THEORETICAL MODELS FOR SUBSURFACE GEOPHYSICAL
PROBING WITH ELECTROMAGNETIC WAVES
by
Leung Tsang
SB,
Institute
Massachusetts
of Technology
June 1971
SM, EE, Massachusetts Institute
of Technology
February 1973
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF
at
PHILOSOPHY
the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
December 1975
Signature of Author
..
Departmen
of Ell
and Computer Science,
.r.
a
-
-...
Engineering
December
12,
.f.Eect
1975
Certified by
Tass
b-6
r
..................
Accepted byQ.
ChairmTh
0 00 0
Departmental Committee on
Graduate Students
ARCHIVES
APR 2 1976
&RAR1ES
THEORETICAL MODELS FOR SUBSURFACE
GEOPHYSICAL
PROBING WITH ELECTROMAGNETIC WAVES
by
Leung Tsang
Submitted to the Department of Electrical Engineering
and Computer Science on December 12, 1975, in partial
fulfillment of the requirements for the degree of Doctor
of Philosophy
ABSTRACT
The electromagnetic interference fringes method
radio or lower frequencies)
(at
can be used to probe subsurface
features of celestial bodies and terrestrial areas with low
In order to interpret the interference patterns,
conductivity.
we have studied theoretically the electromagnetic fields due
to a horizontal dipole antenna on the surface of a stratified
n-layered uniaxial medium.
Three approaches are used to cal1. geometric optics apculate the interference patterns:
3. fast Fourier Transform
proach,
2. mode approach, and
approach.
saddle point
residue
The geometric optics approach makes use of the
method.
The mode approach makes use of the
series method and the modified saddle point method.
The fast Fourier Transform approach is designed to treat general complicated cases when the above two analytic approaches
Numerical results from these three approaches
are not useful.
The advantages and disadvantages of
agree with one another.
Theoretical results also
each of the approaches are discussed.
compare favorably with data obtained from a model tank experiment.
Theoretical models are constructed for the microwave
passive remote sensing technique which can be used for geophysical subsurface exploration of large geographical areas.
By using a laminar structure (A laminar structure has variations in one dimension only.), simple analytic formulas are
obtained for the brightness temperatures.
The problem of
microwave thermal emission from a half space random medium
with a nonuniform temperature distribution is studied with
the radiative transfer approach. Modified radiative transfer
(MRT) equations are derived for a two layer random medium
from the Dyson equation by using the nonlinear approximation
and from the Bethe-Salpeter equation by using the ladder approximation.
It is found that the nonlinear approximation
instead of the more popular bilocal approximation should be
used for the case of bounded medium.
A two variable expansion procedure is used to solve for the mean Green's function
which is then substituted into the Bethe-Salpeter equation
to derive the
MRT equations.
The
MRT
equations give rise
to simple solution for the emissivity of the two layer random
medium. The fluctuation-dissipation approach is used to solve
for the brightness temperature of a stratified medium with a
temperature profile.
Because a stratified model can approximate a continuously varying model by including more layers,
our solution can actually apply to a stratified and/or continuously
varying medium.
Thesis Supervisor:
Jin Au Kong
Title:
Associate Professor of
Electrical
Engineering
4
ACKNOWLEDGEMENT
Professor Jin
graduate schooling.
Au Kong helped me throughout my
He provided the motivation for the
work in this thesis and gave me the encouragement and
advice I needed throughout the research.
I also wish to thank Professor Gene Simmons for
his interest and for valuable assistance in this thesis.
I would also like to express my appreciation to the
other members of my thesis committee, Professors
Robert Kyhl and David Staelin for their helpful suggestions
and criticisms.
Thanks are also due to Mr.
Eni Njoku for many
stimulating discussions.
Finally
I
would also like
to thank Miss Cynthia Kopf
for her excellent typing of this thesis.
The research was conducted at the MIT Research
Laboratory of Electronics.
DEDICATION
TO MY PARENTS
6
TABLE OF CONTENTS
TITLE PAGE
................................................
1
ABSTRACT ...................
2
............
4
.................
5
..........
6
............
10
ACKNOWLEDGEMENT
DEDICATION
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF PRINCIPAL SYMBOLS
12
..
.........
21
Chapter 1.
INTRODUCTION
Chapter 2.
DYADIC GREEN' S FUNCTION FOR STRATIFIED
.......
UNIAXIAL MEDIUM
Sec.
29
1 Dyadic Green's function for
medium
Infinite space
Sec. 2 Dyadic Green's function for
.........
a
......
*00..
3 Propagation matrices
...35
39
....................
...
30
stratified
..............
medium
Sec.
i.
Sec. 4 Reflection coefficients expressed in
.......
continuous fractions ...
Chapter
3.
DIPOLE RADIATION IN THE PRESENCE OF
.. 45
STRATIFIED MEDIUM ......
Sec.
1 Electromagnetic fields for horizontal
electric
Sec.
43
dipole
2 Transmitted fields
........
for
h alf space medium
*.. 46
..49
7
Chapter 4.
RADIATION PATTERN FOR HORIZONTAL ELECTRIC
DIPOLE.
....................................
Sec.
Radiation patterns ........................
Sec.
Graphical determination of k vector
Chapter 5.
.......
52
53
59
GEOMETRICAL OPTICS APPROACH .................61
Sec.
Mathematical development
..................
62
Sec.
Physical interpretation ...................
68
MODAL APPROACH
............................
73
Chapter 6.
Sec.
Modes contribution
........................
75
Sec.
Inhomogeneous wave contribution ............
78
Sec.
Direct wave contribution
..................
79
Sec.
Numerical results and discussion ...........
84
FAST FOURIER TRANSFORM APPROACH ............
89
Chapter 7.
Sec.
Mathematical preliminaries .................. 90
Sec.
Aliasing ..................................
93
Sec.
Numerical results and discussion ...........
99
Chapter 8
MICROWAVE PASSIVE REMOTE SENSING
.......... 104
Sec.
Reciprocity approach ...................... 105
Sec.
Radiative transfer approach ................ 106
Sec.
Chapter 9.
3 Fluctuation dissipation
approach
BORN SERIES APPROXIMATION
PROBLEMS
..............................
Sec.
1 Formulation ...........................
Sec.
2 Born series
...........................
..........
108
TO SCATTERING
....
110
....
111
..117
8
Sec. 3 Special cases ............................ 121
Chapter 10.
BRIGHTNESS TEMPERATURE OF A
HALF SPACE
RANDOM MEDIUM WITH NONUNIFORM TEMPERATURE
PROFILE .................................. 123
Sec. 1 Formulation
.............................. 124
Sec. 2 Scattering phase functions ................ 128
Sec.
Chapter 1l.
3 Solution of RT equations .................. 134
EMISSIVITY OF A TWO LAYER RANDOM
MEDIUM ----
MRT APPROACH .................. 143
Sec. 1 Introduction ............................. 144
Sec.
2 Renormalization method ----
Dyson's
equation and Bethe-Salpeter equation
Sec. 3 Mean Green's function and mean field
..... 146
.....
153
Sec. 4 Derivation of MRT equation from Bethe-
Salpeter equation ........................ 166
Sec. 5 Solution to MRT equations ................. 180
Sec. 6 Conservation of energy
Chapter 12.
................... 187
THERMAL EMISSION FROM STRATIFIED MEDIUM
WITH NONUNIFORM TEMPERATURE PROFILE
...... 191
Sec. 1 Brightness temperature of stratified
medium ................................. .. 192
Sec.
2 Numerical results
and discussion ....... .. 198
9
CONCLUSIONS AND SUGGESTIONS FOR FURTHER
Chapter 13.
STUDY
BIBLIOGRAPHY
...
.................................. 205
..................
.................... 208
BIOGRAPHICAL NOTE............................................222
10
LIST OF FIGURES
uniaxial
36
medium ......................
2-1
Stratified
4-1
Radiation pattern of HED over half space
medium .........................................
58
4-2
Graphical determination of k-vector .............
60
5-1
Interference
5-2
Interference pattern of 'a two layer medium .....
5-3
Physical interpretation of GOA ................... 69
5-4
Graphical determination of k-vector .............
71
6-1
Complex 0-plane for mode analysis ...............
76
6-2
Comparison of results
83
6-3
Interference pattern for thin layers ............ 85
6-4
Comparison of theoretical results ----
pattern
for
half
space medium
.....
by OSP, MSP and GHQ ......
67
Simpson's
rule, mode and GOA .............................
6-5
66
86
Comparison of scaled model tank experimental
data with GOA and mode .........................
88
7-1
Aliasing ....................................... 95
7-2
Comparison of Simpson's rule and FFT for a
three layer model .............................. 101
7-3
Comparison of Simpson's rule and FFT for a
six layer model ................................ 102
7-4
Comparison of FFT, mode and GOA for a
two
layer model .................................... 103
11
9-1
Two layer random medium ....................... 112
10-1
Geometry of the problem of thermal radiation
from a half space random medium ............... 125
10-2
Brightness temperature for
shelf glacier and
compared with that obtained by Gurvich et al
10-3
42
.137
Brightness temperature for continental glacier
and compared with that obtained by
Gurvich et al 4 2
10-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Brightness temperature as a function of
frequency ..................................... 141
10-5
Brightness temperature as a function of viewing
angle for TE and TM waves
..................... 142
11-1
Emissivity of a two layer random medium ....... 186
12-1
Stratified medium with temperature
distribution .................................. 193
12-2
Brightness temperature as a function of viewing
angle ......................................... 199
12-3
Brightness temperature as a function of
frequency ..................................... 200
12-4
Brightness temperature calculated with the
stratified model compared with that calculated
with the WKB method ............................ 202
12-5
Reflectivities calculated for the stratified
medium and for the WKB method ................. 203
12
LIST OF PRINCIPAL SYMBOLS
Numbers enclosed in parentheses are the pages on which the
symbols are defined.
We use superscript single prime to denote the real part of
a
denote the
double prime to
and superscript
quantity
imaginary part of a quantity.
Angular bracket <
> is used
to denote ensemble average.
A
: coefficient of upward going TE waves in the Z th region.
(37)
anisotropy factor of permittivity.
a:
a(f):
Fourier transform of g(k
(93)
(94)
version of a(f).
aliased
a p(f):
).
(54)
B :coefficient of downward going TE waves in the k th
region.
(37)
B, Bo, Bh : emission sources. (106),
Bf(
,
)
defined in text.
(117)
b : anisotropy factor of permeability.
b(
)
function of dielectric
correlation
uations.
C
C
: defined in
:
(135)
(54)
constant fluct-
(129)
text.
(160)
coefficient of upward going TM waves in the k th
region.
(37)
c : speed of light in vacuum.
(108)
13
defined in
D
D
(175)
text.
: defined
in
text.
(175)
defined
in
text.
(115)
D2
: coefficient of downward going TM waves in the Z th
D
region.
d
(37)
:depth of the k th layer.
lu'
(36)
u 'd
ld :fluctuating parts of the field in region 1 of
upward and downward going waves respectively.
(z')>
<,,(z)E
: covariance of the fluctuating part of the
electric field
(z)
E.m
mean field in
region X.
field.
(118),
0
e
e
(118),
(30),
(45)
components of electric field in cylindrical
(47)
electric field of incident wave.
(111)
: unperturbed electric field in region k. (116)
emissivity of the medium.
(105)
: unit vector in the direction of electric field for
TE wave in the k th region.
erfc
: error
F0
Nyquist frequency of a(f).
F
k.
(115)
coordinates.
E.
(151)
(147)
: electric field vector in region Z.
(E , E , Ez)
E
region Z.
in
fluctuating part of the field in region
E9 :scattered
E
2FO.
FFT : fast
(166)
function.
(34)
(81)
(93)
(93)
Fourier transform.
(23)
(147)
14
: defined in text.
f(z)
f 2 : def ined in
G
.
ijm0
, G
.
:
ijml
(113)
(181)
text.
respectively
zeroth order and first
order
mean Green's functions with source in region
j and observation point in region i
. .
:dyadic
Green's function with observation
region j.
region i and source in
=t
G
point
Gaussian quadrature formula.
: geometric optic
GOA
0
G..
approach.
(155)
in
(35)
transpose of the dyadic Green's function.
GHQ
.
(37)
(79)
(23)
: unperturbed Green'n function with observation point
1J
in
region i and source in
g : defined in
text.
: def ined in
text.
(177)
(176),
(91)
. (96)
gp (k ) : aliased version of g(k
(1) : Hankel function of first
kind and order n.
Hn
HED : horizontal electric dipole.
(H9, H,
Hz)
coordinates.
TM wave in
I
: electric
I(
,
)
the direction of electric
the Z th region.
dipole moment.
:
intensity
in
cylindrical
(47)
(34)
specific intensity. (106)
IZ
(47)
(58)
: magnetic field components
: unit vector in
(113)
(158)
gf, gb : defined in text.
g(k )
region j.
(46)
operator.
(151)
field for
15
: correlation between fields of upward and downward
I
(168)
going waves.
I
: denoting respectively the poles, branch point
, Ib' Is
and saddle point contributions of the integral
occuring
:
Iu
in
chapter
6.
(74)
(124)
1. upward propagating specific intensity.
2. magnitude square of the fluctuating part of the
electric field of upward going wave. (167)
1.
Id:
(124)
downward propagating specific intensity.
2. magnitude square of fluctuating part of the electric
field of downward going wave.
I
)
( ,
)
I2
,
)
into
source of scattering
J
I d.
direction
of I.
(106)
)
: respectively covariances of fields of
(167)
upward and downward going waves.
J
'
'
,
c2
covariance between fields of upward
)
and downward going waves.
function of
J
( )
K
: Boltzmann's constant.
: Bessel
text.
first
Ak
: sampling interval of g(k ).
kz)
(47)
(93)
(31)
: wave number in
(kr, k y
n.
(178)
ks : defined in
k : wave vector.
kind and order
(167)
(106)
Ks,
k
(171)
: defined in text.
(124)
di '
)
J'
)
respectively sources of scattering into Iu and
:
d
Ju'
,
14 (
(167)
k th region.
(39)
: wave vector in cartesian coordinates.
(31)
16
k zZ,
wave vector components of the k th region in
k z
z direction for type i
(33)
respectively.
E
,
k
:
wave vectors of the Z
:
(34)
integration variable occuring in integral representof electromagnetic
ations
k l m: mean wave number
Ll,
th region for type I and
waves respectively.
type II
k
and type II waves
L2
in
fields
in region 1.
: defined in text.
(158),
EIF method.
(46)
(113)
(161)
t : correlation length of fluctuations of permittivity. (117)
za'
s
: respectively absorption and scattering lengths.
M
: mass operator.
M
, M2
: defined
(148)
in
text.
(159)
(28)
MRT : modified radiative transfer.
MSP : modified saddle point method.
N
: number of data points in FFT.
Nl, N2
(153)
: defined in text.
(79)
(92)
(159)
saddle point method.
(79)
OSP
: ordinary
Pf
Pb : respectively forward and backward scattering phase
PfI
functions.
(124)
pb : defined in
text.
(179)
Q(,
)
RT
radiative
RTE,
RTM : reflection coefficients for TE and TM waves
: mass operator.
transfer.
respectively.
(149)
(27)
(35)
17
R
reflection coefficient for TE wave in region
k,
caused by boundary separating regions Z and 9+1.
r : reflectivity of the medium.
:
r
radius
vector.
and k+l.
0,
(r,
$)
(105)
(30)
reflectivity
r,(9 +1)
of boundary separating
(119),
Sr
S
T
T
(53),
(55)
reflection coefficient for TM wave in region
Z,
caused by boundary between regions k and Z+l.
(41)
: Poynting vector.
,
k
spherical coordinates indicating location of
radial component of Poynting vector.
TE
regions
(126)
observation point in EIF method.
Sk(k+1)
(55),
(55)
(59)
TM..
: transmission coefficients for TE and TM waves
respectively.
(38)
T : 1. period of g p(k ) ; (96)
2.
TE
TB ,
temperature of the medium.
TM
TB
:
(z)
(105)
brightness temperature of TE and TM waves
respectively.
T
(41)
(109)
temperature profile in region
T(z)
: temperature profile.
To'
h : defined in text. (134)
(126),
9.
(192)
(134)
t 9I(k+l): transmissivity of boundary separating regions
and
Z+l.
(183)
k
18
=TE
=TM
V (Z+1)'
(Z+1)k : forward propagating matrices from
to region
region k
Z+l for TE and
TM waves respectively.
X.. : 1 + R...
(49)
Y..
(49)
:
1 + S. ..
(40),
(41)
y1 , y2 : quantities used to test conservation of energy.
(187)
a
:1. defined in text;
(56)
2. effective extinction per unit length;
3.
am
defined in
text.
(181)
denoting location of the saddle point of the m th
image
source.
defined in
(63)
text.
(56)
long distance scale.
6:
6(
(134)
(155)
variance in fluctuations in dielectric constant.
)
: Dirac delta
function.
(117)
(31)
I ,1 n2 : effective propagation constants in random medium.
(161)
permittivity tensor in
medium. (30),
(36)
: permittivities of the uniaxial medium in region
CZ, ez
9.
Cer :relative
Ce(r)
the Z th region of the uniaxial
(30),
(36)
permittivity.
(106)
imaginary part of permittivity at position r
region 9. (108),
(192)
in
19
(111)
im
: mean permittivity in region 1.
if
: fluctuating part of the permittivity in region 1.
F
for wave
coefficient
reflection
medium.
(112),
y
defined in
K,
K:
text.
(134)
downward propagating
region for type I
Ka
: absorption per unit length.
K
: 1.
scattering per unit length;
scattering wave number.
2. defined in text.
1
,
K2
z in
the
waves
(126)
(106),
(126)
(106),
(153)
: 1. total extinction per unit length
K
K
and type II
in
(34)
respectively.
2.
onto random
incident
(115)
wave vectors
k th
(111)
;
(126)
(178)
: two times the imaginary parts of
1 and n2
respectively. (175)
: permeability tensor in
P
y
region k.
(30),
(36)
, y9z : permeabilities of the uniaxial medium in region
k.
(30),
defined in
(36)
(91)
text.
v
:
o
: angular frequency.
: 1.
2.
(30)
scattering albedo;
defined in
text.
(135)
(181)
20
(p,
$,
z)
components of cylindrical coordinates
denoting location of the observation point
in
60
1.
EIF method.
refer to
(r,
(47)
60, f);
(53),
(55)
2. viewing angle of radiometer.
6
(124)
1. complex integration variable in the plane of which
contour integration is carried out;
(73)
2. direction of specific intensities Iu and Id*
(124)
21
Chapter 1
Introduction
Electromagnetic methods have been used widely for
geophysical exploration.1-2
Until recently, the main thrust
of interest lies in probing areas of relatively high conductivity.
Surface effects are dominant in such cases and rough
surface scattering behavior has been studied extensively.
Subsurface probing of such areas is chiefly carried out by
1 3
1 2
the induction method, - the antenna impedance method, ' -4
and the wave tilt method,5 etc.
Recently there is an interest in subsurface probing of
highly resistive geological environments, 6,7 such as ice and
snow areas, glaciers and deserts.
The surface layers of many
planetary bodies e.g. the moon, are also of very low conductivity.
feasible.
For such cases, electromagnetic wave methods are
In 1955, radio frequency interference fringes
(EIF)
were used for geophysical prospection of underground water in
the Egyptian desert.8-9
Recently the technique was developed
for use during the Apollo 17 mission to measure the subsurface
electromagnetic properties of the moon.
The
EIF
method consists of a
transmitting antenna which
22
radiates
at
radio or
lower frequencies
and a
receiving
antenna
which measures wave amplitudes of the various wave components.
With the receiving antenna transversing surveying areas, the
measured data can be plotted as a function of distance from
the transmitter.
Due to the interference of various wave com-
ponents generated, the result is an interference pattern.
The
peaks and troughs of the pattern conceal information about the
subsurface electromagnetic properties of the medium.
The problem of using dipole antennas in geophysical
An excellent
probing has been studied extensively. 1,10-28
review of the half space case is contained in the book by
Sommerfeldl0 and in the monograph by Banos.11
Propagation and
radiation in stratified medium are treated by Wait, 1
Brekhovskih,17 and Budden.29
2
Ward, 1
3
Wolfl8 and Bhattacharya19 con-
sidered the case of dipoles on a two layer earth.
Wait 2 0 - 2 1
solved the problem of electric and magnetic dipoles over a
stratified isotropic medium.
The case of an anisotropic half
space was studied by Chataev22 and Wait.23
and Bhattacharya,
25262
and Sinha2 6 -27
Praus,24 Sinha
treated electric and mag-
netic dipoles over a two-layer anisotropic earth.
Wait 2 8
formally solved the case of a horizontal dipole over a
fied anisotropic medium.
strati-
All these works are carried out by
means of Sommerfeld's Herzian potential functions and the
primary interest is concentrated in the limits of high conductivity.
The theory of the
EIF
method, which applies to low
23
conductivity areas such as ice, snow, deserts, and lunar
surface,
is
lacking.
In
this
thesis,
oretical basis for such a method.
stratified
we will develop a the-
The model chosen is a
medium.
Kong 3
0
abandoned the conventional method of using poten-
tial functions and chose the electric and magnetic field along
the principal axis to be the fundamental scalar functions in
terms of which all other field components are expressed.
Geometric optic approach has been used to interpret interfer-
.31-35
ence fringes of a two layer isotropic medium.
In this thesis, we use dyadic Green's function to relate
the electromagnetic fields to the source.
In chapter 2 Dyadic
Green's functions for stratified uniaxial medium are derived.
In chapter 3,
integral representations for electromagnetic
field components for a horizontal electric dipole are obtained.
Three approaches are used to solve the integrals--the geometric
optics approach
(GOA), the mode approach, and the Fast Fourier
Transform
approach.
(FFT)
The geometric optic approach interpret the interference
fringes as generated-by-the interference of waves given rise
by image sources.
To facilitate physical interpretation of
the geometric optics approach, we plot the radiation pattern
of the horizontal electric dipole over a half space uniaxial
medium in chapter 4.
In chapter
5,
the geometric optics ap-
24
proach is used to solve the interference pattern for the
dipole over a two layer uniaxial medium.
A combination of
saddle point method and branch cut contour integration is
used.
The geometric optics approach yield simple formula for
the various wave contributions.
It is useful if the slab is
thick and lossy when only a few images are needed.
In chapter 6, we discuss the mode approach.
The residues
method, the modified saddle point method, and the GaussianHermite quadrature method are exercised.
The mode approach
interpret the fringes as generated by the interference of the
modes and the direct wave from the antenna.
It is useful when
the slab is thin so that only a few modes are excited.
results of
GOA
The
and mode are compared with each other and
also with data from a model tank experiment.
Analytic methods are not useful when the number of layers
gets large.
On the other hand, a direct numerical integration
consumes a lot of computer time.
In chapter 7, we introduce
Although this method con-
the Fast Fourier Transform method.
sumes more computer time than the analytic approaches, it can
treat cases of a general nature.
Since a continuous profile
can be approximated by a stratified model by including more
layers, interference pattern in the presence of continuously
varying medium can also be handled.
We also compare numeri-
cal results from the three approaches--GOA,
and
FFT.
mode approach,
25
Geophysical subsurface exploration of large geographical
areas can be handled by the method of microwave passive remote sensing with a radiometer aboard aircraft or satellite. 3
6
In this area of passive remote sensing with microwaves, considerable effort has been spent in collecting data from spacecraft and satellites.
While voluminous data keep on piling
up, theoretical models are lagging behind.
Emphasis on theory
has been largely restricted to scattering effects due to rough
surfaces. 3 7 - 4 1
Theories are few for the case of low conducti-
vity areas, such as snow, ice covered land or water, desert and
lunar areas, where absorption, subsurface layering and
scat-
tering, and nonuniform temperature effects are dominant factors.
Assuming uniform temperature distribution, Gurvich et al 4 2
derived expressions for the brightness temperatures of a half
space random medium with a laminar structure in the single
scattering approximation.
in one dimension only.)
(A laminar structure has randomness
England43 examined emission darkening
of a half space containing randomly distributed isotropic point
scatterers by employing the radiative transfer approach.
With
the model of a vertically structured medium, Stogryn44 formulated the solution for the brightness temperature of a vertically structured medium in terms of a two-point boundary value
problem by solving a second order ordinary differential equation, together with the evaluation of an integral.
He also
26
studied scattering by random dielectric constant fluctuations
in the low conductivity limit by using the distorted Born
.45
approximation.
Extending the half space model of Gurvich et al, 4
2
Kong46 considered a composite model, which accounts for absorption,
layering,
tropy.
submerged
scattering
The composite model was used 4
from satellites
ments.50-51
and from aircraft
The objective
in
centers
7
~4
9
and aniso-
to match data collected
with ground truth
this
thesis
is
measure-
to develop
theoretical models that are practical and useful in data
interpretation.
'The half
space model of Gurvich et
al42 does not simulate
the practical situation because the Antarctic area exhibit
nonuniform temperature distribution. 5 2 -5 3
In addition, for
highly resistive areas, e.g. the shelf glaciers, multiple
scattering effects are important.
The layered model of Kong46 assumes that due to scattering
only the noncoherent part of the field is needed.
It does not
simulate the practical situation because there is significant
coherent effect
5 1 ,5 4
in a layered model.
In chapter 8, we outline the three approaches that have
been used to obtain the brightness temperature--the reciprocity approach, the radiative transfer approach, and the fluctuation-dissipation
approach.
27
In our consideration of scattering models, we have
This
chosen the laminar structure used by Gurvich et al.42
solutions
choice enables us to obtain simple analytic
for the
brightness temperatures in many cases.
In chapter 9, we discuss the Born series approximation
to scattering of waves by a two layer random medium.
series is a result of
The Born
iteration of the integral equation for
the scattering problem.
The series converge very fast if ab-
sorption dominates over scattering.
In chapter 10, we use the radiative transfer approach to
solve the brightness temperature of a half space random medium
with a nonuniform temperature profile.
By assuming far field
The radiative trans-
interaction, phase functions are derived.
fer approach
(RT) can deal with multiple scattering effects.
In the case of low albedo, the results from
RT
reduce to
that of Born series approximation.
In
chapter 11,
random medium.
two layer
adequate.
medium
we study the emissivity of a two layer
There is
so that
significant
coherent effect
a
transfer
radiative
A wave approach is needed.
in
a
approach is
in-
The renormalization
method has been used widely to study wave propagation in random unbounded medium. 5 5 - 5 6
for the mean field
It gives rise to the Dyson equation
and the Bethe-Salpeter
covariance of the field.
The bilocal
equation for the
approximation and the
ladder approximation are usually applied to the Dyson's
28
equation and the Bethe-Salpeter equation respectively.
We found that the nonlinear approximation,
57-5 8
55
-5
6
instead of
the more popular bilocal approximation, should be used for
the case of bounded medium.
We use a
two variable expansion
procedure to solve for the mean Green's function which is then
substituted into the ladder approximated Bethe Salpeter equation to derive a set of modified radiative transfer equations
(MRT).
They are modified because, in our derivation, we have
included the correlation effect between waves going in the
The
upward and downward directions.
to simple analytic solutions.
equations give rise
MRT
Conservation of energy59 is
shown to be preserved under the nonlinear approximation.
the case of
low scattering albedo, the
that of the Born series approximation.
MRT
In
results agree with
For the half space case,
they agree with those from radiative transfer in chapter 10.
In chapter 12, we use the fluctuation-dissipation approach
to find the brightness temperature of a
nonuniform temperature profiles.
of chapter 2 is used.
stratified medium with
The dyadic Green's function
The solution to this problem is an im-
portant one because only very few special profiles can be approximated and solved in analytic forms.
By including more
layers, a stratified model can approximate a continuously
varying model.
29
Chapter 2
Dyadic Green's Function for Stratified Uniaxial Medium
For linear fields, wherein the superposition principle
is applicable, the basic problem is that of determining the
field excited by a point source.
source is the Green's function.
for
the vector electromagnetic
The field due to a point
Green's functions are dyadic
fields.60
Since stratified
medium is the basic geometry in this thesis, dyadic Green's
function
G
for stratified uniaxial medium will be derived
first.
The Fourier transform approach61 will be taken to derive
the dyadic Green's function in the absence of stratified medium.
The dyadic Green's function in other regions of a
stratified medium is then obtained by matching boundary conditions.
With the aid of propagation matrices,
61
wave
amplitudes are easily calculated in terms of those in any
other region.
A closed form solution for the reflection
coefficient is obtained in the form of continuous fraction.
30
2.1
Dyadic Green's Function for Uniaxial Medium--Infinite
Space
In
for
this
section we derive the dyadic
an infinite
the
z
uniaxial
axis.
Green's function
medium with optic
axis
parallel
to
The Fourier transform method is adopted.
From Faraday's law and Ampere's law, it
that the wave equation for the electric field
(V -y
- _-: Er) -
= We
-V
-
=
iw
can be shown
E
-
~) --
is
(2-1)
where
Fl/p
0
0
1/p
0
0
0
(2-2)
0
1/py
is the inverse permeability tensor,
E=
is
0
0
0
6
0
0
0
S
the permittivity
(2-3)
z
tensor,
I
denotes the identity
matrix,
31
and
V
V - A = V x A
is defined by
for any vector
A.
The dyadic Green's function satisfies the following
equation
(V
=-o
V
-
-
E:) - =
G (r- ,
r'-
=
-- I6(r
f
=i
r'r)
(2-4)
.
We write the Fourier transform pair of dyadic Green's function as
G'
r')-
r G(r,d 3dr
ik - r
e
(2-5)
-Co
00
G(r,
r')
=
1
d 3 k G(k,
-f
8'r3
=
(2-6)
_0
On substituting (2-6)
G(k, r')
- r
-)ik
r'
e
into
-iw[k
(2-4),
=1
- y
we obtain
k + W
2=
2_
1
e
ik
-r'
(2-7)
where
0
-k
z
-k
y
z
k
0
-k
k
0
x
After introducing
y
(2-8)
x
(2-2) ,
(2-3) ,
and
(2-8)
into
(2-7),
32
and performing the matrix inversion, we obtain,
-k k
k 2
G (k,
r'
-
=-
.-.-
= -
ZZ
WE z
k
x
2
+ k
y
k
k
+ 1
D2 (k
x
2
+ k
k
x
_
zk
2
+k
x
_
-k k
2
(k
2
2
-
.r'
e k
2
+ k
k
2
)D
2
x
2
2
+ k
2
2
2
W 2PICz
x
k
2
x
+ k
k
y
x
k k
_
2
0
0
k k
y
k k
x
z
2
zk
2
+ k 2
2
kk
y
y
z
y
k k
z x
0
y
y
y x
k 2
x
2
k2+
01
y
x
-
y
x
-
[
( )
+ k
y
yZkk
x
y
W2
z y
k
2_
2-
(2-9)
where
D 1 (k)
=
2
k
+
-E- (k
2
+ k
2
2V
-
y
(2-10)
E)
z
D 2 (k)
=
k 2 +
(k
z
2
x
+ k 2 _ W2P
).
(2-11)
Z
To evaluate the integration over
kz
in
(2-6),
we find that
33
D2 (k),
corresponding to
k m
z
2
=k
-
k
=
and
k
k
and
D 1 ()
the poles of the integrand occur at the zeros of
k ze
z=
where
(2-12)
+ k y 2 )/P z
(kx 2
and
k ze -
/2
E(ka2 + k 2)/E
x
y
-
(2-13)
z .&
They correspond to the type I and type II characteristic waves
in
the medium.61
Assuming the medium is
slightly lossy,
then
we find
C
87f 2
x
0
-
(E
r')
-
ee e
k
y
e
e
)h(k zj )e
z
kze
ik
A
t
dk dk
1e
h(k
+
G(r,
~
ff
--
r
e
for
z'
< z
r')
00
8ir
+
2
1
kze
z
iK
dk dk
ff
&
(r - r')
em
^
^
e
i7e
h(-k e )h(-k
)e
z
z
for
z < zI
(2-14)
where
34
k
K
= x k
m
= x k
e
+ y k
x
+ y k
x
A
K
e
= xk
m
y
+ z km
(2-16)
y
- z ke
(2-17)
z
z
A
A
+ yk
x
y
=(x
x
)
2
=
+ k
y
z
(2-19)
- y k )
2
x k ]
(2-20)
[e x K ]
(2-21)
[e
-
(2-18)
- z km
k
k
h (k
(2-15)
y
x
e
ke
z
+ z
+ y k
=x k
k
k
e)
h(-k
)
k
z
and
direction
1A
-
k = w/pE.
We see that
of the electric
field
e
is a unit vector in the
for
TE
waves and
in the direction of the electric field for
TM
h
waves.
is
35
Dyadic Green's Function for Stratified Medium
2.2
Consider a point source located above a stratified
medium.
Let region
(Figure 2.1)
0,
The field in region
for
0
be free space.
z < z',
Thus, using equa-
of both upward and downward going waves.
tion
(14),
must be composed
the dyadic Green's function in this region must
assume the following form,
G
00
(r, r')
= -
8T2
f/
1
dk dk
-{(R
TB"
e (k)
e
ik
-r
z
+ e(-k
)
e
1K
r
z
+ (R TM hh (k)
) e(-k z ) e
-1K
*
e
}
) h(-k ) e
z
+ h(-k ) e
z
for
z < z'
(2-22)
In
writing dyadic Green's functions,
subscript to denote the region in
is
located
and the second
subscript
which the source is located.
RTM
we use the first
which the observation point
to
denote the region in
In equation
(2-22),
RTE
and
are reflection coefficients the values of which are to
36
Figure
Stratified
2.1
Uniaxial Medium
z
Region
0
p,
S
z= 0
,
Region 1
z
=-d
z =
-d
Region k
Region n
yn'
z
=-d
z
=
-1
n
z =
Region t
-d
it' St
-d
n
37
be determined by boundary conditions.
the dyadic Green's
for
relation
(r,
S
00
(',
G0')
00
=
where the superscript
it
Using the
symmetry
function,
(2-23)
r)
t
denotes the transpose of the matrix,
immediately follows that
CO~
S
(r,
F')
f
-
oo
x
802
-iK
- r'
Z
1
ykz
"ikr
) e
{e(k
(R
TE "
e (-k
z
z
-r'
~-ik
+
e(k
) e
+
(RTM h(-k ) e
+ h(k ) e
+ h(k
-
dk dk
) e
)
.
z > z'
(2-24)
In order that the boundary conditions can be matched at
z = -dn,
n =
0,
1,
2,
...
,
for all
the dyadic
r',
Green's
functions in other regions must assume the following form,
iK m
r')
S (r,
to
(,2
=
-
JL
j
dk dk
{(Az e (km
z)
-z
.- m
+Be
( k
)
e
-
) e(-k)
e
38
+ (Czh (k)
e
in region
~ e
e
)
+e
+ zhk(-k Z)
e
)
h(-k )
e-
K
- r
(2-25)
and
Z,
.- m
G to
(r,
r'
//
E
8Tfr 2
=
e(-kz)
+TTM
..
TTM
e
-1K
h (
t
The coefficients
dk dk
x
Y k
_00
TE
,
t
Z
( -km
tZ
et
-
R
TM
,
}.
h(-k ) e
t
e
tz
R
{TTE e
A,,
Bt,
Ck,
Dk,,
T
TE
(2-26)
and
are related to one another through boundary conditions.
We note that the
TE
wave corresponds to type I char-
acteristic wave because the electric field is perpendicular
to the optic axis.
In the same way,
TM
wave corresponds
to type II wave because the magnetic field is perpendicular
to the optic
axis.
39
Propagation Matrices
2.3
The boundary conditions for the dyadic Green's functions
are the continuity of
-d
at each boundary
z x V x G.
continuity of
Using equation
z x G
(2-25),
and the
we find
that
-ikm
d
zz k + B e
e
A
ikm d
z
k
e
ik
e
+ Bk
km
k
[A9 e
ik ' d 2
(k
d
-ikm
(9 + 1) z k
)d
+ 1) z z
(2-27a)
km
ik z d
- B91e
+ 1)
-ik
[At
ke
e
(k
d
d
+ 1) z A
ikm
e
B
-
k
kz
-ik
12
+ 1)z
k (k +1)
(0C
--
(2-27b)
ke
iked
D 91z k.
-ike d
9,z z
(Cz e
d
(9. + 1) z 2],
[CR e
+ 1 e
-ik ed
Z
2
d
ik
+
d
(2.+ 1) z k. - D9
1) z 2.9(-2a
+
(2.
e
+ 1
+ D e
ik e d
z
k
-
(=.+1)
11(z
+ 1)
2-2 8a)
40
-ike
A
with
= 1,
D
B0 =
= RTE,
Ct
=
d
0,
Dt
=
ee
(9
Bt =
+ 1) z
TE,
d
9]
(2-28b)
C0 = RTM
TM
(2-28),
we can express the
in terms of those in region
Z + 1
wave,
-ikm d
+1
+ 1)zd
e-ik
Z + 1
(2-27) and
wave amplitudes in region
TE
+ D
At = 0,
1,
and
Using equations
For the
ike
(P + 1) z
[Ck + 1 e
-TE
V(k + 1) 9
(
B 9,+1e
where=TE
V (9+
where
+ 1) zdk + 1
1)z
B~e
TE
is called the
ikm d
Z
(2-29)
forward propagation
matrix and is given by
=TE
V(9 + 1)
k
eik
R(
9,
-
1.(~
+ l
2
p
+ 1) z (d
+ 1
+ l)(
ik
9,
y- + 1) kmz
(,
km
k (
-
+ 1)z
+ 1),
dz )
)e(Z +
R~
t
+ 1
-
d
)
e
dk)
1
1) z
(9 + 1)z (d9, + 1
-
d )
(2-30)
41
where
1-y
+ 1)
1
km(9, + 1) z /1 ((9 + 1) km
9,
k
9+ km
(9,+ 1) z
/P1
is the reflection coefficient for
(9,+ 1)
TE
km
waves in region
caused by the boundary separating regions
For
-ik e
+ 1) zd 9, + 1
ik e
+ 1
9 +
and
1.
ike d
9,
z 9,
C e
=
e
9
waves, we have,
TM
+ 1 e
C
(2-31)
m
Mk
d
VTM
V(k + 1)t
9
(91 + 1) z 9, + 1
D e
ike d
9z
(2-32)
where
=TM
V(91 + 1)9
(9,+ 1)kk
2
K
1 +
-ik e
+ 1 - d )
(9,+ 1)z (d9
SZ +
,e
ke
F-9
y k(z + 1)
e
ke
i(9, + 1)
(9 + 1)z)
(9
, + 1),e
ik e
(d
)
9,+ 1 - d 9,
(9, +1) z
e
+ 1)z(d z + 1
+ 1)z
i(Z
(d
-
- d
dt)
)
(2-33)
and
Eke
S9k(k + 1)
- k (9
+ 1)z
z + 1
ke
9,Z
E
/
1
+
ke
(9 + 1)z
E9
k z
(2-34)
42
is the reflection coefficient for
TM
waves in region
caused by the boundary separating regions
k
and
k
k + 1.
43
Reflection Coefficients Expressed in Continuous Fractions
2.4
By using equations
(2-27) and (2-28),
reflection coefficients
RTE
and
RTM
we can cast the
in the form of con-
tinuous fractions.
RTE
--
-i2k m d
1
(1
Rl
1
+
/
/
1 ---
1K
e
-i2km
lzd
-i2k
R12
(d2
-
1+.... 000*
R12
+
1
--- e
d1 )
-i2k mz(d 2
-
d1 )
e
+
+
R(n
+ Rnt.
-
1)n
R2
(n
-
-i2km
nz (d n - d n - 1
1)n)
F
/
-i2km (d
nz dn
1
R(n -
~ nd
n
-
l)n
(2-35)
(
44
/
-i2k
d
1
RTM
--
1
K
+
Sl
1 i 2 ke d
S~l
S12
-e
S01
-i2k e(d
1
- d
2
)
)e-
- --
Sl 2
+
+
-i2ke
1
- d1 )
(d 2
-e
S12
1
1
+
S (n
+
e
1-'
-i 2 ke (dn - dn n
nz
1)
S (n-Lnl-
)n
-i2kenz (dn - d
n -
1e
S (n
1)
)n
(2-36)
+ Snt.
The procedure is to calculate
using equations
Ci,
Dy,
TTE
(2-35) and
and
TTM
(2-36).
RTE
and
RTM
The values of
first by
A.,
B ,
can then be obtained through the
forward propagation matrices.
45
Chapter 3
Dipole Radiation in the Presence of Stratified Medium
Once the dyadic Green's function is determined, the
electromagnetic fields for arbitrary source distribution can
be found.
region
0,
E
(r)
For a given electric current source
the electric field in region
=
f/f
d 3 r'
G o(r,
r')
Z
j(r)
in
is given by
J(r').
(3-1)
In this thesis, we illustrate the use of horizontal
electric dipole for the
EIF
method.
In this chapter,
single integral representations are obtained for the electromagnetic fields of this horizontal dipole in the presence
of stratified
medium.
46
3.1
Electromagnetic Fields for Horizontal Electric Dipole
Consider a horizontal electric
origin and pointing in the
x
dipole placed at
the
direction with current dis-
tribution
j(r)
= I k 6(r).
(3-2)
Using equation (3-1), the electric field in region
k
is
given by
E=
G
(r, r' = 0)
- x.
In the zeroth region, the
(3-3)
z
component of the electric
field is, on substituting the appropriate values in equation
(2-24)
00
=
E
it
8Tr
2
ff
dk dk k
(F
x00
y
e
ik
- r
(1 -
R
).
(3-4)
We can convert the above expression to cylindrical
coordinates by setting
= k
k
k
y
= k
cos k
(3-5a)
sink
(3 -5b)
p$
47
x =
(3-6a)
p cos $
y = p
sin
$ .
(3-6b)
The location of the observation point is expressed in
cylindrical coordinates,
p,
#
and
z.
On using the following Bessel function identities, 6 2
=
n
2r
1
2Trin
eiB cos 6 + in
6 dO
-vyri H
H(1) (
( )
(3-8)
we can further simplify the expression
k
8rs
(3-7)
0
-Co
2
p
dk
p
eikz
in
(1 - R TM)
equation
H
1
(3-4)
(k p) cos
p
.
(3-9)
The procedure outlined in equations
repeated for other field components.
in cylindrical coordinates with
and
H
denoting
[H9,
H ,
Hz]*
E
(3-4) to
(3-8) can be
We express our fields
denoting
[Er,
E,
Ez
48
TM ik z
ikzk
k
-oo
\
-
z
i(
dk
TM =f
(1
z
H
RTM
ikzz
H
RTM
ik z
z H
RTMp
(kp) sin #
1p
p
8TrE/
k
p
(1
-
cos $
(k p)
(k p p) cos $
1
(3-10)
1 (1
-
RTM)
k. (1 -
RTM)
ik z
H
(k p)
sin $
p
00
-TM
dkp
(It/8r)
p
e
H (1) (k p) cos $
1
p
j
0
(3-11)
l
Z
kp
(.1
+ RTE
eikz
(1
+ RTE)
ez
dk
-o
(Iwp/8Tr)
-p
1
(k p) cos
p
ik z,
k
FTE = f
H
ep
z
H
(k
p)
sin
kz
0
(3-12)
ik
ik
(1 + RTE
e
p
sin $
pk
i(1 + RTE
-TB7
p
kP2
k
(1
+ RTE
ik zz
H ()(kp)
1H
ik zz H~1
p
Cos$
) si
(k p) sin
1 p
I,
(3-13)
49
3.2
Transmitted Fields for Half Space Medium
The fields in other regions can be calculated with the
help of equation (3-3).
In this section, we illustrate this
by calculating the transmitted fields of the half space medium.
For a half space medium,
RTE = R
(3-14)
RTM = S
(3-15)
TTE
(3-16)
=
XOl
lk Y
TTM
1
(3-17)
yk 1
where
X..
1JJ
=
Y..
=
lJ
(3-18)
1 + R..
J
+ S...
(3-19)
IJ
Substituting equations
(3-14) to
(3-17) into
(3-3),
using equation (2-26), we find that, fields in medium
are given by, in cylindrical coordinates,
1
50
00I
f
-ik
k k
dk
p lz
_
~p
-0
ez
e
lz H
1
01
Bro
(k
p) cos
$
p
e
-TM
I9
dk
f--Od
L1
dk
(
It
/
1
zz H(
87 -we pz
00
f
kz
1
y0
\
k
p
z
lZZH ()(k
-ik
Y01 e
-i
(k p)
1
p
sin
$
p)
cos
$
lz /
(3-20)
r
00
dk
-00~
00
HTM =
f dk
-00
(iz1
81T p
)
-ik e
Y0
1
e
eeik
lz
Ir
k
\8
7T P)
y
e
01 e
-iklzz H (k p)
H
1
p
1
(kp)
1
p
sin $
cos$
0
(3-21)
51
It
00
f
-00
p
00
-TE
EB
f
-O
dk
op
T
-
dk
TTE H
(k p) e
- ikm z
lz
Cos
8T kzP)
It
op kpT
(l
-ik mz
l
P T TEH ()(k p) e
87r k
p
p
sin $
0
(3-22)
6
IPk km
co
f
dk
-o
It
00
flTE =
f
-00
f
dk
(87
dk
-00
k p
lz
8' kz
i
p
I
1
1
km y
k1
kz
-ik m z
T TE H
)
(k p) e
lz
sin#
p
-ik m z
TTE H
(k p)
e
lz
cos #
l
TTE H (
(k p)
-ikm z
lz
e
sin#
87r k1
(3-23)
52
Chapter 4
Radiation Pattern for Horizontal Electric Dipole
To facilitate physical interpretation of the interference pattern through the geometrical optics approach, the
radiation patterns for the horizontal electric dipole over a
half space medium are calculated.
The mathematics in this
chapter and the next, with appropriate scaling, follow closely
that of reference 31.
the first
order saddle-point contribution of the integral.
The directions,
are noted.
The radiation field is equivalent to
in
which maximum radiation energy are launched,
These directions will later
on be related to the
geometrical optics interpretation of interference patterns.
In section 2 of this chapter, the direction of propagation of wave and the direction of Poynting power flow from
the dipole are related graphically.
53
4.1
Radiation Patterns
By using the saddle point method of integration,
asym-
totic evaluation of the integrals can be carried out for large
radial distance.
In reference 31, it was shown that, for a
typical integral,
o F(k
ik2
)k
- k
2z
H(1) (k
0
/k2
--
-
k
p
the saddle point occurs at
k
p = k sin k 6
kg = 60
0
point contribution to the integral
2 eikrF()
s
i
4fr
p)
dk
p
(4-1)
2
on making the transformation
= 2 -
p
+
tan~
Is,
(F"(6)
and
p/z.
k z = k cos k,
The saddle
to second order, is
+ F'(6) cot
6)
(4-2)
2ikr
where
r=
p2
Z2.
Inside the uniaxial medium, from equations
(4-3)
(3-20) to
(3-23), we observe that a typical integral assumes the form
54
k P2
0
F (k )dk
I: = f
- --
z
f
e
p
p
-00
k2
i
H ()(k
0
P
p)
k 2
dk
p
(4-4)
2
k2
f
where
a,
f
=
a
=
b
=lz
b
lz
1
(4- 5a)
1.
(4- 5b)
Equation (4-4) is of the same form as equation
[F
k
we identify
k
and
z
-+
(4-1) if
Therefore, the saddle
z/v'.
point occurs at
VE k lp
k
km
km
=
lz
p
p
for
2
+-
k z
1
=
bp
z
b
=
;k
2
p
+-
2
a
TM
+
Z2
k z
1
Vak lp
for
(4-6)
waves and at
TE
k
2
waves.
ke
lz
=
fap
+
(4-7)
z2
55
By following the asymtotic method procedure in references
It is found
31 and 32, the radiation fields can be determined.
that the Poynting vectors are in the radial direction both in
the isotropic and uniaxial medium.
the upper medium,
In
Sr -
(I/4)
which is
P k[X2
2
2
free
sin
(0
space,
2
$
+ cos
2
G
010
r
Y
o
2
(0 ) cos 2p]
10o
(4-8)
where
2 y k cos
X01(0
(4-9a)
=0
2
k
pk 1
2
sin G
0
2
~b0
11
k2
2E
Y1
0
(
0
)
00
k1 2
++ 1 lk cos Go
sin 2 G
a
-
(4.9b)
=
k
2
2
sin G
E+
1 k cos
2
k
e
a
Inside the uniaxial medium,
Sr
1 (I/4 )2
+
2
r
+
1
oelz
(a
sin
20(b
ask 1
sin2 6
bk 1
(sz
3
cos 2 e0
52
+ COS
3/2
2
Ix1 0 12 sin 2 4
0)
cosl
(4-10)
+ cos
2G0
)
56
where
a
= tan
P
tan-
(v5 tan 0 )
(4-1la)
(Va
(4-1lb)
IzI
= tan
=/ap
tan
tan
0)
zI
X10
2P
yk
1
cos a + y
2 E
Y0
In equations
-b
k
/k 2
-
(4-8) to (4-12) ,
k
are shown in
the broadside direction
(the
a k
00
2
sin 2
is used to denote the
Figure
y
sin2a
(4-12b)
observation angle measured from the
The results
2
2
- a k
cos 6 + c
s k
(4-12a)
z-axis.
(4.1).
direction,
We note that in
5 = 90*),
the
radiation patterns are affected by the anisotropy in permeability because the magnetic field
is
in
the plane determined
by the optic axis and the direction of propagation.
By the
same argument, radiation patterns in the endfire direction,
$ = 0*,
(the
x
in permittivity.
direction) are affected by the anisotropy
We also observe that power couples more into
the half space medium when the medium is positively uniaxial
(a > 1,
b > 1).
The opposite is true when the medium is
57
negative
< 1,
(a
uniaxial
b < 1).
In Figure 4.1, the quantity
4Tr 2
G(6,
G(6, $)
defined by
S
r
$=
Pt
is plotted.
Pt
and is equal to
is the total radiated power in free space
/y'j/E
(kIk) 2 /(12-r).
58
z
= 3.2
4
4
4
b =0 .8
b =1
8
Sb =1 .2
HED
Broadside,
#-
7T
2
z
4
4
a = 0.8
a=1
4
-a
=1.2
8
HED
Endfire,
# = 0
Figure 4.1
Radiation Pattern of
HED
Over Half-Space Medium
59
4.2
Graphical Determination of
The wave vector
k
Vector
is normal to the ray surface and
is normal to the wave surface.61
S
the Poynting vector
k
In the case of dipole radiation, inside the uniaxial
medium, the Poynting vector is always radial from the antenna
to the observation point.
To determine the direction of
k
vector for the observation point graphically, we can construct
a ray surface around the point.
For the uniaxial medium, the
ray surface is governed by the equation
s
s
2
f = a, b
z-
2
P
f
k
(4-13)
2f
which is an ellipse.
A line from the dipole source to the
observation point intersects the ray surface at one point.
The normal to the ray surface at this point is in the same
direction as the
R
vector.
(Figure 4.2)
This fact can be
checked mathematically by using the values in equations (4-6)
and
(4-7)
for the
k
vector.
In the broadside direction, the direction of maximum
radiation power
(Figure 4.1) is at the Type I wave critical
angle of the
vector.
k
This corresponds to
k
= k.
In
the endfire direction, the radiation power is null in the
direction of the critical angle of the
II wave.
k-vector for the type
60
Figure 4.2
Graphical Determination of
E-vector
z
dipole source
-P
N
N
N
N
N
surface
observation
point
A
R vector
S vector
61
Chapter 5
Geometrical
Optics Approach
Due to the interference of the various wave components,
the field distance plot is an interference pattern.
In order
to obtain explicit expressions for the six field components,
the following three different approaches are used to evaluate
the integrals of the fields:
proach
(GOA),
(i) the geometrical optics ap-
(ii) mode approach, and
(iii) FFT
(Fast Fourier
Transform) technique of numerically calculating the integrals.
In this chapter, we illustrate the geometrical optics
approach by the two layer case.
The geometrical optics ap-
proach consumes very little computer time and permits a very
simple interpretation in terms of ray optics.
cussion of the validity of the
31.
GOA
A detailed dis-
can be found in reference
62
5.1
Mathematical Development
The reflection coefficients for the two layer case are
I
1 R1
RTE
1
1
Rl0
lz
R1 2
01
1 A +R12exp
(i2k d)
and
RTM
S1
1
01
_
S 01
1
S01
(5-lb)
+ Sl2 exp (i2k
1 + R 1 2 exp (i2k
TM
zd)
(5-2a)
X0
1 + R0 1 R1
1 -R
d)
d)
So 1
TEB =
l + R
1
-
(5-la)
2
exp (i2km d)
S12 exp (i2ked)
=
1 + S 0 1 S 1 2 exp (i2k ed)
(5-2b)
Y100
In the geometrical optics approximation, we expand
1
-RTM
and
1 + RTE
1 + RTE
in
a series.
exp (i2k~nmd) I
+Rml
m = 1
010
12 em
(5-3a)
63
00
RTM
1 -
_
1
1 Y 01
11
M =E 1
221
exp (i2k
md)}.
(5-3b)
The first term in equation
solution.
(5-3) is the half space
The summation term corresponds to contributions
arising from image sources.
In the summation term, the saddle
point occurs at
k
for
p
= Va k
1 sin am ,F
a=
-l p/a
tan
2md
TM
wave, and at
k
p
a
=
/
=tan
k
TE
-l p/E
wave.
m = 1,
2,
3,
.
(5-5)
(5-6)
1 sin am ,i
2md
for
(5-4)
m = 1,
,
2,
3,
.
(5-7)
We have assumed that both the slab and the
bottom media have the same factors of anisotropy
a
and
b.
We next apply the saddle point method and the branch cut
contour integration method31 to our integrals.
results for
E
pz
and
H ,
The asymtotic
are, when the observation point is
64
at
z = 0
4Tr
P
iVE kip
2ak 1
ikp
2
kp 2
EIzAoy
k 2 p 2 Elz
i/H7 k
m
k 2 M=
1 Va- k ke
y 1 ~y 0
Yz 0 01
z
1
m e
12
rn-i
0
R
1mm
(5-8)
R
m
where
2
R
H
2md)
a
+
(5-9)
k2
2
_9I
z
47 T
2 (bk 1
2
-
k 2)
00~ 2k'~
-
01X10
1-
k 2
e
ik R
1x
m-m
k z k1
im
i/F k P)
11ezikp
P
R
2
(5-10)
R
m
where
R
mF
=
2
(2md)
+
2
(5-11)
b
As discussed in reference 31, the saddle point method
as used in the
point
am
G.O.A.
approach is not valid when the saddle
is close to the branch point due to
kz.
the problem will not be serious if the slab is thick.
However,
65
In Figure 5.1 we plot the interference pattern for
|E
for a half space medium and in Figure 5.2, we plot the inter-
ference pattern for a two layer medium.
The plot is normalized as follows.
E
I
= 20 log 1 0
IZ&p
norm.
4-F
k
is taken to be
2w.
66
Figure 5.1
Interference Pattern of Half Space Medium
Observation point at
z =
E
Pnorm
3.2(1 + i.01)E 0
C=
a
=
.8
-15 -
-30 -
-45
-
-60
0
0
3
6
Distance in
9
wavelengths
12
15
0
67
Figure
5.2
Interference Pattern of a Two Layer Medium
E
Pnorm
observation
I
d = 3X
= 3.2(1 + i.01)E
1
a
-15
= Po
=
.8
F2 = 81 9
-
first peak due to
reflection
-30
-
-45
-
I
-60
0
3
I
9
6
Distance
in
z = 0
point at
12
wavelengths
15
2
=
a
=
.8
68
5.2
Physical Interpretation
In
GOA,
the interference pattern is determined by
three kinds of waves:
1.
the direct wave from the antenna
2.
the inhomogeneous wave excited by the antenna on the
surface, and
3.
the wave reflected in the subsurface following ray optics
paths
(Figure 5.3)
The first two kinds of waves are in the classic half space
solution.
The direct wave corresponds to the saddle point
contribution
(equation 4-2) and travels with wave number
k.
The inhomogeneous wave corresponds to the branch point contribution due to
ke
lz
travels with wave number
TE
wave and
TM
for
/lT k
for
km
lz
TM
for
TE
wave and
It
wave.
V
k
for
wave.
The third kind of wave corresponds to the summation
series in equation (5-3).
Each term in the series is attributed
to a particular image source.
The wave that arrives at the re-
ceiver after one reflection from the subsurface can be traced
back to the first image and is identified with the first term
in the summation series.
the receiver after
sented by the
nth
n
Similarly the wave that arrives at
reflections from the surface is repre-
term
(image)
in
the summation series.
69
Figure 5.3
Physical
Interpretation
of GOA
receiver
transmitter
o'
Eo
1dJ~,
C1
d
1.0
d
first
image
2d
second
image
t'
~t
70
The ray direction is in the same direction as the
Poynting vector
S.
To obtain the
R
vector, a ray surface
can be drawn around the point of reflection at the subsurface
(Figure 5.4) and the
K
vector is the normal to the surface.
The phase of the wave at the receiver from the
source is then given by
nth
image
k p + klz (2nd).
If there is no subsurface reflector, the interference
pattern is generated by the first two kinds of waves.
spatial wavelength of the interference pattern
Xint
The
is re-
lated to the dielectric properties of the medium by
V/F
=
1 +
k
where
0
f
a,
b
(5-12)
int
is
X
from Figure
the free space wavelength.
This can be observed
(5.1).
When there is a subsurface reflector, we first note the
direction of the maximum radiation energy
the peak due to the first
reflection occurs at
then the thickness of the layer,
d =
y
d,
(Chapter 4).
If
a distance
pco'
is given by
(5-13)
.
c
2 tan y
We can observe this effect in Figure
(5.2).
The geometrical optics optics approach yield simple
71
Figure 5.4
Graphical Determination
of
k
Vector
receiver
transmitter
o '
o
d
Ole%
't'
Ile.
d
first
image
t
72
analytic formula for the various wave contributions.
It is
useful if the slab is thick and lossy so that a ray optics
analysis is valid and also only a few image sources are needed.
73
Chapter 6
Modal Approach
In this chapter, we illustrate the model approach by
solving the
component of
Hz
iItkp2
o
H
=
z
f
dk
-
p
P
8irk
the two layer case.
TE
(1 + RT
ikzz
)
e
Z
zH
(k p)
p
sin
$
z
(6-1)
In
order to solve the integral
in
(6-1),
we make the
transformation
k
= k sin 0
(6-2)
so that the two Riemann sheets due to the branch point
are unfolded.
branch point at
k
We then have four Riemann sheets due to the
km
lz
and
km.
2z
Inside the integrand in
(6-1),
we can write
, (1)
H1
(k pP)
= H
(1)
(k p)
exp (-ik
and insert an exponential factor
p)
exp(ik p)
(6-3)
which is in fact
74
the factor for the Hankel function at its asymptotic limits
when
k
p
tends to infinity.
there is a
plex
saddle point at
0 =
Corresponding to this factor,
ff/2.
We detour, on the com-
6-plane, the original path of integration to the steepest
descent path passing through the saddle point.
to the integral is then composed of three parts,
H
(1)
z
Ip
=I
is
p
+ I
27i
b
+ 1
The contribution
(Figure 6-1).
(6-4)
s
times the residue series of the poles
enclosed between the new and old paths.
(2)
I
b
is the branch cut contribution due to
km
lz
and
km
2z'
(3)
the saddle point contribution.
I
is
Ip
corresponds to the modes excited.
geneous wave contribution, and
bution.
Is
Ib
is the inhomo-
is the direct wave contri-
In the following sections, we elaborate on these three
kinds of waves.
The mode approach interpret the interference fringes as
generated by interference of these three kinds of waves.
75
6.1
Modes Contribution
A pole location plot on the complex
2.16(1 + iO.0022) c
wavelength with
1
(6-1) for layer of depth
trated in Figure
permittivity
6-plane is illus-
over a perfect conductor.
Each pole can be interpreted as a normal mode pertaining to
The original path of integration and the
the layer medium.29
new steepest descent path are also shown in the same figure.
The number of normal modes excited by the antenna depends on
the thickness of the slab.
If the slab is sufficiently thin
no mode will be excited at all.
1 + RTE
The poles of
are determined from
R 1 0 R 1 2 exp(i2 k z d) = exp(i2Z7).
k = 0, 1, 2,
Setting
In
...
actual computation,
R10
=
yields positions of the poles.
we set
exp(i)
exp(i
R12
(6-5)
(6-6a)
2
(6-6b)
)
so that
2k M d + $
z
+ $2
=
2.
k =
0,
1,
2,
...
(6-7)
1X
2.16(1 + iO.0022),
s
Osurf
ace wave modes,
)
A
leaky wave modes,
unexcited modes
a "
perfect conductor
X
original path
of integration
A
steepest
descent
x
A
path
A
X
6'
'T/2
0
A
A
A
A
Figure 6.1
Complex
6
plane for Mode Analysis
Tr
77
The pole locations are obtained through iterating equation
(6-7)
by first
neglecting
$
2*
and
Each pole corresponds to a wave mode.
wave vector
components
Re(k )
and
modes.
< 0
Im(k )
k < Re(k
> 0
)
< k1 ,
(Figure 6-1)
The modes with
Im(k )
> 0,
are surface wave
These surface waves are associated with the guided
waves of the slab region.
They decrease in magnitude exponen-
tially as they leave the surface.
number of them excited.
There are only a finite
They are the significant ones when
distance is large from the transmitter.
modes are the leaky wave modes.
components
Im(kz) < 0.
Re(k )
< k,
Im(k P)
The other group of
(Figure 6-1)
> 0,
Re(k z)
The wave vector
> 0,
and
They correspond to waves that reach the receiver
from beneath the surface and increase in magnitude exponentially
as they leave the surface.
There are an infinite number of
leaky wave modes being excited and they decay very rapidly with
distance from the transmitting antenna.
only
in
the near and intermediate ranges.
They are important
78
6.2
Inhomogeneous Wave Contribution
Corresponding to the two branch points at
kP = k
and
Re(k
Vb,
) = 0.
and medium
we chose two branch cuts
k
.
The integrand is
IF
k
Thus the branch point at
can be calculated
by detouring
the
to the path of steepest descent.
references 31 and 32.
L(l
and
to
does not contribute
km
+ e
R
2
a
path parallel
is
The result
2
k
2
-k
2
b1 / 2
i2k zd
1
The mathematics is similar to
(1 + R 0 l) (1 + R 10 ) k 2e
2 Tr
and
an even function of
branch cut to
-2
I-
= 0
Re(kM)
The contribution due to the branch point
the integral.
that in
VIE
k1
same value of anisotropicity
to assume the
the same permeability.
=
we allow medium
For the sake of simplicity,
2
k
km k
lz z
1
d
eik2P/jE
p2
k
= k2
(6-8)
Physically this corresponds to an inhomogeneous wave that
decays away from the bottom surface with the exponential factor
exp(-2
k 2 2 - k1
2
d).
Therefore, except in cases when the
slab region is very thin, the effect due to the branch point
k2
is not observed on the surface of the layered medium.
*S.
79
6.3
Direct
Wave Contribution
In the absence of any stratified medium, the saddle point
contribution from the transmitter corresponds to the direct
wave from the dipole.
When there is stratified medium present,
the evaluation of the saddle point contribution is complicated
by the poles in the neighborhood of the saddle point.
Besides
the pole effects, the saddle point contribution is also affected
by the rapidly varying exponential
reflection
when
coefficient.
d
is
large.
exp(i2k z d)
The oscillation
of this
factor is
fast
We thus evaluate the saddle point contri-
bution for the following three cases:
use the ordinary saddle point method
(1) When
(OSP).
we use the modified saddle point method1 6,63
p ~ d,
factor in the
p >> d,
(2) When
(MSP).
we
p > d,
(3) When
we resort to numerical evaluation by using the Gaussian
Hermite Quadrature
(GHQ)
formula. 6 2
In applying the saddle point method, we transform the
integral by setting
sin 6) = x 2 /2
ikp(l -
(6-9)
so that
I
=
S0
where
f
0
(x) exp(-x 2 /2)
dx
(6-10)
80
iI
k
(x)=
8Trk
=
x)
TE
(1 + RTE)
"11
H1
(k p) eikp dk p.6-l
P
z
(
dx
in a Taylor series.
O (x)
We expand
2
0~
x
A 2
(6-12)
M = 0
p >> d,
When
there are no poles near the saddle point, the
does not oscillate very fast.
4l(x)
function
saddle point method is applicable.
The ordinary
The asymtotic series solu-
tion is obtained by directly substituting equation (6-12) into
(6-10) .
We denote the solution by
If
there is
a pole,
say at
to the saddle point, i.e.
saddle point method.
M
= lim
x*x
= OSP.
x = x0 ,
x 0 << 1,
equation (6-12) is ill-behaved.
Is
which is
very close
then the Taylor series in
We then apply the modified
Let
cl(x) (x - x 0 )
(6-13)
0
and define
(x=
(x)
x
which is
(6-14)
M
well-behaved.
-
x0
81
Since,
fir Me
x
-o
-
erfc
//
(-ix
)
Imx
>
0
d/2
- Me_
p
0
x
-x 2/2
Me
S-ir
_
erfc (ix
< 0
/2) Imx
0
0
(6-15)
therefore, on substituting equation
(6-14) into equation
(6-10)
and applying ordinary saddle point method to the well-behaved
function
p (x),
we obtain, taking into account the effect of
all the poles,
I
=
MSP = OSP +
E
{W
-
[W ]}.
(6-16)
poles
[W ]
p
dently, as
denotes
x0
W
>> 1,
p
in
inverse power series of
Evi-
the modified saddle point result is
identical to the ordinary saddle point result.
in
x .
The summation
(6-16) extends over all poles near the saddle point.
For the case
p < d,
the
MSP
fails to apply.
Because
in addition to the effect of the poles, the integrand oscillates rapidly due to the presence of the
in the reflection coefficient.
exp(i2km d)
factor
In this case, the Gaussian
Hermite Quadrature formula is particularly useful in evaluating
the line integral along the steepest descent path.
82
Figure
(6-2)
illustrates
layer with permittivity
e1 = 2.16(1 + 0.0022) S0
OSP,
MSP,
and
GHQ.
GHQ
coincide for
considerably for
GHQ,
For
for
p > 1OX.
p < 1OX.
1OX < p < 10OX,
p > 10OX,
over a
It is seen that all
three methods approach the same value as
and
dielectric
4X
The saddle point contribution is calcu-
perfect reflector.
lated by using
the case of a
OSP
p > 10OX.
MSP
The
All three methods depart
Thus for
p < 1OX,
one can use either
is sufficiently accurate.
we recommend
MSP
or
GHQ.
83
Figure 6-2
Comparison of results by OSP, MSP and GHQ
Vertical scale is
z=0
30 dB/divison
d=4X
E =2.16(1+i.0022)E
perfect reflector
OSP
MSP
GHQ
S.
1
10
100
Distance in Wavelengths
500
0
84
6.4
Numerical
Results and Discussion
In this section, we illustrate the mode approach by
calculating the interference patterns as a function of distance from the transmitter.
When the slab is sufficiently
thin, no mode is excited and the integral is due to the saddle
point and the
k z
branch cut.
pattern due to a layer with
0.1
c
Figure
(6-3) shows interference
= 3.0(1 + iO.001) 6
and depth
wavelength on top of a dielectric half-space with
62 = 5.0(1 + iO.002) c0.
The saddle point contribution is
When the layer
calculated by the ordinary saddle point method.
becomes thicker, modes will be excited.
depth
.5
and
1
wavelength, only
1
For the same media with
to
3
surface wave modes
are excited.
The integral in equation
(6-1) can be readily computed by
However, the compu-
using the Simpson's rule of integration.
tation time is large because the Hankel function is a rapidly
oscillating function for large
p
and thousands of divisions
are needed to provide the required accuracy.
In
Figure
(6-4),
we show
Hz
as a
function of distance
calculated on the basis of three different approaches
mode approach and Simpson's rule).
The model consists of a
single layer, four wavelengths in thickness, with
0.01,
and loss tangent
fect conductor below.
at a height of
3
(GOA,
1
' = 3.3E
between free space above and a perThe fields are calculated for a receiver
meters above surface at
8 MHz.
Field
85
Figure 6-3
Interference Pattern for
Thin Layers
z=
0
Vertical scale is 20dB
per division
d
C
= 3(1+i.001)£ 0
E2 = 5(l+i.002)£
d =1
0
X
HI
d = 0.5 X
d
=
0.lX
For purpose of comparison,
neighboring curves are separated
by one divison
0
2
6
4
Distance
in
Wavelengths
8
10
Figure 6.4
Comparison of Theoretical Results--Simpson's Rule, Mode
and
GOA
1.2
Numerical
~...
(Simpson's Rule)
Geometrical Optics
Mode
1.0
\. 0
z = 0 . 08 X
4X
E
1
=
3.3(1 + iO.01)6 o
0.8
perfect conductor
0.6
-
0.4
0.2
0.0
4lO6
8X
Distance in
Free Space Wavelengths
10x
12X
87
amplitude is normalized with factor
IZ/4TX
2
.
Inspection of
the figure shows that the results from mode approach agree
well with those obtained from Simpson's rule.
from
GOA
The results
are inferior for the reasons stated in chapter 5.
In Figure
(6-5),
we further compare
GOA
and mode ap-
proach with scale model tank experimental data 3 4 - 3 5 for various
thicknesses.
We note that all three agree for large layer
thicknesses.
Note specifically that for shallower depths the
geometrical optics approximation fails to account for even
the gross features whereas the mode approach fits the experimental data excellently.
The general features of the model
tank experiment were described by Rossiter et al. 3 4
We conclude that
plement each other.
thicknesses.
is large when
exercised
5,
GOA
GOA
and mode approach serve to com-
Mode approach is valid for all layer
However, the number of surface wave modes excited
d
is large.
in that case.
In addition,
GHQ
has to be
On the other hand, as noted in chapter
is particularly appropriate for thick and lossy layers.
88
Figure 6.5
Comparison of Scaled Model Tank Experimental Data
with GOA and Mode
Experimental Result
.
---
Geometrical Optics
Approximations
d
e=
2.16(1 + iO.0022)
a
Mode Approach
+o
vertical scales
(15 dB/division)
d
=2.5X
*0
0
So
0410
t
d
=5X
.-
0
3
6
Distance in
9
*.
12
Free-Space Wavelengths
15
18
0
89
Chapter 7
Fast Fourier Transform Approach
Though analytic methods yield simple solutions and allow
physical interpretation of the results, they are not useful
when the number of layers gets large so that the reflection
coefficient becomes extremely complicated.
is sought to handle more complicated cases.
A numerical method
Due to the rapid'
oscillation of the Bessel function inside the integrand, a
direct numerical integration consumes a lot of computer time.
64
To save computer time, we have developed a fast Fourier Transform
65,84
'
method.
technique.
In this chapter, we elaborate on this
90
7.1
Mathematical Preliminaries
By using Bessel function identities, the Hankel function
in equation
(6-1) can be changed into a Bessel function and
the range of integration can also be changed to
integral in
f
z
o.
-
The
(6-1) becomes
dk
00
2
ik
dk
H
0
i
ik z
(1
I
47r
k
+ RTE
z
J
1
(k p)
sin$.
Z
(7-1)
As
k
p
+
R
o,
TE
+ 0
and
exp (ik z)
also tends to
z
0.
The Bessel function oscillates rapidly for large arguments.
The integrand will converge rapidly if we choose
z /
Physically, this choice corresponds to the observation
0.
point being above the free surface of the stratified medium.
Separate the integral in (7-1) into two parts
12.
I
and
The first part corresponds to solutions in the absence of
any stratified medium, and the result is given by the identity
ikr
-
=-
I
1
.
3p
r
-k
-
2
-
2
kP 2 e
k
ik z
(k p) dk.
z
1
The second part includes all effects due to
coefficient
RTE
and is
p
p
(7-2)
the reflection
91
iltk
0
12 =f
dk
0
2
9
P
R
TEikzz
e
Z
(k p)
4Trk
sin
#
which is the integral that we want to solve with
In applying the
(7-3)
J
FFT,
FFT.
we use the formula66
-- vk
f
e
J 1 (k p) dk
P
9
0
1 -
=
p
P
2
+
92
Re{v + ipl> 0.
V = VR + iv 1 ,
(7-4)
The integral is written in the following form
-v k
1
= f
dk
g(k )
e
pJ
(k
(7-5)
p) dk
where
k
2
ik z + v k
=
g(k)
9
The aim of our
e
Rp
VR < z.
(7-6)
4'r kR
z
FFT
method is to develop an algorithm such
that when calculating the field as a function of distance
we can make' use of the formula in
calculate the integral in
The
FFT
(7-4) and do not have to
(7-5) for every individual
p.
subroutine computer program, for a given set
p,
92
of data of
N
points,
dk,
k =
0,
1,
2,
j
= 0,
1
...
N
-
1,
returns
N
-
l.
(7-7)
the result
27rjk
N
T.
=
1N
k = 0
The subroutine
d
k
e
.
(7-7) is available through the Mathlib of MIT
Information Processing Center.
In the following section, we will show how to cast the
integral in (7-5) into discrete form so that equation (7-7)
is
directly applicable.
93
7.2
Aliasing
We define
a(f)
g(k ).
to be the Fourier transform of
00
=
a(f)
f
(7-8a)
g(k ) exp(-2Trifk ) dk
g)
g (k )
f
(7-8b)
a(f) exp(21'rifk
Suppose the Nyquist frequency of
a(f)
for
~ 0
(Figure 7-la).
Ak
Next, we sample
a(f)
IfI
is
F 0 ,o
i.e.
(7-9)
> F
g(k )
with sample interval
where
Ak
=-
-
p
F
(7-10)
2F
0
(7-8)
Then, from equation
00
g(jAk )
a(f) exp(2Trifj/F)
= f
j
k = -C
-00,
...
...
-1,
0,
(k + 1)F
00
E
=
df
f
a(f) exp(27rifj/F)
kF
df
1, 2,
00o
94
F
=
f
a
a
(f)
(f)
(7-ll)
exp(27rifj/F)
where
00
p
=
The function
sion of
a
a(f)
.
(Figure 7-ib) is called an aliased ver-
(f)
and is a periodic function.
g(jAk )
(7-11) identifies
ap (f)
Therefore,
F j =-00
We next sample
Af = F_
N
as the Fourier coefficients of
(7-13)
j/F)
exp (-2T if
g (jAk p
with interval
ap (f)
where
Af,
1
NAk
(7-14)
P
1
(nAf) =
a
The relation in
we can write
a (f)
P
(7-12)
a(f + kF).
E
k = -00
F j = 0
P
00
E
-k = -00
g (jAk
n = -o,
=-
1N -
P
+ NAk
..
exp(-2rijn/N)
P
-1,
0,
1, 2,
. *
.o
1
NE
g (jAk ) exp(-2Trijn/N)
p
F j = 0
(7-15)
95
a(f)
-F0
0
a
-2F
-F
Figure
-F
f
F0
Figure 7-lb
(f)
0
0
7-la
F
F0
g (k
2F
)
Figure 7-ic
I
6
____fn
-T/ 2
T/2
\..
(k
1
k
)
A
Figure 7-ld
NIo
,0
3T/2
\,,000
-T/ 2
Figure
j
\000
7-i
iT
T/ 2
Aliasing
2
X kp
96
where
00
g (k ) =
P p
0
is the aliased version of
T
We choose
p
(7-17)
such that
~ 0
for
(Figures 7.lc and 7.ld).
for
and is periodic with period
.
N
g(k )
is
g(k )
where
T = NAk
for
(7-16)
g(k
+ ZT)
p
CO=
Z
k
lying between
chosen such that
k
> T/2,
gp (k ) = g(k )
We notice that
-T/2
and
(7-18)
Ik | > T/2
T/2.
Therefore,
if
vR
is essentially zero
gp (k ) exp(-vRk p)
then
T/ 2
f
gp (k ) exp(-v k)
J
(kPp) dk
p
T/ 2
g(k
) exp(-v Rk ) J.(k
p)
dk
0
00
0 g (k P) exp (-vRk P
J(k P )
dkP
=
2*
(7-19)
97
By switching the roles of
a(f)
g(k ),
and
an equation analogous to equation
we can derive
(7-13),
0n
(k)
=
Z
-
T
..
=
a(nAf) exp(2Trink P/T).
(7-20) into
Substituting equation
equation
(because
(7-19), and making use of
(7-4), we have
00
00
2
(7-20)
--
-
a (nAf) f
Z
T n
a(f)
is almost zero outside
1 N/2 - 1
T n
exp(-k p (yR
2'rin/T) )
-
= --
a(nAf) f
-N/2
=
0
exp(-kP(vR -
2'iin/T))
NR -
NAk9 n
=
-N/2
)
-
dk P
F )
N/2 - 1
a(n/NAk
J 1 (k P )
J 1 (k~p) dkp
i2ffn/NAk )
1 -
Sp
/(R
-
2p p
i2'rn/NAk )2 + p
(7-21)
Equation (7-21) is the final answer for the integral
12'
We list the numerical computation procedure as follows:
98
g(jAk )
(1)
First
(2)
gp(jAk P)
(3)
a P(nAf)
is obtained directly from the integrand.
is then calculated by aliasing
is
then evaluated by using equation
is identical to the
(4)
a(nAf)
FFT
N,
(7-15)
subroutine in equation
is then obtained by de-aliasing
The proper choice of
a
p
which
(7-7).
(nAf).
the number of samples, is
essential for the best performance of the
and should be some integral power of
etc.).
g(jAk ).
2
FFT
(i.e.
subroutine
21,
22,
23,
99
7.3
Numerical
In Figures
pattern for
Results and Discussion
(7-2) and
Hz
(7-3), we show the interference
calculated with
FFT
approach and Simpson's
rule for a three layer and a six layer case respectively.
height of the observation point is at
responds to
2
z = 0.213X
meters for a frequency of
The
which cor-
32 MHz.
The two
methods, as seen from the figures, yield identical results.
The computation time for
cases.
FFT
is about
2.5
minutes for both
For Simpson's rule, the computation time is about
minutes for the three layer case and about
the six layer case.
40
32
minutes for
The computation was carried out on the
IBM 360/65 computer.
One must also bear in mind that
FFT
consumes a lot of storage space of the computer when the Nyquist
frequency and the number of sampling points are big.
The parameters
in
Figure
(7-1)
are
E1 =
(3.3) (1 + iO.01)E
0
d 1 = 1x
E2 =
(5.0) (1 + iO.02)E
0
d 2 = 2X
£3 = (8.0) (1 + 10.04)e
0
'
The parameters in Figure
(7-2) are
(2.0)(1 + iO.01)E
0
d 1 = 0.5X
+ iO.02)E
0
d2
£3 = (4.0)(1 + iO.03)E
0
d3 = 2A
1 =
£2 =
(3.0) (1
=
lx
100
=
(5.0)(1 + iO.04)s
0
d
e5 =
(6.0) (1 + iO.05)s
0
d5 = 4X
Et =
(8.0) (1 + iO.06)e 0 '
E4
In Figure
from
GOA
very little
(7-4), we compare
and mode approach.
time,
FFT
= 3X
method with results
The analytic methods consume
but they are applicable only in
regions according to the nature of the method.
other hand, can treat the most general cases.
lacks physical interpretation.
certain
FFT,
on the
However,
FFT
Figure 7-2
Comparison of Simpson's Rule and FFT
for a three layer model
Vertical scale
is 10 dB/divison
The curves have been displaced
20 dB for ease of comparison
FFT
10 dB
IH zI
SIMPSON'S
I
0
1
RULE
I
I
2
3
I
I
4
5
Distance in Wavelengths
I
I
6
7
8
H
01
H
Vertical Scale is
10 dB/divison
Figure 7-3
Comparison of Simpson's rule and
FFT for a six layer model
FFT
10 dB
IHZI
SIMPSON'S RULE
The curves have been
displaced 20 dB for
ease of comparison
0
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
Distance in Wavelengths
H
C0
to
Figure 7-4
Comparison of FFT, mode and GOA for
a three layer model
Vertical Scale is
10 dB/divison
4X
E
= 3.3(l+i.02)E
0
E 2 = 6.0 (l+i.04)E0
FFT
1z
MODE
H
GOA
10 dB
The curves have been
displaced vertically
20 dB for ease of comparison
1
2
3
4
5
6
7
Distance in Wavelengths
8
9
104
Chapter 8
Microwave Passive Remote Sensing
Geophysical subsurface exploration of large geophysical
areas can be handled by the method of microwave passive remote
sensing with a radiometer.
In this chapter, we list the three
approaches that have been used to solve for the brightness
temperature of a medium--the reciprocity approach, 3 7 the radiative transfer approach,67,71 and the fluctuation dissipation
approach.
68
neglected in
,83
Atmospheric effects and cosmic radiation are
our development.
105
8.1
Reciprocity Approach
If the medium under observation is of constant temperature
T,
the brightness temperature
T
e
where
reflectivity.
=
=
e
TB
is given by
eT
(8-1)
1 - r
(8-2)
is the emissivity of the medium, and
The reflectivity
from boundaries as well as
r
is the
includes effects of reflection
scattering
from the medium.
106
Radiative Transfer Approach
8.2
In microwave thermal emission, the specific intensity of
radiation of a given polarization,
I,
measured in free space,
I
by 6 8
TB
is related to the brightness temperature
B
(8-3)
-2
where
K
1.38 x 10-23 Joules/*K
=
is Boltzmann's constant.
At local thermodynamic equilibrium, energy conservation requires
that the changes in specific intensity of radiation
given polarization
dI
where
=
K
-
ds
state
a
Ids
-
K
I
of a
be given by
s
Ids + K
a
Bds +
K
Jds
s
(8-4)
is the distance the wave propagates through a die
electric medium of relative permittivity
r
,
K
are the absorption and scattering coefficients,
due to absorption,
direction of
I,
K
K
Bds
Ids
and
a
-
K
Ids
K
s
is
is due to scattering out of the
is due to emission, and
due to scattering into the direction of
Jds
K
is
I.
At microwave frequencies, the emission source
B
is
given by
E
B
r
KT
(8-5)
107
where
T
is
the local temperature
By solving for
I
from
(8-4),
in
ds.
the corresponding bright-
ness temperature can be obtained from equation
radiative transfer approach is simple to apply.
(8-3).
The
However, it
neglects the wave nature of radiation and thus only yields
useful results in cases where coherent effects can be ignored.
108
8.3
Fluctuation Dissipation Approach
dissipation
According to the fluctuation
theorem,
thermal radiation is generated by an equivalent current
J(r,
w),
4
with expectation value
W')>
W) J(r',
<J(r,
-
4
-
s"(r)
4
,
source
6 7
KT(r)I
6(w -
w')6(r - r')
Tr
(8-6)
where
is the imaginary part of the permittivity.
e"(r)
< >
angular bracket
The
is used to denote ensemble average.
The expected value of the energy density of the radiation
with polarization
U(r)
p,
is, in free space,
(8-7)
<lp -12>
=
where
00
(r, t)
=
dw f d 3 k E
f
w) exp[i(k
wk,
r- - wt)].
(8-8)
0
The energy density is related to the specific intensity, in
free space, by,
00
df f dQ
U~r
0
(8-9)
c
109
(8-7)-(8-9), we have,
Combining equations (8-3),
T
/"(7)
C/
T B(k, w)
2100
(c/)
00
f
cE
-
E
{e -
)
W
i(-
E*k'
k')
r
-
') >
i(w -
for the horizontal polarization or
TTM (k, w)
{h - <E(k,
i(E -
w')t}
TE
(8-10)
waves and
k 2 dk f
dw' f
0
0
2
K
,
f
(c/w)2 1- c
23
d 3 k'
-0
0
0
2
K
3
200
k 2 dk f
dw' f
dsk'
-C
w) E*(k', o')>
'
-
-
i(w
for the vertical polarization or
Once the electric field
-
TM
E(k, w)
w')t}
8-11)
waves.
in free space has been
determined, the brightness temperature can be found from equations
(8-10)
and
(8-11).
The determination of
E(k, w)
in
the presence of the current source in equation (8-6) and the
medium properties poses a well defined electromagnetic boundary
value problem.
110
Chapter 9
Born Series Approximation to Scattering
Problems
In this chapter, scattering of waves from a two layer
random medium with a laminar structure, is solved with the
Born series approximation.
Emissivity is calculated by using
reciprocity.
By using Green's function for unperturbed problem, inAn iteration procedure 6 9
tegral equations can be formulated.
can be applied to solve these integral equations, leading to
the Born series.
scattering,
In the case that absorption dominates over
the Born series
give useful
converge very fast in such a case.
answers as the series
The answers from Born
series approximation can also be used to compare results from
other approaches within their common regions of validity.
111
9.1
Formulation
Consider a two-layer medium with boundaries at
and
z = -d
(Figure 9-1).
z = 0
The slab between the two boundaries
is a random medium with a permittivity
= 6lm +
Fl
(z)> =
<i
(9-1)
if (z)
0.
(9-2)
The mean permittivity
Elm
is independent of depth.
random part of the permittivity
Ef (z)
is real and has a
zero ensemble average as indicated by (9-2).
ing in the
z
direction only.
The
It is also vary-
The variance of the fluctuating
part of the permittivity is also assumed to be small.
A monochromatic plane wave is normally incident upon the
two layer medium
E
= exp(-i k z).
The time dependent factor
(9-3)
exp(-iwt)
has been suppressed.
The governing equations are the one-dimensional wave equations
in each region.
d2 + k2
0
\dZ2
in
region
0,
E
o
=0
(9-4)
112
Figure 9-1
Two-Layer Random Medium
z
L
e-ik 0
z
I
reik z
region 0
z = 0
1
= Elm +
region 1
random medium
z = -d
region 2
62
(Z)
113
+ k2
(dZ
in
f'(z)
E1
(9-5)
and
1,
region
d 2
in
=
im a
+ k 2 2) E 2 = 0
where
2,
region
2
:- mk
k
lm
(9-6)
(9-7)
0
0
f (Z)
;i f (Z)
-
k
2
(9-8)
0
0
(9-9)
0
The Green's function for the unperturbed problem
E5 f (z) = 0)
is first determined.
(i.e.
It satisfies the following
equations:
(2
+ k 2)
0
(
G01(z,
2
k2
G1
(z,
z')
= 0
z'I)
=
6(z - z')
(9-1Oa)
(9 -1Ob)
114
D' 2
+ k2
G21(z,
Sz 2
We use superscript
o
=
0.
(9-1Oc)
to denote unperturbed quantities.
Solving equation (9-10)
G0
z')
gives
(z, z I)
x1
0
[exp (-iklmz')
+ R12 exp(iklm(z'
1k z
+ 2d))]
(9-la)
0
e
2iklm
+ R1
[exp(-ik lmz'
[exp(iklmz)
G
(z,
z')
=
-R
0 1
+ 2d))]
exp(iklm(z'
exp(-iklmz)]
for
1
2ik lmD
2
z > z'
2
[exp(iklmz')
-R
[exp(-iklMz)
+ R12 exp(iklm(z + 2d))]
exp(-iklm
for
z <
z'
(9-lib)
115
G2
(z
X 12
') =
2ik
[exp
(-iklmz')]
D2
- exp(-ik 2 (z + d) + ikimd)
(9-11c)
where
D2 = 1 + R0
1
R1
The scattered
Es (z)
field
= r
(9-12)
exp (2iklmd) .
2
in
region
0,
E
(Z) ,
is
given by
(9-13)
z) .
exp (ik
r
is the reflection coefficient and is a random variable.
It
can be shown easily
that
E
satisfies the integral
(z)
equation
0
E9 (Z)
= Es
(z) + f
G0
(z,
z
f (z ) E
)
(z ) dz,
(9-14)
whr-d
where
ik
E
s
(z)
=
r0 e
o
z
0
is the unperturbed field in region
(9-15)
0
and
116
2ikld
r
0
R01 + R12 e
=
The field in region
(9-16)
1
satisfies the integral equation
0
E
(z) = E
1
1
(z) + f
G 0(z, z1 ) f(z 1 )
~-da
1
E
(z1 )
dz 1
(9-17)
where
E
is
-
ik lm(z
-iklmz
X01
+
e
the unperturbed
field
r12ei
in
region
1.
+ 2d)
I
(9-18)
117
9.2
Born Series
To form the Born series for
(0)
(0) + G0
E
=
1
0
we apply the method
(z),
(01) f(l) E
0
(9-17).
(l)
0
+ G 0 1 (01)
In equation
S
(9-14) and make use of
of iteration to equation
E
E
(9-19),
f(l) Go
(12)
f(2)
E
(2)
...
(9-19)
.
a contracted notation is used.
Integration
is implied for repeated-integer arguments.
We use
B
(Z,
6
where
z1 )
=
<f(z) f(z
)> = 6k 4 exp(-[z -
z1
I/k)
(9-20)
represents the magnitude of fluctuations and is a
small number.
is the correlation length of the fluctua-
Z
tions.
Multiplying
(9-19) by its complex conjugate, taking the
6
ensemble average, and keeping terms to order
r
find the reflectivity
r
=
<rr*> = <E
= <E 0(0) E
only, we
(0)
E *(0)>
(0)> + G01(01)
G
(02)
B
(12)
118
E
(1) E
+ 2 Real
(2)
(E
s
(0)
G0
01 (01)
G
11 (12)
B
f1(12)
E
(9-21)
(2)).
We let
E (z)
<eS
where
(z)>
(9-23)
0
=
Esm (z)
ating field.
(9-22)
(Z)
(z) +
=E
6
is the mean field and
We see from equation
(z)
(9-19)
is the fluctu-
that the first
and
third terms in equation (9-21) represent the contribution of
the mean field and the second term represents the contribution
of the fluctuating field.
Substituting equation (9-20) into
(9-21), and carrying
out a brute force integration, we obtain
-2K
6k 14klX
r =
r01
01X1 0
2
1
+I
41klM 12I D2
-
K
) e
1 + 2a
K
Ka
-2K
(1
d
e
(1 + r2
1
a
2
a
d
-2K
+ 8r1 2 d e
1
d
a
-+
Ka
-1-
2a|
2
119
2ikld
+2
Real
1 - a*_
1 - 2a*
2
2ikm #-
2R 1 2 e
ik*
2ikmd + 2 Real
e
1 + 2a*
2 ad
R12
2r
-l
ikim
-
(1
(1 + 2ik
-
ilm
e -4ikmd (1
-
-
2ik{Z)
+ 2a*)
2a) (1
_
-2 ik*
+ R*
R*
-
26k lm , Real
9)
(1 + 2a)(1 -
2ik
(1
2a)
d
4ik
R 2
12
+ 2a
(1 - a)
-2 iklmd
2 Real
-2ik* d
m (1 +a*)
01
2iklmd
I
d
m
e
12
01 X10
2
2
3 - 4ik
9
8iklmD
2a*)
2
3 + 4ik
9
4iklmR1
e
(1 -
(R 0 1
2ik lmd)
R 01 R 12 e
S12
1 + 2ik lm
2iklm
R3
2iklmd
-2)
1 - 2a
+e
(2
R12
1 -
2a
1
120
2(3
2a)R
-
1
-
2
(1
+ 2(3 + 4a)
01 + 2R
+ 2a
R{2
1 + 2a
(1
-
2R
1R12
1 + 2a
-
4a)R 0 1 R 1
1 -
2a
4iklmd
2(3
4a R 1
2
2a
R3
(3 -
8a + 8a
(1
2a)
-
+ 8a + 8a 2 )
+ 4a (3
R
(1
+ 2a) 2
2(3 + 2a) R01R12 +
e
1 + 2a
11
2
2
2
)
U
R2
01 12
6ik lmd
(9-24)
where
a = ik lm,
(9-25)
-
121
9.3
Special Cases
We examine the following special cases:
(a)
Half space case,
r
=
IR 0 1 1
-
o:
I|X
6k
2
j201
-
+
4|k
R
R
Real
Since
d
1
all - 2a1
X-(
01
CR
8ik'(1 - 2a)
0 1X 1 0
a
km
12K
2
26k' 4 z
lm
2)
(9-26)
the third term in equation
(9-26) is
much less than the second term and our results are identical
to that of Gurvich et al.42
We conclude that he essentially
used the Born approximation.
(b)
For
equation
klmd >> l,
i.e. the slab contains many wavelengths,
(9-24) reduces to
-2K
6k i2|X01X12
|
r
+
(1 + rl2
11
m
4|klm 2D 2
1
K
-
(1 -K
a
al
Z)
e
1l
d
d
1-2K
d
d e
+ 8r
a
2
a
Ka
-2K
1+
e
+ a12
12
a
122
-
26k '4 Z Real
lm
,.
R*
i
lm
12
2ik
m
3
1
-2ik* d
m
2
2
2
4ik
z
-2ik
X
2
D
X
0110
z
2ik d)]
A
3 +4ik
-
|D
8ik3
t
4iklmd R
e
+ R
R0lRl2 e
(9-27)
.
1 + 2ik l
im
We end this chapter by making the following remark.
The Born series
tude of
f(z)
(9-19) can be ill-behaved.
Though the magni-
is small, if the terms of the integrand inter-
fere constructively, an integration over a large
domain may give a significant contribution.
(or infinite)
On the other hand,
if absorption dominates over scattering, then there is an
exponential decay involved, and the Born series converges.
we will discuss in chapter 11, the
can be interpreted as the
nth
nth
As
term in the Born series
order scattering.
123
Chapter 10
Brightness Temperature of a Half Space Random Medium
with Non-Uniform Temperature Profile
In this chapter, we use the radiative transfer approach
to solve the problem of microwave thermal emission from a half
space random medium with a laminar structure and non-uniform
temperature profile.
The randomness is in one dimension.
Scattering phase functions are derived and are then substituted
into
the radiative
transfer
the absorption and scattering
(RT)
equations.
the case that
are constants
coefficients
dependent of space coordinates, the
In
RT
in-
equations can be
solved exactly.
We note that while the Born series approximation
truncation) treats a finite order of scattering,
with multiple scattering effects.
RT
(upon
deals
124
10.1
Formulation
We assume the half space random medium to have the same
In addition, we allow
characteristics as that in chapter 9.
T(z)
the medium to possess a nonuniform temperature profile
A radiometer is sensing at an angle
(Figure 10.1).
the polarization and the observation angle
lm
00.
Let
Iu
and
denote intensities for the upward and downward radiation
inside the medium which make angle
E"
from
The resultant brightness temperature is dependent on
nadir.
Id
60
<<
c ',
lm
terms of
6.
The
RT
d
and
Since
are related simply by
G
In what follows all results are expressed in
Snell's law.
from equation
equations are,
(8-4),
I
cos 6
=
K
-
dz
cos
6
the angles
with nadir.
6
6
d
e
I
e
d
u
+ K
a
a B
B + K
-
Ks
J
s
d
u
(10-1)
(10-2)
dz
with
f
2
I
Id
+
2
(10-3)
125
Figure 10-1
Geometry of the Problem of Thermal Radiation from
a Half Space Random Medium
z
to radiometer
10
Region
0
C 0 ' 10
Region t
Random medium
t
lm
+
1
f (Z)
126
J
f
~-
I
+ -
2
subject
to
(10-4)
I
2
the boundary condition
(10-5)
=rot Iu
Idz
Z
wz = 0
where
b
0
is the Fresnel reflectivity at the boundary which
r ot
depends on
6,
e0,
polarization, and the properties of the
medium.
In equations
B
lm
K
=-
(10-1) and
(10-2),
(10-6)
(z).
-
0
is
K
the loss per unit length of intensity due to absorption
K
K
e
a
=
(10-7)
2 k"
lm
is the total extinction per unit length caused by absorption
and scattering.
K
e
=K
a
+
K
(10-8)
s
The scattering loss per unit length
Ks
and the scattering
127
phase functions
Pf
(forward phase function) and
ward phase function) for
following section.
TE
and
TM
Pb
(back-
are derived in the
Scattering is caused by the randomly
fluctuating part of the permittivity
1
f(Z) .
128
10.2
Scattering Phase Functions
We imagine an incident wave with
=A E e
E.
1
0
0
(10-9)
shining on a volume with fluctuating permittivity
c + E (r)
(10-10)
k = wVP.
Es
The scattered electric field
of incident field
can be expressed in terms of total field
E
E.
by the wave equa-
tion
V X V x Es()
-
k 2 Es (r)
=
w 2 .P
f (r) E(r).
(10-11)
In terms of the dyadic Green's function we have
Es(r)
=
-
i
f d 3 r' G(r, r')
Inside the integrand in
-
f (r')
E(r'
(10-12)
(10-12), we write,
E(r') % E.(r').
Using far-field approximation, we obtain,
(10-13)
129
e
ES2
Es C)
ks k )
(-
=
s
- A
*4
0
i(Ek
f d 3 r'
ikr
e
4ir
E s ) - r'
-
f(r') e
(10-14)
and
k64 E
<
Er)2
2
sin2
16Tr2 62r
d 3 r2
f d3 r
2
1
f
s
(f2)>
r2
1
(10-15)
where
X
is the angle between the incident polarization
and the scattered direction
<Ef (r 1 )
f*(r
2 )>
Letting
ks.
= AE 2 b(r 1
-
p = r1
- r
2
r2 )
A0
and
(10-16)
we find the bistatic scattering cross section per unit volume
to be
q(k,
This result
k
=
sin 2 X
A
f d 3 p b(p)
e
checks with that obtained by Gurvich et al42
(10-17)
(1973)
130
A = 4ano2
(1961) when we identify
and Tatarski70
To calculate scattering loss per unit length, we consider
the case when
b(p)
(10-1 8)
=e-IzI/k.
x -
Let the plane of incidence be the
z
(10-1 9)
+ z k cos 0
k = x k sin e
s+
=x k sin 0s cos
y k sin 6
+ z k cos
The scattering loss per unit length
K
=
27
IT
d$
0
sin $s
(10-20)
s
K
s
is
q (k, kS)
f dQ
= f
We write
plane.
f
s
d
0
s
sin
sin2X
k4 A
s
^n
6(k sin 6 - k sin es cos $s)
Within the range of
2k
2
1 + k (cos 6 - cos
6(k sin es sin $s).
s) 2 Z 2
(10-21)
integration, the Dirac delta functions
give nonzero results at
(1)
$s
= 0
and
es
=
e
which cor-
131
responds to forward scattering, and give rise to forward phase
function
Pf
and
(2)
$s
= 0
as = f -
and
0
which corre-
sponds to backward scattering and give rise to backward phase
function
This finding is
Pb.
essentially dictated by phase
matching for the laminar structure
The integral in
[Kong,61 19751.
(10-21) is evaluated by changing variables
(10-22)
u = k sin Gs cos $s - k sin 0
s
v = k sin 0s sin
which gives the Jacobian
(10-23)
1/Ik 2 sin 6s cos
s
.
Before we list the final answers for the scattering phase
function, we make the following remarks.
The assumption of
Scattering contributions
far field interaction has been made.
to intensity at a point mainly arise from scattering that are
far away.
Scattering phase functions, thus, are well defined.
The variance of the fluctuation
proximation
(10-13) is valid.
A
is small so that the ap-
(This is contrast to the
Rayleigh phase function in which a quasistatic approximation
is made for the field inside the small particle.)
We make the correspondence
section and
A
with
6
k
with
of chapter 9.
klm
of the previous
Because the randomness
is in one dimension only, there are only forward phase function
P
and backward phase function
Pb.
From equation (10-21), we
132
observe that
(1)
TE
For
=
K
waves,
k2
6k
lm
l
s 2 case6
X = 7/2,
1 +
and
(10-24)
1
The first term is from forward scattering and the second term
from backward scattering.
The forward and backward phase
Pf + Pb = 2.
functions obey the relationship
(2)
For
TM
waves, we have for forward scattering,
sin2X = cos
and for backward scattering
loss per unit length is
s
K
k m 6
2lm e
2 cos
26.
The scattering
calculated to be
cos
1 +
1
l
2
X = f/2
2
26
4k 2 m2
1 +
(10-25)
cos 2O
lm)
Again the first term is from forward scattering and the second
term from backward scattering.
In summary,
(1)
For
K
=
s
TE
waves
k2
6k
lm
cos
1 + 2k 2
£2
lm___
e
1 + 4k m
_
2
cos
__
_
2
0
_
cos26
(10-26a)
133
1 + 4k 2 m2 cos 2 e
P
lm
Pf
(10-26b)
1 + 2k2
2
lm
Pb
(2)
For
cos
G
1
=
(10-26c)
1 + 2k 2lm £2 cos 2 6
TM
waves
k m 6Z
1 + 4k m
2cos
2
cos 2 0 + cos2
20
(10-27a)
1+4k 2 k2
lm
2
2(1 + 4km
lm
P
1 + 4k 2
im
p=
b
2
£2
2
cos
2
0)
(10-27b)
cos 2 0 + cos
2
20
2 cos 2 20
l + 4k
£2 cos 2 0 + cos 2 26
1m.
(10-27c)
2
We notice that as frequency increases,
from
1
to
2
while
Pb
decreases from
proportional to frequency squared.
Pf
i
However,
increases
to
0.
Ks
cannot be
allowed to increase arbitrarily because the condition
klm
must hold in order that
RT
is valid.
is
K
Ks
«<
134
Solution of
10.3
Equations
RT
The task now is to solve equations
(10-1),
(10-2),
and
Once the upward intensity
(10-5) for given medium properties.
is obtained, the brightness temperature
TB
as measured by
a radiometer is given by
TB
(1
,x
K
(10-28)
u
- rot
-lm
z= 0
We assume a nonuniform temperature of the type
T(z)
=
T
found to be
The solution is
I
(10-29)
+ Th eyz
Td
=p
6+B 0
eZ/C5
=P eaz/cos
+ B
d
0
u
a
-
a
+ K
Ka
+ a2 + Ka yCOS 6
+ a
_
2
ez/cos6
+B
a
Cos
2
Cos2
+a2
0
2
(10-30a)
Bh e
Ka. y cos 6
2 Cos2e
z
(10-3 Ob)
where
a
=
Ke
(1f- 2)
1 -
2
P
/2
+
Pb
I
(10-31)
135
(a* + K) )
Pa
(a
(1-
+ K
( 2
rot
+ KY cos
(
6)
-
B
o
= K s'
im Th
lm
r)
at
r ot
(a2
_ KY COS
B0
(10-32)
B
=K~'B2(10-33)
B
Bh = K
-
2 COS 2
_
CL2
a
-
T /c X
a
(10-33)
o
2
(10-34)
,
a
and the scattering albedo
(10-35)
W = Ks K e.
The brightness temperature is determined from (10-28)
(10-30b)
and
T
2
Ka (
Ka
a+
=
B
a
{T
-
Ka)
+
a
a
+ y cos
T
.
J
(10-36)
can be interpreted as the effective extinctive rate
of the medium.
a
rt
rot a
Since there is forward scattering, therefore
is always less than
K
.
We now examine several
special cases.
136
(a)
For medium with uniform temperature distribution and no
T
B =
Ks =
Th = 0,
scattering,
(1 - r
0,
and
a =
K
Thus
.
ot ) T .
The emissivity is
(10-37)
(1-7
1 - rot.
seen to be given by
This result
also agrees with that obtained by reciprocity arguments.
(b)
For medium with no scattering,
Ks
=
0,
the result
is
(1 B
r t) (T 0 +
o
o
K
K T
a
h.
+ Y Cos 6
T
=
It
is interesting to note that
(10-36) also reduces to
which occurs at high frequency.
Pb = 0
(10-38) when
(10-38)
In our
model, the overall scattering effect on the brightness temperature is zero when there is no backward scattering.
when
(c)
Pf =
2,
a =
Also,
Ka'
For medium with scattering and uniform temperature dis-
Th = 0,
tribution,
2K
=
T
B
and we find
(1
-
rt)
a
(
+K)
a
- r
ot
ot
(a
T .
(10-39)
- K )
a
Numerical results are presented in Figures (10.2) and
(10.3)
137
Figure 10-2 Brightness Temperature for Shelf Glacier and
Compared with that Obtained by Gurvich et al
(broken line)
Gurvich et al's
Notations
Shelf Glacier
F-im
n. = 2.5 x 10~4
=1. 8 E,
lm = 6.7 x 10 -
FLm
ss Temperature
Br
6
=
.002
9
=
2 mm.
z
= 2 mm.
P=
2.7
7
8
= 2580K
Ta
260
42
250
240
230
220
210
200
190
180
170
_
-
160
150
0
1
2
3
4
5
6
Wavelength in cm.
9
10
138
Figure 10-3
Brightness Temperature for Continental
Glacier and compared with that obtained
(broken line)
by Gurvich et al
Gurvich et al 4 2
Continental Glacier
'm =
Brightness
Temperature
0i
Notations
1.8 E,
lm
= 5.4 x
6
=
.0006
k
=
.6 mm.
10
4Ec
n.1
=2 x 10~ 4
zO
=
P
= 1
.6
mm.
= 233 0K
250
240
230
220
210
200
190
180
170
160
I
150
0
1
1
2
11
1
1
1
3
4
5
6
Wavelength in
w
7
cm.
8
9
10
139
(solid lines) and compared with that obtained by Gurvich et
al42
[19731
(broken lines).
[Figure 10.2],
with our model.
We see that for the Shelf glacier
higher brightness temperatures are predicted
This is because their results are applicable
to low scattering cases while the scattering albedo for the
Shelf glacier is rather high.
glacier
[Figure 10.3]
In the case of continental
where the scattering albedo is quite
small, both models agree quite well.
(d)
In the case of uniform temperature distribution and small
scattering albedo, we expand to first order in
o
by noting
that
(1 K U
%e
t
Pf /2)
and
Ka
Ka
K
e (1-)
The result is casted in a form to compare with that obtained
by Gurvich et al42 [1973].
(1 - r
T
=
B
)
(1-r
ot
1
)
ot
8k"
lm
2
lm
6k
cos e (1 + 4k 2 k2 cos2O)
T
lm
(10-40)
140
This expression differed from their result in which there is
a
cos Go
in the numerator instead of a
nominator as shown in (10-40).
6 = 60
= 0,
(9-26) from Born approximation.
(10-4), we plot the brightness temperature for
= 222 + 81 e0.51z
T(z)
In the case of observation
(10-40) is identical to Gurvich et
al's42 results as well as
In Figure
in the de-
(10-40) can be confirmed by
using the Born approximation.71
at nadir,
cos 6
by using equation
-
88 e0.66z
(10-36).
We observe that the brightness temperatures in Figures
(10-2),
(10-3) and
2k lm
(10-4) have a minimum at
= 1.
The
overall scattering effect on the brightness temperature is
the product
Ks Pb.
we observe that
Ks
related to
(10-27),
with frequency while
Pb
From equation
(10-26)
and
is monotonically increasing
is monotonically decreasing with
frequency.
Numerical results are given in Figure
TE
and
TM
(10-5) to compare
waves as a function of radiometer viewing angle
0 .
0
In the case of non-constant absorption and/or scattering
coefficients, the differential equations of
RT
can be cast
in the form of integral equations which can be solved by an
iterative
approach.
71
141
Figure 10-4
Brightness Temperature as
a function of frequency
Brightness
Temperature
im = 1.8c0
Fm
=
Q
= 2 mm.
.00054c0
230
220
6
=
.0001
6 =
.0005
6
.001
210 .
200
=
_
190
180 -
170
10
20
Frequency
30
in
40
GHz
50
142
Figure 10-5
Brightness Temperature as a function of
Viewing angles for TE and TM waves
T(z) = 222 + 34 e0.81z
Frequency = 20 GHz
Brightness
Temperature
e '
lm
=
1.8t.
Elm
=
.00054t
lm
240
6
=
0
220
TM
200
TE
6=.002
180
6=0
160
TE
6=.002
140
120
100
80
60
40
0
10
20
30
40
50
60
70
Observation angle in degrees
80
90
2 mm.
143
Chapter 11
Emissivity of a Two Layer Random Medium--MRT Approach
In this chapter, we will
study the emissivity of a
two layer random medium by employing a wave approach.
In
a two layer medium, there is significant coherent effect
between waves going in the upward and downward direction.
A
RT
approach is inadequate.
The Born series approxima-
tion, as pointed out in chapter 9, works only for cases of
low scattering
albedo.
144
11.1
Introduction
The problem of random medium has been treated in the
past with the renormalization method 5 5 - 5 6 which deals directly
with field quantities.
It gives rise to the Dyson equation
for the mean field and the Bethe-Salpeter equation for the
covariance of the field. 5 5 - 5 6
In solving these equations, the
bilocal approximation is usually applied to the Dyson equation
which is then solved by mathematical techniques such as the
A
Fourier transform method.
the Bethe-Salpeter
of
iteration.
ladder approximation is made on
equation which is
then solved by the method
In the case of multiple wave scattering, the
method of iteration involves solving many integrals and leads
to complicated results after one or two iterations.
the assumption of far field
Under
interaction and incoherence among
waves in different directions for unbounded medium, radiative
transfer equations have been derived from Bethe-Salpeter equa-
.72-77
tion to study multiple scattering.
In the past, attention has largely been restricted to
cases of unbounded medium.
Rosenbaum studied the coherent
wave motion (i.e. the mean field) in the case of a half space
laminar random medium by using the bilocal approximation to
Dysn's78
Dyson's equation.
Stogryn, 45 by employing dyadic Green's
function, generalized Rosenbaum's approach to treat three
dimensional variations.
He also iterate the Bethe-Salpeter
145
equation once
(which corresponds to single scattering of the
mean field) to obtain the
coefficients.
scattering
static
scattered intensity and the bi-
In this chapter, we investigate the scattering of waves
by a slab random medium with a laminar structure
(randomness
in one dimension only) and bounded by different dielectrics
on both sides
(i.e. same model as chapter
that the nonlinear approximation,
9).
It is found
57-58 instead of the more
popular bilocal approximation, should be used in the Dyson's
equation for the case of bounded medium.
tions
yield
identical
results
for
variable expansion technique
79-80
The two approxima-
the unbounded
case.
A two
is applied to obtain the
mean Green's function, which'is then substituted into the
Bethe-Salpeter equation to derive a
transfer
(MRT) equations.
set of modified radiative
They are modified because in our
derivation we include correlation effects between waves in
different directions which are important in the case of
bounded medium.
The
MRT
two layer random medium.
equations are then solved for a
Conservation of energy is shown to
be preserved under the nonlinear approximation and the results
for the case of small scattering albedo agree with that obtained from the Born series approximation.
give rise
to
simple analytic
solutions.
The
MRT
equations
146
11.2
Renormalization Method--Dyson's Equation and BetheSalpeter Equation
(9-17),
If we iterate equation
E
(0)
=
E
0
(0)
+ Go
1
f (1) E
(01)
+ Go
(01)
f (1) G'
+ Go
(01)
f (1)
Go
we obtain the series
(1)
(12)
f (2)
E 1 0 (2)
(12)
f (2)
Go
(23)
f (3)
E 1 03)
(11-1)
Using the notation of Feymann's diagram,55-56 representing
G0
by
f
by
E 0
by
we rewrite equation
E
+
......-.
(11-1)
as
+
1+
S
+
...
.
0
S
S~-
(11-2)
147
f(z)
is assumed to be a gaussian random process with
correlation function given by
average of
(11-2)
+
Em
1M
Taking the ensemble
(9-20).
+
*
=
4
+
(11-3)
+6ee
denotes the two point correlation between two
where
f's.
To give an example, the last figure in equation
is
equivalent
G0
(01) Go
(11-3)
to
(12) Go
(23) Go
(34) E 0(4)
B
(13) B
(24) .
The fields in each region are separated into two parts;
one
part is the mean field, the other accounts for fluctuation.
E = Em +
with
<E> = E
(11-4)
and
<6> = 0.
The field intensity is
148
<EE*> =
E 12
+
<C(*>.
(11-5)
We next list the following definitions:
(i)
A diagram without terminals is a diagram which has
been stripped of its
(ii)
external
solid
lines.
e.g.
A diagram without terminals is strongly connected if it
cannot be cut into two or more diagrams without cutting
(Th
any
e.g.
lines.
connected:
the following diagrams are strongly
("
The following diagrams
are weakly connected:_
(iii)
The mass operator denoted by
M
or the symbol
*
is
the sum of all possibly strongly connected diagrams.
+
Equation
operator.
...
(11-3) can be resummed by making use of the mass
(11-6)
149
+
Elm
+
-~
p
+
0
0
we obtain the Dyson's equation,
Therefore,
Elm =+
-
(11-7)
Elm
Or,
0
(d2
dZ
where
2
+ k')
E 1 m(Z)
-d dz 1 Q(z,
z 1 ) Elm (z)
(11-8)
-d
Q (z,
z1 )
the mean Green's
is
the mass operator.
function
G 1 1 m(z, z 1 ),
Writing in terms of
the Dyson's equation
becomes
0
+ k
)
dz 1 Q(z, z 1 ) G 1 1 m(Z1 , z')
G 1 1 m (z,
-d
+
6(z -
z').
(ll-9)
150
Following the same procedure as outlined
for Dyson's equation,
we obtain the Bethe-Salpeter equation for the covariance of
the field by resummation,
<E 1 (z)
E 1 *(z')>
=
+
+
+
+
.
0
.a(11-10)
By defining the intensity operator as the sum of all strongly
connected operators,
+
+ .
.
+
.
the Bethe-Salpeter equation takes the following form:
(11-11)
151
0
< <(z) &*(z')>
=
1
f
0
-d
G*
I(z 1 ,
z2 '
dz
2
-d
(z',
[Elm(z
where
0
f
dz 1
2
z
0
')
z 2 ')
dz2'
dz ' f
-d
1
-d
I(z 1 , z 2 ' z
) EIm(z 2 ')
z 1 ',
f
+ < 6'(z
Gllm(z,
imZ
2
z
)
1
', z 2 1 )
2
2*(z)>
1
(11-12)
is the intensity operator.
(11-12) involves the mean
Since the integrand in equation
Green's function, thus the Dyson equation must be solved before one can solve the Bethe-Salpeter equation.
Both the Dyson equation and the Bethe-Salpeter equation
are exact equations.
However, the mass operator and the
intensity operator are in the form of infinite series.
We notice that the weakly connected terms are more secular
than strongly connected terms.
Thus the most popular approxi-
mation to the Dyson equation is the bolocal approximation which
of replacing
consists
~
the mass operator
by its
first
---
term,
(11-13)
Another approximation for the Dyson's equation is the
nonlinear approximation in
Q(z,
z)
Qn(z, z1 )
which,
=
Gll(z, z ) B
(z,
z1 ).
(11-14)
152
The usual approximation to the Bethe-Salpeter equation
is
the ladder
I(z 1 , z 2 ;
approximation,
z1 ', z 2 ')
y B
(z
1
,
z')
6(z
-
z2 )
6(z'
-2
(11-15)
In this chapter, we will use the nonlinear approximation
and the ladder approximation to derive the
MRT
equations.
These two approximations are energetically consistent for
bounded medium
(Section 11.6).
153
Mean Green's Function and Mean Field
11.3
Before proceeding to the solutions, we define some
parameters.
real
Let
k
lm
=
k'
+ i
lm
k"
lm
with
k'
im
and
k"
lm
both
(we use prime and double prime to denote the real and
imaginary parts of a quantity respectively).
wave number
Ka
and the absorption length
The absorption
Za
are defined
to be
K
=
a
ka =
The
2k"
lm
(11-16)
2ff/K a
(11-17)
scattering
wave number
Ks
and the scattering
length
Zs
are defined to be
K
=
6k k'2
lm
(11-18)
=
2
(11-19)
s
zs
where
k
7T/Ks
is the correlation length of the fluctuation of
the random permittivity.
taken to be
Aim = 2/k
The wavelength inside medium
1
is
.
Our basic assumptions
(which actually are quite natural
154
assumptions for the purpose of
(1)
K
a'
Ks
<<
k
.
subsurface
For subsurface
sensing)
are:
probing by waves,
must be much larger than the rate of absorption
the rate of scattering
(2)
>> Ka' Ks.
1/k
Ks
k'
Ka
and
in the medium.
Absorption and scattering should not
completely distort the wave within one correlation length
of
the fluctuation.
d >> Xalm*
(3)
The slab must contain many wavelengths so that
the fast varying part of the wave can average out and a
modified radiative transfer approach is adequate.
(4)
There must be enough statistical
d>> Y.
samples in
the slab.
The Dyson equation under the nonlinear approximation is
0
d
dz
2
+ k2
Gl(z,
z')
In equation
dz
6k 4
Gllm(z, z 1 )
' -d
I
e-|z
-
z
Gl1 m(z 1 ,z)
+ 6(z
-
z').
(11-20)
(11-20), we have used the exponential correlation
function of chapter 9.
to
=
(Our approach can easily be generalized
include other correlation functions.)
This equation can be
155
solved by using the two variable expansion procedures.
define
C
long distance
= 6z 1 .
d2
dz
= 6z
From
92
=
2
3z
= 6z,
scales
32
+ 26
2
=
C'
6z',
We
and
it follows that
+ 62292
-z35
(11-21)
352
The mean Green's function is
expanded
in
perturbation
series
; z',,
GllM~z (
= G llmo (z,
C')
+ 6 Gl l m l(z,
Substituting
(11-21) and
on balancing terms to
k2
+ k2
C;
z',
C;
z',
(11-22) into
z',
C')
...
(11-22)
.
(11-20),
zeroth order in
Gllmo(z,r;
+
E')
C')
we find that
6,
=
6(z
-
z')
(11-23)
Bz2
and to first order in
2
+ k
2
Gllm
(z,
6
; z',
') = -2
2
Gll (z,
; z', , ')
156
0
dz 1 G 1 1mo(z,
+ km
z1 ) e
G llmo(z
,
-m
d
z').
(11-24)
The solution to
(11-24) is written as
Gllmo (z,
C')
z
for
C;
z',
> z'
=
[A(C)
e-iklm zI
[U(C')
eik lmz' + W(') e- ikm zI
[A(E')
eiklm
(ll-25a)
and
=
E')
z',
Glmo(z, C;
[U(C)
z < z'.
for
+ B(E)
e ikm
In
+ B(E')
eklmZI I
(ll-25b)
eiklmz + W(C) e-iklmz
(11-25) note the symmetry property of the
Green's function.
Following the two variable expansion procedure, we substitute
(11-25) in
observe that
A(
(11-24) and eliminate secular terms.
),
B(
1
),
U(
1
)
in the integrand are varying on the
They vary much more slowly than
1
at
z =
z
and decays on the
and
ka
W(
and
1
)
ks
exp(-|z - zjI /Z)
Z
scale.
We
as variables
scale.
which is
These terms can
157
be taken out of the integral sign and substituted with their
values at
=
(.
We find that in order to eliminate the
secular terms, the following equations must be satisfied:
dA(C)
+ g A(
)
B(C) U(C)
d
2
+
i3k 1 m,k
l1 + i2k lmZ
(1
+ A(E)
dB()
-
g B(E)
A ()
W(C)
W()
ik1m
1 (1 - i2klmZ
2
-
\1 -
dL
+ B( ) U(E)
l
ik
0
(11-26)
0
(11-27)
i3klm Z)
i2klmZ/
l + ik
1 + i2k lmz
dUI(E)
+
gU(
)
B(
) U(
)
d
dC
/
(1 + i2k
+ A( ) W(C)
dW(E)
l + iklm
- g W(C)
A(C) W(E)
91
(2 - i3k
lm0
2m-1
1 -
i2k lmZ
1 -
iklm
l
- i2k lm
)
/Z
(11-28)
158
+ B(C)
U()
2 + i3k=
1 + i2kl
(11-29)
0
-k
where
Combining
(11-30)
.
g = ik
with (11-29),
(11-26)
and
(11-27)
with (11-28),
we obtain
W(c)
dA(C)
d
+ A(C)
B(C)
dU(()
dE
+ U(
(11-31a)
dW(C) =0
dE
and
) dB(C) =
dC
(11-31b)
0.
Thus
where
A(5) W()
=L
B(E) U()
=
L
and
Substituting
(11-32a)
(11-32b)
2
L2
are constants independent of
(11-32) in
(11-26) to
C.
(11-29), we can solve for
159
U(
B()
A~g),
A(g)
C
=
g (N
L2
U(g)
=
1 + N2
-g(N
1
(11-33)
(11-34)
+ N2
(11-35)
e
--
W(C).
-g ( M2 + M 1 )
e
C2 e
B
and
), r
C2
L
g (M2 + M1)
e+
Cl
W(E)
(11-36)
where
m
1 -
L
(1 -
M2 = L2
N2
=L
2
(11-37)
2iklmZ)
(<
2 + i3k lm Z
(11-38)
1 + i2kl m
2
IN L
ik lm k
-
i3k lm)
(1 -
i2k lm 91
1 +
ik lz
lm~ £
1 + i2km
(11-39)
(11-40)
160
Substituting in (11-25) yields
G llmo(z, z')
+ M1 )6z
-g(M2
= {C e
g(N
+ N2)6z
e
ik MZ
-iklmz
ik mz'
+ N 2 )6z'
-g(N
{L2e
L1
+ -
e
-ik
g(M 2 + Ml) 6z'
mz'
e
}m
(11-41a)
C
for
z
> z'
and
+ M 1 )6z'
-g(M2
G llmo(z, z')
ik lmz'
e
={C e
-iklmz
g(NI + N 2 )6z'
+ e
}
e
-g(N
1
+ N 2)6z
ikmz
{L2e
g(M
L
2
+ Ml)6z
-iklmz
}
+e
(11-41b)
C
for
z < z',
where
C = C 1 /C
be determined together with
2
L
in
(11-41) is a constant to
and
L2
by the boundary con-
ditions.
The Green's functions in
(11-41) are continuous at
161
z =
-
z '4
By
(11-20) we must also have
[dGllmo/dz] z = z
L
-
L2 =
'
= 1
which gives
(11-42)
1/i2klm
We next match the continuity
z =
0
and at
[dGllmo/dz]z = z +
z = -d.
of
At
z =
G llmo
0,
and
dG llmo/dz
at
we have
(11-43)
C = 1/R10
where
R10
z = -d
is
the reflection
coefficient
z = 0.
At
we have
C L2
g[N
+ N2 + M
+ M
e
where
at
] 6d.
1
R12
i2kM d
im
e
is the reflection coefficient at
k
R10
2
k lm
k
k
(11-44)
z = -d.
- k
0
+ k
k2
kl1M + k 2
R 01
(11-45a)
(11-45b)
162
We define
11
1)2
From
i'
+
T 2'
+
-
2"1
K
-
=
(1
k
-
k2
=-j
T
S
+ n
=
+R
view of
(11-46b)
(11-47)
2
3 -
In
+ M2)).
6 Z(M 1
(l1 -4 6a)
(11-44), we find
(11-42) and
-n1
6k (Nj + N 2 ) )
T1 " = klm(1 - k2
i
2k
-
1 0R 1 2
(11-41),
kl
6m L
-
i4k lm
i2k 1 MZ
(3 + i4k mk'\ i (
1
e
\i + i2k 1
I)
(11-42)
and
1
R 1 0 12 e
(11-4 8)
(11-4 9)
i
-
2)d
(11-44) , therefore
1
i2k 1 m(1
+
+ n 2 )d
)
Thus the zeroth order mean Green's function as determined from
(11-41) reads
163
Gllmo (z, z ') = L 1 (e
+
( (n
T2
2+
R 10e-1z)
)d + in
-in 2z '
z
(ll-50a)
(R1 2 e
for
z >
z'
and
Gl
(z,
z')
(R1
e
= L 1 (e
i(
for
2
z < z'.
1+
From
2z'
n 2 )d
+ in 11z
and
(11-47)
tive propagation constants
conditions
as well as
z'
+ R10 e-in
-in
+e
(11-48)
and
1
q2
the properties
2z
)
(11-5Ob)
we see that
the effec-
depend on the boundary
of the media.
Under the nonlinear approximation, the mean field satisfies the equation,
dd/
dz
0
+ k2
2
1lm;
Elm (z)
Mi
=
6k4
It can easily be shown from
= 2ik G lM(0,
dz
GllM(z,
z
)
-d
e
E lm(z)
f
-Iz
-
ZlI/k
E M (z
(11-20) and
z) .(
).
(11-51)
(11-51) that
(ll-52)
164
Hence,
(l1-50a),
using
Elmo (z) = i2klm
1
+ R12 eiI
For the
L
+
1
2
space case as
half
(11-53)
)d + i 1 z).
d -*
L2
1
-
2
(e
X01
then
o,
0,
2ik1
'n
= kl
+
idt k 2
lm
2
k2
lm
i6.
n
kim +
lim
2
_
-1
K
-
2ik iRz
lm-
I
2
k
I~
im
-
3ik
lm
k
2 iklmA -
- in
Gl
(z,z') =
2ik
1
-
(e
+ R10
z
e-in
2
z'
e
+
for
z
>
z'
for
z <
z'
2z'
(e
+ R10e
2ik1
165
This answer
78
is different from that given by Rosenbaum
who applied the bilocal approximation.
and
n2
In his results,
n
are the same and are both given by the above ex-
pression for
12'
For the case of unbounded medium,
G llmo (z,
=e
z')z')
1
2ikl
el2Iz
1
-
R1
0
=
0,
and
z'|
1 .T
This answer is identical to that yielded by the bilocal
approximation.
7
8,81
166
Derivation of
11.4
from Bethe-Salpeter Equation
MRT
MRT
We now proceed to derive
Under the ladder approximation
Bethe-Salpeter equation.
(equation
(11-15)),
equations from the
(11-12) becomes
<6 (z) 6 *(z')>
0
0
=
6km f
f
dz
-d
(Z"-1Z
G*
G
z2
Gllm(z, z1 ) Gilm(z',
{El(z ) E
+ < 6(z
(z 2 )
dz
-d
1
2
1 -
e
<e (z1 )
part
The fluctuating
§ 1 *(z
2
)>
of the field
(11-54)
) &1 *(z 2 )>}.
Note that in the right hand side of equation
only need to know
z2 1/k
(11-54), we
|z
for
is
-
written
z2
|
=
as a
O(A).
sum-
mation of upward and downward propagating waves,
e
(z)
1lu =
(z) e ikimz +
1d (z)
(11-55)
e -ikim
We then have
<e (z)
1
61U*(z')>
=
Ju(z, z')
e ikjlm(z
-
z')
167
-ik{m(z
+ Jd(z,
+Jc
C2
z')
z')
-
e
+Jc
e
(z, z')
-ikim(z
(z, z')
e
ikjm(z + z')
i
+ z')
(11-56)
where
(Z,
J
Jd(z,
Jc
z')
=
<d
z')
=
< l(z)*
61d (Z
z') = <
(z,
Jc(z,
z') =
C2
(z'>
(11-57a)
d (z')>
(11-57b)
(Z) $
lu
<e-lid
(z)cc*d(z'>
1d
(11-57c)
*
(11-57d)
z)
l
(z')>
The case to be dealt here is different from that of unbounded
medium because there are correlations between the upward and
downward going waves.
J
and
l
J
These correlations are denoted by
2
We wish to derive radiative transfer equations that govern
the behavior of intensities
I u(Z)
=
Id(Z)
=
u(z,
z)
Jd(z, z)
(11-58)
(11-59)
168
I (z) = J
C
c
(z, z) = J* (z,
c2
The derivation
of
MRT
is
z).
(11-60)
long and complicated.
We
first list the observations on the Bethe-Salpeter equations
as expressed in
1.
(11-54).
The quantities of interest are
Ic (z).
I
(z),
Id (z)
Thus, in view of the statement beneath equation
(11-54), we conclude that we only need to
)
<6(z
and
when
'*(z')>
Iz - z'| = O(k
solve for
Xlm)
or
in
both
the left hand side and right hand side of equation
(11-54).
2.
We foresee that
d'
u'
Ic
d'
c
and
are varying with the rate of the order of
Ka
and
and thus are varying much more slowly than
k'
or
J
Ks
Iu'
1/9.
3.
Iu'
In view of 2, the quantities
be close to the boundary on the
and yet can be far
(This fact
is
away on the
very important later
boundary conditions for
I
and
and
Id
1/K a'
or
Xlm
1Ks
Z
Ic
can
scale
scale.
on when we match
Id'
169
Next, we outline the steps in the derivation of
MRT.
1.
Substitute
dz
2
(11-56) into
The integral over
(11-54).
can be easily evaluated because of observation
2 above.
2.
The integration over
can be approximated
dz
following argument.
Since
d
>>
Xlm'
by the
therefore only
constructive interference terms contribute when the
integration over
is carried out over many wave-
dz 1
lengths.
3.
(11-54) can
The left and right hand sides of equation
be balanced by their phase factor dependence on
z
and
z'.
Integral equations are then set up with
Jc
and
J
on the left hand sides of the integral equations.
c 2
4.
By setting
z'
-+
z,
solely in terms of
5.
the integral equations are then
I u,
Id
and
Ic'
Finally, the integral equations for
are converted
Consider the
wavelength
scale which is
much less
Id
and
Ic
equations.
into differential
separation of
Iu'
z
and
than
z'
ka
to be on the
and
Z
scale,
170
|z
«
z'I '" O(x)lm
-
lz'
|z + d|,
are far
+ d|,
away from the two boundaries on the wavelength scale
However,
and the correlation length scale.
can be close to the boundary on the
z'
so are
z'
and
z
namely both
d >> Xlm'
Iz'l,
|z|,
Consider also that
s.
ka'
J
,
we consider
J
J
and
(11-54) for
z'
On substituting
s
ka'
and
scale, and
Without loss of generality
z > z',
and then let
phase factors
.
J
z
both
balance terms by their
z.
+
(11-50) into
(step 1) we find
(11-54)
that,
<
1
(z)
-R
1
= {e
*(z')>
n 2z
2 z + irl *z'
l
01
-i(TI
* +
-
in
-
2
-iriiz
*z'
z -
-mir
R0e
.12 *)d + in
+ ir
*z'
+r01e
2
z
-
in
1
in2*z
} 1 1 z,
z')
*z
12
z + ir 2*z'
- Rl01-in
- R0R12 e-i
+
{R01
i
hri
(T1 * + n2*)d
iT12 z + ir 2*z'
-
+ q 2 ) d + in1 z
in 1 z - ir 1*z I
-
} 1 2 (Z'
z')
2z + ir *z'
~
~i0
ir12*z '
01
171
-i1 1 2 z
ill 2 *z'
-
e
R*1R
01 12
+ q 2 )d + inlz + ifl*z
i(r)l
+
-(1
K
}
3 (z,
z')
z - inI *z'
2)d + in
+ {rl2 e
-in
2z
+ R*12 e
z
jfl*2
+ R
e
12
+ n2*)d - il2z
-i(fl*
+ - 2 )d + ir 1 z +
i ( i1
2 *z'
+ ii
-
ir1 *z'
(11-61)
z ')
(z,
} I1
where
2
6 k
= D1
z')
[E
-da
* +
-i (r
[R* 2
fd
2
4|
zI
z
T'2 *)
e
(z1 ) E*
(z
dz 1
2
)
f
-d
2
[R 1
2
2
)d + iI
+e
e
d - i-n 1* z2
+ <e (z
+
i(r I
dz
z 1-
I exp
(11-6 2a)
) 6' *(z2)>]
lm
6 k
12 (z,
fd
z')
4D 12
(ei
(R 1 2 e
dz 1
-d
+ n 2 )d
-i1
0
z
f
dz
z'
+ il
z
2
+e
(e
12z
1
2
z2 1
*z 2
ill1*z
R
z1
) exp
2
Z21
172
[Elm(z
) E mz 2
6 k2
3 z
z')
=
0
2
-i(a
z
* + n2*)d
E m(z 2 )
=
41D1
(e -1n2
*
[E
l(z)E*
-
(e
2
in
integrands
in
(z1 , z 2 ),
+ <d
0
-R 01
*z2
-
R
2*z2
*(z
-2 (
)ex
z
)>]
dz
(e
2
- z2
(11-62c)
- R01
z
(z2 ) + <f
(z
(11-53) and
since
J c2(z 1 , z 2 )
scale, we replace them by
respectively.
z2 1
-z 1
)l*Z2
exp
e
(11-62),
and
T1 1 Z 1
0
z
1
E
(z )
dz
mf
2
We now substitute
)
dz
-d
z
6 k2
I 4 (z, z')
(11-62b)
6 * (z2) >]
+ e
[Em (z)
Ic*(z
)
e
(R1 2
Jc
(z
dz 1 f
m f
41 D
+ <f
(11-62d)
*(z2)>.
z
1
)
(11-56) into
J
(z
,
z2
'
(11-62).
d(z
,
In the
z2)'
k
are slowly varying on the
Iu(z 1)
Id(z
),
Ic(z )
Because of the range of
z
and
and
z',
most of the contributions of the integrals in (11-62) come
from terms of constructive interference
(step 2).
We conclude
173
then that
z')
12 (z , z ' )
and
and
I3
z,
z')
are small compared with
1 4 (z, z').
Balancing terms on both sides of equation (11-61) by
their
respective
phases
(step 3) ,
we arrive at
the following
equations
ik{
J
(z, z')
(z -
e ik'l
-
+ rl2
(z -
I1 +
z')
Sd
(z ,
z')
ie
+ iT
n 2 )d
(z -
-ik{
phase factor
z')
e
phase factor
1
2z
z
-
in2*z'
z')
- in
*z'
14 (z,
(ll-63a)
z')
z')
e
e-ik m(z
-
-inIz + in *z
z')
= r01
Il (z,
z')
-i1n2 z + irn2*z'
14 (z,
phase factor
eik
(z + z')
z')
(ll-63b)
174
(z
ik
z')
S(z,
c1
+ z')
R*
=-
e
+ ir
:11111 + n 2 )d
1
e
ir2z
+ irl1*z
-ikl
(z + z')
e
-ik{
(z + z')
=--Rl e
-i(1* + n2 )d
i1*z
-
z'
+
z
in
equation
- (K
=
1 1 (z, z)
r0 1 e
i2k'
z')
2z
-
(11-63d)
4 (z, z') .
(z) = e
Id(z)
1 (z,
e
Setting
I
ir12*z '
-ir1 z -
z') e
+ R*12
(11-63c)
14 (z, z')
phase factor
Jc C2
(z, z')
z + ir12*z
+ R12
(z,
I
I
(z,
z
+
2
+
K2)d
(step 4)
-
K1 z
I
gives
(z
z)
(11-64)
(11-65)
(z, z)
I 42z
+ n *)z
I
01
1
1
e
2
z) + e
i (r12
Ic (z) e
+ R1ei(I
+ r1
(11-63)
)d + i(q 1
(z,
z)
+ n2*)z
I4 (z
z)
(11-66)
175
where
=
1
2r
K 2=
s
1
2
2
|D1
K
K
D
+
K
"j = K
+
-
a
-
2a +
sk2lm
2
D
i
D
1
D
= 1 -r0l
+
0 1
12
6 k
-(z
4
gf rl 2
(1
+ 4k2
2)
(3
+ 8k2
2
)
(11-67b)
(1 + 4k
2
2
(11-68a)
dz
e
(11-67a)
2 )d
K2 )d
+
1
(11-68b)
-d
2
X1
1
|D12
-
+
+ gf r12
d(z
+Kd
-i
1
-2(K
+ K2 )d
1
2 2 +
u (z ) e-K
Ic (z ) e
R*
f12
2
-
[9b rl2
2K2
+ (KK 21 )d
+ gb rl2
+ g
2)
lm
z
4 1D 1
+
2
+ 8k2
r1 2 e
D
=
+
2
(3
e
- (K
Z)
D
K
K
2
K
-
Z
+ gb
d (z )
e
+ gK
u
z
ez
+ q2*)d + i(2k'
-
TI
-
2)z
2K 1 z 1
(PTL-TT)=
zz
{
(0OL-TTV
T
T
.I
a (T Z) I TO~ ;f,
*U+
-
T
z(* ZLt
-
LI)
T
UIT
Ll
q_
a (( TZ)P
G
+ *T U)T-
+ T >1)
z (Z
G
T
T 11TO
*
*
(Tz)
+
(Tz) n,
+
1
p(U+
p(
T LI)
T
-
TI
T
+ TN) -
p (ZN
q5]
T
O _I +
;aZT 11T011 J f
T:1TO : + T)
I111t
zI TOxIJI
fz -Z
I9c
0
(69-TT)
T Z)
T (WT
9LT
-
ZU+
+5
+ T 1) z
Tz(ZN + TA)
)
(Tz)n I q.6) +
Tb +
a
OJ qB +
DI TO~ 11J
a
Z
a ((Z)P
P(* L
=
T U)T + P( ZU + T U)T
Tu-.
(Z
i Z) :vI
177
29.,
g_
and
r01
=
express
(11-71b)
1 + 4k2m
-
1R0
r1
,
=
2
in terms of
I
i(2km
2
11
2
-2
|R1 2 1 2 .
From
I
Id'
l*)z
-
and
U
{
ce
1
R
R1
2
+ R*
01
r 01
(z) e
r12
(Z)
I12z
ie (
+ 12
In view of
(n
+ T 2 )d + K2z
K~z
2
(K 1
+ K 2 )d
-
K1 z
e
e
+ T 2 )d -
(11-64),
(11-65),
r 0 1 1u(0)
(-d)
e
K 1z}
d
Id
I
(z)
0
01
D
-
I
(11-66), we can
=
r 1 2 Id(-d).
(11-7 2)
(11-69) and (11-70)
(11-73a)
(11-73b)
These are the boundary conditions satisfied by the intensities
178
I Uand
65) in differential form
dI
--
I
K
(step 5) , we obtain,
I
(p
Pf+
5
K
dz
K
I
ed
-(K
erl21-4
2
-
s
2
(Pf Id
+ pb Iu
+
K 2 )d
K z
1
+
1
+
-
e
+
/
(Ks
X0 1
+D 2L
s b e
+
d)
+ pb
2
+K
(11-
K2z
K
--
+
dz
dId
(11-64) and
Writing the integral equations
Id*
K2 )d
2
x01
-
r1
ID12
k pf
Pf
K1 z
(11-74)
K pb
sb
2
2
K2 z
(11-75)
2
where
K
k
(1 + 2k 2
im
=
S
1 + 4k
1D+
2
)
(11-76)
2Z2
D.
=
K
K
K
e
=
=K
K
a
2
k
+ K
(11-77)
S
S
(11-78)
179
p
1 + 4k 2 - 2
klm
=
2 k2
1 + 2k
(ll-79a)
=
(11-79b)
lm
p
1
1 + 2k
lm
2
(11-75) together with the boundary con-
(11-74) and
Equations
ditions in
(11-73) are the modified radiative transfer equa-
tions derived from the Bethe-Salpeter equation under the
ladder approximation.
In the case of a half-space random medium, we let
d
+
co
and the modified radiative transfer equations reduce to
kp
k
dI
u
=
-
K
I
Iu + pb
d)
+
s
2
2
dz
d
(p
+
(Pf Id + Pb
e Id
dz
u
_
s
K Z
0112 e e
x0 1 12 e e
2
2
We remark that if we use bilocal quantities in the
derivation, the
MRT
equations so obtained lack symmetry and
do not conserve energy even for the half space case.
The
principle of energy conservation is preserved with the use
of the nonlinear approximation
(Section 11.6).
180
MRT
Solution to
11.5
Equations
The modified radiative transfer equations (11-74) and
(11-75) are simply two simultaneous first order differential
equations with constant coefficients.
They are readily
solved and written in the following form:
d
d
01
12
(P ez
I
01
(
e-a (z
+ Q f
f2
+ 2d)
z + Q e-a(z + 2d)
(ll-80a)
2z)
r
-K
z
-
(K
+ K 2 )d
ID
(ll-80b)
where
(1
(1
-
f 2 r 0 1 ) (1
-
-
f2 r 1 2)
f2 rl2)D
-
(f2 - r 0 1 ) (f2 - r 1
2
e-
(ll-81a)
Q=-P
2
1 -f
- r2
2
r1 2
(ll-81b)
181
a-
K
f
Ka
+
a
C5
2 P
r%1/ 2
(-W)
cK
W = Ks Ke
and
(11-82)
a
f2
that
(11-83)
2
is the albedo.
of the two layer medium,
e
To calculate emissivity
we note
pb 1/2
+
by reciprocity
(11-84)
e = 1 - r
where
r
is the reflectivity.
The field in region
0
can
be written as
-ik
E
0
z
0
=e
ik
+ Fe
The reflectivity is
<E E *>
By
and
(11-53) and
r
z
08
<PI*>.
=
|Eom (0) 12
(11-85)
=
lElm(0)
We observe that
2
at
<E E *> =
the boundary
z =
0.
(11-85), we have
iRr+R
R01 + R12
n2)
(11-86)
D
182
<E E *>
0
By
(11-56)
<E 1 E
= 1 + <r> + <1*> +
12 +
JElm(Q)
=
<L
=
1 + <F> + <r*> +
I<r>|
=
1 + <F> + <F*> +
|<r>| 2
and
I<r>|
= <rP*> =
R0 1 + R12 e
2
2
+ Iu 0)|X
ih1 + -2)d
{ (1
-
+
(1 -
f
2
~r
r
0 1 ) (1
(11-88)
)X 1 0 12
2
2
t
2
01
-r(K
1d
2
-
-
1 2 )(
2 -
1
1
+
r 0 1)
+
K2
)
2 -
r12
K 2 )d
-2ad
-2ad
2 12)
-
+ Ic*(0)
1 01
-(K
(f
0
Iu
+ Ic(0)
ID1 2
f2 r12
f2 r 0 1 ) (1
+ Id(0)
+ Iu(0)
+
-
0
=
we obtain
(11-88)
D 12
(1
z
11
0
(11-87)
(11-87)
<Fr*>.
and (11-66) , we have
(11-64)
,
z=
Equating
r
0
f2 r 1 2)
(11-89)
- f2 -
r 01)(
2
-
r
1 2
)e-
2a
d
183
where
t01 = 1 -
-90)
01
We now consider
1.
A
several
cases:
special
half space random medium.
We let
d
+
oo,
and
(11-89)
becomes
t2
r
= r01 +
2
1(11
1 -r
This is
the
2.
and
RT
identical
f2
to that
in
equation
(10-39)
as yielded
by
approach.
Small albedo.
a
01
-91)
K
(1
-
w
We let
p f/ 2 ).
+
0,
Equation
-2K
e
2
4K
d
e
e)(
-2K
+ 8 (p /pb )Ke
(11-89) becomes
t2
k p
+
01
s
b
1D1 4
|D12
+ r2
p b
f
2
1
R0 1 + R12 e
{ (1
which yields
r 1 2 de
-2K
d
e
d
e }1.2
(11-92)
184
It
can
be shown that
easily
this
identical
is
yielded by the Born series approximation.
to
(9-27)
as
It is useful to
note that the result obtained by the Born approximation is
valid when
k
d
>>
1
Ka
s
and
Xlm'
and the
k
«
result is valid when
MRT
ka'
ks.
Thus the two results
complement each other and they agree within their common
domains of validity.
3.
The second layer is perfect conductor.
and
r
r1
2
= 1.
Equation
i(h1
+ n2)
i(1
+ -2
(11-89)
R 12=
We have
-
becomes
2
=
t2
t01
+
(1
f
d
e
R
-(K
+ K2 )d
2
+
-e
(1
-
- f 2 r 0 1 + (f2 - r 0 1 ) e-2ad
fe
l-
1
+ K 2 )d
i 01
-2ad
+ T)2 )d
I-
ROl e
(11-93)
It is interesting to note that as
f2 + 1
and from (11-93)
r
+ 1.
W + 1,
we have
a
-+
0,
Thus all the incident power
185
is reflected.
In Figure 11.1, we illustrate the emissivity for a slab
medium with
6 = 0.1
and
0.2.
We see that scattering dam-
pens the interference pattern and decreases emissivity in
general.
The existence of the interference patterns depends
on the location of the subsurface and the extinction loss of
the random medium.
Figure 11.1
00
Emissivity of Two Layer Random Medium
Emissivity
1.0
k = 2 mm
20 cm IEm = 3.2(1 + iO.05) 0
E2 =
81
E0
.9
4op.00
0.1
4000
do
dop
400
am
do
.00
000
.8
6
=
0.2
.7
.6
.5
2
3
4
5
6
7
8
Frequency in
9 10
GHz
20
30
40
50
187
11.6
Conservation of Energy
Because the fluctuating part of the permittivity is
real, we can obtain from equation
(9-5) the energy conserva-
tion relationship
E *
dz
2
E
- E
1
1 dz2
E * + 2i(k 2
1
)"|E 1
lm
2
=
0.
(11-94)
We use superscript double prime to denote imaginary part of
a quantity.
Taking the ensemble average of
E*
d2
lm dz 2
+ lim
Z'+Z
+
lm
-E
lm dz
2
d2 <e(Z)
<
dz
2i (k2)
E*
lm
*(z')>
-
2
+
<
(z)
Taking the integration of
d2
dz
2
| 2 >}
=
2
"{ IEl 1
(11-94), we have
y1
<
* (Z)
(z')>
(11-95)
0.
over region
1,
we
define
y2
0
-d dz y
= 0.
(11-96)
188
y1
Note that
and
are defined for the purpose of
y2
testing energy conservation.
For energy conservation to the zeroth order, both
and
of
y2
must be of order
0(6)
does not imply
integration of
y1
We remark that
0(6).
is of
y2
from
z
=
for a half-space medium, may yield a
y2.
of
being
the
0 (6).
For instance,
0,
which is the case
to
-o
y1
y
significant value for
Thus there should be no constructive interference terms
0(6)
in
y
antee that
From
E
This condition then is sufficient to guar-
is of
2
d
2
L
E
0(6).
we obtain
(11-51),
m dz 2
+ 2i
y1 .
d
llm dz
lm
2
2
2
E 1M = -2i(k lm )1"|Elm
0"
Gll(z, z ) B
-d
(z,
z1 ) E
1m
(z) E
m
l
(z)
dzl.
(11-97)
In
view of
d <6(z)
lim
z-*z'
dz
2
(11-20)
6*(z')>
and
(11-54)
(z)
(z')>
189
=
-
0
0 dz
l(Z) |2> + 2i
2i(k m
Gllm(z,
1
z 1 ) B f(z, z1 )
l-d
i
)
+ <e (z) el*(z
[El(z) E*m(z)
0
+ 2i
/
dz
Gll
(z, z
) B
(z,
z )
(z )
(z)>
-d
(11-98)
Substituting
(11-97) into
Thus energy is
(11-98), we find that
y1
=
0(62).
conserved to the required order.
If we had used the bilocal approximation to Dyson's
equation rather than the nonlinear approximation, i.e. instead
of
(11-14), we use
Q(z,
z1 )
G 0(z,
G 0(z, z1 )
where
z1 ) B
(z,
(11-99)
z1 )
is the Green's function for
the unperturbed
problem, we would have obtained
0
y,
= 2it
[E
0 dz
(G lm(z,
(z) E m(z)
z1 )
+ <&
*(z)
G lm(z,
z ))
1)>0
B
(z, z1 )
(ll-100)
(11-1(z
190
We use superscript
If for
|z -
G
(z,
then
101)
y
z11
b
to denote bilocal quantities.
< 0(k),
b
as given by
,
)=
(11-100)
0(6),
is
of
(11-101)
0(62).
Equation
(11-
is in general true for unbounded medium but not for
bounded medium.
191
Chapter 12
Thermal Emission from Stratified Medium with Nonuniform
Temperature Profile
In this chapter we apply the fluctuation-dissipation
approach to find the brightness temperature of a
medium with nonuniform temperature profiles.
stratified
(Scattering by
random fluctuations of dielectric constant is not included
in the model of this chapter.)
According to the fluctuation
dissipation theorem, thermal radiation can be treated as
generated by an equivalent source with a certain expectation
value. 4 4 ,67
The dyadic Green's function for stratified medium
serves to relate the electromagnetic field to this equivalent
current source.
The propagation matrix formalism is then
applied and the brightness temperature is solved in closed
form.
Since a
stratified medium can approximate a contin-
uous profile, brightness temperature for a stratified and/
or continuous profile can be calculated.
192
Brightness Temperature of Stratified Medium
11.1
From equation
(8-6),
2th
the current source in the
layer (Figure 12.1) is given by
<3 rw)
J
±E
" KT
=- 4
(',o)
-E I6(o
=K-T (z)
W')
6(r
(12-1)
- r')
iT
Using dyadic Green's function, the electric field in region
0
is given by
E,
t
E
O) =
k =
No
where
t = N + 1
f
dx' f
dy' f
(12-2)
(,
')
-
P
')
and
d
+
.
By the symmetric
of dyadic Green's functions,
the superscript
dz'
-00
-W
1
t
oz (r, r')
denotes transpose.
properties
(r', r),
=
In view of
where
(2-25),
we find
iiiz - r-
00
5
2
ok
E')
r
=
-
8(2r,
/
d 3 k 6(k
z
-/2p6
-
k
x
2
-
k
2)
y
e
k
k
Z
193
Figure 12-1
Stratified Medium with Temperature
Distribution
z
L
Region 0
0.1O, O
z =0
Region 1
Region
k
Region n
Region t
T 1 (z), I
, E1
z
=-d
z
=-d
z
=-d
z
=
-d -l
z
=
-d
T z(z) , p ,
Tn (Z) , pn'
Tt(z),
pt'
n
t
n
194
-ik
(-k z)
(kz )[A
{e
(k z
e
+ D kh k(k zZ
e
+ B e
e
II
-ik
+ h(kZ) [C9,h
(-k kz) e
] I.
(12-3)
k
and
-k
Introducing
into
(12-2),
E(k,
o)
is made in arriving at
y
(12-3).
k
y
-k
x
(12-3)
-d
00
E
ff
dx'dy'
-00
k 28 k, = 1
-
_/W 2
6 (kz
x
we obtain
t
=-
,
to
A change of variable of integration from
e
-
k k22-
k 2)2
1
f
dz'
-d
{~
{e(kz) [A k e k (-kz)
1K
e
z
-i]
+ h(k )[C
+ B ze 91(k z ) e
-ik0
A
+ Dk h
Using
h z(-kz
9, ) e
9
(k zz
(12-1),
e
(r'
.
(12-4)
(8-10) and (8-11), we find the brightness
temperature to be
r
195
TE^
TT
(k, o)
OS8
= k cos0
t
"-d
E
f
B = 1
1
[A e
1
ik
(-k z)
1 dz'
-d
E
-ikk
z'
e
T k(z')
+ B e
(k
zz
2
}
e
z)
(12-5a)
z
t
TM ^
TB
S
-d
"
(k,
A)
COS G
= ks co
09
1
[C h
ik
(-kk z
e
T
(z')
-
-
1
1 dz'
3
-ik
z'
Z
+ D h 9,(k9,
9, z)
z'
2
.
}
9,z
e
(12-5b)
z
In the derivation of
-
/2yE
k
2
-
k 21
-1
for
= U(k ) cos 0
z
y
x
o
U(k ) = 0
and
> 0
k
(12-5) we use the relation
6 (k
-
for
w/IpI)
forward.
where
-
U(k z )
< 0.
k
The calculation of the integral in
,
6 [kz
(12-5) is straight-
Without loss of generality, we let the plane of
observation be the
perature
T
(z')
x
-
z
plane and set
is a constant in region
= 0.
k
9,
The tem-
(the case of
non-constant temperature can be generalized easily) we find
that for horizontal polarization
TE
TB (a
B
n
0
)
6 k"T
A9
e -{k
k
=
_k
OSb
U0o
X,
A
=1
2k'Z
zd Z
2z
196
-2kz "(d
k9zdZ2
k(19
B ke
d )
-
-1
( 2kk"(d
-
- e
1 - d )
)
2k"
-ik
(A
d
ikz d9
S~1) (B , e
Z)*
e
(1
- e
-i2kz' (d9
-
d)
1
)
2ik
d
ik
-iktzd
(A9 e
e
)*(B
(12ik
k
cos
Similarly,
TTM
B
(12-6)
9
2kt"
0
0
for vertical
cos
-ik
C
L
e
1
Z=
0
a
9Pz d
(
1k
2
-2k
9zi
"(dz
2
+ k 2)
-
-9
)
e
"
ik d
ID9e k z 2
2
2k
(1 - e
2kZ"
we obtain
polarization,
(1
2k
n
Tt,
t
k
0
1 - dz)
-
e)
'
-2ktz
j TE 1 2 e
+
(dk
i2k Z'
)
+
"(d
- d
9
)
k'z
Ik
-k
|
2
+ k
197
-ik
zd
e
(C,
ik
e)*
)
d
1-Ze
i2k k
(d
_
- d
)
)
2ikz'
-ik
d
ik
(C e
)*(D
d
)
e
2ikt'
k
Cos 6
+
)
- d
'(d
( -i2k
2
£t (|ktz
+ k
2
)Tt
ITTM
2
e-2ktz dn
(12-7)
|olkt
0
.
2ktz
For homogeneous half-space with constant temperature,
we find
TTE
(l
B
(12-8)
TE ) T
and
'TM
.5.
TM
= (1
In the derivation of
identities
2
t'
k tz
2k
'
+
Et
(12-9)
Tt.
(12-8) and (12-9), we made use of the
ktz
ktz ")/ktz
t
and
I tz
2
+ k 2
198
12.2
Numerical Results and Discussion
In
Figure
(12-2)
we examine the angular
the brightness temperature.
1 GHz,
At
TTE
dependence of
and
TTM
are
plotted as a function of observation angle for the following
profiles:
61(z)
1
=
9.0(1 + iO.3)
-
(5.5 +
i2.5)
a
eaz
(12-10a)
0
et
(12-10b)
1(-d)
T(z) = T
+ AT ebz
(12-10c)
Tt = T(-d)
with
and
(12-10d)
a = 0.02 cm ~,
AT =
± 204K.
b =
0.05 cm~,
d = 30 cm,
T0
= 280*K
It is interesting to observe a maximum for
vertical polarizations similar to the Brewster angle for a
uniform half space medium.
In Figure
(12-3), we plot the brightness temperature of
a radiometer looking from nadir as a function of 'frequency
with
and
a =
0.02 cm~,
b = 0.05 cm
-1
d = 30 cm,
and
-1
0.1 cm~.
T0
= 280*K,
AT
=
20*K,
In the calculations the
temperature and the permittivity profiles are stratified into
Brightness
Temperature
Figure 12-2
Brightness Temperature as a
function of viewing angle
280
270-
TM
AT =20 0K
270
TM
260
AT =
-20
0K
250
240
230
TE
220
AT
200K
210 _E
200-
190-
180.
-1"7A
Observation angle in degrees
AT
200K
200
Figure 12-3
Brightness
as
Temperature
a function of frequency
Brightness
Temperature
.4
b =
270
.05
b =
265
TO = 280
260
-
255
-
250
_
245
-
0.1
AT = 20 0 K
0 K,
AT =
T0 = 2800K,
-
20 K
b = 0.1
240
b =
235
I
2
I
4
1
6
8
0.051
1,__
1
1
1
10
12
14
16
Frequency in GHz
18
20
201
350
layers from
0
z = -d.
to
We see that at low fre-
quencies the subsurface temperature affects
at high frequencies.
TB
more than
At very low frequencies the brightness
temperatures for different parameters of
the same value dictated by
T0
a
and
b
approach
and the effective emissivity
of the medium.
The profile in (12-10) can actually be solved exactly
in terms of Bessel functions. 8
cal
comparison.
The result
2
,8 3
calculated
We have made the numeriwith the stratified
model is found to agree very well with that calculated with
the exact formulas.
At the high frequency limit, we can use the
od 4 4 ' 8 3 which may require less computer time.
(12-4), we plot the
WKB
WKB
In Figure
result to compare with the result
obtained from the stratified model.
We note that the
result is quite accurate in the high frequency side.
low frequencies, the
fects.
meth-
WKB
fails
WKB
For
to predict interference ef-
This is due to the fact that the reflectivity is a
constant for the
WKB
approach,
whereas for the stratified
model, the reflectivity exhibits the interference effect.
The reflectivities are plotted in Figure 12-5.
in using the
IBM 370
CPU
time
for calculation of the brightness tem-
perature in Figure 12-4 is approximately
the stratified model,
The
0.10
0.12
minutes for
minutes for the exact solution,
Brightness Temperature
280
KB
Brightness Temperature
Figure 12-4
Calculated with the Stratified
Model compared with that calculwith the WKD method
270
260
Stratified Model
250
240
230
0.1
1.0
Frequency in GHz
10.0
C
[
Figure 12-5
Reflectivity
Reflectivities calculated
for the stratified medium
Lb
and for the WKB method
0.15
0.14
0.13
-
0.12
0.11
.
0.10
.
stratified
0.09
-
0.08
-
0.07
-
model
WKB
0.06
0.1
1.0
Frequency in GHz
10.0
C
LbJ
204
and
minutes for the
0.03
WKB
for Figures 12-4 and 12-5 are
The parameters
approach.
.
C1(Z)
=
(2.88 + iO.34)
e-az
E:
0
Et
(z
T(z) = 300
a = 2 m
b
=3 m~-
=
-
-1 m)
20 ebz
-00
<
z
<
0,
205
Chapter
13
Conclusions and Suggestions for Further Study
In this thesis, a theoretical basis of the EIF method
has been developed by using the model of a stratified medium.
Three approaches are used ----
the geometric optics approach,
the mode approach, and the fast Fourier
Transform approach.
Each of the three approaches has its advantages.
They serve
to complement one another.
For lossy media and large layer thicknesses, the geometric optics approach gives simple and accurate results.
The
interference patterns calculated with this approach can be
easily interpreted in terms of ray optics.
When losses are
small and layers are thin, the mode approach is most attractive.
The results can be interpreted
modes of the layered medium.
in terms of the normal
Analytic methods are not useful
when the number of layers gets large.
The fast Fourier trans-
form method is a numerical approach designed to treat general
cases of arbitrary number of layers with various properties.
Though the fundamental work on the EIF method has been
completed, there are still many loopholes to cover.
We have
206
used only the stratified model.
In many situations the
interfaces between layers are sloping instead of being flat.
They may also have random undulations.
The medium may con-
tain inhomogeneties acting as scattering centers.
ing of electromagnetic
tain a
waves of a
dipole
source
Scatter-
(which con-
spectrum of plane waves) by the above obj.ects is a
difficult problem.
However,
simple studies can be made on
scattering by a finite number of large objects with the
ray optics approach.
Scattering by small objects can be
handled by a perturbation
analysis.
In this thesis, we have also developed theoretical
models for subsurface passive microwave remote sensing.
using a laminar structure,
simple analytic formulas
By
are
obtained for the brightness temperatures.
The difference between our results
Gurvich et al 42,
and the results of
is that they studied only single scattering
while we have been able to analyse multiple scattering
effects.
We also included a nonuniform temperature profile
in the half space random medium.
A
two layer medium exhibits coherent effects.
We
derived modified radiative transfer equations from the Dyson's
equation and the Bethe-Salpeter equation.
Then we solved
for the emissivity of a two layer random medium.
We also
found that the nonlinear approximation must be applied to
the Dyson equation in order to obtain MRT equations from
207
ladder approximated Bethe-Salpeter equation.
The fluctuation-dissipation method is the wave apprWe use
oach to treating nonuniform temperature effects.
it to solve for the brightness temperature of a stratified
medium.
Because a
stratified model can approximate a
continuously varying model by including more layers, our
is
solution
applicable to a
stratified
and/or continuous
medium.
The task of developing theoretical models is by no
means complete.
We have concentrated on laminar
structures.
Lateral variations need to be considered in the future.
Surface roughness has so far been neglected.
Scattering by
a composite model, including surface roughness and random
fluctuations of dielectric constant, is an important and
challenging problem.
The fluctuation-dissipation approach
should also be extended to include scattering effects.
Finally, more attention is needed in the area of interpretation
of data for both the EIF method and the microwave
remote sensing technique.
just
When we consider
for a two layer lossy medium,
independent
parameters,
the difficulty
the fact
there are at
least
that
five
of interpretation
and the need of a good inversion scheme become obvious.
208
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222
BIOGRAPHICAL NOTE
Leung Tsang was born in
Hong Kong on July 7,
1950.
Having finished high school at Wah Yan College, Kowloon,
Hong Kong in 1967,
Canada, as a
as a
he went to McGill University, Montreal,
freshman.
sophomore
in
In September 1968, he came to MIT
the Department of Electrical
from which he received the
SM and EE degrees in
graduate
and a
student,
teaching
SB degree
February
1973.
Engineering
in
June 1971 and the
In
the years as a
he has served both as a research assistant
assistant.
Mr.
Tsang is a member of Tau Beta
Pi and Eta Kappa Nu.
Publications
1.
L. C. Shen, L. Tsang, and J. A. Kong, "Multifrequency
Excitation of a Wire Antenna for an Invariant Radiation
Pattern", IEEE Transactions on Antennas and Propagation,
vol. AP-20, no. 6, November 1972, pp. 784-785.
2.
L. Tsang, J. A. Kong and Gene Simmons, "Interference
Patterns of a Horizontal Electric Dipole over Layered
Dielectric
Media", Journal of Geophysical Research, vol.
78,
3.
no.
17,
June 10,
1973,
pp.
3287-3300.
J. A. Kong, L. Tsang, and Gene Simmons, "Geophysical
Subsurface Probing with Radio Frequency Interferometry",
IEEE Transactions on Antennas and Propagation, vol. AP-
22, no. 4, July 1974, pp. 616-620.
223
4.
L. Tsang and J. A. Kong,
"Electromagnetic Fields
due
to a Horizontal Electric Dipole Laid on the Surface
of a Two-Layer Medium", IEEE Transactions on Antennas
and Propagation, vol. AP-22, no. 5, September 1974,
pp. 709-711.
5.
L. Tsang, Raymon Brown, J. A. Kong and Gene Simmons,
"Numerical Evaluation of Electromagnetic Fields due
to Dipole Antennas in the Presence of Stratified Medium",
Journal of Geophysical Research, vol. 79, no. 14, May
1974, pp. 2077-2080.
6.
L. Tsang and J. A. Kong, "Application of the Radio
Frequency Interferometry Method to a Stratified Anisotropic Medium", IEEE Transactions on Antennas and
Propagation, vol. AP-23, September 1975, pp. 725-728.
7.
L.
Tsang, E.
Njoku, and J.
A.
Kong,
"Microwave Thermal
Emission from a Stratified Medium with Nonuniform
Temperature Distribution", Journal of Applied Physics,
in press, 1975.
8.
L. Tsang and J. A. Kong, "The Brightness Temperature
of a Half Space Random Medium with Nonuniform Temperature
Profile", Radio Science, in press, 1975.
9.
L. Tsang and J. A. Kong, "Microwave Remote Sensing of a
Two-Layer Random Medium", IEEE Transactions on Antennas
and Propagation, in press, 1976.
Conference Papers
1.
J. A. Kong, L. Tsang, and Gene Simmons, "Lunar Subsurface Probing with Radio Frequency Interferometry", URSI
Symposium, Williamsburg, Virginia, 1972.
2.
L. Tsang and J. A. Kong, "Microwave Remote Sensing of
Bounded Random Medium", URSI Symposium, Boulder, Colorado,
1975.
3.
J. A. Kong, W. C. Chan, and L. Tsang, "Geophysical Subsurface Probing with the Electromagnetic Interference
Fringes (EIF) Method", URSI Symposium, Boulder, Colorado,
1975.