THEORETICAL STUDIES FOR MICROWAVE
REMOTE SENSING OF LAYERED RANDOM MEDIA
by
MICHAEL ANTHONY ZUNIGA
SB, Pennsylvania State University
March, 1973
SM, Drexel University
June, 1975
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January, 1980
Signature redacted
Signature of Author
-...........
..
.....
:.
Departifent of Physics
January 9, 1980
ODIY~ktUIt: I t: UdL; t: u
Certified by .......
...
*
.
0**..
.0..............
Jin-Au Kong
Thesis Supervisor
Accepted by ........
Signature redacted
.......
............
George F. Koster
Chairman, Department Committee
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OF TECHNOLOGY
FEB 28 1980
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115
2
Theoretical Studies for Microwave Remote
Sensing of Layered Random Media
by
Michael A. Zuniga
Submitted to the Department of Physics on
January 9, 1980, in partial fulfillment of the
Requirements for the Degree of Doctor of Philosophy
Abstract
In the microwave remote sensing of earth terrain, the
model of a layered random medium can be used to account for
volume scattering effects due to random permittivity fluctuations. Applying the first order Born approximation,
analytic results for the bistatic scattering coefficients
and the backscattering cross-sections have been derived
for active remote sensing of a two-layer random medium with
arbitrary three-dimensional correlation functions.
It is
found that as a result of the second boundary, the horizontally polarized return, a
can be greater than the vertically polarized return, whgieas for a half-space random medium
a
is always greater than a .
The theoretical results are
iYustrated by matching backscntering data collected from a
vegetation field. The bistatic scattering coefficients are
used to obtain the emissivity of a two-layer random medium
and in the case of thin, low loss layers the emissivity is
shown to exhibit strong coherent behavior in the spectral
dependence.
As a more realistic simulation of earth terrain for
active remote sensing analytic expressions for the backscattering cross-sections are derived for a stratified random
In
medium by applying the first order Born approximation.
the special case of a three-layer random medium two maxima
are found in the spectral dependence of the backscattering
due to resonance scattering within each random layer. The
3
theoretical results also are found to compare favorably with
data obtained from vegetation and snow-ice fields. The Born
approximation is carried to second order to obtain backscattering cross-sections that account for depolarization effects
In the half-space limit, adin a two-layer random medium.
ditional wave effects are found which are not accounted for
by the radiative transfer theory nor by the Bethe-Salpeter
equation in the ladder approximation.
The mean dyadic Green's function for a two-layer random
medium has been obtained by applying a two-variable expansion
technique to solve the non-linear Dyson's equation. The
coherent wave is found to propagate in the random medium as
in an anisotropic medium with different propagation constants
for the characteristic TE and TM polarizations. The effective propagation constants obtained in the zeroth order
solution are compared with the scattering coefficients of the
radiative transfer theory.
Modified radiative transfer (MRT) equations for the
electromagnetic field intensity are derived from the ladder
approximated Bethe-Salpeter equation together with the zeroth
order solution to Dyson's equation under the non-linear approximation. The MRT equations contain significant wave-like
corrections not accounted for by phenomenological radiative
transport theories due to the presence of the bottom boundary.
The MRT equations are solved in the first order renormalization approximation and comparisons are made with the results
obtained in the first order Born approximation. A method for
resumming an infinite sequence of terms in the intensity
operator is presented. A renormalized Bethe-Salpeter equation is derived which takes the form of a pair of coupled
integral equations.
Thesis Supervisor:
Jin-Au Kong
Title:
Associate Professor of
Electrical Engineering
Author:
Michael A. Zuniga
4
ACKNOWLEDGEMENT
I would first and foremost wish to thank my wife whose
loving patience, encouragement and support has been unfailing
during the years of my graduate studies.
I expecially wish to thank my advisor, Professor Kong,
for his continual guidance,
advice, and high standards of
encouragement throughout the course of my research.
In ad-
dition, a special debt of gratitude is due to Dr. Leung Tsang
for many valuable discussions and suggestions.
I also wish to acknowledge excellent discussions with
my colleagues and,
in particular,
I thank Mr. Tarek Habashy
for his programming of the three-layer model.
I wish to
thank my parents for their love and encouragement as well as
my wife's parents for their very special love and support
over the years.
Finally,
I would like to thank Miss Cynthia Kopf for
her excellent typing of this thesis and extend my appreciation
to the other members of my thesis committee, Professors
Bernard Burke and John Joannopoulos.
5
DEDICATION
TO
JOSEPH R. MERCADO
6
TABLE OF CONTENTS
Title Page ......
1
Abstract .........
2
Acknowledgement
4
Dedication ---...
5
Table of Contents
6
10
.
List of Figures
List of Principal Symbo ls
12
Chapter 1.
Introduction .................................. 14
Chapter 2.
Dyadic Green's Functions for Two-Layer Media --
Sec. 1.
Sec. 2.
Chapter 3.
Sec.
1.
21
Dyadic Green's Function with Source and
Observation Points in Different Regions .......
22
Dyadic Green's Function with Source and
Observation Points in the Same Region......--..- 27
Active Remote Sensing of a Two-Layer Random
Medium in the First Order Born Approximation
34
Far Field Dyadic Green's Function by Saddle
Point Method ...................................
35
40
Sec.
2.
First Order Scattered Intensity ...............
Sec.
3.
Bistatic Scattering Coefficients and Backscattering Cross Sections ......................... 47
Sec.
4.
Rough Surface Effects
Sec.
5.
Data Matching and Discussions
......................--.................
52
55
7
Chapter 4.
Emissivity of a Two Layer Random Medium .......
62
Sec. 1.
Emissivity Using Reciprocity Concept ..........
64
Sec. 2.
Emissivity of a Two-Layer Random Medium .......
65
Sec. 3.
Angular and Spectral Dependence of Two-Layer
Emissivity ....................................
69
Active Remote Sensing of Stratified Random
Media .........................................
73
Backscattered Intensity for a Multilayered
Random Medium in the First Order Born
Approximation .................................
75
Chapter 5.
Sec. 1
Sec. 2.
Sec. 3.
Backscattering Cross Sections for a Three
Layer Random Medium ...........................
Application to Data Matching and Discussion ...
Sec. 4.
Appendices
(i)
84
90
Appendix A ............................... 100
Chapter 6.
Depolarization Effects in the Active Remote
Sensing of a Two-Layer Random Medium .......... 102
Sec. 1.
Depolarization Backscattering Cross Sections
in the Second Order Born Approximation ........ 103
Half Space Limit and Comparison with BackScattering Cross Sections of Radiative
Transfer Theory ............................... 113
Discussion and Application of Results to Data
Matching ...................................... 115
Appendices
Sec. 2.
Sec. 3.
Sec. 4.
(i)
Appendix A.
Mean Field Cross Terms ....
119
(ii)
Appendix B.
Derivation of.Backscattering of Cross Sections
123
(iii)
Appendix C.
List of Coefficients for
Appendix B ................ 138
8
Chapter 7.
Sec. 1.
Sec. 2.
Renormalization Method in the Active Remote
Sensing of Random Media........................
144
Introduction-- The Dyson and Bethe-Salpeter
Equations.....
145
Approximations to the Mass and Intensity
Operators..... .. 0 . . . . . . .. . . . . .
.
.
151
Chapter 8.
Mean Dyadic Green's Function of a Two Layer
Random Medium ................................. 154
Sec. 1.
Zeroth Order Mean Dyadic Green's Function Using
the Two Variable Expansion Technique .......... 157
Sec. 2.
Effective Propagation Constants for TE and
TM Modes........................................ 185
Sec. 3.
Comparison with Scattering Coefficients of
Radiative Transfer (RT) Theory ................ 193
Sec. 4.
Half Space Limit-Comparison with the Result
of Tan and Fung ............................... 198
Special Case of a Laminar Structure ........... 201
Sec. 5.
Chapter 9.
Sec.
1.
Modified Radiative Transfer (MRT) Theory for
a Two-Layer Random Medium .....................
211
Derivation of MRT Equations from BetheSalpeter Equation .............................
212
Sec.
2.
Boundary Conditions for MRT Equations.........
235
Sec.
3.
Comparison of MRT and RT Equations and
Solutions in the First Order Renormalization
Approximation ...................................
239
Illustration of Backscattering Cross Sections
246
Sec.
4.
Sec. 5.
Comparison with First Order Born Backscattering Cross Sections .........................
Sec.
6.
251
Appendices
(i)
Appendix A.
Coefficients of Mean Dyadic
Green's Function .......... 254
(ii)
Appendix B.
Amplitudes of Zeroth Order
Mean Fields ...............
(iii)
Appendix C.
Listing of Terms:
z',
) .....................
I.
256
(z,
257
9
(iv)
Appendix D.
(v)
Appendix E.
(vi)
Appendix F.
Matrices in the MRT
Equations ................. 271
(vii)
Appendix G.
MRT Equations and Boundary
Conditions in Standard
Form ......................
278
Chapter 10.
Renormalization of the Bethe-Salpeter Equation
283
Sec.
1.
Physical Significance of Cross Terms ..........
284
Sec.
2.
Resummation of Cross Terms in the Intensity
Operator .........................................
287
Renormalized Bethe-Salpeter Equation and
Discussion .....................................
293
Conclusions and Suggestions for Further Study
298
Sec. 3.
Chapter 11.
Listing 2f Terms:
and
M. (g,, k.)
N jn(, k)................ 265
Summation of Residues..... 269
Bibliography ...............................................
302
Biographical Note .......................................... 315
10
LIST OF FIGURES
2.1
Stratified medium
3.1
Scattering geometry of two-layer random medium
3.2
Two-layer medium with rough surface
3.3
ahh
and
ac
as a function of angle at
9 GHz
3.4
ahh
and
a
as a function of angle at
13 GHz
3.5
ahh
and
ac
as a function of angle at
16.6 GHz
3.6
ahh
as a function of frequency at
4.1
Emissivity as a function of frequency at nadir
4.2
Emissivity as a function of angle at
5.1
Scattering geometry of
5.2
ahh
and
5.3
ahh
as a function of frequency at
5.4
ahh
as a function of angle at
9 GHz.
5.5
ahh
as a function of angle at
13.0 GHz
5.6
a hh
as a function of angle at
16.6 GHz
a
and
0*
30*
10 GHz
M-layered medium
as a function of frequency
30*
and
60*
11
5.7
CT hh
as a function of frequency at
30*
and
50*
5.8
ahh
as a function of frequency at
300
and
50*
6.1
ahh' avy
and
0 hv
6.2
ahh' av
and
ahv
6.3
ahh'
a
vv
and
8.1
rTE
and
rTM
as a function of angle at
10 GHz
8.2
rTE
and
rTM
as a function of angle at
10 GHz
8.3
1TE
and
TM
9.1
ahh = a vV
as a function of frequency at nadir
9.2
ahh =
as a function of frequency at nadir
9.3
Constructiv e interference path lengths
9.4
avv
9.5
Constructive interference path lengths for second
vV
0 hv
as a function of angle at
5.9 GHz
as a function of angle at
35 GHz
as a function of angle at
35 GHz
as a function of frequency at
20*
as function of variance
order backscattering
12
LIST OF PRINCIPAL SYMBOLS
In the following list, a superscript single prime is
used to denote the real part of a quantity and a superscript
double prime to denote the imaginary part of a quantity.
R.
reflection coefficient of
interface
TE
wave at
i
-
j
S..:
reflection coefficient of
interface
TM
wave at
i
-
j
X. .:
transmission coefficient of
i - j
interface
TE
wave at
Y..:
transmission coefficient of
i - j
interface
TM
wave at
D2 :
defined in text
F2 :
defined in text
G01'
11:
dyadic Green's functions for source in region 1
and observation points in regions 0 and 1,
respectively
Gllm
mean dyadic Green's function
d :
depth of
E lm:
mean electric field
E0 ,n 1:
n :
electric fields in regions 0 and 1, respectively
e:
unit vector in direction of electric field for
TE wave
h:
unit vector in direction of electric field for
TM
layer
order electric field in region
wave
-
n-th
Z-th
Z
13
wave vector in region 2Z
transverse wave vector
C (r
-
r 2 ):
(kg, k
)
j.
correlation function of random medium
spectral density of correlation function
bistatic scattering coefficient
Y
backscattering cross-section per unit area
permittivity of region
2
mean permittivity in region
(r):
E
2
random permittivity fluctuation
6:
variance of permittivity fluctuations
2,:
lateral correaltion length
p
vertical correlation lengths
z
e:
I,
emissivity with polarization
JL:
p
intensity matrix and Stokes vector, respectively
z-component of effective wavavector
P
:
scattering phase matrix for incoherent intensity
scattering phase matrix for mean intensity
Zv' 1h:
long distance scales for
TM
and
TE
waves
14
CHAPTER I
Introduction
In recent years, active and passive microwave remote
sensing techniques have proved to be a useful tool in the
study of earth terrain, such as snow-ice fields,1-30 vegeta-
.31-45
tion coverage,
and meteorological
oceanographic 4 8 phenomena.
46-47
as well as
The majority of the work per-
formed in remote sensing has been experimental in nature
with theoretical developments lagging far behind.
Although
past theoretical emphasis has been largely restricted to
rough surface scattering,49-53 recent theoretical models
have been proposed to account for volume scattering effects
in low loss media such as snow, ice and vegetation.
Stogryn 5 4
considered the scattering of electromagnetic waves by a halfspace random medium whose dielectric constant contains a
small random part and a non-random part which can vary as a
function of depth.
Using first order perturbation theory of
Karal and Keller,55 Stogryn derived bistatic scattering coefficients by assuming that the correlation lengths are small
compared to the wavelength.
The cross-polarized scattering
coefficients were shown to vanish in the backscattering direction.
This is expected since only first order terms were
15
considered and contributions to the cross-polarized backscattering comes from the higher order terms.
Tsang and Kong56 also studied the electromagnetic scattering by a half-space random medium with three dimensional
correlation functions.
The scattered fields in the non-random
region are obtained by using the dyadic Green's function for
a half-space medium and the Born approximation where the field
in the random medium is replaced by the unperturbed field.
Following Peake's57 definition the bistatic scattering coefficients are derived from the scattered fields.
The cross-
polarized backscattering coefficients also vanish since the
Born approximation is a single-scattering approximation which
is valid only when the albedo is small.
More recently Tsang and Kong58 investigated the problem
of scattering by a slab of random medium with a laminar
structure.
A two-variable expansion technique is used to
solve for the zeroth order mean Green's function from the
scalar Dyson's equation under the non-linear approximation.
The mean Green's function is then used to derive modified
radiative transfer (MRT) equations from the Bethe-Salpeter
equation under the ladder approximation.
The MRT equations
are solved for a two-layer random medium with laminar structure and the reflectivity at normal incidence is determined.
Tsang and Kong59 extended the renormalization method to the
16
case of a half-space random medium with three dimensional
correlations.
The zeroth order Green's function is solved
from the scalar Dyson's equation under the non-linear approximation and MRT equations are derived from the ladder approximated Bethe-Salpeter equation.
In the limit of a laminar
structure two effective propagation constants are found to
exist.
Tan and Fung60 also employed the two-variable expansion
technique and solved the non-linear Dyson's equation for the
zeroth order mean dyadic Green's function in the case of a
half-space random medium.
Tan and Fung retain terms only to
lowest order in correlation lengths, and the resultant vector
solution contains only a single propagation constant for all
components in the Green's dyadic.
The first order renormalization method also has been
employed in the study of electromagnetic scattering by random
media.
In this method the incoherent scattered intensity is
obtained from the ladder approximated Bethe-Salpeter equation
by neglecting the scattering of the incoherent field.
Fung
and Fung61 applied the first order renormalization method to
a half-space characterized by a random permittivity with a
cylindrically symmetric correlation function.
Fung62 extended
the first order renormalization method to study the scattering
of a vegetation layer characterized by a correlation function
17
which is cylindrical laterally and exponential vertically.
Finally, we mention the work of Tsang and Kong,63 who
used radiative transfer theory to calculate the bistatic
scattering coefficients and the backscattering cross-sections
of a half-space random medium with lateral and vertical fluctuations.
They solved the radiative transfer equations
iteratively through second order and showed that non-vanishing
cross-polarized backscattering coefficients result.
All these past works were carried out either with the
model of a half-space random medium or with a two-layer random
medium in the case of scalar wave propagation.
Therefore the
objective of this thesis is to study and develop electromagnetic
scattering models which are applicable to the interpretation
of active remote sensing data of earth terrain such as vegetation coverage or snow-ice fields.
To this end, we consider
the model of a two-layer random medium with arbitrary threedimensional correlation functions.
In Chapter 2, we review
the dyadic Green's functions appropriate for a two-layer medium
where the source and observation points are located in different regions.
We then derive the dyadic Green's function
where both source and observation points are within the same
region.
In Chapter 3, we solve the problem of scattering by a
18
layer of random medium with three dimensional correlation
functions with a wave approach by applying Born approximations.
Carrying to first order in albedo, bistatic scattering coefficients are derived which reduce to previous results56 in
the limiting case of a half-space.
In Chapter 4, we study the emissivity of a two-layer
random medium with three dimensional variations.
Using the
results of Chapter 3 for the bistatic scattering coefficients,
we calculate the emissivity for arbitrary correlation functions.
The coherent behavior in the spectral dependence of the twolayer emissivity is illustrated.
In Chapter 5, we extend the first order Born approximation
to the case of backscattering by a stratified random medium.
Analytical expressions for the backscattering cross-sections
are derived for an arbitrary number of random layers.
The
results are illustrated in the special case of a three-layer
random medium, with correlation functions which are gaussian
latterally and exponential vertically.
In Chapter 6, we carry the Born approximation to second
order to obtain backscattering cross-sections that account for
depolarization effects.
The results are reduced to the half-
space case and wave-like effects not accounted for by radiative
transfer theory are discussed.
In order to account for multiple scattering of the
19
electromagnetic field renormalization methods are necessary.
The renormalization approach has been widely used to study
wave propagation in random unbounded media.6 4
55
It gives
rise to the Dyson equation for the mean field and the BetheSalpeter equation for the covariance of the field.
In Chapter
7, we review the derivations of the Dyson and Bethe-Salpeter
equations and the various approximations which are applied
to the mass and intensity operators.
In Chapter 8, we solve the non-linear Dyson's equation
for the zeroth order mean dyadic Green's function for a twolayer random medium.
The propagation of the coherent wave
in the random medium is similar to that in an anisotropic
medium with different propagation constants for the characteristic
TE
and
TM
polarizations.
The effective propaga-
tion constants obtained in the zeroth order solution are
compared with the scattering coefficients of radiative transfer theory by *aking the limit of a half-space.
Comparisons
are also made with Tan and Fung's60 half-space solution for
the zeroth order mean dyadic Green's function.
The special
case of a laminar structure is considered and two effective
propagation constants for each polarization state are found
to exist.
In Chapter 9, modified radiative transfer (MRT) equations
for the electromagnetic field intensity are derived from the
20
ladder approximated Bethe-Salpeter equation together with the
zeroth order solution to Dyson's equation under the non-linear
approximation.
These approximations have been shown65,66 to
be energetically consistent and therefore appropriate in the
development of a radiative transport theory.
The MRT equations
contain significant wave-like corrections not accounted for by
phenomenological radiative transport theories due to the presence of the bottom boundary.
The significance of these additional
contributions is discussed in the context of backscattering by
solving the MRT equations in the first order renormalization
approximation and comparing with wave solutions obtained in
the first order Born approximation.
In Chapter 10, the physical significance of the cross
terms in the Neumann series for the field covariance is discussed.
A method for resumming an infinite sequence of terms
(including cross terms)
in the intensity operator is presented.
A renormalized Bethe-Salpeter equation is derived which takes
the form of a pair of coupled integral equations.
21
CHAPTER 2
Dyadic Green's Function for Two-Layer Media
The Dyadic Green's function technique of treating
electromagnetic boundary-value problems was first formulated
by Schwinger in the early 1940's.
Since that time the subject
matter has been subsequently discussed by Morse and Feshbach, 6 7
C. T. Tai68 and Tsang et al.69
Since a two-layer medium is
the basic geometry in this thesis we will review dyadic
Green's functions appropriate for a two-layer medium.
By matching boundary conditions we then derive the twolayer dyadic Green's function,
G 1 1 (r, r')
where the source
and field points are located within the same region.
22
Dyadic Green's Function with Source and Observation
2.1
Points in Different Regions
Consider a point source located within the
of a stratified medium (Fig. 2.1).
Z-th
layer
Let region 0 be free-space.
The field in region 0, is composed of upward going waves only,
and has been derived69 in the form
d 2 k.
O (r, r') =of8
)
(k
e
)
e
e
r.'
x
dk dk
=xk +yk, d2kwhere
h
z'
k
z x k
(k z
kz
h(k
h
)[C
)[A z e (-kZ)
z
(-k
-ik
)e
e
z'
Z
(
lz
y
A^
)
Oz
z
ik
]+
x
A
e(k
{e (k
.e-(2.1)
x
y
and
-
h (k
k
z'
-ik
+ D
e
7r2
-ik
+ B
z
ik
Sik
e 2 (kZz
(2.2)
x
(2.3)
23
z
j
'
E0
L
z = 0
0
1
z = -d
z = -d
p-1
; = -d
z
0
=-d
n
z = -d
O0
n +1
Stratified geometry
Figure 2.1
n
24
k
and
(k
=
k,,2 =
- k 2)1/2
2
Izys
1,
et
(2.4)
is in the direction of the horizontally
polarized electric field vector and
h
is in the direction
of the vertically polarized electric field vector.
amplitudes
AZ, BZ, CZ
D
and
propagation matrix formalism.70
The
are determined through the
In (2.1) the first subscript
of the dyadic Green's function indicates the region of the
observation point while the second subscript indicates the
region of the source point.
Taking
k = N = 1,
we obtain
from (2.1) the dyadic Green's function for a two-layer medium.
The result is
G0 1
'
Jd k
(rL
ik _L
>
2
1) =
01
(k
-r.)
z, z1 ) e
(2.5)
where
> Io1
9
.k z,
z1 ) =
-
1
-
8Tr2
k
X 10 (k )
D 2 (k )
)[R
e(k
oz
(k
e i2k lzd
25
-ik lz z 1
2
e
k 1Y 1 (k _)
-k
F (k
+
,
^
e
[S12(k ) e
h(k lz
-ik lzzi
+ e
e 1 (k)lz]
i2k l d 1ik lz 1^
e
l
h(-k lz
ik lz
(k lz
2
2
(2.6)
.
iklzzi
and,
=
k.
i
k.
-
z
(k ) =
Y..(k
(2. 7b)
kjz
(2.8a)
(k
13 +
= 1 + R.
=
1 + S..(k
(2.8b)
)
13
k
i jz
(
X..(k
s.k.
3 iz
E kiz +
)
S.
(2.7a)
Jz
lz
)
(k k +k.
)
R..(k
.
R
D 2 (k )
=
1 + R 0 1 (k ) R 1 2 (k ) e i2klzd 1
F 2 (k4 . )
=
1 + S 0 l(k)
S 1 2 ( .)
e
i2k
d
1z 1
(2.9a)
(2.9b)
26
The subscripts
2.
i
and
j
in (2.7a)-(2.9b) denote
We note that the portion of
the unit vectors
e
corresponds to
same way, the portion of
vectors
h
G 0 1 (r, r 1 )
corresponds to
301 (r
TM
r 1)
TE
0, 1
or
which contains
type waves.
In the
which contains the unit
type waves.
27
2.2
Dyadic Green's Function with Source and Observation
Points in the Same Region
In this section, we derive the dyadic Green's function,
G
where both source and observation points are within region
1 of the two-layer medium.
We first write
3 1 1 (r, r1 )
as the
sum of two parts
G1 1 (r, r1 ) = GDll(r, r ) + GRll(r, r )
where
G
1) -
k1 2
D
x 7 x 3Rll (r,
1) -
k1 2
Rll(r,
Physically,
GR
satisfy the vector wave equations
V xGDll(r,
x
at
GRll
and
r1
GDll
(2.10)
(r,
1)
= iW46(r -
r1 )
) = 0.
(2.lla)
(2.llb)
represents the direct response to the source
and does not contain boundary effects.
Alternatively,
represents the response to the image sources produced
by the boundaries at
GRll
reason that
tion whereas
equation.
GDll
z = 0
and
z = -d
It is for this
satisfies the homogeneous vector wave equasatisfies the inhomogeneous vector wave
The solution to (2.lla) is just the free space
ik
z-ik
z
1 (k
z)]
28
dyadic Green's function, which has been derived by Tsang. 6 6
The result may be written in the form:
DI
)= -
'.
Dll
1
-
~WE 1
(r - r
+ G
l1
where the free space radiating portion of
in
GF11 r,
r 1 ).
r)
Combining
= -
--
(2.10) and
(r, r)
(2.12)
is contained
GDll
(2.12) we obtain,
) + G
1
(2.13)~
r, r 1
where
GSl (r, r 1 ) = GFll(r, r ) + GRll(r,
(2.14)
r
In view of (2.5), (2.6) and in order to match boundary condiz = 0,
in (2.13)
takes the form:
G
(r, r1 )
-ik
+
e
the dyadic Green's function
-W111
2
d 2k
=
GSil(r, r1
)
tions at
iklzz
-{[a e
e (k
lz
z
1
ez
1(-k lz
lz 1 e (-k1z) + a e
e
+
29
+ 6 e
h1(k)lz
ik lz z
ik
-
h1(-k)lz
(r
-
r
1S(-k
lz
-ik lz zi
e
+
[TI e
-ik lz z
-
z
ik
[y e
h 1(k lz
'L
(z
The boundary conditions to be satisfied
by
z = 0
>
G0 1
(2.15)
z ).
and
at
G
are:
0 1(r,
1
(2.16a)
z x G 1 1 (r, r 1
)
z xG
x G 0 1 (,
z x
Substituting (2.5),
conditions
1
=
x
x G1 1
(,
).
(2.16b)
(2.6) and (2.12), (2.13) into boundary
(2.16a) and (2.16b), we find the result
a =X1
01
(2.17a)
(2.17b)
xo01
-1
k
(2.18a)
30
kk1 SS10
(2.18b)
ko Y
01
p
x01 12. i2klzd1
=
e
(2.19a)
D2
x0
1
(2.19b)
k Y S
k o 01 12
k
k
(2.20a)
F2
Y
(2.20b)
k 1F2
The terms which appear on the right hand sides of
are defined in equations (2.7a)-(2.9b).
11(k z, z 1)e
(2.17a)-(2.20b)
Combining equations
(2.15) and
(2.17a)-(2.20b), we find the dyadic Green's function
OGil
(z > z )
G
for
( ,
r1)
=
d k
to be:
,
(z
where:
>
z)
(2.21)
31
1
z1
ik lzz
1
Se
D(k8T2klz (k)
D2 (k I.
+ R 12(k
e
)
)
+ R10 (k
-ik
z
e
Jz
1
-ik
zz
lz
e(-klz
i2k lzd 1ik lz
zie
[e
h (k)lz
(klz
ik
z
(-k^
ezi1e(-kilz)]}
ik lz
+ F
1
+ S 1 0 (k ) e
-
,k z,
)
S
h(-klz
F 2(k L)
-ik lz
(eh
kl
1(k l) + S 1 2 (k )
e
i2k lzd
e
ik lzz
h1 (-k lz) ].
For
z <
(2.22)
we use the symmetry condition for dyadic Green's
z
- functions
,
1
where the superscript
T
denotes the transpose of the matrix.
The dyadic Green's function
from
(2.21)-(2.23)
to be
(2.23)
r)
l'
G=
Sil
for
(z
< z1
)
S l
is found
32
- rl.L)
(z
(2.24)
< z1
)
<)= dzk ,kz )e(
2
where:
-- -- 11 -iklzz k
- -
, z, z1
)
l
8ir 2 klz
D2(k )
S
+ R 1 2 (k)
e i2klzd
Lz
+ R 1 0 (k)
-ik
e i~lz
z
ik lzz
e
e
ik lz zi
e 1(k lz )e1
-ik
1
(klz)
+
[e
1
(-k lz
zz
hlz(-klz
)
F 2 (k
)
+ S 1 0 (k
e
1-iklz 1
ik
e
ik lz
z
z
h (-k lz
hz
h 1(k lz )Il
(2.25)
h1 (klz)].
The net dyadic Green's function,
G 1 1 (, r
)
i2k d
e z 1
+ S 1 2 (k )L
for both source
and observation points within Region 1, may be written as
r)
=
-
-E-
zz 6(r -
r1
)
G
33
-
(r.
-
.)
(z
> z1
(z
< z1
2k_
g1
(k L, z,
(r
J
-r
)
.kz ) e
)
-
+ fd
ik
~
=>
(ki-, z, z ) e
)
91
(2.26)
with
g1 l
given by (2.22) and (2.25).
34
CHAPTER 3
Active Remote Sensing of a Two-Layer Random Medium in the
First Order Born Approximation
In this chapter, we make use of dyadic Green's functions
to solve the problem of scattering by a layer of random medium
with three dimensional correlation functions by applying Born
approximations.
Integral equations which govern the scattered
field are solved by iterating to first order in albedo.
Making
use.of a correlation function that is Gaussian latterally and
exponential vertically, we find backscattering cross-sections
for the two-layer problem that reduce to previous results 6 3
in the limiting case of a half-space.
A brief discussions of
rough surface scattering effects upon the backscattering is
presented.
The first order Born results are illustrated by
matching experimental data.
35
3.1
Far Field Dyadic Green's Function by Saddle Point Method
Consider a layer of random medium with permittivity
<E 1 > + E 1 f(r)
El()=
where
E (r)
is a real function of
position characterizing the randomly fluctuating part whose
amplitude is very small and whose ensemble average is zero.
The layer of random medium has boundaries at
z = -d
e0
and
The upper region is free space with per-
(Fig. 3.1).
mittivity
z = 0
and the bottom medium is homogeneous with
permittivity
E2.
same permeability
All three regions are assumed to have the
p%.
The formal solution to the scattering of electromagnetic
waves with time dependent factor
e-iwt
by the two-layer
random medium can be cast in terms of dyadic Green's functions.
We have
(O)
-1
0) (r) +
0
=3
d 3 r1 G 0 1 (r, r1 )1 Q(r )E1(r
1
)
- -
E (r) =
V1
0ip
(3.la)
(0)
=
E
-3
(r) +
d 3r
1
iWP0
1G
(r, r1 )1 Q(r )E1(r
)
-
E
VI
(3.lb)
36
z
0.
iOi
110 F
0
0
z= 0
0i~
<F_1
+
if~r
e.
Scattering geometry of a two-layer random medium
Figure 3. 1
37
where the integrations extend over region 1 occupied by the
The first and the second subscripts of the
random medium.
dyadic Green's functions
G 0 1 (r, r )
and
G1 1 (r, r )
refer
to the regions of the field and source points, respectively.
The random fluctuating part is accounted for as a source distribution with
Q(r)
=
k
The superscript zero in
2
E:if (r)/<E 1 >
E
01
(r)
and
where
E
(r)
k
1
=
2
refers to the
solutions in the absence of the random fluctuation part which
are also the zeroth order terms in an iterative series soluWe shall use parenthesized superscripts
tion.
1,
2,
3
etc.
to denote higher order terms.
Since we are interested in the scattered far field, we
evaluate
G 0 1 (r, r 1 )
point method.
G 0 1 (r, r 1 )
which appears in
From Equations
(3.la) by the saddle
(2.5) and (2.6) we may cast
into the form:
G 0 1 (r, r 1 )
dk
((,
e
z
) e
where
-ikozz
-
)
(3.3)
F(k
and
k
=
z
k
=
e
+ z k oz.
g01
,
Z
We introduce the transformations
(3.2)
38
k = k cos
x
p
k
= k
y
p
oz
(3.4a)
sin
(3.4b)
0
(3.4c)
p
and
(3.5a)
$
x = p cos
y = p sin$.
(3. 5b)
Making use of (3.4a)-(3.5k ),
CO
G01'
r
1
2 7.
k dk
=
equation (3.2) becomes
-iii
[F(kL, z ) e
27r
*
ik p cos(
d
-
e
5) + ik
z
OZ
rL
(3.6)
The
c
and
k
$
= k cos
x =p
k = k
sin i
y
p
k
integrations are performed by using the
saddle point method.
function is simply:
The resultant far field dyadic Green's
39
ik r
G0
1
(r, r1)
iW 0
0
e
-i+
Fr
+e
{H e
}
(3.7)
4Trr
where
H=-
= 01
e (k
D2
) e (k)lz
k
+ -k
k 01
=12 e(k
) e
2
ei2k lzd
k
K
k
Y 01
01 h(k ) h (k)lz
2
zF
(3.8a)
koY 01A
(-k lz)
1
+
s12 h(k
1
2
h (-klz
(3.8b)
+ z klz
(3.9a)
-
(3.9b)
z klz.
40
3.2
First Order Scattered Intensity
Solutions to (3.1) can be obtained by iteration.
Sub-
stituting (3.lb) in (3.la) we find the total solution in region
0 in the form of the Neumann series
=
0
where the
-(n) E9 (r)
(r)
0
n-th
r
E
+
n = 1
order field
(3.10)
E(n)(r)
0
-(n)
En
i
ie by:
y
is given
-
1=
=
W
. 11 . n
Physically, the
n f dr
-
l'
n-th
"' d r n -G 01(r,
...
n
1
-
=
r 1)-G 11(r , r2)
0n
'''L
order field represents the
tering of the incident field
tivity fluctuations.
-
E(
1
n)
(in
-
E (
...
n) (3.11)
n-th
scat-
by the random permit-
In (3.11) it is understood that each of
the volume integrations extends over the layer of random medium.
Forming the square of the absolute value of
averaging, we obtain the intensity in region
E (r)
0
and ensemble
41
00
<L!ECo(2>y
+
<
0
E
n= 1
It
of
2Re{E(0)(r)
Th2 +
. (0)
2n)
0
0
nl
CO
C(n)
0
m=l1
E (m) (r>
0
(3.12)
is to be noted the ensemble average of all odd order moments
Q(r)
vanish in the cross terms.
The first order scattered intensity in (3.12) is given by
(1)
1E
(-)2>
(2)
+ 2Re{E (0)*
0
0
0
It can be shown that the mean field term
specular and is much smaller than <|EC
1
2>(1)
(Wp
01
0
d r
)
r2)
The unperturbed fi eld
V
d r
G
(r
1 2 0 1jr
r
<Q(r 1 ) Q*(r2
)>
1
*
(0)
1
(r)
1
- (0)
-
(r)
in the low con-
2>
We thus have, after making use of (3.11)
ductivity regime.
< IE
is
(2)
-
<(Er,(r)I2
Cr0)
r)
is given by
(3.13)
42
li
-
-r
K
e
-A
(r) = E[f
*
(0)
E
i
+
B.
e
k
e . + --
Y
.S
I
(3.14)
where
.R
.
i2k
e
A
.id
1
e
D.
k
21
h 1(k lzi
k
A
-
e
e
.d
21
(3.15a)
h 1 (klzi
k1
f = e 11 +
i
YOli
+
2
The subscript
i2k
e
h1 (-klzi)
X
B
F
1
.
.
X
f
h
(k
(3.15b)
1 (-klzi)
F2i
(3.16)
.).
denotes the incident direction, and the
vector components denote the fractions of the vertically and
horizontally polarized components of the incident wave.
Making use of the unperturbed field
far field approximated Green's function
E(O)(r)
1
and the
5 0 1 (r, r 1 ),
tain from (3.13)
<
r2>1
1
~~f
d 3 r 1 d 3r 2
Es
Z , i s,s , -
lp*p,p
we ob-
43
+ s'k
i(k . -
e
k )
-i(pk*
1
.)z
1e
lzi
lz
r
-
-i(k
d'-e
.
i(sk
+ p'k* )z
2
lzi
-k
3
)
r 2 -L
C (r1 -
2)
(3.17)
where
s, s',
p
and
take values of either
p'
-1
or
+1,
and
=
4E7
--
(f
-
=-H
=
-
=~
-
=.
(3.18a)
A.)
*
(3.18b)
.
E0
-
47
$4=
=
--
F=
F1f -A
-
(3.18c)
(-
=
)
E
Eo
-
- (f
---
S=
(3.18d)
The correlation function
C(r 1 -
r2
<Q(r 1
Q*(r2)>
may be
(r 1 -
r2
6k
4-i(
fCd (D2
- ( 1 - r
)
expressed in terms of its Fourier transform
(3.19)
44
where
6
Re(k 1 ).
is the variance of the fluctuations and
k1 ' =
Substituting (3.19) in (3.17) we may perform the
integraions over the spatial variables.
One of the transverse
spatial integrations yields a delta function and the other
transverse spatial integration yields the illuminated area
A
of the horizontal plane.
enables the
z
out the
3daz
<IE 12
Sj
(1)
(1)
Furthermore, the delta function
integration to be performed.
and
After carrying
integrations, we obtain
z2
4
4r 2 6k' A
_
E
E
s"s' P.P'I
r2
prp
so's
f
zC
1
(k
k-.
-i [ (sk l
-
z
[1 + e
(sk 1z + s'k lzi
z
pk* ) +
1z
-
(s'k
-i [sk lz + s'kzi - $ZId
. -
Sz)+
-z
p'k* .)d
1zz
i[pkiz + p'k
-e
pk1 k+
l
i
i
-
z Id
z11
(3.20)
The
z
integration is performed by noting that for a
low conductivity random layer containing many wavelengths the
45
p = s
dominant contribution occurs at
the poles at
az = pklZ + p'klzi.
p' = s'
and
for
It is to be noted that
there also exists an additional significant contribution only
in the backscattering direction for the case
p' = s = -p = -s'.
To obtain the residues at these poles we must carefully note
the positions of the poles on the complex
example when
s = +1
and
s' = -1,
Sz-plane.
the pole may lie on
either the upper or the lower half of the complex
according to whether
Pz
Im(klzi).
Im(klZ)
For
Sz-plane
Carrying out the
integration in this manner, we find:
2
<|E 01
+ 4d1 A #(2k
+ s'k
G(sk
.)
s,s,
r
, 0) Re[
1
*
]
(3.21)
where:
,
*-
G(sk
lZ
+ s'k
.). = Li -.
lzi
l 1z + klzi.
sk"
lz
[e
Here,
A
2(sk" + s'k" )d
lz
lzi 1-
sk
+
s'k
.)
+ s'k".
lzi
].
(3.22)
is one in the backscattered direction and is zero
46
for all other scattering directions.
density
Thus given the spectral
(D of a correlation function, the first order scat-
tered intensity from a two layer random medium is readily
determined from (3.21) and (3.22).
47
Bistatic Scattering Coefficients and Backscattering
3.3
Cross Sections
The bistatic scattering coefficients are defined by
Peake 57as:
y
(
y
where
(3.23)
A-*.o cos 6 .i(I.)
is the incident wave intensity with polarization
(I.)
is the scattered wave intensity with polarization
(I)
1y,
4 r 2 (1)
.,k ) =lim
is the incident wave vector at the incident angle
and
and
= 0,
$
,
and
r
to the surface.
Considering spectral densities with an even
dependence, we find from (3.21) and (3.23)
letting
and
f = 1
e
by letting
f = 0
e
6
is the distance from the observation point
and
Tr 26 k' 4
k Z
101l2 -I +- l elzi. 1z 1
Yhh=z
2
f = 0,
m
y vh and
vhvv
and
Yhh
and
Yhv
by
y
f = 1.
m
IX 10X0
cos
e0
Ikizi 2
[1 +
IR
2 e
R
12121
ID 2 D 2 i
ij 2
2
-2(k".
IZI
_
k
-2(k"
()
+ k
[
e
+ k" )d
1z
1]
+
z
6e0
is the scattered wave vector at angles
k
v,
k u, .
lzi
-k ,,
lz
+ k" )d
48
-2 (k"
lzi
-k"
.
[1 -e
-4k d
e
z 1]
2
+ [R 1 2
)d
lz
1] [|R
2
2
-2(k"
lzi
e
cos 2p + A47r 2 dk ' 4
+ k" )d
lz 1
Ixoli
1
ID 2~i14
e
0)
-4k"
d
lzi 1(
ko1I 2
_ r2 6k{ 4
Be i
2
{
10F~I
k2
F2F 2
+ k" )d
-2 (k"
lz
lzi
1][
k
lz j
+ k"
lz
.
Cos
~
+
k
[1
IS1 2 R 1 2 i 1 2 e
+
-2 (k".
lzi
e
-
[l -e
d1
(3.24)
Y 1 0 X 0 li
2
Yhv
2
+
f (2k
IR 1 2 i
-
k" )d
lz
-2(k"
lzi
+ k" )d
1
lz
1
-k
[I R1
2
-2 (k"
e
li
+ k")d
1+
2
e-4k"
|s 1 2 1
d
sin
2
(3.25)
Tr 26k'
Yvh
Cos
4
oz
eoi
|klz
2lzi
2
k 2
0
I1
Jx 1 0 yoli2
ID 2 F 2
i
2
Ik"
+k
lz i
+ k"
1z
49
+ k
)d
[z-IRz 11+
2S
+
k-
[J
-2 (k"
+ k" )d
iz1
e )-zi
[
e
-2 (k"
izi
e
-
-
-2(kizi
[1
-
1 2 i1 2
k"lZ )d
1]
- k
klzi
IR12
}
-4k"l d
lz 1
.
+ k" )d
-2 (k"
lz 1 +
izi
12 e
[(IS 12i
2
sfn2
(3.26)
Ik 0 I2
k
0
k1 1
6 0 i Iklz I 2
cos
012
lYloY
10 oli2
4
-2 (kizi + k
lzi-e
IF F2F2il
k
+
Tr 2 6k' 4
lz
kzi
2
-2 (k"
)d
[1 +
IS1 2 S1 2 iI 2
e
+ k"
)d
lz1
izi
2
sin G . sin
03.
k0 2
izi
e-2(k"izi. -
-
[1
e
+ is121 2
+
cos $
-
klzi klz
k") lz d I
-4k" d
lz 1
e
k
[IS 1 2 i
.k
lzi lz
lz
-2 (k
+ k
e izi.
)d
2
2}
cos
+ sin eo
sin e
((2k
e
0
+ A47r 6k{
4
0
k1
Is
4
F2 i
12i 1
d,
,
k
2
0)
-4k" .d
lzi 1
50
k2
2
lzi
k
S
where
-
sin26.
2
(3.27)
o
0
=
+ z(k
- k
-Li-Lzi
(k.
k' )
12
and subscripts
h
and
denote, respectively, horizontal and vertical polarizations.
v
We consider a correlation function which is Gaussian
laterally and exponential vertically:
Q*(r)>
<Q(r.)
1
j
6k
=
e
1
-lI.
-
2 /k
9..|j
2
-Iz.
-
zj/
e
(3.28)
The corresponding spectral density is
4()=
P
4Tr 22
2(1
e
+
(3.29)
2Z2
Substituting in (3.24)-(3.27) and letting
+ $.,
area defined by
= Y
a
pq
6k'4{
hh
.
and
$ =
we obtain the backscattering cross section per unit
4
2
IX
cos 6
pq
0 ij4 k
ID 2 i
4
2i
k
..
01
. 4
izi
.
Tr
e= e
-k
2Z
2
sin 2 O
51
e -4ki zidzi
.
(1 + 4k' 2 ,
+ 8diIR 12i
Sk' 4 tj
2
avv
1 - e
-4k"
d
lzi l
2
-4k"
d
e lzi 1
jYj
IF 2 iI
4
{
2
Lzi
e
)
2k"
1lz
(1 + IR1 2 i
I4
k
(3.30a)
4
-k
ozi
.
-
1
e
o
2Z
p
2sin
2
e
oi
4
-4k"
d
lzi 1
-4k"
(1 + IS1 2 i
4
e
d
1 )zi
.
2k"
(1 + 4k'
z2)
izi
lzi
2
k2
klzi
+
sin 2 e95
+ 8d11.S 1 2 i
2
e
d
-4k"
1
lzi
0
2}
k2
lzi
k 2
It
sin 2 eo.
01
is to be noted that
(3.30b)
ahv
S
=vh0.
Thus there is no
depolarization effect in the first order scattering theory.
52
3.4
Rough Surface Effects
The backscattering cross sections derived in the preceding
section are based upon the assumption of flat boundaries.
This
assumption may not be realistic in many situations of interest
such as vegetation-ground and snow-air interfaces.
A more
physical model for such situations is to consider the surface
to be irregular and characterized by a surface correlation
length,
Zs
(or
Isy
2.sx for anisotropic rough surfaces)
and R.M.S. height deviation
a .
In Fig. 3.2 we illustrate
a two-layer random medium with a rough surface at the bottom.
Such a model would simulate, for example, a vegetation layer
with irregular ground.
The scattering by a rough surface which
separates two homogeneous, dielectric media has been solved
only in the cases of
Z
>> X
or
<< X.
Scattering by a
composite rough surface and random medium (e.g. Figure 3.2)
is a problem still unsolved at present.
However, in order to account for observed rough surface
effects in some fashion we may incoherently superimpose the
backscattering of a very rough surface
backscattering of the random medium.
(Is >> X)
with the
It is shown in Section
3.5 that such a superposition of intensities matches experimental data remarkably well.
The backscattering cross section
of a very rough surface-is given by Barrick71 and may be
53
z
0'
0
z= 0
S<1
rough surface
(as s 9
> + s if
'1
E2
Two-layer random medium with rough surface
Figure 3.2
54
written as
ahh
sec 4 G
R 0 1 (0)1
2
e -tan 2
0
/S
2
(3.31)
vv
where
s2
(3.31) with
is the mean square slope of the surface.
Comparing
(3.30) it is clear that the rough surface back-
scattering falls off much faster with angle than does the
volume backscattering.
Physically this is due to less rough
surface and more volume being seen at large incident angles.
55
3.5
Data Matching and Discussions
We have obtained, for a two-layer random medium, the
bistatic scattering coefficients in
(3.24)-(3.27) and the
backscattering cross-sections in (3.30) for a correlation function which is Gaussian laterally and exponential vertically.
It is noted that these results can be derived by using a
Fourier transform method instead of applying the saddle point
method to the evaluation of the dyadic Green's functions.
In
the case of scattering by a half-space random medium, we let
d
+
4
and find
Sk''
a
=
hh
a_
vv
=
9, 2
k
1
8k" .(1 + 4k 2
lzi
lzi
6k' 4 9, 2
1
8k"I
(1 + 4k2
z 2)
lzi
lzi
-k
eo
2
2
p
Ixi4
4
oz
-k
e
2
2Z
o
sin 2 e
p
01
k2)
lzi
1i
k
.
k
.
lzi
4 k2
izi + sin2e
k
2
2
0i
i
sin 2 o
01
These are exactly the results obtained with the radiative
transfer theory by Tsang and Kong.63
When plotted as a
56
function of incident angle,
av
is always larger than
ahh
This is because more vertically polarized wave components
are transmitted and backscattered by the random medium.
For wave scattering by a two-layer random medium, the
results in (3.30) suggest the possibility of -av
than
ahh
when the first terms in the curly brackets become
smaller than the second terms.
In this case
IR 1 2 ilI,
and
aV
,, Jy 1 QiJ4 j 1 2
i
2
'12i <
ahh
i
Brewster angle effect at the bottom interface.
though
smaller
Yl1 iJ _ JX 1 0 ij,
10i
due to the
Thus even
it is possible to have
avv < Chh*
Physically this is because more horizontally polarized wave
is reflected by the bottom boundary resulting in more horizontally polarized wave in region 1 to be backscattered by
the random medium.
The results in (3.30) have been applied to the interpretation of remote sensing data collected in vegetation and
snow-ice fields. 42,1
In order to account for rough surface
effects, which dominate over volume scattering in the angular
region about nadir, we have incorporated the backscattering
cross-section of a very rough surface in an incoherent fashion
(Section 3.4).
In Figs. 3.3-3.5, we have matched the back-
scattering data of a corn field with a height of
three different frequencies.
The letters
V
and
2.1 m
H
at
represent
57
experimental
avv
and
ahh
In order to match these data, we
the theoretical results.
assigned the values
s2 = 0.03
surface.
where
s2
Z
while the continuous curves depict
=
0.32 cm,
z = 1 cm,
6 = 0.361,
and
is the mean square slope of the rough
As seen from these figures, the match between the
experimental and theoretical results are very good.
scattering dominant portion of the curves for
8
The volume
> 20*
ac-
counts for frequency change very well which the rough surface
dominant portion for
e
. < 20*
does not due to the frequency
insensitivity of the very rough surface result.
It is inter-
esting to note that in order to match these backscattering
data, it was necessary to choose
Z > Z
p
.
In Fig. 3.6, we
consider the same corn field and fit the frequency dependence
of the backscattering cross section at nadir and
30*.
The
slight frequency variation in the theoretical result at nadir
is due to the fact that volume scattering coes contribute at
this angle, even though the frequency independent rough surface result
dominates.
58
o (dB)
-- a vv
ahh
v
-4
H
v
\
H
-6
-8
v
H
-10
-12
H
-14
H
-
\
-16
0
10
20
30
o
ahh
and
a
40
50
60
.(DEG.)
as a function of angle at
Figure 3.3
9 GHz
70
59
a
(dB)
vv
hh
-4
\v
-6
-H
-8
-10-
H
H
-12--\
v
\
-14
-16
0-
10
20
30
40
50
60
8 .(DEG.)
ahh
and
avv
as a function of angle at
Figure 3.4
13 GHz
70
60
S(dB)
vv
ahh
-4
-H
H
-8
-
-6
v
H
-10
H
-12 -\H
v
-14
-16
0
20
10
40
30
8
ahh
and
a
50
60
.(DEG.)
as a function of angle at
r
Figure 3.5
16.6 GHz
70
a(dB)
I
-2
H
H
H
H
-4
H
H
H
H
H
H
H H
0
0
-6
H
-8
H
01 r3 30*
H
-10 - H
H
H
H
H
H
H
H
-12
8
p
p
9
10
11
12
13
16
15
14
17
Frequency (GHz)
hh
as a function of frequency at
0*
and
300
0~
H
Figure 3.6
62
CHAPTER 4
Emissivity of a Two-Layer Random Medium
The study of earth terrain can be effected with the
method of microwave passive remote sensing.
In this method,
a radiometer aboard aircraft or satellite records the microwave thermal emissions from naturally occuring media such as
snow-ice fields, vegetation coverage and sea-ice.
The volu-
minous data collected.and its interpretation has been the
subject of considerable theoretical effort in recent years.
Gurvich et al. 7 2 first derived expressions for the
emissivity of a half-space random medium with laminar structure in the single scattering approximation.
England 73ex-
amined emission darkening of a half-space containing distributed isotropic point scatterers by employing a radiative
transfer approach.
Tsang and Kong 6 9 ,5 6 ,7 4 have considered
thermal microwave emission from a stratified medium with
non-uniform temperature distribution, the emissivity of a
two-layer laminar random medium as well as thermal microwave
emission from half-space random media.
In particular, Tsang
and Kong56 employed a wave approach in the first order Born
approximation and calculated the emissivities of a half-space
random medium.
In this chapter, we follow a wave approach
63
similar to Tsang and Kong's56 and obtain emissivities of a
two-layer random medium with arbitrary three dimensional
correlation functions.
64
4.1
Emissivity Using Reciprocity Concept
Peake57 has defined the emissivity of a natural surface
as:
e (k) = 1
-
f
rh
4 7r
e (k.) = 1
where
rh
-
r~
and
-
r
--
d
fda
[Yhh(ki
[y
k)
+ Yhv (Ki
k )]
(k., k ) + yh (kk)
(4.la)
(4.lb)
are the Fresnel reflectivities for hori-
zontal and vertical polarizations for a homogeneous two-layer
medium.
The integrands of (4.la) and (4.lb) are the bistatic
scattering coefficients of a two-layer random medium and the
angular integrations extend over the upper hemisphere.
65
4.2
Emissivity of a Two-Layer Random Medium
The bistatic scattering coefficients of a two-layer random
medium have been derived in Chapter 3 and are given in (3.24)(3.27).
In order to develop an expression for the emissivity
in as general a form as possible, we expand the spectral
density in harmonics of the scattered azimuthal angle,
cD(K L .-
siis
, k
k
izi
lzs
) =
$s
im
E
M = -CO
s
e
m
(k
ps
, k ,k
lzi
pi
k
lzs
(4.2)
Substituting
(4.2)
into
(4.la) and (4.lb), the
may be performed directly.
s
integration
After some algebra, we obtain
emissivities in the form:
+
+
h4
I
4
10 s
(2)
k0
2
is
k
2
0
de
0
kJ
sin e
+ A
N
+
2)
s
N-
s
f
P
h
A
1 0s
Ik 1 z5
+
{lk::::
2a.()
.i1 2
(1)
2
is
is
M
+
h
M
-
1 h/2f
(4.3a)
66
v -
ev
-
1 7r/2
2
+ IY 0 ls j k 0ais L 2k
de s sin 6s Pv A+
'Jklzi 2 iklzs
-2 (2)
k2
I
2
I(1)
2
M+
4
+ 2 sin2 e6 0 sin 2 es
D0 (+)
sin %, sin 6
-
2Re (k* k
izi lzs
Ixosi
+ A
-
+
k
k2.
12k lz
2
I~l 0sK
2
k2
2(1)
2
0
N
is
klzi
-(2)
1 1klzs
2k
0
s
k 2
0
M_ + 2 sin 2Ss
4
sin 6 i sin e
+ 2Re (k* k
lzi lzs
+
1
[
1 (-)
+
+
(-)
ij
D0
U.
(4. 3b)
where
STr26
k'
4
= 1
2 cos 6 0 i
2
_X_
li
ID 2 i
2
k2
ozs
(4.4a)
67
k2
ozs
2
'olil
1rzck{
p=
A+
(1
-
cos6
| IE2 i
k
Iklzs
2
2
2
0
(4.4b)
1k 114
(ki izi + kis)di
e-2
(kii
izi
-
(4.5)
k is
kz5
1+R1
-22J
(1)
is
(kit + kit)d 1
(4.6a)
ID 2s1
-2(k". + k"
1+s
2
12i 1 2
e
)d
lzs
izi
1
(2)
ais
ID 2 s
2
(1)
i-s
(4.6b)
2
-2(k" . + k"
)d
e
zi
lzs 1 + |R
1 2 s12
F2 s
-4k"
d
e zs 1
2
(4.6c)
IR1 2 i
2
IF 2 sj
-4k"
+
(2)
is
-2(k" . + k"
)d
e
lzi
lzs
1
i~s 2
2
e
d
zs 1
2
(4.6d)
-(1)
1 + IR 1 2 sS
1 2 iI2
ais
ID
e
+ ki"izs )d
z
(ki
(4.7a)
2s
68
jD- 2s
(2)
is
-(2)
is
O
Is 1 2 i
s()
(4.7b)
1F2
2
2
)d
-2(k" . + k"
lzs 1 + IR1
lzi
e
Sis
1D2s1
2
d
-4k"
lzs 1
e
2
(4.7c)
-2(k"
is12i1
(2)
is
. + k"
lzi
e
lzs
)d
1 +
N
=
()
259(
( ) n=
The subscripts
+ (2
-
)
62
and
d
-4k"
lzs 1
(4.8a)
(4.8b)
-2
n
n (k s, k p .,
s
e
+ (-
i
k1 i
k lzs
)
= 2(
2
[2
IF
M
is1 2 5
n = integer.
(4.9)
denote that a term is to be evalu-
ated at the scattered and incident wavevector angles, respectively.
69
4.3
Angular and Spectral Dependence of Two-Layer Emissivity
In
(4.3a) and (4.3b) we have obtained the emissivity of
a two-layer random medium with arbitrary correlation function
in the first order Born approximation.
To illustrate the
emissivities, we take the spectral density to be
2,
=z 4112(1
-a
2
+
z
2/4
2z,
9
e
(4.10)
2 22
which corresponds to a correlation function that is Gaussian
laterally and exponential vertically.
(4.2) we solve for the amplitude
Substituting (4.10) in
Dm
-k 2 t
2
D (k , k ., k .
lzi
pi
m ps
(
k
Ips
M
where
Im(x)
k lzs ) =
k .Z
Pi
4T2[1
2/4
k2
-
(k
lzi
+
.Z
2 /4
pip
PsP
)292]
k
lzs
2
pj
(4.11)
2
is the modified Bessel function of order
deriving (4.11) we also have taken
$.
= 0,
m.
In
due to the azimu-
thal symmetry of the assumed correlation function.
Inserting
70
6
(4.11) into (4.3a) and (4.3b), the
completed numerically.
We illustrate the results in Fig. 4.1,
where we plot the emissivities
function of frequency for a
6 = .0005,
<E1> =
=
(1.8 + is m )
.005
.0005.
=
Z = .0021,
-20 cm
2
.
integrations can be
e
e ev = eh
20 cm
= .0l'0
1
at nadir as a
thick random layer with
=
2
(6.0 + iO.6)e
and
The dashed curve corresponds to
whereas the oscillating curve corresponds to
The coherent effects due to the boundary at
E m
Es
=
z = -d
are apparent in the low loss case where the emissivity
oscillates as a function of frequency.
As the loss of the
medium is increased, the bottom boundary is seen less.
is demonstrated in Fig. 4.1 where for
Elm = .005
This
the oscil-
latory behavior has disappeared, in the spectral dependence
of the emissivity.
and
eh
In Fig. 4.2 we plot the emissivites
as a function of angle at
10 GHz,
for a
e
20 cm
thick random layer with the same parameter values as Fig. 4.1
and
E"lm = .005.
The emissivity of the vertically polarized
wave exhibits a slight maximum due to the Brewster angle
effect.
The horizontally polarized wave has no Brewster angle
effect so that
angle.
eh
decreases monotonically with increasing
e
E:"1
= .005
E"
= .0005
lm
1.0
.9
.8
lm
.7
.6
8.0
9.0
10.0
11.0
12.0
GHz
Emissivity as a function of frequency at nadir
Figure 4.1
I-j
e
.9
.8
.7
.6
eh
ev
.5
.4
.3
.2
10
20
30
40
50
60
70
Emissivity as a function of angle at
Figure 4.2
80
10 GHz
90
a .
(DEG)
73
CHAPTER 5
Active Remote Sensing of Stratified Random Media
In active microwave remote sensing of earth terrain the
model of a layered random medium has been applied to account
for volume scattering effects.
Using the model of a half
space random medium, backscattering cross sections have been
calculated with an iterative approach 5 6 ,6 3 and with the firstorder renormalization method.61
To improve the simulation of
earth terrain for vegetation coverage, snow-ice fields or
culture targets,
two-layer models have been developed, with
the random medium bounded by air above and earth below. 6 2
In these past models, a correlation function with variance
and correlation lengths is specified for the entire volume
scattering region of interest.
However,
in remote sensing
applications a more realistic model might consist of partitioning the entire scattering region into sub-regions or
random media each with a characteristic correlation function.
For example, in vegetation cover such as forest terrain the
sub-regions would be the leaf and trunk regions of the trees,
where the respective lateral and vertical correlation lengths
are significantly different.
Similarly,
in the case of snow-
ice fields, one usually finds a complex layered structure in
74
which the lateral and vertical correlation lengths may vary
significantly from layer to layer.
In this chapter, we use an iterative wave approach to
solve the problem of backscattering by
N-layers of random
media with three dimensional correlation functions, arbitrarily
distributed within
(M - N)
homogeneous layers.
Carrying to
first order in albedo and making use of correlation functions
which are Gaussian laterally and exponential vertically, we
find backscattering cross sections for a three layer problem.
The results are illustrated by matching with experimental data
collected from vegetation and snow fields.
75
5.1
Backscattered Intensity for a Multilayered Random Medium
in the First Order Born Approximation
Consider a vertically stratified medium consisting of
random layers and
M - N
boundaries at
0,
m-th
layer
bility
=
0
z
(m = 1, 2,
(M I N)
homogeneous layers, with
-d1 , -d 2 ,
...
,
U
and permittivity
or
1
M)
...
,
-dM
Mf(r)
[Fig. 5.1].
The
is characterized by permea-
em = <Em> + AmEmf (r)
according to whether the
geneous or random, and
N
m-th
where
layer is homo-
is a real random function of
position whose magnitude is small and whose ensemble average
is.zero.
The formal solution to the scattering of time harmonic
electromagnetic waves by the stratified medium can be cast
in terms of dyadic Green's functions.
The first order scat-
-(
_ d3r Gom(r, r )-Qm(r 1Em (rs) (5.1)
where
W0A
-
1
M
=N-
-
-
-(0)
-
tered field is written as
1
iWOm
=
1
Gom (r, r 1 )
m Emfr 1'
Vm
is the dyadic Green's function,
and the superscript zero in
Em
Qm(r 1
(r )
to the solution in the absence of random fluctuations.
integration in (5.1) extends over the
m-th
refers
The
layer occupied by
76
z
18
Z=Q
Po, 1E-
o-p 6 2
<e,>+A,
<E 2 >+A.
Ef ()
2
fC
Z =-d 2
Z=-dt....1
/Ol
EM
<M >+AmEMf(T)
Z= -dM
/.J 00M+I
Scattering geometry of
M-layered medium
Figure 5 .1
77
random media and the summation is over the
unperturbed field
r . mi
()
m
(r) = E
o
(0) (r)
m
ff - c.
e
mi
M-layers.
The
is given by
mi
+
Mi
(5.2)
where
mi =Ki
+ z kmzi
(5.3a)
mi = k
-
(5.3b)
The subscript
i
z kmzi-
denotes that a quantity is to be evaluated
at the incident wavevector angles.
The amplitudes in (5.2)
are determined through the propagation matrix formalism 7 0
(Appendix A) and
TE
and
TM
i
is a vector which denotes the fraction of
components of the incident wave.
The first and
second subscripts of the dyadic Green's function
Gom (r,
1
refer to the regions of the field and source points, respectively.
The plane wave representation of
derived elsewhere.69
The result is
Gom
' r)
has been
om (r,
r
om (K., z, zi) e
1k
--
ik
-
0
.
i
-
78
(5.4)
where
)
-ikz
m
e
m
(k
mz
)
+ B
m
(k
The amplitudes
mz
)
m
h
e
oz
)[A
m
eikmzz1
e (-kmz
m
oz
z1
e
-ik
+ D
{e(k
)
= ---87r 2 k
z1
] + h(k
z
mz 1].
oz
)[C
m
h
(-k
m
mz
)
Z
e
ik z
mz 1
(5.5)
-
am (k L ,
gom
and
D
are also determined by
m
using the propagation matrix formulation. Taking the obserAm' B' m C
vation point in the far field,
is evaluated with
)
Gom (r, r 1
The result is:
the saddle point method.
-ik
Gm(r,
r1 ) = i
W0m
e
z
mz 1 +
e
m
ik
ik r - ik
z
mz 1
* r
e
47r
(5.6)
where
79
) + D
(5.7a)
) hM((kmz
=
B
e(k
F
=
A
e(k oz ) ein (-kmz ) + Cin h(koz ) hm (-kmz
i
e(kmz
W21Em
m(
h(k
oz
(5.7b)
(k
m
(5.8a)
xm
= -- e
m
mk
-
mz
m
k
k
h (k)
k
mz
oz
)
in
m(
)
M
km2 m=
) e (k
H
(k
mz
)
x
k
(5.8b)
m
- k 2)1/2
L
2
(5.9)
>, e is in the direction of the horizontally
polarized electric field vector, and
h
is in the direction
of the vertically polarized electric field vector.
Forming the absolute square of (5.1) and ensemble averaging, the scattered intensity in region 0 is given by
s
->
2
M
M
Z
En
m =1n
2 12
a
= 1
d3 r d3 r
V
M
fV
1
2
om
(
r-
1
)
<|E(
1
n
-(0)
- m
(r )
-
(r, r2)
En
E
(r2)
m(rl) Qn* (r2)>
(5.10)
80
Assuming statistical homogeneity throughout the layered medium,
the two point correlation function
only upon the difference of
assume that
- d
Id
r1
- 1
n *(r
<Qm Cr 1
and
r2 .
)>
depends
Moreover, if we
m = 1, 2,
zM,
2
...
, M,
that
is, the thickness of a typical layer is much'greater than any
of the vertical correlation lengths of interest, then it is
clear that random layers separated by one or more homogeneous
layers contribute negligibly to the first order scattered intensity (5.10).
n = m + 1, m -
In the case of adjacent random layers, with
1,
the range of integrations
in
(5.10)
is
restricted such that contributions only come from boundary
layers on the order of a vertical correlation length thick
straddling the
(m, m + 1),
However,
Idm
since
-
dm -
and
1
(m, m
z M
1)
-
m = 1,
interfaces.
2,
...
,
M,
the
volume scattering of the boundary layers is much smaller than
the scattering of individual random layers.
Therefore,
we
approximate the two point correlation function as
<Qm(rl)
Qn (r2)> = 6mn Cm(r1 -
r2 ).
(5.11)
Making use of the unperturbed field
far field approximated Green's function
from (5.10):
r()
3
om
(r, r
and the
we obtain
w
81
<|IE 1
s
M
- mE
2
E
Ppp I
r2z m = 1 s,s'
i(sk
(m)
-*
e mz
+ s'k
.)z
mzi
:
V
1 e
d3 r
2
-(m)
sfs
m
-i (pk* + p'k* )z
mzi 2
mz
)
pp
Cm r 1 - r
2
fV
d 3 r1
E
e
e
-i (kL
- k)
22.
(5.12)
where
s,s',
p
E
-Cm)
and
0
H
4w
p'
take values
+1
or
and,
(5.13a)
mi)
m
-1
E
Hm
47r
-
(5.13b)
(f - ami)
-(m)
E
=-J
0:
* 47T
Em
SCf
*
(5.13c)
mi)
E
1,1
47r
The correlation function of random layer
in terms of its
(5.13d)
n
Fourier transform:
m
may be expressed
82
r2
-
6
where
mk'
<< 1
d6a
(aL
z) e
1
is the variance of the fluctuations,
Re(km)'
and the factor
density
0 M('
).
,z
A
(5.14)
2
)
Cm(r
km
=
has been absorbed into the spectral
Substituting
(5.14) into
(5.12) and pro-
ceeding as in Chapter 3, we find
Sm
M
m
+ P
where
4
= 1
2
r
d-
(dm -
P = 1
3
A(M)
s,s'
) 0(2k
,
0)
2 G (sk
Re(
for backscattering and
tering directions,
A
m
ss
1
P = 0
*
+ s'k
mz
.)
mzi
)]
(5.15)
for other scat-
is the illuminated area on the hori-
zontal plane,
G (sk
+ s'k
[e
.)
mza.
=
D (i
m
ai
. -
, sk
'
mz + s'kmz i.)
2(sk"
+ s'k" .)d
2(sk" + s'k"
.)d
mzi m _e
mzi m-1
mz
mz
(5.16)
83
and
k"
mz
=
Im(k
mz ).
Thus, given the spectral densities of the
correlation functions the first order scattered intensity of
a stratified random medium is readily obtained from (5.15)
and (5.16).
84
5.2
Backscattering Cross Sections for a Three Layer Random
Medium
The backscattering cross-sections per unit area are given
by
cos e . y
a
where
y
(Ki, K 2 )
(k, -K .)
is the bistatic scattering coefficient as
given by Equation (3.23).
v
(5.17)
The subscript
u
represents
h .or
for horizontal or vertical polarization and we have noted
in (5.17) that there is no first order depolarized component
in the backscattering direction.
Combining (5.15), (5.16) and
(5.17) we find the backscattering cross sections for the stratified random medium:
=
P
M
E
m=
+ S(m)
Pll
where
6 k'4
m m
(D
m
R(m)
(2k .
2k
Limzi
k"
mzi-
(d -d
m
M
)
S
I
) + (2K
Ii'
m
0)
(5.18)
85
R(m)
Nh
2
-mi
TE A
r2[
.1 2 +
mi
aT
mi
B.j 2
mi
TE B*.
mi mi
+ 2Re(A
TE*
mi
(5.19a)
S
11
(m)
hh
2
[aT
mi A.j
Mi
2
4k"
.d
mzi M _e
(e
R m) = 2 7r2 [T
vv
k2
mi
k2
-
.
p1
C . I
mi
mzi.
+
I T
mi
B .
2
miI
e
-4k"i
mzi
m(d+ dm m
1
I
4k" .d
mzi m -5.)b
2
+
(5.19b)
TM D*.
mi mi
.i2 + 2Re(C
IaTM D miI
mi
2
(5.20a)
km
-4k"
-
vV
-
S (m)
2
[!ci Cmi 2
4k"
.d
mzi m
(e
The coefficients
D
.
and
mi
4k"
_e
mz3
aTE,TM
mi
D
. 12
e
.
mzi
(d
+
m
m
dM - 1
.d
M
(5.20b)
TE,TM
mi
as well as
mi
mij
are given by the propagation matrix formalism in
Appendix A.
mi
86
As a special application of (5.18) we consider two adjacent
random layers bounded by air above and earth below.
M = 2
case
In this
and the amplitudes appearing in Eqs. (5.19)
and
(5.20) are easily obtained through the method in Appendix A.
The results are
X Oli
51 ki4 Tr 2
=
0) IR12i + R23i
Sd
D2i
(d2
+ R12i 23i ei2k2zi
21
-4kii
d
e lzi 1 + (I
R12i + R23i
D1 ( 2k,
,
I
2k to
0) 1R 2 3 i
-4k"
+
(JR 2 3 1
4
e
)
i2k2zi (d2 - d
2k lzi
+ 6 2k 2
e
-(d
2zi
2
1
Oli
l21i
2
-
d
1
1) (1
-e
dzid
-4k"
d
lzi 1
-4k"
+
d
1
e
-d)
2 i(d
- d1
2
14
-4k"
(1
ID 2 i
2
12
-
1l + R
R
.12i 23i
d1
-4k'f
i2k 2 z i (d2 -d1
)
+
-
)
- d1
)
ei2k2zi (d2
)
yhh
(d - d
1 ))
2zi ( 2
87
e
-4k"
d
izil1
2k 2 'zi
]
(5.21a)
2k"
2z ii-
k 4
8d
)
- d1
-4k izid
14
(d2
i2k 2
+
d
1I4)
11 + s12i 23i
1
8(d
-
2
k2
-
d1 )
k2
+
62k
D 2 (2EL
.2
y1 2 i 14
d
zi 1
k04
42
I E2i
e
Ik 2
(1
yoliI 4
(2K L I 2klzi
2k
lz_
0) |S 2 3 i 1
2
,
(
-4k"
-e
e
14
- k 21
-4k"
(d
2zi 2
-4k"
d
1zi1 + (is23i 1
e
-1
(d
-4k"
2zi
- d
)
)
ei2k 2 z i (d 2
+ s
(IS 1 2
)
k
2
.
k2
j14
+
-
l
-
k2.
2
+
-4k"
d
izi 1
- d1
i2k2zi (d2
+ Se12i 23i
2|
)
- d1
ei2k2zi (Cd 2
0) Is12i + s23i
k1
1E 2il
e
~
0
)
4
1 0i
ki 4T 12
v
1+ 1)
88
-4kidizi
(d - -di)
-4k"
1)
2zi 2
,
1 (D2 (2k-
)
e
2 k2z i
2k"
2zi
(5.21b)
where
R..
13
k.
=z
-
k.
+ k.
Iz
<E .>k.
-
<
<E .>k.
+
<E.>k.
1Z
J
X..=1+
1J
(5.22a)
Jz
J
S.
13
k.
Jz
R
Iz
.>k.z
1
(5.22b)
JZ
(5.23a)
ii
(5.23b)
Y..=1 + S.
1)
ij
e
i2k2z (d 2 - d 1 )
2 -
1
)
e i2k2z
1 + s2s
12 23
d1 )
-
i2k2z (d
2
3
i2klzd1
D 2 = 1 + R 1 2R 2 3 ei2k2z(d2 - d1)l
e
+ s01 [S 1 2 + S 2
ei2klz d
(5.24a)
+ R01 [R 1
2
+ R2
3
"
E2
(5.24b)
89
To illustrate the results as given by (5.21a) and (5.21b) we
shall consider correlation functions, which are Gaussian
laterally and exponential vertically:
m
-4r.
(r. - r.) = 6 k'* e
ii
i
where
j
m
- r.
mm
1, 2
/z
2
-
pm
jz.
I
-
z. 1/9.
3zm
(5.25)
and the corresponding spectral densities are
-$
A
m(
2
ji
2
2;2
/4
pm
e a
.
m zm pm
472(l +
2
(5.26)
zzm
)
C
The choice of correlation functions of the form (5.25) are
advantageous in that we may vary the variances as well as the
correlation lengths in the lateral and vertical directions
independently for each random layer.
90
5.3
Application to Data Matching and Discussion
We have obtained, for a stratified random medium, the
backscattering cross sections in (5.18) for arbitrary correlation functions.
In (5.21a) and (5.21b) we have taken the
special case of two adjacent random layers, bounded above and
below by homogeneous media.
Such a case is of considerable
interest, in the active remote sensing of vegetation covers
and snow-ice fields where the correlation lengths describing
the upper scattering region may differ significantly from the
correlation lengths characteristic of the lower scattering
regions.
To illustrate
and
TM
the theory, we plot in Fig. 5,2
backscattering cross sections,
ahh
and
the
TE
a
as
given in (5.21a) and (5.21b) as a function of frequency.
Note
that the spectral variation of the backscattering cross sections
exhibits two maxima due to resonant scattering within each
random layer.
This phenomenon of double resonance
(or multiple
resonance in the case of many random layers) may explain the
spectral behavior observed in some backscattering data.
Fig. 5.3 we have matched
TE
falfa field.
ahh
The letters
backscattering cross section as
30*
a function of frequency at
H
In
and
(or
h)
60*
for a
50 cm
al-
represent experimental
while the continuous curves depict the theoretical results.
91
As seen from this figure the model of the two-layer random
medium can account for the observed minimum in the frequency
dependence of the backscattering cross section.
In Figs.
5.4, 5.5 and 5.6 we match the observed angular dependence
of the
GHz
TE
and
backscattering cross section at
16.6 GHz
9.0 GHz,
for the same alfalfa field.
13.0
In order to
account for rough surface effects at the ground-vegetation
interface, we have incoherently superimposed the backscattering
cross section of a very-rough surface with mean square slope
s2
and the backscattering cross sections of the random media.
As seen from Fig. 5.4 the backscattering exhibits a rough
surface effect, which dominates over volume scattering near
nadir.
In Figs. 5.5 and 5.6 the backscattering data does
not manifest ground-vegetation rough surface effects due to
the shielding effect of the random layers at
13.0 GHz
and
16.6 GHz.
In Figs. 5.7 and 5.8 we match for both morning and afternoon
30*
TE
and
backscattering data as a function of frequency at
50*
for a
27 cm
snow field.
In order to account
for the diurnal change in the collected data, we model the
snow field as two-random layers:
and a bottom layer
23 cm
thick.
A top layer
4 cm
thick
The oscillations in the
theoretical curves are due to coherent effects of the top
92
4 cm
layer, and are more evident in the morning than in
the afternoon.
In order to match the afternoon data, only
the top layer parameters have been changed, the parameters
of the other layers are maintained at the same values used
to match the morning data.
It was found that the most signi-
ficant parameter to vary in matching both the morning and
afternoon data is the imaginary part of the top layer mean
permittivity.
The increased value of
Im<e >
required to
match the afternoon data, is consistent with the expected
increase in the free water content of a surface layer due
to the solar illumination.
The increased free water content
causes the surface layer to appear more specular thereby
decreasing the amount of backscattering especially at the
higher frequencies.
93
4. =
%
<Ej>=(I.2+iO.001) Eo
8 =0.05
tp =2.0 cm
c-(db)
tz = 0.1 cm
Z=-30 cm
< 2 >=(1.2+iO.003) co
3 =0.09
tp =0.06 cm
tz =1.0 cm
Z =-100 Cm
E 3 =(6.0+iO.6)eo
-6
-T
HH
-10
vv
-12
-14
-16
I
10
2
chh
and
I
20
a
30
40
I
50
as a function of frequency
Figure 5.2
,
-18
60
GHz
94
'HH
(db)
32 = -0,
<e >
8
z=o
(1.4+i0.07) c 0
0.2
tp = 1.0 mm tz = 0.6 mm
Z =-I cm
<E 2 >= (1.4+iO.02)eO
8 = 0.09
Zp
= 6.0 mm
tz = 10.0 mm
E3
-6
Z = -50 cm
= (6.0+ i 0.6) 4e
10
hh
hh
-fIL
H
H
H
H
-18
8
ahh
H
= 600
H
I
I
I
i
I
moo
10
12
14
16.
18
GHz
as a function of frequency at
Figure 5.3
30*
and
60*
95
(db)
I
Z=0
<e>= (1.4+ i0.07) eo,
S=
4
0.2
= 1.0 mm Iz = 0.6 mm
<F >=(
0 -
8
co
=0.09
= 6.0 mm
=10.0 mm
I
-4
1.4+10.02)
I
Z=-I cm
Sz =0.00
Z =-50 cm
C3
(6.0+iO.6) Eo
-8
H
-12
H
H
H
-16
-
0
H
20
10
ahh
30
40
50
60
as a function of angle at
Figure 5.4
70 0
9 Gaz
96
crHH
(db)
~z=0
I
<e, >=(1.4+iO.07)
Eo
= 0.2
Ip = 1.0 mm Iz = 0.6 mm
8
<e 2 >= (1.4+1i0.02) eo
a = 0.09
0
ep
I
-4
-8
=
mm
= 10.0 mm
6.0
s 2 =0.001
Z=-50 c m
S3
=(6.0+ i0.6) co
H
H
-12
H
H
H
H
60
70
-16
C
10
ahh
20
30
40
50
as a function of angle at
Figure 5.5
13 GHz
8
97
(db)
Z =0
< E>= (1.4+ 10.07) eo
a=
Ip
0.2
= 1.0mm Iz=0.6mm
Z=-l cm
<e> =(1.4+i0.0 2) co
8 =90.09
1
= 6.0 mm
0
tz=10.0 mm
-4E3
S2 =0.001
Z=-50cm
=(6.0+iO.6)e 0
F H
H
-12
H
0
I
I
10
20
hh
30
I
.40
I
50
as a function of angle at
Figure 5. 6
I
I
60
708
16.6 GHz
98
e2
001i4
S2
<
rHH
1
=
>=(1.6+i0.012)E0
8 = 0.3
tp =2.5 mm tz = 1.5 mm
(d b)
0-
Z -4cm
<e2>= (1.9+i0.004) c 0
8 = 0.45
ZP = 0.9 mm
Z =1. mm
Z =- 27 cm
8.
3 = (6.0+iO.6)e 0
4
0
-
H
-4
-8-12
10
6
Chh
14
18
22
26
30
as a function of frequency at
Figure 5.7
30'
34
and
GHz
50 0
99
s2 =
0.0027
i
<E , >= (I. 8 +i0.07) E0
8 = 0.3
= 0.45 mm Zz =-0.45 mm
THH
(d b)
<C2>= (1.9+i.004)6 0
8 = 0.45
ep = 0.9 mm
-2
tz = I. mm
Z=-4cm
Z= - 27cm
3 = (6.0+i0.6)EO
-
-
*0
h
-6
-10
H
-14
-18
-22
I
H
I
I
10
6
ahh
I
I
14
I
I
I
I
18
22
26
as a function of frequency at
Figure 5.8
I
30
30*
I
34
and
GHz
50*
100
5.4
Appendices
Appendix A
The amplitudes
A , B , a TE
M
m
M
determined through the
-ik
e
TE
ik
d~
mz m
Sr-ik
d
mz m
L-
in each layer are
propagation matrix formalism:
m
m e
TE
m
M
and
m,m
-
1
Lnm
--j
m
e
Lik-1
.n-
d
l)z m
-
d
1 )zm -
~
-
J1
where
and
-
1
Xm,m
e ik mz
-
1
m -
R
m,m
m
is given by (5.23a).
-ik
mz
(d - d
m
~
M
-
ik
mz
(d - d
m
-19
1
e
m
represents
M
A
or
)
Ci
d -d
m
m- 1
-
z
-
R
m,m
1
)
e
m,m
-
A
-
_
TE
m,m -1
101
TE '
aTr
m
m
mm represents
r = 1.
TE
The
B
TE
m 'I
m
or
o = R TE ,
n
0
0
= 1
and
reflection coefficient has the continued
fraction representation:
-i2klzd1
1
R-d +-i -i1 klz
1
RRTE =
R01
The
TM
0
+
-i2k
2
(l/R0 1 )
amplitudes
Cm'
TM
Dm, am
_
d
TM
and
m
in each layer,
are obtained from the above results by letting
and by making the replacement
E
case
D
m
or
represents
TM
TM
m
Cm
or
Rmfm
TM
am
SMm -
-
and
Tm
r = km
/km
In this
represents either
102
CHAPTER 6
Depolarization Effects in the Active Remote
Sensing of a Two-Layer Random Medium
For the problem of wave scattering by a half-space random
medium, backscattering cross-sections have been calculated with
a radiative transfer theory.63
The radiative transfer theory
was derived by a wave approach making use of the nonlinear
approximation to the scalar Dyson equation and the ladder
approximation to the Bethe-Salpeter equation.
59
Using the'
technique of Born approximations and evaluating the dyadic
Green's function with the saddle point method,
in Chapter 3,
we calculated to first order in albedo the backscattering
cross-sections for a two-layer random medium.
In this paper we carry the Born approximation to second
order to obtain backscattering cross sections that account
for depolarization effects.
Instead of using the saddle point
evaluated dyadic Green's functions, we apply Fourier transform
methods in the calculation of the bistatic scattering coefficients.
The results are reduced to the half-space case and
discussed in the limit of radiative transfer theory.
The
backscattering cross-sections are applied to matching experimental data collected from vegetation and ice-snow fields.
103
6.1
Depolarization Backscattering Cross Section in the
Second Order Born Approximation
The second order scattering intensity takes the form
(Chapter 3):
+
(2) (-)(
>
2Re{<E
0
-E(3)*
0 r)>
(r)
(4)* (r>}.
S<
(0)
0
2
-
2 (2)
z
+
<
(6.1)
(r>
0
It can be shown that the terms in the curly bracket are negligible [Appendix A].
We can concentrate on the term giving
rise to depolarization effects
( 2 ) (0
01
1
01
3
<Q(r 1)
Q (r2)
[
r)
1
12
W4 11 0
d 3 r d 3 r d 3 r dar
4
G1 1 (r, r 2 ) -E
1 1 (r 3
( 3)
'
r4 )*
4
Q*(4)>
1
fV1
2
)
<
(0)
1
.
2 > (2)
<JOI
v 1V
2
3
4
]Iv
6.4
(6.2)
104
u. to denote the polarization
Here we use the first subscript
v
of the incident wave and the second subscript
the polarization of the scattered wave.
to denote
We shall consider
U # v.
the case of
The fourth order moment of
Q(r)
in (6.2) may be expanded
in clusters:
+ C(r 1 -
where
C(r.
-
Q*(r4)> = C(r 1 - r 2) C(r 3
r 3 ) C(r 2
-
r4
-
r 4 ) + C(r 1 - r 4 ) C(r 2
)
(r2) Q*(r3
-
(6.3)
r3
)
1
is the two point correlation function for
r.
SJ
the random medium.
Note that the first term of
(6.3) when
substituted into (6.2) gives the square of the second order
mean field which can be neglected in our calculation of
I2
(2)
-
1
4
= 1 1 (r
- 1 , -r2 )
-G
dardar2
LW
V
1
2
9
3
r3 3
4[
d
r 4G0
-1
Er
G11(r
2)()y~V -- [G01(r, r3) - =3, r4 -
)
(2)
<
We thus have
01
r
=
-
-
depolarization effects.
105
-
-
(C(r1 - r
3
) C(r 2 -
r ) + C(r1 -
r
)
-
(r4 )]
-
(0)-
E
(6.4)
3 )].
C(r 2
The unperturbed incident field may be cast in the following
form
1
( )
(0) (z)
l
li
(6.5)
e
where
ik
-(0)
= E [f - A . e
(z)
E
1
0
ii
f,
and
A.
and
B.
-ik lziz
z
B
(6.6)
have been defined in Chapter 3.
Intro-
ducing the expressions for the dyadic Green's functions and
the correlation functions of Chapters 2 and 3, we first
perform the integration over transverse spatial variables
which yield delta functions useful in the evaluation of
transverse wave vector variables.
2>(2) =
<|9(2)
0
yv
V
f
d
We obtain
J
k({k
41
4
f.
d2
f.00
_00
dS da
z z
106
0
=<
-1 gL~
-(0)
'
z 2)
z(z
(k
(z 2
(k ,
1
z,
z3
(z
z 3 , z)
-
-E
z4
- g0
- g1 1 (k '
e
=
11 (k-L
-
z 3 ) eiaz(z2 -
-
(z
)
)
, z
dz dz 2dz 3dz
af -d
z ) e iaz(z 2 -
=>
z3
g 0 1 (k , z, z3
+kL , z3, Yg - i E ()]IV.
(0)
(6.7)
(
=
-z
- [gol (k ,
[eiz
.',
,
-
)
-z) DL i
The integrand of (6.7) is related to the bistatic scattering
(Fung and Fung 61).
k = -
we set
01
.
coefficients
and obtain
4r(2k
y (2)
pv
oi
oi
' K..+ R~.,
In the baskcattered direction,
1
')8 6 2 k
L4
0
E0
4
a ) c?(Rji
2
-
cos
1
|2
RKL'r
i
f
f
id
j
-00
dS da
Z
Z
107
- {2
Uv
(-
-k.L,
4.
,
+ 1*
(I, -k i ,
Faz yv J. J i'
az
(6.8)
az, a )]}
where
(k.
, -k
,r0az' az) =
- g1(k-L ,1
Z
2)
dz2
= >
01(-k
-iazz2
azz
(
+
E(0)
dz
f
Ei
(2
,Iz, z 1)
dz 2 (
1
01
-iaz 2 -
(0) (z9)
-
I
z
(-k
, z,
iS z1
Pv
=<
-
-
g1 1 (k , z1 , z 2 ) - -(
zy
e
z
)
0
-I
2
(6.9)
Carefully carrying out the integrations over
az' SZ
and the
spatial coordinates, we obtain, after considerable manipulations (Appendix B],
2) ()-
hv
oi
oi
h
Yh
oi
oi
o
-
oi
K
2
a
E
1
(
i
co
th (i
cos 2Q.
1
2 ID
2
IF 2i
2i
2i
i
tv
108
sin 6 sec 8
de
(sec e. + sec 6
0
W32(6,
6
W 1 (e,
a
(sec2 e.
8
)
,
sec2 ) W
-
1(6
6.) W3 (6
W2 (6,
sin e sec 8
(sec e. - sec e)2
d
+
0
9i)
) m (,
sin 8 sec e
+ fr/2 d6
M 2 (8,
2
6
)
f 7r/2
VY%
141
W 2 2(e,
8i)
) m3(,
6
(6.10)
where
k
'
W
) =
,
2~,
2
(sin 2 e + sin 2 8.)
12
2-
2
e
(6.11a)
2
Z2.
W2(,
8)e =
47r 2 [1
+
k 2
2
2
0
(cos
e-
cos
(6.l1b)
o.)2
Z2
W 3 (6, 8 ) =
2
47 [(1
p
2
2
+ k'j Z (COS
(6.11c)
8 + COSe
2
-4k'1
M (,
8)
=
-
4
cos26
IR12i2S2i2
e
d
izi1
R
S
COS 2 e
D12 +
12
D 2
F2
2
109
2
Sl cos O
10
+
D2
1
+ k") d
lz
1) 2
-2 (ki
(e - e
F2
cos e
2
i
I
S 1i2
R12i
R12i + S 1 2 i
+
I
S10S1 2 cos 2 e
10 12
S12i
10S121
2
F2
cos 2 0 sin 2 0 sin2 i]
F21 2
+ kz(e
[2d 1 (kit
+ k
-2 (kI
1Zi
-e
-4k" d
lz 1
)d1
-4k"
d
1
lzi
)e
-2kIid1
) (e
- e
-2k"
d
-2kd
z1) e
d
(6.12a)
M2 (e,
o
1
2 il
2
lR 1 2 i
2
e
-4k"
d
lzi 1 + 1)
1I COS
)
(IS
os 2 9
L-
2
F2
+
(k"
lzi
+
k")z
e
1
-4k"
d
lzi 1 -
[(k
-
ki)
12
-2 (k"
+ k" )d
lz 1
lzi
.
+
cos 2 e sin26 sin 2 e i
[
2
.
Cos2 2
2k".
lzi
e
110
(k"
sin 2 e sin2.
. + k"1
12i
1
lz
IR 1 2 i
2
-
)
cos 2G
lzi
2
F2
k"
-
G
e
1z
-4 (k"
lzi
-2(k"
r
2
+ k"o )d
lz
. + k" )d
lz
lzi
e
1
2
S
cos 2G
12
+
1
I
1
2
F
2]
-0
Is12 12
+
+
S10S.
F2
-4k" d
lz 1 - 2k"
lzi
4
D2
2
i
[k".
Sl 2 if2 L cos 2 e
10 cos
R10 +
s10
e
2
F2
D2
)
4
+
12
R 10
cos.
+
e
S 1 0 S1 2 cos 2
-
cos 2 6 sin 2 e sin2e
-2 (k".
]
e
lzi
+ k" ) d
lz
1
I F2 12
[e
3 ( ," e.)
-2k"
d
izil
=-cos2e.
4
-2k" d
-e
1
I
|S 12i |
-2k"
d
e
izi 1
R 12i
|2
e
-4k"
d
lzi l
(6.12b)
cos 2 Q
S
+
10
F
2
2
ill
-
+
2]
2
-2k" d
d
S2 cos G
I
+ 12
(e -2k izi1 -e
z 1
F
D2
2
2
1
2
Cos 6
IR 1 2 i
-
2
s 1 2 i.
2
+
1
F2
D2
IR 1 2
+ s1 2
I2
cos 2 6 sin 2G sin 2 e.'
F
F
. lzi
[2(k"
1
2
2
co s
cos 2
4
k" )d
e
1z
1
J
2-
-2k"
d
izi l
R12i
-2k" .d
e zi l
S1 2
-2k"
d
-2k" d
izi 1 _e
1z
cos2e 2
S S10s12
R10 12
F2
D2
cos 2 6 sin 2 6 sin 2e.
+
IR12i
e
-e
-4k"
d
1zi 1
i
12i 12
-2k
F2 12
dd
-2k" d
-2k" .d
lz 1 + e
izi
[2(ki
- k" )d
lzi
lz1
10
s1 2
d
-2k"
(6.12c)
(6
12c
112
k 12
where
area,
area,
ty (
)
=
0
k1 2
= 2k"
(2)
(2)
Ka
IX1 0 i
2Oli
2
Ix
1~
2lII~~
12
(6.13)
k0
The backscattering cross-section per unit
]I=Vy cos 6
readily follows from (6.10)
.
)
k 2
th (8
113
Half-Space Limit and Comparison with Backscattering Cross
6.2
Sections of Radiative Transfer Theory
The backscattering coefficient (6.10) reduces to that for
a half-space when we let
Y
=y
,)
(
d1
+.
0.
We find
(2)
=
-
(,
(6.14)
+ y (2)
where
(2)
Y+
2tv(6
cosO.
K
W1(e,
+
e
+ S10 cos 2 e
3
2 cos 2
y (2)
e
)
j
de
0
){W2 (o, 6
sin 2 e
e
sec.
+ W(a
cos 2 0 sin2
The first term
) th
) W 3 (e,
/2
)
t
4 7r20 cos 6
=
i
sin 6 seC e
sec 0 + sec a
I cos 20
4(sec e. + sec 8)
sin 2 e
[IR
2
jS 1 0 12 cos 2 6 sin26 sin 6])
1 0
(6.15)
is also obtained from the radiative trans-
114
fer theory derived with the nonlinear approximation to the
Dyson equation and with the ladder approximation to the BetheSalpeter equation.
59
The second term
y
(2)
is an additional
contribution not accounted for by the ladder approximation nor
by the radiative transfer theory.
The physical significance
of the additional contribution especially in the context of
renormalization methods will be discussed in Chapter 10.
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115
116
a (dB)
vv
hv
0 1
V
H
V
H
-5
HH
vH
Vv
v
V
C
C
Cb
-15
c-
C
-
C
-
.
b
-20
0
10
20
30
40
50
60
a
ahh' avv
and
ahv
as a function of angle at
Figure 6 .1
(DEG)
5.9 GHz
117
Ihh
-- -a VV
vv
-
(
a dB)
hv
H
-
-0
v
H
vV
v
C
C
H
-
-10 -4
H
C
v
v
V
--
H
C
-
-20
-"""-
---
CC
----......
v
H
C
-30.
C
-40
1
-50
0
10
20
30
40
50
6,0
70
S-. (DEG)
01
ahh, avv
and
ahv
as a function of angle at
Figure 6.2
35 GHz
118
hh
a (dB)
avv
ah
-a-.-
ahv
10
V
0
V
H
HH
VH
C CE
C
C
U
V
CC
'.
-M
-- %0 ..
C
C
-
-30
-40
20
10
0
30
40
50
60
9O02. (DEG)
0 hh'
avv
and
ahv
as a function of angle at
Figure 6.3
35 GHz
119
6.4
Appendices
Appendix A
The Cross Terms of Equation (6.1)
According to Equation (6.1) the cross terms are
(r)
2Re{<E
0
- EE
3
.
The first cross term,
-(0) (r)
*-)
(r)
0
0
<E
(r)
()
.<
(4)*r)>}.
(A.1)
0
- E( 3)*()>
does not contri-
bute to the depolarized backscattering according to the
following argument.
field
El
It is well known that the first order
has no depolarized component in the backscattering
0
direction.
Therefore, even though
ized components the product
( r)
<E-(1)
(r)
- E-()0
0
may have depolar(r)> is easily
seen to produce no depolarization effect in the backscattering
direction.
The fourth order scattered mean field is given by equation (3.11):
13r
W411
'
< E (4 ) (
<E
r
4
f
1 2
4 01
r1)
1
2
120
<Q(rQ
1
2)
3
- E1(0) (r4
)
ll(r3 , r )
-
- , r
-(r
2
3
=
G
(A.2)
Q(r 4 )>.
Substituting Equation (6.3) as well as Equations
(2.5), (2.26)
1
-
(r)> =
f
- 90 1 (k , z,
11
(k
z1 )
(0i
e
(k ', z1 , z 2
9
" z3 , z ) f dcd 3
I,
-
e(r
-la
11
dak dak 'd 2 k "d2k
-
(z4
(r2
e
-
r 2)
(r2
* (r
(r
(r_1.
3z
e(
12.)
z2 , z3
-()
- a - (r3 - r 4
r4
iE) -
F
S(r
)
-
d 3 rydar 2 dar 3 d3r
-
<
)
-(4)
'
and (3.14) of the preceding chapters into (A.2), we obtain
r - re) -ia - (r
2
i
-(r,.
3L
-
r41)
-
-
r3
r3
22L
Li
(A.3)
121
It is understood here that
actually consists
gll(k , z., z.)
z. > z.. Performing the trans1
J
verse spatial integrations first we then may carry out the
of two parts corresponding to
integrations over
c ,
,
R "
and
kJ'".
The result takes
the form:
E
0
e
.
(
4
=
()
0
a
+ ik oziz
r
(A.4)
..Li
where
-
(k
Az g Li, z2, z3
dazd8f dz dz2dz3dz4
W 411 04
Z.
-11
(k I-
k
z
f
12
90 1 (k
az)
'
,
z,
z
,
-i z(z
z
1
-
z 2)
-iaz(zc
2-z)
3
-
- g 1 (kL, z2 , z3
)
-
'
k
_Lf
1 Z2
,k
Sg 1 1 (k , z 3 1 z4 ) e
+
.
-ik ziz _>
e
k ,z)
-
f
k L,
z 3,
z
) e -i,
z (z1 - z 3
)
d2k d2 k
0
)
2
3
(k({'
)(2)
3
122
(k
z2'z3
e
-iaz (Z2 -
z 4)
+ 0 (k
-ikL', az)ll
-iS(z1 - z )
e
91 1 (k', z 3 , z) e
-
li
(z
4
,
-iaz(z
2
-
z3
(A.5)
).
-ik
It is to be noted that the term
in (A.5) is independent of
z.
e
.z
iz
jo>(k j, z, zl)
It is clear from (A.4) that
the fourth order scattered mean field is specular and can only
contribute to backscattering at normal incidence.
However,
even for normal incidence it is important to recognize that
E
0
has no depolarized
-.
(4)*
()2Re[EO (r) - <E*
(r)>]
-
the zeroth order reflected field
component.
Therefore the cross term
00
does not contribute to depolarization in the backscattering
direction.
123
Appendix B
(6.10)
Derivation of Equation
The integrand of Equations
(6.9) may be cast into the
form:
=g 0>
(A< ($)) S
5',5
pp
prpI
i(sk
izi. + s'k lz )z 3
of
+1
The indices
and
-l
-(0)(z
+ p'klz)z
e i (pk lzi
(B. 1)
(2.6),
s, s',
p
(2.22), and (2.25)
and
and the amplitudes
Substituting (B.1)
in Appendix C.
)IP
ik
.z
e(Blozi
where we have used Equations
Chapter 2.
1"
Z3
,I
g 11(k
)1
z
'z,
p'
of
take the values
(A< ($))
V
y-
into (6.9)
ssp
,
,
are listed
we form the
following expression:
I V(k
i'
=
E
s,s'
z
E
p,p'
Z
,
yv
ppp
L'
-k
{(>Pv (I))
z'
ss'pp'
z
(A V
l.v
))
--
ssp
(13
(i,,'s
I
IT z)
1 d "d)a
1zp
Z
z
0
0
*1
z
p
Iz
0\I
(xrt
dd, ss((
0
,dda ss
((') t
+
z
Z
OZ
s
(51731
liz:o
I T z)
Izz
zz
0
z
It
zzp
dl d)
Iddiss(( 4 ' )t
'Z)
'f.
0
~a
d'fd)
Izp
0
V\)
1
m\T
ddiss
+
Zu
(z
2
~
,z
I S.1 s
d'd.
z)
a zp
0
0
dd, ss
'f
j t,
Ez
z
Iidld)*
I p-
T zp
zz
r
.1
0
rt
(
- dd ss
>Y)
+
( (4)
(z
(1,s s% idd)
z
dz 70
s
JZZ ",z)
dS
s
Id
i9f
d)a
EZ
zz
0
125
s ,Is')
(z , z
,
(B.2)
z
z
)
F* (p, p,
where:
s
,s')
z 2,
az'
F(PP',
i(pklzi + p'k lz -
az)z1
z
-
(ziI
.
i (sk
e
+ s'k
-c
:z
)z2
- (B. 3)
LL
~
ss'
p,E-
1 (-k
liv
,
E
fs'I P
'
z
0
dz 1
-di
--d
11 v
-k
($k))
0
dz 2
1
dz
I a
Z, aZ
*L
kL 4,
ss'Ipp'
z3
dz
3
4
.
s
a
F* (pp,
(z3 , z
4
(A>
1v
az)
(w))
s s pp,
+
+
z
'-v
($+7
))
s
--
'p
-
(A
Iv
(>
+
-R
,
(k
*
Similarly, we obtain,
F(p, ', s,s')
(zi, z 2' az Jz
ss Ipp
126
0
0
dz 1
-d 1
F
3
<
s
1iV
0
s p)'
p
11V
z4
(ss
0
(v
0
dz 3
1
F* (pp' ,
d1
4
Ss')
(z 1 , Z2'
2z a
.
z
dz
4
F(Pz
, ss')
(z , z 2'
a
SZ
Z
s')
z 4 , az
az)
Substituting Equations
integrations.
1-d
Jd1
'p',
0
0
dz 2
l-d
perform the
F
3 dz
azF az)
p
dz 1
(z3
r
+
(A<
Cz' azl
OS',
,
(z3
'Pps "s'I)
, z 2'
a Z, 8z
(B. 4)
-
F* (P,p
(z
+ iT)),
Sv($
0
dz 1 J:
dz 2 lid dz
z1
1d 3
f-d
+
-d
2
F
dz
ss')
r 8
az)
,
(z
f-d
,
(pI
dz 2
(B.2) and (B.4) into (6.8), we then
integrations before carrying out the
z
The only contributions must come from the poles
127
of the spectral densities.
However, in order to determine the
direction in which to close the contours in the complex
and
z
Sz
az
planes, it is necessary to subdivide the domain of
integrations further.
In Equation (B.2) we must break
the integrations into regions in which
z2 < z
z
and
< z3'
In Equation (B.4) we break the integrations into regions in
which
z
z4
<
and
z2 < z3.
For example, consider the first
term of Equation (B.2) when substituted into
Y
1IV
(2)
.-
oi'
k
'.)
01
first term
(6.8)
j
, - - , - p,p s,s
rp,p
ss
d2 k g
of (B.2)
-d
dz 1
+
(z3 ' z4,
-d1
(k
k,
a'
f3
dz 3
fzl
-d 1-ddz 2
)
-
k
dz4 daz d8z
(), ) zF' sIs')
,
ss
-
(A<
11V
(
)
ss
,
,.
pp
(B.5)
We then break up the integrations over
z,
as
128
dz 1
-d 1d
dz 2
z1
1
Z2
dz 3
0
-d
dz 1
Z
o
-d
Z
d
dz 3
J
dz 3
Z2 dz4
d1
dz 4
rZ
dz3
d
dz 2 fZ dz3
z2
z2
0
dz 2
dz 1 f-l dz 2
d1
z2 dz 4
-d
z2
Z
dz
=
dz
-d 1
z
-d 1
3
dz
dz2 f
dz 2
1
-d
-d
-d
d
Sdz
+
dz 3 f
z 3 dz 4 +
d
d
+
f -d
3
z
dz4
(B.6)
z2
when (B.6) is substituted into (B.5) the direction in which
to close the contours in the complex
defined.
We denote the poles of
where the superscripts
+
and
D
-
a
Sz
planes is well
m = 1, 2,
as
signify the location of the
pole in the upper or lower half complex plane, accordingly.
Therefore, upon performing the
Equation (B.5) becomes
az
and
z
integrations,
129
y (k .,-k
.)
first term
of (B.2)
[Res '(Di
E
m,n
s~
1,iiv
+ Res
w2,
-
(
,
m
(S
m
-',
-,
S
n
Res
(P,P', S,s',
-
SI~
Sm)
Res
5 (m +
,m+ Res c(k.
p',,
(k
-
k J-P
-
k
a
;,O
Sn)
3,lv($M + .a-)
Res
D(k
.1
w4,Iv(S,m
+jE., k
.1i
gn
)
w
r, '
1
Sn
n
+ k.,
(k
jv
r
Res @( K. x
m
) Res
(.
.L1
- k
+
w
m, -)
.1.
(2ri) 2
Ppp
J'
n
)
(P,p',
+
s' P",' Ss,'
)
.'
pp Ess pEp1
s,
)
E
01
)
01
'
lJV
-
(2)
130
D(k
+ .Li 8 +)
.i.
Al m
(P,P', Ss',
W 5,1yV m ' I
Res 0(k
n )+
k'
.
+ Res
n
I
where the summations over
m
(B.7)
and
n
extend over the poles of
(D and
dz 1
1
(A
1,V
dz 2
dz 3 (3
1
ss) )p, ,ss((
ss pp,
PV
2.
dz 4
1
) - p- F (Pzp , s t
ss'pl'p(z ,2' a z'p az
)
(p,p', s,s', p0p')
W
(aZ' az)
(B.8a)
(pp',
Ss',
W 2,yvv a az z)
,
)
F (z ," z ,' a , $Z))
0
f-d
d rzi
dz 1
dz2
d
(zi
1 dz3
z2
2 dZ4
d
(P8 0E)
(z
I
z
(is's
z)
,ddss
M %rt)
I dld)a
-;c
czp
zpEf
Tzf
T-
Tp
s z
di d)
(as IsI
*ddss
(M( rt
p
zp
TZ
0f
(js's
d
( s ' ,)
did I'is's ild'd)
)
(08 *
zI
sM &rt
did)a
di d)
,dd ss ((a)
Tzp
zzp
0
V)I
zs
(JS
s ',d'd)
(Iss#di
Z.'
(Isis
>)
'
Z f
0
(Z 5
(rt
,dd ss
*
('E'~I
z
z
(5173
i- P-
Ivzp
~zz
(q80S~)
z
jzz
z
dIz)
(is isI
A' cz)
id id) a
0rt
1 ddss
TET
132
sts'
W5,uv (aZ'
, , ,z, ;
'
(p,p'II
dz
az)
1
d
{(A> ($))
PV
,
d
(AlV ())
0,
ss pp
)
z
z
dz3 f3 dz
z2
- --
FIP'P ', ss')
ss'pp'
(zi, z 2' az'Jq Z
y
(z3 , z 4 , az' z
J.
dz 2
(B.8e)
-J
The next step is to perform the
z
integrations retaining only
those terms which are dominant in the low conductivity regime.
The result is:
K
z
E
-z-zS's, p"p, sts' Prp
oi
oi
first terrh
of (B.2)
6 -
,-,
ss 6 ss
6 -
pp
6
-
pp
II(>
lv
(4))
ss'pp
2
4K"
ss'
Res (D(k
Res
+ k
.K -
c(
~)x
(K*
ss
,
-
a
n
k
,
) (K*
,
pp
n
-
8
m
)
4V
)
Y(2)
133
Res
(k
8
) Res 0 (kL
k
+
+
Res
m
,
pp
am+
+ k, S
ss
-
n
M
+
(K*
8)
(k-k
pp
+Li
n
(k
-
Res D(k
.
-+
8+
n
(*,ss
m-
(k
)
+ kL,
+
(
(
Res
*
8 npp)
( ss,-
(B.9)
where
M(
P
is defined below in (B.lla).
Following
p
steps analogous to those which led to
(B.9), we include the
remainder of the terms in (B.2) as well as all of
evaluating
(2) (k
IV
(6.8).
The result
-k .)
oi,
is:
(.k' 4 .)2 k
41 (27T)
4 14
0
0
m
m,n s,s'
+
(R< (c))
yIv
pfp'I
11
(
I ,1* N (S',
ss pp
(B.4) when
(k p )
2
cos
e
2k-"
(2
k
''-'2
0
+
ss'ppy
p,p')
+ (K>
+(V
tv
))
))
ss
pp ss
ss'PP'
spp
134
* [(A<
1Jv
(t:~+
M(s,s'
x
(k
+
pp ss
p,p')
(W
U v
(4))
ss'pp
I*
Imn
)
4 K"
p
(B.10)
,I
where
2
+ K"
ss. ,)d 1
pp
e
2 K"
,d
ss
2K"I Id
ep PP
1(1
)
M(s,s,
(k
(K" ,
1 - e
pp')
p
(K"
ss ,)
+
,
pp
K"
K"
PP'
(B.lla)
+
2K"I
,)
K"
(K"
ss
Res D (k
+
+ K"
pp
pp
,
m ) Res 0 (k
-
n
- k ,
mn
(K*
s
Res
D (k
+.,
. J..
(K*
pp
m
n
m
6) Res P (k . -
I
$n
4~ .' n
+
Res D (k
*
n + )(K pp
-
ss
., k
+
Ll
ss
m
a
n
+) Res
-(
m
(k.
I*
Npp I
J.i
-p
-k
m
I
_
J'
n
)
(K
+
J
,
+
p
d
(B.llb)
)
N(SSPP')
(k
s-S ,)d 1l
Kt
1 - e
135
Res D(k
,
+
+)
+
(.
- k ,
The term
Z JMn
m,n
argument.
Let
,
ca
-
a)
L
(z
p
., az)
-
)
(
da
(
(B.llc)
may be simplified by utilizing the following
00
z
.+
n
m
np
(K*s
where
Res
m +i
Z
(B.12)
z
vanishes everywhere on the circle at infinity
and has an equal distribution of poles
{a m}
m
the upper and lower half complex planes.
z
and {a
(B.12) we may close the contour either up or down.
> 0
p
(k., Zp)
Closing up
( (k, a +]
(B.13)
+
-
To evaluate
we find:
Res
(27ri)
in
m
(zp -
am
+
Im z
}
is a complex
number which lies anywhere in the complex plane.
and assuming
n
and closing down, we find
Res
I =
-
(kL, an)
(B.14)
27ri E
n
(z
-an
136
Therefore, it follows that
(B.15) by
n
4(k ,
(z
Res
p
(z9
z )
k(D',
-
)
an
yields
(
'c,
. +)
n
) Res $(D ', a
a +)(z
s
m
-
(B.15)
a
-
ct)
n
) Res N(k
-a)(z
m
( kL, a
s
- a
) Res
',
n
a
+
Res
p
,.
~Res t(K., am+) Re.s.
(z -a
(+
p
m
s
( (k, a
(z
(
_n
am +)
(IfmZs)kL
Res
Res
)
Multiplying
(zp -
a
+
m
( (k
+
Res
(k ',
an
(B.16)
(zp -
a
)
(z
-
a
This result would not change if we had assumed
rather than
result
Im(z p) > 0
as we did.
(B.16) we may sum equation
Im(z p) < 0
Therefore, utilizing the
(B.llc) as:
137
z
=(k
J
+
,)
m,n
.L
-
k., K
(B.17)
,).
Substituting (B.17), Equation (3.29) of Chapter 3, and the
from Appendix C, into (B.10) we obtain
)
(A
after some algebra; Equation
(2)
Yhv (Koi, -K oi
(2)
-,-
vh
k
(6.10)
0. cos 6 01
,
-1)
th(6) tv(6i) sec 6
=
W1(e,
+
7/2
0
2 F 2 i2
e
de
a2
0
ei ). M (0,
) W3;1(e,
sin 6 sec 9
(sec 6
-
W3 (6,
) M 2 (6, e)
W 1 (e,
2
W 22)
(6,
6
e
sin 6 sec 6
(sec 6 + sec 6) 2
)
i
W 1(o, e ) W 2(O,
6
sec 6)
+
J7
r/2
0
) M 3 (6,
d6
e
)
2
Cos
2
j1r/
de
-D
E
)
values of
sin 6 sec e
sec 6)
(sec 6 -
2
(6.10)
138
APPENDIX C
The Coefficients.
Ah
= Fv ei2klzd1
$))vyyy
- klzi
hv
-FV e
klz i R12 cos
lz d 1 (R1 i
e
2
i2k
1iz..d 1)
sin $ + klz S12
D2
D 2k
>
hv
, p, p
klz S 12
k12
sin2
i2k
,
s'?s
ss
(R1 2 i S 12 i ei4klzid
R12
os2
WX(
uyV
liv
2
7
e i4k lid1
11,11-1
i2klzd
v
klzi
12i
12i
10R12
klz
D2
10 S12
F_2k2
k-.
-4
D
2
lzi
CO
sin
+
klz
Fk 2
-
-
I
139
(A
v
v
1,-1,,1
zid
.
- k11l2. cos $ sin
i4k
12i S12i
++
za+O
F 2k 12
Di
klzi R 1 0 R1 2
ei2klzd
c
+ k lzS10S12
s
D 2
klzi
v (S12i
R12
i2k zid
sin
F v (R1 2 i e
1,1, -1,-
i2k
k lzF
lzd
S12
F 2k 12
D2
(-v
klz
lzi
i2k z d
izi 1
R10R12
klz S10S12
F2:1
-z
Lz D2
cos s21
i2k lzd
+
v (0))-lylflfl
F 2 k1 2
k2
140
i2k lzid
lzi
cos $ sin
+
+
-
D2
F2k 1
i2klzd
klz
10 12
cos $ sin $ +
e
klz S10 S12
z+01
2
-
e
F2k
klzi
i2klzd
R
12
2
s $ s
VD
+ klz S12
F2k
2
i2k
-1v1-
i2k
v
lzd
(S1 2 i e
klzi R10 R12
e
.d
.zi 1
klz
cos $ sin $_+__$)
L
k
.k
_ lzi cos
D2
2
$ sin
10 S 12
F 2 k1
+
lz
F 2k 12
2
141
i2k
11-11-1,1
k lzi
=F v
R 12e
Cos $ sin $
z
-
D2
F 2k
i2klzd 1Flzi
.d
izi.1
2
klz S
R1 0 R1 2
F 2k
D2
i4k
~1,-1,1,-1
k lzi
10
=
v (R 1 2 i
Cos $ sin
5 12i
lz
-
D2
2
.d
izi. 1)
e
S10
F2k}
_
w)
10S 2
=
_
,-{klzi
k
_
10
Cos sin
+
D2
(Nv W it -1
r (R1 2 i
i2klzid
10
F2k
klzi
D2
+ klz S10 a
F2k 12
zS 1
lz
R10
cos
sin
$
-<
142
=
k
.
c s $ sin $. -z
k
(
lzi
2
Fkk
S Lcos
zi R
10 R12
k~~
.
D2
k
-
2
klz
ocos
l
lz
Fk
211
klz 010
F k 2$
F21y
2
sin
1 0S
12
-
i2kelzd
$sin$-_$
Sz S
F21
c
143
(
(k
-
k
. cos $
z izi
k k .)
sin
op
$
where
y 1i xOli
F 2D
2i 2i
h (k ozi)
e (kozi
k k
k
Lo 1zz
for scattered
TM
waves
for scattered
TE
waves
(Avh(ss')p,
The coefficients
.
0"2
P-
are obtained from the rela-
,
(A ( ))
,
vh
ss pp
=h(s
v
+ 7))
,
pp ss
.
tion
144
CHAPTER 7
Renormalization Methods in the Active Remote
Sensing of Random Media
In the preceeding chapters the problem of electromagnetic
scattering by a two-layer random medium is solved with an
iterative approach.
However, if multiple scattering effects
are to be included renormalization methods are necessary in
which the Neumann series for the mean and covariance of the
electromagnetic field are resunmmed.
Tatarskii64 employed a
Feynman diagrammatic technique to develop the Dyson equation
for the mean field and the Bethe-Salpeter equation for the
covariance of the field.
He also considered the bilocal and
ladder approximations to the Dyson and Bethe-Salpeter equations,
respectively.
Rosenbaum65 also investigated the coherent wave
motion by applying a non-linear approximation to the Dyson
equation.
In this chapter we review the development of the Dyson
and Bethe-Salpeter equations and.discuss the various approximations made in the solution of these equations.
145
The Dyson and Bethe-Salpeter Equations
7.1
In Chapter 3, we obtained the integral equation which
governs the scattering of electromagnetic waves by a random
G==
(r,-- =(0
r) 0)
=G
G
(0)
-"
(r
-r' '-
=
Gr', G
Qr'
r
-
) + fV d3r'
(r, r
)
In the case of a point source, we have:
-
medium.
(7.1)
where
= wps
Q(r)
(r).
Iterating (7.1)
leads to the in-
finite Neumann series given by:
=
-
G(r, r 0)
0
*
()r, r)
+
G
d 3 r'
(r, r ) +
0
f
SQ(r') Q(r")
d 3 r" G(
d 3r' G
r, r') Q(r')
- G
,
Cr,
=(0) (r", r) +.
G
Ensemble averaging (7.2)
(7.2)
yields
0) <G(r, r 0)> == -G-=(
(r, r ) + fVd r'
(r
f
dr"
G.O
(r,
r
r')
r0
)
(0)
--= 0
-
-=0)--
146
(0)
() C)
+
(7.3)
where it should be noted that all odd order moments of
vanish and all even moments of
Q(r)
Q(r)
are cluster expanded
in terms of the two-point correlation function,
C(r - r').
A convenient method of handling cumbersome equations of the
type (7.3) is through the use of Feynman diagrams.
The fol-
lowing symbols are used in constructing the Feynman diagrams
<G (r, r0)>
(0)
411*
0
C(r -
-
G (r, r0
r')
Evertex over which integration is implied.
Thus (7.3) may be expressed as
++
+
4T~+
1/00000
_
4 1A
147
(7.4)
where, for example:
=
=
dar'ddr" G(r, r' ) - G(r'
,
r")
(7.5)
-
) C(r'
r
(r
-=*-
-s
-
-4 1Th~--
We now define diagrams (minus end connectors) as strongly
connected if it is impossible to bisect the diagram without
All other diagrams are
breaking the correlation connections.
called weakly connected.
The sum of all strongly connected
diagrams defines the mass operator, given by
.
+
+
(7.6)
We note that with (7.6), Equation (7.4) may be written in
+
j---
---
-
+
-
_____+
+
-
-
-
-
the form
(7.7)
148
+
-
--
E[---------+-
(7.8)
_
where (7.9) is the diagrammatical representation of the Dyson
equation for the mean Dyadic Green's function.
form (7.9)
reads:.
<G(r, r
-)>
-Q(r'
Here
Q
f
(, (r + d3r'd3r"G
=G
r"
In analytic-
(r, r')
G(r", r 0)>
(7.10)
is used to denote the mass operator.
In a similar fashion we introduce the symbol for the
field covariance:
-<FQC,0 0 >
]IL
E
G(,r0
P0
0
(7.11)
149
The Neumann series for the field covariance has the following
diagrammatical representation:
ILL -2Z~22
+Ix
(7.12)
where for example
C - r').
= (0)
r', IG 1 )( 0*') 1,,r
= (0) (r , . r
1'0
(r1 -
1,
(7.13)
The sum of all strongly connected diagrams is defined as the
x
=
:,00
0
+
intensity operator:
+
I
dIr d r ' ["3() , Fl) - =G ( ) , F)I
B 1 ''Vl I'
=
::I-
This allows the infinite Neumann series
as:
(7.14)
(7.12) to be written
150
K
(7.15)
*
X
+
+
X
X
6
'' 9(7.16)
L
X
+
X
Equation (7.17) is the diagrammatic representation of the
Bethe-Salpeter equation, which will be given in the next
section in analytical form.
(7.17)
151
7.2
Approximations to the Mass and Intensity Operators
In (7.10),
Q(', r")
denotes the mass operator, which
must be approximated if Dyson's equation is to be solved
analytically.
The most popular approximation to the mass
operator is the bi-local approximation, which consists of
retaining only the first term in the series for
Q(r',
")
(0) (r
Q,
that is,
r") C(r' - r").
(7.18)
Physically the bilocal approximation corresponds to a single
scattering of the mean field as can be seen by substituting
(7.18) into (7.10).
The validity of the bilocal approximation
has been discussed by Tatarskii,64 and Rosenbaum65 has shown
that the bilocally approximated Dyson equation together with
the ladder approximated Bethe-Salpeter equation do not lead
to an energy conserving formalism.
Another approximation to the mass operator which circumvents these difficulties is the non-linear approximation, in
which an infinite sequence of terms in (7.6) is summed.
S~+
+
...
(7.19)
Substituting (7.19) into (7.10), we obtain a non-linear integral
152
<G(r,
r
-
)
<G(r',
(r, r)
r")>
-
+
d r' d r" G
<G(r", r )>
C(r' -
(r
-
equation for the mean dyadic Green's function
r")
(7.20)
Physically, the non-linear approximation accounts for multiple
scattering of the mean field and in this regard is superior
to the bi-local approximation.
The usual approximation made for the intensity operator
is the so called ladder approximation in which only the first
term of the series (7.14) is retained.
That is,
(7.21)
In which case the Bethe-Salpeter equation reduces to:
1E
or:
(7.22)
153
<(
0
)
*(r',
i)>
0
0\
=
V d'rld3 rl' C(r
<
, i9> >')
r0
1
0
-
(7.23)
where
P(r, r ) 2
o
(r,
r ) on.
mean dyadic Green's function.
<G(r, r0 )>,
is the incoherent
154
CHAPTER 8
Mean Dyadic Green's Function of a
Two Layer Random Medium
The study of the mean dyadic Green's function for a two
layer random medium is of special importance in the fields of
scattering, radiation, and diffraction of electromagnetic
waves as applied to optical communications in the atmosphere,
radar backscattering from earth terrain, and active remote
sensing of the terrestrial environment.
It is well known
that the coherent wave motion in a random medium can be described by Dyson's equation, which is an exact equation for
the mean field.
Dyson's equation expresses the coherent
field in terms of a mass operator
Q
which is in the form
of an infinite series and must be approximated.
commonly used approximation to
Q
The most
is the so called bilocal
approximation which follows by retaining only the first term
in the infinite series representation for
Q.
Solutions to
the bilocally approximated Dyson's equation have been the
6 4 ,6 5
subject of extensive investigation in the literature.
However, as pointed out by Rosenbaum,65 the bilocal approximation not only leads to solutions with potentially severe
range restrictions; due to the omission of higher order terms
155
in the mass operator, it also leads to solutions which are
energetically inconsistent with the Bethe-Salpeter equation
under the ladder approximation.
An approximation to the
mass operator, which circumvents these difficulties is the
non-linear approximation
65 in which an infinite sequence of
higher order terms in the series for
Q
is summed.
This
approximation to the mass operator results in an intractable
non-linear integral-differential equation for the coherent
field.
Rosenbaum 6 5 ,7 5 found approximate solutions to the
non-linearly approximated scalar Dyson's equation, for unbounded random media in the limit of large and small scale
fluctuations.
Tsang and Kong58,59 using a two variable ex-
pansion technique have solved the scalar Dyson's equation in
the non-linear approximation for the cases of a one-dimensional
two-layer laminar structure and a three dimensional half-space
random medium.
In the limit of a laminar structure they found
the coherent wave motion to possess two effective propagation
constants.
More recently, Tan and Fung60 also employed the
two-variable expansion technique and solved the non-linear
Dyson's equation for the zeroth order mean dyadic Green's
function in the case of a half space random medium.
Their
vector solution contains only a single propagation constant
for all components in the Green's dyadic.
In this paper we employ the two variable expansion tech-
156
nique to obtain the complete zeroth order solutions for the
mean dyadic Green's functions of a two layer random medium
with arbitrary three dimensional correlation functions.
It
is found that the coherent vector field in general propagates
in the random layer as if in an anisotropic medium with different propagation constants for the characteristic
TM
polarizations.
TE
and
Moreover, in the limit of a laminar struc-
ture two propagation constants for each polarization state
are found to exist.
157
8.1
Zeroth Order Mean Dyadic Green's Function Using the
Two Variable Expansion Technique
Consider a two-layer random medium with boundaries at
z = 0
z = -d 1
and
[Fig. 3.1].
The random medium has a
permittivity consisting of the sum of a mean part
<E1 ()>
and a random part
vanishes.
e
(r)
whose ensemble average
The media in the regions
nonrandom having permittivities
e0
z > 0
and
E
(r)
z < -d
and
E ,
All regions are characterized by permeability
fluctuations
el,
are
respectively.
y9.
The random
will be assumed to be statistically
homogeneous so that the two-point correlation function of the
fluctuations is a function only of the difference in the two
points.
The coherent dyadic Green's function of a point source
imbedded in the random medium satisfies Dyson's equation which
under the nonlinear approximation takes the form
11m
+
where
frd
k2
o)
mllm
o
o
'r
2 mr2 1i 2P' ) C(i - 2) (8.1)
=
E2o1m
and the spatial integration extends over
the layer of the random medium.
The first subscript of the
158
dyadic Green's function indicates the region containing the
observation point, the second subscript indicates the region
containing the source point, and the third subscript indicates
that the dyadic Green's function is the mean dyadic Green's
function.
We introduce the Fourier transforms of the mean
dyadic Green's function and of the correlation function:
i~c*
G
1rn
(r,
r
0
) =
d2 k Lg
____
1
2
(27r)
J
.
urn
(k
.
(r
z ) e
z,
,
49
-r
OJ.
0
(8.2)
-ia
d 3c
r2 ) =
C(r-
A
where
k
-(r
- r2)
e
(8.3)
A
x k
X
.1.
D(a)
+ y k
y
d2k
and
IJ
= dk dk
-
x
Substituting
.
Y-
(8.2) and (8.3) into (8.1) and performing the transverse
we obtain
spatial integrations,
il
+ k{
(k , z, z
-11MlLk , z, ) = -I S(z
)
da dz2 (k - k,
T
d k
dr 00
-
-
ik
+ z
i
+ z
z) - (2r) 2
r 0
d
_-iaZ
(z -
z2
159
(kL ',z, z 2
glm
(8.4)
* 9 1 1 M(k, z 2 , z)
where
k
lmZ
=
[k 2
im
-
(8.5)
k 2 1/2.
L
To solve equation (8.4) we make use of the two variable expansion technique, which has been used to solve for the long
distance behavior of the wave propagation in a random medium
with laminar structure. 58
Following the procedure as in
Tsang and Kong 5 8 ,5 9 and defining a bookeeping parameter,
0
we find that to zeroth order of s
lmz
(k ;
- gllmo (k , z, r
and to order,
32
---
s
z,
,
,
for
-
) = -I 6(z -
,
;
z > Z
( J. ; z,
;
,f
= -2
.
+ z
'3
(8.6)
z)
2
g1 m
ik
3z
2
+ klmz
+ z
i
0
.0
1llmo
,
+ km]
3Z2
s,
160
,(;
ll1MO
0o &0)
0
f -d1
*
gllmo
dZk
'
<
(k ' -
k,
z
2
)
da
dzz2
-00
(kL', z, Ef z2'
2)
9lmo ( .V
Z2'
E2; Zo,
(9)
(8.7)
E = sz,
where
E
= sz,
and the subscripts
o
and
2 = sz 2
1
are long distance scales
which follow the subscript
m
on the mean dyadic Green's function denote, respectively, the
zeroth and the first order solution.
In deriving (8.7) we
have used the divergence relation for the mean dyadic Green's
function in Fourier transform space, which for
z < z0
iki
z > z0
and
is given by
+ z {-+ s
+ sgilml
From which it
*
-i.'Z
gllmo (k ; z, (;
E; Z 0 ,
follows that:
O)
+
..
z0 ,
=
0.
E
(8.8)
161
Zeroth Order:
ii
+ z
-
-
llmo (,
z, (;
z,
y
= 0
(8.9a)
=
-
^ 9 llmo (k,, z,
-
-lml
(,,
E; zo,
E
z,
z.,
,
+ z
)
ik
First Order:
(8.9b)
The zeroth order Fourier transform mean dyadic Green's
function satisfies an equation identical to that satisfied by
the Fourier transformed dyadic Green's function of the nonrandom problem.
This is not surprising since in the limit of
vanishing random fluctuations the zeroth order solution obtained from (8.6) must reduce to the dyadic Green's function
of the corresponding homogeneous problem.
Moreover, the two
variable expansion technique carried to zeroth order, accounts
for the random fluctuations essentially by introducing corrections to the phase of the unperturbed dyadic Green's function.
6(z -
g1mo (k
,tz, E; zo, E) = zz
lm
z
)
Therefore the zero order solution takes the form
9 1 1 mo(k.1
z,
z0 ,f
;
(z
)
> z
)
162
ilmo k,
Lz, ; ,
(z
0O
<
z
)
(8. 10a)
where
,k ,
911mo
0
)
I [A 1 (k
ik
z
e(klmz) elmz + A
()
-ik
2 -L
) e(-k
lmz
z
) e
mz
k
[B 1 (k ,
) e(-klmz) e
+ B2 (k'
) e(klmz) e
I
+
z; z ,f
ik
[C (kL,L) h(k lmz) e
z
+ c2(k, ()h(-klmz
-ik
ik ,z
[D 1
h(-k lmz
kL' Z,
Z;of,
EO)
-ik
+ D 2(k
e
(z >
911mo
z
lmz
z0)
o
h(klmz
z
e
(8.10b)
163
ik
[B (k ,
(k
) e
-ik
+ B2(k
ik
z
) e(-k lmz
,
z
lmZ
-ik
z
lmz
E ) e (klz
+ A 2 (k. ,
lmz
e (-klm
O)
z
lmz
+
[A1 (k ,
)
ik mz z
[D 1 (k ,)
h (k lmz
+ D2 (k
e
ik1
[C 1 (k , EO) h(-klmz)
-ik
e
mz
z
e
-ik
+ C2 (k , g)
(z
The unit vectors
) h (-klmz
,
z
h(klmz)
e
< z0)
e(ky )
z
lmz
(8.1Oc)
and
h (klmz)
directions of the electric field for the
waves, respectively, we next substitute
TE
point in the
and
TM
polarized
(8.10a)-(8.10c) into
(8.7) and eliminate secular terms independently for each
polarization.
This yields four differential equations per
,
polarization in the variable,
for
C1 , C 2 ' D1 ' D 2 .
for
A1 , A , B1 , B
and
After careful manipulations, the differ-
ential equations may be cast into the following compact form:
(k , a) +(2iri)
+n
i[-sk'lmz + pk
lmz
n
, d2k
Rak
s
m s T
(
,
E)[[
f
(k
',
)
p
(
0 = -2ipklmz
'
i[-sk'
f
-s
-,
.
L
d 2 k ' Res 0(k
a
'n
+ pklmz + an
p
i ,
' -
_
-
,g
T~i
a +)
n
lm
z
T (k
)
)
, A
Tl(-k
s
+
k 2mn
d
+
( L
D(k
m,
+
p
T (Res
+
164
0 = -2ipk
T C-
mz~
, p)
p
Res
(
'
(2Tri)
+
S
S
+
(-R L ',
E
2
f
d 2k
f
n s T'
-
K
a.-)
Res $(k
)] +
()fc
d 2 k'
-s
-T
K.
d 2 k,' Res t(k
0
_
i [-sk i
',
T)
-
4
'
-k.
a+
-L
n
+ pklmz + an
p
'
-s
.
s
i[-sk'
lmz + pk lmz + a n ~]
-T (-k, (
L.
p
ZEE
'
(8. 11a)
S
k , a +
(-k , m)
z
^
*
nk
5
lm
(8. l1b)
165
where
,
A (k
ct~
()
T = TM
(8.12b)
()
e (-klmz)
T = TE
(8.13a)
C 2 (k,
E)
h
(-klmz)
T = TM
(8.13b)
B
,
,
) h (klmz)
()
e (-k lmz)
T = TE
(8.14a)
D (k,
()
h (-k lmz)
T = TM
(8.14b)
SB
(ks,
()
e(klmz)
T= TE
(8.15a)
D2 (k,
()
h(klmz)
T = TM
(8.15b)
(k
.
=
2
-
(k, ()
2
(
(=
,
SZ_
(8.12a)
(k,
A
1T
T = TE
(k~,
C (k
a
e(klmz)
166
In (8.lla) and (8.llb) the indices
or
-1,
for either
TE
of
+1
p
and
s
whereas the superscripts
or
TM.
take the values
T
and
T'
Moreover we have performed the
integration using residue calculus, with
denoting, respectively, the
n-th
an+
poles of
the upper and lower halves of the complex
case of laminar structures, for which
(8.lla) and (8.11b) are different.
az
klm P
a)
plane.
-+
C,
az
a
and
N(,
stand
in
In the
equations
Treatment of this case is
deferred to Section 8.5.
Solutions to (8.lla) and (8.llb) have the form:
--
,
)
=
a
T
-
(k ) e
ipx
P
-
p
T T (k ,
=
a
e(k
-
T
-
b T)(k ) e
Jp
T(k )L
T
ipX
(8.16a)
-
-T
a(k
(k )(
(8.16b)
where
a
a
)
T = TE
(8.17a)
h (klMz)
T = TM
(8.17b)
A2 (k I) e(-klmz)
T = TE
(8.18a)
C2 (k ) h (-klmz)
T = TM
(8.18b)
(kd)
(k(
167
B 1(k )
(-klmZ)
T = TE
(8.19a)
D(k I)
h(-klmz)
T = TM
(8.19b)
T = TE
(8.20a)
T = TM
(8.20b)
) e(k lmz)
B2(k
b1
{
(k )
D
In
(k ) h(k
(8.17a) through
)
(8.20b), we have redefined the coefficients
in the right hand side of (8.12a)-(8.15b).
i-dependence of
Al, A 2 , B1 , B 2, C 1 , C 2 , D,
First note that the
and
D2is
exponential in behavior and thus may be combined with their
ik
multiplying exponentials,
e
z
ilmz
We then factor out the am-
dyadic Green's function (8.10a).
plitudes of
A
C1
and
which appear in the mean
which depend only on
combine these with the other amplitudes in
km,
(8.10a).
defines a new set of coefficients which are given in
and
This
(8.17a)-
(8.20b) and are determined through the boundary conditions on
the zeroth order mean dyadic Green's function.
(8.16a) and
(8.16b) into (8.lla) and
0 = 2X Tk ) klmz ap
(k ) + (2Tri)
.a.~~~
Substituting
(8.11b), we obtain
d 2 kL'
E
sm pT '{
168
(k
-
'
k
M
)
Res
a (k ')[b
s
L
-s (kI ')
i[-sk'
+ pk1
+ an
lmz
n
1mz
' -k
Res.(k
n
+a d 2 k'
(k)] +
p
i[-ski
+ pk
+ an
lmz
lmz
n
b
(-k ') [a
,
-T
+!
-s
(-
d k ' Res
k2
lm
- T
b
lmz
0(K
'
-
i
'
-k,
(27ri)
n
+ ca
(-kc )] +;d2 k',
(
- T
(-k ) +
pns
i[-sk'
+ pk
lmz
-b
(k
a
+) z za
(8.21a)
,
Res
a T
p
')
)
-T
a
(k ,
0 = 2X T (-k)
k
J
n
~]
E E
{I d k
2
'
s
-T'I
-j
-( ')
a
(k ')[b-T'I (k
-s iL
i
s
Res
(k '-
k. , an)
i[-sk'
lmz + pk lmz +
nt
n
J.
2
+
k
2
1m
-s
(-T
-b
')
(-k
.
k' Res
p
.
[a
(-k '
bT
s
)
169
,a
-
zz
(-
n'
p
1
(8.21b)
It is to be noted that equations
(8.21a) and
X (K ).
sistent, yielding-the same result for
seen as follows.
In
dot with the vector
(8.21a) let
p(k ).
p
-p
+
-T
a-, (Y ).
and
In (8.21b), let
then change integration variables as
dot with the vector
(8.21b) are con-
'
This may be
s
+
-s
k+
then
-k.,
- finally,
-k ', and
We also make use of the fol-
lowing properties of the spectral density, which are valid for
a large class of physically interesting correlation functions:
(a
)=a(-a ,a )
,
Res @((a,
a
n
=
-a
n+
c
(8.22a)
) = - Res D(a , a
)
(8.22b)
(8.22c)
Upon performing the described operations on (8.21a) and (8.21b),
170
=
-T
k lmz [ap( k L)
-
T (
-p
Z ZI
n s T'
-r T -
(k )
d2 k
'
we find the consistent result:
(
)]
Res
(
'
-T'
( k) -a
(k)] [b
,
[a_
-T
(sk mz
-TI (k ) - b s (
-T
[a
(sk
-
pklmz +
n+
-Tv--Ts-T
(
')][a
') - b)
- pklmz -an
(k '-k
d2k' Res
k
(-k1 ') - b
,
an
lmz im
[z
*
-T
T
+
(sk
]
ap (kL)][z - b T(k
L
..
- pklmz - an
Utilizing (8.17a)-(8.20b), we evaluate
T=TE
and
(8.23)
T=TM.
The results are:
(8.23) for the cases
TE
2r)
-dk
Res
imz
B+a(k
lmz
A 2 (k
')
B
-
+
+)2
-
n
kz
lmz
(k
(k
I a
l
') (kkmz + an+
-
Cos'
B2 (k ')
0 (k '
Ai
j
k lzn
+
171
lmz - an
-
imz
-
an+)2
ni
k
2
lmz
k2
2
lmz sin 2(
k2
(k
-
lmz
+ a
n
+
D 2 (k ')
+
lm
k2
+ a+2
(k
imz
n
lmz
k'2
C 2 (k ')
2
mz
D (k ')
(
(k
-
-
lm
-
(k
xTM
k27
E
(8.24a)
an )zz- k2
d 2 k'
Res
(k
+ an +)
-
, a
+(
lmz
km
lm
(k{im
sin2 Gp
-
')
-
k2
B2 (k
a
+
+)2
-
n
k2
lmz
k2
A 2 (k') B (k
lz(kl
+) sin2-
-
km
(k'
imz
a +)
n
2
-
n
k2
lmz
172
(k'2 k2
cos 2(
') +
-
k
2 k, 2
)
2
+
lm
(k
+ a +)2
n
lmz
k 2 k'
k k ' cos($
lmz lmz ..
(k'j
+ a +)
+
n
.
I I
-
lmz
-
2D2 (k)
k2
(k'
lmz
C2 (kL)D (k')
k*4 1
(k'mz
-
+ a +) 2
k2
+(k2
mz lMz
-
- 2
n
lMz
cos 2 (
')+
kkLZ)
-~
lm
k2
D1 (k ') 'mz k
2C 2 (k
k4
k k ' cos(4
lMzL
k
#
D 1and
2
d 2 kL' Res c4 k
-
k 4Lf1
$'
-
k , an+).
are the azimuthal angles subtended by
respectively.
D2
'
(8.24b)
flL
lm
and
'
Here,
lmz
k
The coefficients,
A 2 , B1 , B 2,
k
C2
,
7
-lk
+
(kjmz - a n+
K
k2
lmz
1M+
+
and
a +) 2 -
-
(kj(k{n-
must now be determined by imposing the boundary
conditions on the zeroth order mean dyadic Green's function.
173
The Zeroth Order Mean Dyadic Green's Function
Combining Equations (8.16a)-(8.20b), together with (8.10b)
and
(8.10c), the zeroth order mean dyadic Green's function may
be written in the form:
(r,r)
limo
o
(2r)
J d 12r
k
2
2
{[e(k
-i(k l
+ A 2 (k )
e(-klmz)
lmz
+ A
) e
i(k
)z
e
i(k1
+
A
)
0
[B 1 (k ) e(-klmz)
+ +xTE)
A
)z
-i (k
0o
elmz
+ B2 (k) e(klmz)
+
G
i (k l
[h (k lmz
+ x TM ) z
e
-i (k
+ C 2 (k) h(-k
1
+
TM)
I
) e
i(k
[DI(kl) h(-km)
e
+ ATM)z
lmz
0
+ AtE )z
IZ
174
) h (klmz)
+ D2 (k
(k
e
+
(z
Gllmo (r,
1
(27)
ro
d2k
+ B2 (k ) e(-klmz) e
[e (-k
lmz
) e
^(k
e (klmz
I
+
i (k
e lmz
e
i (klz+ x TM)z 0
[h (-k lmz ) eilmz
+
TE) z
m
-i (k lm+
h (-klmz)
(
- r
04.
(8.25a)
+ ATE )z
e-i (kmz +
[D1 (k) h (k)lmz
ik
> z0)
{ [B1 (k
-i(k l
+
)
0 11
i (k] mz +
xTE)z 0
+ A 2 (k ) e (klmz)
+ D2 (k
)zTM
lmz
TM)
TM )z
I
i (k
e
+ x TE) z
175
-i(kimz + x TM)z
+ C 2 (k ) h(k mz
e
i.l
0.-
(.L
e
(z
(8.25b)
< z 0 ).
The boundary conditions which the zeroth order mean dyadic
Green's function must satisfy are:
At
z = 0:
z
x
G
Olmo
z x V x G
At
(r,
r
o
limo
imo (r,
Olmo (r,
r) = z x V
(r,Pr)
= zx
G
r 0)
a
z x V x G limo (r,
o
(8.26a)
(r, r 0
x
G
r ).
o
(8.26b)
z = -d 1 :
z xG
2imo
0
z x V x G 21mo (r,
From
) = z x G
(8.6)
it
follows that
llmo
(,r)
(8.27a)
o
r o).
(8.27b)
176
lamo
- k
lin _ gllmo
-k=l
az
-11.
Z-)*Z
ik
z
+ 911lmo -
0
ag
A
) -
limo -
+
lim
LZ-)ZO
3z
+
1mz
(8.28a)
-
0
e (k
-
+
k, -w 1im
k-1
allmo
lim
z*z0
3z
0
e (klmz ) = -1
where
k
E k /k .
(8.28b)
In region 0, the Fourier transformed mean
dyadic Green's function assumes the following form in order
to match the boundary conditions at
ik
z) =
i(k
e
lmz
e(k
) e
OZ
(B1 (k )- e (-klmz)
+ xTE)z
-i(k
o + B2 (k ) e(klmz) e
+ xTE
)
oz,
z
+
ik
r2 h2(k oz ) e
z
OZ
[D 1 (k) h (-k
1linz
) e
i (k
+ +Xo
TM
)
g
z = 0:
177
Similarly,
)
h(k)lmz)
X
)TM
)
+
+ D 2 (k
e- i(klmz
(8.29)
in Region 2, the mean dyadic Green' s function is
taken to be:
-ik
TE
0 + A2
(k
e
z
A
fe (-k lmz)
2z
e
e (k)lmz
-i (k
+
lmz
xTE) z
0
+
lmz
3 e (-k2z)
)
i(k
e
=
)
921mo (k
z, z
+
,
z
2z
h
(klmz)
e
lmz
e
+
TM
lmz
z
+ +Xo
TM
)
i(k
[h(-k
)
e
-i(k
)
+ C 2 (k
)
-ik
r4 h(-k 2
0
.
(8.30)
Applying boundary conditions (8.26a)-(8.28b)
to (8.25a),
(8.25b),
(8.29) and (8.30), results in the following equations for the
unknown coefficients contained in (8.25a),
and (8.30) :
(8.25b)
(8.29)
178
r1 = 1 + A
2
(8.31)
k
(-1 + C 2
0
k
klm
(z
= klmz
k
(1
(8.32)
)
2 =
A2
(8.33)
ki0r 2 = km(i + c 2
(8.34)
-
)
r
)
oz
ik 2zd
F3 e
-
[1 e
1E3
k 2F 4
B2 -
ik zd
e 2z 1
-iK
By e
k
e ik2zd 1
lmz
[Bm e
=
iK
B2 e
d
im
(8.35)
TM d
lm 1
iK
D
e
- iKTEd
1
ik d
e2z 1 =k[De
lmi
A2B
d
m+
klmz k2
- [D
k2 z kim1
-iK TMd
lm 1 + D
iK
B
2
e
e
e
d
lm 1
(8.36)
d
m 1
iK TMd
lm 1
(8.37)
(8.38)
(8.39)
-
IE4
=
2ikx lmz
D2 -
D
-
1
2iklmz
(8.40)
(8.40)
179
where
KT
1M
k
lmZ
.
+
Equations (8.31)-(8.40) are easily
solved for the unknown coefficients.
The resulting zeroth
order mean dyadic Green's functions are:
S(r, r
limo
0
(27r)
1
d 2 kL e
2
2 ik, m
+
xTE)z
[e (klmz) e
)
D2 (k
+ R10 (k ) e (-klmz)
)
i2K
[R 1 2 (k
e
-i(k
)z
I
d
m 1 e (-k lmZ
i(k
e
lmzo
e mz + X T) 0
+
+ e (k)lmZ
+ A
-i (k
em
(k
+ A
)z
) e
1[h (k
lmz
F2 (k )
-i(k l
+ S 1 0 (k ) h (-k
) e
+
ATM )z
I
+ A
) 0
180
)
(k l
)
i2K TMd
[S 1 2 (k
e
I h (-klmz
e-i (klmz + X TM) Z0
r
Gllmo (r,
(27r) 2
I.
e
)
(z
(r
0
r
.1-
e
>
(8.41)
z0)
1
0
2ik1
D2 (k)
d
im 1 e (k lmz)
+ TE)
-i (k
+A
)]z A
lmz
e
] [e
+ XTE )z
i (k
e
-klmz
lmz
e
+ xTE) z
i (k
0
lmz
)
e(klmz)
I
ij
d2k
i2K
[R1 2 (k4
A
i (k
+ X TM) z
e mz
TE z
O
)
+
-i (k
elmz
e (k)lmz
+
)
+ R 1 0 (k
)
S12 (k
h (k)lmz)
+ x TM)
i (k
elmz
)
F 2 (k
i2K Td
im
e
-i(k
+ h (-k)lmz)
.+
S 1 0 (k)
10m
e
h (klmz
+ A
lmz
e i (klmz
zTM)z
] [h (-klmz
+
TM)Z]
ei (k lmz
I
(z
+ x TM) z
< z0)
0
(8.42)
181
d2k
1
-L
e
-
(r
G0mo (r,
2ik 1
i2K imdi
e(koz )R
D 2 (k )
O
12
(k ) e
+ )TE)z
k
I
) e
i2K
(S1 2 (k ) e
TMd
lm 1 h(-kz)
h(klmze
e
i(lmz +A)z
e
e(-klmz
m
Y 10 (k)
m+
k9 F2 (k )
A
h(k o)
z
i (k
+ +xTM)
A
lmz
)
imz
+
h+ ATM )z
i (k
1mz
+ h(kimz) e
'(8.43)
(r
1
f d2 k e
(2W) 2
L
G
(r,r)=21nmo
0
X1
(k)
-r
) e
.L o.
-ik z
2z
2ik1
i (k
mz + X
) 0
)
e (-k2z) [e (-klmz
D 2 (k
z
z
TE
(
X
X10.( _)
+ e (k
eik
e
+ AT Ezz0
-i(k
+ R
(k ) e (kz) e
lmz
10
lmz
k
Y 1 2 (k
+ -k2
F2 (k
F 2 (k)
A)
A
+ S10 (k ) h (klmz) e
+
Ai(k
h(-k 2z)[h(-k
2ziz
)e
ATM)z
1mz
+ xTM) z
-i (k
lmz
If
(8.44)
182
where:
f1
d2k
TE ( L
ik
n
imz
cos 2 (,
k'
-
(k'
0 ')
D (k')
lmz2.
Res D(k
'
-k,
+ an
n
+ a +)2
(k'
- mz
n
R 10(k ') R1 2 (k ')
(k'lmz
e
+
a
a
k2
-
lmz
i 2 K , TEd
lm 1 (kjm
F
-
Un+
-
k2
-
an)]
- k2
lmz
n )2
sin2
(k
+ an
+
(
kjm
'
k lm F 2 (k')
-L
) (k'
- M:
S10 (k,') S 2(k ')
(k'
imz
f*
(
iT
ik lmz n
d2k
1
n
-
imz
k2
lmz
Res +(
z
' -
(k
lmz
k,
a +)
n
+ an
)
-
2
k lmz sin (
2
i2K'
d
im 1(k
e
n
'
xTM
+ a +
lm
k'
R 1 0 (k ')
D (k ')
mz2
R1
(k
1
(k')
imz
-
(k
mz
+0(C+ n
i2K'E d
m 1 (k'
e .
lmz
- k2
an+)2
n
lmz
-ka
lmz
- a+
-
k 2.
}
(8.45)
183
+
1
k'
4
k lm
(k4Lkgcos 2 (
F2 (ka-
'
lmz
4') + k2 kv 2 ) (k
-
1~
S 1 2 (k
1)
.
k k'cos(o - ')
2
i2Ki,TM dT1
im
e
+
')
+n) + 2k'k]
2
i+ an+
n
(I,
S 1 0 (k
++
kmz km F2
k
2kIjI
k! c'
cos(4-~)-(j
-)
+
- a
(kI
lmz
n
L
k
-
k
lmz
2L
k
d 2 k'
E
-
4
um
)z
'
k -k)
- k
ltnz
Res <D (k'
n
kl= Cos2(
-k
,
n
(8.46)
.
.~
(k:
)
-
and
k.
-
k.
lz
kz
k.
+ k.
LZ
(8 .47a)
]z
(k)
E
3
.k.
lz
.ki.
+
3Jlz
X.
(k ) = 1 + R
-
e.k.
1 3z
(8 .47b)
c kj
:Ljz
(k .
)
S
(8.48a)
184
Y i(k
) = 1 + S..
(8.48b)
D2(k
1 + R01 12 e
i 2 KTEd
lm 1
F 2 (k )
1 + S
0 1 S1
e
i2KTd
lm 1
K'
1m
k
lm
+ X
In (8.47a)-(8.49b)
where
(kj)
klz
E1
and
respectively.
2
k ')
i
T = TE,
and
j
(8.49a)
(8.49b)
TM.
take the values
are to be interpreted as
Equations
(8.50)
0, 1
klmz
and
and
2,
Elml
(8.41)-(8.49b) represent the complete
zeroth order solution for the mean dyadic Green's function of
a two layer random medium in the non-linear approximation to
Dyson's equation.
185
8.2
Effective Propagation Constants for
TE
TM
and
Modes
The two variable expansion technique has been applied to
Dyson's equation in the non-linear approximation to obtain
the zeroth order mean dyadic Green's function for a two layer
random medium.
The results as given by (8.41)-(8.49b) indi-
cate the anisotropic effect of the random medium to vector
fields resulting in effective propagation constants which in
general are different for the characteristic
modes.
TE
and
TM
Physically, this is not surprising if we consider wave
propagation within a random medium with distinctive vertical
and lateral correlation lengths.
Waves with vertical polari-
zation will experience an effective decay differing from that
However, for correla-
of waves with horizontal polarization.
tion functions with azimuthal symmetry, and for wave propagation along the. z-direction, we -expect idential propagation
characteristics for both the
TE
and
TM
This ob-
modes.
servation is substantiated by the results given in
(8.46) by letting
k
-+
TM
X
in which case
0
(8.45) and
TE
XT
To illustrate the results given by (8.45) and (8.46), we
take the spectral density to be
6k 4
(z
4=m
4
2z
Z
P
e2P
2
/4
(8.51a)
186
which corresponds to a correlation function of the form:
-~r
1
C (ry -
I- 2/Z 2
-
r2
=
kl4
-
z2
-Z
e
and k denote the lateral and vertical correlation
p
is the variance of the fluctualengths, respectively and 6
Z
where
tions.
Taking the residue of (8.51a) at the pole
(i/Z),
where
6n,l
6 n,l
is the Kronecker delta symbol, we find
Sk 4
i)=
Res
an+ =
z
2
Zz
m P
e
2 /4
P.
(8.52)
8 7r2
Substituting (8.52) into (8.45) and (8.46) yields
x
(k
26
=
-k
km
8
k k
-
2/4
k'
2
z.
2
/4
e
d2k
-m
22.
7Tklmz
'2
2
P
cos(
-
')
2
cos 2
{ mz
+ kmz sin2 (
k2
F (k ')
lm
2
J_
)
M (k'
2
-
M
(k ')
2
(
TE -
(8.53a)
187
and,
4
Zk
=p
lm
8 7k lmz
k k 'p
2
.1*
2Z
cos (
')
lm
(M 3 (
+
k4
lmzlm
k 22 2/4
L p
F
2
(k ')
-
2
lmz sin (p
k2
k'
2/4 -
e
2
9
2
e
-k'
d 2k'
-
2,
TMX
(kQ)
')
k'
z
M 1 (ks')
K.I
D2
+ m4
of)
))
(8.53b)
where
~pJi
m (k
- R10 (k
') 3kjmz
(
i2k'
d
lmz1
R 12(k
')
k2
lmz
lmz
(8.54a)
-2
lmz
-
lmz
k mz
+
e
-
'
)
lk lmz
M 2 (k
-
k lmz
+
2
-
k2
lmz
S1 0 (k ') S12 (k ')
188
i2k'Mzd
e
i
klmz-
-k
kimz -
10 (k') S12 (k') e
M 3 (k
(8.54b)
Z'
1z __
i2k'
d
lmz 1
4
2k.k
k k
cos-
(kv 2 k2
cos2 (
+')k k'2) k
+
-
k2
k'
lmz
lmz
(8.54c)
In (8.53a) and (8.53b),
and
k'.
cos $',
$
-
$'
is the angle between
If we make the transformation
k ' = k lm sin
portion of
e'
sin $'
k-space, we obtain
k ' = k
x
k
sin
lm
e'
and neglect the evanescent
d2k
=
k2
e'
cos
sin e' dQ'd$ ',
where the angular integrations extend over the hemisphere defined by
0 < 4' < 2w
and
0 < 6'< r/2.
The
may be performed directly.
However, the
e'
only be carried out numerically.
*'
integration
integration can
The numerical results are
illustrated in Figs. 8.1, 8.2 and 8.3 where we have plotted
FT = 21m[XT (k )]/k"
-L lz
versus angle and frequency for
T = TE
189
waves.
In Fig. 8.1 we have plotted
of propagation angle,
6 = .1
for a
rTE =.rTM.
6,
at
10 GHz
Z = .1 m
= .01 m,
thick random layer.
40 cm
as a function
and
Note that at nadir
This follows froni the azimuthal symmetry of the
greater than
TM
Z
with
correlation function used.
the
r
r
r
At large angles,
becomes
indicating a grater scattering loss for
,
wave than for the
TE
wave.
This is because
Z >
which presents a greater scattering cross section for
TM
waves than for
Z
TE
waves.
To illustrate the case of
,
TM
and
> Z,
TP
we have plotted in Fig. 8.2
40 cm
= .009 m,
Z
angle, with
> rTM
rF
and
at
as a function of
6 = .1
and
for a
It is interesting to note that in
r
decrease with angle with
TM
wave.
random layer.
maximum around
2
We have taken
Both
r
27 GHz,
and
TE
In Fig. 8.3 we have plotted
as a function of frequency for a
20*
5 = .0005.
10 GHz
indicating a greater scattering loss for the
wave than for the
T
at
t = .0004 m
thick random layer.
this case, both
r
FT
P
p
=
.95 m,
40 cm
thick
2 = .002 m
and
are seen to exhibit a broad
and a slow decay at higher frequencies.
This behavior is due to the effect of resonant scattering, upon
and
X
TM
(k )
.
TE X (k ) ,
-
the propagation constant corrections,
190
)}
2Im{h
kj"
.17
.15
TE
.13
.1S114
.09
20
10
30
40
SO
60
8 (DEG)
rP
and
T
as a function of angle at
Figure 8. 1
10 GHz
191
m(X TO)
1.0
-
.9
-
.8
.
1z
TE
.7
TM
.6
.5
-
21
.4
I
20
10
30
I
40
I
50
I
60
8 (DEG)
STE
and
rTM
as a function of angle at
Figure 8.2
10 GHz
192
1.0
.8
.6-
TE
TM
.2
5
10
FREQUENCY
rTE
and
r
20
15
25
30
(GHz)
as a function of frequency at
Figure 8.3
200
193
8.3
Comparison with Scattering Coefficients of Radiative
Transfer Theory
It is of interest to compare in the half space limit the
imaginary parts of the effective propagation constants
as obtained from the zeroth order solution to Dyson's
XTE
equation with the scattering coefficients
Kh(e)
as given by the radiative transfer theory.
space limit of
(8.53a) and
TE 6k~42.
2z
im p
87 cos a
Im[BTE ]
k
g
2
2
lm
-
f
Taking the half
(8.53b), we obtain
do'
(sin2 e + sin2 e'
-
2 sin
e
sin e' cos($
2
e
[l + k
(1 +
=
2 2 (cos 2 e +
k2
2
im Z (cos e + cos e')2 [1
{cos2 (4
Im[X T
63
K ve)
and
-
and
XTM
-
c')
kim2
87 cos e
+ cos 2 e' sin 2 (c -
do'
+
cos 2 ')]
k 2im Z 2 (cos e
4')}
-
cos
6')2
(8.55a)
194
k2
-lm
2
z
p
(sin2 e + sin 2 e' - 2 sin e sin e' C()
-
4'))
4
e
[1 + k2 Z2 (cos
lm
e + cos
2
e )
+ km 92 (cos
][1
ha
e
-
-
41))
cos
z2
(0
cos 2 6 cos 2 6'
,
+ 4k 2
2
$')
2
+ cos 2G sin (0)
sin e sin O' cos(
(8.55b)
-
+ cos 2 e cos 2 e' cos
-
{[1 + k{ Z 2 (cos 2 e + cos 2 e')][sin 2 e sin 2 eI
The scattering coefficients as defined in radiative transfer
Sk 4
Tr
I.
dO' [D(', $ ';
lm
2
Kh(6)
2 +
[(v'
6k4
K
(8)
-
+
c(7
T
im
2
-
(h'
e, 4)
8,
o',
$'; 8,
$)[(h'
))]
(8.56a)
I.
4';
+ $(D
h)2]
h
dW' {t(D', O'; e,
8',
63
-
theory are given by Tsang and Kong:
(h'
V)*
+
(v'
v)2
+
- v)
2
(v
*
7r -
e'
V) 2]
(8.56b)
195
$' ,;
e,
k
2,
2
-
p
e
22,
2,
(sin 2 e'
+ sin26
-
@(D'e ,
$
where
2 sin e' sin e cos(O - $'))
4
47 2 [1 + k 2 Z 2 (cos 6 - cos 6')21
im
(8.57)
(v'
= sin26 sin 2 e + cos 2 e cos 2 6' cos 2 (
V)
+ 2 sin e sin e'
(h'
-
(v'
*h)2
2
(h' - h) 2
cos
e cos e' cos(,
= cos 2 e sin 2 ($ -
= cos26' sin 2 (4,-
= cos 2 (4 -
4,')
$')
0').
Combining (8.56a)-(8.58d), we obtain
-
$')
-
,)
(8.58a)
(8.58b)
(8.58c)
(8.58d)
196
6k 4 2 22
lm p
4T
-
(
Kh
k
2,
1*
2
(sin2 e + sin 2 e' -
Z2 (cos 2 e + cos 2 6')]
[1 + k
[1 + k 2
9 2 (cos 6
im
ha
cos e')2][1
-
-
[cos 2 e' sin2 ()
6k 4
K (6)
k
(cos
e + cos e')21
(8.59a)
$'1)]
2
(sin 2 e
+
sin 2 6' -
e
-
cos 6')2][1
4
[1 + k im
-
2
z 2z
f d'
7r
=
2,
kim
4') + cos 2 (4)
2
(cos
{[1 + k 2
lm
2 sin 6' sin e cos($ - 4'))
+ k2 k2 (cos 6 + cos e')23
un
z 2 (cos 2 6 + cos 2 6')][sin 2 6
sin 2 6,
+ cos 2 6 cos 2O'
cos 2 ($
+ 4k2
cos 2 e' sin 6 sin 6' cos($ - $')}.
Z2
COS2e
-
e
-0))
4
+
e
2 sin 6' sin e cos($
$')
+ cosz e sin 2 (4
-
V))]
(8. 59b)
197
Comparing (8.59a), (8.59b) with (8.55a),
(8.55b), we find that
2 Im[k TE
cos
e
= Kh(e)
(8.60a)
2 Im[X TM
cos
e
= K V(6).
(8.60b)
It is clear from (8.60a) and (8.60b) that the scattering
coefficients as defined in the radiative transfer theory are
related to the imaginary parts of the effective
TE
and
TM
propagation constants obtained in the zeroth order solution
to the non-linear Dyson's equation for a half space (or unbounded) random medium.
In the case of a two (or more) layer
random medium the terms in (8.53a) and (8.53b) which contain
Fresnel reflection coefficients, are present in the effective
propagation constant and represent coherent effects due to
the presence of the bottom boundary.
198
8.4
Half-Space Limit-Comparison with the Result of Tan and
Fungy
As another interesting case, we consider the limit of
small scale fluctuations for a half space random medium,
characterized by a correlation function of the form:
where
e_L
./ime
- 2Iz1-z
- i1
211(
6k
- - r2)
k
.
)
C(r
= Re(k
(8.61)
The spectral density of this correla-
tion function is:
a
2Z
6k iAZ
lm P
O(D ,
L )=
4..
z
27r 2 (1 +
29
p
2)3/2
(8.62)
(1 +
z
2
2)
Substituting (8.62) into (8.45) and (8.46), and retaining terms
through first order in the vertical correlation length we obtain, the leading corrections to the propagation constant,
1
.
k
A TE
i
lm
3k1
(8.63a)
199
6k 3 Z
TM
22
k4
imZm
k
4k
6ZSk+ im
P (k+k2
im
P
2
,
k' 4
2
6k
lm
lmz
lmz
(8.63b)
We again observe that in the limit of small scale fluctuations
the propagation constant corrections as given by (8.63a) and
(8.63b) are distinct, but reduce to an identical result for
propagation in the
The results (8.63a) and
z-direction.
(8.63b) should be compared to those of Tan and Fung,60 who
for the same correlation function,
(8.61) obtained a single
propagation constant correction for all components in the mean
They found the zeroth order mean
dyadic Green's function.
g.1J.(u,
v, z, z') =
+ r. .(u,
[C. .(u, v)
13
(z
e
-
dyadic Green's function to be:
v) e-iy'[Z + Z']
(8.64a)
with,
k
y ' (u, v) = y
1 -
-
(u
2
+ v
)
_
4
a.2
(k
'
)2
(8.64b)
ka
L- 4y2
y= (ka2 - u2
2
2)V21/
2
(8.64c)
200
where
ka
is the mean wavenumber,
a2
is the variance of
fluctuations and the transverse Fourier transform variables
are denoted by
u
and
v.
Comparing the second term in the
square bracket of (8.64b) with the first term of (8.63b), it
is clear that Tan and Fung's res'ult represents the lowest
order correction to the real part of the
constant.
The terms of order
k 3 2.
lm p
2Z
TM
propagation
in (8.63a) and (8.63b)
correct the imaginary parts of the propagation constants and
account for the decay of the coherent field as it propagates
in the random medium.
In the case of laminar structures, with
k MP
-
co,
it
is found (Section 8.5) that additional secular terms arise
in the first order equation and that elimination of the secularities results in two propagation constants per mode of
polarization.
The existence of two propagation constants also
has been found in the case of scalar wave propagation in a
half-space random medium
ture is taken.
59
when the limit of a laminar struc-
However, in the limit of an unbounded, laminar
random medium, we recover one propagation constant per polarization in the mean dyadic Green's function.
201
8.5
Special Case of a Laminar Structure
In the limit of a laminar structure, we have
(D (a
2
) = (27)
,
5(a )
(8.65)
Z)
Substituting (8.65) into (8.7) and eliminating secular terms
results in four coupled differential equations per polarization to be solved for the
0 =-2ipk
1
dependence of the coefficients:
(kR,
k) - (27r)2 (27ri)
--
Z
-TSs
k ,_
, ()
i[(p
-T
Ss
_) - (k,()
+
-
(-
(2Tr)
2
-
_-T
L(
s)k 1mZ + a n
--
) a-
k ,
i[(p -
s)k1
(27ri)
- T
- U)
Z Res D(an
n
T
+ a,
-
- T
a s (k
)
n,s
I
Res
D(an+
,
202
pT
a
(-k
()[a
(k
)I
Ian
- T -
- T(k
P
,
(k
-_
4-
-P
,
(k
a
ian
p
T
(
,T
-s
s
i[(p
(2n) 2
(27ri)
)
,
-s)klmz
-T (
T(RL EC-k_
- T
E) -
(K
i[(p
-
Z
oo2-
(2.)
2
p
(8.66a)
(2Tri)
-k
-
-
T(
a
T
(
+
n
2ip k mz
-
n
+ an
-TT (-k
L
P
s)klmz + an
Res
(a
n
-T
-
-T
)[a
-k-
la n
)
-T
-
-
+
0 =
zz
(a
)
lm
Res
.
i(2)
k2
-
+
(-k
E Res (D (a +y
n.s
203
-T
--T
-
T
ian
i(2) E Res (
+
lm
k2
-T
a
where
(8.66b)
zz
n
and
example with
T
+
are given by
-T
T = TM,
(8.12a)-(8.15b).
For
and after careful manipulations, the
coupled differential equations to be solved take the form:
0
=
-kmz
C
31 + Cy Z Res D(an
3
1 n
4Lr 3
kL 2-
1
+
I -3_++
2D1
i(2klmz + an)+
ian
i (2k l
+
(2w7) 2 k
)}
+
in
2
lm
(k
j
2
-
k2
lmz-
(8.67a)
$
+ D2 Z Res
2
1
(an) FD 2 C
Ln
n
i (2klmz + an+}
k
21
-3_
+
n
+
+
C D
+
+
i (2k
+ a n)
2
-
0 = k1
47 3
I(27r ) 2 k 2 (k 2
lm
.
-
k2
lmz-
(8 .67b)
204
aCr
-3
2 + C ZRes (D(an) D C
21
2 n
3t
+ CDD ---- +}
21
ia n
n
1
{in+
i
(2klMz + an
k 2
.L
i(2k
+
lmz
4ir 3
lmz
+ a
n
(27r) 2 k 2 (k 2
lm I
)
k
0 =
-
k2
lmzj
(8.67c)
k lmz
+ D
47r 3
E Res
1n
3E
1 -+ +1
n
D C
2 1
{
-3
i2klmz + an
ct)
k
2
kim (k
2
+
+ C 2 D1
(a )
-
0 =
ian
i (2klmz + Cn
(2 7)
2
.1.L
-
2
kI
lmz
(8.67d)
Solutions to
C
(8.67a)-(8.67c) have the form
E gn(MTM + M)
,TM, e
n in
2
= C1 e
TM
c 2 c2
e
T
=
n(N TM+ N TM
n in
2n
(8.68a)
(8.68b)
205
D, =
TM
1
+ N
E g(N
2n'
n m in
e
)
LTM
(8.69a)
2
D
D2
L TM
1
(M TM + M
I~n
n
)M
2n
(8.69b)
TM
C.,
where
= TM
1
1
i2lmz +
-n
TM -3
.
2
MTM =
1
in
-3
L
2n
Here,
-
LT
. (2klmz + an
2
2 k{
(k
2 -
k2)
-1
(8.70b)
k
2
(k
2
2+
1ian+
N TM
g
_
n+)
(8.71a)
(2r)
i (2k lmz + an+)
1
1TM
+
NTM
= TM
(27r)
1
+
2n
(8 .70a)
+
in
+
MTM
ian
i (2k
= D2
TM
2
k2
-k
(8.72b)
(8.73)
(an).
Res
Here TM CD
+ an
lmz
2
=
C 2 D.
A similar set of equations
206
and corresponding solutions follow from (8.66a) and (8.66b)
for
T = TE.
Substituting these results as well as
(8.68a)-
(8.73) into (8.10a) and (8.10b) and applying boundary conditions (8.26a)-(8.28b), yields the zeroth order mean dyadic
(- -
=
1
)
i(r
d2 k
G lmo(r, r )
llmo
o(27r)
4e
-i(k
1mz
e
xTE
U
+
(elmz
+ R 10(k ) e(-klmz
)
D 2 (k
)
[e(k
1
2ik1
2
i(k
1
-r
)
Green's functions for a laminar structure:
+ e (k
+ xTE
D
)m e
([R12(k
) e
F 2 (k)
U
i(k
lmz
lmz
0
+
TM
u )z
lmz
-i (k
+ S 1 0 (k
i(k
TE)
imz
[h(k
+
+
-i (k1
)m e
) e (-k
i 2 kTEd
1m 1
h(-k
) e
e
+
x TM
D
I
12
(k ) h(-klMz
+ xTE)
D
0
207
i (k
+ ADT )z
A
+
mz
0 + h(k
im
) e -i (klmz
+ +A0)
TM
)
i2K TMd
e rn
lmz
r0) =
limo0(2wr)
(-
f.*
d kg
2
2
(8.74)
(r.
e
D
(k )
(k )e
12 J..
(k
-i (k
+ e (-k
lmz
) e
lmz
)e
(k
)e
+ xTE)z
u
] [e (-k
(klm
e
+ xTE)
lmz
l
(k
) e
lmz
) e
+ x TE
u
lmz
0
0
i2KTMd
K1 1 I
(k
1 mz
+ ATM)z
D
lmz
F 2 (k)
+h(-k
D
m
[S 1 2 (k ) h (k
+
i(kz + xTE)
D
lm1e
Ai
lmz
-i (k
+ R1
1
r
2ik1
i2KTEd
(R
-
)
Gllmo (r,
> z0
)
(z
) e
-i (k1
lmz
+A TMz
u
A
] [h(-k
) e
i (k
+ ATM)z
u
0
lmz
208
+ xTM)z
-i (k
D
lmz
(z
1
< z0
(8.75)
ik
.1*
z
d 2k
(27r) a
.1
X1 0 (k) ^
e (ko ) [R (k
oz12 .L
D 2 (k
2ik m
i2K
e
lmz
d
i(k
+ xTE z
o
D
lm 1 e
lmz
lmz
u
e
0]
k
+ -i
ko
(k
Y
10 (kF2(k)
h(k z
i2 KTMd
i(k
+ XTM)z
lmz
D
0
lmi e
-i(k
+ XTM)z
0
u
lmz
)mze
+ h(k
1
G2lmo (r, r)
(27r)2
f
-
r
-ik z
) ei2z
e
2ik 1
i(k
e(-k 2 ) [e(-k
)
D 2 (k)
(8.76)
(r
d 2k
)
x12(k
X
12(
h(-k
)
)
[S 1 2 (k
TE
)
+
-i(k
lmz
e
+ e (klmz)
)
)
)
e (-k
)
)
G Ol
Olmo (r, r 0
0
)
+ S 1 0 (k ) h(k)lmz) e
lmz
e
+ ATE) z
lmz
u
0
209
+ R10 (k.
k
+ xTE)
-i(k
D
e
imz
)
lmz)
Y 1 2 (k ) ^
i(k
F (k ) h(-k
F2( L
k+
k0
2
)[h(-k
-i(k
+ S
0 (k.)
h(k
) e
1
) e
+ xTM)
u
0
+ xT)z
imz
D
0]
(8.77)
where
g(MT
T
U
n
T
D
T
T
ln
2n
n
g(NT
n nln
TM
L1
1
2iklm
TM
2
+ NT
2n
(8.78b)
(8.79)
F2(k)
S1 0S 1 2 e
2ik
(8.78a)
i2K
d
lmi
F2(k)
imz 2 -i.
L TE
(8.81)
)
2iklmz D2(k
(8.80)
210
i2 KTE d
lmi1
LTE
10 R12 e
101
(8.82)
JjU
2
Klmz
L)
The two effective propagation constants per polarization of
a laminar structure follow from (8.78a) and (8.78b).
S
T2
(8.83a)
) = klmz
(KL) = k
+
XD
T
(8. 83b)
211
CHAPTER 9
Modified Radiative Transfer Theory for a
Two-Layer Random Medium
Modified radiative transfer (MRT) equations appropriate
for electromagnetic wave propagation in bounded random media
are derived from the Bethe-Salpeter equation in the ladder
approximation together with the solution to the non-linearly
approximated Dyson equation.
The MRT equations are solved in
the first order renormalization approximation to obtain analytical results for the backscattering cross-sections, of a
two-layer random medium with arbitrary three-dimensional
correlation functions.
The coherent effects of MRT theory
are illustrated and comparisons are made with backscattering
cross sections obtained in the first order Born approximation
to the wave equation.
212
9.1
Derivation of MRT Equations from Bethe-Salpeter Equation
The Bethe-Salpeter equation for the dyadic field covariance,
under the ladder approximation may be written as
E
1
- E
1
d 3 r 2 C(r 1
(r, r
+ <Gllm(
where
f
d 3 rl
=f
*(r')>
r
1 1(r
( ,r)
2
E1 *
Elm.= <E >
and
electric field in Region 1 (Fig. 3.1)
Dyson's equation.
r
'
1
(r2
*
E1 (r) - E lm(r)
(r)
2 ){ 1 1
(9.1)
r2
is the mean
and is a solution to
The zeroth order mean field propagating
within the random layer, due to an incident plane wave
E
,
<1()
takes the general form:
=[E
.ue
[E
. e
+
e(k
.)z+ E
h(k
.) + E
d
.de
inhiz
inie(-k1
ik~
.)]i
e
.)]i
e
+
E lm(r)
ienhiz
in vi
vuii
lmzi
. e
vdi
viz
iki
h(-k
(9.2)
- rs
213
where the subscript
i
denotes that a quantity is to be
evaluated at the incident wave vector angles.
The unknown
amplitudes which appear in (9.2) are determined by matching
boundary conditions on the mean fields to zeroth order, and
The effective propagation constants
are listed in Appendix B.
'n
and
are defined in (8.50).
nh
We first decompose the incoherent field into a spectrum
of upward and downward propagating plane waves:
s 1 (r)
=
d 2
E
e.
-i5'
+ Ed(z, 5 ) e
(z,
e
lmz
z
(9.3)
m
where
hu (z,
d
) e(
d
imz) + E(z,
d
L ) h( S
)
.
)=
Eu (z,
(9.4)
In (9.3) and (9.4),
tained in
E
(z, $_)
lmz
and
=
Re($
lmz ).
Ed (z, S )
characterized by the quantities
{Im(lh) }-.
kv
The
z-dependence re-
is a long distance scale
=Im)
()}
and Zh
Therefore-, two points can be close together on
214
Zv
by
lm
Zh
or
lm
scale and yet far apart on a scale characterized
z
(Z
being the vertical correlation length).
z
z
We will assume that for points
Z '
the
h
<Ej
(z, a)k
<jd
(z,
<s.
ju
(z,
<jd (z,
close together on
z'
scale:
=
(
-
) Jjku(z, z',
(z, z',
))Jkd
)
c ) E*(z',
kd
a ) E
j,
where
and
k = h
(z9,-
or
)> = 6(
-
)>
-
..
v
=
(
) J.(z,
i
9
)
'(9.5a)
c)
(9.5b)
z',
c)
Jjkc2(Z, z',
)
jkcl
(9.5c)
(
the
(9.5d)
for horizontal or vertical polarizations.
Thus incoherent fields with different transverse directions of
propagation are uncorrelated.
However, for a given transverse
direction there exist correlations between upward and downward
propagating incoherent fields due to the presence of reflecting
dielectric interfaces at
z -= 0
and
z = -d .
tions are represented by
Jjkcl
and
Jjkc2
These correla-
in (9.5c) and
(9.5d).
Combining Equations
(9.3) and (9.5a)-(9.5d), the left hand
side of the Bethe-Salpeter equation (9.1) may be written as
=
f
e iimz(z
-
z')
r
+
(z,
-iS'
e
d 2 Q$
e
{'f u(z, z',
+ rd(z,
e mz (z
-
z')
i6 Iz(z + z')
mz
e
z',
-i+
e
z ,'
+ Pc2(z, z',
a
)
s*(r')>
<sr
)
215
(z + z')
(9.6)
}
mz
where
) = Jhhu(z,
z',
+ J (z,
e( jmz)
) e(5mz) e(lmz)
z
(amz)
) h(Smz) h
h( mz
+ J vhu(z,
z',
+ Jhvu(z,
z',
)
z',
h(almz) e(Simz)
)
U(z,
(9.7a)
Fd(z,
z',
+ J vvd(z,
=
Jhhd(z-, z',
z',
) h(-a
S.)
mz)5
e (- lmz)
h
-a mz)
e
mz)
(PL "6)
Lz
)ZL
c+
T4 ( z
oz )
5 );g
ZUr[
T4 '1Z
.5%'
(ZUIT
T
ZUIT5
ZUIf
-
ZUIT
(
ZUT5)9
++
IZ))A
~( 5)
%e,.
(T 5
IZ)' Z
'z
D
=T1
r
I &1 Z
'z)
LI
(O'L 06)
zuIT
ZUITS
'a
(ZUIM
lz) TOIA
1
a 1
(ZUIT5)ea
.
-,Z)TZAr
(751liz
(ZM 9-) ti ( UIT[5)t
(ZUIlg)
z AfZ)T4f
lz
=*
+
)
+
1( 5
Z) ToAtlP +
S'z
-.
iZ) TJ
(qL 06)
TU-
(ZT
)T4( zu-)e
1Z
jZ) PA~I
+
J)tIc+
)
ZUIT5
(5 i 1 z
9TZ
217
The right hand side (RHS) of the Bethe-Salpeter equation (9.1)
may be expanded as:
lm
+
()G*
(r,
(lm
z
,
<G
-d
dz
E*m
n
(r'r)
r)
f d 2 r 2 1 "f z
V
-d
**
2)
'
G
s1 r)
dz
G
im
1
(r)
Illm(r,
-d
(r2)
+ <Gllm(r,
r
r) 2
s*(r
12
-
m
)>]C(r
1
lm ('
2
E 1 (r) G *(r, r
2
(r2 )>] C(r 1 - r2
+
=>
Sdz 1f-d1
ll
1 lurn
-
n (r
2
) + im<G 2
(r,
'r
r)
1
Elm (r 1
(
) G
G * (rlim
(
'
I
r2
r
r
-
dzz'
dz 2G llm
1
2
(r',
(r'
(r, r1
dz1
)
E1
d2r
'
*E
=
)
R.H.S.
2
-
r)
2
218
2*(r)>1
dz
z dz
+
respectively.
1
1
11m
G11m
r2
(r
Elmri1
(r, r)
dz 2 (G
1
2
1
fl
z
1 *(r 2)>]
where
C(r 1 -
and
C(r 1 -
11m
r)
(r',
r
'i
(r
im2
*
2
2
llm
1
G
r2 )
(9.8)
-
1
are defined by (8.25a) and (8.25b)
In the limit
z
-
z'
it can be shown by sub-
stituting (8.25a), (8.25b) and (9.2) together with (9.3) and a
typical correlation function into (9.8) that the second and
third terms in (9.8) do not contribute as significantly as the
first and fourth terms.
The same conclusion may be reached by
the following argument.
The correlation functions which appear
in (9.8) restrict the integration variables
z
and
z2
to be
separated by not more than the order of a vertical correlation
length,
Z .
Therefore,
in the limit
z
-+
z',
it is clear that
in (9.8) the first and fourth terms contribute over the entire
range of the
z, z 2
integration whereas the second and third
219
terms can contribute only over a narrow range on the order of
z = z'.
a correlation length which straddle the point
Thus,
we approximate (9.8) by retaining only the first and fourth
terms.
Introducing the Fourier transform of the correlation func-
tion
-i(a
C(r1 -
d3c
r2 ) =
We substitute
-
@(a) e
(r
-
29.a
(9.9)
(8.25a), (8.25b), (9.2), (9.3) and (9.9) into the
approximated form of (9.8) and make use of (9.7a)-(9.7d).
Equating this result to (9.6), we next balance terms of similar
phase and polarization.
hhu (z, ',
l1mz(z
z')
iaimz (z -
z')
a) elm
S (z, z',
+ 15 (z,z',
5 ) +
)+
B
I5 (z, z',
inhz - inh*z
I
(z,
)
i
e
-
Phase Factor
This yields:
z',
[ I A 1 (a L )12
84)
(9.10a)
220
in
z')
)
1 2 (z, z',
1
+ ID
iS'
S
3 (
+ 1 7 >(z,
z',
4
Z
in hz -
z')
z',
)
z',
) + 17 <(z,
,
+ 8> (z, z',
+ B 1 (S)
(z
-
in Z
V
8 ) + D 1 (L) B 1*(5L)
+
8 <(z,
)
z',
y
V
|2
*z'
[A
z',
1(
i
)
)]
z',
z')
in
13< (z,
D*(L
lmz
89
[IC 1 (
(9 .10b)
h
is'
Jvhu (z,
(z -
*z
V
-aI)
e
.) e
in
12 Cz, z',
+ 1 6< (z. z',
hvu(z, z', 8.)
z V
89].
C 1* (
)
-
)
Jvu~z
(z
m
Sf e
(9 .10c)
-
in hh*z
[C
I 4 <(z, Z,
(
)
iS'
z',
8)
(9.10d)
221
-i$'
6.) e
z
hhd(z
[A2 (
+ I
(z -
1 1 >(z,
12
-i5I
Z,
Jvd(z
[|C 2 (9
+ I
0
) e
+ I>
)
z',
+
z',
(z
-
z')
-invz
+ inv z'
+
(z -
I 2 <(z, z',
)]
-inhz -
z')
(9. llb)
inV*z'
lmz
Z2*(S3)
(z, z',I
(z,
(9.lla)
1 2 1 2 >(z, z',
e
I
9
(z, z', 5) + I< 0 (z, z',
Z ,
[A2(A
-inhz + infhZ
lmz
-iS'
Jhvd(z,
z')
-i8im(z -
(z, Z',z ) +I <(z, z', )]
9
z')
m
e
)
Phase Factor
I+
) + I<
(z,
(z,
z',
z',i f
<)+
(9.llc)
222
-iS'
[C 2 (
+ ih *z'
-i)
=e
e
) A 2 *(
) + 14< (z, z',
14 >(z, z',
)]
(9.lld)
.
+ I>2(z, z', 5 ) + I<2(z, z',
+2.I12(
z
12
)
89
Jvhd (z, z',
(z - z')
where the coefficients
I. <
J
(j = 1,
respectively.
2,
...
,
A1 , B 1 , C1 , D1 , A 2, C 2
12)
and the terms
are listed in Appendices A and C
A similar set of equations may be obtained for
the cross correlation terms, Jhhcl'
hhc 2
...
by balancing
phase factors of the type
e amz(z + z')
e -smz(z
and
+ z')
The resultant set of equations, however, is not required in the
development of MRT equations for three dimensional random media.
In the case of a one dimensional laminar structure in which
k 1M
lm p
+
co
the set of equations for the cross correlation terms
is required in the derivation of MRT equations.
In order to
see this point more clearly and moreover to illustrate the
evaluation and reduction of the
will consider
I5> (z, z' , 51)
I.
J
terms in Appendix C, we
in detail.
223
z'
dz 2 e
-icc
z
(z 1
@(D
z2 )
- T.c a)
F(zi,
-d
k L)
Zf, z2'
+ rd(zi,
z2' kj) e
-
z2
-ik'
(z
lmz 1
-
z2
ik'jm
+ rc 2 (z1 , z
2
k.)
(z1 + z 2
'
'
+ Tcl(z1 , z
2
R L) e
)
ik'
(z
elm
)
-d1
-{u
daz f d 2 k
(27rr) 4
=
dz
z
takes the form:
I5
)
',
(z,
S
)
5
(z, z',
-
The term
15
)
The Term
where
F(zi
k'
)[B
*(z 2
(9.12)
A.
= Re(klmz )
) = A 1(
-ikjm (z, + z )
2
1
e m
()
and
e
inh z 1
e(-Slmz) + e
-i
hz
e(Slmz)I.
(9.13)
224
We note from Equations
rc2
and
zI
vary on the long distance
cl
l , rd'
(9.7a)-(9.7d) that
z2
and
There-
scales.
fore, in (9.12) due to the extremely sharp peak in the correlation function relative to the long distance scale, we may
replace
T c2.
z2
by
z1
in the arguments of
Next we take the limit
z
integration into the intervals
The
az
-+
z'
ru' d' cl
and
and break up the
-d 1 < z2
z1
z2
zI < z 2 < Z.
integration then is performed using residue calculus
followed by the
z 2-integration.
In performing the
z2
integration we retain only constructive interference terms
of the form
or e
e
,
where
n
-
Im(n ).
Destructive interference terms of the form
i(2n' + an )z1
en
or
i21'z 1
e
(where
n' E Re(nh
or
Re(n v)
will contribute a small fraction
and
an
is explained below)
n"/(2n' + an )
or
n"/n'
of the constructive interference terms when integrated over
z1.
Upon performing the operations discussed above, the ex-
225
15 (z,
z,
becomes:
15
pression for
)
=
i(27)5
-d
-d
n
-~~
i(an
d 2k
E
hhuz,
n
-
f Z )[
l
(
-
1
dzi Res c4(k
L , L [ln
R+
6,a
-
- IK-) )e
-2n
hz1
. J
2 nhz
k )e
L,F
]
+
+ M2n
,
zi,
k1. )[M3n sJ
-2nh toz 12
e
, k)
+ M4
Th ifz1
e
I
+
Jvvu(z
kj)
k
J vhu(zi,
zi,
k
M )[5n (
e6n
2
, k ) e
h" z 1
-
z,
+ Mn
+ M8n
e
,
k. )
e 2 nh1z
+
k
)[M7n iL,
)
+
Jhvu (zi,
-2hz 1
I kL) e
Jhhd(zi, zi, k)[Nln
'I
-2T
e
z
_
+ N2n (,
K ) e
2
h" z
]
+
226
hvd(z
,
e-2h1z
+ N 4n
S) -2 e hZ
+ N6 n
K
)
(zi, zj, L) [N3n
k
2nhzI
k.
e
+
ivvd
E) [Nn (.L'
zL
k4 ) [N 7 n ( Lf
,
+
zl,
+ N8n (.L'
h
)
e
.)
+
Jvhd (z1,
Shhcl (z ,I zl
k )e
2n
L)[Wln(T_..'
-i(h
A..
+ qh* -
e
i(nh + h nh*
h + 2k 'mz )z :1
2k'
)z1 I m
+
+ W
,
,
Jhhc2 (z1
+ W4nL.'
zLr ,
)[w3 n(.
IL3n
-i(
k ~heh
+
k)
e
(-h +
h * + 2k' mz )z 1]
hfl*
-
2k'
)z
lmz
1
(9.14)
227
where
an+
and
an
represent the
n-th
in the upper and lower half complex-a
M in
inN.in
W.
and
(i = 1, 2,
jn
...
plane.
(c.,ctz)
The functions
j = 1, 2, 3, 4)
8
,
poles of
listed in Appendix D, for reference.
are
In deriving (9.14), we
have made use of the following properties of the spectral density which are valid for a large class of physically interesting
correlation functions:
n+
n
-
(9.15a)
n
Res =-cta
Res #(ct.,
ct
=.RsDaLa(91b
)
=
-
Res Nc(a ,
n
(9.15b)
).
It is important to note in (9.14) that the terms
which appear in the phases are function of
nh
8 .
and
nh
However, for
laminar structures the spectral density attains a delta function
dependence
6(k
-.L 8 . )
so that in
(9.14)
k lmz
+
8lmzo.
In this
case a typical cross correlation term in (9.14) takes the form:
Jhhcl(z1, z1 , 8)
W 2 n8, a)
e
-n -(
+ nh*(,) - 21Jz]
h
-
-
(9.16)
A typical equation obtained by balancing phases and pol-
228
arization terms (as was done in (9.10a)-(9.lld)) and then letting
z
+
z'
is:
hhcl(z,
= A (a.) A *(a
1
..
2
+ I> (z, z,S
13
2.) + nh*(a ) -
h
) e
z,
.
)I
)+
(z,
1
I <(z,
13
2ajm )z
.) + B (<
) I(z, z, S.)
1 .. 1
z,
z,
.).
(9.17)
It is clear that equation (9.17) may be used to express (9.16)
each of which contains
2nv 1Z
or
constructive interference terms of the form e
in terms of
e 2 h''z1.
IS , I
,
I
and
13
However, for general non-laminar structures, the
cross correlation terms of (9.14) do not have the form (9.16)
and therefore cannot be expressed in terms of contructive interference terms by means of equations like (9.17).
Therefore,
as discussed above, the equations obtained by balancing phases
and polarizations for cross-correlations are not required in
the development of MRT equations for three dimensional random
media.
Finally, the residue of the spectral density which
appears in
(9.14), may be reduced by the method described in
Appendix E, and the final form of
I5>
is:
229
(z,
=
z,
d2 k
(2r) s
)
15
L) B 1 ()
dz 1{[ A 1(
I2
1
1+
) 12 F-hu
IA 1 (
,
(hu
h"z
e
(z
-
I
+
JA
, k )j + [A 1 (a) B 1 (
(.
2
)
~hd~L
2n
hlz
-L'
-2nh" zI
k1
)
-2n
k)
2nh"z
where,
h( K
P hu
,
2
hdY
and
Phd
k ) =
-
K
1
(9.18)
I(Z,
)
(S9)
are row matrices, defined as:
,
S ) [{e (TS
- e(k mz2
m)
{e (~lmz)
{e (Talmz)
k-L.
'
hd
~(k
(A
e k mz
e ;lmz)
a+.
h (k lmz)},
lmz)
e(-k
(9.19a)
01
lmz
'
-
230
{e( 61m)
lmz
e(-k mz) 01
- h(-k
h (-klmz )}2'
{e( 6 lmz)
h(-k lm)}e(
h
U
E kmz +1mz
where
)m
1
and
Id
(9.19b)
e
are column matrices given
by:
z, k)
jhhu (z,
d
J
I (z, k ) =
d
vvu(z, z,
d
(9.20)
2Re{J vhu(z, z, R ) I
d
21m{Jvhu(z, z, K )}
d
listed in Appendix C, may be reduced and
I.
J
term (9.18).
cast into a form similar to that of the 15
The other terms
We illustrate the development of MRT equations by considering equations
(9.10a)-(9.10d) in the limit
+
Z,"
IB 1 (L)12
e
2 fnh"z
1
=
1<(z,
1A 1 (S)1
z,
2
1 1 >(z,
) + [I5
(z,
z,
6
)
J hhu(z,
z,
i
)
Equation (9.10a) may be written as
z'
-+
z.
231
(z, z,
(9.21)
)
+ I
Operating on both sides of (9.21) with
d/dz,
we obtain
dJhhu(z, z,
= - 2nh
)
hhu(z, z,
dz
+ e
dI >(z,
Z,
)
- 2 nhIz
[fAl(8L)| 2
dz
+
2
)|
|B
d_ I1< (z,
z,
) +
dz
---
(I> (z,
z,
dz
+ 15 < (z, z
(9.22)
The transport equation for
for
)
Jhhu(z, z,
the replacements
Equations
h
J
(z, z, a )
is similar to that
and can be obtained from (9.22) by making
-+
11<
v,
16 'I A
+
C
and
+
D1 ..
(9.10c) and (9.10d) may be cast into the following
2 Re{Jhu(z,
dz
)}
z,
vhu
=
-
21m[KVh
vh
vhu(z, z,
vh
)
forms:
d
B
232
+riKVhz
+ eA
+ e
dI3
z,
1 C1 *
+ B D * d
11
dz
iKvhz
dI
C A *
v
z, 5,)dI3
(z, z,
*
dI
3
(,z
dz
<(z,
z,
+ D B 1*
dz
iKvhz
d
>
dz- (17 (z, z, 5 ) +
d
+ e
>
--- (I8>(z, z,
) +8
-
vh
+
dz
(,,
(z,
)
-i*z
)L
z,
(9.23)
dz
S
=
2Re[Kvh
vhu(z, z,
)
- 21m{J vhu(z, z,
dzvh.Lv
vh
-i*z
+ i e
vhA C
i*h
zd
- e
-
where
vh
r
h
Tfl
v
) +
B
* I
dz
dz
dz
dz7
C A *-
e
Kv
I3>(z, z,
(z, z, )
(I 8 (z, z,
+ 17
I
(z,
(z,
(z, z
z,
z,
S))
))
(9.24)
8 *d8
-
~hJ
Reducing the terms
I.<
listed in
233
I5
Appendix C, according to the method outlined above for the
term and then substituting into (9.21)-(9.24), we may cast the
MRT equations governing the upward propagating intensity into
matrix form:
u (z,
) = -
9z,
|Slmz I
(Z
)
ISlmzt2
dz
+ Q
UU
(a
L
k
I
11
+
1 Qcl .L j, k)Li
-
U (z,
-Li
mu
mcl (z
k ) + Pud a
kj
ud
k
L
+
-Li
LI
md
dzk[F
(3
Id(z, k )].
,
)
. (z,
k.) +.,k ) (z, k.)
(9.25a)
Similarly, equations (9.lla)-(9.lld) may be reduced to matrix
form governing the downward propagating intensity:
d~
d
= -
-~
-a
-~mI
-
dz
+ Q dd(
k)
dd
i
Imd (z, k .) +
lmd
a
d
(
du
a
,k
.)
ii .
- Imu (z, k .)
s
234
~Lcl (6. r,.1) imc2-
- d (z,
E)
+ Pdu
1 c
,,
..1.1. ) ++
)
where in (9.25a) and (9.25b),
(j
=
1, 2, 3, 4),
e [fh
occur.
[F (S ...
4dd
,I
- I (z, k )]
A
= 1
(9.25b)
if
=
i
and is
Mathematically this arises in the reduction of
zero otherwise.
I.
J
d 2k
(z, k
)
- 1 Qc
-
where terms of the form
i)Izl
h (. ..
It is clear that these terms are of the constructive
interference type only for
destructive interference type.
=
+. .
and otherwise are of the
Discussion of the physical sig-
nificance of this result is deferred to Section 9.5.
The
Q
and
P
matrices in (9.25a) and (9.25b) are scat-
tering matrices for the mean and incoherent field intensities
respectively and are defined in Appendix F, together with the
other matrices in (9.25a) and (9.25b).
In order to complete
the derivation of MRT equations we must obtain boundary conditions satisfied by the intensity matrices,
Iu
and
Ido
235
9.2
Boundary Conditions for MRT Equations
The boundary conditions appropriate for the modified
radiative transfer equations are obtained from Equations (9.10a)
by first taking the limit
-(9.lld),
z = 0
and
-d1 .
and (9.lla) at
z'
-+
z
and then setting
To illustrate the technique, consider (9.10a)
z = 0.
We have
A
(0, 0,(0, 0,
)A= A2Il> (0, , ) +I5hhu
(9.26a)
0,
Jhhd 0,
A 2 (S )
)
2
1
>(0,
0,
)
+ 1 9(0,
0,
(9.26b)
scribed above for
15
>
(0
0
5
and
I
according to the method de-
15>
it is easily shown that
-A|A2-
) =
>
I
A2 (
From Appendix A, we obtain
2
JA 2
(0, 0,
(9.27)
9
2
=
|R 1 0 (81 21A1
2
,
I5
,
Upon evaluating
which
236
upon combining with (9.27),
Jhhd(0,
o,
(9.26a) and (9.26b) yields
(9.28)
hhu (0, 0,
.
R10
) =
A similar procedure may be applied to the other equations in
(9.10a)-(9.lld), so that the complete set of boundary conditions
I
and
Id (0'
=
,
takes the form:
.
U
R1 2
'01
(9.29a)
1d(-di,
(9.29b)
)
Iu(-d
10
Id
)
satisfied by
where:
0
0R.
.12
0
0
0
0
~
3-J
0
R..(S
where
is
.|12
(9.30)
) =
i
= 1
0
0
Re (R*S. .)
0
0
Im(R*".S..)
1] ii
and
j = 0
1J iJ
or
-Im(R*.S.)
R JIJ
Re(R*.S. .)
iJ
ij-
2.
The incoherent intensity transmitted from region 1 into
237
region 0, is given by
Iou (
a
=
T1 0
(.L
u (f 0 a)
(9.31)
where:
X10
2
0
TI
0
2
--Yb0
0
0
0
0
10
-
R-
IM-
0
0
Re
Y 1-1 0
-Im --
0
0
Im
Y 1 0 10
ReL
Y10 10
10 1Oki
-(-.32
-
-1
1
(9.32)
and
ohu(0,? 8i ) oh(O,
<S
I (0,
ou
(0,
.j.
.
) s* (0, S.)>
ovu
(9.33)
=
2Re<E
) E*
(0,
ovu
L
ohu
(0,
238
Boundary condition (9.31) is obtained by noting that the plane
wave components of the incoherent fields in regions 1 and 0
satisfy the boundary conditions
E
ovu
(0,
Eh(O,
-o
K)
= x1
Y (a.) E
(U,()
(9.34a)
10 ._.
lvu
.J.
Ell
.L
E lhu(,
).
(9.34b)
Upon forming the appropriate ensemble averages, boundary condition
(9.31)
results.
239
Comparison of MRT and RT Equations and Solutions in the
9.3
First Order Renormalization Approximation
(9.25a)
It is of interest to compare the MRT equations
and (9.25b) with the conventional RT equations in current use.
For example the RT scattering phase matrix coupling the four
Stokes parameters of the scattered wave in direction
Q
to
the four Stokes parameters of the incident wave in direction
0'
is given in Ref. 63 and may be cast into the notation of
this thesis as:
uu
k.'
..
2..'
(e - e')
(h
*
Z lmz
2
e')
2(h - e') (e - e')
- klmz
(e - h')
(e - e') (e - h')
0
(h- h') 2
(h - e') (h
h')
0
(h - h') (e - e')
0
2(h - h') (e - h')
+(e - h') (h
0
0
e')
0
(e - e') (h - h')
-(e
h') (h - e')
(9.35)
where
e = e ($ lmz
h
h (,lmz
e' E e(k lmz)
and
h' 2 h(k
)mz'
240
Comparing (9.35) with (F.16)-(F.20),, we note that the MRT
scattering phase matrix has additional terms not present in the
wave like corrections due to the bottom interface at
z = -d
.
These additional terms represent
RT scattering phase matrix.
This can be seen by removing the effects of the bottom boundary
(i.e. by letting
ac2
-
0
R1 2 ' S1 2
-*
0)
in which case
and the MRT scattering phase matrix
ah, av' acl
(F.16) reduces
identically to the RT result (9.35).
We now consider the MRT scattering coefficient matrix as
given by (F.1) and (G.13).
The RT scattering coefficient
matrix is given in Reference 63, and in the notation of this
thesis is given by
Kh(e)
0
0
0
0
Kv(6)
0
0
0
0
+ Kh())
2
0
0
0
(Kv(6)
0
(K (6) +K()
h+GK
v )
2
(9.36)
In Chapter 8 it was shown that,
241
2 Im[X TE
cos e = Kh(e)
21m[X TM
cos 6 = Kv
(9.37a)
(e)
(9.37b)
where
(
= nh (KL)
(9.38a)
- kImz
)
X;TE-(
(K
= n
(.)
(9.38b)
Therefore, it is clear that (G.13) reduces to the form
Kh(e)
0
K v(6)
0
0
0
0
K (0)=
(Kv (e)
0
+
Kh(e)
0
2
(Kv(.e) + Kh(e))
0
0
-(nv
h
2
(9.39)
Comparing (9.39) and (9.36) it is apparent that the MRT result
(9.39) exhibits additional wave-like correction terms of the
242
(n
form
-
h'
Physically these terms arise from the
fact that TE and TM polarized waves in general propagates in
the random medium with distinct phase velocities, thereby
affecting their cross correlation.
However, should the random
medium be such that TE and TM polarized waves have identical
phase velocities, then the MRT result (9.39) reduces to the
standard RT result
(9.36).
As an application of the MRT equations
and to illustrate the significance of the
(particularly the term with factor
A1 )
(9.25a), (9.25b)
Q
matrix terms
we compute the back-
scattering cross section for a two-layer random medium by applying the first order renormalization approximation.61
In
the first order renormalization method, the scattering of
the incoherent intensity is neglected in the Bethe-Salpeter
equation.
dz
( , k.) I I(z, k.)
i
.2 md ..
(Z,'
=
-
-
(
Iu (Z,
-
f
) +
uIlmz
I
(z, k .) +
mu
.u1
+ A1
cl (.L.
Ikmcl
Ud
.
1
=
{Qu
-
d
-- -y
In this approximation the MRT equations reduce to
(6 , k .)
k
(z, k i)}
(9.40a)
243
dlmz
-md (z,
k .)
+
md .1
i Ocl QL'
-
of (9.40a) and
Id
Jk
Iu(
h
(1
0
u
-
4d1
=
=1.
u
2 I
I
The results are
f7r 2F(
, 0) IR12i
-
.i
.)
lmzi
-4n". d
e
1 i
) (1 + |R
thi
, 2k
2
.i
e
d
(2K
uv
.
, -1
-4T11
e
4
Imzi
(9.40b)
.L)}
(9.40b) is known, we can readily solve for
in which case
.i2
+
1mc2(z 1
.Li
Moreover we consider the backscattering direction,
-k
lmz i
Im
mu (z, k iL1.)
in conjunction with boundary conditions (9 .29a),
(9.29b).
=
(uL , k 1.)
z-dependence of the terms within the curly brackets
Since the
and
d
d
.
k
2 k
2
0
e
hi 1y
hi 1
(9.4la)
2
2
I F
lm
14
2i-
m
2
(2
L
.
i
,
2k
.)
2k lmzi
244
(1
-4n" d
e e vi 1 )
-
a
+
1e-
s
4n".id1
1)
+
4d11
(2K
. ,
0)
vi
(k2 - k 2 .)
Lik 2 lmzi
2
is
-4n".d~
e
vi 1
where
I
and
elements of
I
signify respectively the first and second
Iuv
.
U
-
lm
(9.41b)
The intensity transmitted into region 0, is
found by applying boundary condition (9.31) to (9.41a) and
(9.41b).
I
=
I
=
ouv
jIX 1 0i
2
-IY
10i.
I
(0, -K L)
(9.42a)
uv (0, -T .Li ).
(9.42b)
The backscattering cross sections per unit area follow from
the relation
Chh
ouh
=
avvi1
2
cos 2 6 .
(9.43)
ouv
and f = 1 in I
f = 1 in I
e
ouh
m
ouv
We now consider a spectral density of the form
.
with
4r k
245
2
Sktk4
Z
km p z
z) =
47r 2
(l +
$
-_
e
22,
2/4
ip
(9.44)
iz
z
which corresponds to a correlation function which is Gaussian
Combining equations
laterally and exponential vertically.
(9.41a)-(9.43), we obtain the backscattering cross sections per
unit area for a two-layer random medium in the first order
renormalization:
Sk{k
2
Z
IX 1 Q.iI
=_4
_
-k 2Z
jkozi|1
_ZD
2iOI
klzi
0
1
d
hil1
-4T1"
2 n"
hi
+ 8d,
(1+
2
2)
(1 + 4k'
lmzi z
|R 12il I e
6ka4
2z
CrvV=m P
sin 2 e
0)
- 4 n"1 d
e
hil)
-4+nghiRd it
(9 .45a)
4 k ozi 4 -kk 2 sin2e0
4IF iI
1k31l4e
F2i 14
z
-4n
(1 -
R 1 2 i1 4
2
p
4T
d1
)s
v1
-4'i" d
(1 + Is e
2 ". (1 + 4k' 2
vi
lmzi z z 2)
+ 8d 1 S 1 2 i 1
2
e 4n".td
vi l
-
) lmzi +sin2e6
k2
kmzi - sin 2 o
0
l
k2
12
k02
1
2
(9.45b)
246
9.4
Illustration of Backscattering Cross-Sections
In order to illustrate the backscattering cross sections
(9.45a) and (9.45b) obtained from MRT theory, we plot in Fig.
9.1
a
ahh " av
20 cm
= a
at nadir, as a function of frequency for
thick random layer.
The coherent effects of MRT
theory are apparent in the oscillatory behavior exhibited by
the spectral dependence of
a.
In Fig. 9.2 we plot
a
as
a function of frequency for the same parameters as Fig. 9.1
but with increased
6.
We see that increased scattering
dampens the interference pattern by shielding the bottom
boundary to a greater extent.
A particularly significant coherent effect arises from
the
A1
As discussed in Sec-
terms in (9.25a) and (9.25b).
tion 9.2 these terms constructively interfere only for
k
(i.e.
forward or backward scattering).
i
=
Physically,
this is illustrated in Fig. 9.3 where we have sketched the
path lengths traversed by the single scattered mean field
and its conjugate in the two cases of forward and backscattering.
Constructive interference terms of this type are
not included in phenomenological radiative transport theories.
In order to gauge the error produced by omitting these constructive interference terms, we set
A
=
0
in
(9.25a) and
(9.25b) and rederive badkscattering cross sections,
ahh
and
247
aV .
The results are similar to (9.45a) and (9.45b) except
that the second terms within the curly brackets are reduced
by a factor of
2.
In the case of thin layers with low ex-
tinction the contribution of the
A1 terms can be significant.
However, for thick lossy layer the error produced by omitting
the
A
terms is minimal.
a
(db)
Figure 9.1
hh
-8
vv
as a function of frequency at nadir
-10
-12
-14
z
-16
S0
<e
z = 0
1> = (3.2 + i0.01)c 0
6 = 0.1
-18
p =0.1
1
C2(80,
8
9
cm
f
(GHz)
cm
z =0.05
t iO, 6 )E
11
10
t
'7
~Th
gJ
12
o (db)
Figure 9.2
-2
as a function of frequency at nadir
-
-4
Ghh = avv
-6
-8
-100
z= 0
<1 > =
(3.2 + iO.01)e0
6 = 0.8
-12
-
= 0.1 cm
p
k
z
=0.05cm
z = -20
62 =
8
9
(80. + iO.6)E 0
11
10
f (GHz)
12
cm
z
z
11
I
/
ft
I
10 1E0
/
0 160
I
Z = 0
t1
> +
ot
if
1 r
Ur'
P 0, <E1> + E if
Z = -d
O
(a)
P
2
(b)
Baci scattering
E
Forward scattering
Constructive interference path lengths
Figure 9.3
110
Ln
0
251
9.5
Comparison with First Order Born Approximation Cross
Sections
As another case of interest, we compare the backscattering
cross sections of MRT theory,
(9.45a) and (9.45b) with those
obtained in the first Born approximation to the wave equation.
In Chapter 3, the Born approximated cross sections for a twolayer random medium with spectral density (9.44) are given in
(3.30a) and (3.30b).
Comparing (3.30) with (9.45), the MRT
results are seen to have the same form as the Born-wave results
a
but with renormalized decay constants
k Mzi.
of the decay constant
v+
k lmzi
fluctuations
and
in place
n"*
In the limit of small permittivity
(small scattering)
6
-+
0
and
ni,
"
and the MRT backscattering cross sections reduce to the first
order Born results.
fixed incident angle
kim,
This is illustrated in Fig. 9.4 where for
6
.,
frequency,
f,
and decay constant
we plot backscattering cross sections,
a function of the variance,
6,
av
(in db) as
in both the wave-Born and
MRT-first order renormalization approximations.
For large permittivity fluctuations the MRT result is
bounded by the asymptote
avv = constant.
Physically, this
is due to a balance between the increased backscattering and
increased shielding of the medium produced by large permittivity
fluctuations.
Alternatively, the Born result is unbounded as
252
6
increases.
This follows from the absence of multiple
scattering and the associated shielding effect in the first
Born approximation.
vv (db)
Figure 9.4
a
vv
as a function of variance
0
Born
-4
MRT
-8
-12
z=0
<e > =
z
-16
p
9.
z
-
(1.8 + iO.005)c 0
= 0.1 cm
0.05 cm
=
k__
E2 =
_=
_0.05_cm_
z = -50
cm
(6.0 + iO.6)e0
A0
.0
.004
.012
.008
'S
.016
tn
254
9.6
Appendices
APPENDIX A
A
a
(k )
(A.1)
R1 0 (k
A 2 (k) = a
2
)
D 2(k
(A. 2)
D2(k)
B (k ) =R12 (k) e
(A.3)
C1 (kl )
(A.4)
=
F2 ( .i)
)
S 1 0 (k
C 2 (k
=
a
(A.5)
F2(k)
D (k ) = S2 (k ) e i21vd
(A. 7)
where:
(A.8)
-
a =
(2-7r)
2
2ik1 m
255
D 2 (k) = 1 + R0 1 (k
R 1 2 (k ) e
i2n hd1
(A. 9)
i2Iv d
F 2 (k
1 + S0
-
k.
R. .=
(A.10)
k.
jz
Z
+
k.
1
(k ) S1 2 (k ) E
(A.l1)
Ek.
iz
JZ
(A. 12)
Es.k.
+ s.k.
i Jz
3 iz
where
i,
j = 0, 1, 2
interpreted as
klmz
and
and
klz
Elm'
as well as
E
respectively.
are to be
256
APPENDIX B
Here we list the amplitudes in Equation (9.2) for the zeroth
order mean field in the two-layer random medium.
Assuming a
unit amplitude incident plane wave, we have:
2
Ehu
=
X
.hR
. = f exO
e 2i
E
d1
iihi nh.d
.2 e i
(B.1)
x
.=
hdi
f
e
O
D
(B.2)
.
E
D2i
Y.
k
E
.
vui
= f m-E -0
mk
F
lm
2i
.2 e
S
v1
(B.3)
12
k
Y
Oli
-c
=f
E
vdi
m k
F.
lm 2i
(B.4)
X
(B.5)
Y
where
.d
i2n
-.
Oli
.
f
e
1 + R
Oli
= 1 + S
and
f
m
(B.6)
i.
denote the fraction of
ponents in the incident wave.
TE
and
TM
com-
257
APPENDIX C
which
(j = 1, 2, ... , 12)
I. (z, z', i)
J
For j = 1, 2, 3,
appear in Equations (13a)-(14d) are listed.
The terms
has the generic form:
I.
J
(z,
z',
)
dz
fz'
-d
=
ei
(z1
2
{5*(z
j
2,
da
(2 )' f
-
z2 )
-
(k i
i z
-d
> (z1
0 E lm(zi
31)
,
4,
,
dz 1
)}
(C.1)
E*(z
E.
lm'2' k.)}
1l
)
where
.)
=Ehui einhiz e(k lmzi) + Ehdi e-ihiz
+ Eviii. e
-in
i-n .z ^
e(-k lmzi)
.z
,
E lm(z,k
h(k 1mzi.) + Evdi. e
h(-k
i
.mi)
(C.2)
The coefficients in (C.2) are given in Appendix B.
-)
I. (z, z',
J
has the same form as (C.1) except the integration ranges
become
z < z
1
< 0,
z' < z
2
< 0
and
1
,
Q.
are replaced
258
by
P <,
Q.<.
>
P (z,
B
)e
inh(S)Z
l2
e
+
)z
nh
-i
e(lmz
(C.3)
a1>
(z,
I
)=
(C.4)
Seinh()z
e
)z
)
z
52 z,
(C.5)
<(z, I
(C.6)
)
52) =
S1 <(z,
>
( ) e(almz)
+ A
P (z, ) = A 1(S9) e (-S m) e
h
)= D 1 (%L)
(5
in
h(-5lm)
e
)z
-in
A
+ h(S lmz
4
V
(
)z
e
(C.7)
2 >(z, " ) =
2<
-
P2 (z, S9)
2 >(z,
k
(C.8)
)S
()
Ain
=
C 1(59)
h(-5 1
) e
V
)z
.1
+ C2 (
) h(a mz
259
-in
(8
)z
(C.9)
Q2 <(z,
8)
2(Z
S)
(C. 10)
P 3 >(z,
=
> (z,
3 >(z,
) =
(z,
S)
(C. 12)
) =
.(Z,
S)
(C.13)
3
(z,
3 >(z,
p4
) =
) = P
(z,
4
(z,
89) =
P4
(z,
)
L
)
5 4 <(z,
(C. 11)
(C.14)
2 (z
(z,
(C.15)
(Z,
(C.16)
= P 2 <(z,
(C.17)
I ).
(C. 18)
260
j = 5, 6, ... , 12
I.< terms which appear in
3
(9.l0a)-(9.lld) have the generic form:
e
(z
2
1
mz
(z
1
ik'
e
- {(z 1 , z 2, k
z2
e
2
-ik'
(z
lmz
1
z2
e-ik{(z
where the dyads
r
R
2,
J
except the ranges of the
< 0,
and
z' < z
R.>.
3
2
,
,
ct
Z
z
z2
z2
1
+ z2
< 0,
>* (z
are defined in (9.7a)-(9.7d).
(j = 1,
I.
j
and
... ,
12)
z1 , z2
F.>
3
and
(C.19)
,
,
c2 (Z,
sion for
)
+
k
i
'
z2
e
)
ikjmz (z1
+ rcl(z ,
1
-
>(
)
+ rd(zl,
-
z2
-
z
+(D
)
'
dz
d2 k
C
)
J0Z
f 00.
-ict
z
-d
f z dz
-id dz
1
da
)
(27r) 4
) =
-
I.> (z, z',I
the
+
For
is identical to
The expres(C.19)
integrations become
H.>
3
are replaced by
z < z
F.
J
261
F 5 >(z,
B)
B1 (5.L) e
A 1 (BQ)
=
emz)
inh(SZ
B1
5 > (AI
5 ) = H (z,
A(Sz
-in h(I)z
e
(C.20)
e (almz
)
+ A 1(a.)
<
F (z,
S
5
+ B1 (
(z,
a)
)inh
, A (aL) B, (a. )
=
(a
e
(C.21)
e $mz)
) A2 (a.L)
(S
in
C
1(
, c (a_)
)z
e
D, (a. ) e
<'z, ) = H6
+ D 1 (L) C2 (L) e
7(
=
V
i
55( ,i
) =
) =
7 > (z
(z
)
( ,
-in
6>
(z,
(z 5
)
)z
)z h
L
h (-a lmz
(C.22)
h .1.. h(
<
F6
v
(
=
R6 )> (z,
-i+)
+
e (-Sy
)zm eh
(i
6 >(z,
Z
)
<z
C )D
1(L)
(
h(..L)
--
e
inv
)
(5
z
h (- lmz
(C.23)
(C.24)
(C.25)
262
p7 (z,
,)
5 ) =
7 <(z,
8
( ,5)
=
6
(Z,
59
=
5
(z,
8 >(z,
;)
R< (z,
9 >(z,
5 )
=
H9
= P *(z,
F< (z,
59
+ B2
) A
1
=
(5) A
1 ()
(C.28)
S)
(C.29)
(C.30)
he
(Slmz)
(5 ) z
(C.32)
e(-Simz)
= A 2(5)
e
S)
(C.31)
2 (5) e
H9<(z,
(C.27)
-I
(z, 5 ) = A2n
in
+ B
S)
S)
6) =6<(z,
8 <(z,
(C.26)
5)
= F5<(Z,
h(
e
e (-a3mz)3
ei(hlz
(C. 33)
263
-inv (8)z
1010
)
=
z
)
D1 (
) e
H
in
+ C 2 (a)
F<(z,
8)
(z,
in (Z
=
(z,8
(8 )z
C8
h )z
h(-81z)
(z,
=
=
2
A
(C.34)
e
(8
)z
lmz
(C.35)
(C.36)
8)
(C.37)
(C.38)
g< (z,
(C.39)
((z, ) =z,
12(z,
v h(8mz)
A
(z,
) =
Hj(z,
2L
-in
e
SQ
e
h (-.lmz)
)
1 (z'
+ C 1 (8)
C2
-<
-
)=10(z )=Vh(
=
(z,
F9 >(z,
8)
(C.40)
..
(C.
)
Fz
41)
264
1~2
(z
<2(z,
L
F,
F~ < F9(z,
0 (z,
Q
f
(C.42)
(C. 43)
265
APPENDIX D
mln
f
i. )L
m2n
mz)
*{e
0
{e(lmz)
- e (k lmz )
(D. 1)
12
* e (klmz) }2
(D. 2)
II
3n
.1
f
{e
f
{e(almz)
(D. 3)
h (klmz
lmz)
A
M4 n
{e
lmz
,
5n (T
-
h (k
e (klmz
(D. 4)
)}2
lmz
h (klmz
(D.5)
{e(lmz)
6n
Se (klmz)}{e( lmz)
h (k
)}
(D. 6)
{e ( mz) h h(klmz)}{e( -mz)
7n
4. -LR
f
-
Se (klmz
(D. 7)
(ST'Q)
{ (ZUrT9) 9
a
(Z'UIgS);aI{
("'iIN-
ZUIT[
=
Tx
17g
UL N
( V'I)
I{(
zu[ -)
T4
(ZUIT ) a{
ZUITX-
Tx
* zurT
ifg)U9
1.
V0
%,
(ET*C)
ZUIT
ZUIT
zuxT
-
T
ZUIT
(T~ 4
~
T
us
N
I1W11,
+
(ZTQ) C
zur 31-) t
ZIT
* (
(T T a)C
)
ZMT
I{(
( zmr
z{
9-) a)
U) N
.6
ZDIT
-)a
_r
B
'-
)UN
X
+
(OT *
(6 0CO
* (
-)
UT[
o-r
5-) 9)
%0
(8 0G)
ZUIT
* ur (UT }( ZUIT[X)X
Sm[ ) 9 0
*(
+
T-
Ir)
usW
99Z
267
A
k
e(-kimz
S-{e ( lmz)
)
N 8 n (L,
(D.16)
where:
(D.17)
(an
kimz
n+ -
lmz
klmz
mz
C+
+
(D.18)
(an
C+
imz
A
C_
(A
k mz
-
lmz
imz
(D.19)
12
B
=lAL)
lmz''c
(D.20)
)12
Also,
k L)
b
{e (-lmz)
e(kimz) }{e(lmz)
e(-klmz
+
Wln ($.
W2 n
-LF
(
)=b* Ic-
-
(D.21)
{e (a mz)
e (klmz
lmz)
- e(-k lmz)
(D.22)
268
L
W3n
k )
{e (-a lmz
b
- e (-k)lmz}{e(mz) Se (klmz)
(D.23)
-J{e ( lmz)
= b*
W4n aj-
e (-klmz
e(lmz
- e (klmz)
(D. 24)
where:
b
=
A(I
)
12
269
APPENDIX E
In this appendix, we illustrate the technique used to sum
the residue terms which appear in (9.14).
Referring to equation
(9.14) and Appendix D, a typical sum over residues has the form:
(an-
z
..L +
)
c
,
-
L
n
) (a
(E.1)
- z
z
where the complex variable
(ki
I-mz
6*z)
-
-(k'
or
lmz
represents
p
+ a*
lmz
imz
cording to the specific term in
Let
Im(z )
> 0
az
the real
1
and
-(k'
lmz
-
a*
imz
lmz
),
)
or
ac-
(9.14) we are trying to sum.
and consider the following integral,
I 0
(k
da
{a
n
J
(ct
-
The spectral density
n
or
lmz
along
axis:
27ri
{an }
)
+ 6
(k
}
cD
-
z
,
-
z
a.
)
n
(k
E -.
)
Res
(E.2)
p
has an equal distribution of poles
in the upper and lower half complex-a
plane and vanishes everywhere on the circle at infinity.
this case we evaluate
(E.2) by closing the contour both up
In
270
plane:
(k
(k _L
p
_
Res
down
Equating
Dk
an
(N
-
a
-z
n
I
-z
(E.3)
z) + E
-8Z,
and
n
Idown,
Z Res
NDk
-p
L
(E.4)
,
Iup =
,
-
Res #(
+
az
and down in the complex
p
we obtain
(n
-S
.
n
(an
n+
~zp)an~ -z)
(E.5)
where we have made use of Equations
(9.15).
unchanged if the initial assumption
Im(z p)
by
Im(z ) < 0.
Therefore,
Result (E.5) is
> 0
is replaced
(E.5) may be used to sum the re-
sidue terms of the form (E.1) which appear in
(9.14).
271
APPENDIX F
Here we list the matrices of the MRT equations (9.25a)
.
and (9.25b)
0
h
0
~1~
0
0
0
0
)
2
0
0
0
0
(n
"
+ -h ")
h
-(nI
I
( TI
v
- Th
h
+
(F.1)
where
n
=h"
v
Quu
and
Im (
h)
v
v
.J
2
Re (TIh
Tht
-5.'
Ii
V
5 lmz -
klmzi
A
Lh
{elmz) - e (klmzi
12
av{h(Slmz ) - e (klmzi) }2
h (klmz i
ah {e1mz)
av{h
lmz
0
0
0
0
0
0
0
0
0
0
0
0
(F. 2)
272
Qdd
Jk.
= same as
j)
= same as
.Li)
and
k
J.. i)
= same as
cl
AL ,,a
.1. .)
=
QUU
.
..L'
(
QUU
kmzi + -k
except let
except let
Slmz
(F.3)
lmz
(F.4)
. + -k
lmzi
.lmzi-
Qdu
Fealmz)
QU
.
L
.
Qud
except let
almz
+
(F.5)
lmz
0)
2
-*e(klmzi.}
- e(-k
{e(- lmz
0
0
0
0
0
lmz i.)}
A
)
{h(6 lmz) -h(k lmzi.)}
0
0
(h (-a lmz)
0
0
0
0
0
0
0
0
(F.6)
273
; 2 n".(5
hi
e
d
)z
.
IE hui
2
)
T2-n".(l
Evui 12 e
(F.7)
d
mu (z,
0
d
0
2
Re{Ehui
hdi
2 n"
Y
z'
hi
-2"z
2Re{E
mcl
(z,
V.L
E*
VU.
vi
y~~ I) e
vi
(F.8)
k .)
J.1
0
0
z
2-n
I2Re{E hui
2Re{E v
hi'
hi
e2n viz
E*
Yvi
vdi
(F.9)
kJL i
)
mc2 (z,
Ehdi
e
0
0
where:
1 ah =
IR 10(
)
12 IR 12(a
) 12
-4nh t()d 1
(F.10)
ID 2 ( L.) 12
274
a
v
=
1 -
iS10
-(.) 12 1S12
.L.)
F2C
-i2n
.
y hi.=R* 12i e
(1.
*d
hi
12
e -4nv It(a.L)
1
(F. 11)
2
i2 nhidi
+ ROliR 2i
(F. 12)
D2 i
i2ri. d
e
y vi. = S*.
12i
S
.
.S
Oli i2i
e
)
-i2n*.d
vii1
(i +
(F.13)
F2 ij
D*
Yhi
R10i
ID
(F.14)
2
F*
Y.
vi
= -
s
.i
(F.15)
2i
IF 2~I
We also have:
uu
where:
k ) =
.L
(k
2
LO
(F. 16)
lmz -kimz
C
D
275
h {e
e (klmz
lmz)
ah {e( lmz
(F.17)
-e (klmz)
2
a v{h (a lmz
* h (klmz)12
)
"v h(lmz)
h
lmz
e (k lmz
e amz)
(klmz)2
(F. 18)
B
e (klmz
(h(almz)
0
(klmz)2
)
a v h(almz
2acl {h(lmz
{e(a lmz)
-2ac2 {h(almz)
{e(a lmz)
e (klmz)}
e (kz
)}
e (k lmz)}
lmz
2acl
{e(a lmz)
-2a c2{h (lmz
c2
lmz)
Sh (klmz
Sh (klmz
Sh (klmz
Sh (klmz
(F. 19)
276
cl
h (k lmz}
h(Mz)
ci
ac2{e(lmz)
e (k lmz
lmz)
h (k lmz
lmz)
h
lmz)
e klmz
lmz) - e (klmz)
A
+ {e(almz) - h (k)lmz
-
lmz) - e (klmz)
c
{h(a mz
-ac2
- e (klmz)
{h(mz) - h (k)lmz}
lmz)
h ( lmz
-e (k lmz
+ {e( lmz) - h (klmz)
{h(klmz) - e (klmz)
lmz)
- h (klmz)
lmz)
- e (klmz)
)
(F.20
..
Sud
k'k.)
..
=
same as
Puu
k)
except let
klmz
-klmz
(F.21)
(du
.L ,k)
= same as
Puu
k..)
except let
Slmz * -almz
(F.22)
277
P ,(I
kL) = same as
and
FlR
SReL
klmz
R
01 12
k
except let
lmz
)
dda..'
-klmz
S* S*
01 12
-a1mz
(F.23)
i2(nh - n
d
e
(F.24)
D2F2
F2
=
-
e i2(n
0 1R 1 2
01 12
-
nv *)d
h v
I
(F.25)
22
where Appendices A, B and C should be referenced for the definition of the variables in (F.1)-(F.25).
278
APPENDIX G
In this appendix we convert the MRT equations
(9.25a)
and (9.25b) to standard form by using solid angles and specific
intensities.
To illustrate we consider the angular decomposi-
tion of the upward Poynting flux associated with the vertical
polarization:
S
d 25
= /7-lm/i
r
/Elm/7
(z,
)
Jvvu(z
z,
<E
d2S
E*
(z,
>
(G.1)
).
Let
x
a
(G.2)
lm
sine
y=
sin
(G.3)
then the integral operator
d2
dm2{ cos 0
= J
s
n/2
f2r
sin ed6
0
d 2Q
cos
lmd4S
becomes
l
(G. 4)
279
3
where we have neglected the evanescent portion of
space.
Therefore (G.1) becomes:
Sd=
7d
vu
[vM COS
(z, z, )].
(G.5)
i
1m
o
We define the quantity within the square brackets of (G.5) as
(z, 0)
the upward specific intensity,
polarization.
of the vertical
Therefore we define upward and downward specific
intensity matrices 4u(z, Q)
as:
d
a.2
cos
) =
u(z,
d
where
1l
=
-
lm
and
(G.2), (G.3), and (G.4)
tions
cos
IU(z,
(G.6)
)
I (z,
d
)
is given by (9.20).
Using
d together with (G.6) the MRT equa-
(9.25a) and (9.25b) may be cast into the form:
-- ju(Z,
dz
+ Qud ('
G)
=
-
K
(z, C)
-
K(0)
-vu(z, Q)
auz'0
)i \.md(z, SI ) + A1 Qcl(Q ' 0i
'h?
mcl (z,
2.)
280
Q') + Nud
2
)
-, Cd(z
,
u(z,
'
+
dM'[fuu '
Q')]
(G.7)
-
(Q OF0i
' u u(z,
(v,
')
=
2')
(z,
I)
1 (8
I
(
2.)
82im1 cos
,
( )(z,
0)
2i) ~a'*md(z,
du
')]
.
)
'
)
2') + Pdd ('
+ Qdd
i)
m mc3 (Z'
)
Q
-
2')
'
-
Q.
5mu (z,
2)
+
' 0 Qi
)
+ Qdu
d (z,
:a
(G. 8)
(G. 9)
i
)
d
-Co s e dd(z, 0)
e
(G.10)
1
UOcl' ( 2
OCl
2
lm
L.' k.Li
COS
s
(G.11)
281
K
=2s"
a =
m
21=
2
Im(8lm)
-
K((Q) = cos 8 rj(a )
where
I
(G.12)
K I
-
(G.13)
is the identity matrix and it is understood that
is to be expressed in terms of
(G.2) and (G.3).
spectively
u
The subscripts
or
d.
U
e
and
v
Combining (G.6) with
the boundary conditions relating
p
and
and
by means of
signify, re(9.29a) and
4d
(9.29b)
are easily de-
rived.
4d (00,
4u
=
(-dl ,
R1 0(M 1)
Olu(
= R 1 2 (Q 1)
(G.14)
Gy)
d (-dl,
(G.15)
).)
Conversion of boundary condition (9.31) to standard form may
be accomplished by first multiplying both sides of (9.31) by
d 2 $_
and then making use of the transformation implied by
(G.4):
y
)=
0
($1 ) -
u(
) d2
$
d 2 L Iou (0,
(G. 16)
282
(S) - I(0, ) 2 COS 6 d%
d2
cos 60
S0
0
ou
(0,
..
)
=
10
1
u
lm
1
1
(G. 17)
E
-
)
u(0
0
Em
cos 60
L
T 10 (21
,&(,
l).
(G. 18)
-SOS 1, no
In going from (G.17) to (G.18) we have used (G.6) and the
result:
d
d~l
'm
cos
elm 'no
Cos e0
Fos n
(G.19)
283
CHAPTER 10
Renormalization of the Bethe-Salpeter Equation
In Chapter 6, it was found that the cross-polarized
backscattering cross-section computed in the second order
Born approximation does not agree with the cross-polarized
backscattering cross section obtained from radiative transfer
theory.
In this chapter we examine the reason for the dis-
crepancy between the wave and radiative transfer results.
It
is found that an infinite sequence of terms in the intensity
operator may be summed resulting in a renormalized BetheSalpeter equation which contain coherent effects not accounted
for by the ladder approximated Bethe-Salpeter equation nor by
the radiative transfer theory.
284
10.1
Physical Significance of Cross Terms
The additional contribution to the depolarized backscat-
tering as shown in (6.14) originates from the last term of
the cluster expansion (6.3) which correlates
r2
with
r3
r1
with
r4
and
To see this more clearly, consider the last
term of (6.4) in the limit of very strong correlations.
It
becomes
{daridsr
2 [G%1 (r,
[01 '2
r%)
-G 1 1 (r1 , r2 ) -.
G11
2'
1
1
(2y
(
V
The backscattering path of the field represented by the first
of the square brackets is depicted in Fig. 10.1 by a solid
line.
The backscattering path of the field contained in the
second set of square brackets is depicted in Fig. 10.1 by a
dashed line.
We may consider each represents the wave vector
of a plane wave.
Thus in the backscattering direction the
two path lengths are equal and the two waves interfere constructively.
In the radiative transfer theory only the ladder
terms are included and the constructive interference is con-
285
z
A
0
z = 0
'E
1
+
lf
r2 ' r
3
r1 , r
4
z = -d
I,
2
Constructive interference path lengths for second
order backscattering
-Figure 10.1
286
sidered due to coincidental ray paths rather than phase fronts.
Thus only the first term of (6.14) is accounted for in the
radiative transfer theory.
01(r, r,
2
01
r2 )
and
E(0)
E1
G*1 (r
22
(r
1(~ and
r)
over the respective
(r, r)
01'
1'
E1(0) (r2 )
-
Mathematically we see that the phases of
as well as
and
G 111'
(r
combine in such a way that when integrated
z-coordinates, a contribution of second
order in albedo is produced.
This argument may be applied to
higher order intensities with the result that additional significant contributions besides the ladder terms in the radiative
transfer theory may be identified.
Therefore in the limit of
low loss and high scattering these additional contributions
may be included in a renormalization of the Neumann series
for the covariance of the field.
We then obtain a Bethe-
Salpeter equation with a renormalized intensity operator in
which the infinite sequence of cross type terms are summed.
287
10.2
Resummation of Cross Terms in the Intensity Operator
The dyadic Green's function for a random medium satisfies
d3 r d3 r
T
1
Vl
Q(r ) Q(r 2 ) G j
Q(r 1) -(0)
G kj
2 GGik ( , r1
(r2 , r)
+.
G
-
(r , r
()r
)
-
d 3 r G (0) (r,- r)
1 ikC,
TV.
r2
)
r
-
.
(r,
,
(0)
)
= G
(r, r)
)
G.
r
-
an integral equation similar to that of the field
+
(10.1)
To simplify the analysis we associate Feynman diagrams with
the various terms in (10.1) according to the following rules
<G i
(r,
GP) -r
G ] (r
r2 )
=
(10.2a)
(10.2b)
-
().
2)=
C(r 1 - r2 )
=
or
(10.2c)
288
<G..(F,
ij
) G*(9', r')>
o
lk
o
=
I
(lO.2d)
The correlation of the dyadic field in (10.1) is given by
+1
Th
+Ix
H-
-1--
(10.3)
Integration is performed over the coordinates of all
internal
vertices and convolution is carried out over all indices of
internal vertices.
The second order ladder and cross terms
which lead to an equally significant contribution are readily
recognized in (10.3).
We consider the sum of the strongly connected diagrams
and identify the "kernel" of the this expression as the intensity
operator:
289
4-
9 0
+
+
x
0
-I--
(10.4)
The approximation usually made is to retain only the first term
of (10.4) resulting in the so-called ladder approximation.
The
ladder approximation reproduces, for example, the second and
fourth terms on the right hand side of (10.3) as well as higher
order ladder terms.
In order to reproduce the cross terms of (10.4) which contribute significantly to the backscattering cross sections, we
renormalize (10.4) by considering the following expression
AM
Al
-H
>I<
+
*..
(10.5)
290
Equation (10.5) resums an infinite sequence of cross terms in
the intensity operator.
We may proceed with the usual develop-
ment of the Bethe-Salpeter equation to obtain
+lll
I
(10.6)
and approximate the intensity operator by
0+
(10.7)
Equations (10.6) and (10.7) lead to a nonlinear integral equation
from the correlation of the dyadic Green's function.
form,
(10.6) and (10.7) may be written as
In analytic
291
<G..(r,
13
+
r)
0
f
G*
Zk
(r', 3r ')> = <G..(r,
03
d'r d 3 r d 3 r'd 3r
S
l
2
2
[C(r 1 -r 1 ') 6(S
1
+ C(r
-
-
'<G.(i,
ip
j) 2
0)
Grk
1
-
zq
j')2
6
o
2, '
I)>
)>
('i
rj)><G*
6( 1'
i
k'
PS 6qr
2 )<G ps Cr, r)
2 G*qr (l 1 '
1 22') C(r 1
Gsj~2'
)><G*(i'
o
r2
(10.8)
0
Consequently the covariance of the scattering field takes the
form
d r d 3 r d3 r
E.*(r')> =f
C
+ C(r
2
r2
1-
-r2'
C(i
[<Es (?2 )><Er
1
-
'
- 2
1
'd'r '<G.
2
(
2
'
-
)
i
<G
)
<s.(r)
2)> + <E
ps
('
ip
il)><G* (i',
1
jq
(i,
')
6
2)
G*r
2
(r2) E r 0 21>
ps
qr
6
i')>
1
rq
'
2
1
109
292
where
C.
= E.
:1.
-<E
293
10.3
Renormalized Bethe-Salpeter Equation and Discussion
Continuing with the renormalization we write (10.8) in
diagrammatic form:
Th
I
--
zz~><
(10.10)
Using the first two-terms on the right hand side of (10.10)
as the zeroth order iteration, and iterating with respect to
the correlation which appears on the right side of the third
+
term in (10.10), we obtain:
jmcj
+
+
AsWt7T1
(10.11)
294
We regroup the terms of (10.11) as follows:
____
S..]
__________________________
rI
I
_
-
Th
+
+
[
+A
+7-
(10.12)
In (10.12) we define the sum of all the terms which appear in
the square brackets by the symbol
proven that
.
It is easily
satisfies the following integral equa-
U x
IC)I
+
tion:
ZAhcCQo
(10.13)
295
in which case
(10.12) takes the form:
-U
(10 .14)
Equations (10.13) and (10.14) are a pair of coupled integral
equations for the correlation of the dyadic Green's function
which analytically takes the form:
(i, P )><Gk
<G
(i
Usj,rk
Fj)><Gq(',
122 '
P
'
0
9k0
-
r 2 ') C(r1
')><G
(pi,
F2
>
r2
-
)
1J
d 3 r dar1 'd 3 r 2 dr 2 ' C(
+
<G
r ') = <G.
G*F,
(E,1 E ) G* (E' E ')> = U..
( '
aj
k
0
1)1,zk'
')
>
(10.15)
o')
j
'
aI!
E
o
)
Ir'
00
'
'
ijrk
)
Uijk
296
f
drd
3r
1
' U ip
rs
' j1
' r') Cr 1 - r1 3
)
+
(ry, r Gs ' )>
<G
(10.16)
G
where we have defined
Uij,2k
as
.
It is interesting
to note that the renormalized equation (10.16) has a structure
similar to the ladder approximated Bethe-Salpeter equation.
In fact,
if
Uijrlk
is approximated by the first term on the
right hand side of (10.15), then (10.16) reduces identically
to the Bethe-Salpeter equation in the ladder approximation.
In order to determine which terms have been picked up in the
resummation of (10.1)
(or equivalently,
(10.6), (10.7)), we
may solve equations (10.15) and (10.16) by successive iteraThe result is:
+-
+
tions.
4W
gd
ARL
=
+
<
0~0
. 0W
297
Ank
x
-H
+--
xx
+4--(10.17)
It is clear from (10.17) that the renormalization includes
ladder terms,
strongly and weakly connected cross terms as
well as weakly connected ladder-cross terms.
298
CHAPTER 11
Conclusions and Suggestions for Further Study
In this thesis, theoretical models have been developed
for electromagnetic wave scattering from
layered random
media with applications to microwave remote sensing.
Applying
Born approximations which are valid for small albedo, backscattering cross-sections have been derived with a wave approach
for a two-layer random medium with arbitrary three-dimensional
correlation functions.
Carrying to first order the backscat-
tering cross sections illustrate the
possibility of
ahh >vv
due to the Brewster angle effect at the bottom boundary.
Previous models of a half-space random medium do not reproduce
the effect of
a hh > avv
which is observed in certain back-
scattering data.
The first order Born approximation also has been applied
to the case of backscattering by a stratified random medium
with arbitrary three-dimensional correlation functions.
It
has been found that multiple resonances occur which may explain
the spectral dependence observed in active remote sensing data.
The multiple resonances are due to resonant scattering in each
random layer and to illustrate this important effect the
special case of a three-layer random medium has been used.
299
In addition the three-layer model also has been found useful
in interpreting the diurnal change of snow-ice field due to
solar illumination.
Cross polarized backscattering cross sections for a twolayer random medium have been derived by applying the second
order Born approximation to the integral equations of scattering
theory.
In the half-space limit the cross-polarized backscat-
tering cross sections do not reduce to previous results 6 3
obtained using radiative transfer theory.
The discrepancy is
due to cross terms not accounted for by radiative transfer
theory nor by the Bethe-Salpeter equation under the ladder
approximation.
In order to account for these additional cross
terms the Bethe-Salpeter equation has been renormalized by
summing an infinite sequence of terms in the intensity operator.
The result takes the form of coupled integral equations for the
covariance of the electromagnetic field.
It is found that the
renormalization accounts not only for cross terms but many
other terms in the infinite series representation for the
intensity operator.
In this thesis we have applied renormalization methods to
study the multiple scattering of electromagnetic waves by a
two-layer random medium.
Due to the presence of a bottom
boundary there exist significant coherent effects in a twolayer random medium.
300
We have solved Dyson's equation in the non-linear approximation for the zeroth order mean dyadic Green's function.
The coherent field is found to propagate within the random
medium as if in an anistropic medium with characteristic
and
TM
polarizations.
TE
The zeroth order solution to Dyson's
equation in conjunction with the ladder approximated BetheSalpeter equation have been used to derive modified radiative
transfer (MRT) equations appropriate for electromagnetic
scattering within a two-layer random medium.
The MRT equations
have been solved in the first order renormalization approximation and significant coherent effects not accounted for by
phenomenological radiative transport theories are found.
The task of developing theoretical models is by no means
complete.
We have considered primarily the case of a two-layer
random medium with three dimensional correlation functions.
Although the first order Born approximation has been applied
to a stratified random medium, the second order Born approximation as well as the renormalization approach should be extended
to multi-layer random media.
The coupled-integral equations
of the renormalized Bethe-Salpeter equation need to be developed
into a wave radiative transfer (WRT) theory which accounts for
the cross terms discussed in Chapter 10.
Finally, scattering by a composite medium including rough
surface and random permittivity fluctuations is an important
301
and challenging problem to be solved especially when we consider
the inability of volume scattering to account for backscattering
data near normal incidence.
302
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BIOGRAPHICAL NOTE
Michael A. Zuniga was born in Brownsville, Texas, on
February 23, 1951.
He received the B.S. degree in Physics
from Penn State University, State College, PA, in 1973 and
the M.S. degree in Physics from Drexel University,
PA,
Philadelphia,
in 1975.
In 1975 he became a graduate student at MIT in the
Department of Physics.
During 1975-1977 he was an IBM Fellow
and conducted research in the kinetic theory of plasmas.
In
the summer of 1977 he was with the Fusion Energy Division,
O.R.N.L., Oak Ridge, TN.
In the years as a graduate student,
he has served both as a Research Assistant and Teaching
Assistant.
Mr.
Zuniga is a member of Sigma Xi.