station separation of 131 km). The solid curve is for rain only and the dashed curve for rain plus melting layer. The latter is not shown above 30 mm/h because the existence of a melting layer above this precipitation rate Is considered unlikely. The terrestrial and earth-station elevation angles of 1.0° and 20°, respectively, place the center of the common volume at a height of about 3.0 km. The bottom of the melting layer is positioned at this height, which approximately maximizes the interference effect of the melting layer. Other parameters are indicated in the caption. As seen from Figure 3, the effect of the melting layer increases with increasing rainrate (and increasing melting layer thickness) until above about 5 mm/h, where the effect of attenuation in the melting layer begins to outweigh the effect of increased scattering cross section. At rainrates of 8, 13, 15, 19, and 29 mm/h, the interference level is approximately 2.5, 2.2, 2.0, 1.8, and 1.5 dB higher, respectively, than it would be if there were rain in place of the melting layer. This compares with melting layer enhancements of 1.7 and 2.7 dB exceeded for 0.1% and 0.01% of the time in the worst season (summer). These values were estimated from a comparison of actual data for summer and winter at two common volume altitudes (data provided courtesy of Rutherford-Appleton Laboratory; see also Revised Recommendation 452-4 (Doc. 5C/TEMP9)). In composite rain climate C,D,E, these exceedances correspond to rainrates of 8 and 19 mm/h on an annual basis, or about 13 and 29 mm/h on a worst season basis. A rainrate of about 15 mm/h is exceeded for 0.1% of the worst month in the same rain climate. The worst season enhancements of 1.7 and 2.7 dB at the 0.1% and 0.01% exceedance levels will also be observed in the annual distributions but at smaller exceedance levels. It Is interesting, but possibly a coincidence, that the annual distributions of interference level predicted for the Chilbolton-Baldock link by the method of Revised Recommendation 452-4 (Doc. 5C/TEMP9) underestimate the measured distributions by about the amount of the melting layer enhancement obtained from the seasonal measurements. As evident from the comparison of model and experimental estimates of the enhancement in interference caused by the presence of the melting layer, the latter is close 141  CCIR Document 12 3129 (Rev. 1) and supplement -  4 to the optimum estimated value (1.7 versus 2.0 dB) at 0.1% of the worst season and greater than the optimum estimated value (2.7 dB versus 1.5 dB) at 0.01%. This suggests that either the height variation of the melting layer in the summer is not large or that the experimental estimates are a little on the high side, at least at 0.01% of the worst season. (The melting layer was known to occur approximately at the common volume height on Chilbolton-Baldock link during the summer months (COST 210, 19911.) It should be noted, however, that it is possible in principal for melting layer and associated precipitation scatter a low precipitation rates to contribute to the interference levels exceeded for smaller percentages of time than the corresponding precipitation rate. There will be some instances, for example, when the melting layer and rain attenuation associated with the scattering will be smaller than that predicted by a model estimating the average attenuation (e.g., if the scatter volume were on the edge of the precipitation cell without much attenuation outside the common volume). In any case, the combined experimental and model results suggest that the effect of the melting layer is too significant to ignore when the scatter volume is at about the height of the melting layer in any given month or season, even at 11.2 GHz. The increase in coordination distance required to offset the effect of the melting layer enhancement of 1.7 dB exceeded for 0.1% of the time in the worst season is 21 km. Results are given in Figure 4 for the same parameters of the Chilbolton-Baldock link, but at a frequency of 4 GHz. Here the height of the center of the melting layer is positioned at the height of the center of the common volume, which approximately 'optimizer the enhancement of the melting layer. The 'optimum' enhancements are 6.6, 7.5, 7.8, 8.3, and 9.2 dB at rainrates of 8, 13, 15, 19, and 29 mm/h, respectively. On the basis of a frequency scale factor of 3.4 derived from the enhancement ratio 7.5/2.2 at a rainrate of 13 mm/h, the measured enhancement of 1.7 dB at 11.2 GHz scales to 5.8 dB at 4 GHz. Similarly, the measured enhancement of 2.7 dB ( optimum' estimated value of 1.5 dB) at 11.2 GHz scales to 16 dB (9.2 dB for 'optimum' value) at 4 GHz on the basis of a frequency scale factor of 9.2/1.5.6.1 derived from the enhancement ratio at 29 mm/h. A more optimistic scale factor of 3.9 derived from the enhancement ratio of 7.8/2.0 at 15 mm/h would reduce the latter figure to 5.9 dB. The increase in coordination distance needed to offset a 5.9 dB enhancement for a rainrate of 15 mm/h at 4 GHz is 66 km. -  In agreement with calculations obtained elsewhere [COST 210, 1991], calculations for higher frequencies using the current model indicate that the effect of the melting layer on interference continues to diminish as a result of increasing attenuation both in the rain and the melting layer. The 2.8 dB 'optimum" enhancement estimated at 1.1.2 GHz from Figure 3, for example, is reduced to about 0.7 dB at 20 GHz. Certain geometries for which the effect of attenuation outside the common volume is reduced should pose problems up to quite high frequencies. One such geometry is that of an intersection or near-intersection between two earth-station beams pointed at elevation angles significantly higher than that of the average terrestrial station antenna. Such a geometry was investigated at 11.4 GHz on a 9.3 km side-scatter link near Graz [COST 210, 1991]. The model calculations for this link, with and without the melting layer, are given in Figure 5. Here the top of the melting layer is positioned 0.1 times Its thickness from the center of the -common volume at 2.9 km height to obtain "optimum" enhancements of 5.8, 6.4, and 2.9 dB at 2, 10, and 30 mm/h, respectively. (Corresponding figures for a frequency of 20 GHz are 1.9, 1.5, and 0 dB.) Rainrates of 2, 10, and 32 mm/h are exceeded for 1%, 0.1%, and 0.01% of the year for composite rain climate F-K corresponding to that of Graz. It is interesting that the annual distributions of interference level predicted for the Graz link by the method of Revised Recommendation 452-4 (Doc. 5C/TEMP9) underestimate the measured distributions by amounts ranging from 12 dB to 6 dB between the corresponding exceedance levels of 1% and 0.01% of the time on an annual basis [COST 210, 1991]. The enhancement caused by the melting layer can perhaps explain much of this large discrepancy.  142  CCIR Document 12-3129 (Rev. 1) and supplement  5  6. Discussion and Conclusions The model results given and supporting comparisons with experimental data suggest that the effect of the melting layer should be taken into account in both coordination distance calculations (such as those obtained from Revised Recommendation 620 (Doc. 5C/TEMP6)) and detailed interference calculations (such as those obtained from Revised Recommendation 452-4 (Doc. 5C/TEMP9)). This is most important for the 4-6 GHz band where the effects of rain and melting layer attenuation are least significant with respect to that of the scattering. Another way of looking at the results is that the apparent increase in the Interference level introduced by the presence of the melting layer is larger than that in changing from one composite rain climate to another (e.g., climate C,D,E to Climate F-K). If the use of such "fine" climatic differences is justified, then the introduction of the apparent effect of the melting layer would seem even more justified. Of course scattering from rain and the dry snow and ice region above the melting layer are only important from an interference coordination viewpoint if there Is a main beam Intersection. Clearly the chance of such a main-beam intersection occurring is very small, which is no doubt one reason that interference due to hydrometeor scatter has apparently not been observed in practice. Even if an interference causing main-beam intersection or near-intersection had occurred in the past, it would not be surprising for the interference to have remained unobserved. Interference due to hydrometer scatter it not generally as long lasting as that resulting from the clear-air mechanisms. Furthermore, performance monitoring has not been generally carried out. Even If a deterioration in performance were observed, there would be no easy way of knowing If It were due to interference or to attenuation of the wanted signal. At first sight the chances of having a main-beam intersection and a melting layer occurring within it at the time of year when precipitation intensities are greatest would seem to be even smaller than having a main-beam intersection occurring within rain. This is no doubt true for short distances between terminals with the main-beam intersections occurring at low altitudes. However, at the 100 km and larger distances for coordination the main-beam intersections occur at altitudes for which the melting layer also occurs in the summer months, at least at temperate latitudes. Thus, it would appear that the melting layer should always have some influence on coordination at these latitudes in the frequency bands including and below the 11.14 GHz band. At low latitudes in rain climates for which convective rain clearly dominates the Interference statistics at the critical time percentages (e.g., 0.01%), the melting layer should not be a factor in either coordination or detailed interference prediction. Composite rain climates L,M and N,P are believed to be in this category. Melting layer scatter is believed to be a factor in composite rain climate F-K because scatter from the less intense precipitation in the melting layer dominates that from some of the more Intense precipitation in convective rain. Although the possible effect of melting layer scatter on the accuracy of detailed interference calculations is mentioned in Revised Recommendation 452-4 (Doc. 5C/TEMP9)), there was insufficient data and other Information available at the meeting of Working Party 5C to propose a suitable modification to the prediction technique for estimating hydrometeor scatter interference levels. Data and other information were similarly lacking to propose a suitable modification to the coordination procedure in Revised Recommendation 620 (Doc. 5C/TEMP6). A crude modification could be carried out on the basis of the information given in this document If it were desired to give designers of earth stations the option of including 100% of possible interference geometries. A difficulty is that there is no corresponding method as yet in Revised Recommendation 452-4. At the very least, the information In this document provides a much dearer Indication than was previously available of the potential risks Involved in Ignoring the effects of the melting layer in interference coordination.  143  CCIR Document 12-3129 (Rev. 1) and supplement  6 REFERENCES COST 210 Management Committee [1991) Influence of the atmosphere on interference between radio communications systems at frequencies above about 1 GHz. Final Report of Cost 210 Management Committee, EUR 12407, ISBN 92 - 826-2400-5, Commission of European Communities, Luxembourg. HULAYS, R. [1991] Precipitation scatter Interference on communication links with emphasis on the melting snow layer. MASc. Thesis in preparation, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. KHARADLY, M.M2. [1990] A model for evaluating the bistatic scattering cross section of melting snow: Phase III. Final Report on Contract CRC 36100-9-0247-101-ST, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. KHARADLY, M.M.Z. [1991a] A model for evaluating the bistatic scattering cross section of melting snow: Phase IV. Final Report on Contract CRC 36001-0-6572, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. KHARADLY, M.M.Z. [1991 b] Private communication. KHARADLY, M.M.Z., and CHOI, A. [1988] A simplified approach to the evaluation of EMW propagation characteristics in rain and melting snow. IEEE Trans. Antennas Propagat., Vol. 36, 2, 282-296. KHARADLY, M.M.Z., and KISHK, A. [1991] Models for estimating melting layer attenuation. Manuscript in preparation, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. KLASSEN, W. [1988] Radar observations and simulation of the melting layer of precipitation, J. Atmos. Sci., Vol. 45, 24, 3741-3753. MATSUMOTO, A. and NISHITSUJI, A. Ed. [1971] SHF and EHF propagation in snowy districts, Monograph series of Res. Inst. of Applied Electricity, No. 19, Hokkaido University, Sapporo, Japan. OLSEN, R.L., ROGERS, D.V. and HODGE, D.B. [1978] The aRb relation in the calculation of rain attenuation, IEEE Trans. Antennas Propagat., Vol. 26, 2, 318-329.  144  CC!R Document 12-3129 (Rev. 1) and supplement  -712-3/29-E  0.001  ^to^au^as frequency (az)  Figure 1. Comparison of model (Mode! I) and Mie scattering ('exact') forward scattering cross sections (m 2/m 3 ) of rain (S•1) as functions of frequency. R.25 mm/h, 0•C water temperature. ,  .  10000  1000  100  0.01  11.001  0^6  10  16^SO^SS^SO  Frequency  (Gliz)  SS  40  Figure 2. Comparison of model (Model I) and Mie scattering •eicacr) forward scattering cross sections (m 2 /m 3 ) of melting snowflakes (S•0.1) as functions of frequency. R.25 mm/h, 0•C water temperature. 145  CC!R Document 12-3129 (Rev. 1) and supplement  Transmission loss in dB  -136.00 -138.00 -140.00 -142.00 -144.00 -146.00 -148.00 -150.00 -152.00 -154.00 -156.00 -158.00  1  ^ ^ ^ ^ 3 10 30 100 Rainrate in mm/h  Figure 3. Comparison of transmission loss with and without the melting layer as a function of rainrate at 11.2 GHz for parameters of Chilbolton-Baldock link. — rain only, -- rain with melting layer; 131 km station separation, 20° earthstation elevation angle, 1.0° terrestrial-station elevation angle, 3.0 km common-volume height, 3.0+Dm km height to melting layer top ("rain height"), 40.5 dB terrestrial antenna gain (1.6° half-power beamwidth), 59 dB earthstation gain (0.18° half-power beamwldth, 55% efficiency). 146  CCIR Document 12-3129 (Rev. 1) and supplement  9  'rransmission loss in -132.00 — -134.00 -136.00 -138.00 -140.00 -142.00 -144.00 -146.00 -148.00 -150.00 -152.00 -154.00 -156.00 -  Wm,  -158.00 -160.00 -162.00 -164.00 -166.00 —  1^3^10^30^100 Rainrate in mm/h  Figure 4. Comparison of transmission loss with and without the melting layer as a function of rainrate at 4 GHz GHz for parameters of Chilbolton-Baldock link. — rain only, — rain with melting layer; 131 km station separation, 20° earthstation elevation angle, 1.0° terrestrial-station elevation angle, 3.0 km common-volume height, 3.0+0.50in km height to melting layer top (°rain height ), 40.5 dB terrestrial antenna gain (1.6° half-power beamwidth), 59.0 dB earth-station gain (0.18° half-power beamwidth. 55% efficiency). .  147  CCIR Document 12-3129 (Rev. 1) and supplement  10  Transmission loss in dB -122.00 -124.00 -126.00 -128.00 -130.00  =MI  -132.00  dm.  -134.00  -136.00  ■I.  -138.00 -140.00 -142.00 -144.00  OEM  -146.00 -148.00 1  ^ ^ ^ 10 3 30 Rainrate in mm/h  Figure 5. Comparison of transmission loss with and without the melting layer as a function of rainrate at 11.4 GHz for parameters of Graz link. — rain only, — rain with melting layer; 9.3 km station separation, 16.8° transmitter elevation angle, 36.3° receiver elevation angle, 2.9 km common-volume height, 2.9+0.1 Dm km height to melting layer top Crain height"), 37 dB transmitting antenna gain (3.0° half-power beamwldth), 47 dB recanting antenna gain (0.6° half-power beamwidth, 55% efficiency assumed).  148  CCIR Document 12-3129 (Rev. 1) and supplement  Long-Path Error Statistics' for COST 210 Links at 1% Path  Rot 724  Chilbolton•Baldock, 131 ChilboltonBaldock, Bb Cap d'Antifar • Chilbohon Fulda, Ft Fulda. Fb Radar Simul., S2 Radar Simul., S3 Radar Simul., S4 Radar Simul., S5  2.0 dB  Rot 724 Mod (4.5 dB/km) -3.9^(-1.9)  6.0  Mean Standard Div.  Rpt 724 Mod (-5.0 dB/km)  Rpt 724 Mod (-4.5 dB/km)  Rpt 724 Mod (-4.0 dB/km)  Rpt 724 Mod (-3.5 dB/km)  -5.4  • .50 (•0.50) -3.00  -2.05 (-0.05) -2.20  • 1 .60 (0.40) -1.40  -1.15 (0.85) -0.6  -3.9  .3.9^(-1.9)  -3.9^(-1.9)  -3.9^(-1.9)  -3.9^(-1.9)  -3.9^(-1.9)  • • 0.0 0.8 3.9 11.0  • 0.0 0.8 3.2 '2.5  • • 0.0 0.8 3.35 4.45  2.7 _^4.7  -1.0^(-0.4) _ 3.42 (2.96)  .0.1 10.5) 3.22 (2.68)  0.0 0.8 3.41 5.11  0.0 0.8 3.46 5.76  0.0  0.8  3.52 6.42  0.4^(1.0) 0.2 (0.7) 0.7^(1.3) 3.23^(2.68) _ 3.27 (2.72) _^3.35^(2.80)  Short-Path Error Statistics' for COST 210 Links at 1% Path Laidschandam, L1 ,• Loldschandam, L2 Laldscharrdam, 1.3 Laidsdi•ndam, 1.4 Laidschandam, 1.5 Laidschandam, 1.6 Loldschandam, L7 Laidschandam, LB Darmstadt, D1 Darmstadt, D2 Darmstadt, D3 Darmstadt, D4 , Darmstadt. D5 Darmstadt, D6 Graz Radar Simul., Si Mean Standard Div.  Rpt 724  Rpt 724 Mod (4.5 dB/km)  10.8 11.4 12.4 14.0 16.1 18.5 21.1 22.9 6.7 11.7 14.0 18.3 7.0 10.8 -0.6 3.1  10.8 11.4^(11.9) 12.4 112.9) 14.0 (14.6) 12.2^(12.7) 10.7^(11.2) 7.5 2.9 6.7 8.5 4.9 2.7 7.0 4.2 -3.9^(2.5) 3.1  Rpt 724 Mod (-6.0 dB/km) , 10.8 11.4^(11.9) 12.4 (12.9) 14.0^(14.5) 13.1^(13.6) 12.5^(13.0) 10.6 7.4 6.7 9.2 7.0 6.3 7.0 -0.7 -3.1^(3.3) 3.1  6.6 (7.1) 5.50 (5.07)  7.9 (8.5) 4.92 (4.34)  12.3^ 6.4  .  Rpt 724 Mod Rot 724 Mod (-4.6 dB/km) (-4.0 d13/km) 10.8 10.8 11.4^(11.9) 11.4^(11.9) 12.4 (12.9) 12.4 (12.9) 14.0 (14.6) „ 14.0 (14.6) 13.7 (14.2) 13.4^(13.9) 13.1^(13.6) ..._ 13.7^(14.2) 12.7 11.7 9.0 10.6 6.7 6.7 9.5 9.7 7.7 8.4 7.5 8.7 7.0 7.0 1,6 0.6 _ -2.9^(3.6) -2.6 (3.8) 3.1 3.1 6.4 (8.9) 4.83^(4.21)  8.8 (9.4) 4.80 (4.14)  Rpt 724 Mod (-3.5 dB/km) 10.8 11.4^(11.9) 12.4 (12.9) 14.0^(14.5) 14.0^(14.5) 14.3^(14.8) 13.8 12.1 6.7 10.0 9.1 9.9 7.0 2.8 -2.4^(4.0) 3.1 9.3 (9.8) 4.83^(4.15)  Combined Long- and Short-Path Error Statistics' (dB) for COST 210 Links at 1%  St:ndard en nw Day.  RIX 724  Rpt 724 Mod (4.5 dB/km)  6.8  4.88 (4.46)  Rpt 724 Mod Rp1 724 Mod (•5.0 dB/km), (•.5 dB/km) 4.39 (3.85)  4.33 (3.75)  Rpt 724 Mod (•4.0 dB/km)  Rpt 724 Mod (-3.5 dB/km)  4.32 (3.70)  4.36 (3.73)  Predicted interference levels minus measured levels. Values in parentheses Include rough  correction for melting layer.  149  CCIR Document 12-3129 (Rev. 1) and supplement  Long-Path Error Statistics' (dB) for COST 210 Links at 0.1% Path  Rpt 724  Chilbohon•Baldock, Bf ChitboltonBaldock, Bb Cap d'Antiler • Chilbotton Fulda, Ft Fulda, Fb Radar Simul., $2 Radar Simul., S3 Radar SImul., $4 Radar Simul., S5  0.0  Rpt 724 Mod (-6.5 dB/km) -5.9^(•3.0)  3.9  -6.5  Rot 724 Mod (•5.5 dB/km) -4.95 (-2.95) -4.90  Rpt 724 Mod (-5.0 dB/km • .50 (•2.50) -4.10  Rpt 724 Mod (-4.5 dB/km) -4.05 (-2.05) -3.30  Rpt 724 Mod (-4.0 dB/km) -3.60 (.1 .6 0) -2.50  Rpt 724 Mod (-3.5 dB/km) •3.15 ( -1.1 6) -1.70  -4.7  -4.7^(-2.7)  -4.7^(-2.7)  -4.7^(•2.7)  -4.7^(-2.7)  -4.7^(-2.7)  -4.7^(-2.7)  • • 0.0 0.0 2.8 8.9  0.0 0.0 2.1 0.4  • • 0.0 0.0 2.20 1.70  • • 0.0 0.0 2.25 2.35  • 0.0 0.0 2.31 3.01  • • 0.0 0.0 2.36 3.66  -1.0^(-0.4) 3.00 (2.44)  -0.7^(-0.1) 3.08 (2.41)  1.6 4.2  Mean Standard Dev.  -1.6^(-0.7) -1.2^(-0.7) •2.1^(-1.5) 3.49 (2.98)) _^3.22 (2.75) _ 3.13^(2.51)  • 0.0 0.0 2.42 4.32^' -0.4^(0.2) 3.12^(2.44)  Short•Path Error Statistics' (dB) for COST 210 Links at 0.1% Rpt 724  Rpt 724 Mod (-6.5 dB/km)  Rpt 724 Mod (•5.5 dB/km)  Rpt 724 Mod (-5.0 dB/km)  Lsidschandam, L1 LAidschendam, 1.2 Leidschendam, L3 Leidsehendam, L.4 Leidschendam, 1.5 LaIdschendam, 1.6 Leidschendam. 1.7 LAidsehendam, L8 Darmstadt, D1 Darmstadt. D2 Darmstadt, D3 Darrnstadt, D4 Darmstadt, D5 Darmstadt, D6 Graz Radar Simul., $1  7.2 7.0 8.9 10.2 12.0 14.6 18.1 20.9 3.3 8.5 9.6 13.3 2.6 10.6 -0.8 2.8  7.2 7.8 (8.3) 8.0 (9.4) 10.2 (10.8) 8.1^(8.6) 6.9 (7.4) 4.5 0.8 3.3 6.9 0.6 . -2.3 2.5 -4.3 • .0^(2.4) 2.8  7.2 7.8 (8.3) 8.9 (9.4) 10.2 (10.8) 8.70 (9.20) 8.00 (8.50) 6.66 3.85 3.3 6.76 1.90 0.10 2.5 -2.05 -3.65 (2.85) 2.8  7.2 7.1 (8.3) 5.9 (0.4) 10.2 (10.8) 9.00 (9.50) 8.60 (9.10) 7.60 6.40 3.3 6.00 2.60 1.90 2.5 -0.90 -3.30 (3.10) 2.8  Mean Standard Dev.  9.3 6.8  3.6 (4.2) 4.66 (4.20)  4.6 (5.1) 4.07 (3.69)  4.9 (5.5) 5.4 (5.0) 3.91^(3.48) _ 3.81^(3.33)  Path  Apt 724 Mod (-4.5 dB/km) 7.2 7.8 (8.3) 8.9 (9.4) 10.2 (10.8) 0.30 (9.80) 0.20 (9.70) 8.65 6.05 3.3 6.25 3.30 2.60 2.5 0.25 -3.05 (3.35) 2.8  Rpt 724 Mod (-4.0 dB/km) 7.2 7.8 (8.3) 11.9 (0.4) 10.2 (10.8) 0.60 (10.1) 9.80 (10.3) 9.70 8.50 3.3 6.50 4.00 3.70 2.5 1.40 -2.80 (3.60) 2.8  Apt 724 Mod (•3.6 dB/km) 7.2 7.8 (8.3) 5.9 (9.4) 10.2 (10.8) 9.90 (10.4) 10.40 (10.9) 10.75 10.05 3.3 6.75 4.70 4.90 2.5 2.65 -2.55^(3.65) 2.8  SA (6.4) 3.79 (3.27)  6.3 (6.8) 3.84 (3.22)  Combined Long- and Short-Path Error Statistics• (dB) for COST 210 Links at 0.1%  Combined Standard Day.  Apt 724  Rpt 724 Mod (-6.6 dB/km)  Rpt 724 Mod (-5.6 d111/km)  Rpt 724 Mod (-5.0 d13/km)  Rpt 724 Mod (-4.6 dB/km)  Rpt 724 Mod (-4.0 dB/km)  Rpt 724 Mod (-3.6 dB/km)  6.1  4.16 (3.87)  3.76 (3.37)  3.62 (3.16)  3.64 (3.03)  8.62 (2.98)  3.56 (3.00)  'Predicted Interference.levebs minus measured levels. Values in parentheses include rough correction for melting layer.  150  CCIR Document 12-3129 (Rev. 1) and supplenient  Long-Path Error Statistics* (dB) for COST 210 Links at 0.01% Path  Rpt 724  Chilbolton•Baldock, 131 ChilbottonBaldock. Bb Cap d'Antifin • Chilbolton Fulda, Ff Fulda, Fb Radar Simul., S2 Radar Simul., S3 Radar Simi., 44 Radar Simul., S5  0  Rpt 724 Mod (4.5 dB/km) • .9^(-3.9)  4.9  Moan Standard Div.  Rpt 724 Mod (•5.0 dB/km) .4.50 (•2.50) -3.10  Rix 724 Mod (-4.6 dB/km) -4.05 (-2.06) -2.30  Apt 724 Mod (-4.0 dB/km)  -5.6  Apt 724 Mod (•.6 dB/km) -4.95 (-2.95) -3.90  -3.60 (•1.60) -1.60  Rpt 724 Mod (-3.5 dB/km) -3.15 (-1 .1 6 ) -0.70  4.3  • .3^(4.3)  4.3^(-4.3)  4.3 (4.3)  4.3^(-4.3)  4.3 (4.3)  4.3 (4.3)  •1.3 0.0 -0.6 •1.6 0.0 6.9  -1.3 0.0 -0.6  -1.6  -1.3 0.0 -0.6  -1.6  -0.7 '^-1.6  -0.61 •0.31  -1.3 0.0 -0.6 -1.6 •0.65 0.35  -1.3 0.0 -0.6 -1.6 -0.50 1.01  0.2 3.8  .2.6^(•2.2) 2.63^(1.92)  -2.0^(-1.5) -2.2^(-1.7) 2.29^(1.60) _^2.24^(1.53)  -1.7^(-1.3) 2.24 (1.53)  -1.3 0.0 -0.6 -1.6 -0.44 1.66 •1.6(•1.1) 2.28 (1.60)  -1.3 0.0 -0.6 •1.6 -0.39 2.32 •1.3^(-0.9) 2.37^(1.73)  Short-Path Error Statistics' (dB) for COST 210 Links at 0.01% Path Laidschendam, L1 Laidschandam, L2 Laidschandam, 1.3 LaIdsc►andam, 14 Loidschandam, L5 Laidschandam, 16 Laidschandam. L7 Laidschandam, LB Darmstadt. D1 Darmstadt, D2 Darmstadt, D3 Darmstadt, 04 Darmstadt, D5 Darmstadt, D6 Graz Radar Simul., Si Mean Standard Div.  Apt ra Mod (-4.6 dB/km) 3.4 4.1 (4.6) 6.6 (6.1) 7.1^(7.6) 6.0 (6.6) 4.4 (4.9) 6.0 4.2 2.9 6.7 2.9 0.6 2.0 -1.7 -2.1^(4.3) 1.3  Apt 724 Mad (-4.0 dE►km) 3.4 4.1 (4.6) 6.6 (8.1) 7.1^(7.6) 6.3 (6.8) 6.0 (6.6) 7.0 6.7 2.9 '^6.9 3.6 1.8 2.0 -0.6 -1.8 (4.6) 1.3  Apt 724 Mod (•.6 dB/km)  3.4 4.1^(4.6) 5.6 (6.1) 7.1^(7.6) 6.4 (5.9) 3.2 (3.7) 3.9 1.1 2.9 6.2 1.6 -1.8 2.0 -4.0 -2.6 (3.8) 1.3  Rpt 724 Mod (-5.0 dB/km) 3.4 4.1^(4.6) 6.6 (6.1) 7.1^(7.6) 6.7 (6.2) 3.8 (4.3) 4.9 2.6 2.9 6.4 2.2 -0.6 2.0 -2.8 -2.3^(4.1) 1.3  2.4 (3.0) 3.09 (2.95)  2.8 (3.4) 2.84 (2.64)  3.3 (3.8) 2.71^(2.45)  3.7 (4.3) 2.64 (2.32)  4.2 (4.7) 2.71^(2.33)  Apt 724  Rpt 724 Mod (-6.6 de/km)  Apt 724 Mod (-4.5 dB/km)  3.4 4.1 6.6 7.1 8.7 9.8 15.4 18.1 2.9 7.9 9.2 11.4 2.0 8.7 0.2 1.3  3.4 4.1^(4.6) 6.6^(6.1) 7.1^(7.6) 4.8 (5.3) 2.0 (2.5) 1.8 -2.1 2.9 4.6 0.1 -4.2 2.0 -6.1 -3.1^(3.3) 1.3  7.2 6.0  1.6^(2.1) 3.69 (3.53)  3.4 4.1^(4.6) 6.6 (6.1) 7.1^(7.6) 6.6 (7.1) 6.6 (8.1) 8.1 7.3 2.9 6.2 4.3 3.0 2.0 0.7 -1.6^(4.8) 1.3  Combined Long- and Shod-Path Error Statistics' (dB) for COST 210 Links at 0.01%  Combined Standard Day.  Rpt 724  Apt 724 Mod (4.6 dB/km)  Apt 724 Mod (-5.6 dB/km)  Rpt 724 Mod (•6.0 dB/km)  Apt 724 Mod (4.6 d13/km)  Apt 734 Mod (-4.0 dB/km)  Apt 724 Mod (•.5 dB/km)  4.6  3.19 (3.00)  2.78 (2.61)  2.69 (2.27)  2.50 (2.13)  2.48 12.05)  2.54 (2.10)  * Predicted interference levels minus measured levels. Values in parentheses include rough correction for melting layer.  151  Long-Path Error Statistics (dB) for COST 210 Links at 0.001% .  Pah  Rat 724  Rpt 724 Mod (4.5 de/km)  Rpt 724 Mod (-5.5 d8/km)  Rpt 724 Mod (-5.0 dB/km)  Rpt 724 Mod (-4.5 d8/km)  Rpt 724 Mod (-4.0 dB/km)  Rpt 724 Mod (-3.5 dB/km)  Rpt 724 Mod (-3.0 d8/km)  Rpt 724 Mod (-2.5 de/km)  Chilbolton-Baidock. BI ChIlbollonBeklodi, Bb Cap frAntiffor CfillboNon Fulda. Ft _ Fulda, Fb Radar Smut.. 62 Radar Shut. &I Radar 8Inst, 64 , Radar Shaul.. 65  0.4  -5.5  -4.55  -4.10  -3.65  -3.20  -2.75  -2.30  -1.85  5.1  -5.3  -3.70  -2.90  -2.10  -1.30  -0.50  0.30  1.10  -5.9  -5.9  -5.9  -5.9  -5.9  -5.9  -5.9  -5.9  -5.9  -1.7 0.0 -2.6 -6.2 -5.2 2.9  -1.7 0.0 • .6 -6.2 -8.6 -5.6  -1.7 0.0 -2.6 -5.2 -5.81 -4.31  -1.7 0.0 -2.6 -5.2 -5.75 -3.85  -1.7 0.0 -2.6 -5.2 -5.70 -3.00  -1.7 0.0 -2.6 -5.2 -5.59 -1.69  -1.7 0.0 -2.6 -6.2 -5.53 -1.03  -1.7 0.0 -2.6 -5.2 -5.46^, -0.36  Moan ,^Standard Ow.  -1.4 3.9  -4.2 2.2  4.6 1.98  -3.5 1.96  -3.3 1.99  -2.9 2.20  -2.7 2.36^_  -2.4 2.56  -1.7 0.0 -2.6 -6.2 -5.64 -2.34 -3.1 _^2.07  • Predicted interference levels minus measured levels. Values in parentheses include rough correction for melting layer.  References  References 1. Awaka, J. [March, 1978] Estimation of rain scatter interference in the case of medium scale broadcasting satellite for experimental purposes (BSE). Journal of the Radio Research Laboratories, Vol. 25, No. 116, 23-48.  2. Awaka, J. [March 1989] A three-dimensional rain cell model for the study of interference due to hydrometeor scattering. Journal of the Radio Research Laboratories, Vol. 36, No. 147, 13-44. 3. Capsoni, C., Fedi, F., Magistroni, A., Paraboni, A., and Pawlina, A. [May-June 1987] Data and theory for a new model of the horizontal structure of rain cells for propagation applications. Radio Science. Vol. 22, No. 3, 395-404. 4. CCIR Document 12-3/29 (Rev. 1) [January 1992] Effect of the Melting Layer on Hydrometeor Interference and Coordination Distance, Canada submission. 5. CCIR Revision Rep. 452-4 [December, 1991] Prediction Procedure for the Evaluation of Microwave Interference between Stations on the Surface of the Earth at Frequencies above 0.7 GHz, Doc. 5/14(Add.1) 6. CCIR Rep. 569-3 [1986] The evaluation of propagation factors in interference problems between stations on the surface of the Earth at frequencies above about 0.5 GHz. Document 5/1051—E. 7. COST Project 210 [1991] Influence of the atmosphere on interference between Radio Communications Systems at Frequencies above 1 GHz. Final Report of COST 210 Management Committee, EUR 12407, ISBN 92-826-2400-5, Commission of European Communities, Luxembourg. 153  References  8. Crane, R.K. [March, 1974] Bistatic scatter from rain. IEEE Trans. Ant. Prop., Vol. AP-22, 2, 312-320. 9. Gibson, C., Kochtubajda, B. and Bergwall, F. [March 1991] Feasibility study for the extraction of melting layer statistics from archived Alberta S-Band weather radar observations. Final report; Environmental Research and Engineering Department, Alberta Research Council, Edmonton, Alberta. 10. Kharadly, M. and Choi, Angela S.-V. [February, 1988] A Simplified Approach to the Evaluation of EMW Propagation Characteristics in Rain and Melting Snow. IEEE Trans. Ant. Prop., Vol. 36, No. 2, 282-295.  11. Kharadly, M. [September, 1990] A Model for evaluating the bistatic scattering cross sections of the melting snow. Final report: CRC 36100-9-0247—/01—ST, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. 12. Kharadly, M. and Kishk, A. [1992, in preparation] Models for estimating melting layer attenuation. Manuscript in preparation, Department of Electrical Engineering, University of British Columbia, Vancouver, Canada. 13. Klassen, W. [1988] Radar observations and simulation of the melting layer of precipitation, J. Atmos. Sci., Vol. 45, No. 24, 3741-3753. 14. Medhurst, R. [July, 1965] Rainfall Attenuation of Centimeter Waves: Comparison of Theory and Measurement. IEEE Trans. Ant. Prop., Vol. AP-13, 550-564. 15. Oguchi, T. [September, 1983] Electromagnetic propagation and scattering in rain and other hydrometeors. Proc. IEEE, Vol. 71, No. 9, 1029-1078. 16. Olsen, R., Rogers, D.V., and Hodge, D.B. [March, 1978] The aRb Relation in the Calculation of Rain Attenuation. IEEE Trans. Ant. Prop., Vol. AP-26, No. 2, 318-329. 154  References  17. Persson, 0., and P. Lundgren [1986] Reduction of melting level effects on radar rain rate estimates. Promis Report no. 5, Swedish Meteorological and Hydrological Institute. 28 pp. 18. Ray, P.S. [August, 1972] Broadband complex refractive indices of ice and water. Appl. Opt., Vol. 11, 1836-1844. 19. Rogers, R. R. [1979] A short course in cloud physics. Pergamon Press, Toronto, Ont.  155  

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