close

Вход

Забыли?

вход по аккаунту

?

Microwave remote sensing of random media using multiple scattering theory

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may
be from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted. Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand corner and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in
reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6" x 9" black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly
to order.
University M icrofilms International
A Bell & H owell Information C o m p a n y
3 0 0 North Z e e b R oad , Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 USA
3 1 3 /7 6 1 -4 7 0 0
8 0 0 /5 2 1 -0 6 0 0
Order Number 9126119
M icrow ave rem o te sensing of ra n d o m m edia using m u ltiple
s c a tte rin g th e o ry
Mudaliar, Saba, Ph.D.
Syracuse University, 1990
UMI
300 N. Zeeb Rd.
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. Also it has been seen in Chapter 3 that anisotropic media have
rather complicated bistatic scattering charateristics. Optimum
polarizations may lend some physical insight into their scattering
properties.
219
BIBLIOGRAPHY
Arfken, G., Mathematical Methods for Physicists, Academic Press,
3rd Ed., 945-948, 1985.
Bahar, E., Scattering cross sections from rough surfaces - Full wave
analysis, Radio Sci., 16, 331-341, 1981.
Barabanenkov, Y. N. and V. M. Finkelberg, Radiative transport equation
for correlated scatterers, Soviet Phys. JETP, 26, 587-591, 1968.
Barrick, D. E. and W. H. Peake, A review of scattering from surfaces
with different roughness scales, Radio Sci., 3, 865-868, 1968.
Bass, F. G. and I. M. Fuks, Wave Scattering from Statistically Rough
Surfaces, translated by C. B. Vesecky and J. F. Vesecky, Pergamon
Press, Oxford, 1979.
Batlivala, P. P. and F. T. Ulaby, Radar look direction and row crops,
Photogrammetric Eng. and Remote Sensing, 42, 233-238, 1976.
Beckmann, P. and A. Spizzichino, The scattering of Electromagnetic
Waves from Rough Surfaces, Macmillan, New York, 1963.
Blot, M. A., Some new aspects of the reflections of electromagnetic
waves on rough surface, J. Appl. Phys., 28, 1455-1463, 1968.
Boerner, W-M, B-Y Foo, H. J. Eom, Interpretation of polarimetric
copolarization phase term in radar images obtained with JPL L-Band SAR
system, IEEE Trans. Geosci. Rem. Sens., vol. GE-25, 77-82, 1987.
Borgeaud, M.,R. T. Shin and J. A. Kong, Theoretical models for
polarimetric radar clutter, J. Electromagnetic Waves and Applications,
73-89, 1987.
Born, M. and E. Wolf, Principles of Optics, Pergamon, 1980.
220
Bourret, R. C., Stochastically perturbed fields with applications to
wave propagation in random medium, Nuovo Cimento, 26, 1-31, 1962.
Brown, W. P., Propagation in random media - Cumulative effect of weak
inhomogeneities, IEEE Trans. Antennas Propag., AP-15, 81-89, 1967.
Brown, G. S., Backscattering from a Gaussian-distributed perfectly
conducting rough surface, IEEE Trans. Antennas Propag., AP-26, 472-482,
1978.
Brown, G. S., A stochastic Fourier transform approach to scattering from
perfectly conducting rough surfaces, IEEE Trans. Antennas Propag., AP-30,
1135-1144, 1982.
Brunfeldt, D. R. and F. T. Ulaby, Microwave emission from row crops,
IEEE Trans. Geosci. Remote Sens., GE-24, 353-359, 1986.
Campbell, K. J. and A. S. Orange, The electrical anisotropy of sea
ice in the horizontal plane, J. Geophys. Res, 79, 5059-5063, 1974.
Chandrasekhar, S., Radiative Transfer, Dover, NY, 1960.
Chang, J. C., P. Gloersen, T. Schmugge, T. T. Wilheit and H. J.
Zwally, Microwave emission from snow and glacier ice, J. Glaciology,
23-39, 1976.
Clemmow, P. C., The theory of electromagnetic waves in a simple
anisotropic medium, Proc. IEE, 110, 101-106, 1963.
Cloude, S. R., Polarimetric techniques in radar signal processing,
Microwave J., 119-127, July 1983.
Cole, J. D., Perturbation Methods in Applied Mthematics, Blaisdell,
Waltham, Mass., 1968.
Delker, C. V., R. G. Onstott and R. K. Moore, Radar scatterometer
measurements of sea ice: the Sursat experiment, RSL Technical Report, RSL
TR331-17, 1980.
221
Dence, D. and J. E. Spence, Wave propagation in random
anisotropic media, in Probabilistic Methods in Applied Mathematics, 3,
edited by A. T. Bharucha-Reid, Academic Press, New York, 1973.
DeSanto, J. A., Coherent multiple scattering from rough surfaces, in
'Multiple Scattering and Waves in Random Media', edited by P. L. Chow,
W. E. Kohler and
G. C. Papanicolaou, North Holland Publishing Co., 1981.
Djermakoye, B. and J. A. Kong, Radiative transfer theory for the
remote sensing of layered random media, J. Appl. Phys., 6600-6604,
1979.
Durden, S. L.,
J. J. van Zyl and H. A. Zebker, Modeling and
observation of radar polarization signature of forested areas, IEEE
Trans. Geosci. Rem. Sens., GE-27, 290-301, 1989.
England, A. W., Thermal microwave emission from a scattering layer,
J. Geophys. Res., 4484-4496, 1975.
Frisch, U., Wave propagation in random medium, in Probabilistic Methods
in Applied Mathematics, 1, edited by A. T. Bharucha-Reid, Academic
Press, New York, 1968.
Fung, A. K., Scattering from a vegetation layer, IEEE Trans. Geosci.
Remote Sens., GE-17, 1-5, 1979.
Fung, A. K. and H. L. Chang, Backscattering of waves by composite rough
surfaces, IEEE Trans. Antennas Propag., AP-17, 590-597, 1969.
Fung, A. K. and M. F. Chen, Emission from an inhomogeneous layer
with irregular interfaces, Radio Sci., 16, 289-298, 1981a.
Fung, A. K. and M. F. Chen, Scattering from a Rayleigh layer with an
irregular interface, Radio Sci., 16, 1337-1347, 1981b.
Fung, A. K. and H. J. Eom, Multiple scattering and depolarization by a
randomly rough Kirchhoff surface, IEEE Trans. Antennas Propag., AP-29,
463-471, 1981a.
Fung, A. K. and H. J. Eom, A theory of wave scattering from an in
homogeneous layer with an irregular interface, IEEE Trans. Antennas
Propag., AP-29, 899-910, 1981b.
Fung, A. K. and H. J. Eom, Application of a combined rough surface and
volume scattering theory to sea ice and snow backscatter, IEEE Trans.
Geosci. Remote Sens., GE-20, 528-535, 1982.
Fung, A. K. and H. S. Fung, Applications of first-order renormalization
method to scattering from a vegetation-like half-space, IEEE Trans.
Geosci. Remote Sens., GE-15, 189-195, 1977.
Fung, A. K. and F. T. Ulaby, A scatter model for leafy vegetation, IEEE
Trans. Geosci. Remote Sens., GE-16, 281-286, 1978.
Furutsu, K., On the statistical theory of electromagnetic waves in a
fluctuating medium, J. Nat. Bur. Stand., 67D, 303-323, 1963.
Furutsu, K., Transport theory and boundary value solutions, J. Opt.
Soc. Am. A, 2, 913-931, 1985.
Giuli, D., Polarization diversity in radars, Proc. IEEE,
vol. 74, 245-269, 1986.
Gurvich, A. S., V. L. Kalimin and D. T. Matveyer, Influence of
internal structure of glaciers on their radio emission, Atm. Oceanic
Phys. USSR, 9, 713-717, 1973.
Hoekstra, P. and P. Cappilino, Dielectric properties of sea and sodium
chloride ice at UHF and microwave frequencies, J. Geophys. Res., 76,
4922-4931, 1971.
Hollinger, J. P., B. E. Troy, Jr, R. 0. Ramseier, K. H. Asmus, M. F.
Hartman and C. A. Luther, Microwave emission from high Arctic sea
ice during freeze up, j. Geophys. Res., 89, 8104-8122, 1984.
223
Huynen, J. R., Phenomenological theory of radar targets,
Ph.D. Thesis, Tech. Univ. Delft, The Netherlands, 1970.
Ioannidis, G. A. and D. E. Hammers, Optimum antenna
polarizations for target discrimination in clutter, IEEE Trans.
Antennas Propag., AP-27, 357-363, 1979.
Itoh, S., Anaysis of scalar wave scattering from slightly rough surfaces:
a multiple scattering theory, Radio Sci., 20, 1-12, 1985.
Ishimaru, A., Correlation function of a wave in a random distribution of
stationary and moving scatterers, Radio Sci., 10, 45-52, 1975.
Ishimaru, A., The theory and applications of wave propagation and
scattering in random media, Proc. IEEE, 65, 1031-1061, 1977.
Ishimaru, A., Wave Propagation and Scattering in Random Media, Vol. 1,
Academic Press, New York, 1978.
Keller, J. B., A survey of the theory of wave propagation in continuous
random media, Proc. Symp. on Turbulence of Fluids and Plasmas, 131-142,
Polytechnic Inst, of Brooklyn, New York, 1968.
Kennaugh, B. M., Polarization properties of radar reflections,
Antenna Lab., Ohio State University, Columbus, Ohio, Project Rep. 389-12
(AD 2494), 1952.
Kong, J. A., Theory of Electromagnetic Waves, John Wiley & Sons, 1975.
Kostinski, A. B. and W-M Boerner, On the foundations of radar
polarimetry, IEEE Trans. Antennas Propag., AP-34, 1394-1404,
1986.
Kostinski, A. B., B. D. James and W-M Boerner, Optimal reception of
partially palarized waves, J. Opt. Soc. Am. A., 5, 58-64, 1988.
Kovacs, A. and R. W. Morey, Radar anisotropy of sea ice due to
preferred azimuthal orientation of the horizontal c-axis of ice
crystals, J. Geophys. Res., 83, 6037-6046, 1978.
Kovacs, A. and R. M. Morey, Anisotropic properties of sea ice in the
50 to 150 MHz range, J. Geophys. Res., 84, 5749-5759, 1979.
Kritikos, H. N. and J. Shiue, Microwave sensing from orbit, IEEE
Spectrum , 16, 34-41, 1979.
Kuga, Y. and A. Ishimaru, Retroreflectance from a dense distribution
of spherical particles, J. Opt. Soc. Am. A., 1, 831-835, 1984.
Kuga, Y., A. Ishimaru and Q. Ma, The second-order multiple scattering
theory for the vector radiative transfer equation, Radio Sci., 24, 247
252, 1989.
Kupiec, I., L. B. Felsen, S. Rosenbaum, J. B. Keller and P. Chow,
Reflection and transmission by a random medium, Radio Sci., 4,
1067-1077, 1969.
Lang, R. H., and Sidhu, Electromagnetic backscattering from a layer of
vegetation: a discrete approach, IEEE Trans. Geosci. Remote Sens., vol
GE-21, 62-67, 1983.
Lee, J. K. and J. A. Kong, Dyadic Green's functions for layered
anisotropic anisotropic medium, Electromagnetics, 3, 111-130, 1983.
Lee, J. K. and J. A. Kong, Active microwave remote sensing of an
anisotropic random medium layer, IEEE Trans. Geosci. Remote Sens.,
GE-23, 910-923, 1985a.
Lee, J. K. and J. A. Kong, Passive remote sensing of an anisotropic
random medium layer, IEEE Trans. Geosci. Remote Sens., GE-23, 924-932,
1985b.
Lee, J. K. and J. A. Kong, Electromagnetic wave scattering in a twolayer anisotropic random medium, J. Opt. Soc. Am. A. 2, 2171-2186,
1985c.
225
Lee, J. K. and J. A. Kong, Modified radiative tranfer theory for
a layered anisotropic random medium, J. Electromagnetic Waves and
Applications, 2, 391-424, 1988.
Lee, J. K. and S. Mudaliar, Backscattering coefficients of a half­
space anisotropic random medium by the multiple scattering theory,
Radio Sci., 23, 429-442, 1988.
Loomis, R. S. and W. A. Williams, Productivity and morphology of crop
yield, edited by J. D. Eastin et al, American Society of Argonomy,
Madison, Wisconsin, 1969.
Maystre, D., Rigorous theory of light scattering from rough surfaces, J.
Optics, 15, 43-51, 1984.
Mudaliar, S. and J. K. Lee, Microwave scattering and emission from a
half-space anisotropic random medium, Radio Sci., 25, 1990a.
Mudaliar, S. and J. K. Lee, Scattering coefficients of a random medium
with a random interface, submitted for presentation at the 1991
Progress in Electromagnetic Research Symposium, Cambridge, Mass.,
November, 1990b.
Moore, R. K., Active microwave sensing of the earth's surface - A mini
review, IEEE Trans. Antennas Propag., AP-26, 843-849, 1978.
Morey, R. M., A. Kovacs and G. F. N. Cox, Electromagnetic properties of
sea ice, CRREL Report 84-2, USA Cold Regions Research and Engineering
Laboratories, Hanover, New Hampshire, 1984.
Nieto-Vesperinas, M., Depolarization of EM waves scattered from a slightly
rough random surface: a study by means of extinction theorem, J. Opt.
Soc. Am., 72, 539-547, 1982.
Njoku, E. G., Passive microwave remote sensing of the earth from space A review, Proc. IEEE, 70, 728-750, 1982.
226
Onstott, R, G., Y. S. Kim and R. K. Moore, Active microwave measurements
of sea ice under fall conditions: the RadarSat/FIREX fall experiment,
University of Kansas Remote Sensing Laboratory, Tech. Report 331-30/578Final, 1984.
Onstott, R. G., R. K. Moore and W. F. Weeks, Surface-based scatterometer
results of Arctic sea ice, IEEE Trans. Geosci. Rem. Sens., vol. GE-17,
78-85, 1979.
Peake, W. H., Interaction of electromagnetic waves with some
natural surfaces, IRE Trans. Antennas Propagat. AP-7, Special
Supplement, S324-S329, 1959.
Rayleigh, J. W. S., The Theory of Sound, Vol. 2, Dover, New York, 1945.
Rice, S. O., Reflection of electromagnetic waves by slightly rough
surfaces, Comm. Pure Appl. Math., 4, 341-378, 1951.
Rosenbaum, S., On the coherent wave motion in bounded randomly
fluctuating regions, Radio Sci., 4, 709-719, 1969.
Rosenbaum, S., The mean Green's function: a nonlinear approximation,
Radio Sci., 6, 379-386, 1971.
Ruck, G. T., D. E . Barrick, W. D. Stuart and C. K. Krichbaum, Radar CrossSection Handbook, Vol. 2, McGraw-Hill, New York, 1970.
Sackinger, W. M. and R. C. Byrd, Reflection of millimeter waves from snow
and sea ice, IAEE Report 7203, Institute of Arctic Environmental
Engineering, Univ. Alaska, 1972a.
Sackinger, W. M. and R. C. Byrd, Backscatter of millimeter waves from
snow and ice, IAEE Report 7207, Institute of Arctic Environmental
Engineering, Univ. Alaska, 1972b.
Shen, J. and A. A. Maradudin, Multiple scattering of waves from random
rough surface, Phys. Rev. B, 22, 4234-4240, 1980.
Sinclair, G., The transmission and reception of elliptically
polarized waves, Proc. IRE, vol 38, 148-151, 1950.
Sobolev, V. V., A Treatise on Radiative Transfer, Van Nostrand,
Princeton, New Jersy, 1963.
Staelin, D. M., Passive remote sensing at microwave wavelengths, Proc.
IEEE, 57, 427-459, 1969.
Stogryn, A., Electromagnetic scattering by random dielectric constant
fluctutations in a bounded medium, Radio Sci., 9, 509-518, 1974.
Stogryn, A., A study of microwave brightness temperature of snow
from the point of view of strong fluctuation theory, IEEE Trans.
Geosci. Remote Sens., GE-24, 220-231, 1986.
Tan, H. S. and A. K. Fung, A first-order theory on wave
depolarization by a geometrically anisotropic medium, Radio Sci.,
14, 377-386, 1979.
Tan, H. S., A. K. Fung and H. J. Eom, A second-order renormalization
theory for cross-polarized backscatter from a half-space random
medium, Radio Sci., 15, 1059-1065, 1980.
Tan, H. S. and A. K. Fung, The mean Green's dyadic for a half-space
random medium: A nonlinear approximation, IEEE Trans. Antennas Propag.
AP-27, 517-523, 1979.
Tatarskii, V. I., Have Propagation in a Turbulent Medium, McGraw-Hill,
New York, 1961.
Tatarskii, V. I., Propagation of electromagnetic waves in a medium with
strong dielectric-constant fluctuations, Soviet Phys. JETP, 19,
946-953, 1964.
Tatarskii, V. I., The Effects of Turbulent Atmosphere in Have
Propagation, NTIS Tech. Report, US Dept, of Commerce, Springfield,
228
Virginia, 1971.
Tatarskii, V. I. and M. E. Gertsenshtein, Propagation of waves in a
medium with strong fluctuations of refractive index, Soviet Phy3 . JETP,'
17, 458-463, 1963.
Toigo, P., A. Marvin and N. R. Hill, Optical properties of rough
surfaces: General theory and small roughness limit, Phys. Rev. B, 15,
5618-5626, 1977.
Tomiyasu, K., Remote sensing of the earth by microwaves, Proc. IEEE, 62,
86-92, 1974.
Tsang, L. and A. Ishimaru, Theory of backscattering enhancement of
random discrete scatterers based on the summation of all ladder and
cyclical terms, J. Opt. Soc. Am. A., 2, 1331-1338, 1985.
Tsang, L and J. A. Kong, The brightness temperature of a half-space
random medium with nonuniform temperature profile, Radio Sci., 10, 10251033, 1975.
Tsang, L. and J. A. Kong, Microwave remote sensing of a two-layer
random medium, IEEE Trans. Antennas Propag., AP-24, 283-288, 1976.
Tsang, L. and J. A. Kong, Theory for thermal microwave emission from a
bounded medium containing spherical scatterers, J. Appl. Phys., 48,
3593-3599, 1977.
Tsang, L. and J. A. Kong, Radiative transfer theory for active remote
sensing of half-space random media, Radio Sci., 13, 763-774, 1978.
Tsang, L. and J. A. Kong, Wave theory for microwave remote sensing of
half-space random medium with three-dimensional variations, Radio Sci.,
14, 359-369, 1979.
Twersky, V., On scattering and reflection of sound by rough surfaces, J.
Opt. Soc. Am., 29, 209-225, 1957.
Ulaby, F. T. and J. E . Bare, Look direction modulation function of the
radar backscattering coefficient of agricultural fields,
Photograimetric Eng. and Remote Sensing , 45, 1495-1506, 1979.
Ulaby, F. T., R. K. Moore and A. K. Fung, Microwave Remote Sensing:
Active and Passive, Vol. 3, Artech House, 1986.
Ulaby, F. T., D. Held, M. C. Dobson, K. C. McDonald and T. B. A.
Senior, Relating polarization phase difference of SAR signals
to scene properties, IEEE Trans. Geosci. Rem. Sens., vol. GE-25,
85-91, 1987.
Valenzuela, G. R., Scattering of electromagnetic waves from a tilted
slightly rough surface, Radio Sci., 3, 1057-1064, 1968.
Vant, M. R., R.
properties of
B. Gray, R. O. Ramseier and V.Makios, Dielectric
fresh water and sea ice at 10 and 35 GHz, J. Appl. Phys.,
45, 4712-4717, 1974.
Vant, M. R., R.
0. Ramseier and V. Makios, The complex
constant of fresh sea ice
in the range 0.1 -
40 GHz,
dielectric
J. Appl. Phys.,
49, 4712-4717, 1978.
Van Zyl, J. J., H. A. Zebker and C. Elachi, Imaging radar
polarization signatures: Theory and observation, Radio Science,
22, 529-543, 1987a.
Van Zyl, J. J., C. H. Papas and C. Elachi, On the optimum
polarizations of incoherently reflected waves, IEEE Trans. Antennas
Propag., AP-35, 818-825, 1987b.
Waterman, P. C., New formulation of acoustic scattering, J. Acoust. Soc.
Am., 45, 1417-1429, 1968.
Weeks, W. F. and A. G. Gow, Preferred crystal orientations in the fast
ice along the margins of the Arctic ocean, J. Geophys. Res.,, 83, 5105-
230
5121, 1978.
Weeks, W. F. and A. 6. Gow, Crystal alignments in the fast ice of Arctic
Alaska, J. Geophys. Res., 85, 1137-1146, 1980.
Weeks, W. F. and S. F. Ackley, The growth, structure and properties of
sea ice, CRREL Monograph 82-1, USA Cold Regions Engineering Laboratory,
Hanover, New Hampshire, 1978.
Zebker, H. A. , J. J. Van Zyl and D. N. Held, Imaging radarpolarimetry
from wave synthesis, J. Geophys. Res., 92, 683-701,
1987.
Zipfel, G. C. and J. A. DeSanto, Scattering of a scalar wave from a
random rough surface: A diagramatic approach, J. Math. Phys., 13, 19031911, 1972.
Zuniga, M. A., T. M. Habashy and J. A. Kong, Active remote sensing of
layered random medium, IEEE Trans. Geosci. Rem. Sens., vol; GE-17,
296-302, 1979.
Zuniga, M. A. and J. A. Kong, Modified radiative transfer theory
for a two-layer random medium, <7. Appl. Phys., 51, 5228-5244, 1980a.
Zuniga, M. A. and J. A. Kong, Active remote sensing of random
media, J. Appl. Phys.,vol. 51, 74-79, 1980b.
Zuniga, M. A. and J. A. Kong, Mean dyadic Green's function for a twolayer random medium, Radio Sci., 16, 1255-1270, 1981.
Zuniga, M. A., J. A. Kong and L. Tsang, Depolarization effects in
the active remote sensing of random media, J. Appl. Phys., 51,
2315-2325, 1980.
BIOGRAPHICAL NOTE
Saba Mudaliar was born in India on 10 August, 1957. He received his
B.S. degree in electrical engineering from University of Madras in
1979.
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.
Документ
Категория
Без категории
Просмотров
0
Размер файла
5 514 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа