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Application of the iterative moment method for dielectric scattering calculations Huang, Xiaomin 1994

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APPUCATION OF THE iTERATIVE MOMENT METHOD FOR DIELECTRIC SCKL1ERING CALCULATIONS by  XIAOMJN HUANG B. A. Sc. Beijing Institute of Technology, 1983 M. A. Sc. Beijing Institute of Environmental Features, 1988  A THESIS SUBMrrriD IN PARTIAL FULFiLMENT 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  THE UNIVERSITY OF  COLUMBIA  April 1994  °Xiaomin Huang, 1994  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 In presenting this thesis  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  ELec,  (J  The University of British Columbia Vancouver, Canada Date  DE-6 (2J88)  Abstract  In this thesis, the calculation of the electromagnetics(EM) scattering of three dimensional, irregular, inhomogeneous and lossy dielectric objects is discussed. The iterative moment method (MM) which can solve problems involving a large number of unknowns than the conventional MM is used. For result comparison and checking, conventional MM is discussed too. Generally, the method also applies for any arbitrarily shaped penetrable body with arbitrary dielectric distributions which will have importance in communication, medical, biological, meterological and military studies. A new melting-snow particle model is presented in this thesis, based on physical observations or assumptions. The EM scattering cross-sections of the new melting-snow model are evaluated at certain frequencies, melting factors and rain rates by the iterative MM. The analyses of the new model give us a better understanding to the EM scattering property of melting snow particles at the early stage and to the effect of outer layer water distributions on the scattering of melting snow. The results obtained are compared with those of two other existing models. The comparison shows that the different melting snow particle models have different scattering properties; the different melting water distributions assumed significantly affect the scattered fields. The validity of the iterative MM for large number of unknowns has been shown. The numerical calculation examples in this thesis prove the equivalence of the 11  two moment methods at special choice of basis functions and weighting functions. The thesis also discusses the computer realization of the two moment methods, compares their advantages and disadvantages. Conclusions regarding the choice of subcell size and the initial guess, the convergence and precision, computer memory, and computing time are obtained which will have academic interests.  111  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  vi  List of Figures  vii  Acknowledgements  x  Chapter I Introduction 1.1 Motivation of The Thesis 1.2 The Set-up of the New Melting-snow Particle Model 1.3 Thesis Overview 1.4 Literature Survey on Methods of Dielectric Scattering Calculation  1 1 2 6 7  Chapter II Theoretical Analysis of Two Moment Methods 2.1 Introduction 2.2 Integral Equation Derivation 2.2.1 The Volume Equivalence Principle 2.2.2 Volume Integral Equation Derivation 2.3 The Conventional MM Solutions 2.3.1 The Principle of MM 2.3.2 The Choice of Basis Functions 2.3.3 Testing Procedure 2.3.4 The MM Solutions 2.3.4.1 The Calculation of Impedance Matrix Z 2.3.4.2 The Calculation of Excitation Matrix V 2.4 The Iterative MM Solutions 2.4.1 The Iterative Algorithm 2.4.2 Conjugate Gradient Method 2.4.3 The Iterative MM Solutions 2.5 Formulating the Scattered Fields  11 11 12 12 12 16 16 19 21 22 22 26 27 28 29 33 36  Chapter III Computer Implementation and Discussion 3.1 Selected Numerical Results 3.1.1 Numerical Results for a Thin Dielectric Layer 3.1.2 Numerical Results for Lossless Dielectric Spheres  40 40 40 44  iv  3.1.3 Numerical Results for Lossy Dielectric Spheres 3.1.4 Numerical Results for Concentric Dielectric Spheres 3.2 Computation Considerations 3.2.1 Modelling Considerations 3.2.2 The Choice of Subcell Size 3.2.3 The Choice of an Initial Guess in Iterative MM 3.3 Discussion 3.3.1 The Convergence of the Iterative MM 3.3.2 Comparison of the Two Moment Methods 3.3.3 The Volume Integral Equation and the Surface Integral Equation 3.4 Conclusion Chapter IV Evaluation of the Scattering from Melting-snow Particle Models 4.1 Introduction 4.1.1 The Three Melting-snow Particle Models 4.1.2 Basic Parameters 4.2 The Computed Results for Melting-snow Particle Model C 4.2.1 The Choice of the Number of Outside Water Drops 4.2.2 The Calculation Results 4.3 The Comparison of Model A, Model B and model C 4.3.1 Forward Scattered Fields 4.3.2 Backward Scattered Fields 4.3.3 The Ratio of Forward and Backward Scattering Cross sections 4.3.4 Rain Rate R 4.4 Summary  49 54 57 57 58 59 60 60 60 62 64  66 66 66 67 70 70 71 79 79 84 86 86 88  Chapter V Conclusions and Suggestions for Further Research 5.1 Conclusions 5.2 Suggestions for Further Research  92 92 94  References  96  v  List of Tables Table I. The Inside E-fields of the Dielectric Layer  43  Table II. The Error in Each Iteration  61  vi  List of Figures Fig. 1.1 Two currently used melting-snow particle models  5  Fig. 1.2 The new melting-snow particle model  5  Fig. 2.1 Replacement of a dielectric body by an equivalent volume current Fig. 2.2 An arbitrarily shaped dielectric body illuminated byaplaneEMwave  13  Fig. 3.1 A dielectric layer illuminated by a plane EM wave  41  Fig. 3.2 (a) The inside E-flelds of the dielectric layer in Fig. 3.1 (b) The scattering cross-section of the layer  42  Fig.3.3. (a) The inside E-flelds of the layer (O=60°) EX/Ei O°) Ey/Ei 6 (b) The inside E-flelds of the layer (O= 45 (c) The inside E-fields of the layer (O=6O°) EzIEi (d) The scattering cross-section of the layer Fig. 3.4 The scattering cross-section of lossless dielectric spheres (a) Er= , ka=1, f=1OGHz 4  47  Fig. 3.5 The scattering cross-section of a lossy dielectric sphere Er41.775J41.204, ka=0.154, f=1OGHz  50  Fig.3.6 The scattering cross-sections of lossy dielectric spheres ’= 75 10GHz) 1204 7 • 41 (r= 4 J (a) ka=3.5 (b) ka=3.8 (c) ka=4  51 52 53  Fig. 3.7 The scattering cross-sections of concentric melting-snow particle model vu  55 56  (a) f=1OGHz (b) f=2OGHz Fig. 3.8 Meshing of a sphere  58  Fig.3.9 Comparison of the number of unknowns between SIE andVlE  63  Fig. 4.1 Relative permittivity of water vs. frequency Fig. 4.2 Scattering cross-sections of the three models (R=5mmlh, S=0.02)  72  Fig. 4.3 Scattering cross-section of the three models (R=Smm/h, S=0.08)  73  Fig. 4.4 Scattering cross-sections of the three models (R=5mm/h, S=0.14)  .  74  Fig. 4.5 Scattering cross-sections of the three models (R = 12.5mm/h, S = 0.04)  75  Fig. 4.6 Scattering cross-sections of the three models (R=12.5mm/h,S=0.12  76  Fig. 4.7 Scattering cross-sections of the three models (R=25mm/h, S=0.04)  77  Fig. 4.8 Scattering cross-sections of the three models (R=25mm/h, S=0.12)  78  Fig. 4.9 Scattering cross-sections vs.S f=1OGFIz)  80  (R=Smm/h,  Fig. 4.10 Scattering cross-sections vs.S (R=Smm/h, f=4OGHz)  81  Fig. 4.11 Scattering cross-sections vs.R (S=0.06, f= 10GHz)  82  vm  Fig. 4.12 Scattering cross-sections vs.S (S=O.06, f=4001—lz)  83  Fig. 4.13 Ratios of forward scattering to backward scattering  87  -  vmi  Acknowledgments  I would like to express my gratitude to my research supervisor, Dr. M.M. Kharadly, for suggesting this project and providing guidance during the course of this work. I am also grateful to Dr. G. E. Howard for many helpful discussions regarding this work. My thanks extend to my fellow student and friend Rafeh Hulays for kindly allowing me to use some of his computer programs to get the comparison results. My thanks also go to all of my friends and those individuals in the Electrical Engineering department who have helped me in various ways through out my studies. Thanks are also due to Dr. A. A. Kishk of the Univ. of Mississippi for the scattering results of model A. Last, I would like to thank my husband for his help, encouragement, understanding and patience during the course of my study.  x  Chapter I. Introduction  1.1 Motivation of the Thesis Electromagnetic (EM) scattering by dielectric bodies has been the subject of intensive investigation. It is very important in problems including propagation through rain or snow, scattering by airborne particulate, medical diagnostics and power absorption in biological bodies, coupling to missiles with plasma plumes or dielectricfilled apertures, and the performance of communication antennas in the presence of dielectric and magnetic inhomogeneities. Many methods have been used for solving EM scattering from dielectric scatterers. These methods can be classified into two types: analytical methods and numerical methods. The former give analytical formulae or expressions to show how the scattering might change with a change in the dielectric distribution, dielectric shape and radar frequency, but they can only solve for dielectric bodies with simpler shapes such as spheres or cylinders with homogeneous dielectric distributions; when the dielectric bodies are inhomogeneous or in irregular shapes, we have to resort to numerical methods. Numerical methods have the advantage that they can be used to solve arbitrarily shaped and inhomogeneous dielectric problems, but for scattering from electrically large or dielectric bodies, most of the numerical methods are impractical because of the computer memory limitations. It is necessary to have a method which fully evaluates the scattering from large size, inhomogeneous dielectric distributions or in arbitrarily shaped three-dimensional dielectric scatterers. 1  In terrestrial and satellite communications, more and more attention has been paid to the scattering by melting-snow particles in order to study the influence of the melting-layer on microwave propagation. Various prediction models are used to evaluate the EM scattering from melting-snow particles. Presently there are two commonly used models to represent the scattering of melting-snow particles, the concentric snow sphere and water shell or the homogeneous water-ice-snow mixture sphere (discussed in next section), which are simple shapes and are easily analyzed by analytical methods. For better understanding of the influence of the melting procedure on EM scattering, in this thesis, we present a new melting-snow particle model which is a three-dimensional, irregular, inhomogeneous and lossy dielectric body. At present there are no analytical methods to obtain the scattered fields  ,  we have to use a  numerical method to evaluate its scattering properties. This study is mainly motivated by the idea of fully calculating the scattering of dielectric bodies and evaluating the newly developed melting-snow particle model.  1.2 The Set-up of the New Melting-snow Particle Model In communications, the incoming signal and interference caused by other systems have to be distinguished in order to establish a reliable system. When an EM wave propagates in a hydrometeor medium such as ice, snow, melting snow and rain, EM scattering as well as attenuation occur; the scattering is one of the causes of interference between communication links operating at the same frequency. Recently,  2  considerable research has been done on interference caused by hydrometeor[1] [2]. Scientists are interested in the scattering by the melting-snow layer[3] [4] because its presence affects microwave communications[5]. In the study of the snow melting procedure, the melting-snow layer is a concept often used. We call the region between where the melting starts and where the melting ends the “melting-snow layer”. In a melting-snow layer, the melting occurs basically because the surrounding air is at a higher temperature. The melting snowlayer extends from the dry-snow region at a higher altitude to rain at a lesser altitude. From the evaluation of the EM scattering from the melting layer one can know its effect on communications and therefore design a reliable communication system. The melting layer is made of numerous melting-snow particles. Usually the scattering of the melting layer is evaluated by considering the scattering of a melting-snow particle. Melting-snow particles are expressed using different melting-snow models for some specifications. Melting snow particles are assumed to have a variety of complex shapes that may be extremely difficult to analyze. Thus in order to look into the problem of the dependence on particle geometry, some simplified geometric shapes which can be reasonably analyzed are considered. At present there are mainly two models[6] of melting-snow particles used in analysis and prediction. One assumes that melting occurs from the outside of the particle, with the water forming a shell around the snow core. The corresponding model is a concentric sphere model where a uniform layer of rain (water) of permittivity  Ew  surrounds a dry snow core of permittivity  3  Es  (as shown in Fig.1.1 (a), model A). The other assumes that the water forming on the outside of the melting particle percolates to the inside, forming a heterogeneous mixture of air, water and ice. The relevant model is a “homogeneous” “artificial dielectric” sphere model with the equivalent pennittivity  Eav  of the water-ice-air  mixture (as shown in Fig.1.1 (b), model B), the 6 av is calculated from the Maxwell Garnett theory or the Bohern extended theory for spherical inclusions. They are analyzed relatively easily by using analytical methods such as Mie scattering theory. The model A assumes that the melting occurs from the outside of the particle, but what happens before the forming of the outside water shell? People have observed that when a snow flake melts, at first there are always several water drops at the snow flake surface. From the physical sense, it is not difficult to understand that the melting starts from a small amount of melting to a bigger amount, therefore, the water shell of model A is formed by a number of water drops. In the other words, when the air which surrounds a snow particle is warm enough, at the outside several small water drops which come from the early melting of the snow particle form at first. After a certain period of time, those water drops become bigger and then merge into a water shell as described in model A. Based on model A and the analysis above, it would be necessary to assume a new melting snow model in order to describe the melting snow in the beginning melting period. The new melting-snow particle model to be set up and evaluated in this thesis is described as a snow core of permittivity c with several even distributions of water (rain) drops of permittivity e outside (as shown in Fig.1.2). In the rest of the thesis,  4  h x -t ur e  snow  wQter Concentric sphere model  (a)  Homogeneous sphere model (b)  Fig. Li Two currently used melting-snow particle models  snow  Wa ter  Fig. 1.2 The new melting-snow particle model  5  we call it model C. It is believed that this model describes the melting-snow particles at the early stage of melting and it has never been analyzed. It is complementary to model A and model B, and will give us a better understanding of the effect of outside water on the EM scattering of melting-snow particles.  1.3 Thesis Overview In this thesis, a new melting snow particle model is described and evaluated. The internal electric fields as well as scattering of a plane wave (either polarization) illuminating three dimensional, irregular, inhomogeneous and lossy dielectric bodies are obtained. Two moment methods, the conventional moment method [7J[8] (we will call it the MM) and the iterative moment method[9] (we will call it the iterative MM or the 1MM), will be used, based on the volume integral equation method employing the volume equivalence principle. The latter will be used to evaluate the scattering of the new melting-snow particle model. The scattered field obtained by the new model is subsequently compared with the other two models. Conclusions are obtained. Once the validity of the iterative MM is established, the method can be applied to any other three-dimensional dielectric scatterers with large size in any shape or dielectric distributions. The rest of the thesis is organized as follows: In Chapter II, the theory of the MM and the iterative Mlvi is developed to obtain the EM scattering from dielectric bodies. The analysis starts with the derivation of the general volume integral equation based on the volume equivalence  6  principle. The integral equation can be applied to any three-dimensional, irregular, inhomogeneous and lossy dielectric scatterers and solved by the MM. Pulse basis functions and delta testing functions are chosen. The iterative MM, based on the conjugate gradient method, is also discussed and applied to solve the integral equation, where the same basis functions and testing functions as in the MM are used. After the unknown interior fields or currents are found, the scattered field as well as the scattering cross-section in the far radiation region are determined. Chapter III discusses the computer implementation of the two moment methods. The computer programs written are tested by comparing the solutions with known solutions and by comparing the solutions obtained by using the two methods. The modelling of the dielectric scatterer, the choice of a subcell size, and the determination of an initial guess in the iterative MM are considered. The convergence and precision of the iterative MM are also discussed. The two methods are compared regarding computer memory, computer time, and the number of unknowns that can be solved. In Chapter IV, the scattering by the new melting-snow model is studied. The results obtained are compared with those of two existing models  (  model A and  model B), from the calculation one can see that the outer layer water distributions affect the scattering of the whole melting-snow particle, different models have significantly different values because of different assumptions. This model gives us a better understanding of the early melting stage of the melting-snow particles. Chapter V, the last chapter, will give some conclusions and the suggestions for  7  further research work.  1.4 Literature Survey on Methods of Dielectric Scattering Calculations Modelling of penetrable dielectric bodies is more complicated than the modelling of perfect conducting bodies because of two reasons: (1) it is needed to deal with the fields of sources radiating in at least two different media, and (2) at the surface of a dielectric body, both the equivalent tangential magnetic current and electric current are not zero while at the surface of a conducting body the tangential magnetic current equals zero. EM scattering from dielectric objects originally started from Descartes rainbow theory[1O] based upon ray optics. The problem was subsequently treated as a boundary value problem and solved by many researchers via the classical separation of variables method. Mie[11] was the first to develop solutions for arbitrary size and homogeneous spheres. Moglich, Aden, Scharfman, Raleigh, Wait, and Yeh [12] —[17] obtained solutions for the scattering and internal fields of homogeneous ellipsoids, dielectric coated spheres, infinite cylinders and elliptical cylinders respectively. Those classical methods are only effective for a body whose bounding surface coincides with one of the coordinate systems for which the vector Helmholtz equation is separable. A perturbation technique which is more effective for bodies that are only slightly non-spherical was also developed by Durney[18] and applied to prolate spheroid models for long wave-length irradiation. A superposition estimation of a concentric-two-layered sphere was suggested  8  by Peters[191. Standard optical approximations such as the physical optics (P0) and the geometrical theory of diffraction (GTD) have also been modified by Peters and Keller[20] so that they may be applied to dielectric bodies whose dimensions are much larger than a wavelength. The rapid development of numerical methods made it possible to solve dielectric scattering problems accurately and conveniently. The most-used numerical methods are the MM[7][81 and the extended boundary condition method (EBCM)[21] which are used to solve integral equations, and the finite element method(FEM)[22] and the unimoment method (UMM)[23] which are used to solve differential equations. Those methods are powerful when the dielectric bodies are small compared to the wavelength. The spectral iteration techniques (SIT) and the boundary element method(BEM) are also used in solving medium size dielectric bodies. For the EM scattering from electrically large dielectric bodies, the use of the above numerical methods by themselves is impractical because of the computer memory limitations. Since 1975, various hybrid solutions, using numerical methods together with analytical methods or even using two different numerical methods together, have been developed to solve the problems of scattering from larger size dielectric bodies. They are the hybrid solutions of the MM and the GTD, the MM and the P0, and the MM combined with the FEM[24]. Hybrid methods can save computer memory, but they will make the calculations much more complicated, and normally they cannot be used for arbitrary irregular shapes. Recently, the possibility  9  of using iterative methods for large bodies has been investigated because they can be carried out without the direct involvement of a large matrix which rapidly exhausts the computer’s memory. At the beginning, iterative methods could only be applied to two-dimensional problems[25]. In 1989 Wang[9][26] used an iterative conjugate gradient method in the MM (it is also called the iterative MM or the generalized MM) and successfully solved a three-dimensional arbitrarily shaped dielectric body under plane wave illumination. The iterative MM can solve problems involving a larger number of unknowns than the conventional MM by at least an order of magnitude. Using the iterative MM, we can now solve for large inhomogeneous arbitrary bodies as long as the computer time allows. Although the newly developed melting-snow particle model in this thesis is not bigger in electric size, it needs more unknowns because (1) outer layer water drops need more unknowns than regularly shaped dielectrics with the same volume, and (2) the snow core is a sphere which needs to be cut into more subcells in order to represent the shape more accurately. That is why we chose the iterative MM to solve the scattered field of our model.  10  Chapter II. Theoretical Analysis of Two Moment Methods  2.1 Introduction The moment method (MM) was applied in EM field calculations by R.F. Harrington in his significant book entitled “Field Computation by Moment Method”[27] in 1968. It is claimed as forming the basis of all numerical tools in computational electromagnetics. In the last few decades, numerous papers and many books about the mathematical basis and the applications of MM have been published. In 1992 IEEE published a special collection[24] in which a selection of key articles about the analytical formulation, numerical implementation and practical applications of MM are given. The publication shows the importance of MM in electromagnetics theory applications. Recently J.J. Wang applied the iterative MM[261 to EM scattering problem in which improvements were introduced to MM. The iterative MM can solve problems involving a large number of unknowns than the conventional MM, but only a few papers for solving three dimensional EM problems have been published so far, further studies are going on. In this chapter, the basic theoretical analysis of the conventional MM and the iterative MM for solving the scattering of a three dimensional dielectric scatterer is discussed. At first, the volume integral equation is derived, and then solved by using the MM and the iterative MM respectively. Finally, the internal E-fields, as well as 11  the scattering cross-section, are obtained.  2.2. Derivation of Integral Equation 2.2.1. The Volume Equivalence Principle[261 The volume equivalence principle states that in calculation, a dielectric body, with volume V, permittivity e(r) and permeability  being illuminated by an EM  wave E’, can be represented by an equivalent volume current 3(r) density in V as follows: (2.1)  E(r) ]c J(r) -jc [e (r) 0 —  where r is a position vector of the point of interested inside the body,  is the angular  frequency of the wave, E(r) is the total electric field inside the body, and 6(r) is the permittivity of the dielectric body at position r. This principle is illustrated in Fig.2.1. Using it we have replaced the dielectric body occupying V by equivalent volume current J(r). As long as we know the J(r), the E-field inside the dielectric body can be obtained from Eq.(2.1). The volume equivalence principle can be applied for inhomogeneous bodies, in which 6(r) is a function of the position vector r. If u does not equal to  there will be an  equivalent volume magnetic current M(r).  2.2.2. Derivation of Volume Integral Equation[26][28] Consider a plane wave illuminating a three dimensional arbitrary shaped dielectric body, as shown in Fig.2.2. The surrounding medium is considered to be free 12  E E  E —  vi  Oh  (a)  \  (b)  —— S  / Ii O.)O  /  Fig. 2.1 Replacement of a dielectric body by an equivalent volume current[26]  2  So  ‘1’  x Fig.2.2 An arbitrarily shaped dielectric body illuminated by a plane EM wave 13  space with parameters  , j.. 0 E  From Maxwell’s equations and the Lorentz condition,  the scattered field inside the volume, due to the equivalent volume current J(r’), is given by the following relation[26]: E s(r) _fJ(r’)ä(r,r’)dv  (2.2)  where 0 is the free-space dyadic Green’s function  (r,r—jpo(7+-4VV g(r,r’),  (2.3)  g is the scalar Green’s function:  ,.  g(r,r )-  1 e -j-F  24  4idr-rl  I is the unit dyadic: (2.5) and  zi(Q)”. k-k 0  (2.6)  Eq.(2.2) shows that ES(r) is singular at r=r’. To circumvent the singularity, we , (V-V ) 0 plus the remainder 0 split the volume V into a small spherical volume V 0 is an open neighbourhood containing the point r=r’. The integral over (V where V 0 so ) can be carried out since G is not singular in this region. We can choose V 0 V  14  small that the current density J(r) can be approximated by its value at the centre 0 tends to ) and take the limit as V 0 , apply Green’s theorem to the region (V-V J(r ) 0 zero. We can obtain the following equation:  Es(r)  +  vf J(r’) G(r,r’) dV  0 3k J(r)  (2.7)  +PVfJ(r’)G(r’)dv’ V  where G is given by Eq.(2.3), and PVJ denotes the principal value of the integral. Now we can write the total unknown electric field E(r) inside the body as the sum of the incident electric field E’(r) and the scattered field ES(r): r)÷E S(r) E(r)-E ( t  (2.8)  Substituting Eq.(2.7) into Eq.(2.8) and rearranging the terms gives the desired  volume integral equation (VIE) of our dielectric scattering problem: E ‘(r) =PVf J(r’) ?(r,r’) dv”+ D(r)J(r)  (2.9)  where  e 1 (r)+2 3j[e(r)-]  15  (2.10)  and £(r)-r(r)eo.  (2.11)  Our goal is to find a solution for the equivalent volume current J(r) density in Eq.(2.9) in terms of the incident field and the physical characteristic of the scattering object. If we can get J(r), the internal E-field and scattered fields of the dielectric body can be obtained. For this purpose, two MM solutions will be developed in next sections.  2.3. The Conventional MM Solutions MM is a powerful method in solving EM scattering or radiation problems, especially when the objects are inhomogeneous or of irregular shapes which cannot be solved by analytical methods.  2.3.1. The principle of MM[27][29] The basic principle of the conventional MM is to expand unknown functions into a series of basis functions so that an operator equation, such as a integral equation, a differential equation, or a integro-differential equation, is transformed into matrix equations which can be solved easily with a computer. The integral equation Eq.(2.9) obtained in the last section has the form (2.12)  16  where, in general, A is a linear integrodifferential operator in a certain domain, and X is the unknown to be solved given a known excitation Y. Assume that the unknown solution can be expanded in terms of known 1 are linearly independent and 1 such that Ax expansion functions (basis functions) x span the range space of A. We then seek XN--an approximation to the exact solution X--of the form: (2.13)  N  XXN-)  I  i—i  where a 1 are unknown coefficients of expansion functions x, i=1,2,...,N. In order to solve for the unknowns a, first we form the residual N  (2.14)  RN=Y-AXNY--  and then choose a set of known functions (testing functions) W (j = 1,2,  ...,  M) so that  the weighting of the residual RN with respect to the testing functions W is zero: -O, j-1,2,...,M <R,W > 1 where M=N or M>N and  <.,.,>  (2.15)  denotes the usual inner product  <R,W>=f R(x)W(x)*dx L  (2.16)  N fL  17  a Ax? W(x)*dx  here the star  *  denotes the complex conjugate and L denotes the domain for the  inner product.  For M=N, the Eq.(2.16) gives rise the well-known matrix equations: ,W 1 <Ax >  W 1 , 2 <Ax >  W> c4x , 1  c4Xj>  ...  ...  (2.17)  W,> <AX <AX , 1 ,WN> 2  ...  <AX,,,WN>  aN  Let  [Z]-  ,W 1 1 <Ax > ,W 2 <Ax >  ...  ,W> <AX22 Ax 12 , W>  ...  <AX W>  (2.18)  ...  ...  > <Ax,W J WN> 1 ,<AX 1  which is called the impedance matrix and  [r]  (2.19)  —  aN  18  which is called the current matrix, and  > 1 <y,w [1’]  (2.20) -  <Y,WN>  which is called the excitation matrix, Eq.(2.17) becomes [Z][IJ  [T’]  (2.21)  which is called the MM matrix equation and from which we can solve for the unknown coefficients aj (i= 1,2,..,N), Then get the solution X in Eq.(2.13).  2.3.2. The choice of basis functions[26J[29][30] The choice of basis functions is the key point of the MM, it will decide the accuracy and convergence of the calculation. From a mathematical point of view, the choice of basis functions does not depend on the choice of weighting functions. In , Wa>, is a double integral, for <Ax , Eq.(2.18) each element of the right side matrix, 1 which it is difficult to get an analytical solution. Therefore the calculation of the Z matrix might need a lot of computing time. When using the MM to solve EM problems, we should choose basis functions and weighting functions carefully so as to get the solution accurately as well as economically. There are many sets of basis functions that can be chosen theoretically. For a given problem, only a few sets of basis functions are suitable. The basis functions  19  chosen for a particular problem have to satisfy the following criteria: (1) They should 1 form a be in the domain of the operator and (2) They must be such that Ax complete set for the range of the operator[29]. In practice, if we have certain pre-knowledge about the solution, we should choose basis functions to (1) be as close to the practical solution as possible; (2) meet the boundary conditions; (3) make the calculation converge fast so as to save computing time; and (4) be simpler so as to carry out the calculation easier. In solving dielectric problems, for different dielectric shapes and integral equation types, normally there are several kinds of basis functions that can be chosen[24]. For example, for bodies of revolution or cylinders in surface modelling, the scatterer surface is cut into small rings and basis functions are triangle wave functions or sinusoidal functions; for arbitrary shaped homogenous bodies in surface modelling, the scatterer surface is cut into small triangle surface patches and basis functions could be related with each triangle’s face, edges and vertices; for arbitrary shaped inhomogeneous bodies in volume modelling, the scatterer body is cut into a set of tetrahedral elements, each basis functions is associated with each face of the tetrahedral model. In arbitrary shaped inhomogeneous bodies in volume modelling, the simplest way might be to cut the object into small cubes, and the basis functions are three dimensional pulse functions. In the following calculation, we will choose this one which will lead to simpler analysis. Dividing V into L small volumes vl,v ,...,vL so that J(r) and 2  E (r)  are constant  in each subvolume. We choose the pulse functions as our basis functions:  20  B,A(r) ..k(r)_j kP,(r) 2  where  Uk  (2.22)  are unit vectors x,y,z, when k= 1,2,3 and  P r)-{  1 0  for r€V,  (2.23)  elsewhere  Expanding unknown current J(r) in terms of above basis functions as  J(r)  kB k(r) 1  (2.24)  -  i-i k-I  31 are unknown coefficients for each subcell v B are basis functions in k , 1 1 where k  Eq.(2.23), k=x,y,z, 1= 1,2,...L. Eq.(2.9) becomes  j2 i-i k-i  E ‘(r) _Fvf  k  jk  l2kPj(r)dv’  (2.25) r) P ( 1  i-i k-i  k 1 j The VIE in Eq.(2.9) has been expressed in terms of unknown current coefficients j the solution of the k known incident field E’ and pulse functions. If we solve the 1  VIE in Eq.(2.9) will be obtained.  2.3.3. Testing Procedure[31] The next step in applying the MM is to select a testing (or matching) procedure so as to generate 3L independent equations which will be solved for the k 1 unknown expansion coefficients j  Generally in MM there are two kinds of weighting functions chosen, based on  21  two matching methods[271, the Galerkin’s method and the point-matching method.  The former is to choose the weighting functions as the conjugate of the basis functions and the latter is to choose the delta functions as the weighting functions. For both methods, the weighting functions must be in the range of the operator. Delta functions are the simplest weighting functions and the related matching method is the point-matching method. We choose it in our testing procedure, in the mathematical form:  w-  p-I n-I  ö(r-r)zI  (2.26)  p-I n-I  Where n is related to x, y, z directions and p is related to the subcells. Taking the product with weighting functions in Eq.(2.26) on both sides of our integral equation (2.25), we can get our MM equation as Eq.(2.21). The details will be discussed in the next section.  2.3.4. The MM Solutions[30j[32J 2.3.4.1 The calculation of impedance matrix Z In rectangular coordinates, the inner product J(r’).G(r,r’) in Eq.(2.9) can be written as:  22  ) t ) G (r,r 1 G (r,r’) G ‘(r,r (2.27)  G(r,r’) J(r’)- G(r,r’) G(r,r’) G3z(r,rI) G (r,r’ G (r’) G (r’)  [J(r’)f  where  G’(r,r”)- -jø jL(8 +.! Ic  )g(r,r’5  (2.28)  aUfl&Sk  here S is a Kronecker delta function and g(r,r’) is shown in Eq.(2.4). From Eq.(2.9), we get the discretized integral equation:  re V  (2.29)  ’(r) 2 1 k Z&_fG(r,r’) B(r’)dv’+D(r) B 3  (2.30)  JkZE ‘(r) i-i k-i  where  =  n-i  [f  G(r,r’)B(r’)d1/]+D(r)Bl”(r)l2k  Letting  G1[k(r) U+D(r)Bi(r)U  (2.31)  where Gi (r)_fG (r,r)Bik(rI)dvI  23  (2.32)  and  (2.33)  z-Ezsl.  Eq.(2.29) can be rewritten as 3  ‘33  ID ID 1—1 k—i  3  J, Z,i2,- EE.’(r)d K—i  rEv  (2.34)  n—i  Taking the product with weighting functions, on both sides of our discretized integral equation (2.34), yields the following 3L equations:  ±f  (2.35)  .i,kz> v;  I-i k-i  where  z>(w’,z,a)  (2.36)  here I correspond to the subcells related to the basis functions and p correspond to the subcells which are related to the weighting functions. As in Eq.(2.29), 1 and p range over all possible values , we obtain the following matrix representation of Eq.(2.25) Z] [ZJ [Z] J] ,] [Z,]. j,j 7 Z,x] [Z Z] [Z,,] [Z1  24  11  Vx]  (2.37)  -  içi  where  is given in Eq.(2.36), k=1,2,3 and n=1,2,3. According to Eq.(2.28),  the element Gi is G1j(r,)-fG (r,r)B,(rdv  (2.38)  pl where  f dv’ 1 AV  (2.39)  VI  Using Eq.(2.28) to evaluate G’(r,r ) gives[30] 1  G ( 1 r)  _f Ioko(e (r?-e)AV,e  e  44 ..cosOf’cosO(3-4+3ja )]  )8 2 [(4-1 _ja  (2.40)  l#p  where (2.41)  (r’—r,  ,  (T’-Tk’)  (2.42)  (2.43)  R,  r,—rff+r,9÷r?f  25  (2.44)  (245  1.. r +r ÷r 3 = r x 1 y 2 z  (2.46)  R4r, -r)  1 is approximately replaced by a small When l=p, as we have discussed in 2.2, v 28][30J, and [v sphere 0  G1  k 8  2)0 1.10  )eth0at_ 1] [(1 ÷jkoa 1  lp  (2.47)  0 3k where  a  (2.48)  3 (M’)l 4it  is the radius of the sphere. Now we have the elements for the matrix Z in Eq.(2.18).  2.3.4.2. The calculation of the excitation matrix V[27J[33J Assuming the incident wave is a plane wave in (Of, E ‘(r) =áEe  )  jkr  direction: (2.49)  where a is the polarization direction of the field, E is the amplitude of the incident E field, k is the propagation factor: ÷9sinO sincp ÷‘cosO ) k=k=k(isinO 1 cosp 1  26  (2.50)  and r is the position vector: r..rP-d+yj÷ze  (2.51)  sinq +zcos0 ) k’r=k(xsinO cosp +ysinO 1  (2.52)  From  Eq.(2.49) becomes:  K ‘(r) =âEe  ÷ysiuOeosp +zcosO -jk(xsinO gcos 1  (2.53)  Therefore, the excitation matrix in Eq.(2.35) is: v=<E Z(r),8(r.,,,)li> E’e  +zcosD ) -jkxsinOcosp +y,,sinO posp 1  (2.54)  where n=1,2,3, and p=1,2...,L. Substitute matrix elements Z in Eq.(2.36) and V in Eq.(2.54) into Eq.(2.35) Jk (k=1,2,3 and and solving the linear equation, we can get the coefficients 1 1= 1,2,...,L) of 3(r). From Eq.(2.24), the MM solution of the volume current density 3(r) in the scatterer is obtained. If L is sufficiently large, the approximation given in matrix (2.35) will give adequate results.  2.4. The Iterative MM Solutions[26] The iterative moment method is the conjugate gradient method (CGM) used in the MM. The CGM is an important special case of the method of conjugate directions. It plays a central role in iterative methods because it terminates with a  27  solution in at most N steps  (  N is the dimension of the unknown vector), if no  numerical round-off errors are encountered.  2.4.1. The iterative algorithm[34] As discussed above, almost all direct scattering problems lead to integral equations of the form, as in Eq.(2.12), Y(r) _fX(r’)K(r,r”)dV’  (2.55)  where X(r’) is the unknown, Y(r) is the known excitation, and K(r,r’) is the kernel function of the integral equation. Express X(r’) and Y(r) in vector forms: X(r’) [x(rl),x(r2),x(r3) .,x(r)]  (2.56)  ] ,y(r ) [y(r ) 7 ) ,y(r,),...,y(r Y(r)— 1  (2.57)  ,..  -  and  , r 1 , 2 where r  ...  ,  are the position vectors in the volume V. In order to get the  solution of X, at first an initial estimation for X is given: ] ,X°(r ),...,X°(r 1 [X°(r ) 2 X’°(r’) ) —  (2.58)  The superscript 0 denotes the first time estimation. Substituting  x° into Eq.(2.55), if x° is not equal to the unique solution X  in Eq.(2.55), the deviation is F°— Y(r) _fX(°)(r’)K(r,r’)dv’  28  (2.59)  The integral square error is defined as: (2.60)  ERR(°)—fIF(°)(r)Fdv  After n steps of iteration, the integral square error is: ERR _fIF(r)Fdv  (2.61)  or dv 42 ERR_fjY(r) _fX(r’) -K(r,r’)dv  (2.62)  In the frequency domain scattering cases, F can be complex value. We note that (2.63) and hence ERR  (2.64)  Only when the estimation equals the solution, does the equality sign hold. The ERR differs from each iteration. The basic scheme of the iterative method is to choose certain estimations and iterations to minimize the integrated square error in the iterative process until a satisfactory result is obtained. The total iteration number N depends on the precision required.  2.4.2. Conjugate Gradient Method[35j[36] In general iteration methods, going from the (n-1)th step to the nth, we take  29  (2.65)  X(r)—X’(r) +i  where  =  (1)(r) is a variational parameter and g(fl)  chosen variational function. The choice of g and  t  =  g (r) is a suitable  depends on the iterative method  chosen. In Eq.(2.61) the deviation in nth iteration is  V  F—F” —  Y(r)_fX(’I)(r’)K(r,r”)dv’  From Eq.(2.65) and Eq.(2.66) we have —i f’(r’)K(r,r’)dv’ F—F 1 1 —F’’—i  (2.67)  (n(n)  where f is the variational deviation:  (2.68)  -f)(r’)K(r,r’)dv’  Now ERR can be written as 1 ff)dv] +Fq ERR—ERR’—2Re[i  Bf”)Pdv  _’ 1 +h where A ()_fF’)f’dv  30  (2.70)  and B(-.f’Fdv  (2.71)  From Eq.(2.69) we can see that if we choose the variation parameter  (2.72) the ERR gets the minimum value L4”P  (2.73)  As long as (2.74) we always have 1 ERR <ERR  (2.75)  Substituting Eq.(2.66) into Eq.(2.70) and interchanging integrations, we obtain (2.76)  A  where r) fv1_1*(nI,th,I =S ( 1  31  (2.77)  is called the gradient vector of the iterative process. From Eq.(2.67) Eq.(2.76) and Eq.(2.77) we can also get (2.78)  A (fLs(1)12dv  Eq.(2.73) states that the iterative solution will be improved after each iteration if Eq.(2.72) and Eq.(2.74) are satisfied. Based on Eq.(2.73), we can see that this condition can be met until ERR is zero if we choose that the variational function g(fl) and gradient vector  to have a certain relationship. In the conjugate gradient  method, the choices of g(fl) are: (2.79)  for n1  g°>(r).’S’°(r) and  forn>1  r) g ( 1  (2.80)  1 A’ It is not difficult to prove that our choices can meet two separate conjugate conditions. One is g are mutually conjugate: =O <Ag,g > 0 The other is that  ij  (2.81)  are mutually orthogonal: (2.82)  32  That is why the method is called as conjugate gradient method.  2.4.3. The iterative MM solutions[9j[26j[27] In general the results between the direct and the iterative MM are different. However, when pulse functions are chosen as the basis functions and delta functions as the weighting functions in the direct MM, and the same basis functions are used in iterative MM, the same numerical results in the two MM are obtained. In the previous section we have obtained the basic method of the iterative MM. We will derive the 1MM solution for our scattering problem in Eq.(2.9). Comparing with Eq.(2.55), we have X(r)—J(r)  (2.83)  Y(r) =E ‘(r)  (2.84)  K(r,r’) G(r,r”) +D(r) ô (r—r’)I  (2.85)  and  As in conventional MM, we first divide the object into L subcells with centre ,r 1 at r , 2  ...,  rL. The Y vector is (2.86)  ) (I=1,2,...,L), 1 Assuming the current at each subcell is J(r (2.87)  33  and the initial guess current is  (2.88)  —  then the deviation is: F°’(r) — [E ‘(ri) —f(r ),E ‘(r2) —f(r 1 )—ftr ‘(r ) ] ),...,E 1 2  (2.89)  f°(r) _fK(r_,r’)J(°)(r’)dv’  (2.90)  where  Whereas taking the same derivation method as in the MM, we have 3  ftr ) m -E i2f(r) i-i  i—i  (2.91)  [E E 1—1  jk  G1(r)+D(r_)J, r2]  k—i  where G1 1 is given by Eq.(2.32). If ERR=f0Pdv  (2.92)  is not small enough, we take next step n=1. Following the general operations as discussed above, the gradient vector S in Eq.(2.77) is:  It)(rM) 4 gs  S(r) _  L  3  3  —ID ID ID i—i  (293)  3  F(r_)  [G1_(rj)]*+[D(r_)]*  k—i i—i  i—i  34  F(Tm) aj  where L  3  ) -F D(rj) 1 F(r  (D D L_lkr,)P.vi)* k-i 1-1  —  3  (2.94)  L  )P.v .frkr 1 E k-i i-i If n=1 (2.95)  g°(r) .-S°(r)  Ifn>1  g(r) —S (r) 1  +  A (n) 1 A’  g’(r)  (2.96)  3  A (n)=fL_1)(r)dvf I S “(r) iiJdv L  3  (2.97)  E —E k-i i-i 3  B=f(r)Fdv_ 1’ IEf(r)z2 dv J  (2.98)  Vk_l  3  L  n)  -  ) .v, 1 (r  k-i I-i  (n)  A (n) —____ L  3  ] 1 £ L’’kr,)P.v  —  k-i i-i 3  L  E  k-i i-i  35  (2.99)  j —J(r) —J’ (r) +ri g(r) 1  1i ‘/‘(r) F(r)—F’(r)—  (2.100)  (2.101)  the integrated square error is  ERR_f  V  71r)Pdv_f I  F(r)i2dv  V(k_l)  (2.102)  P.v, LFkr ) 1 k-i i-i  In calculations, the normalized integrated square error is often used which is defined as:  ERR)  ERR  I L°(rP  J V  (2.103)  ERR  f[i(°)(r’)F If ERR is not small enough, those steps are repeated until satisfactory precision is obtained.  2.5. Formulating the Scattered Fields[33] [37] [38] Although the accuracy of internal field values is a more stringent test of a computational method than is the accuracy of scattered fields, it is difficult to accurately measure the fields inside most dielectric objects. Furthermore, many applications require only scattering data and not internal field data. 36  Once the system of linear equations(2.35) has been solved, the unknown coefficients are found, the volume currents in the dielectric body are obtained, from which the scattering fields of the dielectric body can be calculated at any point in space. To simplil’ the calculation, it is convenient again to divide the dielectric body , 1=1, 2, J(r ) 1 and the current 1 into L small cube cells, each of volume v At the receiving point P(r, O,  ),  ...,  L.  the far-field, due to a current source J(r)  in a region v, is is  s  1 (r) ,(r)——E 4 H 0  (2.104)  r) --jøA ( 0  is  S  0 (r).—E,(r) H  (2.105)  - -jø A(r) where A is the vector magnetic potential, generated by the current source J(r) in the region  A(r) =  e  4irr L  c*  and its  e  .fJ(r’)e ‘dv’ V  1 -jkrr e —J(r,)e  (2.106) 1 ,v  and p components are eJk  (r’)e” dv’ 0 (A 0 r)=____.J J 4tr V L  1 -JkrT e cx 1 v, r (e 0 —J ) 1-1 ‘,3CT  37  (2.107)  1 e  )e’dv” 1 A,(r)=.-__-_.J J,(r 4itr L  v  (2.108)  ..•.  !__.J,(r)e z- 4r  u  As Eq.(2.52)  e  -jk(x,s1nO,cos 1 +yphiOsincp,+zposO)  (2.109)  Since in the analysis above we have the current in Cartesian form: (2.110)  J(r) 1(r) +9J(r) ÷U(r)  From the relation between Cartesian and spherical coordinates: (2.111)  Ô=icosOcosp +9cosOsinp -sinO  -  (2.112)  -sinp +9c0s(p  we can get its U and p components r -JposO 1 ( 0 .1 )  —JsinO -Jos0 sinp 3  (2.113)  (2.114)  -Jposcp s—J1n J(r ) 1  From Eq.(2.104) Eq.(2.105) and Eq.(2.106) we can get the scattered fields Es and the scattering cross-section of the dielectric object:  2 a(O p ,,Op )—lim 47tr r-oo  38  )P  (2.115)  where r is the distance between the source and the object; Es is the scattered field 1 is the incident field at the strength at the receiver due to the target scattering and B target, (Of,  ) is the direction of incident field and  (8,  )  is direction of scattered  field, and p and q are either 0 or p, respectively. When 0  =  O and p = p, we call a the monostastic scattering cross-section 5  which is when the transmitter and receiver are in the same direction with respect to the scatterer, we also call the monostastic scattering cross-section, the backscattering or radar cross-section (RCS) which is a very important concept in radar systems; The scattering cross-section for angles other than backscattering is called the bistatic scattering cross-section. In later calculations, we are most concerned about two important scattering directions, backward scattering cross section (0 =cp) and forward scattering cross section (O  =  ir- O and  =  O and  =  Now we have the MM and 1MM solutions of the volume integral equation in Eq.(2.9) and the scattered fields.  39  Chapter III. Computer Implementation and Discussion  General computer programs, based on the two moment methods described above, were written and used to solve the inside E-field intensity and scattering crosssection for arbitrary dielectric distributions objects. A set of numerical simulations to test the analysis methods and programs are performed on a Sun Sparc station. At first, the validity of the computational procedure was verified by making calculations, using both the MM, of internal E-fields and scattered fields for a simple dielectric layer and comparing the results with those which have been published. The first example shows the equivalence of the two MM. After this test, the validity check was extended to lossless and lossy spherical shapes which will be useful in the calculation of our melting-snow particle model later.  3.1 Selected Numerical Results 3.1.1. Numerical Results for a Thin Dielectric Layer In order to compare our results with those given in [30], we consider a thin homogeneous lossy dielectric layer illuminated by a 300 MHz plane wave travelling in the positive Z direction and the E-field perpendicular to the plane of the layer, as shown in Fig. 3.1 where Er= j The layer dimension is 5 O 7 (Er 9 . 59 O c=lmho/m ). 7 3 . In the calculation, the layer is cut into 96 subcells of O.5x0.5x0.5 cm 3 O.5x4x6 cm each. The layer structure geometry is then defined for the computer model in terms  40  E  z  )9  Fig. 3.1 A dielectric layer illuminated by a plane EM wave[30]  of the subcell centre in Cartesian coordinates. For identification, each subcell is numbered 1, 2,  ...,  96 from the upper right to the lower left. Fig.3.2 (a) shows the  magnitude of the calculated E-fleld component E inside the layer (E=E=O) when the MM is used. The E-field values of the calculation and the results given by [30] are shown in Table I. It can be seen that the two results are almost the same. Since the layer is symmetric, only a quarter of the results are shown. In Fig. 3.2 (b), the scattering cross-section results of the layer via 0 (0 =00 =  00,  are shown where 0  =  180°), in the vertical plane  0° is the forward scattering direction and 0  =  1800 is  the backward scattering direction. Although the iterative MM was developed primarily for large-body problems, it is easier, as the first step, to demonstrate the method and its accuracy for small and  41  0.01 em  0.04 0.01 0  z  Sc.LtLriD5 Cr..*  .0.02  (rPI2) z 1O  U—  —  I  I  5.50  LOC 7.50 7.00 6.50 Lw Lw 5.00 Lw 4.00 Lw Lw 1.50 1.00 Lw Lw Lw 0.00 0.00  50.00  100.00  15000  Fig. 3.2 (a) The inside E-elds of the dielectric layer in Fig. 3.1 (b) The scattering cross-section of the layer 42  Table I. The Inside E-flelds of the Dielectric Layer cell No.  results in [30]  our results  cell No.  results in [30]  our results  1  0.0210  0.02105  2  0.0155  0.01549  3  0.0160  0.01604  4  0.0158  0.01581  5  0.0158  0.01579  6  0.0158  0.01577  13  0.0155  0.01549  14  0.0108  0.01077  15  0.0112  0.01120  16  0.0110  0.01101  17  0.0110  0.01098  18  0.0110  0.01096  25  0.0161  0.01607  26  0.0112  0.01122  27  0.0116  0.01164  28  0.0114  0.01143  29  0.0114  0.01140  30  0.0114  0.01138  37  0.0159  0.01587  38  0.0110  0.01105  39  0.0115  0.01146  40  0.0112  0.01125  41  0.0112  0.01122  42  0.0112  0.01120  simple objects. The iterative MM is also used to solve the scattered and internal E fields of the layer shown in Fig 3.1. When ERRN<0.001, after 18 iterations, we get the same scattering cross-sections as that got from MM. However the E-flelds obtained are a little bit different from those obtained by using the conventional MM. After three more iterations, the same E-fields as the MM solutions are obtained. As discussed in Chapter II, the internal E-field is more sensitive than the scattered field. That tells us if the results of internal field are required, more iterations are needed. The second example shows the induced E-field inside the same layer as shown in Fig. 3.1 when the incidence wave is 60° from the layer plane surface. The two  43  moment methods also get the same solutions. The B-fields obtained by the 1MM are shown in Fig. 3.3(a)  —  3.3(c) and the scattering cross sections are shown in Fig.  3.3.(d). As expected, when the plane of the layer is not perpendicular to the incident E-field, all the components of the E-field--E, E and  are induced. This example  proves that the intensity of the induced B-field inside a dielectric body depends heavily on the body’s orientation with respect to the incident wave.  3.1.2. Numerical Results for Lossless Dielectric Spheres For three dimensional dielectric bodies, very few scattering problems can be solved analytically. Since the scattering of homogenous spheres and layered, inhomogeneous spheres can be determinated analytically, those structures will be considered as examples in current and later sections. The bistatic scattering cross section of a dielectric sphere with 4 = and ka=1 (a is the radius of the sphere) illuminated by a 10 GHz plane wave is calculated by using the 1MM. The number of unknowns is 80x3. The results for two polarization cases, the vertical polarization (VP) and the horizontal polarization (HP), are obtained as shown in Fig.3.4(a). The results agree well with those obtained by using Mie scattering theory. The same calculation is done for a real snow core sphere Er = 1.2, ka =0.17, and illuminated by a 10 GHz plane wave. The results are shown in Fig 3.4(b) and also agree well with Mie scattering solutions as shown in Fig.3.4(b).  44  0.04  0.020.02  0.04  0.01 0 -0.02 y  .0.02  .0.04  2  Fig. 3.3. (a) The inside E-fields of the layer (O=6O°) Ex/Ei (b) The inside E-fields of the layer (O=6O0) Ey/Ei  45  0.01  0.02  0  0  •0.01  -0.02  z  y  &attering Crau-.ectcc (rP1*m2)  x 1O  (d) The scattering cross-section of the layer  46  Scattering Cross-section (cnY’2) 1.50—I 1.40  —  (ie-HP -  —  ie-VP  1301.20-  —  110  —  —  k  1.000.90  -  —  “.  —  0.80-  EE 0.40  —  -  \  N -  0.30-  4.  41  —  4. 4.  0.20  /  —  0.100.00  /  .‘  —  ...‘.  Ni.. -..--  -  0.00  —  50.00  100.00  —  Theta(degree) 150.00  Fig. 3.4 (a) The scattering cross-section of lossless dielectric sphere , ka=1, f=1OGHz 4 6r  47  Scattering Cross-section (cm’2) x iO 500.00 ET —  L4...  450.00  m—  —*—.—.#  —  ‘IMM:Rp  -  TMMP 400.00  350.00  300.00  250.00  200.00  150.00  100.00  50.00 0.00  -  -  -  -  -  -  -  -  -  0.00  50.00  J. 100.00  Theta(degree) 150.00  Fig. 3.4 (b) The scattering cross-section of lossless dielectric sphere r’  ka=O.17, f=2OGHz  48  3.1.3. Numerical Results for Lossv Dielectric Soheres  Fig. 3.5 shows the bistatic scattering cross-section of a rain drop under 10 GHz incident plane wave. Its radius is 0.1467 cm and  41.775-j41.204. The result agrees  r 6  well with those of Mie scattering solution as shown in Fig 3.5 too. For larger dielectric lossy spheres, the iterative MM also gets good results compared with Mie scattering theory. Fig. 3.6(a) shows the bistatic scattering cross as above and ka=3.5.  section of a dielectric lossy sphere with the same  When the size of the sphere becomes bigger, the unknowns increase, therefore the computing time increases quickly. We calculated spheres at ka=3.8 and ka=4.0 respectively. The results are shown in Fig. 3.6(b)  —  3.6(c). When ka=3.5, unknowns  are 1188x3, CPU time is about 95 hours. When ka=3.8 and 4.0, the unknowns are 1790x3 and 2175x3 respectively, while the CPU time are about 185 and 294 hours respectively. From those calculations we can see that the bigger the size is, the more unknowns there are, the more error will occur, ie. the round-off errors increase when the number of unknowns increase as in all of the numerical methods. Fortunately, in the calculations later, we do not need to handle so many unknowns, so we can get enough precision. The calculations here show that using the iterative MM, we can solve problems with many unknowns. Of course here we only show the validity of our method, if we only want to get scattered fields from spheres, we do not have to use this method here, instead, we can use other methods such as Mie scattering method or use the MM in surface modelling which can solve larger size spheres than this method when the same number of unknowns are involved.  49  Scattering Cross-section (cm’2) x 10-6 40.00I 3800  -  I  \-  36.00 —  34.00 32.00 30.00  ——a-.-,.  -  irip  -  -  28.00I,  26 00  24.oo  I’  -  -  ‘I  22.00  1’  -  Is  20.00  -  I  18.00 16.00 14.00 12.00  -  —  -  \  -  I /  10.00 -  /  8.00-  0.00  50.00  100.00  I 150.00  Fig. 3.5 The scattering cross-section of a lossy dielectric sphere  Er  41.775j41.204, ka = 0.154, f= 100Hz  50  Theta  Scattering Cross-section (cm”2) 170.00 160.00 150.00 TMM-VP 140.00 130.00  “I”  120.00 110.00 100.00 90.00 80.00  k  70.00 60.00 50.00 40.00 30.00 20.00 10.00  ‘1—--•• 0.00  I 0.00  HP  I 50.00  100.00  I 150.00  Fig.3.6 (a) The scattering cross-sections of a lossy dielectric sphere c r41775J41204 f= 10GHz ka=3.5  51  Theta(degree)  Scattering Cro.e-section (cm’2) I  240.00 220.00 200.00  Mle-Vt’  MM:Vp -  -  180.00  -  160.00  -  140.00 120.00  -  100.00  -  80.00  -  60.00  4O.OOL 20.00  00 . 0 L -  0.00  50.00  100.00  Theta(degree)  150.00  Scattering Croas-section (cm2) I  24O.00  I  Mle-W  UP  220.00 200.00  —  180.00 F 160.00’ 140.00 120.00 100.00 80.00 60.00 40.00 20.000.00  Thet.a(degree)  —  100.00  50.00  0.00  150.00  Fig.3.6 (b) The scattering cross-sections of a lossy dielectric sphere  €=  41.775-j41.204 f= 100Hz ka= 3.8  52  Scattering Cross.secüon (czn2) -  280.00  Mie-VP  -  -  260.00 240.00 220.00  -  200.00 180.00 160.00 140.00 120.00  -  -  -  100.00 80.006000 40.0020.00 0.00 r Theta(degree)  0.00  50.00  100.00  150.00  I  I  Scattering Crees.sectioD (cn2) I  I  280.00  1 -  260.00 240.00 220.00 200.00 180.00  160.00 140.00 120.00 100.00 80.00 60.00 40.00 20.00 0.00 Theta(degree) 0.00  100.00  50.00  150.00  Fig.3.6 (c) The scattering cross-sections of a lossy dielectric sphere 2 704 • 41 r= 4 J 75 1 f= 100Hz ka= 4  53  3.1.4. Numerical Results for Concentric Dielectric Spheres From Mie scattering we can get backward and forward scattering crosssections of the concentric melting-snow particle model A. As shown in Fig.1.1, the inside of model A is a snow core and outer layer is a water shell. We used the two MM to calculate the scattering of a concentric melting-snow particle model A, the outer radius is 0.061 cm and the inner radius is 0.041 cm. The  two MM get the same solution. Fig. 3.7(a) and Fig. 3.7(b) show the bistatic crosssection of the particle when frequencies are 10 GHz and 20 GHz respectively. Although the exact values of scattering cross-section for scattering angles between 00 .1800 were not available, from Mie scattering theory we know that the forward 2 and the backward scattering cross-section scattering cross-section is 0.120755E-04 cm 2 at 10GHz (r is 0.110149E-04 cm  41.775-j41.204), our solutions are 0.12378E-04  2 respectively; while at 20 GHz( 6 2 and 0.110855E-04 cm cm r= 8.867-j30.75), the Mie scattering solution in forward and backward scattering cross-sections are 0.202278E-03 . 2 2 and 0.17863E-03 cm , our results are 0.2098E-03 cm 2 2 and 0.1718507E-03 cm cm In the two important directions, the iterative MM solutions meet Mie scattering theory solutions well. For the concentric sphere model, since the outer layer rain shell is much thinner, we cut a concentric sphere into a set of nonuniform subcells: at the outer layer water shell, the subcells are smaller than those inside.  54  Scattering Cross-section (cm”2) x io-6 I  I  0.00  50.00  ft  12.00 11.00 10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 Theta (degree) 100.00  150.00  Fig. 3.7 (a) The scattering cross-sections of concentric melting-snow particle model f= 10GHz  55  Scattering Cross-section (cm’2) x 106 220.00 210.00 200.00 190.00 180.00 170.00 160:00 150.00 140.00 130.00  I  I  100.00  150.00  H?  -  -  -  -  -  -  -  -  -  120.00 110.00 100.00  -  -  90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00  -  —  —  —  —  —  —  —  —  -10.00  Theta(degree) 0.00  50.00  Fig. 3.7 (b) The scattering cross-sections of concentric melting-snow particle model f=2OGHz  56  3.2 Computation Considerations In order to get accurate results in dielectric scattering calculations, we have some special considerations as follows:  3.2.1. Modelling Considerations The first step in numerical calculations is to mesh the dielectric body into a set of small subcells so as to accurately represent the physical body. The meshing is related to the basis functions and the object modeffing, and will affect the complexity and accuracy of the later calculation. Normally, as discussed in Section 2.2, for objects of rotation or cylinders, small rings are cut, while for cube cylinder block, small cubes should be cut. For arbitrarily shapes, triangle patches are chosen as subcells for surface modelling and cubes or tetrahedral elements are chosen for volume modelling. The cube elements are the simplest elements in MM calculation, and one of its primaiy advantages is the ability to model arbitrary shaped inhomogeneous objects  .  Our melting-snow particle model to be evaluated is inhomogeneous and  irregular, we use small cubes as subcells in all of our calculations. For spheres, meshing is even more important. In our program, a sphere is meshed automatically: given the radius a of the sphere and the centre coordinate  0;  at first, a cube with the length D  =  2a, and the centre at o is set up, then it is cut  3 subcubes with length d into N  =  D/N, in the modelling each subcell centre is  checked by the program, if the distance rj (from the centre of the subcube to o) is smaller or equals to a (eg. point A in Fig. 3.8), the subcell is in the sphere; if r is  57  bigger than a (eg. point B in Fig. 3.8); the subcube is out of the sphere. The more subcells are cut, the more accuratç the modeffing might be.  D=2a  V  Fig. 3.8 Meshing of a sphere  3.2.2. The Choice of Subcell Size When using a pulse function expression in MM, it is important to establish an upper limit on the dimension of the subvolumes. In theory, for a scatterer, the smaller the subcell size is, the more accurate the calculation will be. On the other hand, when the subcell size is getting smaller, more unknowns are needed, and bigger computer memory and longer computer time are needed. The bigger calculation will also raise the computer round-off error. So there is a trade-off for the size of subcell and the computer time. To arrive at the limit of subvolume size for our methods, we have performed several tests. Our results shows that the upper limit of subcell size depends on the method used, the permittivity and shape of the dielectric object, and the accuracy required.  58  If we want to get the scattering cross-section instead of internal field, we can have bigger upper limit. According to [30], for a block or a cube, the length of a subcell smaller than )/4 is accurate enough and it did not mention the relationship with permittivity. But our dielectric bodies are spheres and have curve surfaces which are more sensitive to the size of subcell, now we have not found publications discussing about the subcell size limitation for spheres. After several calculations, we get the conclusion that for our objects and the range of permittivities, when a spherical shape dielectric object is cut into small cubes, the small cube size should not exceed 0.08  in order to yield  reliable data.  3.2.3. The Choice of an Initial Guess in the Iterative MM In Chapter II we have developed some iterative schemes to arrive at the iterative MM solution. The only freedom we have is the choice of an initial guess. In principle, the initial guess  can be chosen arbitrarily, eg.,  =  0, 1,  because of its convergence property. But in practice a reasonable guess considered to be close to the correct result is made, which will save the calculation time in certain extent. Sometimes a pre-knowledge is needed. For example, in the calculation in 3.1.1, when E° equals to 0 and E /2 respectively, the CPU time is 128.768 and 1 146.800 seconds respectively, j(O) is obtained from the volume equivalence principle in Eq.(2.1).  59  3.3. Discussion 3.3.1. The Convergence of the Iterative MM In the iterative MM, our choice of several parameters and functions guarantees that the method will converge in a finite number of steps. The solution 1 converges to the exact solution X in a finite number of steps. The error is reduced X at each step. For example, in our calculation in Fig.3.2, at the 1st iteration, the normalized integrated square error ERRN is 8.30826E+06, and that for the 2nd iteration is 7.958E+05. As shown in Table II, at the 18th iteration, the error comes to 1.628E-02 and after 22nd iterations, the error is as low as 9.0186E-04. The general consensus in one- and two-dimensional problems is that N/6 iterations are usually adequate, where N is the number of unknowns or basis functions. In all of our three-dimensional calculations, we have found that N/6 are usually sufficient and that the rapidity of convergence depends on the geometry of the object, the polarizations, the angle of incidence field and the initial guess.  3.3.2. Comparison of the Two Moment Methods From the analysis in Chapter II, we know that the two MM are identical for certain choice of basis functions and the weighting functions. However each method has its own advantages and disadvantages. One of the advantages of the conventional MM is that once the matrix elements Z in Eq.(2.21) are computed, they can be stored and reused again for different excitations. The iterative MM does not have this advantage of a stored and  60  Table II. The Error in Each Iteration Iteration No.  ERRN  Iteration No.  ERRN  1  8.3083E+06  2  7.9581E+05  3  1.7123E+05  4  3.8529E+04  5  1.1796E+04  6  2.8775E+03  7  3.6454E+02  8  9.5679E+01  9  3.8084E+01  10  1.7555E+01  11  1.0386E+01  12  3.3566E+00  13  1.4770E+00  14  5.8008E-01  15  2.0035E-01  16  8.0714E-02  17  4.8032E-02  18  2.1182E-02  19  6.4703E-03  20  3.8127E-03  21  1.6793E-03  22  9.0186E-04  reusable matrix, in each iteration every element has to be calculated individually. It will take much longer computer time. For example, in our first calculation in 3.1.1 and Fig 3.2, the computing time for the conventional MM is 111.500 seconds but for the iterative MM the computing time is 175.420 seconds, 1.5 times more than the former. The Iterative MM requires much less computer storage than the MM because it is not involved in a large matrix as in the MM, for example, if we have N  2 space while the 1MM needs unknowns, the computer storage for the MM needs N only about 5N space[34]. Therefore the iterative MM can handle more volume cells, or, the unknowns, an order of magnitude larger in number than the MM. As discussed in [26], on a CDC Cyber 855, the direct MM can handle up to 80 cells, or 240 basis functions, while the iterative MM can handle 3666 cells or 11000 basis 61  functions. For the MM, even after all the terms have been computed to yield the solution, one can not decide the accuracy of the solution; while for the 1MM, the error  ( or ERRN)  at each step of the solution is known and after each iteration the  quality of the solution is known, and an incremental increase in the computational time will provide a better result.  3.3.3. The Volume Integral Equation and the Surface Integral Equation For solving EM scattering problems, either the volume integral equation (VIE) or the surface integral equation (SIE)[37][39] can be applied. As in Chapter II, the VIE is principally based on relating the induced polarization current to the corresponding total fields consisting of the scattered and incident fields. By associating an unknown polarization coefficient either with a cubic or with a tetrahedral cell inside the scatterer, the operator form of the integral equation is converted into an equivalent matrix equation in the MM. It is suitable for arbitrary shape and inhomogeneous scatterers. Normally, in the MM it is more difficult to get convergent results using VIE than using STE. The STE approach works well in analyzing homogenous dielectric objects or objects made up of homogenous layers. The usual procedure of this method is to set up coupled equations in terms of equivalent electric and magnetic currents on the surfaces of the homogeneous regions. When the surface of the scatterer takes on arbitrary shape, modelling the surface geometry becomes  62  complicated. The SIE needs less unknowns than the VIE in modelling the scatterers in most cases. For conceptual simplicity we consider a homogenous square cylinder and a cube as shown in Fig.3.9[41]. If each side of the square cylinder in Fig 3.10 is divided into N segments, then the SIB method requires two unknowns  ( one for the  electric current and the other one for the magnetic current) for each segment and results in a total 8N unknowns. The VIE method, however, requires one unknown for each subsquare resulting in a total N 2 unknowns. For the cube in Fig 3.10, the SIB method requires four unknowns  (  two for the electric currents and two for the  magnetic currents) for each subsquare on each face, resulting in 4N 2 unknowns for each face thus 24N 2 total unknowns. The ViE method, however, requires three unknowns for each subcube resulting in 3N 3 total unknowns.  SiE  8 N UNKNOWNS  VIE N  UNKNOWNS  SIE  24 N UNKNOWNS  VIE  3 UNKNOWNS 3 N  Fig.3.9 Comparison of the number of unknowns between SIB and VIE  63  It is also necessary to mention that for dielectric bodies with very low permittivities, the STE will lead to worse results[41], while calculation using the VIE method as shown in this chapter yields good results.  3.4. Conclusions This chapter has been devoted to some illustrative numerical results obtained by the MM and the 1MM discussed in the last chapter. The results obtained have good agreement with published or known results, which substantiates the validity of the methods and the reliability of the programs. In most of the examples, the same permittivities of snow or water (rain) and practical sizes are used, which will give us the confidence in the evaluations in the next chapter. The chapter has compared the two moment methods as well as the two integral equations which are often used in solving dielectric scattering problems. From the analysis we have known that the iterative MM needs less computer storage therefore it can handle more unknowns than the conventional MM, but it does not have a reusable matrix so it spends longer computer time than the MM. In the iterative MM, the quality of the solution is known and controllable. Although the  1MM needs longer computer time, we can save certain computing time by a reasonable choice of the initial guess after some simulations are performed. Concerning the two integral equations which are often used in solving dielectric scattering problems, the VIE can handle any inhomogeneous and irregular scatterers with any permittivities, but it has to solve more unknowns than the SIE.  64  The SIB needs fewer unknowns than the VIE for a same size scatterer, but it is not quite effective for solving irregular shape and lower permittivity dielectrics[41]. Now we can get the conclusion that using the iterative MM and volume modelling is a good method for solving large, arbitrary and inhomogeneous dielectric problems. In the next chapter, the evaluation of melting-snow particles model will involve lower permittivities and irregularly shaped dielectrics. The analyses in this chapter shows our choice of usin2 iterative MM to solve VIE is reasonable.  65  Chapter IV. Evaluation of Scattering from Melting-snow Particle Model  In this chapter, the newly developed melting-snow particle model (model C) is evaluated. The scattering cross-sections at different frequencies, different rain rates and at low melting factor are obtained by using the iterative MM. The results are compared with those of model A and B. From the calculation we have found that different models give significantly different values because of different assumptions.  4.1 Introduction 4.1.1. The Melting-snow Particle Models[6][42]--[44] Various prediction models used to evaluate the scattering of melting-snow particles have been developed. The different models have different assumptions based on different emphasis, therefore they give different scattering values. In order to look into the scattering of melting-snow particles, some simplified geometric shapes which can be reasonably analyzed are considered. As mentioned in Chapter I, two main models are used to represent melting-snow particles. Model A is a concentric-sphere model, where a layer of water surrounds a dry snow core of density, and model B is a homogeneous sphere model, where water is assumed to progressively percolate inside the snow as the particle falls, until saturation is reached. The average permittivity of the melting-snow particle model B is discussed  66  in [2]. Some other models, for example concentrated ellipse model[43], are assumed too. Based on the discussion above, in this chapter, a new model to represent the melting-snow particles is considered. The model is a snow core of permittivity several even distributions of water drops of permittivity  €  with  at outside. From  Chapter.I we have enough evidence to show that the model C provides a reasonable assumption.  4.1.2. Basic Parameters Melting occurs basically because of the higher temperature of the air which surrounds the melting particles. The rate of melting also depends on the humidity of the air. The melting process, melting rate, and the average permittivity of the melting particle in model B are beyond the scope of the thesis, we only simply give several parameters that will be used in our later calculations.  4.1.2.1. Melting Factor S[4][5] We define a quantity S as the ratio of melted volume of water to the total volume of the melting snow particles. For model A  S_1_(_!) 3 2 a  67  (4.1)  2 is the outside radius of the particle. 1 is the radius of the unmelted core, a where a In the melting layer, S is a function of a depth h into the melting layer, where h is the position in the melting layer measured from the top of the layer. At the top of the melting layer (h=O), S=O, dry snow particles present; at the bottom (h=1), /a decrease with the melting-snow 1 increases,a S=1, rain drops present. As S 2 particles fall through the melting layer.  4.1.2.2. Particle Size All the sizes of particles to be evaluated are available in some related previous work. Here we only give a formula of the radius of the representative melting snow sphere given by [5]: gj P V 1  (4.2)  ä-[(O.1+O.9S)E  where p is the factor related to rain drop with radius a, rain drop of radius a, and  Vmj  VRi  is the fall velocity of  is the fall velocity of corresponding spheres with a  degree of melting S and  Q,— 1.5)sinq 1 Vmi 1.5+(v  (4.3)  4.1.2.3. The Permittivity of Rain or Snow The dielectric property of rain depends not oniy on frequency but also on temperature. For wavelengths longer than 1 mm, the dielectric property of water is 68  due to the polar nature of the water molecule, whereas for wavelength shorter than 1 mm, it is governed by various kinds of resonance absorptions in the molecule. The value of c used in this thesis are based on the empirical model developed by Ray[1] for the complex refractive index of water. Fig 4.1 shows the frequency characteristics of the complex permittivity of water at 00 C. The relative permittivity of snow is 1.2 and the density of it is 0.1. 20.00.1  1—  I  I  66.00  -  80.00•  -  75.00  -  10.00  -  66.00-  eo.oo• •  66.00-  c’  -  -  50.0045.0040.00  -  66.00-  25.00  -  20.00 15.00  -  —  -  10.00  -  6.00  I  0.00  20.00  40.00  10.00  6000  100.00  Fig. 4.1 Relative permittivity of water vs. frequencies  4.1.2.4. The Rain Rate The intensity of rainfall is measured in terms of a rain rate R. If rain consists of spherical drops of uniform radius a, falling with velocity v, the rain rate R is  R-itavN  69  (4.4)  where N is the number of drops per unit volume. When R is measured in mm/h, a , 3 in cm, v in m/s, and N in cm N* io R48ita v 3  (4.5)  4.2 The Computed Results for the Melting-snow Particle Model C Using the theory and computer program discussed above, we calculated the scattering cross-sections of the melting-snow particle model C at three rain rates, six frequencies and seven melting factors.  4.2.1. The Choice of the Number of Outer Layer Water Drops The model C is derived from the model A (the concentric model), in order to compare with model A, it is not unreasonable to assume that the total volume of the water drops at outside in model C equals to the outer layer water shell in model A in the later calculations. Assume the surrounding air is at the same temperature, the water drops should be evenly distributed and of equal size. From the physical point of view, the water drops could not be too big because of the surface pressure, otherwise they can not stay at the surface of the snow core. So the model C is only valid when S is smaller, ie. the outer layer rain drops are not too big. For large S, this model does not conform to physical principle. How many water drops should be chosen in model C? What is a reasonable number? Our program can arrange the outer layer water drops with even 70  distributions on the surface of a snow core. As an experiment, we calculated the forward scattering cross-sections of model C at S frequencies f  =  =  0.04, rate R  =  5mm/h,  10 GHz and 20 GHz, the outside drops number are 6, 12, 24 and 48  respectively. As compared with those of model A, at the two frequencies, the scattered fields of model C are always smaller than those of model A, the smaller the number of chosen outer layer water drops is, the smaller the scattered fields are. The more water drops are chosen, the closer to those of model A the scattered fields will be. It is believed that if we choose even more water drops at outside which will be closer to model A physically, the result should be closer to that of model A, when the outer layer is full of water drops, the model C comes to model A. Considering the computer time and memory, we chose 24 as the number of outside water drops in later calculations.  4.2.2. The Computed Results for Melting-snow Particle Model C The bistatic scattering cross-sections of model C are calculated at S 0.04, 0.06, 0.08, 0.10, 0.12 and 0.14, the frequencies are at f  =  =  0.02,  10, 20, 30, 40, 50, 60  GHz, and the rain rate at R=5, 12.5 and 25 mm/h. The selected results of forward and backward scattering cross sections vs frequencies at different S are shown in Fig. 4.2  Fig. 4.8.  71  Scattering Cross-section (R=5, S=O.02) Ln (Scattering Cross-section (cm”2)) I  2  —  -  1.5  Model A, (forward)  ‘M1?rd -  le-03  -  ••iY•  7  —  5  •W  ,‘  Mo&1B (r dT c (Eakar  —  _—:  • /  32  F,  —  ——  —,  1.5  ,  _  —-  ..•.  —  _  •/  / ,,— ‘‘I  • le-04 —.  -  /  .,  7  _  /  5  —  / /  3 2 1.5 I  le-05  7 5  I  /  3  I  Frequency (GHz) 10.00  20.00  Fig. 4.2. Scattering  30.00  50.00  40.00  cross-sections of  60.00  three models (R=5inmlh, S=O.02)  72  Scattering Cross-section (R=5, S=O.08) Ln (Scattering Cross-section 7—I  (cm”2))  I  I—  odefA, (fthdY  5  -  3-  MO&1B (SakvrJ) MdTc akar -  2 1.5  0  le-03  7 5-  32 _/  1.5 le-04 7  5— 3 2 1.5  —  10.00  20.00  30.00  Frequency (GHz) 40.00  50.00  60.00  Fig. 4.3. Scattering cross-sections of three models (R=5mm/h, S=O.08)  73  Scattering Cross-section (R=25, S=O.14) Ln (Scattering Cross-section (cm”2)) I  I  I  I  I  Model A, (forward)  3  - - -  Mfa  2  MO&1B (S1ra)  1.5  -  -  (a&Var le-02 7 -  5 — __-‘I—  3 —  2  —  —  .—  — *%  —  %%  %  ‘I  1.5 le-03 7 5 3 2 1.5 le-04 7 5 10.00  20.00  30.00  40.00  I  I  50.00  60.00  —  Frequency (GHz)  Fig. 4.4. Scattering cross-sections of three models (R=Smm/b, S=O.14)  74  Scattering Cross-section (R=12.5, 5=0.04) Ln (Scattering Cross-section (cm”2)) 7  ModeI  5 -  3—  oceB (avrJ) ‘MdT akar -  2 1.5 le-03  -  —  ——4—  7— —  a.  5-  _  — — ‘K  3-  /  2 1.5 le-04 /  7  I /  5  I I /  3  / / /  2  /  1.5 —I 10.00  I  I  I  I  I  20.00  30.00  40.00  50.00  60.00  Frequency (GHz)  Fig. 4.5. Scattering cross-sections of three models (R=12.Smm/h, S=O.04)  75  .  Scattering Cross-section (R=12.5, S=O.12) Lii (Scattering Cross-section (cm’2)) 2 Model A, (forward) ‘M&Ji ri1  1.5 le-02  - -.  fA,airaj ‘MO&IB (backwrJ)  7  -  -  ‘Model ( bakar  5 3-  _.___•  2 -  1.5 le-03  -  7 5 3 2 1.5 le-04 7 5 3  —  I  10.00  Frequency (GHz) 20.00  30.00  40.00  50.00  60.00  Fig. 4.6. Scattering cross-sections of three models (R=12.5mm/h, S=O.12)  76  Scattering Cross-section (R=25, S=O.04) Ln (Scattering Cross-section (cm”2))  le-02  Model A, (forward) Mocel C, (forward)  -  7-  - - -  5—  MO&1B  M&I akarr 32 1.5 .4 —  -  le-03  -  —4 — —  7— 5— /  3— 2  1.5 le-04 I  75  _r  3 -d  / I  /  Frequency (GHz) 10.00  20.00  30.00  40.00  50.00  60.00  Fig. 4.7. Scattering cross-sections of three models (R=25rnm!h, S=O.04)  77  Scattering Cross-section (R=25, 5=0.12)  Ln (Scattering Cross-section (cm’2)) I  I  3  Model A, (forward) M1rd  2  -  1.5  MO&1B (Eara) !vIdT (akarj  le-02 7 5 ———  —  3-  a.  — ‘—  . %.  2  .  -..  1.5  •  /  4%  /,  le-03 7 5 3 2 1.5 le-04 7 -1 5 10.00  I  I  20.00  30.00  40.00  I 50.00  60.00  Frequency (GHz)  Fig. 4.8. Scattering cross-sections of three models (R=25mm/h, S=O.12)  78  4.3 The Comparison of the Model A, Model B and Model C Now we have obtained the forward and backward scattering cross -sections of model C. We also have known the forward and backward scattering cross-sections of model A as well as the backward scattering cross-sections of model B by using Mie scattering theory. Fig 4.2  —  Fig 4.8, show the forward scattering cross-sections of  model A and backward scattering cross-sections of model A and model B at different melting factor S and frequencies. As may be seen from these figures, differences in specific forward and backward scattering cross-section values as well as the ratio of the scattering cross-sections in two directions occur by applying the different models considered above. Fig. 4.9 and Fig. 4.10 show the scattering cross-sections of model A and model C vs. S at R=5 mm/h, f=10 and 40 GHz respectively. Fig. 4.11 and Fig. 12 show the scattering cross-sections of model A and model C vs. R at S =0.06, f= 10 and 40 GHz respectively. Comparing with the three models, we can get some conclusions in several aspects:  4.3.1. Forward Scattered Fields Compared with model A, we observed that at the same S, the same rain rate R and the same frequency f, the forward scattering cross-sections of model C are always smaller than those of model A. The difference increases with S increasing. For example, when R  =  12.5 mm/h, at S  =  0.06 and f  79  =  20 GHz, the forward scattering  Scattering Cross-section (R=5, f=1OGHz) Ln (Scattering Cross-section (cmA2)) le-04  A, (forward)  •M&di  8  -  A  1W - ‘i(o&favr  6 5-  4 3.5 3 —  •.D  -  2-  .  —  .•‘  —  ...._ —  1.5  —  I,  1e-o:  -  Ii  —  t 7  -.  6b 0.00  50.00  100.00  Fig. 4.9 Scattering cross-sections vs. S (R=5 mm/h, f=1O GHz)  80  3 Sx10  Scattering Cross-section (R=5, f=4OGHz) Ln (Scattering Cross-section (cm’2))  3  ‘MoJi C 1 (forward)  2.5  - - -  2 .4  MO&1C Skvr)  -4-4.--  1.5  —x  4  le-03 p.  A—  8 —  F  —  .4  6 5  I / I  4  /  1/ I, 4.  I I,  3  I, ‘I  2.5  /  I, I, It,  2  I  1.5  le-04 0.00  50.00  100.00  Fig. 4.10 Scattering cross-sections vs. S (R=5 mm/h, f=40 GHz)  81  S x  -  -  Scattering Cross-section (f=1OGHz, S=.06) Ln (Scattering Cross-section (cm”2)) I  I  I  Model A, (forward)  le-04  - -  9 8-  od1C,(backwa,rJ5  /  -  —  7—  I 0  6-  —  I, I ‘I  /  /  I—  5  / ,,  4.5  /  _  _  /  ,  —.  I  4  ——  I I  /  —— / _  I /  3.5  I  i  I  I I  3-  I I I I  2.5  I?’  ‘I  -  I I  0/  ‘I it  2-  /1 I,  /, ‘II 0/  1.5 I  0.00  -  5.00  I  I  15.00  10.00  I  I  2000  25.00  Fig. 4.11 Scattering cross-sections vs. R (S=O.06, f=1O GHz)  82  R(mm/h)  Scattering Cross-section (f=4OGHz, S=.06) Ln (Scattering Cross-section (cmA2)) ModeI A, (forward) Model C, (forward)  5  3-  -  21.5  --  v a 0 a  4 -  -  1akaN3  -  0  -  -  a  -  le-03  -  S.,  — — —  7-  —  —  —— -—  —  —  _  5 4 3— 21.5  —  le-04 I  I  I  0.00  5.00  10.00  R(mmlh) 15.00  20.00  25.00  Fig. 4.12 Scattering cross-sections vs. R (S=O.06, f=40 GHz)  83  cross-section of model C is about 0.75 of model A, when S  =  0.08 and f  =  20 0Hz,  it becomes about 0.73 of model A. At most frequencies, the difference increases with frequency F increasing. eg., at S  =  0.06 and f  =  30 0Hz, the forward scattering field  of model C is only about 0.71 of model A while at f  =  20 GHz it is 0.75.  From electromagnetic scattering theory we know that the forward scattered fields of a lossy dielectric body depend on the size, dielectric permittivity and dielectric surface area. The particles of model C and model A have the same snow core, the difference is mainly because of the outer layer lossy dielectric distributions. In model C, the outside lossy dielectric drops have bigger surface area than the water shell in model A which could make the scattered field in forward direction becomes With the increasing of S, the volume of water increases, the effect of  smaller.  dielectric would be more significant, the difference in forward scattered fields could become larger. When frequency increases, the electric size becomes large, the effect of dielectric could become more significant.  4.3.2. Backward Scattered Fields In backward scattering, the results are very consistent. The ratio of backward scattering of model C and model A increases with S decreasing and with f increasing. For example, when R  =  25 mm/h and f  =  40 0Hz, at S  scattering of model C is 0.94 of that of model A, while at S when S  =  0.04 and f  =  =  0.02, the backward 0.04 it equals to 0.92;  30 0Hz, the backward scattering of model C is 0.88 of that  of model A.  84  It is noted that at lower frequencies, the backward scattering cross-sections of model C are smaller than those of model A, but compared with forward scattering cross-sections, at the same S and same f, the difference of backward scattering crosssections between model A and model C is smaller than that of forward scattering cross-sections, ie, the backward scattering cross-sections of model A and model C are closer to each other than forward scattering cross-sections. That shows after splitting the outer water shell (in model A) into small drops (in model C), the change of scattering fields occurs. But in backward direction, both the surface penetration and reflection of dielectric exist, so the backward scattering of model C may not affect the scattering so much as in forward direction. It should be also noticed that at higher frequencies, the scattered fields of the two models are comparable even at lower S the backscattering of the model C is bigger than that of model A. In these cases, the higher the f is, the bigger the difference is; the lower the S is, the bigger the difference will be. For example, when R  =  5 mm/h and f  =  60 GHz, at S  =  0.02, the  backward scattering cross-section of model C is 1.29 of that of model A, while at S=0.04 it equals to 1.09; when S  =  0.04 and f  =  50 GHz, the backward scattering  cross-section of model C is 1.06 of that of model A. Because at higher frequencies (ie, for larger electric sizes), the effect of reflection of dielectric may play more important role than penetration, more reflection is generated, the backscattering would become higher. Compared with model B, at certain lower S and lower frequencies, the backscattering of model C are higher than that of model B, while at higher S and  85  high frequencies, the backscattering of model B are higher than those of model C and model A. These results show the scattered fields will change from different models.  4.3.3. The Ratios of Forward to Backward Scattering Cross-sections The forward and backward scattering  cross-sections are the two most  important scattering directions in evaluating the scattering of scatterers because it is easier to receive and measure for a radar in the two directions. The ratio of the forward to backward scattered fields is also an important parameter to evaluate the dielectric scattering properties. We should always consider the scattering in both directions. Fig.4.12(a) =  25 mm/h and S  =  0.02  —  Fig.4.12(d) only show the rates at R  0.08 respectively. From the results we have found the  ratios of forward and backward scattering cross-sections in model C are closer to 1 than those of model A, ie, at the same S and f, the scattered field curve for model C is more “smooth” than that of model A.  4.3.4. Rain Rate R We have calculated the scattering cross-sections at three rain rates R  =  5  mm/h, 12.5 mm/h and 25 mm/h respectively. For the three rates, at the same S and the same frequency f, when the backward scattering of model C is bigger than that model A (at higher frequencies and smaller S), the scattering cross-sections at R 5 mm/h are the closest to those of model A, whereas at R is the biggest. At R  =  =  25 mm/h, the difference  5 mm/h, the ratios of forward scattering and backward  86  =  0  0  1o  GHz  GHz  0  0  60  60  GHz  GHz  Fig.4.13 Ratios of forward to backward scattering (R=25 mm/h)  87  scattering cross-sections are the smallest in model C as well as in model A, at R  =  25mm/h, they are the biggest. On the other hand, when the scattered fields of C are smaller than those of model A fields at R  =  (  at lower frequencies and bigger S), the scattered  25 mm/h are the closest to model A, whereas at R  =  5 mm/h, the  difference is the biggest, ie. compared with model A, the scattering cross-sections of model C are relatively bigger than those at R  =  25 mm/h and R  =  12.5 mm/h.  Those results show that the rain rate does affect the scattering fields because it affects the melting snow particle size as in Eq.(4.4).  4.4 Summary The backward and forward scattering cross-sections of model C at R 25 mm/h, frequency f  =  =  5, 12.5 and  10, 20, 30, 40, 50 and 60 GHz, the melting factor S  =  0.02,  0.04, 0.06, 0.08, 0.10, 0.12 and 0.14 are calculated respectively. The results are consistent, reasonable and conform to the scattering principle. The scattered fields obtained are compared with those of model A and model B. As may be noted from the above results, the three models differ in their predictions of scattering levels in the melting layer. From the comparisons at the same S, same f and the same R, we have found: (1). The forward scattered fields of model C are always smaller than those of model A. The difference increases with S increasing and with f increasing at most frequencies; (2). Compared with model A, at all frequencies the relative values of  88  backscattering of model C increases with S decreasing and f increasing; (3). At higher frequencies and lower melting factor S, the backscattering crosssections of model C are higher than that of model A, in these cases, the higher the f is, the bigger the difference is; the lower the S is, the bigger the difference will be; (4). At lower f and higher S, the backscattering cross-sections of model C are lower than those of model A, in those cases, the higher the f is, the smaller the difference is; the lower the S is, the smaller the difference will be; (5). The backward scattering cross-section of model C is closer to model A than forward scattering cross-section; (6). When S is smaller, the backscattering cross-sections of model C are higher than that of model B. When S is higher, the backscattering cross-sections of model B are higher than those of model C and model A. (7). The ratios of forward scattering cross-section and backward scattering cross-section in model C are smaller than those of model A; (8). At higher f and smaller S, when R  =  5mm/h, the scattered fields of model  C are the closest to those of model A, where at R  =  25 mm/h, the difference is the  biggest, vice versa; and (9). At R  =  5 mm/h, for both model A and model C, the ratios of forward  scattering cross-section and backward scattering cross-section are the smallest, whereas at R  =  25 mm/h, they are the biggest.  Those aspects show the different melting snow models have different scattering properties, the different melted water distributions assumed (model A and model C)  89  significantly affect the scattering fields. These conclusions show the new model is important in the evaluation of the scattered fields of melting snow particles. The model can be used to represent the melting snow particle to a certain extent. It is suitable when we consider and analyze the effect of outer water layer on the electromagnetic scattering. It is a good complement of model A and model B. It might replace model A when S is smaller. The EM scattering phenomenon is a complicated one, from Eq.(2.104)  —  Eq.(2.108), at a far field point, the total scattered field of a scatterer is the summation of all the individual scattered fields generated by the subcell currents in the scatterer. It is a vector superposition procedure, the total scattered field not only depends on the magnitudes of individual scattered fields, but also their phases. When the individual phases are cancelled, the total scattered field is smaller, when the individual phases have same signs, the total scattered field is enhanced. A subcell current depends on the incident field, the dielectric permittivity distribution and dielectric size. In our melting-snow model calculations, when frequency changes, not only the electric size changes, but the permittivity changes too. It is difficult for us to give a complete explanation of the results obtained above or to predict the scattered fields by theoretical analysis only. Although the numerical method used is only a numerical experiment, it generates no analytical formula or expression in which to gauge how the result might change with a change in dielectric distribution, dielectric configuration, body orientation or radar frequency, nor does it state how various scattering mechanisms come into play. Nevertheless, like any well designed  90  experiment, that insight can be gained by running the experiment again and again, with judicious parameter changes from one to the next, until a sufficient body of experimental observations has been made. When we can not analyze our problems by using analytical methods, a numerical method is the only choice.  91  Chapter V. Conclusions and Suggestions for Further Research  5.1 Conclusions In this thesis, a new melting snow particle model, a snow core with several even distributions of water drops at the outside, is developed, based on physical observations or assumptions and the current commonly used snow particle models. The electromagnetic scattered fields of the melting-snow model are evaluated at certain frequencies, melting factors and rain rates. The results obtained are compared with those of two existing models (model A and model B). The comparison shows that the different melting snow particles have different scattering properties so that the different melting water distributions assumed significantly affect the scattered fields. Usually the new model has lower scattered fields and a more “smooth” ratio of forward and backward scattered fields than model A and higher backward scattered fields than model B, at the same frequency f and the same melting factor S. The analysis of the new model gives us a better understanding to the EM scattering property of melting snow particles at the early stage and to the effect of outer layer water distributions on the scattering of melting snow. Therefore we can predict the scattering of melting snow layer and then analyze its effect on microwave propagations further. The model C is complementary to model A and model B. At present, all of the assumed melting snow particle models are based on 92  different emphasis or assumption for the reasons of simpler and easier calculation. The new model is the most complex model assumed so far. For such a three dimensional, irregular, inhomogeneous and lossy dielectric scattering problem, no analytical methods can be applied. Normally such problems can be solved primarily by the conventional moment method (MM), but because of the computer memory limitations, it can only be used when the dielectric objects are smaller in size or in regular shape, in other words, with fewer unknowns to be solved. The iterative moment method, which is derived from the conjugate gradient method and the MM, can solve problems involving a larger number of unknowns than the conventional MM. In this thesis, the iterative MM is used to calculate the scattered fields of the new melting snow particle model. For result comparison and checking, the conventional MM is discussed too. For cubic dielectric blocks, spheres and concentric spheres, good agreement between the two moment methods, as well as between the iterative MM and published results, are obtained. We have demonstrated the feasibility and, in fact, the applicability of the iterative MM solution to the volume integral equation formulation of three dimensional, irregular, inhomogeneous and iossy dielectric scattering problem. Generally, the method and our computer program also apply to any arbitrarily shaped penetrable body with both lossy or lossless high or low dielectric materials. We have also shown the validity of the iterative MM for a large number of unknowns. In [26], it only solved the problem with 423x3 unknowns while we have solved the problem with more than 2175x3 unknowns.  93  The numerical calculation examples in this thesis prove the equivalence of the two moment methods for the special choice of basis functions (pulse functions) and  weighting functions (delta functions). The thesis discusses the computer realization of the two moment methods, compares their advantages and disadvantages. Conclusions regarding the choice of subcell size and the initial guess, the convergence and precision, computer memory, and computing time are obtained which will have academic and pratical interests.  5.2 Suggestions for Further Research There is some work, both in theory and experiment, that needs to be done on the subject of evaluating the melting-snow particle and calculation of the dielectric scattering in the future. We have some suggestions for the further work: 1. The model is only an assumption to the melting snow particle, more real observations to the melting snow process should be made. For example, by changing the surround air temperature in a lab, the changes of the outer layer of a snow core should be recorded and compared with the model assumed. 2. The model C is only valid at lower melting factor S, for bigger value of S, a corresponding model should be presented and evaluated in order to analyze the whole melting procedure. 3. Use the computer programs obtained to study more diverse geometries and a wider range of parameters. For example, a concentrate ellipse snow core with even distributions water drops at the outside or a real snow flake should be studied.  94  4. The scattered fields of the model have never been calculated before, the quantity of the calculation precision to the new model is unknown. Some other methods, if available, should be used to evaluate the model in order to compare and know the precision of the results obtained.  5. The 1MM has its limitations. From the calculations in 3.1.3, although the computer memory can handle many unknowns, the round-off errors increase with unknowns increasing, when the number of unknowns is big enough, it cannot get satisfactory results. Although the iterative MM can solve a larger number of unknowns than the conventional MM, it needs more computer time. There should be further work to improve the method. In order to save computing time, some work can be done mathematically or in computer programming. As discussed in [40], a better method is being developed.  95  References  1. T. Oguchi,  “  Electromagnetic wave propagation and scattering in rain and other  hydrometers, Proc. IEEE, Vol. 71, pp. 1029, 1983. 2. Y. M. Jam and P. A. Wasson, “Attenuation in melting snow on microwave and millimetre-wave terrestrial radio links,” Electron. Lett., Vol. 21, pp. 68, 1985. 3. M. M. Z. Kharadly and N. Owen, “Microwave propagation through the melting layer at grazing angles of incidence,”, IEEE Trans. on AP, Vol 36, pp. 1988 4. M. M. Z. Kharadly and A. S.-V. Choi, “A simplified approach to evaluation of EMW propagation Characteristics in rain and melting snow,” IEEE Trans. on AP, Vol. 36, pp. 282, 1988. 5. R. A. Hulays, Precipitation Scatter Interference on Communication Links with Emphasis of the Melting-snow Layer. M.A.Sc. Thesis, U.B.C., Canada, April 1992. 6. R. A. Hulays, M. M. Z. 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