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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 METHODFOR DIELECTRIC SCKL1ERING CALCULATIONSbyXIAOMJN HUANGB. A. Sc. Beijing Institute of Technology, 1983M. A. Sc. Beijing Institute of Environmental Features, 1988A THESIS SUBMrrriD IN PARTIAL FULFiLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(The Department of Electrical Engineering)We accept this thesis as conformingApril 1994THE UNIVERSITY OF COLUMBIA°Xiaomin Huang, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)__________________ELec,Department of (JThe University of British ColumbiaVancouver, CanadaDateDE-6 (2J88)AbstractIn this thesis, the calculation of the electromagnetics(EM) scattering of threedimensional, irregular, inhomogeneous and lossy dielectric objects is discussed. Theiterative moment method (MM) which can solve problems involving a large numberof unknowns than the conventional MM is used. For result comparison and checking,conventional MM is discussed too. Generally, the method also applies for anyarbitrarily shaped penetrable body with arbitrary dielectric distributions which willhave importance in communication, medical, biological, meterological and militarystudies.A new melting-snow particle model is presented in this thesis, based onphysical observations or assumptions. The EM scattering cross-sections of the newmelting-snow model are evaluated at certain frequencies, melting factors and rainrates by the iterative MM. The analyses of the new model give us a betterunderstanding to the EM scattering property of melting snow particles at the earlystage and to the effect of outer layer water distributions on the scattering of meltingsnow. The results obtained are compared with those of two other existing models.The comparison shows that the different melting snow particle models have differentscattering properties; the different melting water distributions assumed significantlyaffect the scattered fields.The validity of the iterative MM for large number of unknowns has beenshown. The numerical calculation examples in this thesis prove the equivalence of the11two 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 ofsubcell size and the initial guess, the convergence and precision, computer memory,and computing time are obtained which will have academic interests.111Table of ContentsAbstract iiTable of Contents ivList of Tables viList of Figures viiAcknowledgements xChapter I Introduction 11.1 Motivation of The Thesis 11.2 The Set-up of the New Melting-snow ParticleModel 21.3 Thesis Overview 61.4 Literature Survey on Methods of Dielectric ScatteringCalculation 7Chapter II Theoretical Analysis of Two Moment Methods 112.1 Introduction 112.2 Integral Equation Derivation 122.2.1 The Volume Equivalence Principle 122.2.2 Volume Integral Equation Derivation 122.3 The Conventional MM Solutions 162.3.1 The Principle of MM 162.3.2 The Choice of Basis Functions 192.3.3 Testing Procedure 212.3.4 The MM Solutions The Calculation of Impedance Matrix Z The Calculation of Excitation Matrix V 262.4 The Iterative MM Solutions 272.4.1 The Iterative Algorithm 282.4.2 Conjugate Gradient Method 292.4.3 The Iterative MM Solutions 332.5 Formulating the Scattered Fields 36Chapter III Computer Implementation and Discussion 403.1 Selected Numerical Results 403.1.1 Numerical Results for a Thin Dielectric Layer 403.1.2 Numerical Results for Lossless Dielectric Spheres 44iv3.1.3 Numerical Results for Lossy DielectricSpheres 493.1.4 Numerical Results for Concentric DielectricSpheres 543.2 Computation Considerations 573.2.1 Modelling Considerations 573.2.2 The Choice of Subcell Size 583.2.3 The Choice of an Initial Guess in Iterative MM 593.3 Discussion 603.3.1 The Convergence of the Iterative MM 603.3.2 Comparison of the Two Moment Methods 603.3.3 The Volume Integral Equation and the SurfaceIntegral Equation 623.4 Conclusion 64Chapter IV Evaluation of the Scattering from Melting-snowParticle Models 664.1 Introduction 664.1.1 The Three Melting-snow Particle Models 664.1.2 Basic Parameters 674.2 The Computed Results for Melting-snow ParticleModel C 704.2.1 The Choice of the Number of Outside Water Drops 704.2.2 The Calculation Results 714.3 The Comparison of Model A, Model B and model C794.3.1 Forward Scattered Fields 794.3.2 Backward Scattered Fields 844.3.3 The Ratio of Forward and Backward Scattering Crosssections 864.3.4 Rain Rate R 864.4 Summary 88Chapter V Conclusions and Suggestions for Further Research 925.1 Conclusions 925.2 Suggestions for Further Research 94References 96vList of TablesTable I. The Inside E-fields of the Dielectric Layer 43Table II. The Error in Each Iteration 61viList of FiguresFig. 1.1 Two currently used melting-snow particle models 5Fig. 1.2 The new melting-snow particle model 5Fig. 2.1 Replacement of a dielectric body by an equivalent volume current13Fig. 2.2 An arbitrarily shaped dielectric body illuminatedbyaplaneEMwave 13Fig. 3.1 A dielectric layer illuminated by a plane EMwave 41Fig. 3.2 (a) The inside E-flelds of the dielectric layerin Fig. 3.1(b) The scattering cross-section of the layer 42Fig.3.3. (a) The inside E-flelds of the layer (O=60°) EX/Ei(b) The inside E-flelds of the layer (O=6O°) Ey/Ei45(c) The inside E-fields of the layer (O=6O°) EzIEi(d) The scattering cross-section of the layer46Fig. 3.4 The scattering cross-section of lossless dielectric spheres(a) Er=4, ka=1, f=1OGHz 47Fig. 3.5 The scattering cross-section of a lossy dielectricsphere Er41.775J41.204, ka=0.154, f=1OGHz 50Fig.3.6 The scattering cross-sections of lossy dielectricspheres(r=41•775J1204’ 10GHz)(a) ka=3.5 51(b) ka=3.8 52(c) ka=4 53Fig. 3.7 The scattering cross-sections of concentricmelting-snow particle modelvu(a) f=1OGHz 55(b) f=2OGHz 56Fig. 3.8 Meshing of a sphere 58Fig.3.9 Comparison of the number of unknowns between SIEandVlE 63Fig. 4.1 Relative permittivity of water vs. frequency69Fig. 4.2 Scattering cross-sections of the three models(R=5mmlh, S=0.02) 72Fig. 4.3 Scattering cross-section of the three models(R=Smm/h, S=0.08) 73Fig. 4.4 Scattering cross-sections of the three models(R=5mm/h, S=0.14) . 74Fig. 4.5 Scattering cross-sections of the three models(R = 12.5mm/h, S = 0.04) 75Fig. 4.6 Scattering cross-sections of the three models(R=12.5mm/h,S=0.12 76Fig. 4.7 Scattering cross-sections of the three models(R=25mm/h, S=0.04) 77Fig. 4.8 Scattering cross-sections of the three models(R=25mm/h, S=0.12) 78Fig. 4.9 Scattering cross-sections vs.S (R=Smm/h,f=1OGFIz) 80Fig. 4.10 Scattering cross-sections vs.S (R=Smm/h,f=4OGHz) 81Fig. 4.11 Scattering cross-sections vs.R (S=0.06,f= 10GHz) 82vmFig. 4.12 Scattering cross-sections vs.S (S=O.06,f=4001—lz) 83Fig. 4.13 Ratios of forward scattering to backward scattering 87-vmiAcknowledgmentsI 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 thiswork.I am also grateful to Dr. G. E. Howard for many helpful discussions regardingthis work.My thanks extend to my fellow student and friend Rafeh Hulays for kindlyallowing me to use some of his computer programs to get the comparison results. Mythanks also go to all of my friends and those individuals in the Electrical Engineeringdepartment 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 thescattering 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.xChapter I. Introduction1.1 Motivation of the ThesisElectromagnetic (EM) scattering by dielectric bodies has been the subject ofintensive investigation. It is very important in problems including propagation throughrain or snow, scattering by airborne particulate, medical diagnostics and powerabsorption in biological bodies, coupling to missiles with plasma plumes or dielectric-filled apertures, and the performance of communication antennas in the presence ofdielectric and magnetic inhomogeneities.Many methods have been used for solving EM scattering from dielectricscatterers. These methods can be classified into two types: analytical methods andnumerical methods. The former give analytical formulae or expressions to show howthe scattering might change with a change in the dielectric distribution, dielectricshape and radar frequency, but they can only solve for dielectric bodies with simplershapes such as spheres or cylinders with homogeneous dielectric distributions; whenthe dielectric bodies are inhomogeneous or in irregular shapes, we have to resort tonumerical methods. Numerical methods have the advantage that they can be used tosolve arbitrarily shaped and inhomogeneous dielectric problems, but for scatteringfrom electrically large or dielectric bodies, most of the numerical methods areimpractical because of the computer memory limitations. It is necessary to have amethod which fully evaluates the scattering from large size, inhomogeneous dielectricdistributions or in arbitrarily shaped three-dimensional dielectric scatterers.1In terrestrial and satellite communications, more and more attention has beenpaid to the scattering by melting-snow particles in order to study the influence of themelting-layer on microwave propagation. Various prediction models are used toevaluate the EM scattering from melting-snow particles. Presently there are twocommonly used models to represent the scattering of melting-snow particles, theconcentric snow sphere and water shell or the homogeneous water-ice-snow mixturesphere (discussed in next section), which are simple shapes and are easily analyzedby analytical methods.For better understanding of the influence of the melting procedure on EMscattering, in this thesis, we present a new melting-snow particle model which is athree-dimensional, irregular, inhomogeneous and lossy dielectric body. At presentthere are no analytical methods to obtain the scattered fields , we have to use anumerical method to evaluate its scattering properties.This study is mainly motivated by the idea of fully calculating the scattering ofdielectric bodies and evaluating the newly developed melting-snow particle model.1.2 The Set-up of the New Melting-snow Particle ModelIn communications, the incoming signal and interference caused by othersystems have to be distinguished in order to establish a reliable system. When an EMwave 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 ofinterference between communication links operating at the same frequency. Recently,2considerable 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 itspresence affects microwave communications[5].In the study of the snow melting procedure, the melting-snow layer is aconcept often used. We call the region between where the melting starts and wherethe melting ends the “melting-snow layer”. In a melting-snow layer, the melting occursbasically because the surrounding air is at a higher temperature. The melting snow-layer 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 itseffect on communications and therefore design a reliable communication system. Themelting layer is made of numerous melting-snow particles. Usually the scattering ofthe melting layer is evaluated by considering the scattering of a melting-snow particle.Melting-snow particles are expressed using different melting-snow models for somespecifications.Melting snow particles are assumed to have a variety of complex shapes thatmay be extremely difficult to analyze. Thus in order to look into the problem of thedependence on particle geometry, some simplified geometric shapes which can bereasonably analyzed are considered. At present there are mainly two models[6] ofmelting-snow particles used in analysis and prediction. One assumes that meltingoccurs from the outside of the particle, with the water forming a shell around thesnow core. The corresponding model is a concentric sphere model where a uniformlayer of rain (water) of permittivity Ew surrounds a dry snow core of permittivity Es3(as shown in Fig.1.1 (a), model A). The other assumes that the water forming on theoutside of the melting particle percolates to the inside, forming a heterogeneousmixture of air, water and ice. The relevant model is a “homogeneous” “artificialdielectric” sphere model with the equivalent pennittivity Eav of the water-ice-airmixture (as shown in Fig.1.1 (b), model B), the 6av is calculated from the MaxwellGarnett theory or the Bohern extended theory for spherical inclusions. They areanalyzed 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 haveobserved that when a snow flake melts, at first there are always several water dropsat the snow flake surface. From the physical sense, it is not difficult to understandthat 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 severalsmall water drops which come from the early melting of the snow particle form atfirst. After a certain period of time, those water drops become bigger and then mergeinto 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 themelting snow in the beginning melting period.The new melting-snow particle model to be set up and evaluated in this thesisis 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,4h x -t ur eConcentric sphere model(a)snowwQterHomogeneous sphere model(b)Fig. Li Two currently used melting-snow particle modelssnowWa terFig. 1.2 The new melting-snow particle model5we call it model C. It is believed that this model describes the melting-snow particlesat the early stage of melting and it has never been analyzed. It is complementary tomodel A and model B, and will give us a better understanding of the effect of outsidewater on the EM scattering of melting-snow particles.1.3 Thesis OverviewIn this thesis, a new melting snow particle model is described and evaluated. Theinternal electric fields as well as scattering of a plane wave (either polarization)illuminating three dimensional, irregular, inhomogeneous and lossy dielectric bodiesare obtained. Two moment methods, the conventional moment method [7J[8] (we willcall it the MM) and the iterative moment method[9] (we will call it the iterative MMor the 1MM), will be used, based on the volume integral equation method employingthe volume equivalence principle. The latter will be used to evaluate the scatteringof the new melting-snow particle model. The scattered field obtained by the newmodel is subsequently compared with the other two models. Conclusions areobtained. Once the validity of the iterative MM is established, the method can beapplied to any other three-dimensional dielectric scatterers with large size in anyshape 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 toobtain the EM scattering from dielectric bodies. The analysis starts with thederivation of the general volume integral equation based on the volume equivalence6principle. The integral equation can be applied to any three-dimensional, irregular,inhomogeneous and lossy dielectric scatterers and solved by the MM. Pulse basisfunctions and delta testing functions are chosen. The iterative MM, based on theconjugate gradient method, is also discussed and applied to solve the integralequation, where the same basis functions and testing functions as in the MM areused. After the unknown interior fields or currents are found, the scattered field aswell as the scattering cross-section in the far radiation region are determined.Chapter III discusses the computer implementation of the two momentmethods. The computer programs written are tested by comparing the solutions withknown 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 thedetermination of an initial guess in the iterative MM are considered. Theconvergence and precision of the iterative MM are also discussed. The two methodsare compared regarding computer memory, computer time, and the number ofunknowns that can be solved.In Chapter IV, the scattering by the new melting-snow model is studied. Theresults obtained are compared with those of two existing models ( model A andmodel B), from the calculation one can see that the outer layer water distributionsaffect the scattering of the whole melting-snow particle, different models havesignificantly different values because of different assumptions. This model gives us abetter understanding of the early melting stage of the melting-snow particles.Chapter V, the last chapter, will give some conclusions and the suggestions for7further research work.1.4 Literature Survey on Methods of Dielectric Scattering CalculationsModelling of penetrable dielectric bodies is more complicated than themodelling of perfect conducting bodies because of two reasons: (1) it is needed todeal with the fields of sources radiating in at least two different media, and (2) at thesurface of a dielectric body, both the equivalent tangential magnetic current andelectric current are not zero while at the surface of a conducting body the tangentialmagnetic current equals zero.EM scattering from dielectric objects originally started from Descartes rainbowtheory[1O] based upon ray optics. The problem was subsequently treated as aboundary value problem and solved by many researchers via the classical separationof variables method. Mie[11] was the first to develop solutions for arbitrary size andhomogeneous 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 surfacecoincides with one of the coordinate systems for which the vector Helmholtz equationis separable. A perturbation technique which is more effective for bodies that areonly slightly non-spherical was also developed by Durney[18] and applied to prolatespheroid models for long wave-length irradiation.A superposition estimation of a concentric-two-layered sphere was suggested8by Peters[191. Standard optical approximations such as the physical optics (P0) andthe geometrical theory of diffraction (GTD) have also been modified by Peters andKeller[20] so that they may be applied to dielectric bodies whose dimensions aremuch larger than a wavelength.The rapid development of numerical methods made it possible to solvedielectric scattering problems accurately and conveniently. The most-used numericalmethods 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 differentialequations. Those methods are powerful when the dielectric bodies are smallcompared to the wavelength. The spectral iteration techniques (SIT) and theboundary element method(BEM) are also used in solving medium size dielectricbodies.For the EM scattering from electrically large dielectric bodies, the use of theabove numerical methods by themselves is impractical because of the computermemory limitations. Since 1975, various hybrid solutions, using numerical methodstogether with analytical methods or even using two different numerical methodstogether, have been developed to solve the problems of scattering from larger sizedielectric bodies. They are the hybrid solutions of the MM and the GTD, the MMand the P0, and the MM combined with the FEM[24]. Hybrid methods can savecomputer memory, but they will make the calculations much more complicated, andnormally they cannot be used for arbitrary irregular shapes. Recently, the possibility9of using iterative methods for large bodies has been investigated because they can becarried out without the direct involvement of a large matrix which rapidly exhauststhe computer’s memory. At the beginning, iterative methods could only be appliedto two-dimensional problems[25]. In 1989 Wang[9][26] used an iterative conjugategradient method in the MM (it is also called the iterative MM or the generalizedMM) and successfully solved a three-dimensional arbitrarily shaped dielectric bodyunder plane wave illumination. The iterative MM can solve problems involving alarger number of unknowns than the conventional MM by at least an order ofmagnitude. Using the iterative MM, we can now solve for large inhomogeneousarbitrary bodies as long as the computer time allows.Although the newly developed melting-snow particle model in this thesis is notbigger in electric size, it needs more unknowns because (1) outer layer water dropsneed 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 torepresent the shape more accurately. That is why we chose the iterative MM to solvethe scattered field of our model.10Chapter II. Theoretical Analysis ofTwo Moment Methods2.1 IntroductionThe moment method (MM) was applied in EM field calculations by R.F.Harrington in his significant book entitled “Field Computation by MomentMethod”[27] in 1968. It is claimed as forming the basis of all numerical tools incomputational electromagnetics. In the last few decades, numerous papers and manybooks about the mathematical basis and the applications of MM have beenpublished. In 1992 IEEE published a special collection[24] in which a selection of keyarticles about the analytical formulation, numerical implementation and practicalapplications of MM are given. The publication shows the importance of MM inelectromagnetics theory applications.Recently J.J. Wang applied the iterative MM[261 to EM scattering problemin which improvements were introduced to MM. The iterative MM can solveproblems involving a large number of unknowns than the conventional MM, but onlya 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 theiterative MM for solving the scattering of a three dimensional dielectric scatterer isdiscussed. At first, the volume integral equation is derived, and then solved by usingthe MM and the iterative MM respectively. Finally, the internal E-fields, as well as11the scattering cross-section, are obtained.2.2. Derivation of Integral Equation2.2.1. The Volume Equivalence Principle[261The volume equivalence principle states that in calculation, a dielectric body,with volume V, permittivity e(r) and permeability being illuminated by an EMwave E’, can be represented by an equivalent volume current 3(r) density in V asfollows:J(r) -jc [e (r) —c0]E(r) (2.1)where r is a position vector of the point of interested inside the body, is the angularfrequency of the wave, E(r) is the total electric field inside the body, and 6(r) is thepermittivity of the dielectric body at position r.This principle is illustrated in Fig.2.1. Using it we have replaced the dielectricbody 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 volumeequivalence principle can be applied for inhomogeneous bodies, in which 6(r) is afunction of the position vector r. If u does not equal to there will be anequivalent volume magnetic current M(r).2.2.2. Derivation of Volume Integral Equation[26][28]Consider a plane wave illuminating a three dimensional arbitrary shapeddielectric body, as shown in Fig.2.2. The surrounding medium is considered to be free12Ii viOh /\ O.)O ——(b)E—S/E E(a)Fig. 2.1 Replacement of a dielectric body by an equivalent volume current[26]2So‘1’xFig.2.2 An arbitrarily shaped dielectric body illuminated by a plane EM wave13space with parameters E0, j.. From Maxwell’s equations and the Lorentz condition,the scattered field inside the volume, due to the equivalent volume current J(r’), isgiven 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:-j-F1,. e 24g(r,r )-4idr-rlI is the unit dyadic:(2.5)andk-k0-zi(Q)”. (2.6)Eq.(2.2) shows that ES(r) is singular at r=r’. To circumvent the singularity, wesplit the volume V into a small spherical volume V0 plus the remainder (V-V0),where V0 is an open neighbourhood containing the point r=r’. The integral over (VV0) can be carried out since G is not singular in this region. We can choose V0 so14small that the current density J(r) can be approximated by its value at the centreJ(r0), apply Green’s theorem to the region (V-V0)and take the limit as V0 tends tozero. We can obtain the following equation:Es(r) +vf J(r’) G(r,r’) dV3k0 (2.7)J(r)+PVfJ(r’)G(r’)dv’Vwhere 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 thesum of the incident electric field E’(r) and the scattered field ES(r):E(r)-Et(r)÷ S(r) (2.8)Substituting Eq.(2.7) into Eq.(2.8) and rearranging the terms gives the desiredvolume integral equation (VIE) of our dielectric scattering problem:E ‘(r) =PVf J(r’) ?(r,r’) dv”+ D(r)J(r) (2.9)wheree (r)+21 (2.10)3j[e(r)-]15and£(r)-r(r)eo. (2.11)Our goal is to find a solution for the equivalent volume current J(r) densityin Eq.(2.9) in terms of the incident field and the physical characteristic of thescattering object. If we can get J(r), the internal E-field and scattered fields of thedielectric body can be obtained. For this purpose, two MM solutions will bedeveloped in next sections.2.3. The Conventional MM SolutionsMM is a powerful method in solving EM scattering or radiation problems,especially when the objects are inhomogeneous or of irregular shapes which cannotbe solved by analytical methods.2.3.1. The principle of MM[27][29]The basic principle of the conventional MM is to expand unknown functionsinto a series of basis functions so that an operator equation, such as a integralequation, a differential equation, or a integro-differential equation, is transformedinto 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)16where, in general, A is a linear integrodifferential operator in a certain domain, andX is the unknown to be solved given a known excitation Y.Assume that the unknown solution can be expanded in terms of knownexpansion functions (basis functions) x1 such that Ax1 are linearly independent andspan the range space of A. We then seek XN--an approximation to the exact solutionX--of the form:N (2.13)XXN-) Ii—iwhere a1 are unknown coefficients of expansion functions x, i=1,2,...,N. In order tosolve for the unknowns a, first we form the residualN (2.14)RN=Y-AXNY--and then choose a set of known functions (testing functions) W (j = 1,2, ..., M) so thatthe weighting of the residual RN with respect to the testing functions W is zero:<R,W1>-O, j-1,2,...,M (2.15)where M=N or M>N and <.,.,> denotes the usual inner product<R,W>=f R(x)W(x)*dxLN (2.16)fLa Ax? W(x)*dx17here the star * denotes the complex conjugate and L denotes the domain for theinner product.For M=N, the Eq.(2.16) gives rise the well-known matrix equations:<Ax1,W> <Ax2,W1> ...c4x1,W> c4Xj> ...(2.17)<AX1,W,> <AX2,WN> ... <AX,,,WN> aNLet<Ax1,W> <Ax2,W1> ...Ax , W> <AX ,W> ... <AX W>12 22 (2.18)[Z]- ... ...<AX1,WN> <Ax,W1J>which is called the impedance matrix and[r]—(2.19)aN18which is called the current matrix, and<y,w1>[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 theunknown 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 theaccuracy and convergence of the calculation. From a mathematical point of view, thechoice of basis functions does not depend on the choice of weighting functions. InEq.(2.18) each element of the right side matrix, <Ax1,, Wa>, is a double integral, forwhich it is difficult to get an analytical solution. Therefore the calculation of the Zmatrix might need a lot of computing time. When using the MM to solve EMproblems, we should choose basis functions and weighting functions carefully so asto get the solution accurately as well as economically.There are many sets of basis functions that can be chosen theoretically. Fora given problem, only a few sets of basis functions are suitable. The basis functions19chosen for a particular problem have to satisfy the following criteria: (1) They shouldbe in the domain of the operator and (2) They must be such that Ax1 form acomplete set for the range of the operator[29].In practice, if we have certain pre-knowledge about the solution, we shouldchoose basis functions to (1) be as close to the practical solution as possible; (2) meetthe boundary conditions; (3) make the calculation converge fast so as to savecomputing time; and (4) be simpler so as to carry out the calculation easier.In solving dielectric problems, for different dielectric shapes and integralequation types, normally there are several kinds of basis functions that can bechosen[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 wavefunctions or sinusoidal functions; for arbitrary shaped homogenous bodies in surfacemodelling, the scatterer surface is cut into small triangle surface patches and basisfunctions could be related with each triangle’s face, edges and vertices; for arbitraryshaped inhomogeneous bodies in volume modelling, the scatterer body is cut into aset of tetrahedral elements, each basis functions is associated with each face of thetetrahedral model. In arbitrary shaped inhomogeneous bodies in volume modelling,the simplest way might be to cut the object into small cubes, and the basis functionsare three dimensional pulse functions. In the following calculation, we will choose thisone which will lead to simpler analysis.Dividing V into L small volumes vl,v2,...,vL so that J(r) and E (r) are constantin each subvolume. We choose the pulse functions as our basis functions:20B,A(r)..k(r)_j2kP,(r) (2.22)where Uk are unit vectors x,y,z, when k= 1,2,3 andP r)-{ 1 for r€V,(2.23)0 elsewhereExpanding unknown current J(r) in terms of above basis functions asJ(r)-kB1k(r) (2.24)i-i k-Iwhere 31k are unknown coefficients for each subcell v1, B1k are basis functions inEq.(2.23), k=x,y,z, 1= 1,2,...L. Eq.(2.9) becomesE ‘(r) _Fvfj2 k l2kPj(r)dv’i-i k-i (2.25)jkP1(r)i-i k-iThe VIE in Eq.(2.9) has been expressed in terms of unknown current coefficients j1kknown incident field E’ and pulse functions. If we solve the j1k the solution of theVIE 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 theunknown expansion coefficients j1kGenerally in MM there are two kinds of weighting functions chosen, based on21two 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 basisfunctions 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 matchingmethod is the point-matching method. We choose it in our testing procedure, in themathematical form:w- ö(r-r)zI (2.26)p-I n-I p-I n-IWhere 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 ourintegral equation (2.25), we can get our MM equation as Eq.(2.21). The details willbe discussed in the next section.2.3.4. The MM Solutions[30j[32J2.3.4.1 The calculation of impedance matrix ZIn rectangular coordinates, the inner product J(r’).G(r,r’) in Eq.(2.9) can bewritten as:22G (r,r’) G ‘(r,r1) G(r,rt)G(r,r’) J(r’)- G(r,r’) G(r,r’) G3z(r,rI) (2.27)G (r,r’ G(r’) G(r’) [J(r’)fwhereG’(r,r”)- -jø jL(8 +.! )g(r,r’5 (2.28)Ic aUfl&Skhere 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:JkZE‘(r) re V (2.29)i-i k-iwhereZ&_fG(r,r’) B(r’)dv’+D(r)B1’(r)2k3 (2.30)= [f G(r,r’)B(r’)d1/]+D(r)Bl”(r)l2kn-iLettingG1[k(r) U+D(r)Bi(r)U (2.31)whereGi (r)_fG (r,r)Bik(rI)dvI (2.32)23andz-Ezsl.(2.33)Eq.(2.29) can be rewritten as‘33 3 3ID IDJ,Z,i2,- EE.’(r)d rEv (2.34)1—1 k—i K—i n—iTaking the product with weighting functions, on both sides of our discretizedintegral equation (2.34), yields the following 3L equations:±f .i,kz>v; (2.35)I-i k-iwherez>(w’,z,a) (2.36)here I correspond to the subcells related to the basis functions and p correspond tothe subcells which are related to the weighting functions.As in Eq.(2.29), 1 and p range over all possible values , we obtain the followingmatrix representation of Eq.(2.25)Z] [ZJ [Z] J] Vx]Z,x] [Z7,] [Z,]. j,j - (2.37)Z] [Z,,] [Z1 11 içi24where is given in Eq.(2.36), k=1,2,3 and n=1,2,3. According to Eq.(2.28),the element Gi isG1j(r,)-fG(r,r)B,(rdv (2.38)plwhereAV1fdv’ (2.39)VIUsing Eq.(2.28) to evaluate G’(r,r1)gives[30]e (2.40)G1(r)_f Ioko(e (r?-e)AV,e [(4-1_ja2)844..cosOf’cosO(3-4+3ja )] l#pwhere(2.41)(r’—r, (2.42), (T’-Tk’) (2.43)R,r,—rff+r,9÷r?f (2.44)251.. (245r=rx+r2y÷3z(2.46)R4r, -r)When l=p, as we have discussed in 2.2, v1 is approximately replaced by a smallspherev0[28][30J, andG1 8k2)0 1.10 [(1 ÷jkoa1)eth0at_ 1] lp (2.47)3k0wherea(M’)l3 (2.48)4itis the radius of the sphere.Now we have the elements for the matrix Z in Eq.(2.18). The calculation of the excitation matrix V[27J[33JAssuming the incident wave is a plane wave in (Of, ) direction:E ‘(r) =áEe jkr (2.49)where a is the polarization direction of the field, E is the amplitude of the incidentE field, k is the propagation factor:k=k=k(isinO1cosp1÷9sinO sincp ÷‘cosO ) (2.50)26and r is the position vector:r..rP-d+yj÷ze (2.51)Fromk’r=k(xsinO cosp +ysinO1sinq +zcos0 ) (2.52)Eq.(2.49) becomes:K ‘(r) =âEe -jk(xsinO gcos1÷ysiuOeosp +zcosO (2.53)Therefore, the excitation matrix in Eq.(2.35) is:v=<E Z(r),8(r.,,,)li> (2.54)E’e -jkxsinOcosp +y,,sinO posp1+zcosD )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)and solving the linear equation, we can get the coefficients Jk1 (k=1,2,3 and1= 1,2,...,L) of 3(r). From Eq.(2.24), the MM solution of the volume current density3(r) in the scatterer is obtained. If L is sufficiently large, the approximation given inmatrix (2.35) will give adequate results.2.4. The Iterative MM Solutions[26]The iterative moment method is the conjugate gradient method (CGM) usedin the MM. The CGM is an important special case of the method of conjugatedirections. It plays a central role in iterative methods because it terminates with a27solution in at most N steps ( N is the dimension of the unknown vector), if nonumerical round-off errors are encountered.2.4.1. The iterative algorithm[34]As discussed above, almost all direct scattering problems lead to integralequations 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 kernelfunction 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)andY(r)—[y(r1),y7,y(r,),...,y(r] (2.57)where r1, r2, ...,are the position vectors in the volume V. In order to get thesolution of X, at first an initial estimation for X is given:X’°(r’) —[X°(r1),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 Xin Eq.(2.55), the deviation isF°— Y(r) _fX(°)(r’)K(r,r’)dv’ (2.59)28The integral square error is defined as:ERR(°)—fIF(°)(r)Fdv (2.60)After n steps of iteration, the integral square error is:ERR_fIF(r)Fdv (2.61)orERR_fjY(r)_fX(r’) -K(r,r’)dv42dv (2.62)In the frequency domain scattering cases, F can be complex value. We note that(2.63)and henceERR (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 iterativemethod is to choose certain estimations and iterations to minimize the integratedsquare error in the iterative process until a satisfactory result is obtained. The totaliteration 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 take29X(r)—X’(r)+i (2.65)where = (1)(r) is a variational parameter and g(fl) = g (r) is a suitablechosen variational function. The choice of g and t depends on the iterative methodchosen.In Eq.(2.61) the deviation in nth iteration is VF—F”— Y(r)_fX(’I)(r’)K(r,r”)dv’From Eq.(2.65) and Eq.(2.66) we haveF—F1—if’(r’)K(r,r’)dv’ (2.67)—F’’—i1(n(n)where f is the variational deviation:-f)(r’)K(r,r’)dv’ (2.68)Now ERR can be written asERR—ERR’—2Re[i1ff)dv]+Fq Bf”)Pdv+h1_’whereA()_fF’)f’dv (2.70)30andB(-.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 valueL4”P (2.73)As long as(2.74)we always haveERR <ERR1 (2.75)Substituting Eq.(2.66) into Eq.(2.70) and interchanging integrations, we obtainA (2.76)where=S1(r) fv1_1*(nI,th,I (2.77)31is called the gradient vector of the iterative process. From Eq.(2.67) Eq.(2.76) andEq.(2.77) we can also getA (fLs(1)12dv (2.78)Eq.(2.73) states that the iterative solution will be improved after each iterationif Eq.(2.72) and Eq.(2.74) are satisfied. Based on Eq.(2.73), we can see that thiscondition can be met until ERR is zero if we choose that the variational functiong(fl) and gradient vector to have a certain relationship. In the conjugate gradientmethod, the choices of g(fl) are:g°>(r).’S’°(r) for n1 (2.79)andg1(r) forn>1 (2.80)A’1It is not difficult to prove that our choices can meet two separate conjugateconditions. One is g are mutually conjugate:<Ag,g0>=O ij (2.81)The other is that are mutually orthogonal:(2.82)32That 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 functionsas the weighting functions in the direct MM, and the same basis functions are usedin iterative MM, the same numerical results in the two MM are obtained.In the previous section we have obtained the basic method of the iterativeMM. We will derive the 1MM solution for our scattering problem in Eq.(2.9).Comparing with Eq.(2.55), we haveX(r)—J(r) (2.83)Y(r) =E ‘(r) (2.84)andK(r,r’) G(r,r”) +D(r) ô (r—r’)I (2.85)As in conventional MM, we first divide the object into L subcells with centreat r1, r2,..., rL. The Y vector is(2.86)Assuming the current at each subcell is J(r1) (I=1,2,...,L),(2.87)33and the initial guess current is—(2.88)then the deviation is:F°’(r) — [E ‘(ri)—f(r1),E‘(r2)—f(r),...,E‘(r1)—ftr] (2.89)wheref°(r)_fK(r_,r’)J(°)(r’)dv’ (2.90)Whereas taking the same derivation method as in the MM, we have3ftrm)-E i2f(r)i-i (2.91)[E E jk G1(r)+D(r_)J, r2]i—i k—i 1—1where G11 is given by Eq.(2.32).IfERR=f0Pdv (2.92)is not small enough, we take next step n=1. Following the general operations asdiscussed above, the gradient vector S in Eq.(2.77) is:S(r)_gs4It)(rM)L 3 3 3 (293)—ID ID ID F(r_) [G1_(rj)]*+[D(r_)]* F(Tm) aji—i k—i i—i i—i34where3 L(D D L_lkr,)*F(r1)-FD(rj) — k-i 1-1 (2.94)3 LE .frkr1)P.vk-i i-iIf n=1g°(r) .-S°(r) (2.95)Ifn>1(n)g(r)—S1(r)+ A g’(r) (2.96)A’13A (n)=fL_1)(r)dvf I S “(r)iiJdv (2.97)3 L—E Ek-i i-i3B=f(r)Fdv_ 1’ IEf(r)z2 dv (2.98)J Vk_l3 Ln)-(r1) .v,k-i I-iA (n)(n) —____3 L£ L’’kr,)P.v1] (2.99)— k-i i-i3 LEk-i i-i35j—J(r)—J’1(r)+rig(r) (2.100)F(r)—F’(r)—i1‘/‘(r) (2.101)the integrated square error isERR_f 71r)Pdv_f I F(r)i2dvV V(k_l) (2.102)LFkr1)P.v,k-i i-iIn calculations, the normalized integrated square error is often used which isdefined as:ERRERR)I L°(rPJ V (2.103)ERRf[i(°)(r’)FIf ERR is not small enough, those steps are repeated until satisfactoryprecision is obtained.2.5. Formulating the Scattered Fields[33] [37] [38]Although the accuracy of internal field values is a more stringent test of acomputational method than is the accuracy of scattered fields, it is difficult toaccurately measure the fields inside most dielectric objects. Furthermore, manyapplications require only scattering data and not internal field data.36Once the system of linear equations(2.35) has been solved, the unknowncoefficients are found, the volume currents in the dielectric body are obtained, fromwhich the scattering fields of the dielectric body can be calculated at any point inspace. To simplil’ the calculation, it is convenient again to divide the dielectric bodyinto L small cube cells, each of volume v1 and the current J(r1), 1=1, 2, ..., L.At the receiving point P(r, O, ), the far-field, due to a current source J(r)in a region v, iss isH41,(r)——E0(r) (2.104)--jøA0(r)S isH0 (r).—E,(r) (2.105)- -jø A(r)where A is the vector magnetic potential, generated by the current source J(r) in theregioneA(r) = .fJ(r’)e ‘dv’4irr V (2.106)Le -jkrr1c*—J(r,)e ,v1and its e and p components areeJkA0(r)=____.J0(r’)e” dv’4tr V (2.107)Le -JkrT1cx—J0(r1)e v,1-1 ‘,3CT37e1A,(r)=.-__-_.JJ,(r1)e’dv”4itr v (2.108)L ..•.u !__.J,(r)ez- 4rAs Eq.(2.52)e-jk(x,s1nO,cos1+yphiOsincp,+zposO) (2.109)Since in the analysis above we have the current in Cartesian form:J(r)1(r)+9J(r) ÷U(r) (2.110)From the relation between Cartesian and spherical coordinates:Ô=icosOcosp +9cosOsinp -sinO (2.111)- -sinp +9c0s(p (2.112)we can get its U and p components.10(r)-JposO -Jos0sinp3—JsinO (2.113)J(r1)-Jposcps—J1n (2.114)From Eq.(2.104) Eq.(2.105) and Eq.(2.106) we can get the scattered fields Es andthe scattering cross-section of the dielectric object:a(Op ,,Op )—lim 47tr2)P (2.115)r-oo38where r is the distance between the source and the object; Es is the scattered fieldstrength at the receiver due to the target scattering and B1 is the incident field at thetarget, (Of, ) is the direction of incident field and (8, ) is direction of scatteredfield, and p and q are either 0 or p, respectively.When 0 = O and p5= p, we call a the monostastic scattering cross-sectionwhich is when the transmitter and receiver are in the same direction with respect tothe scatterer, we also call the monostastic scattering cross-section, the backscatteringor radar cross-section (RCS) which is a very important concept in radar systems; Thescattering cross-section for angles other than backscattering is called the bistaticscattering cross-section. In later calculations, we are most concerned about twoimportant scattering directions, backward scattering cross section (0=O and=cp) and forward scattering cross section (O = ir- O and =Now we have the MM and 1MM solutions of the volume integral equation inEq.(2.9) and the scattered fields.39Chapter III. Computer Implementation and DiscussionGeneral computer programs, based on the two moment methods describedabove, were written and used to solve the inside E-field intensity and scattering cross-section for arbitrary dielectric distributions objects. A set of numerical simulations totest the analysis methods and programs are performed on a Sun Sparc station.At first, the validity of the computational procedure was verified by makingcalculations, using both the MM, of internal E-fields and scattered fields for a simpledielectric layer and comparing the results with those which have been published. Thefirst example shows the equivalence of the two MM. After this test, the validity checkwas extended to lossless and lossy spherical shapes which will be useful in thecalculation of our melting-snow particle model later.3.1 Selected Numerical Results3.1.1. Numerical Results for a Thin Dielectric LayerIn order to compare our results with those given in [30], we consider a thinhomogeneous lossy dielectric layer illuminated by a 300 MHz plane wave travellingin the positive Z direction and the E-field perpendicular to the plane of the layer, asshown in Fig. 3.1 where Er=7O c=lmho/m (Er7Oj59.95).The layer dimension isO.5x4x6 cm3. In the calculation, the layer is cut into 96 subcells of O.5x0.5x0.5 cm3each. The layer structure geometry is then defined for the computer model in terms40EzFig. 3.1 A dielectric layer illuminated by a plane EM wave[30]of the subcell centre in Cartesian coordinates. For identification, each subcell isnumbered 1, 2, ..., 96 from the upper right to the lower left. Fig.3.2 (a) shows themagnitude of the calculated E-fleld component E inside the layer (E=E=O) whenthe 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. Sincethe layer is symmetric, only a quarter of the results are shown. In Fig. 3.2 (b), thescattering cross-section results of the layer via 0 (0 =00 180°), in the vertical plane= 00, are shown where 0 = 0° is the forward scattering direction and 0 = 1800 isthe 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)941Sc.LtLriD5 Cr..* (rPI2) z 1O —U—5.50LOC7.507.006.50LwLw5.00Lw4.00LwLw1.501.00LwLwLw0.0015000Fig. 3.2 (a) The inside E-elds of the dielectric layerin Fig. 3.1(b) The scattering cross-section of the layer0.040.01em0.01z .0.020I I0.00 50.00 100.0042Table I. The Inside E-flelds of the Dielectric Layercell results in [30] our cell results in [30] ourNo. results No. results1 0.0210 0.02105 2 0.0155 0.015493 0.0160 0.01604 4 0.0158 0.015815 0.0158 0.01579 6 0.0158 0.0157713 0.0155 0.01549 14 0.0108 0.0107715 0.0112 0.01120 16 0.0110 0.0110117 0.0110 0.01098 18 0.0110 0.0109625 0.0161 0.01607 26 0.0112 0.0112227 0.0116 0.01164 28 0.0114 0.0114329 0.0114 0.01140 30 0.0114 0.0113837 0.0159 0.01587 38 0.0110 0.0110539 0.0115 0.01146 40 0.0112 0.0112541 0.0112 0.01122 42 0.0112 0.01120simple objects. The iterative MM is also used to solve the scattered and internal Efields of the layer shown in Fig 3.1. When ERRN<0.001, after 18 iterations, we getthe same scattering cross-sections as that got from MM. However the E-fleldsobtained 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. Asdiscussed 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 shownin Fig. 3.1 when the incidence wave is 60° from the layer plane surface. The two43moment methods also get the same solutions. The B-fields obtained by the 1MM areshown 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 incidentE-field, all the components of the E-field--E, E and are induced. This exampleproves that the intensity of the induced B-field inside a dielectric body dependsheavily on the body’s orientation with respect to the incident wave.3.1.2. Numerical Results for Lossless Dielectric SpheresFor three dimensional dielectric bodies, very few scattering problems can besolved analytically. Since the scattering of homogenous spheres and layered,inhomogeneous spheres can be determinated analytically, those structures will beconsidered as examples in current and later sections.The bistatic scattering cross section of a dielectric sphere with =4and ka=1(a is the radius of the sphere) illuminated by a 10 GHz plane wave is calculated byusing the 1MM. The number of unknowns is 80x3. The results for two polarizationcases, the vertical polarization (VP) and the horizontal polarization (HP), areobtained as shown in Fig.3.4(a). The results agree well with those obtained by usingMie 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) andalso agree well with Mie scattering solutions as shown in Fig.3.4(b).440.040.04Fig. 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/Ei0.02-0.020.010-0.02y.0.02 .0.04 245(d) The scattering cross-section of the layer0.010•0.01y0.020-0.02z&attering Crau-.ectcc (rP1*m2) x 1O46Scattering Cross-section (cnY’2)1.50—I—(ie-HP1.40 -—130- ie-VP1.20-—110 ——1.00- k —0.90 -“.—0.80--EE \ N0.40 —-4.0.30-—414.4.0.20— /—0.10-.‘ /—Ni.. ...‘.0.00 --..--—Theta(degree)0.00 50.00 100.00 150.00Fig. 3.4 (a) The scattering cross-section of lossless dielectric sphere6r4,ka=1, f=1OGHz47Scattering Cross-section (cm’2) x iO500.00 ET450.00 -400.00 -350.00 -300.00 -250.00 -200.00 -150.00 -100.00 -50.00 -0.00 -Fig. 3.4 (b) The scattering cross-section of lossless dielectric spherer’ ka=O.17, f=2OGHz—L4... m— —*—.—.# —‘IMM:RpTMMPTheta(degree)J.0.00 50.00 100.00 150.00483.1.3. Numerical Results for Lossv Dielectric SoheresFig. 3.5 shows the bistatic scattering cross-section of a rain drop under 10 GHzincident plane wave. Its radius is 0.1467 cm and 6r 41.775-j41.204. The result agreeswell with those of Mie scattering solution as shown in Fig 3.5 too.For larger dielectric lossy spheres, the iterative MM also gets good resultscompared with Mie scattering theory. Fig. 3.6(a) shows the bistatic scattering crosssection of a dielectric lossy sphere with the same as above and ka=3.5.When the size of the sphere becomes bigger, the unknowns increase, thereforethe computing time increases quickly. We calculated spheres at ka=3.8 and ka=4.0respectively. The results are shown in Fig. 3.6(b) — 3.6(c). When ka=3.5, unknownsare 1188x3, CPU time is about 95 hours. When ka=3.8 and 4.0, the unknowns are1790x3 and 2175x3 respectively, while the CPU time are about 185 and 294 hoursrespectively. From those calculations we can see that the bigger the size is, the moreunknowns there are, the more error will occur, ie. the round-off errors increase whenthe number of unknowns increase as in all of the numerical methods. Fortunately, inthe calculations later, we do not need to handle so many unknowns, so we can getenough precision. The calculations here show that using the iterative MM, we cansolve problems with many unknowns. Of course here we only show the validity of ourmethod, if we only want to get scattered fields from spheres, we do not have to usethis method here, instead, we can use other methods such as Mie scattering methodor use the MM in surface modelling which can solve larger size spheres than thismethod when the same number of unknowns are involved.49Scattering Cross-section (cm’2) x 10-640.00I I3800-\-36.00—34.00-——a-.-,. irip32.00-30.00-28.00-I,26 00- I’24.oo-‘I22.00- 1’Is20.00-I18.00 -16.00—14.00-12.00- \ I10.00- //8.00-I Theta0.00 50.00 100.00 150.00Fig. 3.5 The scattering cross-section of a lossy dielectricsphere Er 41.775j41.204, ka = 0.154, f= 100Hz50Scattering Cross-section (cm”2)170.00160.00150.00TMM-VP140.00130.00 “I”120.00110.00100.0090.0080.00k70.0060.0050.0040.0030.0020.0010.000.00‘1—--••HPI I I Theta(degree)0.00 50.00 100.00 150.00Fig.3.6 (a) The scattering cross-sections of a lossy dielectricsphere c r41775J41204 f= 10GHz ka=3.551Scattering Cro.e-section (cm’2)240.00 I220.00 -200.00 -180.00 -160.00 -140.00120.00 -100.00-80.00 -60.004O.OOL20.000.0L- Theta(degree)0.00 50.00Scattering Croas-section (cm2)24O.00I220.00200.00 —180.00 F160.00’140.00120.00100.0080.0060.0040.0020.00-0.00— Thet.a(degree)0.00 50.00Fig.3.6 (b) The scattering cross-sections of a lossy dielectricsphere€= 41.775-j41.204 f= 100Hz ka=3.8Mle-Vt’MM:Vp100.00 150.00I Mle-WUP100.00 150.0052Scattering Cross.secüon (czn2)Fig.3.6 (c) The scattering cross-sections of a lossy dielectricspherer=41•775J204f= 100Hz ka= 4- Mie-VP-280.00-260.00240.00220.00 -200.00180.00160.00 -140.00 -120.00-100.0080.00-600040.00-20.000.00 r0.00 50.00 100.00 150.00Theta(degree)I I I I 1-Scattering Crees.sectioD (cn2)280.00260.00240.00220.00200.00180.00160.00140.00120.00100.0080.0060.0040.0020.000.000.00 50.00 100.00 150.00Theta(degree)533.1.4. Numerical Results for Concentric Dielectric SpheresFrom Mie scattering we can get backward and forward scattering cross-sections of the concentric melting-snow particle model A. As shown in Fig.1.1, theinside 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-snowparticle model A, the outer radius is 0.061 cm and the inner radius is 0.041 cm. Thetwo MM get the same solution. Fig. 3.7(a) and Fig. 3.7(b) show the bistatic cross-section 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 forwardscattering cross-section is 0.120755E-04 cm2 and the backward scattering cross-sectionis 0.110149E-04 cm2 at 10GHz (r 41.775-j41.204), our solutions are 0.12378E-04cm2 and 0.110855E-04 cm2 respectively; while at 20 GHz( 6r= 8.867-j30.75), the Miescattering solution in forward and backward scattering cross-sections are 0.202278E-03cm2 and 0.1718507E-03 cm2, our results are 0.2098E-03 cm2 and 0.17863E-03 cm2.In the two important directions, the iterative MM solutions meet Mie scatteringtheory solutions well.For the concentric sphere model, since the outer layer rain shell is muchthinner, we cut a concentric sphere into a set of nonuniform subcells: at the outerlayer water shell, the subcells are smaller than those inside.54Scattering Cross-section (cm”2) x io-612.0011.0010. (degree)Fig. 3.7 (a) The scattering cross-sections of concentricmelting-snow particle model f= 10GHzI I0.00 50.00 100.00 150.0055Scattering Cross-section (cm’2) x 106220.00210.00 -200.00 -190.00 -180.00 -170.00 -160:00 -150.00 -140.00 -130.00 -120.00110.00 -100.00 -90.0080.00-70.00 —60.00 —50.00 —40.00 —30.00 —20.00 —10.00 —0.00 —-10.00 Theta(degree)Fig. 3.7 (b) The scattering cross-sections of concentricmelting-snow particle model f=2OGHzI I H?0.00 50.00 100.00 150.00563.2 Computation ConsiderationsIn order to get accurate results in dielectric scattering calculations, we havesome special considerations as follows:3.2.1. Modelling ConsiderationsThe first step in numerical calculations is to mesh the dielectric body into aset of small subcells so as to accurately represent the physical body. The meshing isrelated to the basis functions and the object modeffing, and will affect the complexityand accuracy of the later calculation. Normally, as discussed in Section 2.2, for objectsof rotation or cylinders, small rings are cut, while for cube cylinder block, small cubesshould be cut. For arbitrarily shapes, triangle patches are chosen as subcells forsurface modelling and cubes or tetrahedral elements are chosen for volumemodelling. The cube elements are the simplest elements in MM calculation, and oneof its primaiy advantages is the ability to model arbitrary shaped inhomogeneousobjects . Our melting-snow particle model to be evaluated is inhomogeneous andirregular, we use small cubes as subcells in all of our calculations.For spheres, meshing is even more important. In our program, a sphere ismeshed 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 cutinto N3 subcubes with length d = D/N, in the modelling each subcell centre ischecked by the program, if the distance rj (from the centre of the subcube to o) issmaller or equals to a (eg. point A in Fig. 3.8), the subcell is in the sphere; if r is57bigger than a (eg. point B in Fig. 3.8); the subcube is out of the sphere. The moresubcells are cut, the more accuratç the modeffing might be.V Fig. 3.8 Meshing of a sphere3.2.2. The Choice of Subcell SizeWhen using a pulse function expression in MM, it is important to establish anupper limit on the dimension of the subvolumes. In theory, for a scatterer, thesmaller the subcell size is, the more accurate the calculation will be. On the otherhand, when the subcell size is getting smaller, more unknowns are needed, and biggercomputer memory and longer computer time are needed. The bigger calculation willalso raise the computer round-off error. So there is a trade-off for the size of subcelland the computer time. To arrive at the limit of subvolume size for our methods, wehave performed several tests.Our results shows that the upper limit of subcell size depends on the methodused, the permittivity and shape of the dielectric object, and the accuracy required.D=2a58If we want to get the scattering cross-section instead of internal field, we can havebigger 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. Butour dielectric bodies are spheres and have curve surfaces which are more sensitiveto the size of subcell, now we have not found publications discussing about the subcellsize limitation for spheres. After several calculations, we get the conclusion that forour objects and the range of permittivities, when a spherical shape dielectric objectis cut into small cubes, the small cube size should not exceed 0.08 in order to yieldreliable data.3.2.3. The Choice of an Initial Guess in the Iterative MMIn Chapter II we have developed some iterative schemes to arrive at theiterative MM solution. The only freedom we have is the choice of an initial guess. Inprinciple, the initial guess can be chosen arbitrarily, eg., = 0, 1,because of its convergence property. But in practice a reasonable guess consideredto be close to the correct result is made, which will save the calculation time incertain extent. Sometimes a pre-knowledge is needed. For example, in the calculationin 3.1.1, when E° equals to 0 and E1/2 respectively, the CPU time is 128.768 and146.800 seconds respectively, j(O) is obtained from the volume equivalence principlein Eq.(2.1).593.3. Discussion3.3.1. The Convergence of the Iterative MMIn the iterative MM, our choice of several parameters and functionsguarantees that the method will converge in a finite number of steps. The solutionX1 converges to the exact solution X in a finite number of steps. The error is reducedat each step. For example, in our calculation in Fig.3.2, at the 1st iteration, thenormalized integrated square error ERRN is 8.30826E+06, and that for the 2nditeration is 7.958E+05. As shown in Table II, at the 18th iteration, the error comesto 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/6iterations are usually adequate, where N is the number of unknowns or basisfunctions. In all of our three-dimensional calculations, we have found that N/6 areusually sufficient and that the rapidity of convergence depends on the geometry ofthe object, the polarizations, the angle of incidence field and the initial guess.3.3.2. Comparison of the Two Moment MethodsFrom the analysis in Chapter II, we know that the two MM are identical forcertain choice of basis functions and the weighting functions. However each methodhas its own advantages and disadvantages.One of the advantages of the conventional MM is that once the matrixelements Z in Eq.(2.21) are computed, they can be stored and reused again fordifferent excitations. The iterative MM does not have this advantage of a stored and60Table II. The Error in Each IterationIteration No. ERRN Iteration No. ERRN1 8.3083E+06 2 7.9581E+053 1.7123E+05 4 3.8529E+045 1.1796E+04 6 2.8775E+037 3.6454E+02 8 9.5679E+019 3.8084E+01 10 1.7555E+0111 1.0386E+01 12 3.3566E+0013 1.4770E+00 14 5.8008E-0115 2.0035E-01 16 8.0714E-0217 4.8032E-02 18 2.1182E-0219 6.4703E-03 20 3.8127E-0321 1.6793E-03 22 9.0186E-04reusable matrix, in each iteration every element has to be calculated individually. Itwill take much longer computer time. For example, in our first calculation in 3.1.1and Fig 3.2, the computing time for the conventional MM is 111.500 seconds but forthe iterative MM the computing time is 175.420 seconds, 1.5 times more than theformer. The Iterative MM requires much less computer storage than the MMbecause it is not involved in a large matrix as in the MM, for example, if we have Nunknowns, the computer storage for the MM needs N2 space while the 1MM needsonly 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. Asdiscussed in [26], on a CDC Cyber 855, the direct MM can handle up to 80 cells, or240 basis functions, while the iterative MM can handle 3666 cells or 11000 basis61functions.For the MM, even after all the terms have been computed to yield thesolution, one can not decide the accuracy of the solution; while for the 1MM, theerror ( or ERRN) at each step of the solution is known and after each iteration thequality of the solution is known, and an incremental increase in the computationaltime will provide a better result.3.3.3. The Volume Integral Equation and the Surface Integral EquationFor 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, theVIE is principally based on relating the induced polarization current to thecorresponding total fields consisting of the scattered and incident fields. By associatingan unknown polarization coefficient either with a cubic or with a tetrahedral cellinside the scatterer, the operator form of the integral equation is converted into anequivalent matrix equation in the MM. It is suitable for arbitrary shape andinhomogeneous scatterers.Normally, in the MM it is more difficult to get convergent results using VIEthan using STE. The STE approach works well in analyzing homogenous dielectricobjects or objects made up of homogenous layers. The usual procedure of thismethod is to set up coupled equations in terms of equivalent electric and magneticcurrents on the surfaces of the homogeneous regions. When the surface of thescatterer takes on arbitrary shape, modelling the surface geometry becomes62complicated.The SIE needs less unknowns than the VIE in modelling the scatterers inmost cases. For conceptual simplicity we consider a homogenous square cylinder anda cube as shown in Fig.3.9[41]. If each side of the square cylinder in Fig 3.10 isdivided into N segments, then the SIB method requires two unknowns ( one for theelectric current and the other one for the magnetic current) for each segment andresults in a total 8N unknowns. The VIE method, however, requires one unknown foreach subsquare resulting in a total N2 unknowns. For the cube in Fig 3.10, the SIBmethod requires four unknowns ( two for the electric currents and two for themagnetic currents) for each subsquare on each face, resulting in 4N2 unknowns foreach face thus 24N total unknowns. The ViE method, however, requires threeunknowns for each subcube resulting in 3N total unknowns.SiE 8 N UNKNOWNSVIE N UNKNOWNSSIE 24 N UNKNOWNSVIE 3 N3 UNKNOWNSFig.3.9 Comparison of the number of unknowns between SIBand VIE63It is also necessary to mention that for dielectric bodies with very lowpermittivities, the STE will lead to worse results[41], while calculation using the VIEmethod as shown in this chapter yields good results.3.4. ConclusionsThis chapter has been devoted to some illustrative numerical results obtainedby the MM and the 1MM discussed in the last chapter. The results obtained havegood agreement with published or known results, which substantiates the validity ofthe methods and the reliability of the programs. In most of the examples, the samepermittivities of snow or water (rain) and practical sizes are used, which will give usthe confidence in the evaluations in the next chapter.The chapter has compared the two moment methods as well as the twointegral equations which are often used in solving dielectric scattering problems. Fromthe analysis we have known that the iterative MM needs less computer storagetherefore it can handle more unknowns than the conventional MM, but it does nothave a reusable matrix so it spends longer computer time than the MM. In theiterative MM, the quality of the solution is known and controllable. Although the1MM needs longer computer time, we can save certain computing time by areasonable choice of the initial guess after some simulations are performed.Concerning the two integral equations which are often used in solvingdielectric scattering problems, the VIE can handle any inhomogeneous and irregularscatterers with any permittivities, but it has to solve more unknowns than the SIE.64The SIB needs fewer unknowns than the VIE for a same size scatterer, but it is notquite effective for solving irregular shape and lower permittivity dielectrics[41]. Nowwe can get the conclusion that using the iterative MM and volume modelling is agood method for solving large, arbitrary and inhomogeneous dielectric problems.In the next chapter, the evaluation of melting-snow particles model willinvolve lower permittivities and irregularly shaped dielectrics. The analyses in thischapter shows our choice of usin2 iterative MM to solve VIE is reasonable.65Chapter IV. Evaluation of Scattering fromMelting-snow Particle ModelIn this chapter, the newly developed melting-snow particle model (model C)is evaluated. The scattering cross-sections at different frequencies, different rain ratesand at low melting factor are obtained by using the iterative MM. The results arecompared with those of model A and B. From the calculation we have found thatdifferent models give significantly different values because of different assumptions.4.1 Introduction4.1.1. The Melting-snow Particle Models[6][42]--[44]Various prediction models used to evaluate the scattering of melting-snowparticles have been developed. The different models have different assumptionsbased on different emphasis, therefore they give different scattering values.In order to look into the scattering of melting-snow particles, some simplifiedgeometric shapes which can be reasonably analyzed are considered. As mentionedin Chapter I, two main models are used to represent melting-snow particles. ModelA is a concentric-sphere model, where a layer of water surrounds a dry snow core ofdensity, and model B is a homogeneous sphere model, where water is assumed toprogressively percolate inside the snow as the particle falls, until saturation isreached. The average permittivity of the melting-snow particle model B is discussed66in [2]. Some other models, for example concentrated ellipse model[43], are assumedtoo.Based on the discussion above, in this chapter, a new model to represent themelting-snow particles is considered. The model is a snow core of permittivity € withseveral even distributions of water drops of permittivity at outside. FromChapter.I we have enough evidence to show that the model C provides a reasonableassumption.4.1.2. Basic ParametersMelting occurs basically because of the higher temperature of the air whichsurrounds the melting particles. The rate of melting also depends on the humidity ofthe air. The melting process, melting rate, and the average permittivity of the meltingparticle in model B are beyond the scope of the thesis, we only simply give severalparameters that will be used in our later calculations. Melting Factor S[4][5]We define a quantity S as the ratio of melted volume of water to the totalvolume of the melting snow particles. For model AS_1_(_!)3 (4.1)a267where a1 is the radius of the unmelted core, a2 is the outside radius of the particle.In the melting layer, S is a function of a depth h into the melting layer, whereh is the position in the melting layer measured from the top of the layer. At the topof the melting layer (h=O), S=O, dry snow particles present; at the bottom (h=1),S=1, rain drops present. As S increases,a1/2decrease with the melting-snowparticles fall through the melting layer. Particle SizeAll the sizes of particles to be evaluated are available in some related previouswork. Here we only give a formula of the radius of the representative melting snowsphere given by [5]:ä-[(O.1+O.9S)EP1Vgj (4.2)where p is the factor related to rain drop with radius a, VRi is the fall velocity ofrain drop of radius a, and Vmj is the fall velocity of corresponding spheres with adegree of melting S andVmi 1.5+(vQ,— 1.5)sinq(4.3) The Permittivity of Rain or SnowThe dielectric property of rain depends not oniy on frequency but also ontemperature. For wavelengths longer than 1 mm, the dielectric property of water is68due to the polar nature of the water molecule, whereas for wavelength shorter than1 mm, it is governed by various kinds of resonance absorptions in the molecule. Thevalue 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 characteristicsof the complex permittivity of water at 00 C.The relative permittivity of snow is 1.2 and the density of it is I I 1—66.00-80.00•-75.00-10.00-66.00-eo.oo• c’-• 66.00--50.00-45.00-40.00 -66.00-25.00-20.00-15.00 —-10.00-6.00 I0.00 20.00 40.00 10.00 6000 100.00Fig. 4.1 Relative permittivity of water vs. frequencies4.1.2.4. The Rain RateThe intensity of rainfall is measured in terms of a rain rate R. If rain consistsof spherical drops of uniform radius a, falling with velocity v, the rain rate R isR-itavN (4.4)69where N is the number of drops per unit volume. When R is measured in mm/h, ain cm, v in m/s, and N in cm3,R48ita3vN*io (4.5)4.2 The Computed Results for the Melting-snow Particle Model CUsing the theory and computer program discussed above, we calculated thescattering cross-sections of the melting-snow particle model C at three rain rates, sixfrequencies and seven melting factors.4.2.1. The Choice of the Number of Outer Layer Water DropsThe model C is derived from the model A (the concentric model), in order tocompare with model A, it is not unreasonable to assume that the total volume of thewater drops at outside in model C equals to the outer layer water shell in model Ain the later calculations. Assume the surrounding air is at the same temperature, thewater drops should be evenly distributed and of equal size. From the physical pointof 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 onlyvalid when S is smaller, ie. the outer layer rain drops are not too big. For large S, thismodel does not conform to physical principle.How many water drops should be chosen in model C? What is a reasonablenumber? Our program can arrange the outer layer water drops with even70distributions on the surface of a snow core. As an experiment, we calculated theforward scattering cross-sections of model C at S = 0.04, rate R = 5mm/h,frequencies f = 10 GHz and 20 GHz, the outside drops number are 6, 12, 24 and 48respectively. As compared with those of model A, at the two frequencies, thescattered fields of model C are always smaller than those of model A, the smaller thenumber of chosen outer layer water drops is, the smaller the scattered fields are. Themore water drops are chosen, the closer to those of model A the scattered fields willbe.It is believed that if we choose even more water drops at outside which willbe closer to model A physically, the result should be closer to that of model A, whenthe outer layer is full of water drops, the model C comes to model A. Consideringthe computer time and memory, we chose 24 as the number of outside water dropsin later calculations.4.2.2. The Computed Results for Melting-snow Particle Model CThe bistatic scattering cross-sections of model C are calculated at S = 0.02,0.04, 0.06, 0.08, 0.10, 0.12 and 0.14, the frequencies are at f = 10, 20, 30, 40, 50, 60GHz, and the rain rate at R=5, 12.5 and 25 mm/h. The selected results of forwardand backward scattering cross sections vs frequencies at different S are shown inFig. 4.2 Fig. 4.8.71Scattering Cross-section (R=5, S=O.02)Ln (Scattering Cross-section (cm”2))21.5le-03 -753-10.0021.5• le-0475321.5le-0575320.00 30.00 40.00 50.00 60.00- Model A, (forward)‘M1?rd-Mo&1B (rdTc (EakarFrequency (GHz)Fig. 4.2. Scattering cross-sections of three models (R=5inmlh, S=O.02)I —••iY•—•W,‘ —F,_—:• /——,——..•. —- _, — _•// ‘‘I -,,—_.,/—. /—//II/I72Scattering Cross-section (R=5, S=O.08)odefA, (fthdY-MO&1B (SakvrJ) -MdTcakar1.510.00 20.00 30.00 40.00 50.00 60.00Frequency (GHz)Fig. 4.3. Scattering cross-sections of three models (R=5mm/h, S=O.08)Ln (Scattering Cross-section (cm”2))7—I I I—53-21.5le-0375-0_/3-21.5le-0475—32—73Scattering Cross-section (R=25, S=O.14)Ln (Scattering Cross-section (cm”2))321.5le-0275321.5le-0375321.5le-04710.00 20.00 30.00 40.00 50.00 60.00Model A, (forward)- - -Mfa-MO&1B (S1ra) -(a&VarFrequency (GHz)Fig. 4.4. Scattering cross-sections of three models (R=Smm/b, S=O.14)I I I I I-—__-‘I—— .———*% %% ——%‘II I —574ModeI-oceB (avrJ)-‘MdT akarle-0475321.510.00 20.00 30.00 40.00 50.00 60.00Frequency (GHz)Fig. 4.5. Scattering cross-sections of three models (R=12.Smm/h, S=O.04)75Scattering Cross-section (R=12.5, 5=0.04)Ln (Scattering Cross-section (cm”2))3—21.5le-03 -7—5-3-21.5——4——— a._ ——‘K//I/II/////—II I I I I75.Model A, (forward)‘M&Jiri1- -.fA,airaj -‘MO&IB (backwrJ)-‘Model ( bakarFig. 4.6. Scattering cross-sections of three models (R=12.5mm/h, S=O.12)Scattering Cross-section (R=12.5, S=O.12)Lii (Scattering Cross-section (cm’2))21.5le-02753-21.5le-03-75321.5le-04753_.___•-— I10.00 20.00 30.00 40.00 50.00 60.00Frequency (GHz)765_r /I/3 -d10.00 20.00 30.00 40.00 50.00 60.00Frequency (GHz)Fig. 4.7. Scattering cross-sections of three models (R=25rnm!h, S=O.04)Scattering Cross-section (R=25, S=O.04)Ln (Scattering Cross-section (cm”2))le-02 -7-5—3-21.5le-03 -7—5—Model A, (forward)Mocel C, (forward) - - --MO&1BM&I akarr.4 — --—4——/3—21.5le-047- I77Scattering Cross-section (R=25, 5=0.12)le-0475I I10.00 20.00 30.00 40.00 50.00 60.00Model A, (forward)M1rd-MO&1B (Eara)-!vIdT (akarjFrequency (GHz)Fig. 4.8. Scattering cross-sections of three models (R=25mm/h, S=O.12)Ln (Scattering Cross-section (cm’2))321.5le-02753-21.5le-0375321.5————— a. .‘— %. .-.. •4%//,-1I I I784.3 The Comparison of the Model A, Model B and Model CNow we have obtained the forward and backward scattering cross -sections ofmodel C. We also have known the forward and backward scattering cross-sections ofmodel A as well as the backward scattering cross-sections of model B by using Miescattering theory. Fig 4.2 — Fig 4.8, show the forward scattering cross-sections ofmodel A and backward scattering cross-sections of model A and model B at differentmelting factor S and frequencies. As may be seen from these figures, differences inspecific forward and backward scattering cross-section values as well as the ratio ofthe scattering cross-sections in two directions occur by applying the different modelsconsidered above.Fig. 4.9 and Fig. 4.10 show the scattering cross-sections of model A and modelC vs. S at R=5 mm/h, f=10 and 40 GHz respectively. Fig. 4.11 and Fig. 12 show thescattering cross-sections of model A and model C vs. R at S =0.06, f= 10 and 40 GHzrespectively.Comparing with the three models, we can get some conclusions in severalaspects:4.3.1. Forward Scattered FieldsCompared with model A, we observed that at the same S, the same rain rate R andthe same frequency f, the forward scattering cross-sections of model C are alwayssmaller than those of model A. The difference increases with S increasing. Forexample, when R = 12.5 mm/h, at S = 0.06 and f = 20 GHz, the forward scattering79Scattering Cross-section (R=5, f=1OGHz)Ln (Scattering Cross-section (cmA2))le-04 A, (forward)•M&di-8 A 1W - -‘i(o&favr65-43.53—- •.D .—2- .•‘—...._—1.5—-I, Ii1e-o:—7t-.6bSx1030.00 50.00 100.00Fig. 4.9 Scattering cross-sections vs. S (R=5 mm/h, f=1O GHz)80Scattering Cross-section (R=5, f=4OGHz)Ln (Scattering Cross-section (cm’2))32.521.5le-03865432.521.5le-04Fig. 4.10 Scattering cross-sections vs. S (R=5 mm/h, f=40 GHz)S x‘MoJiC1 (forward) - - -- -MO&1C Skvr)-.4-4--4.--4—xp.FA———.4I/I /1/I,4.I,I,‘I /I,I,IIt,I0.00 50.00 100.0081Scattering Cross-section (f=1OGHz, S=.06)I I I/ _/ ,,, —./ _II ——I /I ——/ / _II iIIIIIII?’‘III 0/‘Iit /1I,/,‘II0/Model A, (forward)- -od1C,(backwa,rJ5 -Ln (Scattering Cross-section (cm”2))le-0498-7-6-/——I0 —I,I /‘I /I—54.543.53-2.5 -2-1.5I I I I I0.00 - 5.00 10.00 15.00 2000 25.00Fig. 4.11 Scattering cross-sections vs. R (S=O.06, f=1O GHz)R(mm/h)82Fig. 4.12 Scattering cross-sections vs. R (S=O.06, f=40 GHz)ModeI A, (forward)Model C, (forward) --v0aa-1akaN3 -Scattering Cross-section (f=4OGHz, S=.06)Ln (Scattering Cross-section (cmA2))- -0543-2-1.5 -le-03-7-543—2-1.5—le-04- a-S.,— — — ——— _— —— -—I I I0.00 5.00 10.00 15.00 20.00 25.00R(mmlh)83cross-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 withfrequency F increasing. eg., at S = 0.06 and f = 30 0Hz, the forward scattering fieldof 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 scatteredfields of a lossy dielectric body depend on the size, dielectric permittivity anddielectric surface area. The particles of model C and model A have the same snowcore, 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 watershell in model A which could make the scattered field in forward direction becomessmaller. With the increasing of S, the volume of water increases, the effect ofdielectric would be more significant, the difference in forward scattered fields couldbecome larger. When frequency increases, the electric size becomes large, the effectof dielectric could become more significant.4.3.2. Backward Scattered FieldsIn backward scattering, the results are very consistent. The ratio of backwardscattering 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 0.02, the backwardscattering of model C is 0.94 of that of model A, while at S = 0.04 it equals to 0.92;when S = 0.04 and f = 30 0Hz, the backward scattering of model C is 0.88 of thatof model A.84It is noted that at lower frequencies, the backward scattering cross-sections ofmodel C are smaller than those of model A, but compared with forward scatteringcross-sections, at the same S and same f, the difference of backward scattering cross-sections between model A and model C is smaller than that of forward scatteringcross-sections, ie, the backward scattering cross-sections of model A and model C arecloser to each other than forward scattering cross-sections. That shows after splittingthe outer water shell (in model A) into small drops (in model C), the change ofscattering fields occurs. But in backward direction, both the surface penetration andreflection of dielectric exist, so the backward scattering of model C may not affect thescattering so much as in forward direction. It should be also noticed that at higherfrequencies, the scattered fields of the two models are comparable even at lower Sthe backscattering of the model C is bigger than that of model A. In these cases, thehigher the f is, the bigger the difference is; the lower the S is, the bigger thedifference will be. For example, when R = 5 mm/h and f = 60 GHz, at S = 0.02, thebackward scattering cross-section of model C is 1.29 of that of model A, while atS=0.04 it equals to 1.09; when S = 0.04 and f = 50 GHz, the backward scatteringcross-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 moreimportant role than penetration, more reflection is generated, the backscatteringwould become higher.Compared with model B, at certain lower S and lower frequencies, thebackscattering of model C are higher than that of model B, while at higher S and85high frequencies, the backscattering of model B are higher than those of model C andmodel A. These results show the scattered fields will change from different models.4.3.3. The Ratios of Forward to Backward Scattering Cross-sectionsThe forward and backward scattering cross-sections are the two mostimportant scattering directions in evaluating the scattering of scatterers because it iseasier to receive and measure for a radar in the two directions.The ratio of the forward to backward scattered fields is also an importantparameter to evaluate the dielectric scattering properties. We should always considerthe scattering in both directions. Fig.4.12(a) Fig.4.12(d) only show the rates at R= 25 mm/h and S = 0.02 — 0.08 respectively. From the results we have found theratios of forward and backward scattering cross-sections in model C are closer to 1than those of model A, ie, at the same S and f, the scattered field curve for modelC is more “smooth” than that of model A.4.3.4. Rain Rate RWe have calculated the scattering cross-sections at three rain rates R = 5mm/h, 12.5 mm/h and 25 mm/h respectively. For the three rates, at the same S andthe same frequency f, when the backward scattering of model C is bigger than thatmodel 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 = 25 mm/h, the differenceis the biggest. At R = 5 mm/h, the ratios of forward scattering and backward860Fig.4.13 Ratios of forward to backward scattering (R=25 mm/h)01oGHz GHz0 0GHz60GHz6087scattering 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 aresmaller than those of model A ( at lower frequencies and bigger S), the scatteredfields at R = 25 mm/h are the closest to model A, whereas at R = 5 mm/h, thedifference is the biggest, ie. compared with model A, the scattering cross-sections ofmodel 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 becauseit affects the melting snow particle size as in Eq.(4.4).4.4 SummaryThe backward and forward scattering cross-sections of model C at R = 5, 12.5 and25 mm/h, frequency f = 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. Asmay be noted from the above results, the three models differ in their predictions ofscattering levels in the melting layer. From the comparisons at the same S, same fand the same R, we have found:(1). The forward scattered fields of model C are always smaller than those ofmodel A. The difference increases with S increasing and with f increasing at mostfrequencies;(2). Compared with model A, at all frequencies the relative values of88backscattering of model C increases with S decreasing and f increasing;(3). At higher frequencies and lower melting factor S, the backscattering cross-sections of model C are higher than that of model A, in these cases, the higher thef 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 arelower than those of model A, in those cases, the higher the f is, the smaller thedifference 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 Athan forward scattering cross-section;(6). When S is smaller, the backscattering cross-sections of model C are higherthan that of model B. When S is higher, the backscattering cross-sections of modelB are higher than those of model C and model A.(7). The ratios of forward scattering cross-section and backward scatteringcross-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 modelC are the closest to those of model A, where at R = 25 mm/h, the difference is thebiggest, vice versa; and(9). At R = 5 mm/h, for both model A and model C, the ratios of forwardscattering 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 scatteringproperties, the different melted water distributions assumed (model A and model C)89significantly affect the scattering fields. These conclusions show the new model isimportant in the evaluation of the scattered fields of melting snow particles. Themodel can be used to represent the melting snow particle to a certain extent. It issuitable when we consider and analyze the effect of outer water layer on theelectromagnetic scattering. It is a good complement of model A and model B. Itmight 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 thesummation of all the individual scattered fields generated by the subcell currents inthe scatterer. It is a vector superposition procedure, the total scattered field not onlydepends on the magnitudes of individual scattered fields, but also their phases. Whenthe individual phases are cancelled, the total scattered field is smaller, when theindividual phases have same signs, the total scattered field is enhanced. A subcellcurrent depends on the incident field, the dielectric permittivity distribution anddielectric size. In our melting-snow model calculations, when frequency changes, notonly the electric size changes, but the permittivity changes too. It is difficult for us togive a complete explanation of the results obtained above or to predict the scatteredfields by theoretical analysis only. Although the numerical method used is only anumerical experiment, it generates no analytical formula or expression in which togauge how the result might change with a change in dielectric distribution, dielectricconfiguration, body orientation or radar frequency, nor does it state how variousscattering mechanisms come into play. Nevertheless, like any well designed90experiment, 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 ofexperimental observations has been made. When we can not analyze our problemsby using analytical methods, a numerical method is the only choice.91Chapter V. Conclusions and Suggestionsfor Further Research5.1 ConclusionsIn this thesis, a new melting snow particle model, a snow core with severaleven distributions of water drops at the outside, is developed, based on physicalobservations or assumptions and the current commonly used snow particle models.The electromagnetic scattered fields of the melting-snow model are evaluated atcertain frequencies, melting factors and rain rates. The results obtained are comparedwith those of two existing models (model A and model B). The comparison showsthat the different melting snow particles have different scattering properties so thatthe different melting water distributions assumed significantly affect the scatteredfields. Usually the new model has lower scattered fields and a more “smooth” ratioof forward and backward scattered fields than model A and higher backwardscattered fields than model B, at the same frequency f and the same melting factorS.The analysis of the new model gives us a better understanding to the EMscattering property of melting snow particles at the early stage and to the effect ofouter layer water distributions on the scattering of melting snow. Therefore we canpredict the scattering of melting snow layer and then analyze its effect on microwavepropagations further. The model C is complementary to model A and model B.At present, all of the assumed melting snow particle models are based on92different 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 threedimensional, irregular, inhomogeneous and lossy dielectric scattering problem, noanalytical methods can be applied. Normally such problems can be solved primarilyby the conventional moment method (MM), but because of the computer memorylimitations, it can only be used when the dielectric objects are smaller in size or inregular shape, in other words, with fewer unknowns to be solved. The iterativemoment method, which is derived from the conjugate gradient method and the MM,can solve problems involving a larger number of unknowns than the conventionalMM. In this thesis, the iterative MM is used to calculate the scattered fields of thenew melting snow particle model. For result comparison and checking, theconventional MM is discussed too. For cubic dielectric blocks, spheres and concentricspheres, good agreement between the two moment methods, as well as between theiterative MM and published results, are obtained.We have demonstrated the feasibility and, in fact, the applicability of theiterative MM solution to the volume integral equation formulation of threedimensional, irregular, inhomogeneous and iossy dielectric scattering problem.Generally, the method and our computer program also apply to any arbitrarilyshaped 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 ofunknowns. In [26], it only solved the problem with 423x3 unknowns while we havesolved the problem with more than 2175x3 unknowns.93The numerical calculation examples in this thesis prove the equivalence of thetwo moment methods for the special choice of basis functions (pulse functions) andweighting functions (delta functions). The thesis discusses the computer realizationof the two moment methods, compares their advantages and disadvantages.Conclusions regarding the choice of subcell size and the initial guess, the convergenceand precision, computer memory, and computing time are obtained which will haveacademic and pratical interests.5.2 Suggestions for Further ResearchThere is some work, both in theory and experiment, that needs to be done onthe subject of evaluating the melting-snow particle and calculation of the dielectricscattering in the future. We have some suggestions for the further work:1. The model is only an assumption to the melting snow particle, more realobservations to the melting snow process should be made. For example, by changingthe surround air temperature in a lab, the changes of the outer layer of a snow coreshould 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 thewhole melting procedure.3. Use the computer programs obtained to study more diverse geometries anda wider range of parameters. For example, a concentrate ellipse snow core with evendistributions water drops at the outside or a real snow flake should be studied.944. The scattered fields of the model have never been calculated before, thequantity of the calculation precision to the new model is unknown. Some othermethods, if available, should be used to evaluate the model in order to compare andknow the precision of the results obtained.5. The 1MM has its limitations. From the calculations in 3.1.3, although thecomputer memory can handle many unknowns, the round-off errors increase withunknowns increasing, when the number of unknowns is big enough, it cannot getsatisfactory results. Although the iterative MM can solve a larger number ofunknowns than the conventional MM, it needs more computer time. There shouldbe further work to improve the method. In order to save computing time, some workcan be done mathematically or in computer programming. As discussed in [40], abetter method is being developed.95References1. T. Oguchi, “ Electromagnetic wave propagation and scattering in rain and otherhydrometers, Proc. IEEE, Vol. 71, pp. 1029, 1983.2. Y. M. Jam and P. A. Wasson, “Attenuation in melting snow on microwave andmillimetre-wave terrestrial radio links,” Electron. Lett., Vol. 21, pp. 68, 1985.3. M. M. Z. Kharadly and N. Owen, “Microwave propagation through the meltinglayer at grazing angles of incidence,”, IEEE Trans. on AP, Vol 36, pp. 19884. M. M. Z. Kharadly and A. S.-V. Choi, “A simplified approach to evaluation ofEMW 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 withEmphasis of the Melting-snow Layer. M.A.Sc. Thesis, U.B.C., Canada, April 1992.6. R. A. Hulays, M. M. Z. Kharadly, Modelling of melting-snow particles forscattering and attenuation calculations. ICAP’93, Edinburgh, U.K, March 1993.7. J. Moore and R. Pizer, MM in EM: Techniques and Applications. New York: JohnWiley & Sons, 1984.8. S. Li, MM in EM Scattering and Radiation, Beijing, Electronics Industry Press,1985.9. J.H.H. Wang, Generalized MM in EM, New York: John Wiley & Sons, 1991.10. H.C. Van de Huist, Light Scattering by Small Particles, John Wiley & Sons, New96York, 1957.11. G. Mie, “Beitrage zur Optik truber Medien, Speziell Kolloidalen Metallosungen,”Ann. Phys. Vol. 25, pp37’7, 1908.12. F. Moglich, “Beugungerscheinungen an Korpern von Ellipsoidscher Gestalt,” Ann.Phys., Vol. 83, PP. 609, 1927.13. A.C. Aden and M. Kerker, “Scattering of electromagnetic waves from concentricspheres,” J. Appl. Phys. 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Dubberley, “Computation of fields in an arbitrarily-shapedheterogeneous dielectric or biological body by an iterative conjugate gradientmethod,” IEEE Trans. on MTT, Vol. MTT-37, pp.1119, 1989.27. R. F. Harrington, Field Computation by Moment Methods. New York:Macmillan, 1968.28. R. B. Collin, Field Theory of Guided Waves, IEEE Press, 1991.29. T. K. Sarkar, A. R. Djordjevic and E. Arvas, “On the choice of expansion andweighting function in the numerical solution of operator equations,” IEEE Trans.on AP, Vol. AP-33, pp. 988, 1985.9830. D. E. Livesay and K.M. Chen, “EM fields induced inside arbitrarily shapedbiological bodies”, IEEE Trans. Microwave Theory Tech., Vol. MTT-22, No. 12,December 1974.31. T. K. Sarkar, “A note on the choice weighting functions in the method ofmoments,” IEEE Trans. on AP, Vol. AP-33, pp. 436, 1985.32. K. Umashankar and A. 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