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Ann Arbor, MI 48106 MICROWAVE REMOTE SENSING OF RANDOM MEDIA USING MULTIPLE SCATTERING THEORY by SABA MUDALIAR B.S-, University of Madras, 1979 M.S., Syracuse University, 1986 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the Graduate School of Syracuse University December 1990 Approved Date /y. mo ABSTRACT The modified radiative transfer (MRT) theory is used to study electromagnetic wave scattering from a half-space anisotropic random medium. Microwave remote sensing is the application which is of interest here. The MRT equations are solved under the first-order approximation. The scattering coefficients and the emissivities are respectively calculated for active and passive remote sensing. We identify the effects due to multiple scattering by comparing our results with those of single scattering. Several numerical data are shown in order to highlight the characteristics of our results. As an application our theoretical model is used to interpret measured passive remote sensing data of multiyear sea ice. In order to study the validity of the first-order approximation the MRT equations are reexamined. For simplicity the isotropic case is considered. We extend our first-order solutions to obtain higher-order solutions and thus express the backscattering coefficients as an infinite series. The second-order solutions are shown to be important for cross-polarized backscattering. Further, while studying the second- order scattering processes the absence of some 'phase' terms is noticed. We offer explanation for this and suggest that the present MRT equations be further modified. Next we consider a half-space random medium with a random boundary and seek a multiple scattering solution. The Dyson equation and the Bethe-Salpeter equations are derived using the Feynman diagram techniques; these equations respectively govern the mean field and the field correlation. The various scattering processes are identified with the help of the Feynman diagrams. We notice the scattering interaction between the random medium and random surfaces. As the final topic the polarimetric bistatic scattering characteristics of layered random media are investigated. First the bistatic Mueller matrix of a half-space random medium is derived. The power received by the receiving antenna is the quantity chosen to be optimized. For the case when the transmitting and the receiving antennas have identical polarizations the optimum polarizations are derived and the results show they include both linear and elliptical polarizations. Also the conditions for maximum and minimum power are obtained. As further examples the above procedure is applied to two other cases. MICROWAVE REMOTE SENSING OF RANDOM MEDIA USING MULTIPLE SCATTERING THEORY by SABA MUDALIAR B.S-, University of Madr-.s, 1979 M.S., Syracuse University, 1986 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the Graduate School of Syracuse University December 1990 Approved Date D<ttwb» /». Iiio ACKNOWLEDGEMENT First and foremost Z must thank my thesis adviser Professor J. K. Lee for his support throughout my graduate studies. Part of the work reported in this thesis is indeed an extension of his previous work. It is a pleasure to acknowledge that my graduate research was in part supported by Naval Air Development Center. My thanks are due to Prof R. F. Harrington, my master's thesis adviser, for his support during earlier part of my graduate studies. I must thank Professors A. T. Adams and P. P. Banerjee for serving as my thesis readers at very short notice - not to mention the fact that the thesis topic is a bit alien to their primary fields of research interest. Their comments and suggestions have been very useful. Although I did not have much contact lately with Professors R. F. Harrington, A. T. Adams and P. P. Banerjee I am sure that they have always been my wellwishers. Finally I should thank the other committee members Professors M. N. Wellner and E . Arvas for readily consenting to examine my thesis. TABLE OF CONTENTS TITLE PAGE................................................... i ABSTRACT .................................................... ii ACKNOWLEDGEMENT............................................... iv TABLE OF CONTENTS ............................................ v LIST OF PRIMARY SYMBOLS ...................................... viii LIST OF FIGURES .............................................. x Chapter 1. INTRODUCTION AND BACKGROUND ..................... 1 Chapter 2. BACKSCATTERING COEFFICIENTS OF A HALF-SPACE ANISOTROPIC RANDOM MEDIUM .................... 15 Sec. 1 Introduction .................................. 16 Sec. 2 Statement of the problem ..................... 18 Sec. 3 Mean dyadic Green's function ................... 23 Sec. 4 The B-S equation and the MRT equations .......... 27 Sec. 5 Boundary conditions ............................ 32 Sec. 6 Solution of the MRT equations .................. 34 Sec. 7 Backscattering coefficients .................... 41 Sec. 8 Discussion of the results ...................... 46 Sec. 9 Conclusions ................................... 62 Appendix A .......................................... 63 Appendix B ............................................ 65 Appendix C ............................................ 67 Chapter 3. SCATTERING AND EMISSION FROM A HALF-SPACE ANISOTROPIC RANDOM MEDIUM ................... Sec. 1 Introduction..................... Sec. 2 Formulation ................................... 70 71 74 Sec. 3 Solution to the MRT equations .................. 79 Sec. 4 Bistatic scattering coefficients .............. 82 Sec. 5 Emissivities ................................. 85 Sec. € Discussions and Applications .................. 87 Sec. 7 Summary ...................................... 106 Chapter 4. SECOND-ORDER BACKSCATTERING COEFFICIENTS OF A TWO-LAYER RANDOM MEDIUM ................. 107 Sec. 1 Introduction ................................. 108 Sec. 2 Problem statement ............................ 110 Sec. 3 First-order solutions ........................ 115 Sec. 4 Higher-order solutions ........................ 120 Sec. 5 Second-order solutions ........................ 126 Sec. 6 Conclusions .................................. Appendix Chapter 5. 138 ............................................ 140 WAVE PROPAGATION AND SCATTERING FROM A RANDOM MEDIUM WITH A RANDOM INTERFACE ............... 144 Sec. 1 Introduction ................................ 145 Sec. 2 Statement of the problem ..................... 149 Sec. 3 Integral equations for the Green's functions ... 153 Sec. 4 The Dyson equation 160 (i) .......................... Bilocalapproximation ........................ 166 (ii) Nonlinearapproximation ...................... 168 Sec. 5 The Bethe-Salpeter equation ................... 173 Sec. 6 Summary and Conclusions 182 Appendix Chapter 6. ...................... ............................................ 185 OPTIMUM POLARIZATIONS IN THE BISTATIC SCATTERING FROM LAYERED RANDOM MEDIA .................... 187 vii Sec. 1 Introduction ................................ 188 Sec. 2 Polarization ................................ 190 Sec. 3 Description of the problem ................... 192 Sec. 4 Solutions .................................... 196 Part I Half-space radom medium................ 196 Part II Two-layerrandom medium ................. 209 Sec. 5 Sec. Chapter 7. Summary and a few comments ................... .. 213 6 Conclusions .................................. 215 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ........................... ‘..... 216 BIBLIOGRAPHY................................................. 219 Biographical note.................. 231 LIST OF PRIMARY SYMBOLS In what follows we provide a list of important symbols and their brief descriptions. The superscripts ' and " are used to denote the real and imaginary parts, respectively, of the quantity. The subscript m is used to denote the mean part of the quantity. The subscripts i and s are used to indicate the incident and the scattered directions, respectively. The subscripts u and d are used to denote, respectively, the upward and downward travelling waves. e1 permittivity of the anisotropic medium 8if(r) random permittivity fluctuation k n wave vector in region n V tilt angle of the optic axis Q, q, Q defined in (2.10) s s GQ1, dyadic Green's functions for source in region 1 and observation points in regions 0 and 1, respectively C(f -fg) correlation function of the random permittivity fluctuations <D(P) spectral density of the correlation function k transverse wave vector T) effective propagation constants P A 0 unit electric field vector for an ordinary wave A , e unit electric field vector for an extraordinary wave h unit electric field vector for horizontally polarized wave v unit electric field vector for vertically polarized wave 1 , Ij u d incoherent field intensities I coherent field intensities mu , I . md extinction matrices phase matrix for the incoherent intensity phase matrix for the coherent intensity reflection matrix of the wave at i-j interface transmission matrix of the wave at i-j interface bistatic scattering coefficient backscattering coefficient normalized variance of permittivity fluctuations lateral correlation length vertical correlation length defined in (3.4) emissivity with polarization P scalar Green's function matrix of the scalar Green's function matrix of the unperturbed scalar Green's function scattering matrix Mueller matrix thickness of the random medium layer Stokes vector electric field in region n X. LIST OF FIGURES Scattering geometry of the problem.......... 19 2.2 Geometrical configuration of the permittivity tensor....... 20 2.3 Wave-3cattering processes .............................. 47 2.4 Incident angle response of Re(SX )/Re(kf ), p- od, ed ..... p lz 51 2.5 Incident angle response of Im(SX )/Im(k? ), p= od,ed ..... p lz 52 2.6 Incident angle response of or „ and a ............... MRT Born 54 2.7 Incident azimuthal angle response of o.„„ and CT_ ....... MRT Born 55 2.8 Comparison of a from MRT and Born resultsas a vv . 2.1 function of normalized variance ................. 56 2.9 Effect of medium loss on a .......................... Born 57 2.10 Effect of medium loss on 58 2.11 Frequency response of 2.12 Frequency response of Im(5X )/Im(kf ), p = o d , e d ...... p lz 3.1 Scattering geometry of the problem...................... 75 3.2 Wave scattering processes .............................. 88 3.3 Yhh versus scattered angle ............................. 92 3.4 Scattering coefficients for various observation angles. I .. 93 3.5 Scattering coefficients for various observation angles. II 3.6 ©mrt anc* eBorn versus observation angle ................. 96 3.7 Emissivity versus observation angle, ei' isthe parameter .. 97 3.8 e”RTand eRorn versus normalized variance ................ 99 3.9 Frequency response of eBorn and eMRT .................... 100 3.10 ©Born and eMRT versus correlation length ................ 102 3.11 Interpretation of measured emissivity datafrom seawater... 103 3.12 Interpretation of measured emissivity MRT and 60 61 . 94 xi data from multiyear sea ice ....................... 104 4.1 Geometry of the problem.................................. Ill 4.2 Scattering processes for 119 4.3 Scattering processes for a 4.4 Scattering processes resulting from "phase" terms (2) ............................ corresponding to S ^ 2 ............................ 4.5 134 Comparison of MRT and Born results of ®vhas a function of 5 (variance of random fluctuation) ..... 4.8 133 Incident angle response of cross-polarized backscattering coefficients by MRT and Born ....... 4.7' 131 Frequency response of cross-polarized backscattering coefficients by MRT and Born ..................... 4.6 129 135. Comparison of first- and second-order backscattering coefficients <!,_ ................................. 137 hh 5.1 Geometry of the problem................................. 5.2 Scattering process ...................................... 183 6.1 General scattering geometry ............................. 6.2 Geometry of the half-space isotropic random medium problem...................... 150 193 197 A 6.3 (a) Location of P ^ x for backward scattering ............. 207 A (b) Location of P ^ x for forward scattering ............. 207 CHAPTER 1 INTRODUCTION AND BACKGROUND Microwave remote sensing of terrain media has been a gradually growing field for the past couple of decades. Earlier aerial photography has been the only means available; whereas, in the study of ocean bed, acoustic imaging has been widely in use. The reasons for the present increasing interest in microwave remote sensing are many. First, clouds which are detrimental to optical remote sensing are transparent to microwaves. Secondly, since microwaves have the capacity to penetrate deep into targets such as vegetation, sea ice, soil, etc., they possess the potential of obtaining valuable information about internal constitution of the targets. Also the capability of dayand-night operation and all-weather operation are added assets. Other notable features which are in favour of microwave remote sensing are highlighted in many review articles [Staelin, 1969; Tomiyasu, 1974; Moore, 1978; Kritikos and Shiue, 1979; Njoku, 1982]. Finally, it should be noted that the information obtained by microwaves is often different from or complementary to the information obtained by other means such as infrared, visible light, sound, etc. In theoretical microwave remote sensing our objective is to electrically characterize the target and then proceed to compute the scattered field given some incident source. In the case of remote sensing of terrain media it is impossible to characterize the targets exactly. Even if this were possible, one confronts the virtually impossible task of having to compute the scattered field. On the other hand, we observe that it is really not the exact solution which is o£ interest to us but rather the average behaviour of the scattered field. Thus to this end it is sufficient to characterize the target by some statistical parameters [Ishimaru, 1977]. There generally exist three ways of statistically characterizing targets, viz., as random discrete scatterers, random continua and random rough surfaces. The above classification is primarily done for convenience and theoretical work has thus far proceeded along these lines. In practice, however, targets have to be modelled as combinations of one or more of the above three. As one would expect, this is a fairly complicated problem and not much work has been reported so far in this topic. We address such a problem in Chapter 5. For now, let us assume that the surfaces we encounter are smooth. Now one can model the medium as a continuum having randomly varying dielectric constant [Gurvich et al., 1973; Stogryn, 1974] or one can consider the medium to be made up of discrete scatterers whose size, shape and position are random quantities [Tsang and Kong, 1977; Lang and Sidhu, 1983]. Although both these models are equally useful, there are several other factors one has to take into consideration. For, in practice, it is very difficult to determine by measurement any of the above-mentioned random parameters. One such example is a vegetation medium which is a mixture of air, water and biofibre. Since one can measure only components of the mixture, the average permittivity of the mixture has to be estimated using a discrete scatterer model [Fung and Ulaby, 1978] . In this thesis we concentrate on the random continuum model and that is what we shall have in mind, hereafter, when we mention random medium. In the active remote sensing we are primarily interested in scattering coefficients. Stogryn [1974] first calculated the bistatic scattering coefficients for a random medium with spherical correlation function. Following a perturbation approach, Tsang and Kong [1976] studied the scattering of electromagnetic waves by a half-space random continuum model. They employed the Born approximation which is essentially a single scattering approximation. Later Zuniga et al. [1979] and Zuniga and Kong [1980b] studied the scattering from a layered random medium using the Born approximation. It is intuitively clear that the Born approximation is appropriate only when the fluctuations are small [Ishimaru, 1978]; if not, some other procedure which involves multiple scattering should be used. In order to include the effects of multiple scattering we can either follow the wave approach or the radiative transfer (RT) theory [Chandrasekhar, 1950; Sobolev, 1963]. Green's function formalism provides a convenient way to study multiple scattering [Frisch, 1968]. Further, use of the Feynman diagram technique enables us not only to manipulate the terms in the Neumann series but also helps us to identify the various scattering processes involved. The selective summation procedure leads us to two deterministic integral equations [Frisch, 1968] : the Dyson equation for the mean field and the Bethe-Salpeter equation for the field correlation. To understand the coherent wave propagation in a random medium the Dyson equation must be solved. The exact solutions to the Dyson equation are impossible to obtain. There are two often-used approximations, viz., the bilocal approximation [Bourret, 1962] and the nonlinear approximation [Furutsu, 1963]. The bilocally approximated mean field has been calculated extensively for various cases by Tatarskii and Gertsenshtein [1963], Tatarskii [1964], Keller [1968], Brown [1967], Rosenbaum [1969] and Kupiec et al. [1969]. The mean Green's function with the nonlinear approximation was first calculated by Rosenbaum [1971] for the unbounded random medium using the Fourier tranform method. Tsang and Kong [1976, 1979] used the two-variable expansion technique to find the nonlinearly approximated Green's function for the two-layer random medium with three-dimensional fluctuation [1979]. Tan and Fung [1979] solved the vector problem to obtain the mean dyadic Green's function (MDGF) for the half-space case; whereas Zuniga and Kong [1981] obtained the corresponding MDGF for the two-layer case. To investigate the behaviour of the scattered field intensity in a random medium, one starts by solving the Bethe-Salpeter (B-S)- equation which represents an exact relation between the second moment of the field and the statistics of the medium. The covariance or the field correlation in an unbounded random medium has been obtained by Tatarskii [1964, 1961, 1971], Brown [1967], Frisch [1968] and others while solving the B-S equation with the so-called ladder approximation by the method of successive iteration. In the case of multiple scattering, the method of iteration involves solving many integrals and leads to complicated results after one or two iterations. Under the assumptions of far field interactions and incoherence among waves travelling in different directions, the RT equations have been derived from the B-S equation to study multiple scattering [Barabanenkov and Finkelberg, 1968; Ishimaru, 1975]. We see that both formulations, viz., the wave approach and the RT approach, have their share of limitations and merits. As an improvement of the RT theory Tsang and Kong [1976, 1979] derived the modified radiative transfer (MRT) equations from the B-S equation for the case of scalar wave propagation in a one dimensional two-layer laminar structure and later in a threedimensional half-space random medium. They are 'modified' because the coherent effects between waves in different directions are included. We point out that the MRT theory was developed by applying the nonlinear approximation to the Dyson equation together with the ladder approximation to the B-S equation. These two approximations have been shown to be energetically consistent with each other and therefore appropriate in the development of a radiative transfer theory. Zuniga and Kong [1980a] developed the MRT theory for the electromagnetic field intensity in a two-layer random medium with three-dimensional permittivity fluctuations. Since all the above-mentioned investigators have used an isotropic constitutive relation for the medium there wa3 no depolarized backscatter in their first-order results. This stands in contradiction to several experimental observations of natural targets where there is a significant amount of depolarized backscatter. In an effort to account for this, Tan and Fung [1979] used a model with an anisotropic correlation function and on a first-order renormalization obtained cross-polarized backscatter. But the magnitude of the cross-polarized term was very small compared to the like-polarized term. By using a second-order renormalization Tan et al. [1980] obtained a much higher level of cross-polarized return. Also, Zuniga et al. [1980] obtained depolarized backscatter from a two-layer random medium using a secondorder Born approximation and asserted that the cross-polarized return is a second-order phenomenon. The above is true only if the medium is isotropic. Several natural objects such as sea ice, certain row crops etc. have been observed to demonstrate intrinsic anisotropic charateristics. Due to the development of brine inclusions inside the ice crystals, it has been found by Sackinger and Byrd [1972a, 1972b] that the dielectric loss of sea ice is greater when the electric field is parallel to the inclusions than when the field is perpendicular to them. Campbell and Orange [1974] observed the electrical anisotropy of sea ice in the horizontal plane. They discovered the dependence of signal amplitude reflected from the ice/water interface on the azimuthal orientation of the polarized antenna used, particularly for the first- year sea ice. Kovacs and Morey [1978, 1979] found the crystal structure of the sea ice to have a horizontal c-axis with a preferred azimuthal orientation, which causes sea ice to have an electrical anisotropy in the horizontal plane. Several measurements of backscattering coefficients of sea ice [Onstott et al., 1979; Delker et al., 1980] also strongly suggest that the permittivity of the medium should be modelled as an anisotropic tensor. There are several other experimental data which corroborate the anisotropic dielectric behaviour of sea ice [Weeks and Gow, 1978, 1980; Hoekstra and Cappilino, 1971; Vant et al., 1974, 1978; Weeks and Ackley, 1982; Morey et al., 1984]. For terrain media such as vegetation fields with row structures and vegetation canopy with preferred azimuthal orientation, the random medium should also be characterized by an anisotropic tensor, as suggested by the following observations. Loomis and Williams [1969] reported that some varieties of maize and sorghum leaves have a preferred direction of orientation in their azimuth. Batlivala and Ulaby [1976] studied the effect of the radar look direction relative to the row direction on radar returns from row crops. Ulaby and Bare [1979] further Investigated the dependence of the radar backscattering coefficients of agricultural fields on the azimuthal looking angle, which Indicates the anisotropic dielectric property of the vegetation fields with row structures. Brunfeldt and Ulaby [1986] showed that microwave emission from vegetation canopies planted In parallel rows confirm their Intrinsic anisotropic behaviour. Thus we conclude that an anisotropic model Is necessary for several remote sensing problems. The subject of electromagnetic wave propagation in anisotropic media is fairly old [Clemmow, 1963]. Some extensive work dealing with various kinds of anisotropic media was reported by Kong [1975]. Dence and Spence [1973] studied the problem of wave propagation in an unbounded randomly anisotropic medium. In fact they presented a fairly comprehensive approach (using a Green's function formulation) to include multiple scattering. For the case of layered anisotropic medium Lee and Kong [1983] derived the dyadic Green's function (DGF) and presented it in a form suitable for applications in remote sensing problems. With the availability of the DGF, Lee and Kong [1985a] proceeded to derive the backscattering coefficients of a layered anisotropic random medium by using the Born approximation. They showed that cross-polarized backscatter is indeed a first-order phenomenon in anisotropic media such as sea ice, for example. Lee and Kong [1985b] also obtained the emissivities for the two-layer anisotropic random medium under the Born approximation. However, the Born approximation is inappropriate for studying wave scattering from objects such as sea ice which have fairly strong inhomogeneities. In such cases we need a multiple scattering solution with an anisotropic random medium model. In Chapters 2 and 3 we obtain such a solution and discuss its applications in active and passive remote sensing. We turn our attention now to random surfaces and examine the methods available at present for dealing with them. The scattering of waves from randomly rough surfaces has been the topic of study for many years; but still a lot of progress remains to be made. There exist two large monograghs [Beckmann and Spizzichino, 1963; Bass and Fuks, 1979] devoted to this topic. Depending on their nature rough surfaces can be statistically characterized in several different ways. For example, one can treat a rough surface as a vertical variability [Rice, 1951], horizontal variability of deterministically defined elements [Twersky, 1957; Biot, 1968], a combination of two types of random surfaces (one superimposed on the other [Valenzuela, 1968; Barrick and Peake, 1968; Fung and Chang, 1969]) or a combination of deterministic and random surfaces. A comprehensive list of various methods of modelling rough surfaces may be found in Ruck et al. [1970]. The disciplines in which one has the need to study wave scattering from rough surfaces are many and diverse. Some of them are: classical optics [Toigo et al, 1977; Maystre, 1984], acoustics [Rayleigh, 1945; Waterman, 1968; Zipfel and DeSanto, 1972], electromagnetics [Rice, 1951; Brown, 1978], particle physics [Shen and Maradudin, 1980], etc. The available literature appears to be overwhelmingly plentiful and diverse both in methodology and applications. However, with respect to analytic closed form solutions, we observe that the major criteria on which most of the analyses are based are two - small scale roughness and large scale roughness. In the first case the amplitude of the surface variations and the slopes of the surface are small. In this limit, one can use the small perturbation approximation (SPA). This method was first used by Rice [1951] to study electromagnetic scattering from perfectly conducting random rough surfaces. The idea of SPA has been used in various different formulations [Nieto-Vesperinas, 1982; Itoh, 1985]. When the scale of roughness is large, but small compared to the correlation length, the method often used is the Kirchhoff approximation (KA) [Beckmann and Spizzichino, 1963]. Under this approximation the field at any point on the surface is approximated by the field that would be present on the tangent plane at that point. Thus we see that the regions of applicability of SPA and KA belong to opposite ends of the frequency spectrum. This restriction on the applicability of the two methods limits the kinds of surfaces that one can study at a given frequency. However, if the surface can be considered as a combination of the above-mentioned types, then one can intuitively use a linear superposition of the two methods. Although there is no sound justification for this superposition criterion the theoretical results thus obtained agree reasonably well with the measured data [Fung and Chang, 1969; Barrick and Peake, 1968]. Still, it is clear that there exists a whole class of surfaces for which none of the above approximations apply. A method which is claimed to encompass the entire range of rough surfaces has been proposed by Bahar [1981]. He shows that SPA and KA results are merely two special cases in his method. All the methods discussed thus far are useful if one needs to consider single scattering only; there does not exist a straightforward way to extend them to include multiple scattering. To remedy this, 10 DeSanto [1981] introduced a Green's function formulation to construct an integral equation (Lippman-Schwinger type) for the coherent wave. He analyzed the scattering mechanisms using a diagram technique. Brown [1982] introduced a stochastic Fourier transform approach for studying scattering from a perfectly conducting random surface but the results are in symbolic notation and are not in readily computable form. Another very attractive method for treating multiple scattering was introduced by Furutsu [1985] and Itoh [1985] . They used the concept of surface impedance and under a Green's function formulation arrived at an integral equation very similar to the one obtained in a random medium problem. Finally, one should mention the matrix method introduced by Fung and Eom [1981a] to study multiple scattering from a large scale rough surface. On using KA and appropriate shadowing correction they numerically verified that their method satisfied the principle of energy conservation. The papers dealing with rough surfaces mentioned so far assumed that the medium bounded by the rough surface is homogeneous. Often this kind of model may be inadequate. A terrain, for instance, should be modelled as a random half-space with a random boundary. For this kind of problem there is no method as yet which can provide analytic closed-form solutions. Fung and Eom [1981b] extended their matrix method [1981a] to this probem. They [1982] proceeded to model snow and sea ice and demonstrated the usefulness of their model. However, their methods do not offer physical insight into the underlying scattering mechanisms. We shall therefore study this problem in Chapter 5 using the Feynman diagram techniques. One of the prime objectives of remote sensing is target identification. While other parameters such as frequency response, incident angle reponse, etc. have been widely used for this purpose, the area of polarization diversity has not been well exploited; it has been restricted primarily to TE and TM polarizations. This has largely been due to technological difficulties although theroetically the potential for polarization diversity has been well highlighted by Huynen [1970] as early as 1970. Thanks to the recent advances in technology the topic of polarimetry is once again drawing considerable attention [Giuli, 1986; Cloude, 1983]. A simple and yet useful quantity called 'polarization signature' has been introduced by a group from Jet Propulsion Laboratory to study the polarization sensitivity of targets [Van Zyl et al., 1987a]. Polarization signature represents pictorially a complete polarimetric return of the target under study. Zebker et al. [1987] used this in terrain imagery. More recently, the phase of the scattering matrix elements is found to possess the potential for discriminating targets [Boerner et al., 1987; ulaby et al., 1987]. Since there are many parameters influencing polarimetric remote sensing it is desirable to obtain an optimal polarization given the target scattering matrix or the Mueller matrix. Investigations into optimal polarizations have been made by Van Zyl et al. [1987b] and Kostinski and Boerner [1986] . As applications they have obtained optimal polarizations for a terrain medium based on measured Mueller matrix. Their results are, from a practical standpoint, very useful. On the other hand the enquiry into the optimal polarizations for a theoretically modelled target is also of equal interest. We shall take up this task in Chapter 6. Before we end this chapter we shall briefly outline the contents of each of the following chapters. In Chapter 2 we consider a half-space anisotropic random medium and seek the backscattering coefficients. To involve multiple scattering the MRT theory [Lee and Kong, 1988] is used. We employ the first-order approximation and obtain the solutions for the intensities. The backscattering coefficents are calculated and cast in a form suitable for physical interpretation. They are compared with those obtained using the Born approximation. We consider several examples and, with the help of computed data, study some characteristics of our results. The passive remote sensing of the half-space anisotropic random medium is studied in Chapter 3. Here the bistatic scattering coefficients are derived. We observe the effects due to multiple scattering. In passive remote sensing, emissivity is the primary quantity of interest. Therefore the emissivities are calculated and compared with those obtained using the Born approximation. Once again we study our results using computed data. As an application our theory is used to interpret the passive remote sensing data of multiyear sea ice. In both previous chapters the first-order approximation is made to solve the MRT equations. But this approximation has been called into question. We therefore investigate this issue in Chapter 4. In order not to complicate things unneccesarily we focus attention here on a two-layer isotropic random medium. First, the first-order approximation is used to obtain the backscattering coefficients. We continue this procedure successively and obtain higher-order solutions. The backscattering coefficients are expressed as an infinite series and this brings the importance of the second-order solutions to notice. Then the second-order backscattering coefficients are cast in a form suitable for physical interpretation. We notice the absence of some 'phase' terms in the second-order results and point out the need for modifying the present MRT equations. The second-order backscattering coefficients are computed for some typical examples and compared with those of Born. In Chapter 5 we consider the problem of scattering from a random medium with a random interface and seek a multiple scattering solution. First the integral equation for the mean Green's function is derived. On assuming Gaussian statistics the scattering processes are discussed with the help of Feynman diagrams. We apply two types of approximations to the Dyson eqaution, viz., (i) the bilocal approximation and (ii) the nonlinear approximation and discuss their implications. We proceed to derive the B-S equation and again use Feynman diagram techniques to simplify our analysis. In this case the ladder approximation is applied to the B-S equation. In Chapter 6 the polarimetric bistatic scattering characteristics of the layered random media are investigated. We first consider a half space random medium and derive the bistatic Mueller matrix underthe Born approximation. The power received by a receiving antenna isthe quantity chosen to optimize. For the case when the polarizations of the transmitting and receiving antennas are identical we calculate the optimum polarization. We also determine the conditions for maximum and minimum received power. As further applied to study two other cases. examples the above methods are Chapter 7 concludes this thesis with a brief summary of the work performed. A few suggestions for future work are also Included. CHAPTER 2 BACKSCATTERING COEFFICIENTS OF A HALF-SPACE • ANISOTROPIC RANDOM MEDIUM In this chapter we study the electromagnetic wave scattering from a half-space anisotropic random medi’m. The ladder approximated BetheSalpeter equation is used in conjunction with the nonlinearly approximated Dyson equation to derive the modified radiative transfer (MRT) equations for wave propagation in the half-space random medium. The MRT equations are solved under a first-order approximation. Backscattering coefficients 1 are calculated and are compared with those obtained using the Born approximation. The first important thing noticed is that the propagation constants in the Born results are changed to effective propagation constants. Secondly, there are some additional terms contributing to the backscattering enhancement which is an important direct result of the MRT theory. Several numerical results are illustrated to compare the MRT and the Born results. 16 2.1 INTRODUCTION Over the last decade, volume scattering from bounded medium has been extensively studied. In a continuous random medium model,the medium has been modelled as one with randomly fluctuating dielectric constant. The two important approaches, viz.,the radiative transfer theory and the wave theory, have both their own advantages and disadvantages. While the radiative transfer theory handles multiple scattering very easily, it nonetheless loses 3ome phase information. On the other hand, in the wave theory, multiple scattering introduces too many complications to handle. As a compromise between the two, Tsang and Kong [1976] introduced the socalled modified radiative transfer (MRT) theory. In the MRT theory the equations for intensities are obtained from the nonlinearly approximated Dyson equation and the ladder-approximated Bethe-Salpeter equation. Tsang and Kong [1976, 1979] have solved the scalar problem. Later, Zuniga and Kong [1980a] solved the corresponding vector problem. Since all the authors considered isotropic media, there was no cross-polarization term in their first-order results. However, several experimental observations have shown that, in many cases, there is a significant amount of depolarization. In an attempt to account for this, Tan and Fung [1979] used a model with anisotropic correlation function and, on a first-order renormalization, obtained cross-polarization terms. But the level of cross-polarization was very small compared to the like-polarized terms. By using a second-order renormalization Tan et al. [1980] were able to obtain much higher level of cross-polarization. Also, Zuniga et al. [1980] obtained depolarization terms using a second-order Born approximation and claimed that cross polarization is a second-order phenomenon. At this stage, one has to remember that when one tries to use one of the above theories to match a set of measured data, one is assuming, without much grounds, that the medium is isotropic. Quite a few objects like sea ice, certain row crops, etc. have- been observed to be anisotropic [Campbell and Orange, 1974/ Kovacs and Morey, 1978; Brunfeldt and Ulaby, 1986]. Hence, it is apparent that the existing theories need to be extended to anisotropic media. Although the study of wave propagation in an anisotropic medium dates back to 1973 [Dence and Spence, 1973], not much further work has been reported. The extension of the MRT theory to an anisotropic medium could not be pursued until Lee and Kong [1983] derived the dyadic Green's functions (DGF) for a layered anisotropic medium. With the availability of the DGF, Lee and Kong [1985a] proceeded to derive the backscattering coefficients of a layered anisotropic random medium by using the first-order Born approximation. They showed that the cross polarization is a first-order phenomenon. Later, Lee and Kong [1988] formulated the MRT theory for a two-layer anisotropic rendom medium; but the MRT equations remain to be solved. In this chapter we provide the solutions for the MRT equations for the case of a half-space anisotropic random medium under a firstorder approximation. The backscattering coefficients are derived and are compared with those obtained by using the Born approximation. 18 2.2 STATEMENT OF THE PROBLEM The scattering geometry of the problem is shown in Figure 2.1. We denote regions z > 0 and z < 0 by region 0 and region 1, respectively. Region 0 is free space with permittivity e ^. Let the permittivity of the anisotropic random medium in region 1 be e1(r) - < e^r)) + elf (r) where ( e^r)) s (2 .1 ) elm is the mean part and elf (r) represents the fluctuating part so that the fluctuating part has zero mean. Also we =* _ as assume that e._(r) is small compared with 8, .We have an incident if lm electromagnetic plane wave E Oi i(kQi-r - COt) e incident at angle 0 ,. Let the scattered angle be 0 as shown in Figure Ux 3 2.1. The incident and scattered propagation vectors are denoted by k ^ and kg, respectively. S3 S Both 8. lm and 8 If (r) are taken to be uniaxial with the optic axis (z') tilted off the z axis by an angle V as shown in Figure 2.2. The permittivity tensors, 8^°^ and 8 lm l3e^ore tilting are given by e1 0 0 0 zx 0 0 0 8 lz (2 .2) 19 Z REGION 0 z= 0 Figure 2.1 Scattering geometry of the problem. 20 } OPTIC AXIS OF f PERMITTIVITY TENSOR OBSERVATION POINT Figure 2.2 Geometrical configuration of the permittivity tenaor. 21 eif(r) 0 ?<°U> 8lf(E) (2.3) 8lzf(E) = qz 8lf(E) (2.4) We assume that where q i s a deterministic constant. This physically means that 8If(r) and elzf(£) have the same statistical properties except for strength of fluctuation. After tilting, the above tensors become e lm 11 0 = 0 ?1£<«> - 0 0 e £ 22 E __ 32 8lf(r) (2.5) 23 E__ 33 1 0 0 c [22 c [32 0 0 (2 .6) q23 q33_ 8lf(r) q where 2 Z.. *■ e1 , 622 11 823 " 832 = (clz'ei) 003 V 3 i n V . = e1 s m 2 2 * E^cos \|f + E^sin V 2 + elzcos ( .7a) (2.7b) (2.7c) 22 (2.8a) 2 2 q33 =■ sin V + q^oos V (qz~ 1) cos \|f sin V (2.8b) (2.8c) Both regions 0 and 1 are assumed to have the same permeability |l. Our aim is to calculate the backscattering coefficients for the medium described above. Let us outline the procedure involved. First, we solve the nonlinearly approximated Dyson equation to obtain the mean DGF. Next, we construct the ladder-approximated Bethe-Salpeter (B-S) equation for the second moment of the field. From the B-S equation we proceed to derive the MRT eqautions for the upward and downward travelling wave intensities. Under a first-order approximation we solve the MRT equation in conjunction with appropriate boundary conditions and henceforth we obtain the backscattering coefficients. 2.3 MEAN DYADIC GREEN'S FUNCTION The mean dyadic Green's function (MDGF) in an anisotropic random medium satisfies the Dyson equation which under the nonlinear approximation [Lee and Kong, 1985c] takes the following form. 5llm<e' V " SU (2' V + J ^*2 Gn < Q(21) •Gllm(El'22) ’ • Gl l m < W (2'9> where Q(r) - C02|x Elf<r) " (02\l 8lf (r) q - Q(r) q (2.10) or in an integro-differential form V X V x Sllm(3f' V ‘ *lm’ Gllm(2'20) " 5 8<e’V I + | d3r2 <0(I) Q(Z2)> 5 • Slln(I,*2)- 5 ■ SllBl<I2.I0) (2.111 The volume integration extends over the half-space of anisotropic random medium. The first and second subscripts of the MDGF indicate the regions containing the observation point and the source point, respectively. The third subscript m indicates that the DGF is the mean DGF. Assuming the random medium to be statistically homogeneous, the two point correlation function of fluctuation depends only on the separation between the points. Hence < QC^) Q<*2)> - (2.12) and it can be written in terms of its Fourier transform or spectral density as 24 4 C(r1-S2) - 8 k^m f 3 ~iO* ^r i ”r 2^ d tt <D(Ct) e J (2.13) The solution for the MDGF for the layer case has been obtained by Lee and Kong [1985c]. We take that result, apply the limit as d •) », and arrive at a solution to (2.11) for the half-space problem as follows: i l l m (r'r 0 ) " ~ m ) T J d % = > *llm p A. r 9 l l m (iy it)O U z o " l 00^ 6 — A + A 2 (V z' V [ A ° . o(k“_) e lz — Cl a V <2 -1 4 > inO Q,z O ° (“kJ«) 6 ♦ A3 ,Ep, i u f t e1" - 2 ♦ V [ B^kJ exp( iSp - < p - p 0 >) y J,*-, } -iT| z° -ill z. . ou . „ - , a eu. eu 0 w“ + B, (kj e(k^“) e 3 p lz ill o 'on z ° <kl « ) e — a + C 2 (V in o ,z od o(“k l.) e A #ireu\ e^ e u z + C4 (kp) a ed e(klz) e^ e d z 1 J + e(klz) t vy r. «r v * „ o . o(kiz) e -in ou z. a 0 + 3 P eu. -in z„ ‘eu 0 e(kiz) e (2.15a) where 26 The n's above in (2.17) are the effective propagation constants in the zdirection of the four characteristic waves propagating in the anisotropic random medium. They are the ordinary and extraordinary, upward and downward propagating waves. o(±k° ) is a unit vector in the direction of lz the electric field for an ordinary wave and e(k?U) or e (kf^) is a unit lz lz vector for the electric field for an extraordinary wave. Physically, we note that o is linearly polarized perpendicular to the plane formed by the optic axis (z'axis) and the propagation vector, and e is linearly polarized parallel to the plane formed by the optic axis and the propagation vector. Explicit expressions for the above-mentioned unit vectors are given in Lee and Kong [1983]. The other variables appearing in (2.15) - (2.17) are explained in Appendix A. Note that there are no downward "source" waves for 33 waves for in (2.15a) and no upward propagating ^ in (2.15b). This is because we do not have a reflecting boundary below the source point (z ■ z^ ) . 2.4 THE B-S EQUATION AND THE MRT EQUATIONS The second moment of the field satisfies the B-S equation which under the ladder approximation (for our half-space problem) takes the following form [Lee and Kong, 1988] where E, (r) a <E,(r)> is the mean electric field and E,(r) m 5 (r) lm l 1 1 E, (r) is the incoherent electric field, lm Because there is no reflecting boundary at the bottom we expect that there are no upward propagating mean waves (coherent components) in the half-space random medium at least in the zeroth order. The mean field then takes the following form: (2.19) where E ^ and are defined in Appendix B. However, the incoherent field has a spectrum of both upward- and downward- propagating, four characteristic waves as: ?!«) - J 4 p -» dPr ■n°' A a O lPlzz A 5od<z' V o(P l z ) e _O 0 (-P l 2 > ,neui .oedi j ^P-. ,t o , A ,Deu. ^-Pi rlz z , t , s . * ,ned. rlz 2 \\ + ^eu(z'Pp) e(Plz) e + ^ed(z'Pp> e(Plz) e / (2 .20) where jjp and Plz are the transverse and z-components of the (unperturbed) propagation vector, respectively. The superscript ' denotes the real part of the quantity. Assuming that the incoherent fields with different transverse directions of propagation are uncorrelated, the field correlations are written as <5ju(2- v C |i ,' V > ' “ v pp ’ Jjku(z'z,' V <5jd(z'V tJd(I,'*P,>' 6<V V Jjkd(z'z,'V <2-21> where j, k = o (ordinary) or e (extraordinary). Following the same procedure as in Lee and Kong [1988] we obtain from the B-S equation in (2.18), in conjunction with MDGF in (2.14) (2.15), the vector MRT equations for a half-space anisotropic random medium as follows: 29 d dz = (z 'Pp) 51 -Tlu (Pp) -Iu (Z/Pp) ■*" Qua (z » P p ^ p l ) ' ^md^z '^pl) Jd kp { Puu (z,pprkp) •Iu (z,kp) ^ua d ‘^<j (z/^p)} (2.22a) ^d^z'Pp) ~ -T1d ^Pp) "^d ^Zf Pp) ■*" Odd ^ZfPp'^pi ^ *^md ^z'^pi ^ J d kp { Pdd(z,^p,kp)•Id (z,kp) (2.22b) ■*■ ^du (z<Pp»^p) *IU where T) is the extinction matrix which describes the rate of wave XX S3 attenuation due to both absorption and scattering. Q and P are scattering phase matrices for the coherent and incoherent intensities/ respectively. The extinction matrices are given as follows: 2TU'u 0 0 2TU'u 0 0 o o ^u-^u-pn'+pr - (Tle*u -Tl0' U- P i“ + Pi ) Ttf u +Ti;-U (2.23a) 30 lid $p> ■ Tlu (ftp) { replace u by d } (2.23b) where the superscripts ' and " denote the real and imaginary parts respectively. The incoherent intensity vectors Iu and Id are defined as Jqou (Zf ztPp) =u <z ,ftp> Jae A u (Zf z r Pp) r 2 Re £ JBOU(zfzfPp) ) 2 lm | Jeou(z,z,j5p) j (2.24) One of the important differences in the MRT equation for our half-space problem and those given in Lee and Kong [1988] for the two-layer problem lies in the absence of certain mean intensity vectors. As mentioned earlier, due to the absence of bottom boundary there are no upward propagating mean fields. Consequently, I intensity vectors, I . and X me I mcz , the upward propagating mean which account for the interference between the upward and downward propagating mean intensities, vanish in our case. Therefore, we have W 1' V ■ 0 I (z, k .) = 0 mcl pi {2-25a) (2.25b) 31 (2.25c) IE edi e (2.25d) 2 lm (E1 edi odi Note that I e is the downward propagating mean intensity vector. The definitions for the scattering phase matrices Q and P are given by Lee and Kong [1988] . Since the specific intensity is an often used quantity in this thesis we provide below a standard definition for it in terms of electric fields (for the case of isotropic medium). I = <E V E* > V (2.25e) 2 Re <E E* > v h 2 lm <E E* > v h 2.5 BOUNDARY CONDITIONS In order to solve the MRT equations we require the boundary conditions which must be satisfied by the. incoherent intensity vectors I and I,. At z ■ 0 we have d V 0' V (2.26) ■ Sxo'V • V 0' *p» where ir oo ir oe r ir r Re (R R ) eo oo -Im(R R ) eo oo r ir r Re (R R ) ee oe -Im(R R ) ee oe eo ee R10(V 2Re (R R* ) 2Re.(R R* ) oe oo ee eo Re(R R* +R R* ) -Im(R R* -R R* ) ee oo oe eo ee oo oe eo 2Im(R R* ) 2Im(R R* ) oe oo ee eo Im(R R* +R R* ) Re (R R* -R R* ) ee oo oe eo ee oo oe eo (2.27) where the half-space reflection coefficients R , R , R and R are oo oe eo ee defined by Lee and Kong [1983] . At z *» -<*> we have v - ' V (2.28) " 0 because there is no reflecting boundary at z « - ». We denote the transmitted intensity vector from region 1 to region 0 as I . Here the subscript 0 in I stands for region 0. Whereas the ou ou first boundary condition at z - 0, (2.26), relates and 1^, the second boundary condition at z ■ 0 relates I and I as follows: ou u Jou<o' sP' - *xo'*p» • v ° ' i y where (2.29) 33 IXoH„t IXeH„l Re(XeH„X*„) oH -ImtxeHX* > oH IX ,12 oV IX „l2 eV Re(X „X* ) ev oV -Im(X „X* ) eV oV S10 <PP» 2R6(XoVX:h > 2Re ^XeVXeH^ Re(XeVXoH+XoVXeH) 2Im(X X* ) oV oH 2Im(X „X* ) ev eH Im(X „X* +X „X* ) Re(X „X* -X „X* ) eV oH oV eH eV oH oV eH (2.30) The half-space transmission coefficients X „ , X r oH oV defined by Lee and Kong [1983] . X „ and X „ are eH eV 34 2.6 SOLUTION TO THE MRT EQUATIONS Under a first-order approximation we neglect the terms with P in (2.22). Now, the MRT equations for I ans 1^ are decoupled. First, the MRT ■equation for 1^ becomes -S V 2'Sp> - ‘V V •V'-V +°ud,2'iVV •W*-%i> (2.31) The homogeneous part of (2.31) is s r V 2' i y + V V (2.32) • V 2- V - 0 The general solution of the above equation for 1^ in the backscattered direction is obtained as V*,Pp~*pi)- fyz,Pp-kpl) .c (2.33) where e ‘odi . 0 •ru,-kpi> 0 2TI" e z 0 az e e az cos cz az -e sin cz , sin cz az e coz cz (2.34) 35 where a - (2'35) ed' o1 c = -Tl' .+ T\' ,+ k, .+ k” . 'edi odi lzi lzi (2.36) and c is an arbitrary constant vector. As before, the superscripts ' and " denote the real and imaginary part, respectively. We have used the identities (Pp=”^pj^ = ” ^0di ' etc* The 3°luti°n of (2.31) must be of the form Iu <z'"kp i > = 'P(z,“kp i ) ‘ U ( z '_kp i ) (2-37) where z 5(*'-kpi) = I 0 dS rl(S,"kpi)'°ud(S'"kp i ' V )'imd (S'kPi) + * (2*38) The constant vector K in (2.38) is determined using the boundary conditions given by (2.28). Finally, the solution for I is obtained as follows. - where I , ul ( ;ul ' ru2 ' 4Iu 3 ■ aiu4 ) . reSkJ4 f IE .. r lm J odi --i 2 k2 I P ozi 1E + — ^ 12•391 , _1 .___ P , exp(-2'n".,z) oud odl I2 — P^ud exp(-2x\”^,z) f (2.40a) 36 ic5k'4 lm Xu2 ' ie .. r odi peud 2k ozi IE I . edi' + -----q " P,-2W _2 P . exp ( eud r (2.40b) X1 ‘ Y1 (2.40c) I . - X, + Y, u4 1 1 (2.40d) u3 Here X1 " "1 2 ~ T 1 Re(EediEodi1 *'q 003 03 + ° 3in 021 + Im(E^ ^ EQH^) (q sin cz + c cos cz) f cos cz e 0<^ e^ (2.41a) f Re (E ^ E^^ ) (q sin cz + c cos cz) q 1 " .2 . 2^ 2% I kozi(q +c > 1 * \ A - ImtBe(j£E0<^) (“3 003 cz + c sin cz) f Q sin cz e ^ o d i^ e d i* Z (2.41b) The various quantities appearing in (2.40) and (2.41) which are not defined so far are defined in Appendix C. 37 Next, the MRT equation for 1^ under the first-order approximation becomes jf-JjU.Spi - -5d 0 p).id (,,|>p) + 5 d<1w,Sp,kp1)-i1-(.,Epl) (2.42) The solution to this can similarly be obtained using the boundary condition given by (2.26) as follows. 7c8k'4 2d - T X 1- ( 2dl > rd2 ■ 2d3 ' rd4 ) ozi ,2.-43> where IE .. r odi dl IE .. I" edi .1 = Podd Tal 2 P ,. T , exp (-2T)" z) odd al r 'edi 2 -1 2 _ . IR I IR I .2 T oo _1 eo 1 IE JJ I ----- P . + P1 eud J odi L p 'oud + IE ., I‘ edi IR I2 IR I2 — 22— p2 +— — — q oud r eud 38 2Re(E ,E* ,) edi odi _A T _ _ _* * 1 ~— r Qud L Re (Reo Roo ) nq - Im(Reo Roo ) c J + -----(q2+c2j 4 r <q2+=2 > • . 1 1 2lC,(* ° ud [ Re<ReoRoo’ c + Im*ReoRoo* «] ) 9 (2.44a) 1E d2 I — A, *—”— Pedd , . Tb3_ exp " ..i z) ^ (—2ti’od IB I2 edi A2 _2 ~ Pedd Tb3 e x p (- 2T1edi z) , . IR |2 , IE I - 2 e _ pl odi L p oud IR I2 _ g e _ pl q eud J o r |Roe |2 _2o IRee I2 _2o ■. +. IE .. .2 I r ---------p . + pJ1 edi L q oud r eud J 2R9(EediEo d i ’ 4 r • . i + ------ 5— ; -------- Q j Re <R R ) q - rm(R R ) c , 2 ud L ee oe ee oe J (q +c 2 .) .4 2. 2,> tq2+c2 r .... l * . _1 \ .Kai1 °ud I '•'V..1 c + rn<ReeRoe) q' I 9 39 (2.44b) I._ - X. cos cz - X- sin cz a3 2 2 (2.44c) I.. = X„ sin cz + Y- cos cz a4 2 2 (2.44d) where X2 “ { Re(EediEodi)[ Qdd3 9p <°2) ” Qdd4 fp <02>] + Ira(EediEodi) t Qdd3 V C2> " Qdd4 V c2}] } eXP' 2 - I 'E ^.l2 f Re(R R* ) P1 . /p + Re(R R* )/q 1 1 odi I oe oo oud ee eo J + IE ^.l2 f Re(R R* ) P2 /q + Re(R R* )/r 1 edi L oe oo oud ee eo J + Re (E .,E* .) . P edi odi _ A | Re(R R* +R R* ) q - Xm(R R*-R R*) c] 2 2 ud L ee oo oe eo ee oo oe eo J (q +c ) Im(E ..E* ) . edi odi QA »A . f Re (R R* +R R* ) c + Im (R R*- R R*) ql f ud L ee oo oe eo ee oo oe eo J J 2 2 (q +c ) exp(tl"ui+% u i >z 40 *2 ’ { M < E .diEodi>[ Q 2d3 V ° 2 > * Qdd4 V « 2 > ] ★ + ImlEediEodi> [ I °dd3 fp W <02 ) * 0dd4 °dd4 V VC2J’] }f “ ■‘^ S d l ' W - < IE ..I2 [ Im(R R* ) P 1 , /p + Im(R R* JP1 ,/q 1 [ odi L oe oo oud ee eo eud J + IE J ,l2 [ Im(R R* ) P2 , /q + Im(R R* )P2 J r 1 edi L oe oo oud ee eo eud J Re (E ..E* ,) . edi. odi A r * * * * Q . Im(R R* +R R* ) q + Re (R R -R R ) + --- . .2 , 2. ud L ee oo oe eo ee oo oe eo (q +c ) Im(E ,.E ) . edi odi A .2 2V ud (q +c ) Im(R R* +R R ) c - Re (R R ■R R ) ee oo oe eo ee oo oe eo ®xp (T^oui+1^eui)z °] ^} ( .45b) The various quantities which appear in (2.44) and (2.45) are defined in Appendix C. 2.7 BACKSCATTERING COEFFICIENTS The backscattering coeffients are defined by Peake [1959] as O-.lim « r 2( l5g<E> I2 > q 2 16| A IE . Iq A-*» where (2 Ol p the incident electric field intensity with -s 2 polarization P, ( lEQ (r)I ) is the mean scattered field intensity with polarization a. Also, a and P can be horizontal (TE) or vertical (TM) polarization. In terms of the specific intensities (2.46) reduces to 4n cos e .a ia (-n ) A-*°o where a ip2;— '---A !?<&) (2-47) ~(X and I^t-Q^) are the incident and backscattered specific intensities, respectively. We assume l|?(£l) - 1 and (2.47) becomes °pa - 4" c°= 9o i «■ «> We note that in our MRT equations X(k^) are not exactly specific intensities as formally defined in the literature [Chandrasekhar, 1960] Hence, we need to modify (2.48). To that end we make the following transformation: 42 k X sin 0 cos <|> k sin o 0 sin $ k 0 m y k (2.49) The Jacobian of this transformation is given by . k ,k J | a W ] I ” sin 0 cos 0 (2.50) Now, regardless the definition, the total power must be the same in both cases, i.e., 4jc O O dfl I(fl) O O w -oo n- y I(kP ) dk —oo 2n d8 J d* 1 (E ) J [ ] 4n dfl I(k ) k P o cos 0 Therefore, I(£D = I(k ) k cos 0 p o (2.51) 43 Then (2.48) becomes °Pa * 4* kocos20oi I“ (-kpi) (2‘52) In terms of the quantities calculated in our problem the backscattering coeffients are given as follows: for h-polarized incident field, (J = 4JC k2 cos20 . I ..(0,-k .) hh o oi oul pi °hv ■ 4,1 ko°0s2eoi ^ou2(0' - V ; <2-53a) for v-polarized incident field, CT , = 4rt k2 cos20 . I , (0,-ic .) vh o or oul pr a = An k2 cos20 . vv o or 1 _(0,-£ .) ou2 pr (2.53b) where I „ and I . are the first and second elements of I given in oul ou2 ou (2.29). After straightforward substitution and algebraic simplification the backscattering coefficients can be finally cast in the following form: 44 V ' [ S1 + S21>+ Sf >+ S23>t S3 'lcnolltBoi' <D(2 kpi''21Clzi) -“"Sdl (1 ) 2 XaoiXBei1 1 ;<3 > - s < 3 1 >+ s2 31)= 2 s < 3 2 >+ — o' ed' *<2 kpi'-klzi+klzi> (2.55b) a ed = A o e(klzi) •<*,°<~k am Q • AH * <&(2 kpi'-klzi+klzi) (2.55c) iJ s < 33) + (2.55d) s<34> Be<*p.iXpoi) Re(XaeiXaoi} I ^ ) izi'l <&(2 k,P1' T1odi+11edi> s'32»- 2 (2.55a) a ed = a o e(klzi).q.o(-lclzi)_ " -^edi^odi* IX ,XQ ,I (2 )_ aei Poi 2 -2 (T|" ,+1)".) 'edi 'odi (2.54) ™«|)ei*goi> [«<l'lzil (2.55dl) (-T3 ) q +c ®<2 CP.> ^odi^edi* (2.55d2) 1 45 s<33>- 2«.<xSeix*o l , ( - J -2 } q +c [a<k! z i ) -5 - ° (-1C°lzi)l 2 * (2V ' T1odi+11e d i ) (2 .55d3) si34)= 2lm(X„ .X* .) Im(X .X* .) [ -q -■ 1 2 pei poi aei aoi I 2 2 J q +c ®(2kpi, IX .Xa J , 2 aei Bei 3 — — ^ r A .. e d . - * . ,2 ed .1 1 elkizi,'<!'e<ltizi1J (2.55d4) ., e d ’. __ . ( ’ » 1 2.8 DISCUSSION OF THE RESULTS We have solved the MRT equations for the half-space problem and obtained explicit expressions for Iu and in the backscattering direction. Also we have obtained the backscattering coefficients. We first take a close look at the expression for the backscattering coefficients «X). The form in which we have cast a is such that the physics behind the various scattering mechanisms can be easily appreciated. For example, the term S1 corresponds to the scattering process where both the incident and scattered waves are ordinary (o) waves. Similarly, we can proceed to explain the other terms. Altogether, there are five scattering processes which contribute significantly to the backscattered power and they are schematically shown in Figures 2.3(a) - 2.3(e), where solid and dashed lines are used to represent the phase path lengths traversed by the wave and its complex conjugate, respectively. The term (3) (see Figure 2.3(e)) demands particular attention. As shown in Figure 2.3(f), it should be noted that this process becomes destructive in other directions and it has constructive interference only in the backscattered direction. Hence it is appropriately called 'backscattering enhancement'. This is perhaps one of the important results of adopting the wave approach to construct the radiative transfer equations. Quite clearly there is no place for such terms in the conventional RT theory. Now, we subject o to various limits in order to compare it with the available in existing literature. First, we check the isotropic limit, i.e., we let e, X following: 8 , , q « 1 and w - 0, Equation (2.54) reduces to the lz z 47 (o) (b) eo S2 Sa’ (c) (d) $? (e) Figure 2.3 (f) Wave-acattering procesaes. 48 4 . (2.56a) 4 or (2.56b) vv (2.56c) where and Y ^ are transmission coefficients for the h-polarized (TE) and v-polarized (TM) waves, respectively, from region 0 to 1. The corrected propagation constants in region 1 for TE and TM waves are denoted by and T]^, respectively. We check our results with the MRT results of Zuniga and Kong [1980a] . In order to facilitate this we need to take the half-space limit of their two-layer results. On doing this, we find that our results agree with theirs. We immediately observe that no cross-polarization exists in the first-order backscattering for the isotropic random medium, whereas in the anisotropic case, there exists first-order cross-polarization. Second, we take the Born limit which corresponds to the single scattering approximation. We let - 0, § - ou, od, eu, ed. Hence “ and c - 0. Thus (2.54) reduces to the following (2.57) where In order to compare the above with those of Lee and Kong [1985a], we let d - °°' V W = qz V W ' V W " qZ C1 (W - 0n doing thiS we find that our results exactly agree with those of Lee and Kong [1985a]. Next we compare our MRT results (multiple scattering theory) with the Born results (single scattering theory). The MRT equations under the firstorder approximation are given by (2.31) and (2.42). The corresponding wave equation under the Born approximation is given as follows: V X V X E 1 - to2)! eim • E1 - Q(r) • i^0) 2 where Q = 0) (I a (r) . — (o) (2.59) 3 is the unperturbed solution to (2.59) when Q » 0. Comparing (2.59) with (2.31) we can draw the following conclusions. In the Born approximation we have single scattering of waves propagating in an 50 'average' medium which is characterized by the mean dielectric tensor (e1tn). In the MRT theory under the first-order approximation we have single scattering of waves in an 'effective* medium. We recall here that the effective medium is a result of multiple scattering phenomena. So the immediate inference is that by replacing the unperturbed propagation constants in the Born results by the effective propagation constants one 0 f » would obtain the MRT results. We observe that, indeed as predicted, k, . lzi ed" and in the Born results are replaced by and respectively, in the MRT results. Besides this, we notice that the nondiagonal term in the extinction matrix (2.23) introduces additional terms of / Q1\ and a slight modification in S£ (32) (33) fO A \ and • Quantitatively, it appears that the contribution from these additional changes are rather small. We can reason as follows. First, we note that the multiple scattering introduces much smaller correction to the real part as shown in Figures 2.4 and 2.5 where Re [8 X ]/Re[k?* ] and Im[8 A. ]/Re[k|? ], p = od or p lz p lz ed, are plotted versus the incident angle. So we observe that c (the nondiagonal term in the extinction matrix) is a very small quantity. However, we hasten to point out that this is an observation which is perhaps true only in our half-space problem. For the two layer problem we predict that there will be relatively more contributions from these terms. In all our numerical computations we have chosen, for the sake of illustration, the exponential correlation function given as ________ ...4 r C(rx-r2) - SkJm e * p [ ‘*i j ---------------- *2 ' lyi j ---------------' y2' - V 1j <-------p P (2.60) z At this point we turn our attention to the half-space isotropic A problem and examine the role played by the nondiagonal term (c =■ ) 51 - -6 HO -12 30 50 90\[Degrees] Figure 2.4 incident angle response of Re(SX )/Re(kf ), p - od,ed. p lz 60 52 Im (S X ) Im(k]X) 2 .4 2 .3 2.2 e1 =(2.8 + i.02)eo eiz s (2.8 + 1.04) Eo 5 = 0.1, qj = 3 Ip = 3 mm,lz = 4 mm, ^=350 20 40 Figure 2.5 50 Incident angle responae of Xm(5A. J/lmdc1? ), p - od.ed. p lz A there. We notice that in the first-order approximation c has no role to play In the results. It Is Interesting to note that the nondiagonal term distorts only the backscattering enhancement terms. Since there was no backscattering enhancement process involved in the half-space isotropic problem there was no distortion in the structure of the results. However, since a two-layer isotropic problem or a second-order approximation to a half-space isotropic problem both involve backscattering enhancements we A predict that c appears there and eventually contributes to the cross polarization. Next, we compare the incident angle and azimuthal angle responses of MRT and <T_ in Figures 2 .6 and 2.7. We observe that both the angle Born 3 responses of follow the same pattern as those of ®Born* However, °MRT l°w©r that ®Born* This is a direct consequence of the increase in attenuation caused by multiple scattering. We next consider the effect of 8 ( normalized variance of random fluctuation) on scattering. From (2.17) it is clear that for 8 < 1 , <T„__ MRT should approach CTBorn* This is borne out by the numerical results plotted in Figure 2.8. Also, it is important to note that in the case of Born, o is unbounded whereas in the case of MRT a approaches a constant value. Physically this is because there exists a balance between an increase in backscattering and an increase in shielding due to the multiple scattering as we increase 8 . Another point of interest is the effect of medium loss on scattering. We observe that MRT is not as sensitive to medium loss as o„ is (see Born Figures 2.9 and 2.10).The immediate thing noticed is that since only e”z is changed there is no change in o.. for this particular case where .* E» 80*. It is clear from (2.58) that ®Born is inversely proportional to hh 6 ox 54 VV-Bom - 1 0 HH-Bom W-MRT HH-MRT -20 HV-Born HV-MRT e-l »(2.8 +1.02 ) e0 el2s(2*8+1.04) e0 8=0.1, q | = 3 Ip = 3 mm,lz s 4 mm, \f/=350 40 20 30 40 50 60 e oi [Degree^ Figure 2.6 Incident angle response of and ®Born* 70 55 -12 H+Born HH-MRT. HV-Bom -1 8 HV-MRT \ -21 9GHz -2 4 ^1 a (2.8+ 1.02) e0 eiz a(2.8+i.04) Eo 5a0.1,q|a3 -2 7 -Ip= 3 mm, lz = 4 mm, t//=35° -3 0 30 40 50 60 80 <£oi [D egrees] Figure 2.7 Incident azimuthal angle reaponse of <r__ and <J_ nrt Born 56 Born MRT f - 9GHz * c i - 75# e-l =(2.8 + 1.02) e0 elz = (2.8 + i.04) £0 tp = 3 mm,Jz = 4 mm, \j/ = 35° ql*3 QO 0.2 Q4 0.6 0.8 NORMALIZED VARIANCE (8) Figure 2.8 Comparison of function of from MRT and B o m normalized variance results as a 1.0 I 57 -10 — HH(€" - .02 or .1) -1 5 -2 5 f - 10GHz d>.-80* *#* S (3* + 1.01 ) E0 -3 0 eiz = (3.2 + i ^ ) e 0 8=0.1,eg= 5 Jp = 1 mm, g = 3 mm, \p=30° -3 5 0 10 20 30 40 50 6^| [O eg ree^ Figure 2.9 Effect of medium loss on o 60 70 80 58 -5 - 1 0 -1 5 -20 -2 5 e 1 ■ (3.2 +1.01) e0 e u * (3.2 + i ^ J e o -3 0 8=0.1,q|=5 Jp= 1 mm,^ =3 mm, iff=30° -3 5 Figure 2.10 Effect of medium loss on a MRT I; =» o, ed. This explains the decrease in a when e" is increased. On the vv lz other hand, we notice from (2.55) that is inversely proportional to HKT pH + ^ ^ ” 0<*f ed" It should be noted that multiple scattering introduces significant scattering losses and since we often deal with low loss media the result is that MRT becomes less sensitive to changes in medium loss. Finally, the frequency response of values of 8 MRT and a Born for different are plotted in Figure 2.11. The initial steep increase in scattering in both cases can be attributed to the Rayleigh scattering 4 phenomenon (where <J « k, ) which dominates at low frequencies. In the im case of Born at high frequencies the spectral density of the correlation function starts decreasing sharply and thus influences a. In the multiple scattering case the correction to the propagation constant increases steadily with frequency as shown in Figure 2.12. Thus G drops faster than CTBorn at high frequencies. It should be noted here that we are now entering the frequency range where our theories break down. Before we end this section, we would like to make a couple of comments on the first-order approximation. While solving (2.22) we have neglected terms with P, i.e., we have neglected the incoherent scattering matrices. It is intuitively clear that this approximation is a fairly good one if the scattering is weak. However, if one wants to draw a more quantitative conclusion regarding the validity of the approximation we need at least to calculate the second-order solution. We report details about this in Chapter 4. 60 < ^[dB ] ^ 3 (2.8 + i.02) £0 £12 = (2.8 +1.04) e0 i|p = 3 m m ,lz = 4 mm, ^=35° 6 Figure 2.11 8 10 12 FREQUENCY [GHz] 14 Frequency response of C„ and o Bom MRT 61 e1 = (2 . 8 + 1.0 2 ) e0 ei2 = (2.u + I.04) e0 5 = 0.1,qi = 3 (p = 3 mm,^z = 4 mm, V=350 FREQUENCY [GHz] Figure 2.12 Frequency response of Xm(5&p )/Im(Jtflz), p - od,ed. 2.9 CONCLUSIONS We have calculated the backscatterlng coefficients for a half-space anisotropic random medium using the multiple scattering theory. Here, the ladder-approximated Bethe-Salpeter equation in conjunction with the nonlinearly approximated Dyson equation is used to derive the MRT equations. Using a first-order approximation we have solved the MRT equations and we have obtained expressions for upward and downward propagating intensities. The backscatterlng coefficients thus obtained are compared with those obtained using the Born approximation. Various interesting properties emerge from.this. The most important of them are as follows: The expressions for backscatterlng coefficients both from the MRT and the Born have exactly similar structures except that the propagation constants which appear in ®Bo n are replaced by the corresponding corrected propagation constants in are some extra term3 which appear in Besides this there Another interesting thing to note is that, with an increase in 8, 0? „ is bounded while <T_ is not. MRT Born 63 APPENDIX A B1 V " 2)T~ oz “h o Ao H B3 (iy r -2T“ oz aH o AeH aH A 0 <ic_) =* R + R — — 2 p oo eo a„ r Ho p (A. 3) (A. 4) *eV A. (k ) = R + R 4 p oe ee 3 x V - R - R — 2— oo eo X „ eV oV A,(k ) 3 p D1(V (A-2) " 2 2 a. He a. Ho i kz oz ®Ve AoV k oz %>e AeV V C_ (k ) - R + R — — 2 p eo oo fl^Q Roe - Ree oV X eV (A.5) (A. 6 ) (A.7) R - R eo oo eH X oH (A.8 ) 64 X eH w (A.9) <0 H C„(kJ = R— + R -— t 4 p ee oe f (i x^ = R - Rr»» xeH "ee "oe X OH (A.10) pa eV X U X „ - X „ X „ oH eV oV eH a.Ho (A.11) oH a. ve (A.12) xoH „ x eV „ - x oV „ x eH „ where X , X , X , X , R , R , R and R are the half-space oH eH oV eV oo oe eo ee transmission and reflection coeffient3 defined by Lee and Kong [1983]. The corrections to the propagation constants X,0U, X0C*, A.eu and \eC* are given by Lee and Kong [1985c]. 2 lz eu ‘lz ed clz 2 1/2 (A.13) (kl " kp > '23 !33 k y ± - e 2 lz 33 1 2 2 1 33 x 1 lz y 1/2 (A.14) 65 APPENDIX B For a horizontally polarized incident wave we have -ik i .= E h <-k .) e oi oi o ozi .z 021 e ik .‘p pi (B.l) ' For a vertically polarized incident wave we have -ik E . -> E . v (-k .) e oi oi o ozi .z 021 e ik ,-p ^ (B.2) ' ' By using the boundary conditions at z = 0 we obtain the solutions for the mean electric field in region 1 as given in (2.19) where E .. and E are odl edi given by E (“>. - E .A„ . - E . X„ (k = k .) odi oi Hoi oi Ho p pi (H) — — E' ' » E ,A„ - E . X„ (k - k .) edi oi Hei oi He p pi (B.3) (V) a i*. a h: , 4 sb “ E odi ■ Eoi4 A,voi oi4 E '^4 x _ _ (ir Vo (kp - k pi.) ^pi' (V) E 4 “ E 4 A,, . - E . X„ (k - k .) edi . oi vei oi Ve p pi where the superscript represents the polarization of the incident wave. The quantities ARq, A^, A^, AVq, XRo, X^, x ^ , Xye are defined by Lee and Kong [1983]. We draw attentiqn to the difference between the quantities mentioned above and the corresponding ones (with subscripts in the reverse order) mentioned in Appendix A. The subscript is as usual used to indicate that the quantity is to be evaluated at k P = k . . Pi 67 APPENDIX C odi q - 2T1" , + 2TV' . ^ 'odi (C.l) 'edi r - 411"., edi (C.2) 4>(2k , -k° ,+k®d .) [ o(-k° .) *q-e (k®d .)1 pi lzi lzi L lzi lzi J 'oud peud - . -«<2V ] ° ‘2V eud C > Q . = ®(2k ., T11 .+T|* ,.) [ o , .o . - a ..ed .V ud pi 'odi edi k l z i ),q,e(kl zi)J r, - 2 ti" . + 2 it' . r. - 2 ti" . + 2 ri" 1 2 oui 'oui (C.3) (C. 4) (C.5) (C.6 ) 'odi 'edi (C.7) A. - 2T1" . + 21)'’ . 1 'eui 'odi A. - 2l\" . + 2TT' . 2 'eui 'edi where where X 70 CHAPTER 3 MICROWAVE SCATTERING AND EMISSION FROM A HALF-SPACE ANISOTROPIC RANDOM MEDIUM This chapter is a natural sequel to Chapter 2 where the backscattering coefficients of a half-space anisotropic random medium were obtained. Here, we calculate bistatic scattering coefficients by solving the modified radiative transfer equations under a first-order approximation. The effects of multiple scattering on the results are observed. We also calculate emissivities and compare them with those obtained using the Born approximation (single scattering). Several interesting properties of our model are brought to notice using numerical examples. Finally, as an application, we use our theory to interpret the passive remote sensing data of multiyear sea ice in the microwave frequency range. We observe a close agreement between our theoretical prediction and the measured data. 71 3.1 INTRODUCTION Passive remote sensing at microwave frequencies has a wide range of applications in such various fields as meteorology, geology, environmental science, oceanography, and astronomy. The type of theoretical model used largely depends on the field of application. Thus a variety of different models exist in the literature. Even in one particular field, depending on the immediate need, the model used may be different. For example, in the passive remote sensing of vegetation, one may be interested in monitoring the soil moisture; another may be interested in observing plant growth; yet another may be interested in discriminating different types of vegetation and so on. In such cases, often semi-empirical models tailored to the need of the user are used. So, there is no one universal model suitable for all purposes. Whenever a theoretical model is to be considered, there always arises a question of how sophisticated the model should be. For instance, in certain applications very simple models suffice; for certain others, even the most sophisticated models known as yet may prove inadequate. In this chapter we consider an anisotropic half-space random medium model. As an illustration of its use, we will apply this to model multiyear sea ice at microwave frequencies. In order to put our proposed model in the proper perspective, we will very briefly scan through some of the past work in the literature related to this topic. Using the radiative transfer (RT) theory, England [1975] derived the brightness temperature of the random medium layer and applied it to determine the thickness of an ice slab. The model used Rayleigh scattering in its formulation. Thus it is limited to low frequency applications. 72 To overcome this limitation, Chang et al. [1976] used the Mie scattering model to study emission from ice. Often the subsurface temperature is an important factor in determining the brightness temperature. A model to' handle this has been proposed by Tsang and Kong [1975]. A more complicated but useful multilayer model using RT theory was proposed by Djermakoye and Kong [1979]. While all the aforementioned models have assumed planar interfaces, Fung and Chen [1981] developed a two-layer model with irregular interfaces for studying emission from snow. Often a closed-form solution to RT equations is obtained by assuming that the albedo is small. A different approach suitable for the case of large albedos was proposed by Stogryn [1986]. We point out that in all the above-mentioned models the scattering medium has been assumed to be isotropic. But it has been observed that many targets both natural and man-made display properties of anisotropy. For example, Kovacs and Morey [1978] have reported that sea ice has a preferred azimuthal orientation and hence is anisotropic. Also, Brunfeldt and Ulaby [1986] have noticed that certain row crops are anisotropic. With these in mind Lee and Kong [1985a] developed a two-layer anisotropic model using the Born approximation. It is well-known that scattering results in some dissipation which the single scattering model fails to take into account. To remedy this we propose in this chapter a multiple scattering model for a half-space anisotropic random medium. Lee and Kong [1985c] solved the nonlinearly-approximated Dyson equation and obtained the modified radiative transfer (MRT) equations [Lee and Kong, 1988] from the ladder-approximated Bethe- Salpeter equation. It is to be noted that in this approach some phase terms which are ignored in the regular RT theory are included. Under a first order approximation we solve the MRT equations and hence obtain the bistatic scattering coefficients. Finally, we calculate emissivities and use this model to interpret some of the passive measurements of sea ice. 3.2 FORMULATION The geometry of the problem is shown in Figure 3.1. Region 0 (z > 0) is the free space and Region 1 (z < 0) is the anisotropic random medium. kci is the incident wave vector with angles (0 oi , scattered wave vector with angles (03 , (Jig) while ka is the . The permittivity of Region 1 is given as 6 i (r) =» < Ei (r)> + E lf(r) (3.1) where < Ex (r)) = Elm is the mean part of Ei<r). It is to be noted that as a Eim is a constant while Eic (r) is a random function of position. We assume that the fluctuating part, Elf(r), is small compared with Elm. It is clear that the random function Elf(r) has zero mean. a = Both £im and £if (r) are assumed to be tilted uniaxial as given below. Ei 0 0 'lz elf(r) £ 11 0 <r) 0 0 tilted by y (3.2) 0 Clf(r) 0 0 eizf(r) tilted by y (3.3) We assume that 8 i*£ (r) “ q* 81 f (r) (3.4) where qz is a deterministic constant. This physically means that Eif (r) and Eizf(r) have the same statistical properties except their strengths of fluctuations. In general, the optic axis of the uniaxial medium can be tilted from the z-axis by an angle y, say, in the yz 75 REGION 0 2=0 REGION I /^(T) - « r , > + r lfm Figure 3.1 Scattering geometry of the problem. plane. Also we assume that both Region 0 and Region 1 have the same permeability p.. Our goal Is first to calculate the blstatlc scattering coefficients of the above-described medium. Let us consider an electromagnetic plane wave with a linearly polarized tlme-harmonlc field 5o l (r,t) - ioi e i(^°l ‘Z-<at) (3.5) Incident on the half-space medium as shown In Figure 3.1. The electromagnetic Intensity propagation In a half- space anisotropic random medium with multiple scattering taken Into account Is described by the MRT equations m (2.22) given as ^z*Pp) * Qud Pp» ^pi ^ + J dicp { puo <z»Pp»^p)’iu<z»^p) "*■ ^ud lzf?pf^p) '^d (z'^p^} (3.6a) d Z dz ^d^z'Pp) ” "*ld^PpJ*^d^ZfPp^ Qdd^z»^p»^pl^’^od^z'^pi^ J* dkp | Pdd (z,$p,kp) *Id (z,kp) + Pd u (z,Pp,kp)-Iu (z,kp)} (3.6b) where Itt and Id are the upward and downward propagating incoherent intensity vectors, respectively, and Imd is the downward propagating mean intensity vector. T) is the extinction matrix which describes the rate of wave intensity attenuation due to both absorption and scattering. Q and P are scattering phase matrices for the coherent and incoherent intensities, respectively. The extinction matrices are given in (2.23) and the expressions for Q and P can be found in Lee and Kong [1988]. The incoherent intensity vectors Iu and Id are defined as d r I„ <z 'Pp> d r 2 Re ( Jeou(z,z,pp) ) 2 where J.ku (3.7) Im £ J8 ou^z,z,^p^ ) are given by { ®ju^z'®p) ®ku^z''Pp^ “ P (Qp”Pp) Jj )cu (3.8) z'•Pp^ where j, k - o or e. We note that o and e refer to the ordinary and extraordinary waves that can exist within a uniaxial medium [Lee and Kong, 1983]. Spq(r) is the incoherent electric field as defined in (2.20). The other quantities not explained thus far are defined in Chapter 2. The boundary conditions at z-0 associated with (3.6a) and (3.6b) are Id (0 ,Pp) -Rio(Pp) * iu (0,pp) (3.9) where Rl0 (j5p) is defined in (2.27). At z - ”*• we have (— »Pp) " 0 (3.10) We denote the transmitted intensity vector travelling from Region 1 to Region 0 as Xou. Here the subscript 0 in Iou stands for Region 0. Iou is related to Iu by 78 Io ,(0,gp) - T10(pp) • Xu (0,jSp) (3.11) where T1 0 (j5p ) is defined in (2.30). t 79 3.3 SOLUTION TO THE MRT EQUATIONS As a first-order approximation we set lu ■Id » 0 inside the integral in (3.6a) and (3.6b). Physically this means that we neglect the contribution to scattering of the incoherent intensities. In other words we have single scattering of the mean intensity in an "effective medium" which has taken into account the multiple scattering effect. Thus the MRT equation for Iu becomes -|^iu(z,Pp) --fiu<&p> • iu(z,pp) + Qud (z,pp,kpi) • imd (z,icpi) (3.12) The homogeneous part of (3.12) is - ^ i u(z,pp) +T1u (pp ) • Xu(z,j5p) -0 (3.13) The general solution of the above equation for Iu is obtained as iu (z,pp) - V(z,jSp) • C (3.14) where r 2T1- z o az cos cz az . ~ -e sin cz az . e sin cz az e cos cz e (3.15) (3.16) 80 « ■ ^u-'Hou-Piz'+Px*' and C is an acbitrary constant. (3.17) The superscripts ' and " denote real and imaginary parts, repectively. It should be mentioned here that the major distiction between the above expressions and similar expressions in Chapter 2 is that we seek solutions at j5p instead of at — . The solution of (1 2 ) should be of the form iu <z,Pp) = l»(z,pp) • U(z,pp) (3.18) where z U(z,pp) m J* ds (s,Pp) "Qmj (s,Pp,kp^ ) •Im<1 (s,kp^ ) + K (3.19) 0 The constant vector ic in (3.19) is determined using the boundary conditions given by (3.9) and (3.10). Finally, the solution for I,a is obtained as follows. i • I„ 2 ' 3 ' 4 ) (3.20) where IUl , IU 2 are the same as in Chapter 2 except the following changes. Kzi -+Kz where P P -» q -» S (3.21) |J - 2r\^at-2 t)Ju <5 - 2 ^ 1 2 (3.22a) ^ (3.22b) for pp * -kpi (backscattering) (3.23) A - ' 0 otherwise IU 3 and IU 4 are the same as in Chapter 2. The other MRT equation (3.6b) in the first-order approximation is given by £ 2, Xd ^z • P p ) m ~Tld ’Xd (z r$p) + Qdd ) ’Xmd ) (3.24) The solution to (3.24) along with the boundary condition (3.9) is obtained as Id " “ 2^177 ( Idi ' where Xd2 • Xd3 ' Xd4 ) (3.25) Xdi “ T h + T 12+ { T1 3 +T1 4 +At1 1 +At12} e2l^"u1Z (3.26a) xd2 - T2 1 +T22+ { T2 3 +T2 4 +At2 1 +At22} e2T1*u1 z (3.26b) Id 3 ■» X2 cos Cz - Y2 sin Cz (3.26c) Id 4 - X2 sin Cz (3.26d) + Y2 cos Cz X2 - A t3i exp{-T)£dl-Ttfdl}z + (T3 1 +T3 2 +At3 2 +At33) exp{l)^ul+T)Jul}z (3.27a) Y2 - A t4l exp{-Ti£d l -T);d l }z + (t4i +T« 2 +At« 2 +At«3) exp{tljul+ii;ul}z (3.27b) We notice that the structure of the above expressions are exactly similar to the corresponding expressions in Chapter 2. The only differences are given as follows. P -> P lU q -* 9 r -» If i,j - 1 ,2 ,3,4 where ? - 2T)Jdl -2t)£u . Of course the appearance of A ( as defined in (3.23)) should also be noted. 82 3.4 BISTATIC SCATTERING COEFFICIENTS The bistatic scattering coefficients are defined by Peake [1959] as ^ r-K» A-*oo A COS 0ol lEol I2r P where lE0 l I2 is the incident electric field intensity with polarization P and ( li“ (r) l2)a is the mean scattered field intensity with polarization a. In terms of intensities, as shown in Chapter 2, (3.28) becomes Yp0 - 4a k0 cos 0oi if(pp) (3.29) The bistatic scattering coefficients can then be written as follows. For H-polarized incident field, Y hh Yhv "4* k° COS 001 “ 0 kQ cos J Iou1 t°*Pp> (3.30a) OjIOUj (0 ,pp) (3.30b) and for V-polarized incident field Yvh -4JI k* cos 0ol IOBl(0,Pp) (3.30c) Yvv m4^ (3.30d) 003 0oi ^ouj(®rPp) where I0 U 1 and I0 U 2 are the first and second elements of Iou given in (3.11). From (3.11) and (3.30) we obtain explicit expressions for bistatic scattering coefficients. To facilitate easy physical interpretation we cast the results in the following foirm. |4 ?P« ' - co/at : s. - ( Si + S2 + S> + S. + 4 S, ) " i p t - h' (3.31) . 1 3 32.1 83 S2 S3 S4 IXQO1 x . a ' 2 * A «i Aoa' xB.i Q| ■q*o(“k°.i)l .edi O oi. kizi‘Piz> x« a |2 rA, o O i . “ *, ,td 1 , 1 l°(Piz) -q-e(klzl) ] GUI .ed i n *u i ^(Xpi ~Pp« k l z 1 -Pi Z ) [e(p;Z s5(1)+ s 5<2)+ s 5(3)+ s 3<4) s5(1)- 2Re(X0Ullx£ol> R®(Xp.i^0 l) ( (3.32b) (3.32c) (3.32d) (3.32e) ) i [•(kT^’j-q-oj-kl ' zl)]2®(2kpi,tlJdi+n.’^) (3.33a) S5(2)- 2Re<x0tolx5o l ) HB<X0.1X(O1) ( ^ r ) [ e ( k ^ i ) - q - o t - k t J i ) ] 2 <fr(2kp i , T\'0di+T|Jd*) (3.33b) S5<3)- 2lm(X0, 1x£o i ) Im(Xpa l X^|o l ) ( ^ r ) [e<lc“*i) -q-ot-ktji)]2 Q(2kpi, tladl+tijdl) s5(4)- 2im<xa a l x£o l ) Re(xpa l x50 l ) ( ) (3.33c) 84 (3.33d) where q - 2 (T)odl " +i\a"d 1 (3.34) (3.35) A A Here o and e are the unit vectors representing the ordinary and extraordinary waves, respectively. <& is the spectral density of the correlation function of Eif (r). X's are the half-space transmission coeffic‘ents. More details of these and other quantities are given in Chapter 2. l 85 3.5 EMISSIVITIES Conservation of energy requires that the absorptivity is equal to one minus the reflectivity. Here, the reflectivity includes both the coherent and incoherent components. But by Kirchhoff's reciprocity theorem we know that the absorptivity is equal to the emissivity. Hence the emissivity at angle (d0 l,<t>0l) with e |3 3 polarization is given by (0Ol'^o l) - 1 - rpc (0ol,4>ol) “ rpi (0ol,<t>ol), P-H or V (3.36) where r-pc <0 Ot , <|»0 ±) is the coherent reflectivity which denotes the fraction of the power reflected in the specular direction and irp^ {0 OA,<J»0 x) is the incoherent reflectivity which denotes the fraction of power scattered over the upper hemisphere. The reflectivities are given by rpc *®°i' 1 ) “ (3.37) I Rpa*®oi'^ol * a -H,v i rpj.(0oi ' * o i > “ n r L^ a -H,v y J J}h sin d0 0 Yp^oi^oi'-e^) (3.38) o where Ypa is given in (3.31) and Rpo is the reflection coefficient given in Lee and Kong [1983], The brightness temperature Tg^is a commonly used parameter used in passive remote sensing. It is related to emissivity as tb P) where T0 -•p(0oi»*ot> To (3.39) is the physical temperature of the medium, which is assumed to be uniform. It should be pointed out here that since we have used a first order approximation in solving the MRT equations, (3.36) is only an t 86 approximate equation. To be more precise, the use of the first order approximation in (3.36) leads to a slight overestimation of emissivitie3 . Nevertheless we conclude that in all situations where higher order terms are negligible our expressions for emissivities are fairly accurate. l 3.6 DISCUSSIONS AND APPLICATIONS A. SCATTERING We have solved the MRT equations for a half-space anisotropic random medium under a first-order approximation to calculate the bistatic scattering coefficients, which are cast in suitable form for physical interpretation. As seen in (3.31) there are five scattering processes. These five terms in (3.31) correspond to Si, S2 , S3, S4 and S5 in Figure 3.2. Solid lines and dotted lines denote the paths of the wave and its complex cor.4ugate, respectively. We point out that the term S5 does not have significant contribution to Ypa except in the backscattering direction because the o-wave and the e-wave have different phase velocities. As shown in Figure 3.2(f), it is clear that such a scattering process is constructive only when j5p * -kpi because only then the wave and its complex conjugate are in phase. All other scattering angles result in destructive interference as shown in Figure 3.2(e). This is a natural consequence and an advantage of the wave approach that we have adopted. Such terms would have been ignored in the regular radiative transfer approach. Reciprocity demands [Peake, 1959] that YPa<^P1 '^P) cos 9oi ” cos 0, (3.40) This relation is readily verified from (3.31). Next, we would like to check the isotropic limit: we let eiz Cizf (r) - elf(r) ( or qz"l ) and in (3.31). The results are ” 63 , 88 S. (a) (b) S3 4 (c) S5 (e) Destructive Interference Figure 3.2 (d) ( f ) Constructive Interference Wave scattering processes. 89 2n-2* 8 COS cos A 2C 2n 8 • m IXp 11 kx 0ol e.i 2(\ h ”l i , »4 k lm cos 0oi o. I2ra 0 1 a i [h <Pl z) ‘h <-lcl z i H - 2 (T|” +T]") 1*0 11 i ol < 3 . 41a) <t>(kp i - i 5 p , - ) c i z l - P i z ) (3.41b) ®(kpi-pp,-lt1,zl-Pi,z) (3.41d) h *10 1 — [v(Pi,)-ht-ki^)] 2 (\ " l r ® ( kpi “ Pp» _lcl zi “ Pi z ) +H"> V '4 2jc25 ,klm 1*011 Xio 1 cos 9oi +Tp) 2 n !, 14 2n 8 k l B 1 * 0 1 ! * 1 0 i Y v v = cos ”! 2(71" + tI l" ") ) tv(Pi,) -vt-k!'^)] cos 00ol ■H 01 2 ‘\ "V l i V We find that these results agree with the MRT results of Zuniga and A A Kong [1980a] in the half-space limit. Here h and v are unit vectors for horizontal (TE) and vertical (TM) polarizations which are the two characteristic linear polarizations in isotropic media. X0 1 and Y0i represent the transmission coefficients for the corresponding waves travelling from Region 0 to Region 1. The above-mentioned quantities are formally defined in Zuniga and Kong [1980a] . And T|h and T]v are the effective propagation constants in a random isotropic medium, which include corrections due to multiple scattering. It is interesting to note that the 'backscattering enhancement' terms have vanished. This should not be very surprising because in our half space problem, under the first-order approximation, the enhancement is due only to anisotropy. For a corresponding two-layer problem this would not be true, because then there will still be some backscattering enhancement due to the bottom boundary. i Now we take the single scattering limit (or Born approximation) and observe the role played by multiple scattering in y. In the single scattering limit, we have the following changes: z u ■* au klz Tlau • Tlo d “kl z (3.42) ed • ’lad z The bistatic scattering coefficients are given by 27t8kim * coa t b b b b . b \ e01 \ s‘ * s* + 5j + s< +4Ss > l3-43> where b denotes the Born approximation and sm = sm-( same changes as in (42) } m » 1,2,3,4 _b 2Re[ Xo,,x Xgex (XpjgAXgQx ) ] ,a ed i. ®* A ol ,2 5 - -- O „o.,adi t . ------- [e(klzl)-q-o(-klzi)] 2 (kx zx~kx zx) 0 (3.44) .01 ..ad i. (2 kpi,-klzi+klzi) (3.45) In order to compare the above results with those of Lee and Kong [1985b], we have to perform the following operations on their results: we let d=°°, C2 (r2-fx )=qz Cx (r2-fx ) and C3 (f2-fx )=qz Cx (f2-fx ) . On doing this we find our results agree exactly with theirs. The primary change noticed in this single scattering limit is that the effective propagation constants l)'s are replaced by the corresponding unperturbed propagation constants. The physical significance of this change is clear. We observe that Im(T)) > Im(klz). This means that the multiple scattering will result in effective dissipation which in turn will result in smaller scattering coefficients as compared with those due to single scattering. In all the examples that follow in this section we use, for the sake of illustration, the exponential correlation 91 function: , », .4 r lxi-x2 l lyi-y2 l lz!-z2 l i C(e2 -5x) - 8 klm exp [ ---- — ------- — ------- --- j (3.46) The first noticeable feature of multiple scattering is the lowering of the scattering coefficients as compared with the corresponding single scattering case (see Figure 2.6). We note that the observation-angle response of bistatic scattering is interesting. For <t>s=<t>0i or <t>oi+7t (i-e *f the incident plane) we notice in Figure 3.3 that the response is symmetric about 0S=O at normal incidence. For 0ol >0, the maximum scattering point shifts towards backscattering direction. Also the magnitude is lowered. Since we are considering TE polarization this behaviour is attributed to smaller Fresnel transmission at larger 0 oi. Next we want to investigate the influence of the tilt angle of the optic axis on scattering. In Figure 3.4 we maintain all parameters fixed except \|f. We look at bistatic scattering coefficients on the incident plane. When the optic axis is vertical (y=0), there is no depolarization, i.e., Yvh^O* But as V increases yVH increases because y introduces coupling between H- and V- polarizations. Thus we realize that the tilt angle of the optic axis has an important role to play in cross-polarized reception. We also notice that as y increases, the overall level of the like-polarized scattering (yHH) is lowered. Another characteristic phenomenon due to anisotropy is the dependence of scattering on azimuthal angles (<|>0l, <|>a) . To examine this we let y*20‘, 00 1-40* and observe scattering versus 08 . As before we restrict our observation to the incident plane. In Figure 3.5 we vary <J>ol while keeping 92 0.20 SCATTERING COEFFICIENT (ym ) 016 // n i ' k 0.12 v / /./ 008 / \ // 0.04 — > i.004)e0 i.006)e0 Ips 2 m m ,lz = 4 mm 0.00 80 60 40 Forward Figure 3.3 20 20 40 60 80 Backward SCATTERED ANGLE (S»)[oEGREE^ Y h h versus scattered angle. The incident angle is used as the parameter. The observation point and the optic axis lie in the incident plane. 93 '/'s0°, H H HH — ^ s2.5°, HH Scotiering Coefficient [dB] - 1 0 •' N -14 -1 8 - / -22 ei = (2.0 + i.004)e0 £iz = (2.01 + i.006) 6o, 8 = 0 .0 5 ,q i a 2 lp » 2 mm,lz » 4 mm -2 6 80 40 Forward ■*-- 20 * Scattered Angle Figure 3.4 20 40 60 80 ►Backward (6s) [degrees] Scattering coefficients for various observation angles. The observation point lies in the incident plane while the optic axis lies in the plane orhogonal to it. y is used as the parameter. £i * ( 2 . 0 + i .004) e0 e , i = (2.01 + i . 0 0 6 ) E 0. V = 2 0 a 5 » 0.004, m 1.5 lp» 2 mm,lz«4 mm _______ I 80 60 Forward I 40 — I_______ |_______ I_______ |_______ l_______ l * 20 0 20 40 60 80 —— — ►Ba c k wa r d Scatlered Angle (08)[degrees] Figure 3.5 Scattering coefficients for various observation angles. The observation point lies in the incident plane. parameter. 1 is used as the 95 other parameters fixed. When <|>ol - 90* the optical axis is in the plane of incidence. In this case there is no cross-polarization because H and V waves coincide with the o- and e- waves, respectively, resulting in no cross- coupling. For <|>oi - 60*, yVH appears but is small as compared with YHH (or Yvv). For <|>=0* where the optical axis lies in the plane orthogonal to the plane of incidence, yVH is almost as large as yHH (or Yvv)• This concludes our discussion on microwave scattering response of the half space anisotropic random medium. B. EMISSION Now let us examine the properties of emissivities with the help of computed data. First we look into the effect of multiple scattering on emissivity. As observed earlier scattering coefficient is reduced due to multiple scattering . Thus we would expect that the multiple scattering would result in increased emissivity. This is corroborated by the data of Figure 3.6, where the emissivities using MRT and the Born approximation are plotted against observation angle. The subscripts in e denote polarization. It is important here to note that although the variance used in this example is very small ( S = 0.05 ), the difference between the MRT and the Born results are notable. This illustrates that even in the case of weak fluctuations the cummulative effects can become quite significant, thereby warranting a multiple scattering approach. In Figure 3.7 we investigate the effect of the dielectric constant of the medium on eH and ev . The Brewster angle effect is clearly noticeable, when Ei' =2.0. This means that although incoherent scattering is present, the coherent reflectivity is more dominant here. When el =2.8, the > 96 1.00 e v (MRT) 0 .9 0 Emissivity 0 .8 0 0 .7 0 0 .6 0 0 .5 0 ei = (2.0 + i.004)e0 £iz = (2.01 + i.006) Eq ,vy = 30 5 = 0.05,q§ = 2 lp = 2 m m ,lz a 4 mm 30 • 40 50 60 70 Observation Angle (0oi)[degrees] Figure 3.6 e*,,^ and eBorn versus observation angle. 80 97 0 .9 Emissivity 0.8 0 .7 0.6 - f* 9 GHz £i = (e( + i.004)eo 0 .5 £iz= (£i + -01 + i.006) e0, = 30° 5 = 0.05, ql = 2 lp = 2 mm,lz = 4 m m 04 0 10 20 30 40 60 80 Observation Angle(0oj)[degrees] Figure 3.7 Emiaaivity versua obaervation angle. 6 i 1 ia uaed aa the parameter. Brewster angle effect is more visible because due to increase in 6i the effect of coherent reflectivity has become more dominant. When ei'» 1.2, however, we notice that the Brewster angle effect has almost vanished. Because, in this case, the coherent reflectivity is greatly reduced and incoherent scattering has become dominant here. Another interesting thing is the dependence of emissivity on variance (8). On first thought one might expect an increase in scattering coefficient due to an increase in variance. In fact, this is what happens in the single scattering case. But as pointed out before, in multiple scattering an increase in v-riance results in an increase in scattering loss (11") and thus the scattering coefficient remains bounded as it should be. This is in contrast with the single scattering case where emissivity decreases steeply and monotonically with variance. Also at very low variance it is clear that multiple scattering is negligible. Thus there is essentially no difference between single and multiple scattering at very low S. These are illustrated in Figure 3.8. In Figure 3.9 we have plotted emissivity versus frequency. At very low frequencies the scattering is so small that there is little difference between single and multiple scattering results. In the low frequency range Rayleigh scattering phenomenon is observed, i.e., e decreases as the frequency increases because y is proportional to f4 . However, since 'correction' is also proportional to frequency the emissivity by the MRT result does not decrease as sharply with frequency as that of Born. In general, frequency responce diplays a resonance phenomenon; but in our example the resonance does not appear to be very distinct. In a way the dependence of y on frequency is closely related to the dependence of y on correlation length (lp). This can be seen by the 99 0.96 0 .9 7 Emissivity 0.96 0.9 5 45' Q 94 Ei = (2.8 + i.004)e0 £i 2= (2.81 + i.006) £0, V = 30’ q§ = 2 lp = 2 mm,lz = 4 mm 0 .9 3 0.92 0 0.0I 0.02 0 .0 3 Variance (8) Figure 3.8 e*RTand eRorn versus normalized variance. 004 100 1.00 Emissivity 0.95 0.90 e,,(Born) 45° 0.85 Ei * (2.0 + l.004)£o e ,x»(2.01 +1.006) e0,V = 30' 8«0.05.qi»2 l p«2mm, l i = 4 mm 0 .80 Frequency [GHz] Figure 3.9 Frequency response of eBorn and e„KT. 101 following argument. As frequency increases, effectively the size of the scattering objects increases from the point of view of the incident wave. Since the correlation length is a measure of the size of the scattering inhomogeneity, it is clear that the frequency response and the correlation length response convey one and the same information. This is seen by comparing Figure 3.10 with Figure 3.9. Finally, we would like to apply our theory to remote sensing of sea ice which is known to be electrically anisotropic. For comparison, we first consider the emissivity data of the open sea water published by Hollinger et al. [1984]. After taking into account the dependence of the permitivity of sea water with frequency [Ulaby et al., 1986], we have selected appropriate parameters for data matching as shown in Figure 3.11. Letter W represents the measured data while • (dot point) indicates the result calculated by our theory. We note that the theoretical calculation agrees fairly well with the experimental data. In both theory and experiment, the emissivity of open water increases with frequency because the dielectric constant of sea water decreases with frequency. Next, we consider the emissivity data of multiyear sea ice [Hollinger et al., 1984]. Since multiyear sea ice is usually of more than 3 meters in thickness and since the skin depth of multiyear sea ice at microwave frequencies is often not greater than a meter [Ulaby et al., 1986] we can use a half-space model. Also since multiyear sea ice is known to be electrically anisotropic and highly scattering, our multiple scattering anisotropic model is a suitable one. In Figure 3.12, we have shown that the theoretical calculations fit in the emissivity data of sea ice at five different frequencies in the range 19 GHz to 140 GHz by assigning the same set of parameters: 8i *■ 102 (MRT) 0 .9 6 Emissivity 0 .9 4 0 .9 0 0.86 45' f=9GHz e i - ( 2 . 0 + i.004)e0 e i z = (2.01 + i . 0 0 6 ) eo , v - 3 0 ’ 5 * 0.05, qf * 2 2.0, 0 .8 2 0 .0 0 3 0 .0 0 5 Q 007 0 .0 0 9 Correlation Length (^>)[m] Figure 3.10 eBocn and eMRT versus correlation length. LQ3 OPEN WATER 1.0 W • 0 .9 EXPERIMENT (Hollinger et al.» 1984) THEO R Y 0.8 Emissivity 0.7 w 0.6 w 0 .5 w * 0 .4 £2 = (”>8 + i29) e0 (16 + 127) £0 (14 + i24) e0 (6 + l9)£0 (5 + i6) £o 0 .3 at at at at at 19GHZ 22GHZ 31 G H Z 90 G H Z 140 G H Z 0.2 0.1 10 20 50 I0 0 200 Frequency [GHz] Figure 3.11 Interpretation of measured emissivity data from sea water. 104 MULTIYEAR ICE 1.0 M • EXPERIM ENT (HOLLINGER et aL, 1964) THEORY 0 .9 M 0.8 M Emissivify M 0 .7 % M 0.6 0 .5 0 .4 ei =(2.7 + i.003)eo =(2.7 ♦ i.004)eo, v*25* 5 = 0.3 , q$»2 lp = 0.15 m m ,li* 0 .3 m m e 1z Q3 0.2 O.l 10 20 50 10 0 200 Frequency [GHz] Figure 3.12 Interpretation of measured emissivity data from multiyear sea ice. 105 (2.7+i.003)eo, elz = (2.7+i.004)eo, 8-.3 , “2, lp-0.15mm, lz-0.3mm and y-25*. Although there Is an amount of freedom In the choice of the above parameters one must realize the inherent limitations. For the parameters must correspond to the observed properties of sea ice. Thus the process of 'trial and error' must be guided by the knowledge of the behaviour and properties of sea ice. Again, the letter M denotes multiyear ice measured data and • the corresponding theoretical fit. We notice that the agreement between the two is quite good. The emissivity of multiyear ice decreases with frequency because of the scattering effect. We draw attention to the parameters that we have chosen: lz > lp and qz > 1. This implies that the vertical correlation length is greater than the lateral correlation length and the strength of fluctuation in the vertical direction is larger than that' in the lateral direction. This is a direct result of the fact that the brine inclusions inside an ice crystal look more like vertically elongated ellipsoids. The value of E chosen is in the range reported in literature. 106 3.7 SUMMARY We have developed a model for active and passive remote sensing of a half-space anisotropic random medium. In order to incorporate multiple scattering, we have used the modified radiative transfer (MRT) approach. We have solved the MRT equations and have obtained the bistatic scattering coefficients and hence emissivities. On comparing the results with those of single scattering, we notice that the important difference here is the appearance of the (multiple scattering) 'correction' to the propagation constants. With the help of examples we have highlighted several interesting properties of the theoretical model. As an application we have used our model to interpret the emissivity data of the multiyear sea ice. As future work we suggest that the exact solution of the MRT equations ■ be obtained numerically. This would help us identify the range of utility of our model which is based on a first-order approximation. CHAPTER 4 SECOND-ORDER BACKSCATTERING COEFFICIENTS OF A TWO-LAYER RANDOM MEDIUM In this chapter we study the application of the modified radiative transfer (MRT) theory in obtaining the backscattering coefficients of a two-layer isotropic random medium. Since exact analytic solutions for the MRT equations are not available, we look for approximate solutions. Using a first-order approximation we obtain the backscattering coefficients. We recognize the merits of this procedure. In an attempt to investigate the appropriateness of this first-order approximation we examine the higher-order solutions. We observe that the second-order solution is important because it is the primary source of depolarization (cros3-polarized backscattering); besides it also helps us to estimate the error in settling for the first-order approximation. We compute the 3econd-order backscattering coefficients and cast them in a form suitable to identify, with the help of scattering diagrams, the various scattering processes involved. We notice the absence of "phase" terms and point out the need for including them in the MRT theory. Using computed data we study some characteristics of the second-order solutions and compare them with the corresponding Born results and also with the first-order solutions. 108 4.1 INTRODUCTION In the study of propagation and scattering from random media there exist two most commonly used models: the continuum model and the discrete model. Although the discrete scatterer model may be appropriate in several situations, the continuum model is more convenient and even more appropriate in certain others. random continuum model. We focus attention in this chapter on the The radiative transfer (RT) theory is often used to study the scattering and propagation in random media. Particularly it is suit'ble to describe multiple scattering. But since it deals with intensities rather than fields it ignores some phase information. In general this has not been a serious drawback. But there are situations (e.g., backscattering case) where negligence of phase information would lead to significant errors. 1 The remedy for this is to take a wave approach. Here the mean field is given by the Dyson equation and the field correlation is given by the Bethe-Salpeter (B-S) equation. These equations are exact and they account for all the scattering processes. has to make an approximation. But in order to solve them one Using the nonlinear approximation for the Dyson equation and the ladder approximation for the B-S equation, Zuniga and Kong [1980a] have derived the modified radiative transfer (MRT) equa tions. It is remarkable that they have started with the wave equation and transformed it into a pair of equations very similar to the radiative transfer equations. These equations are called the MRT equations because some additional scattering processes ignored in the conventional RT theory are included here. Like RT equations these are also a pair of coupled integro-differential equations for upward and downward travelling wave intensities. Usually these equations are solved using various numerical schemes But when one is interested in analytic solutions/ one has to use some appropriate approximations. approximation. The simplest one is the first-order Lee and Mudaliar [1988] and Mudaliar and Lee [1990] have used this approximation to obtain analytic solutions for scattering coefficients of a half-space anisotropic random medium which in spite of the simplicity of this approximation lend considerable physical insight into the various scattering processes involved. The usual rationale for this approximation is that if the medium is not too strongly scattering, then this approximation is a fairly good one. The objective of this chapter is to investigate this a bit further In this process we obtain the second-order solutions and study some of their characteristics. follows: The contents of this chapter are organized as In Section 4.2 we describe the geometry of the problem and give the MRT equations and boundary conditions associated with it. In Section 4.3 we solve the MRT equations using a first-order approximation and derive the backscattering coefficients. In Section 4.4 we obtain the higher-order solutions to the MRT equations and express the backscattering coefficients as an infinite series. In Section 4.5, we calculate the second-order backscattering coefficients and, aided by scattering diagrams, we examine the various scattering processes involved. We consider some numerical examples in order to illustrate some of the characteristics of the second-order solutions. Finally we conclude this chapter with Section 4.6. 110 4.2 PROBLEM STATEMENT The geometry of the problem is given in Fig.4.1. free space. Region 0 (z > 0) is Region 1 (-d < z < 0) is the random medium whose permittivity is given by V ;' i*'1' where elm is the mean and elf<r) is the randomly fluctuating part of e^r). Region 2 (z < - d) is a homogeneous medium with permittivity E^, three regions have the same permeability All the The incoherent wave intensities in the random medium are governed by the following MRT equa tions [Zuniga and Kong, 1980a]. - " W 2 • V 1- V + Quu (5 Kp', icp i.) • ♦ W V V + 4 Viy V *!J I mu <z,' icp i.) • w - Vi1 •W * ' V s 2i>« I »uu r p p L *-> p • I u u , it.) p + 5ud(^p' V • V * - Ep>] l4-2> Ill Region 0 fji* e, (?) = e im+€,f (f) Region I ------------------------------------------------------------------------- z = -d /*» € Z Figure 4.1 \ Region 2 Geometry of the problem. i 112 " W 1 £ v z' V - •vz-V + Qdd'^p' S i ’ • + Su'S' w z' S i ’ S i ’ • S u |z- S i ’ - A Q (B , ic ,) • I 0 (z, k ,) c Hp' pi mc2 pi' -f d!S tSd'S' S’;Slz'S’ +Su<S' S’•S(z'S’] 1 / (i = - k . p pi 0 , otherwise l4-31 t where Here I and I are the incoherent field intensities of the upward and u d downward travelling waves; I and I . are the corresponding coherent mu md field intensities. Q and P matrices are the phase matrices for the demands some attention. c This represents the scattering process which involves oppositely travelling waves. in the regular RT theory. The matrix no coherent and incoherent field intensities, respectively. Such terms would have been ignored We refer to Zuniga and Kong [1980a] for m quantitative definition of the phase matrices. T| is the extinction matrix given as n<&p> = 2n» 0 Here and ^ 0 0 0 0 o 0 0 0 (4.4) denote effective propagation constants for horizontally and vertically polarized waves, respectively. As mentioned before these are obtained by solving the Dyson equation under the nonlinear approximation. Explicit expressions for these propagation constants are given by Zuniga and Kong [1981]. We shall denote real and imaginary parts of a complex quantity by superscripts ' and ", respectively. For our two-layer problem (Fig.4.1) the boundary conditions associated with (4.2) and (4.3) are given as follows: v 0' pp> - “x o ' V • v 0' V (4.5) (4.6) Here 0; is the reflection matrix for a wave travelling from region 1 to is the reflection matrix for a wave travelling from region 1 to 2. Expressions for these reflection matrices in terms of half-space reflection coefficients are given in the Appendix. 114 Since exact solutions for (4.2) and (4.3) in closed form seem formidable, we will first make the first-order approximation to derive the backscatter ing coefficients. We will investigate the legitimacy of this approxima tion by deriving the second-order backscattering coefficients and studying their characteristics. I 115 4.3 FIRST-ORDER SOLUTIONS Under a first-order approximation we let 1^ ■ I integrals in (4.2) and (4.3). “ 0 inside the Now these equations along with the boundary conditions can be readily solved and the solutions are given as follows. -2 I ™ U. i!p> - ¥<., ?p) m lm3' . { J d, r V . pp> . 3u (,, fp, i:pi) + ^ 1,(?p)} ,fv) dS *(5, i5pl . 5d (3, 0p, kpi) + ii^1’ -S (4.7) (4.8) where ¥(2,13 ) is defined in the Appendix and r J (s, 5 , k .) = Q (5 , k .) • I (s, k .) u p pr uu rp pi mu pi + Q , (B , it ,) - I ^(s, k .) ud Kp' pi md ' pi +AV*p' V *W3' V (4.9) Vs' V V ="°ddV V • W3' V - Q . (B , k ,) • I (s, k ,) du 'p' pi mu ' pi + A V*p' V • W 3' V (4.10) 116 We shall use superscript (n) to Indicate the nth order term. From (4.7), (4.8), (4.5) and (4.6) we have C ' V - I*"1'-*. ?p> -1 -I12(V •S'"1<-»<?»> •w-V •%<iu •^<0'Pp>] f J 153 " d {512(fp> ■ ’, ‘1 (-d' V • 5'<3' *p' • V ^ p ' V 0 - ¥(-d, Pp) . J ds ^ ( S , $p) . ju (SrPppkpi)| (4.11) 0 K^1)(Pp) - $ (0, gp) . R10(^p) • *<0,Pp) K ^ t f p ) (4.12) Now the first-order backscattering coefficients can be written as follows (Lee and Mudaliar, 1988). For horizontally (h-) polarized Incident field - 4nk2 cos20 , o oi i (1! (0, -E .) oul pi (4.13a) 01|1> - 4«k2 cos20 . hv o oi I(H(0, -E .) ou2 pi (4.13b) a ™ hh and for vertically (v-) polarized incident field vh - 4«k2 cos2e . o oi I(1J(0, -k .) oul pi (4.14a) 1 117 (4.14b) where Explicit expression for the transmission matrix !,,,($„) in terms of half-space transmission coefficients is given in the Appendix. subscripts 1 and 2 in I I , respectively. The denote the first and second components of Using (4.7), (4.8) and (4.15) in (4.13) and (4.14) we obtain explicit expressions for the first-order backscattering coefficients. To facilitate easy physical interpretation we cast the results in the following form: (4.16) where (4.17a) (4.17b) (4.17c) (4.17d) (4.17e) 8 is the Kronecker delta and all the other quantities are defined in the P<1 Appendix. The subscripts p and q denote the polarizations of the incident and scattered waves respectively. The subscript i denotes here that the corresponding quantities are evaluated at the incident direction. We notice in the first-order solution that there is no depolarization, i.e., 0p q- 0 for p * q. We also note that these results are in complete agreement with those of Zuniga and Kong [1980a]. But they are here couched in a form, as we physical interpretation. 3 hall see, more suitable for our purposes of There are five terms in (4.16); each term corresponds to a scattering process shown in Fig. 4.2. While the solid lines denote the propagation paths of the fields the dotted lines denote those of their complex conjugate fields. In fact, we note that S_ 5 consists of two terms corresponding to the diagrams O and O . As mentioned before these are the additional terms introduced in the MRT theory. It is important to note that these are of the same order as the other terms in the backscattering region. 119 s, S4 (b) (o) Figure 4.2 Scattering processes for . P<? 120 4.4 HIGHER-ORDER SOLUTIONS Although the major appeal of the first-order approximation is the simplicity of the solutions, one may naturally question the appropriateness of this approximation. On physical grounds one can conclude that if the scattering is not too strong then the higher order scattering is relatively small and thus a first-order solution is a fairly good approximation. But one is often interested in quantitatively assessing the situations under which the first-order approximation is applicable. One way to answer this question is to examine the higher- order solutions. To this end let us first look at the second-order approximation to (4.2) and (4.3). MRT equations under this approximation are given as follows: ■ Ju|z'V V “' M 5uu<5P ' V • !u 1 > , z - v + 5 „ d ' 5 p - v • !d1 , |z ' y 3 < 4 -i 8 > 5d (z' V v ■^pPdd'VV •id11<z'Ep> +fd»tfp'V •!u1,(*'V] (4-19’ where (4 .20 ) Using (4.7) and (4.8) we can readily obtain the solutions to (4.18) and (4.19) as follows: = (2) 7(1) Id (,^ p ) " "d (z' V (4 .21 ) (4 .22) (2 ) + Xd (Z' V where {z d3 I d**P ^"1(3,^p) 5(1) p' -p' * i:‘,(S'k-) u p' [ 5UU(*p'*J + 5»d‘i V V • + 5»2,|V / x “ ’ <z,5p > - r l („|jp ) t ^ r 2 • 1 1 jd) Ad — p* p- + » * < V V Since that (*rPp) and • (4. 23) d» | d"Ep p' «■<v l + <*P >} (*'$p) satisfy (4.5) and (4.6) it is clear (z,Pp) and I^2* (*»Pp) also satisfy (4.5) and (4.6). We then .24) have the following expressions for Rl0(P ) • *(0,8 )] (j5p) and IP lmz ((5p) -2 -d ds <?«> * ^_1 (~dr P„> d2k 1R1 2 (JP *<■> *p> • [ w v v • !dll,s'V * K'du . j rp L ' kpJ ■ i " ’ l3- y ] -d - *<-d, Pp) dsI d k ^ l i".?p> • [ y A p-- pV- * 5« d ' V V • Jd “ • i I3'V l3 ' V ] j i ‘2> (Pp) - *(0, Pp> • Rl0(?p) • *<0,iSp, • itf’ (Pp) (4.25) (4.26) We can continue further along these lines and obtain higher-order solutions. The complete solution for 1^ and 1^ will then take the following form 123 v ^ p ' - r d where J ‘n,<zjp, - iplmz|-2 (4.27) c i(2'5P' n-i d (J ds K d% * <s' V (n-1) u •• p' [5uu(*p'V • C ‘ ♦ w i v y • i dn‘ i) ■ , r l | * ^ p > ' l 1iM ' V ♦c '"2 • { | i , <v} »■ / d ,Ep (4.28) * ’'ft o • [?d d « V v + ?d u < V V t • (4.29) • iun ’ «»'Ep>] where it“ (5 ) . [ty-d, jt ) - r 12(5 ) . f(-d,S ) . 4>(0,5 ) . -2 -a { 5 1 2 (Pp i . V 1 (-d, 5p ) j 0 • Pdd'iv V • i'i'1' <»'V d. | d2kp ?(S ,5p ) 124 + sd u < *pp 'pV • C l’ « * > v ] -d ¥(-d, Pp) • J ds 0 -<n-l) u p- p' p' (4.30a) + 5ud(*p'V * - (n) (4.30b) The nth-order scattering coefficients can now be readily written as follows: For h-polarized incident wave, CT*"* - 4« k2 . I (n! (0, - k ,) hh ozi oul pi (4.31) 4jt k2 , l <ni(0, - k .) ozi ou2 ' pi ^hvn> For v-polarized incident wave, •vh ‘h “ 4jt k2 . I (n! (0, - k .) ozi oul pi (4.32) a'"’ - 4ji k2 . I ' (0, - k J w ozi ou2 pi where C 10' fp» ■ 5pi (4.33) Thus the complete solutions for the backscattering coefficients now become pq - ^T .(h) pq (4.34) 1 125 Now the representation for crp q is meaningful if the series converges. The convergence of this series is guaranteed if CT(n+D lim ~pS n-»«o o < i (4.35) pq But this appears to be a task too complicated for us to perform. We therefore agree to be content instead with the following milder version of the above test. We define the ratio <5^ J a*1* as R. pq expect the series to converge rapidly. pq If R « 1 we In such a situation we can con clude that our first-order solutions of (4.16) would be a meaningful approximation. (2) In order to perform this test we need to calculate CT pq . We undertake this task in the following section. i 126 4.5 SECOND-ORDER SOLUTIONS As shown in (4.31) and (4.32), to obtain the second-order (2 ) -(2 ) scattering coefficients <X , we need to calculate I , which through pq ou (2 ) (4.33) is linearly related to 1^ (2 ) expression for Iy in (4.23). . We have obtained a formal Our task now is to evaluate it and use (2) (4.31), (4.32) and (4.33) to obtain <X . After performing these operations and simplifying we cast the results in the following form which, as before, is very convenient for physical interpretation • £ ’ - - “- I f k u i i! s? + " W ! (4-36> where 5? ■ J k i {[ V i i ' % ' 6/ * "h 'w * - 2»;i>'1[°(2’i"qi - 2»;i> - <3<2i"qi - *&>] + “ hi S'2";! - 2V Sj - sj G <2’1qi - 2TIm >)} « - 27»> {Replace h by v} (4.37b) lmzl + + \ x \ x Dhi Gi-2,'hI - - 2V r2 ' Gl2^ i * 2^I»] gi2v * 'hi')} (4-: (4-37°> 127 S4 " S3{RePla c e h by v) 5? - J ^ V {[ V l l ' (4.37d) W * *1.1 Dhi G<-2^ i Sg ■ !1 *11 " - ^i' s'2’’,! - W I'2 '4 -37®> Sg{Replace h by v} (4.37f) {[vwvfi-i +I ,n'^i-£-i»]'fii-*i»*ii'ki„i >2 " h i DhJ 2 - ^ i l - O I 2^ + 2H J,)} Sg - 8" {Replace h by v} 8? - 8? J J Pi \ Replace ^pi I wlx W1I 2 ''2 (4.37g) (4.37h) ■* P-i "* _T1pi -> W2X -* W2I , j = 1,2, ...,8 (4.38) Here h and v are the unit vectors (3ee the Appendix) of the electric field with the horizontal and vertical polarizations, respectively. Fur ther we have used a notational simplification to represent them here. For instance, we denote h(-k 1RIZ3 ) as h ™8 for s * i or I, and so on. The subscript .1 represents the direction of the scattered wave propagating in the random medium layer. We observe that there are sixteen terms in (4.36) and each term 128 represents two scattering processes. The above representation immediately enables us to identify each term by its corresponding diagrammatic descrip tion given in Fig.4.3. We note that there are 32 scattering processes altogether that are taken into account in the second-order solutions. In fact this is precisely what we would expect if we enumerate all possible second-order scattering mechanisms for the given geometry of the problem. Thus without any further calculations we can readily predict that the number of scattering processes involved in the nth order scattering coef ficient as 4 x (8)n 1.But we note that this number would still be the same had we used the regular radiative transfer (RT) theory. The motivation for using the MRT theory has been, as mentioned before, to include additional significant "phase" terms which are normally ignored in the RT theory. We shall see that these terms become important particularly in the backscattering regime. Thus the MRT theory correctly* takes account of these "phase" terms for the first-order scattering. is manifested by the appearance of This in (4.16) (see also Fig.4.2). But for higher-order scattering it appears that all the phase-terms are ignored. For example, in the second-order scattering the "phase" terms are shown diagrammatically in Fig. 4.4. We observe that these terms are equally ignored in both RT and MRT theories. The reason for this is clear. We recall that in deriving the MRT equation Zuniga and Kong [1980a] have used the ladder approximation to the Bethe-Salpeter (B-S) equation. What we call the "phase" terms correspond to the cyclical terms in the Feynman diagram representation of the B-S equation [Tsang and Ishimaru, 1985]. Because these terms are ignored in the ladder approximation the MRT theory also fails to account for them. Thus it is clear that the present MRT equations need to be further modified by taking into account all the 129 V2 A A h/v 3/4 h/v h/v h/v h/v £ ,4 h/v S u °7/8 Figure 4.3 (2 ) Scattering processes for <J pq Conjugate fields are not shown here for the sake of clarity. 130 Figure 4.3 (contd.) Figure 4. Scattering processes resulting from "phase" terms corresponding to corresponding to each • There are similar diagrams of Figure 4.3. 132 cyclical terms. Only then will it be able to properly describe the special characteristics of backscattering phenomena as observed by Kuga and Ishimaru [1984]. Putting this issue aside for now, we proceed to study some of the characteristics of the second-order results that we have obtained here. The most noticeable fact here is that the second-order scattering gives rise to depolarization in the backscattered direction. for p * q, we note that a*1* =0, while a*2^ * 0. pq pq In other words, This is a well-known property and has been studied by Zuniga and Kong [1980b]. We n^w use numerical results to compare the second-order MRT results with those of second-order Born. By "Born"we mean that the multiple scattering effects on the mean wave are ignored. Thus our Born results are different from the second-order results obtained by Zuniga et al. [1980]. Their results include the cross-scattering terms while ours, as mentioned earlier, do not. The Born results account for the double scattering of the "unperturbed" wave, while our MRT results account for the double scattering of the "mean" wave which includes the multiple scattering effect in it. In Figs. 4.5, 4.6 and 4.7 we compare the second-order backscattering coefficients of MRT and Born. In Fig. 4.5 we have plotted the cross-polarized backscattering coef ficient versus frequency. All the relevant parameters are shown in the inset. The steep climb at the low frequencies is identified as a pheonmenon similar to Rayleigh scattering. It is clear that multiple scattering introduces effective attenuation and consequently reduces scattering. * Thus or is lower than a MRT Born In Fig. 4.6 we have o^h versus incident angle. Once again the MRT 1 133 Born Backscattering coefficient (q,h)[ dB -10 N/RT -20 -30 -40 20 Frequency ^Hz] Figure 4.5 Frequency response o£ cross-polarized backscattering coefficients by MRT and Born. -2 5 Born -35 f= 10GHz -40 z=0 Back scattering coefficient (o^h) J dB -30 €a * ( 60+ i ' 6)*o -50 20 0 Figure 4.6 40 60 Incident angle (0O,)[degreesj 80 Incident angle response of cross-polarized backscattering coefficients by MRT and Born. Backscattering coefficient (oj,h )[dB ] 13.5 Born -5 -10 -1 5 MRT -20 -2 5 im = ( l.8 + i- 0 0 5 K 7 = 2 mm ljo=5mm -3 0 z = -5 m Q02 Q 04 006 0 .0 8 0.0 0.12 Normalized variance( 8 ) Figure 4.7 Comparison of MRT and Born results of G ^ 33 a function of 5 (variance of random fluctuation). 136 results are lower than that of Born and fall off as the angle of incidence increases. In Fig. 4.7 we examine the sensitivity of to 8, the variance of per mittivity. Since multiple scattering effects become negligible for small 8, MRT and Born results are indistinguishable there. But as 8 increases, we note that <T increases monotonically which is unphysical. corn d _ MRT reaches a plateau at large 8. In contrast, Physically this is attributed to the shielding phenomenon introduced by multiple scattering. In Fig. 4.8 we compare the first-order and second-order scattering coefficents for various incident angles. ference between the two is approximately10 dB which corresponds to an R of 0.1. As observed in thisexample the dif From the earlier discussion we thus conclude that the error in using a first-orderapproximation here is of the order of an error can be tolerated then we can be content with the approximation. 10%. If such first-order * On the other hand in situations where the investigation of the phenomenon of depolarization is of high priority, the second-order solutions are of primary importance. This is because depolarized back- scatter is essentially a second-order phenomenon in isotropic random medium. In practical applications such as remote sensing depolarized back- scatter is of late attracting increased interest because of its ability to discriminate between targets. second-order solutions. This places added importance on the 137 BacKsc ottering coefficient (cr^J^dBj -10 -20 hh -25 f- 106Hz -30 z=0 (1.8+i 005)^o lz =* 2 nnm I^s5mm -35 -40 Figure 4.8 S ■ 008 20 40 60 Incident angle (9oi)[degreesj 80 Comparison of first- and second-order backscattering coefficients O. . . 4.6 CONCLUSIONS We have studied the application of the modified radiative transfer theory in obtaining backscattering coefficients of a two-layer random medium. Since exact analytic solutions to the MRT equations appear to be formidable we have looked into possible approximations. Using a first- order approximation we have obtained the backscattering coefficients. We identified the merits of such a solution. In order to determine the situ ation when such an approximation would be appropriate we examined the higher-order solutions. In this context we have found the importance of second-order solutions. We calculated the second-order backscattering coefficients and cast them in suitable form to facilitate physical interpretation. While examining the various scattering processes involved we have noticed the absence of some of the "phase" terms which are impor-* tant in backscattering. Only in the first-order does the MRT theory account for these "phase" terms. We have thus pointed out the need for modifying the present MRT equations. We have also observed depolarization which is primarily caused by second-order scattering. With the help of computed data we have studied some characteristics of cross-polarized scattering coefficients and compared them with the corre sponding "Born" results. Also we have compared the relative magnitudes of first-order and second-order backscattering coefficients. As a final note we draw attention to the recent work by Kuga et al. [1989] which deals with the second-order solution to the radiative transfer equations for the discrete random medium. This is in connection with the study of scattering from a half-space medium containing randomly distributed spherical particles. Since there is an 139 equivalence between our isotropic random continuum model and theirs it is not surprising that some of their comments on depolarization and cyclic terms coincide with ours. I 140 APPENDIX IR±;.|2 0 0 0 IS,.! ij' Ri j <5p ’ 0 0 R e j S ±j) -Im(R*jSij) 0 0 ImtR^S^) Re <RJjS±j) (A. 1) where R.,'s and S.,'s are Fresnel reflection coefficients given as i: 1] k. IZl - k. is. Rij " k + k iz jz ' ifj “ t0'1'2} e,k. - e,k. . 3. .i J a____i_i£ ij 8 .k. + e.k. J ] 1Z i (A’la) (A. lb) ]2 “2TlhZ -2ll"z 'v -az 0 e cos cz -az . -e sin cz e -az . sin cz -az e cos cz (A.2) where 142 i2\ i d Ehui= (X01i/D2i) R12i 6 (A.8a) Ehdi " X01i/D2i (A. 8b) i2\ i d Evui = (V klm> <Y01i/F2i> S12i e (A.8d) Evdi= <k0 /kl») <Y01i/F2i) “h = L1 - |R10R12|2 ® (A.8c) ]I/ ID2 12 (A. 9a) J| '/IF, “ 2''2 (A. 9b) -4T|"c I2 e. av = I1 " IS.nS.ol2 10 1 2 1 e ’V 12 t) "i2T1hid J Yhi = R12i ® 1 + R 01iR12i e s?~. 121 ® 'vi V1... f1 + snlJs,., e O i l 121 -i2T)*.dr ^ .dL J /|D2i'2 (A.10a) vi,dj /IF,. I2 (A.10b) 12T| ’ 21' " ' R10iD2i/|D2i'2 (A.11a) vi " - S10iF2i/|F2i'2 (A.lib) *h |xio|2 —i Y k 10 O Dh - ID2 I (A.12a) (A.12b) (A.13a) 143 Dv “ If 2 I (A.13b) -4iV*d *h " |R1 2 |2 ® (A.14a) -4Tl"d K = 1S1212 e \ " V (A.14b) lR 1 0 lZ (A.15a) Rv = IS i q |2 (A.15b) W1 ' I ® <i:pi ' 5P' P lr»z *kl»zi> <A -16a’ W2 - I ““‘V - *p> ^lmz +W <A -1Cb> W 3 ‘ 2 ® (kp i ' 5 P' Fj = Wj { j ^ -> i^} Fj = W. {i^ 01 ,A-16o) , j = 1,2 (A.18) -> k x} , j - 1 , 2 (A.19) h(,Clmz) "IT (iky • <A-20a) P ^ l m z * " " I lm T T "p (ikx + ■* + * k6” lm (A-20b) CHAPTER 5 WAVE PROPAGATION AND SCATTERING FROM A RANDOM MEDIUM WITH A RANDOM INTERFACE In this chapter we study wave propagation and scattering in a half- space random medium with a random interface. In order to involve multiple scattering we treat surface randomness and volume randomness with equal importance. We assume the random processes to be Gaussian. Also we assume that the random fluctuations of both the medium and the surface are small. These enable us to use approximations and physically identify the various scattering processes. We derive the Dyson equation and with the help of Feynman diagrams we observe the scattering mechanisms. We apply the bilocal approximation and the nonlinear approximation to the Dyson equation and compare them. Following similar procedures we also derive the Bethe-Salpeter equation and apply the ladder approximation to it. The Feynman diagram technique affords us to clearly identify the scattering interaction between the random surface and the random medium. These terms which are usually ignored in a single scattering approximation become important in a multiple scattering solution. 145 5.1 INTRODUCTION In the study of wave propagation and scattering in a random environment the approaches and analyses have hitherto largely depended upon the nature and type of the random environment under consideration, in the referred literature we note that the random environment is classified as random continua, random discrete scatterers and random surfaces. Having made this classification people have taken separate routes and developed theories which are very characteristic of the classifications concerned. Among the three categories we note that on physical grounds the random continuum model is just a convenient approximation of the more general random discrete scatterer model. Thus although the approaches taken in the above two cases are different it is clear that they are compatible with each other and hence this classification is rather artificial. We might as well regard both of them under the category of random media. On the other hand the theories developed for random surfaces appear to be entirely different from those developed for random media. But when it comes down to applications it is very difficult to make unambiguous classification of random media and random surfaces. In nature random media and random surfaces are so integrated into each other that it is rather difficult to decide which of the theories would be most appropriate. Of course it is true that theories for random media and theories for random surfaces have found useful physical applications in active and passive remote sensing. But this has only been possible because there are several situations in practice where scattering due to random surfaces is more dominant than that due to random media and vice versa. But in a general situation when the effect 146 of both random quantities are equally important it is apparent that the existing theories are inadequate. Some [Zuniga et al, 1979; Lee and Kong, 1985] have treated such problems by breaking them into two subproblems - one having random media and deterministic mean surface and the other having deterministic mean media and a random surface. Thus the combined problem has been assumed to be equivalent to the sum of two subproblems. In other words the scattering from random media and random surfaces are assumed to be incoherent. But apart from the obvious convenience there appears to be no firm rationale for such an assumption in a general situation. However, if we are only interested in single scattering, then the scattering from random media and random surfaces are indeed incoherent [Mudaliar and Lee, 1990]. In such a situation we can follow totally separate methods to calculate the scattering due to surface randomness and the scattering due to volume randomness and finally sum them to obtain the total scattering. But if the random quantities involved are not small, multiple scattering becomes important. This immediately leads to interactions between random surfaces and random media. This phenomenon can only be analyzed by a theory having a unified formulation which treats both surface randomness and volume randomness on an equal footing. Fung and Chen [1981b] and Fung and Eom [1981b] realized the importance of this and developed methods for analyzing such problems. They have given several numerical examples and have clearly demonstrated the need for such a unified approach. Unfortunately their approaches eventually lead to a numerical solution which does not, in our opinion, reveal immediately all the physics behind such scattering interactions. Later Furutsu [1985] has studied a similar problem and has given a rather abstract treatment 147 which although comprehensive does not seem to be application-wise illuminating. It is appropriate for us here to examine what mathematical issues are involved in taking a unified approach. Stripped of the physical details we have essentially a set of partial differential equations with a set of boundary conditions. In the random medium problem the coefficients of the the partial differential equations are random quantities. On the other hand in the random surface problem the randomness is enclosed in the boundary conditions. Thus from this point of view the two problems do not appear to be very different. So on fir3t thought a unified treatment seems to be straightforward. But a3 we shall see later there are several difficulties when a physically transparent closed-form solution is desired. Most of the difficulties of course stem from the fact that the quantities involved are random. To summarize, for a multiple scattering solution of the general problem of random media with random surfaces a unified theory is essential. Further it is desirable that this unified approach affords physical insight into the various scattering processes. We have organized this chapter as follows. In section 5.2 we describe the geometry of the problem and state the partial differential equations and boundary conditions associated with it. In section 5.3 we derive the integral equations for Green's functions. We proceed in section 5.4 to obtain the corresponding integral equations for the mean Green's function. Further with the help of Feynman diagrams we inspect the various scattering processes involved. In section 5.5 we derive the BetheSalpeter equation for the field correlation. We apply a renormalization scheme to the B-S equation and apply the ladder approximation. We conclude this chapter in section 5.6 with a brief discussion and summary of the salient features of the chapter. 149 5.2 STATEMENT OF THE PROBLEM The geometry of the problem is shown in Figure 5.1. Region 0 has a medium of density Vq while Region 1 has a random medium of density v^r) where (5.1) Here vlra is the mean part of v^r) and vlf (£) ia the fluctuating part of v^(r). We assume that the magnitude of vu(*) is very small compared with v1m . Region 0 and Region 1 share a common boundary £ given by z - £ (p) where £ (p) is a zero mean random function. Two reference planes z ■ 0 and z = -d are chosen such that £ is enclosed between them. Application-wise the analysis we present here describes scattering and propagation of acoustic waves. The scalar Green's functions, which are what we are primarily concerned about in this paper, satisfy the following equations. (5.2a) G10(f,r') - - ^1!) Ol0«,*'> (5.2b) (5.2c) V2Gn (r,f) + G^S,*') - - 8(f-f') - qf (r) G11(r,f') where kg is the propagation constant in medium 0 and k propagation constant in the mean medium in Region 1. Thus qf (r) - <D2 Vlf(r)/ B is the (5.2d) 150 Region 0 — Region Figure 5.1 Geometry of the problem. z=-d 151 where B is the bulk modulus of medium 1. The first subscript of Green's function denotes the region where the observation point is located while the second subscript denotes the region where the source is located. The boundary conditions satisfied by Green's functions are given as follows. 7 T 0 5 7 0 a i r air G00(P'^;f,) “ Gio(P'S;f,) (5.3a) S0i<P^;*,) “ G1;1(p,C;r') (5.3b) - 5 7 lm a i r < 5 ■ 5 7 a r O n «>.«;*■) lm - 3 <=’ <s.3d> t g where denotes the derivative normal to the boundary £ and the direction of the normal is into the region 0. Equivalent to the four equations of (5.3) we can obtain the following alternative set of four equations 5 J - O 00(p, 0,f> + Z00(f) O00<p, 0,I<> + Z0l(P) G10(p,-d,*M - 0 aS" O0l(p,+0l»') 4 Z00<p> O01<p,+0,»<) + Z01(P) Gll<P'"d;2,) 3n” G10 (P' “d;r ’) + Z1Q{p) GOQ(p,+0;r') + Z^tp) G10<p,-d;E*> - 0 (5.4a) (5.4b) (5.4c) 152 G 1 1 (p,-d;E') + Z1 Q (P) G 0 1 <p,+0;r'> + Z1;L(P> G ^ ^ - d ; * ' ) - 0 3 where Z's can be derived using (5.3). In (5.4a) and (5.4b) ^ an in (5.4c) and (5.4d) <5.4d) 2 — while az . Thus using the notations listed in the appendix, (5.2) and (5.4) can /respectively, be written as follows. H G(r,r') + q(r) G(r,r’) + I 5 (r-f') = 0 9 a =a ^-G(p,s;r') + Z(P) G(p,s;r*) - 0 (5.5) (5.6) Now the task that concerns us in this chapter is to obtain (G(r,Eg)) and (G(r,rg) G(r,,fo')> . The angular brackets here stand for statistical averages. 153 5.3 INTEGRAL EQUATIONS FOR THE GREEN'S FUNCTIONS Consider the situation when the medium 1 is homogeneous and the boundary £ is planar. We shall refer to this situation as the unperturbed situation. The Green's functions for this unperturbed situation satisfy the following equations. X G°(r, r') + 8 (r-r') X - 0 (5.7) gjp G°(p,s;r') + S°(P) G°(p,s;r') = 0 (5.8) The superscript o indicates that the quantity concerned corresponds to m -» the unperturbed situation. We can thus define Z(p) as surface impedance. Z<P) = Z°(P) + z(p) (5.9) where z(P) is the random part of Z(p). We assume that z(p) is very small compared with Z°(p). Consider now the Green's functions G(r',r^) satisfied by the following equations. X' G(r',rQ) + q(r') G(r',r0) + 8 (f’-r0> I - 0 G(P',s;r0) + Z(p’) G(p',s;?0) - 0 (5.10) (5.11) By changing X to X' in (5.7) we obtain (5.12) 154 where the superscript T denotes matrix transpose. By premultiplying (5.10) 9*0 by G (r,r') we get G°(E,r') X* G(r',r0) + G°{r,r') q(r’) G(r',r0) + G°(r,r’) 8 (r'-rQ) I - 0 (5.13) Postmultiplying (5.12) by G(f',fg) we get f =* anT \T \ X' G <r,r')j ** * _ =3 G <EI'E0> + 1 5(f-f') G(r',rQ) =■ 0 (5.14) Subtracting (5.14) from (5.13) we obtain J - S (r-r') G(r',rQ) - G°(r,r') 8 (r'-f0) - G°(r,r') q(r’) G(r\f0) (5.15) where J ■ G°(r,r') X' G(r’,r^) - { X' G°T (r,f')}T G(r\r ) Joo Joi J 10 J 11 (5.16) The matrix elements of J are given as follows. 00 V 1- { O°0 (E,E') V' + V.. { Goo<E,'Eo> - Goo(E’'Eo’ v'G00 (E'E''} V G <!•,*„) - S10<B',E0I V o j j «,!•>} (5.17a) 155 IM IM IM IM O O O C D OO o o s r* H « rl • m in IM V IM O O «H C D > IM IM v IM r* o «e> > I M o Or H C D > o o IM IM w IM IM IM w IM IM IM IM O T” o « “ c C D o H C D C D C D I I l I 1 IM IM IM IM W IM IM IM IM JM IM 0 0 C 0 o r- r" > > O O • » f" c c 0 0 C D > > > IM IM IM IM IM IM IM IM IM v O OO C D H Oo C D o O« H C D 0 ^ 0 IM • c 0 C D 156 (5.18a) 3 Here g^ 7 - 9 ^ 7 in the first integral while 3 ^ 7 - - 3 ^ 7 in the second integral. In short our normal derivative hereinafter will represent the partial derivative with respect to z in the direction into the medium under consideration. Region 1 respectively. and v denote the volumes of Region 0 and represents the surface z - 0 while represents the surface z - -d. Proceeding similarly we obtain the following. 157 (5.18b) I v a3*' J10 ■ J “V { Goo«’,-°!V 9 ^ G!oIEji, ''0> so - s!o<I'|,,'0, 9 ^ Goo«, , '0,Eo)} * I ^ { Gio",''-d' V 9 ^ sn IE'-f,'-dl si - Gjjd.-p'.-d) gfr G 1 0 (p',-d.-r0)} (5.18c) I d3f' Ju - J ^ v {S01«,’'»'V 3^ G?0(E'P''°» so - G 1 0 <E''5''0) 9 ^ Goi(',,'0 ,Eo>} + J dV { Gu (l,''-d ’Eo) a! 7 G?i<Ei< S1 - G^IE.-PW) j S7 8 u (ff’. - 4 , v ) (5.18d) Using (5.8) and (5.11) we can express (5.18) in the following compact form. Jd 3 r' J - J dV { G°(f;p',s) Z(P’) G(p',s;f0) - G°(r;P',s) Noting Z°V) that 2(p)is symmetric, (5.9) and J d3 r 1 J - Jd 2 G(p»,s;r0) } (5 19) (5.19) lead to p ' G°(r;P',s) z<p>) G<p,ra;S()) (5 .20) But from (5.15), I Jd 3 r' J - G(r,rQ) - G°(r,rQ) - Jd 3 r' G°(f,f') q(f') G(f',rQ) (5 .21) Thus from (5.20) and (5.21) we get G(r,rQ) - G°(r,rQ) + +J d 2 Jd px 3 rx G°(r,f]L) qfr^ Gfr^fg) G°(r;pif3) z(Px) Gtp^s.-r^ (5 .22 ) We note that where D(z) is defined in the appendix. Thus G^r.-p^s) z(Px) Gtp^s/Eg) - G°(r;r1) ^(r^ Gtr^rg) (5.24) where f(rx) = Dtz^ z (Px) D(Zl) (5.25) Using (5.24) in (5.22) we get G(r, rQ) - G° (r,rQ) + Jd 3 rx G ^ r ^ ) { q(r1)+ f ^ ) } Gtr^Eg) V1 (5.26) Equation (5.26) is the required integral equation for the Green's functions. In the next section we shall proceed to obtain the integral equation for the mean Green's functions. 5.4 THE DYSON EQUATION We start by iterating (5.26) repeatedly and thereby obtain the following Neumann series G(r,rQ) - G°(r,r0) + +J ^1 Jd 3 rx G°(r,r1) QJE^ G^E^Eg) J d% G°(Siff2) Q(£2) ®°<*2* V + . . . . (5.27) where Q(E) - q(r) + y(E) (5.28) We can interpret (5.27)as a multiply scattering series.The n'th term in the series corresponds to the wave travelling from point E' to E on being scattered in the process by n- 1 scatterers located at l E n- 1 2 Taking the ensemble average of (5.27) we get <G(E,Eq)> - G°(E,E0) + +J d 3 Ex Jd Jd 3 3 Ex G°(E,E1)< QfE^) G^E^Eg) E2 G°(E,E1)< Q(E^) G ^ E ^ ) Q(2 2)> S°(22/£ J + . . . . 0 (5.29) Regarding the properties of the statistical quantities involved we make the following assumptions: (a) We write q(E) * qf (E) P and assume that q^(E) obeys Gaussian statistics. (b) We assume that j-(E) can be written as 161 f <r) - ^(P) 2 (z) v •'V where 2 (z) is a deterministic two dimensional matrix and % (p> is a zero mean random function obeying Gaussian statistics. This means that every element of f(r) is generated by the same random function 4 (?) and they vary only by some deterministic constants. Since f (r) is in fact primarily dependent on the random function ( (p> which describes the boundary, our assumption is physically convincing. (c) We assume that the random medium is statistically homogeneous. In other words, < qf<*2» - cv(12rV> where Cv <r) is the autocorrelation function describing the random medium. (d) We also assume that the random surface is statistically homogeneous. In other words, 1 < 4 (IV 4<P2» - V ' f W ’ where C (p) is the autocorrelation function characterizing the random s boundary. (e) Finally we assume that the random quantities qf (f) and 4 <P) are statistically independent; i.e., < q ^ ) $<P2)> - 0 Hereinafter we shall denote r^ by i for brevity. Under the above assumptions we obtain the following. < Q(l)> - 0 ( Q (1 ) G° (1 ,2 ) Q{2 )> - C (1 ,2 ) (5.30) P G°(1 ,2 ) P + C (1,2) 2(1) G°(1,2) 2(2) s (5.31) 162 < Q (1) G°<1,2) Q (2) G ° (2,3) Q(3) > - 0 <5.32) < Q (1) G°(l,2) Q (2) G°<2,3) Q(3) G°(2,3) Q(3) > ^ P G°(1,2)P G°(2,3)P G°(3,4)P + Cv (1,2)C3 (3/4) P G°(1,2)P G°(2,3)2(3)G°(3,4)2(4) + Cv (1, 3) C 3 (2, 4) P G ° ( 1 , 2 ) 2 ( 2 ) G ° ( 2 , 3 ) P G ° ( 3 , 4 ) 2( 4) + Cv ( l , 4)CS (2, 3) P G°(1,2)2(2)G0 (2,3)2(3)G°(3,4)P + CV( 2 , 3 ) CS (1, 4) 2 ( 1 ) G ° ( 1 , 2 ) P G° (2, 3) P G°(3, 4 ) 2 ( 4 ) + Cv (2, 4) C3 (1, 3) 2 < 1 ) G ° ( 1 , 2 ) P G ° ( 2 , 3 ) 2 ( 3 ) G ° ( 3 , 4 ) P + Cv (3, 4) C a (1,2) Z ( 1 ) G ° ( 1 , 2 ) 2 ( 2 ) G ° ( 2 , 3 ) P G ° ( 3 , 4 ) P + T2 2(l)G°(lf2)2(2)G0 (2,3)2(3)G°(3,4)2(4) (5.33) where T1 =* Cv (1, 2 )Cv (3, 4 ) + CV (1,3)CV (2, 4) + CV <1,4)CV (2,3) (5.34a) T2 - Cg(l,2)Cg(3,4) + Cg(l,3)Cg(2,4) + Cg(l,4)Cg(2,3) (5.34b) Similarly we can proceed to evaluate higher order correlation functions. But the emerging pattern is clear now. Since we have assumed Q(r) to be a Gaussian process all odd-order correlation functions vanish. Also it follows that all even-order correlation functions can be expressed in terms of two point correlation functions. Thus (5.29) becomes ^ ( Z ^ q ) - G°(2,fQ)+Jd3lJd32 + G°(2,l) <Q(1)G°(1,2)Q(2)> G0 (2,rQ) Jd3lJd32Jd33|d34 G°(r,l) <Q(l)G°(lr2)Q{2)G0 (2,3)Q(3)G°(3,4)Q(4)> G0 (4,rQ) 163 + all even-order terms (5 .3 5 ) where sm a G (r,fQ) a < G(r,fQ)> (5.35a) Our objective now is to manipulate (5.35) and try to find a solution for ssjfl G (r,fg). But we note that higher order terms of (5.35) are becoming increasingly complicated. One convenient and elegant technique for handling and manipulating such cumbersome expressions is the use of Feynman diagrams. We use the following symbols to construct the Feynman diagrams. * m r m i 3 n \ n '<■ = s qf (En) ^ (5.36a) §(Pn)Z(zn) (5.36b) G°(r ,r ) m n (5.36c) < G(f ,rQ) > (5.36d) P Cv (rm - rn ) . Z(. ) O P (P - P (5.36e) ) 2(« ) C (r - r ) P G°(r ,r ) P v m n m n (5.35£) (5.36g) 164 Using (5.36), (5.31) and (5.33), (5.35) can be diagramatically represented as r r0 r r0 r i r0 2 R - r » A i n 3 2 4 - ^ ? V X; R r0 ♦ (5.37) Here the notation R stands £or the following replacements. (5.38) This means that we also include in (5.37) those terms which result from R. We define a mass operator mr^r,*) as shown below. 0 . - , ^ R + [ + + S7x \ « h ig h e r o rd e r te rm s (5.39) where (5.40) We note that the mass operator as defined in (5.39) is the sum of all 'strongly connected'diagrams. Let us consider now — comparing this with - =n.. _ .. . On as defined in (37) we observe that the following terms need to be added to - These terms are generated by cascading be written as follows. , h ig h er o r d e r te rm s . Thus (5.37) can Mathematically (5.41) is given by 5 V , * 0> - g V / * 0> + I* J (5.42) where M(r ,r ) - < Q (1) G°(l,2) Q<2)> 1 oo + < Q(l) G°<1,2) Q(2) G° (2,3) Q (3) G°(3,4) Q(4) > + . . (5.43) Equation (5.42) is the so-called Dyson equation which is essentially an integral equation for the mean Green's function. Since the mass operator 3 M(r^,fJ in (5.43) is an infinite series the solution to the Dyson equation t can only be obtained by making an approximation to M(r_,r ). There are two 1 OO standard approximations used in practice, viz., the bilocal approximation and the nonlinear approximation. (i) BILOCAL APPROXIMATION Bilocal or Bourret approximation is one often-used approximation in problems involving propagation and scattering in random media. Here we approximate the infinite series in (5.43) by its first term; i.e., we let M(2 ,2) * < Q (1) G°(1,2) Q(2)> 1 oo In order to see what this approximation means let us examine the Dyson equation under this approximation In the diagramatic representation, (5.44) becomes r r0 r r0 ♦ . r I 2 r0 (5.45) On iteration (5.45) yields ro ' ro r 1 2 r0 R 3 + 4 h ig h e r o r d e r te rm s (5.46) Thus the entire series is made up of two point correlation functions. This is true for non-*Gaussian statistics as well. Hence this approximation bears its name bilocal. On comparing (5.26) and (5.44) we notice that the bilocal approximation mathematically amounts to the following assumption. < Q (1) G° (1,2) Q (2) G(2,rQ)> - < Q(l) G°(l,2) Q(2)> Gm (2,r(J) (5.47) In other words we have assumed that Q(l) G°(l,2) Q(2) and G(2,r^) are weakly correlated. Further on comparing (5.46) with (5.37) we immediately notice that under the bilocal approximation we have ignored terms of the following type. 168 R and similar higher order terms (5.48) There exist several mathematical propositions that are meant to serve as criteria to determine the range of validity of the bilocal approximation. But all of them have been brought into question and they are still open for debate. Nevertheless this approximation has proved to be quite useful in many applications. We therefore avoid further discussions on the issue of validity and conclude this section by making the following remarks regarding the physical meaning of the bilocal approximation. First on \ observing (5.45) we infer that the bilocal approximation can be interpreted as the single scattering approximation of the 'mean' wave. Secondly from (5.36) we see that under this approximation only two point correlations between adjacent scatterers are taken into account. This means that the bilocal approxiamtion would be a fairly good approximation if the scatterers involved are not strongly scattering. (ii) NONLINEAR APPROXIMATION The important point to note in the procedure leading up to the bilocal approximation discussed in the previous section is that we had effected a selective resummation of the Neumann series and then approximated the series by its first term. Now there are several ways in which one can perform the resummation, each thereby leading to different approximations. In this section we shall consider one such resummation technique which will eventually lead us to the so-called nonlinear approximation. As before we shall proceed diagramatically. From (5.37) we obtain the following equations. -*— V (5.49a) /* * * + \ — 4— •— *- + *— *■ (5.49b) X— 4* (5.49c) (5.49d) Similarly we consider all terms of the following type He now define a mass operator denoted by - ^ R ^ as . ^ 7!»■\ "■ ~ 'n».R (5.50) Thus from (5.49) and (5.50) we have — © ----------------- C l! + / 7 ♦ T \r .\ + ~ . a x '*—R + (5.51) Comparing (5.51) and (5.37) we conclude that by cascading we can genarate all the terms in (37) . Thus ' ^0 ^ *'""" 171 (5.52) Equation (5.52) is our Dyson equation as before but the mass opearator has a different definition. We now approximate the mass operator by the first term in (5.50). The Dyson equation (5.52) under this approximation becomes p = r *o + — r r 1 * r0 (5.53) The mathematical equivalent of (5.53) is Gm (r,r0) - G°(r,f0) + J d3f j d3 f2 G°(r,r1) {Cv (fl'f2 > P S ” l'l'?2> P + 2 ( Z i 5 "<'2 - *■o 0* z 2> S"<f2,t0) } (5.54) By the nonlinear nature of the above integral equation this approximation is appropriately called nonlinear approximation. To examine the meaning of this approximation we expand (5.53) by iteration. _____________________ o r r0 r r„ r 1 L * . r„ ^ r 1 \ * » R 4 r0 (5.55) On comparing (5.55) with (5.37) we notice that under the nonlinear approximation we have Ignored terms of the following type ~ r r r \ * , . (5.56) However comparing (5.55) with (5.46) we observe that in the nonlinear approximation we have included more terms than in the bilocal approximation. The additional terms included are of the following type (5.57) Thus the nonlinear approximation is widely regarded as a better approximation than the bilocal approximation. However, it should be mentioned that the nonlinear approximation leads to rather difficult integral equations to solve. 173 5.5 THE BETHE-SALPETER EQUATION Since in scattering problems we are often interested in scattered intensity, we now take up the derivation of the second moment of the Green1s functions. To this end we would like to first compute G(f,rQ) G^(r',Tq '). Consider (5.26). G{r,fQ) - G°(f,r0) + J d3^ Gt(r',f0*) -G°+(r',r0') + G°(f,f1> Q ^ ) Jd ^ ' GtE^) (5.58a) G ^ r ' ^ ' ) Q1-^ ' ) G°t ( r ^ ,rQ') (5.58b) We now expand (5.58) and express the results on omitting the integral signs involved for brevity. G(f,rQ) - G°(f,r0) + G°(r,l) Q(l) G°(1,E0) + G°(r,1) Q (1) G°(1,2) Q (2) G0 (2,rQ) + G°(r,1) Q (1) G°(l,2) Q{2) G°(2,3) Q(3) G°(3,r0) + G°(f,l) Q (1) G°(1,2) Q (2) G°(2,3) Q(3) G°(3,4) Q<4) G°(4,r0) (5.59) 174 Gt(r',r()') - G°t<r',£0') + GOtd',r0') Q+d') G°+<rM') + G°t(2'fB •)' Q+(2') G°+(l,,2’) Q+d') G°t(fM') + G°t(3',r0’) Q+O') G°t(2,,3') Qt(2') G0 t(l,,2') Q+d') G°t(rM*> + G°t(4',r0') Qt(4') G°t(3',4') Q+(3') • G0 t(2',3') Qt(2') G0 t(l',2') Qt(l') G0 t(r',l') (5.60) Besides (5.36) we need the following additional symbols /WVVXA r 5,- S V ro (5.61a) G(r' V I t - < G(r,rQ) Gt(r',f0')> (5.61b) (5.59) and (5.60) can be diagramatically represented as follows. a a /v W a + r 0 • | r + o • • 2 0 r R + — • • r i 2 o R + — -e— *--- 4 . (62a) /W\A/w , t <6 “ r' o' + r' + — ’ ? I o' + o' + r' R •— •— •— + — — R *— ► •— •— *— + p — « X (62b) Using (5.59) and (5.60) and assuming Gaussian statistics we obtain < G(r,fQ) Gt(r',r0')> - G°<r,?0) G°+<f*,r0 ') + G°(f,l) { Cv (l,2 ) Pg°(1,^0) G°t(l',r0') P + Ca (l,2) 2(1) G°(1,rQ) G°t(l',r0') 2( 1)} G°+ ( f ,1 ' ) + G°(f,f0) G°t(2',r0') { Cv (l',2 ') P g ° + ( 1\ 2 ') P + C (1', 2') 2(1') G°t(l',2') 2(2')} G°t(f,,l') + G°(r,l) { Cv (l,2 ) P G°(l,2 ) P + C 3 (l,2) 2(1) G°(l,2) 2(2)} G°(2,Z0) G°t(£'fr0') + . . . . (5.63) Below we represent (5.63) by a two level diagram notation where the top level is used for the matrix space and the bottom level is used for the Hermitian conjugate space. r-r\ + , I r\ r t R o t 2 3 4 ,▼ + /r\\ / ~ i ,R t + + t + R r ? v \ I s~\ + R - R + M I* + R - < » R + - * - * r x > n = n * i x : “H— *- x t * - ^ - t .... (5.64) Consider now G^r^r^) Gmt(r,frg'). Using (5.37) we can express this o. O diagramatically as follows. r ------- o r -----O' + Rt▼ (5.65) + ... R We define an intensity operator m as the sum of all strongly connected diagrams as shown below. s -j i T+ (5.66) We have used the notation E r-/" (5.67) ?n Combining (5.65) and (5.66) we get R + ~ r \ R | X I 4 - V - | X T u (5.68) From (5.64), (5.65) and (5.68) we have X r 0 X _ + 0 I o' O' (5.69) Equation (5.69) ia the diagramatic representation of the following mathematical equation < G(r,fQ > Gt(r',r0 ')> * - Gm (f,rQ) ^ ( r \ r Q ') IA JA JA, I A (5.70) where I(S^, fn ;f ^ ,fn ') ia the intensity operator defined diagramatically in (5.66). Equation (5.70) ia the desired integral equation for the second moment of the Green's functions. An equation of this type is referred to in the literature as the Bethe-Salpeter ( B-S ) equation. B-S equation like the Dyson equation is an exact integral equation. But since the intensity operator is in the form of an infinite series we have to introduce an approximation. One often-used approximation is the so-called ladder approximation. Here the intensity operator in (5.70) is approximated by the first term of the series in (5.66). In other words we let - p v * i ' V > P 5 <Er E-’ ^ v - v ) + 2<I1l W V ’ *1 ■) S<P1-P„) SfPi'-Pn'l (5.71) Under this approximation the B-S equation becomes < G(£,Z0) Gt(£',£0 ')> - Gmt(f,£0») + J d3tl J d3^' ^"(r^) t p cv (tr Ei'> p (5.72) In the diagram notation, (5.72) is given by .0 r ■o' “ r = o r, o ' + r' (5.73) On iterating (5.73) we get 180 i (5.74) The reason for naming this approximation as the ladder approximation is immediately evident on just observing (5.74). In the past,the mean Green's function evaluated under the bilocal approximation has often been used in the ladder-approximated B-S equation. But it has been found that the bilocal approximation is inconsistent with the ladder approximation in the sense that they violate the principle of conservation of energy. On the other hand the nonlinear approximation and the ladder approximation are found to be of the same order and consistent with each other. Using the nonlinearly approximated mean Green's function given by (5.55) in the ladder approximated B-S equation given by (5.74) we obtain .0 i i r r' o o‘ + r— r I R — o i* o' I + r— — r' r\ v 7 I R R — o r. T + 2* o' r'' i 4- O' iQ ir | 1. I 4 - — * — 1 (5.75) 181 Comparing (5.75) with (5.64) we observe that in our approximations the terms that are neglected are of the following type sr. rs ;tc. (5.76) Jt 182 5.6 SUMMARY AND CONCLUSIONS We have considered the problem of scattering from a random medium with a random interface. We have derived an integral equation for Green's functions. Assuming Gaussian statistics we have proceeded to obtain the Oyson integral equation for the mean Green's functions. Using the Feynman diagram technique we have introduced two types of approximations to the Dyson equation. These are the bilocal approximation and the nonlinear approximation. Further we have used the Feynman diagrams to interpret the various scattering processes involved. Perhaps the most important result of this exercise has been the explicit revelation of the interaction processes between volume scattering and surface scattering not withstanding the fact that we have assumed that the two random quantities are statistically independent. This result further enqphasizes the need for a unified approach to scattering problems where random media and random surfaces are involved. The schematic diagram of the scattering process is shown in Figure 5.2. We have proceeded to introduce two types of approximations to the Dyson equation. These are the bilocal approximation and the nonlinear approximation. Time and again we have used the Feynman diagrams to manipulate and interpret cumbersome mathematical expressions. Thus we have studied the physical meaning of the two approximations. Using the integral equation for the Green's functions we have derived the Bethe-Salpeter equation which is an integral equation for the second moment of the Green's functions. We have employed the ladder approximation and again used diagram techniques to interpret the physical meaning of the approximation. Further details about the solution to the integral 183 Region 0 Region I Figure 5.2 Scattering process that includes interaction between volume scattering and surface scattering. 184 equations and the associated scattering coefficients are suggested as a future work. i 185 APPENDIX V2 + k2 H (A. 1) + k lm V'2 + k2 o X' V'2 + k2 lm (A.2) fo (r) f (r) (A.3) fx (r) q(r) qf (r) - 5(f - r1) = 8 (r - r1) V<r) - (A.4) (A.5) ▼oo<f) V 0 1 (£) V 1 0 (£) Vn (5> (A. 6) 186 Z(P> - Z0 0 (P) Z0 1 <P) *10<P) zu<P) J (A.7) Goo(f'f,> Goi(E'z,) G 1 0 (r,r') G 1 ;L(f,f*) G00<p, 0;r1) G q 1(P, 0;f) G1 0 (p,-d;r’) G^p^d/r*) G(P,s; r') V<P,3) - (A.9) v 0 o (p' 0) v 0 1 (P, y 1 0 <p,-d) v 1 1 <p,-d) 0 ) (A.10) 8 D(z) (A.8) (z) 0 - (A.11) 0 8 (z+d) 187 CHAPTER 6 OPTIMUM POLARIZATIONS IN THE BISTATIC SCATTERING FROM LAYERED RANDOM MEDIA In this chapter we investigate the polarimetric bistatic scattering charateristics of layered random media. On applying the Born approximation we have calculated the bistatic Mueller matrix of a half-space random medium. The power received by a receiving antenna is the quantity chosen to optimize. The variables of the problem are the polarizations of the transmitting and receiving antennas. For the case when the polarization of the transmitting and receiving antennas is identical, we have calculated the optimum polarization and we have found that the optimum polarization includes both linear and elliptical polarization. The conditions for maximum and minimum received power are also obtained. In the backscattering case we have considered the situation when the transmitting and receiving antennas have independent polarization. For a two-layer problem we have observed the influence of the thickness of the layer in the classification of the optimum polarization. 188 6.1 INRODUCTION Perhaps the most important objective of remote sensing is the extraction of target information. Often, the target we are looking for is amidst other unwanted objects which we designate in this context as clutter. As far as the radar is concerned the target and indeed the clutter are characterized by their scattering matrices. Now, the scattering matrix depends on several parameters such a3 frequency, aspect angle, polarization, etc. The ta3 k of the radar engineer i3 to make a judicious choice of the parameters so that the target is identified with minimum ambiguities. Although this sounds like an inverse scattering problem we do not address the issue from the standpoint of the very difficult inverse scattering theory; rather our approach is phenomenological in the spirit of Huynen [1970]. As mentioned earlier there are various parameters on which the scattering property of the target depends. Although theoretically all of them can be controlled and varied there may exist several constraints in practice. Since polarization diversity has become viable these days,we will concentrate in this paper on the influence that various polarizations have on radar detection. This topic is widely known .today as polarimetry. From a theoretical standpoint polarimetry has been"studied as early as in the 1950s [Kennaugh, 1952; Sinclair, 1950]. Among the later developments one should mention the almost comprehensive study by Huynen [1970] . An excellent report on all the latest activities in this topic is given by Giuli [1986]. For a beginner the review paper by Cloude [1983] would be a brief introduction. It is the main objective of this chapter to find the optimum polarizations of both the transmitting and receiving antennas that would yield an optimum (maximum or minimum) received power from the scatterers modelled here as layered random medium. Optimum antenna polarizations were recently considered by Ioannidis and Hammers [1979] for target discrimination in the presence of background clutter. They maximized the signal to clutter ratio. The problem of optimization of the received voltage for a target of known scattering matrix has been addressed by Kostinski and Boerner [198 6 ]. In the case of randomly fluctuating targets the Stokes vector formalism was used to solve the problem of optimum received power [Van Zyl et al., 1987b]. Kostinski et al. [1988] used the coherence matrix to study a similar problem. In Section 6.2, we define the polarizations of antennas and waves to avoid possible confusion that often occurs in the literature. In Section 6.3, we describe the problem of optimization in detail, the solutions of which are be found in Section 6.4. There are two steps involved in obtaining the solutions: (1) Calculate the bistatic Mueller matrix of a layered random medium and (2 ) find the antenna polarizations that optimize the received power, given the Mueller matrix computed in (1). Sections 6.4 and 6.5 include some discussion on the results and we we conclude with Section 6 .6 . 190 6.2 POLARIZATION Before we can delve any further into our topic we need a few definitions for describing polarizations of waves and antennas. An electric field <SCr, t) can be described as £(r,t) - (Ehh + Evv ) e x p [ i r - (0 t) ] (6.1) where h and v are horizontally and vertically polarized unit vectors, respectively. We describe the polarization as i6 . ' Eh ' 1E h 1 • Ev • lE.le^ (6 .2 ) where \|f = 8 v-8 h. We note that in calculating the power the overall phase term drops out. Thus we have three parameters IE^I, IEv | and y to describe the polarization state (PS). PS is suitable to describe fully polarized waves. But for partially polarized waves it is convenient to use the Stokes vector I defined by <1Eh 12 > + <IEV I2> < 1Ejj12> - <IEV I2> 2Re <EhEv*> 21m <EhEv*> 1 a I 0 Ul u2 u3 where the angular brackets indicate the time average. 2 2 2 ux + u2 + u3 5 (6.3) In general, (6.4) 1 The equality holds when the wave is fully polarized. Thus we have, as before, three parameters to describe the polarization of a fully polarized wave. In this chapter we choose the reference points (needed to characterize 191 the polarizations) as follows. For defining the polarization of the antenna we choose the reference point at the location of the antenna when it is in the radiation mode. For defining the polarization of the scattered wave we choose the reference point at the location of the scatterer. The polarization of the incident wave is the same as that of the transmitting antenna. I 192 6.3 DESCRIPTION OF THE PROBLEM We describe the transmitting and receiving antennas by their polarization states, E and E , respectively. T i\ The target is characterized by its scattering matrix f(£^, £^): fh h <£V Q i) fh v <£s' fli} (6.5) f . (£1 , ft ) vh where ft and (Figure 6.1). s i f vv (£1,0.) s i are the incident and scattered angles, respectively The scattered field Ea is *hen given by I 3 = f(ft,, ftj.) I± (6 .6 ) where E^ is the incident field and is equal to ET . The power received by the receiving antenna in the direction P = |I3 is given by • IRV (6.7) * This expression is the same as in Kostinski and Boerner [1986]. The conjugation of ER is made to conform to our reference system. In the Stokes vector representation the equation for the received power should read P = IRt is /2 (6 .8 ) The equivalence of the above two expressions is readily verified by explicitly expanding each of them according to the definitions provided earlier. Now the problem is to extremize P and find the optimum polarization for transmitting and receiving antennas. In the case of randomly fluctuating targets we have partially polarized scattered waves and hence we need the Stokes vector formalism to attack the optimization problem [van Zyl et al.,1987b]. definedthe Stokes We have already vector which is actually the equivalent of the PS. We 193 RCVR Figure 6.1 General scattering geometry. 194 now need a corresponding equivalent description of the target scattering matrix. To obtain that we will proceed as follows. Using (6 .6 ) we obtain ’< 1Eha 2>" "< 1fhh 12 > <|fh v l2> <fhhfhv> <^hv^hh> IEhi•2 <IEv s I2> <lfvhl2> <lfv v l2> <fvh^vv> <fvvfvh> 1Evi t2 <EhsEvs> <^hh^vh> <^hv^vv> <^hh^vv> <^hv^vh> EhiEvi <EvsEhs> <^vh^hh> <fvvfhv> <^vh^hv> <fvvfhh> EviEhi Let us denote the above equation as (6.9) Jg =» F JlJ (6 .10) It is clear that J and I are related by the transformation S as follows: j- sI (6 .11 ) where 1 1 1 - 0 1 0 0 0 0 0 1 0 0 1 (6 .12) i -i Under this transformation I3 = S- 1 F S IT (6.13) where IT and Ig are the Stokes vectors for the transmitter (same as the incident wave) and the scattered wave. The power received by the receiver is given by p - -J- Ir * M It where M is called the Mueller matrix [Huynen, 1970] given by (6.14) The problem of optimization is the same as before; the only change is in the representation. We pause here to say a few words about the motivation behind all this. By finding the optimum polarizations we are in effect extracting a quantity from the scattering matrix which will characterize the target. Recently several authors [Ulaby et al., 1987; Boerner et al., 1987; Durden et al., 1989] have come up with heuristic methods to characterize the target. But the underlying philosophy behind all of them remains the same, i.e., to extract some quantity derived from the scattering or Mueller matrix which can be used as a good discriminator for the target under consideration. In this chapter the object under study is a layered random medium. In Part I of the next section we will consider a half-space random medium and illustrate our optimizing procedure with some simple illuminating examples. Later in Part II we will turn our attention to a two-layer random medium. Physical examples of such problems are vegetated and snowcovered ground. 196 6.4 SOLUTIONS PART I; HALF-SPACE RANDOM MEDIUM The geometry of the problem is shown in Figure 6.2. Region 0 is the free space with permittivity e0 . Region 1 i3 the target under consideration; it has a permittivity e^Cr) - 8 im + e^f(r) where Eim is the mean and e^f(r) is the randomly fluctuating part. is small compared to We assume that the magnitude of CifCr) For illustration purposes we choose the following correlation function to characterize the medium: |xr x2 ' <elf<rl> elf(r2 >> = where 8 lelm> 8 is the variance of and 1yry21 ' W / ^z are the correlation lengths in the lateral and vertical directions, respectively. in both region 0 and region 1 (6.16) The media have the same permeability )l. The electric fields, Eg(r) and E^(r), in region 0 and region 1, respectively, satisfy the following equations: V X V x V x Eq (r) - k* EQ (r) - 0 (6.17) V (6.18) X I^r) - k^m E^ (r) - QtrJE^r) where k. is the wave number of Region 0 and e lm Q(r) - lm 2H eif (r) 01 (6.19) (6 .20) The solutions to (6.17) and (6.18) can be written as Eg(r) - Eg(0) (r) + El (¥) " El <0)(7) + d r^ G01<e,r1) Q t r ^ E ^ r ^ (6 .21 ) d3rx G11(r,r1) Q t r ^ E ^ r ^ (6 .22) Region 0 Ah « | ( 7 ) - < « i ( 7 ) > + f „ ( r ) Figure 6.2 Region I Geometry of the half-space isotropic random medium problem. 198 where the superscript (o) denotes the unperturbed solution; G^Cr,1 ^ ) and Gn(r,r^) are the dyadic Green' 3 functions for the half-space medium [Zuniga and Kong, 1980b]. The first subscript denotes the region of observation point and the second subscript denotes the region of source point. By substituting (6.22) in (6.21) repeatedly we obtain the solution for (6.21) as an infinite series. Under a first-order approximation, widely known as the Born approximation, we neglect the second and higher order terms in this series. This approximation is fairly good for small permittivity fluctuations. Thus the scattered field Ea (r) a e q (*) " En U (r) is given by Ea (r) - f% d3^ G()1 (r,e1) Q (r^ Bj_(0> (r^ (6.23) i From (6.5), (6 .6 ) and (6.21) we obtain the following ikr (6.24a) vh s' i ) exp(-iklg* r^) (6.24b) 199 ike f, (a ,a.) hv 3 1 _ d37l f e " X013 V ’W e*P<-ikl3* *x> ‘ Q(rl) l T r Y0 1 ivli<_ klzi) exP<ikii' ri> lm ikr fVV (Q,a.) 3 1 d’ 'l « T (6.24c) _ if1 Y01» ' lm i V e«Pl-ikl3' ? 1> ' Q(rl> k7 Y0 1 iVli(- klzi) exP<ikii- ri> lm (6.24d) From (6.24) and (6.9) F (£2 , £1.) = G 3 1 1 a 12 £ Ibl2 $ab* Sa*b b 12 c 12 be* b*c ab* Sbc* ac* S Ibl2 1 1 a*b Sb*c S lbl ‘ a*c (6.25) t where <&(k Pi G = 7C2 8k '4 lm P - a * W o n b ' X013 C " ( k Tlm 0, ] lm ,k' .+ k' ) PS lzi lzs k? . + k" lzi lzs (6.26) cos 0 ! -3COS 0J (6.27) (6.28) coa">3 - +1> (6.29) Y 01i °OS ®i 5ln(*3 • V Y0 U Y01S (3in 9i 3in 9S " C0S <*S ‘ V 003 9 iCOS GS I The superscripts ' and " denote the real and imaginary parts of the (6.30) 200 complex quantity. <&(k) is the power spectral density (or the Fourier transform) corresponding to the correlation function given in (6.16). The quantities not defined above are given in Zuniga and Kong [1980b]. From (6.12), (6.15) and (6.25) we have Min .a.) - -r 3 1 Z |a|2 +s|b|2 + |c|2 Ia 12 —1 1 b 12 — Ic12 2 <ab*+£bc*)' -2 (ab*+^bc*)" Ia |2 +t|b|2 — |c|2 |a 12 —s|b|2 + 1c |2 2 (ab*-§bc*)' -2 (ab*-$bc*)" 2 (£ab*+bc*)' 2 ($ab*-bc*)• 2(\|b|2 +ac*)' -2 (ac*)" 2 ($ab*+bc*)" 2 (^ab*-bc*)" 2 (ac*)" 2 (ac*-lj|b12 ) 1 (6.31) wht. e s 1 + (6.32) t 1 - ¥ (6.33) Borgeaud et al [1987] have computed the monostatic Mueller matrix for a two-layer random medium. Ours is the more general bistatic case but for a half-space medium. In order to compare our Mueller matrix with theirs we take the monostatic limit of (31), i.e., we let k The result is given as follows. M(£2 ,£2.) - 3 1 Z Ia |2 + 1c |2 Ia 12 — |c 12 Ia |2 - 1c |2 |a 12 + 1c |2 0 0 0 2 (ac*)' 0 0 2 (ac*)" 0 -2 (ac*)" 2 (ac*) ' (6.34) We now take the half-space limit (d -* <» ) of Gq. (6.16) of Borgeaud et al. [1987]. We also note that they have used a slightly different definition for the Stokes vector. After making the necessary modifications to take this into account we find that their Mueller matrix is in complete agreement with (6.34). Thus our Mueller matrix is a generalization of theirs 201 and it is particularly useful while studying bistatic scattering properties in remote sensing problems. It is to be noted that as opposed to the monostatic case all the 16 elements of the Mueller matrix are nonzero in the bistatic case. t 202 OPTIMIZATION Our task, as mentioned before, is to extremize P ■ I M I„,/2. R T We first consider the case when both the receiving and transmitting antennas have identical polarizations; i.e., " I a Cl ux u2 u ^ (6.35) Since the transmitted wave is fully polarized, we have the constraint that Ul2 + U22 + U32 * Thus using the method (6.36) 1 f Lagrange [Arfken, 1985] we have to extremize P(u) - P(u) + p(l - u-u) where p is the Lagrange multiplier and u ■ [u^^ u^ (6.37) “3 ^ * In other words we have to set P' ■ 0 and solve for u. To this end we parametrize u^ by letting u. = k.t + c. , 1 i 1 i =* 1, 2, 3 (6.38) where k^ and c^ are arbitrary constants and t is the parameter. Thus s. s dt * 3u^ dt 3u2 dt 3u^ dt or kl 3ux + k2 3u2 + k3 3u3 " 0 (6-39> Since (6.39) should be true for any k , k2, k^ we conclude that !“ = 0 , i - 1, 2, 3 (6.40) i By solving (6.40) subject to (6.36) we can obtain the extrema of the problem. To determine the nature of the extrema we have to examine the sign of P". The extremum will be a maximum or a minimum according to whether P" is 203 negative or positive, respectively. In our problem d2P dt* a2p 32P 2 32P + k h 3u2 + k2 3ui 3 3u + 2 32P k. k + l"‘2 du^du^ 2 32P k„k + 2 3 3u2 3u3 2 k. k 1 3 32p (6.41) 1. Approximate Solutions Since closed form solutions seem intractable in the general case we make a simplifying but very plausible approximation by assuming that the random medium has negligibl’ small loss, i.e., Im[e, ] « lm Re [e ]. lm The optimal solutions obtained under this approximation are as follows: ■ID ±(a-c)/£ ±b3/C (6.42) 0 (c2 -a2 ±b3Tp/C2 ■[ (a+c)b3±(a-c)T)]/C2 if b2 32 > 4ac (6.43) 0 iHI where (c2 -a2)/TJ -b3 (a+c) /TJ ± [Tj2 + (a+c) j;2 ] c if TJ2 > (a+c)C2 > b2 C2 (a+c) /T| (6.44) - [(a-c)2 +b2 32 ] 1 / 2 (6.45) - (b2 32 -4ac) 1 / 2 (6.46) tl = C2 - b2 C2 (6.47) 3 - 1 .+ % (6.48) Z - 1 - % (6.49) 204 We note that under our approximation a, b and c are real quantities here. It appears that there always exist two solutions to the problem, viz., I ^ a n d I ^ w h i c h are both linear polarizations. When b2 32 > 4ac we have two more linear polarizations as solutions. But when the condition TJ2 > (a+c) £ 2 > b2 f2 la+c) is satisfied we have two elliptical polarizations as solutions. In addition we note that I ^ a n d I ^ a r e orthogonal to each other while I ^ a n d I ^ have identical polarization ellipses but of opposite sense. The nature of the extrema are determined by a set of conditions as given below. With the definition of a discriminant Da(a+c)C, the results are: * a) -» pmax if D > max ( “b2 32 , -<a-c)2) -» pmin if D < - b2 32 * <2> pmax if D < min ( b 2 32 , (a-c)2) i <2) pmin if D > C2 + b2 s2 -13)-* pmin 1 (6.50a) (6.50b) ’ (6.51a) (6.51b) always (6.52) _(§) I -» Pmin if 0 3 =0.; otherwise we have a saddle point X (6.53) Some of the implications of the above results are worth noting. The polarization thatwill correspond to Pmax For the case when has to be either I ^ o r max [ -b2 32, -(a-c)2] < D < min ( b2 32, (a-c)2] both I ^ a n d I ^ w i l l leadto Pmax . These being local maxima we have to proceed to search for the global maximum. On the other hand there is no possibility at all for, both I ^ a n d I*2* to correspond to Pmj.n . Also it should be noted that these conditions are fairly conservative estimates. Part of the purpose of this example is to illustrate a working procedure. But it is remarkable that we are able to obtain such simple solutions to a fairly complicated problem. 2. Exact Solutions We now consider a set of examples where, without making the low-loss approximation, it is possible to obtain simple closed-form solutions. Example 1. Backward Scattering Here we let <t>3 = <|>i+n; this is the situation where the incident and scattered planes coincide. However 03 and 0^ remain distinct. The optimum solutions are -(1 ) i' ' - HP a [1 1 0 0] t ; -(2) i' - VP a [1 -1 0 IcI - 1 al: lc-al2 u2 U3 j(3) 0] t (6.54) (6.55) where u2 + u3 R I c - a l 4- ( l c l 2 - l a l 2 ) 2 (6.56) Ic-a14 _ (3) We also find that I always corresponds to Pmin while both HP and VP A correspond to Pjjiax* However, the global maximum Pmax can be determined by A the following condition. HP will correspond to Pmax IXA,.X., I > Oli 01s lY0 1 sY 0 1 i ‘ 0 0 3 (0 3"ei) (6'57> lm A Otherwise VP will correspond to Pmav . By recalling the fact that for a half-space medium the reflectivity of a TE wave is larger than that of a t 206 TM wave for all incident angles we draw the conclusions for Pmax and they are displayed in Figure 6.3(a). We note that in the case of backscattering (or when 0a - 0^ ) VP A corresponds to Pmax , thus agreeing with a well-known fact that the backscattering cross section for the vertical polarization is always larger than that of the horizontal polarization «J W Example 2. Forward Scattering > (J ). HH (<|> =<)>,) S 1 Since this is also the case when the incident plane and the scattered plane coincide, the optimum solutions are again givri by (6.54) and (6.55). But the conditions which determine the nature of I ^ a n d I ^ a r e different. There are two cases to consider. Case A 0.+0 < rc/2 x s ------ Here HP corresponds to Pmax if X., I > Olx 01s . K k 2 lY0 1 sY0 1 i' lm Otherwise VP will correspond to P,max Case B ---------- 0.+0 X3 003 ^s+OiJ <6-58> > n/2 A Here HP corresponds to Pmax if IX.,. X-, I > Oli 01s K k 2 lY0 1 sY0 1 i' 3in <9 s+9i-*/2> lm A Otherwise VP will correspond to Pmay . <6-59> The above results are displayed in Figure 6.3(b). Example 3. Independent Polarizations We now treat the polarizations of the transmitting and receiving antennas as independent and consider the special case of backscattering <®s “ 9 i» $s “ <t)i+7t > • Here the power received by the receiving antenna is given as 207 max HP VP VP/HP backscattering (a) t max VP HP VP/HP VP VP/HP O (V*.) (b) A Figure 6.3 (a) Location of Pjnax for backward scattering. (b) Location of P^ v for forward scattering. A 208 (6.60) where -r 11 U1 U2 U3] '■ [1 '1 V1 ’5 V2 ■?' V3] (6.61) (6 ■ 62 ) The procedure for optimization remains the same as before; however, the number of variables is doubled. The optimum solutions obtained are as follows: (1) T (2) - ij1'- ^T 2> - (3) T (4) VP (6.63) HP (6.64) = HP ; = VP ; *<4) T (6.65) (6.66) i i « - HP (5) ^T5> 1 lcl2+ lal2+2a 1a 12-1 c 1^ u2 U3 i\ - l cr(lcl2+ lal2)+2lacl2 1a 12— tc 12 v2 V 3 (6.67) where | - [lal2+lcl2] ± £(lal2+lcl2]2 -16lacl2+ 4R2 (lal2-lcl2)j | (6.68) 2 ^ R1 * 12lacl2-(lcl2-lal2)2 4(lalz— Icl2) ^ I l F - l c i 4) [ +2 Iac 12+q (Ia 12+ 1c 12)] 2q2+2 Iac12+q(Ia 12+ 1c12) £ 0 (6.69) (6.70) (6.71) Solution (1) corresponds to Pm a v solutions (2) and (5) correspond to saddle points; solutions (3) and (4) correspond to Pm.jn . We note that 209 the results regarding solutions (3) and (4) are indeed to be expected; because cross-polarized backscattering is always smaller than the likepolarized one. Note also that lal 2 £ Icl 2 for the backscattering case. PART II: TWO-LAYER RANDOM MEDIUM Our target here is the layer (-d S z S 0) of random medium of permittivity £^(r). The random medium layer is denoted as Region 1. Region 0 is free space as before while Region 2 (below Region 1) is a homogeneous medium with permittivity e2 . The rest of the parameters have been explained in Part I. It is straightforward, albeit tedius, to proceed as before and obtain analytic solutions for each of the examples considered in Part I even in the case of two-layer problem. The complication, however, arises when we try to 3pell out the conditions that identify the maxima and minima; this i3 primarily because of the influence of the bottom interface. So, for the sake of illustration we will consider the simple case of backscattering and examine the changes due to the influence of finite d. By following exactly the same procedure as before we obtain the m Mueller matrix M for our two-layer problem in the monostatic case, i.e., for the case when <J>3 - <(»j_-MC and 0a - . The results are as follows. Mt-kpirkpi)' = \ a+fJ a-p 0 0 a-p a-p 0 0 0 0 Y' Y" 0 0 -Y" Y' (6.72) 210 where a Oil Ya, Oli. k.0 . k,lm 21 2 Y - 2 (6.73) < V V °2i (T3+T 4) x°u 1 Y 01i ^0_ °2i F2i klm J (6.74) * 2 < V V (6.75) T1 ’ 8l[ 1+IR12i'* e«Pl-«;siil] (6.76a) T2 - V ' W ' (6.76b) (6.76c) T. - G_I S.A,12 (sin29 . - cos20..)2 ’4 2 121 11 11 ' (6.76d) T5 * -8l[ 1+R12iS12i e*p(-4kL l d)] (6.76e) T6 - °2 R12i S12i <cos!0li-3in>eu ) (6.76f) <D(2k ,2k* ) p G, - «28 k '4 ------ J i -----i 2 i _ 1 - exp( 1 lm 2k? . L lzi (6.77a) G„ => n28k'4 <6(2k .,0) 8dexp(-4k" .d) lm pi lzi i (6.77b) The quantities not described above are defined by Zuniga and Kong [1980b] 211 We proceed, as before, to seek optimum solutions. The optimum solutions are - found to be HP, VP and I (31 where a+|J-Y1 (3) (6.78) u3 . r2 - (6.79) (a+p-y) The nature of these solutions is determined by the following conditions. HP VP -> ■max ?min if y ' < 2a (6.80) P.max if Y 1 < 2P (6.81) > ?min i (3). -max ?min if Y' < a+P (6.82) > There are two limiting cases to consider. Case A . Thick Layer For k^d » 1, Y ' i-3 less than both 2a and 2P. Thus in this case both HP A and VP correspond to Pmax. However, we find that VP corresponds to Pmax (global maximum), thus giving the same result as that in the case of half space random medium. Case B. Thin Laver For k'd < 1 and k' L > 1 we obtain 1 lz z a Oli 52i G2 'R 1 2 i '2 (6.83a) Now it is clear from (6.80) and (6.81) that there exists a possibility in case A B for HP to correspond to A results in Pmax . whereas in half-space problem VP always 213 6.5 SUMMARY AND A FEW COMMENTS We have sought optimum polarizations for a layered random medium. We derived the Mueller matrix for the bistatic case using the wave theory under the Born approximation. Using this Mueller matrix, we have proceeded to obtain the polarizations that will lead to optimum received power. First we have considered a half-space random medium. On keeping identical polarizations for transmitting and receiving antennas, we have obtained optimum polarizations in analytic form, out of which four are linear polarizations while the other two are, in general, elliptical polarizations. The case of independent transmitting and receiving antenna polarizations has also been solved. Later we have considered a two-layer random medium and obtained similar optimum solutions. Often in practice the Mueller matrix is determined by measurement and a then the measured M is used in the optimization problem [van Zyl et al., 1987b; Kostinski et al., 1988]. On the contrary, we have a theoretical model to start with and hence our results are of interest in a different perspective. Since we have analytic solutions they offer considerable physical insight. However, the theoretical model should be an appropriate one, otherwise the 'analytic' solutions would be of little use. Also if the 'appropriate' theoretical model turns out to be too complicated, then the closed form analytic solutions would not be possible and we have to seek perhaps numerical solutions which subsequently will deprive us of much physical insight. We have chosen specific examples not only to illustrate our optimizing procedure but also to. verify the plausiblility of the results with our intuitive reasoning. Some of the results obtained agree with our Jt 214 expectation and some of them are interesting surprises. Besides, this exercise has certainly enhanced to 3ome extent our understanding of the behaviour of the layered random medium. We also have obtained conditions for Pmax and Pmj.n . Obviously the polarization for Pmax is the one appropriate to enhance the target detection. But often in practice the random medium is chosen as a good model to represent clutter. In that case the polarization for Pmj.n is used for the purpose of clutter rejection. Such a situation occurs when there is a need to detect targets in the presence of clutter. Here if we identify the polarization which corresponds to the minimum received power from clutter, then by using that polarization we can effectively enhance target detection capability. On the other hand, if one wants to study the statistics of the radar clutter signal, one can select the polarization in clutter measurements that will give the maximum received power from clutter. In summary, we have highlighted the merits of optimum polarization in the case of targets which can be modelled as a random medium. However, one should carefully weigh the merits against the inherent limitations. The concept of optimum polarization alone is not going to lead to a marked improvement in target detection capabilities. But along with other existing techniques optimum polarization concept can be a useful device. * 6.6 CONCLUSIONS We have considered a layered random medium and characterized it by deriving its bistatic Mueller matrix. Thus we have extended the results of Borgeaud et al. [1987] to the bistatic situation. We then considered the problem of optimizing the power received by an antenna due to the electromagnetic scattering from a random medium. Since the problem is bistatic we have in general independent transmitting and receiving antennas. We have considered several simple examples to illustrate the procedure involved in finding optimum polarizations and their subsequent classifications. For the case when the polarizations of the transmitting and receiving antennas are identical, we have calculated the optimum polarizations and we have found that they include both linear and elliptical polarizations. We have proceeded to find the conditions for maximum and minimum received power. In the backscattering case we have also considered the situation when the transmitting and receiving antennas have independent polarizations. For a two-layer problem we have obsereved the infuence of the thickness of the layer in the classification of the optimum polarizations. These examples demonstrate that a theoretical ' investigation of optimum polarizations can be useful in certain problems of remote sensing of random medium. CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK In this thesis we studied the wave scattering and emission from a half-space anisotropic random medium. In order to involve multiple scattering we used the modified radiative transfer theory (MRT). The MRT equations were solved under a first-order approximation. The backscattering coefficients were derived and expressed in a form suitable for physical interpretation. These results were compared with those obtained by the Born approximation and thereby the effects of multiple scattering were identified. Several numerical examples were given to illustrate the properties of our results. We next proceeded to calculate the bistatic scattering Coefficients for the half-space anisotropic random medium and studied their characteristics. In order that our results might be of use in passive remote sensing the emissivities were calculated. With the help of several numerical examples our results were compared with corresponding single scattering results. As an application our theoretical model was used to interpret the passive remote sensing data of multiyear sea ice. As mentioned earlier the first-order approximation was used to solve the MRT equations. In order to study the validity of this approximation we derived the higher-order solutions and expressed the backscattering coefficients as an infinite series. It was pointed out that the second-order solutions are important for interpreting cross polarized backscatter. We noticed the absence of some 'phase terms' in the second-order solutions and explained the reasons for that. When the random medium has a random boundary a multiple scattering analysis is quite complicated both analytically and conceptually. To study such a problem a Green's function formulation was used to derive the Dyson equation for the mean field. Use of the Feynman diagram techniques lent considerable physical insight into the various scattering mechanisms. We employed two kinds of approximations, viz., the bilocal approximation and the nonlinear approximation and compared their respective implications. Further, using similar methods we derived the Bethe-Salpeter equation for the field correlation. After renormalizing the B-S equation using the Feynman diagram techniques the ladder approximation was applied. Throughout the analysis we noticed the scattering interaction between the random medium and the random boundary. Most targets are sensitive to radar polarizations. This property leads to useful applications in remote sensing. The target of interest here is a layered random medium. We first derived the bistatic Mueller matrix of a half-space random medium. The power received by a receiving antenna was chosen as the quantity to optimize. In the case where the polarizations of the transmitting and receiving antennas are identical the optimum polarizations were calculated and it was found that they include both linear and elliptical polarizations. The conditions for maximum and minimum received power were also obtained. We applied similar methods to study two other examples and obtained optimum polarizations. Finally it is appropriate to end this thesis with a few suggestions for future studies as a continuation of the work reported here. We have obtained first-order and second-order solutions to the MRT equations. It will be instructive to obtain exact solutions (numerical, perhaps) and compare them with ours. This will enable us to comment on the validity of our lower-order approximations. The procedure outlined in Chapter 4 is one way of estimating the error in the first-order and second-order solutions. It will be very useful to 3eek a physically transparent analytic expression for the error. In Chapters 2 and 3 we have restricted our work to half-space anisotropic random medium. Several targets (first-year sea ice, for example) require a two-layer anisotropic model. Thus our analysis should be extended to the two-layer case. In Chapter 5 the integral eqautioris for the first and second moments of the Green's funtions have been derived. The obvious task that remains to be done is to seek their solutions. In Chapter 6 we have sought optimum polarizations for layered isotropic random media. Since anisotropic media are more sensitive to polarizations it will be of interest to extend our work to anisotropic media. 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From 1979-1984 he worked as a research assistant at Indian Institute of Technology, Bombay. He then came to Syracuse University as a graduate student and received his M.S. in electrical engineering in December 1986. In the years as a graduate student he has served as a research assistant and as a teaching assistant.

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