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Microwave polarimetric backscattering from natural rough surfaces

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O rder N u m b e r 9423280
Microwave polarimetric backscattering from natural rough
surfaces
Oh, Yisok, P h.D .
The University of Michigan, 1993
UMI
300 N. ZeebRd.
Ann Arbor, M I 48106
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MICROWAVE POLARIM ETRIC
BACK SCATTERING FROM
NATURAL ROUGH SURFACES
by
Yisok Oh
A dissertation subm itted in p a rtia l fu lfillm e n t
of the requirements for the degree of
D octor of Philosophy
(E le ctrica l Engineering)
in The U niversity of M ichigan
1993
Doctoral Com m ittee:
Assistant
Professor
Professor
Professor
Assistant
Professor K am al Sarabandi, Co-Chairman
Fawwaz T. Ulaby, Co-Chairm an
Sushil K. A treya
A nthony W . England
Professor B rian E. G ilch rist
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To m y wife, E uiJung Yee, for her patience and constant support; to m y son,
Saroonter, who already grew up to be a fourth grade student; and to m y parents
who inspired me to pursue the doctoral degree.
ii
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ACKNOW LEDGEM ENTS
I would like to express m y g ra titu d e to the members o f m y com m ittee and to
the Radiation Laboratory for supporting my research.
Special thanks are due to
Professor Kam al Sarabandi and Professor Fawwaz T. Ulaby for th e ir constant advices,
encouragements, and generous financial support throughout, the course o f this work. I
consider myself very fortunate to have these professors as my advisors for m y doctoral
work.
I would also like to thank the follow ing friends and colleagues for th e ir help, friend­
ship and encouragement d uring m y graduate studies: Roger DeRoo, R ichard A ustin,
J im Ahne, Raid K h a lil, A d ib Nashashibi, Jim Stiles, John Kendra, Leo Kem pel, D r.
Leland Pierce, and Craig Dobson.
F inally, none o f this would have been possible w ith o u t the constant love, patience,
and sacrifice of m y wife, E uiJung Yee, and my son, Saroonter.
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TABLE OF CONTENTS
D E D IC A T IO N ................................................................................................
ii
AC K N O W LED G EM ENTS.........................................................................
iii
LIST OF F IG U R E S.......................................................................................
vii
LIST OF TABLES..............................................................................................xiv
LIST OF A P P E N D IC E S..............................................................................
xv
CHAPTER
I. INTRO DUCTIO N....................................................................
1 .1
1.2
M otiva tion s and O b je c tiv e s ...................................................
Thesis O v e r v ie w .......................................................................
1
1
4
II. REVIEW OF CONVENTIONAL MODELS FOR BACKSCAT­
TERING FROM RANDOMLY ROUGH SURFACES . . . .
8
2.1
2.2
2.3
I n t r o d u c t io n ..............................................................................
8
Small P ertu rb a tion M e t h o d ............................................................
K irc h h o ff A p p r o a c h ..........................................................................
2.3.1
Physical O ptics M odel ..................................................
2.3.2
G eom etrical O ptics M o d e l ...........................................
11
23
27
33
III. NUMERICAL SOLUTION FOR SCATTERING FROM ONE­
DIMENSIONAL CONDUCTING RANDOM SURFACES
. 38
3.1
3.2
3.3
3.4
3.5
I n t r o d u c t io n .......................................................................................
Random Surface G e n e ra tio n ............................................................
Solution by the M ethod o f M o m e n ts ...........................................
N um erical R e s u lt s .............................................................................
E valuation o f Theoretical S cattering M o d e ls ..............................
3.5.1
Small P ertu rb a tion M e th o d ...........................................
3.5.2
3.5.3
Physical O ptics M odel .................................................
Phase P ertu rb a tion M ethod .......................................
iv
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38
39
45
52
61
61
62
63
3.6
3.5.4
Full Wave M e th o d ............................................................
3.5.5
Integral E quation M ethod ...........................................
3.5.6
N um erical R e s u lts ............................................................
C o n c lu s io n s .........................................................................................
IV. AN IMPROVEMENT OF PHYSICAL OPTICS MODEL
4.1
4.2
4.3
4.4
4.5
4.6
64
65
66
67
.
75
I n t r o d u c t io n ......................................................................................
Form ulation for a Two-dim ensional D ie le ctric Surface . . . .
E valuation for a One-dimensional D ie le ctric S u r f a c e ..............
E valuation for a One-dimensional C onducting Surface . . . .
N um erical R e s u lt s ............................................................................
C o n clu sio n s.........................................................................................
75
78
83
88
89
100
V. A NUMERICAL SOLUTION FOR SCATTERING FROM
INHOMOGENEOUS DIELECTRIC RANDOM SURFACES 101
5.1
5.2
5.3
5.4
5.5
I n t r o d u c t i o n ........................................
S cattering From In d iv id u a l H u m p s ...............................................
M onte C arlo S im ulation o f Rough Surface S c a tte rin g .............
N um erical R e s u lt s .............................................................................
C o n c lu s io n s ...........................................................................................
101
105
108
117
129
VI. MEASUREMENT PROCEDURE - RADAR CALIBRATION
FOR DISTRIBUTED T A R G E T S ............................................... 131
6.1
6.2
6.3
6.4
6.5
I n t r o d u c t i o n .......................................................................................
T h e o r y ..................................................................................................
C a lib ra tio n P ro c e d u re .......................................................................
E xperim ental Procedure and C o m p a r is o n ..................................
C o n c lu s io n s ...........................................................................................
131
135
142
149
157
VII. MICROWAVE POLARIMETRIC RADAR MEASUREMENTS
OF BARE SOIL SURFACES, AN EMPIRICAL MODEL
AND AN INVERSION T E C H N IQ U E ...................................... 160
7.1
7.2
7.3
IN T R O D U C T IO N .............................................................................
160
E X P E R IM E N T A L P R O C E D U R E ............................................... 162
7.2.1
S c a tte ro m e te r................................................................... 162
7.2.2
Laser Profile M e t e r ......................................................... 166
7.2.3
D ielectric P r o b e ............................................................... 166
E X P E R IM E N T A L O B SE R V A TIO N S A N D C O M P A R IS O N W IT H
C L A S S IC A L S O L U T IO N S ............................................................. 169
7.3.1
E xperim ental O b s e rv a tio n s ........................................... 169
7.3.2
Comparison w ith Classical S o lu tio n s .......................... 177
v
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7.4
7.5
7.6
S E M I-E M P IR IC A L M O D E L ( S E M ) .............................................
7.4.1 D e ve lo p m e n t.....................................
7.4.2 Comparison W ith Measured D a t a ...............................
7.4.3 Comparison W ith Independent D ata Set ..................
7.4.4 Comparison W ith 60 GHz D a t a ..................................
IN V E R S IO N M O D E L .....................................................................
C O N C L U S IO N S ...............................................................................
VIII. CONCLUSIONS AND RECOMMENDATIONS...................
8.1
8.2
S u m m a r y ............................................................................................
Recommendations for Future W o r k ...............................................
186
186
191
194
196
196
201
203
203
205
A P P E N D IC E S...................................................................................................208
BIB L IO G R A PH Y .............................................................................................227
vi
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LIST OF FIGURES
Figure
1 .1
Investigation of backscattering from random ly rough surfaces
2.1
Geometry o f the scatter problem for a two-dimensional rough surface
2.2
Backscattering coefficients o f a rough surface w ith ks = 0.2,k l = 2.0,
and t T = (10,1) using the SPM; (a) polarization response of the
surface of an exponential correlation, (b) a°lh and a°v, and (c) the
ra tio o'hv/^lv
2.3
2.4
2.5
2.6
2.7
...
^he surface w ith two different correlation functions.
Backscattering coefficients at un-polarization using the SPM for the
surface of an exponential correlation, (a) k l = 2 .0 , eT = ( 1 0 , 2 ), and
the various values o f ks, (b) ks = 0 .2 , eT = ( 1 0 , 2 ), and the various
values o f k l, and (a) ks = 0 .2 , k l = 2 .0 , and the various values o f er .
5
13
19
20
The ratios o f the backscattering coefficients at uu-polarization using
the SPM for the surface o f an exponential correlation; (a) o°lfJa °lh
and (b) c r^ /c r^ for ks = 0 .2 , k l = 2 .0 , and different eT, and the
ra tio Ohyjcrlh. f ° r (c) k l = 2 .0 , t T = ( 1 0 , 2 ), and different ks, and (d)
ks = 0 .2 , er = ( 1 0 , 2 ), and different k l........................................................
21
Backscattering coefficients of a rough surface w ith ks = 1 and k l = 8
using the PO model; (a) <7 ,°^ and <7 °t) for two different correlation
functions and er = ( 1 0 , 2 ) and (b) the ra tio
for the various
values of t r w ith an exponential correlation fu n ctio n .............................
31
Backscattering coefficients at /i/i-polarization using the PO model for
a surface of an exponential correlation, (a) k l = 8 , er = ( 1 0 , 2 ), and
the various values o f ks, (b) ks = 1 , er = ( 1 0 , 2 ), and the various
values of kl, and (a) ks = 1 , k l = 8 , and the various values of er .
.
32
Backscattering coefficients of a rough surface at v v ( = /j/i)-po la riza tio n
using the GO model; (a) for er = ( 1 0 , 2 ) and the various values o f m
and (b) for m = 0.4 and the various values of eT.....................................
36
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2.8
The v a lid ity regions o f the classical models which are the SPM , the
PO model and GO m odel..............................................................................
37
T yp ica l sections o f height profiles for (a) S -l, (b) S-2, and (c) S-3
surfaces...............................................................................................................
42
The height d is trib u tio n of the generated surfaces (dots) as compared
w ith Gaussian p ro b a b ility density functions (solid lin e s)....................
43
The autocorrelations o f the generated surfaces (dots) as compared
w ith Gaussian functions (solid lines)..........................................................
44
The slope d is trib u tio n of the generated surfaces (dots) as compared
w ith Gaussian p ro b a b ility density functions (solid lin e s).....................
44
3.5
G eom etry o f the scatter problem ....................................................................
46
3.6
B ackscattering from a flat conducting s trip o f the w id th of 14A, (a)
the current d is trib u tio n at 0 ° incidence, and (b) the backscatter echo
w id th ...................................................................................................................
51
B ackscattering from a flat conducting s trip w ith resistive cards, (a)
the current d is trib u tio n at 0 ° incidence and re sis tiv ity d is trib u tio n ,
and (b) the backscatter echo w id th .............................................................
53
3.8
The extension o f the random surface w ith resistive cards.......................
54
3.9
The solution by the m ethod o f moments compared w ith the sm all
pe rtu rb a tio n m ethod for the random surface, S -l, o f k s = 0 . 2 1 and
k l= 2 .2 ..................................................................................................................
54
T he solution by the method of moments compared w ith the physical
optics solution for the random surface, S-3, of ks=1.04 and k l= 7 .4 .
55
The solution by the method o f moments for the random surface, S-2 ,
o f ks=0.62 and k l= 4 .6 .....................................................................................
55
The d is trib u tio n o f the phase difference between crjj/, and a°v o f the
surface, S-2, (ks=0.62, kl= 4 .6), (a) at 20° and (b) at 50° incidences.
57
The d is trib u tio n o f the phase difference between cr°lh and a°v o f the
surface, S-2, (ks=0.62, kl= 4 .6 ), (a) standard deviation and (b) mean
values...................................................................................................................
58
3.1
3.2
3.3
3.4
3.7
3.10
3.11
3.12
3.13
v iii
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3.14
The statistics o f the phase difference between a°lU and a°v o f the
surface, S-2, (ks=0.62, k l= 4 .6 ), (a) the degree o f correlation, a and
(b) the coherent phase-difference, £............................................................
59
The degree o f correlation, a, (a) for three different k l values at a
fixed value o f ks = 0 .6 , and (b) for three different ks values at a fixed
value of k l = 4.5................................................................................................
60
3.16
The v a lid ity regions o f the scattering models.............................................
67
3.17
Comparison o f models w ith an exact num erical solution fo r (a) ks =
0.21 and k l = 2.2, (b) ks = 0.62 and k l = 4.6, and (c) ks = 1.04 and
k l = 7.4 fo r b oth o f vv- and /i/i-p o lariza tion s...........................................
71
Comparison o f models w ith an exact num erical solution for (a) ks =
0.21 ancl k l = 2.2, (b) ks = 0.62 and k l = 4.6, and (c) ks = 1.04 and
k l = 7.4 for /i/i-p o la riz a tio n ...........................................................................
74
4.1
Illu stra tio n o f the development o f an exact K irch h off solution.
77
4.2
A typical exam ple o f surface height distrib utio n s measured from n a t­
ural rough surfaces...........................................................................................
80
Comparison between a Gaussian correlation coefficient and its deriva­
tives in case o f I = 0.5m .................................................................................
81
3.15
3.18
4.3
...
4.4
Illu stra tio n of the shadowing correction in backscattering direction.
86
4.5
Comparison between the exact physical optics model and the m ethod
o f moments solution; (a) illu s tra tio n o f the roughness conditions and
the backscattering coefficients for (b) ks = 0.62, k l = 4.6, (c) ks —
0 .6 , k l = 6 , (d) ks = 1 , k l = 6 , (e) ks = 1 , k l = 8, and (f) ks = 1 ,
it/ = 10................................................................................................................
93
Comparison between the exact physical optics model and the method
o f moments solution for the roughness o f ks = 1 , k l = 8 for (a) a
conducting surface and (b) for a d ielectric surface o f e = ( 1 0 , 2 ) for
/i/i-p o la riza tio n ..................................................................................................
95
4.6
4.7
Comparison between the exact physical optics model and the method
o f moments solution for the roughness o f ks = 1 , k l = 8 for (a) a
conducting surface and (b) for a dielectric surface o f e = ( 1 0 , 2 ) for
/i/i-p o la riza tio n .....................................................................................................
ix
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96
4.8
Comparison between the exact PO solution and the approxim ated
PO solutions for a one-dimensional conducting surface of (a) ks = 1 ,
k l = 6 and (b ) ks = 1, k l = 10.....................................................................
97
The ra tio o ^ / c r°„ o f the exact PO solution and the approxim ated PO
solutions for a one-dimensional dielectric random surface o f ks = 1 ,
k l = 8 , and eT = (1 0 ,2 )...................................................................................
98
The backscattering coefficients o f the exact PO solution for a vari­
ous values o f the integration lim its for a one-dimensional conducting
random surface o f ks = 1.2, k l = 6.1..........................................................
98
5.1
G eom etry of the scatter problem for a two-dim ensional rough surface.
104
5.2
Flow chart o f the Monte Carlo sim ulation for the rough surface scat­
te rin g problem ..................................................................................................
109
5.3
H um p types for the rough surface considered in this chapter.
Ill
5.4
P ro b a b ility density function o f the co-polarized phase angle <f>c =
<f>hh — <f>w for a fixed value of £ and four different values o f a.
...
4.9
4.10
...
116
5.5
B is ta tic echo w id th of a squared-cosine hum p o f ej = 15 + z3, W =
0.72A, H = 0.07A over an impedance surface of // = 0.254 —i0 .025 at
(a) 0{ = 0° and (b) 0, = 45° at / = 5 GHz for E- and H-polarizations. 119
5.6
M u ltip le scattering effect on the backscatter echo w id th o f a surface
segment consisting of hump-4 , hum p-5, and hum p-3, corresponding
to the roughness o f ks = 0.36, k l = 2.2, w ith t \ = 15 + *3 over an
impedance surface of 77 = 0.254 — f0.025 at (a) hh-polarization and
(b) vv-polarization at / = 5 G H z ..............................................................
120
A random surface generated using squared-cosine humps, (a) a sam­
ple surface profile, (b) the autocorrelation function of the surface as
compared w ith a Gaussian and an exponential function w ith identical
correlation le ngth .............................................................................................
122
Backscattering coefficient cr° of the random surface w ith ks = 0.12,
k l = 2.13, and ei = £2 = 15 + *3 as computed by the SPM and the
num erical technique; (a) H H -polarization and (b ) V V -po la riza tio n.
124
5.7
5.8
5.9
Degree of correlation a of the random surface w ith ks
k l = 2.13, and ej =
62
=
=
0.12,
15 -f- z’3..............................................................
x
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125
5.10
5.11
5.12
5.13
6.1
6 .2
Backscattering coefficient cr° o f the random surface w ith ks = 0.42,
k l = 7.49, and t i = £2 = 15 + i3 as computed by the PO model
and the num erical technique fo r H H -polarization...................................
126
Backscattering coefficient cr° o f the random surface w ith ks = 0.22,
k l = 3.8, (a) £j = 6 + i0.6 and e2 = 15 + t'3 , (b) ej = 12 -f i2.4
and £2 — 15 + z3 for V V - and H H -polarizations....................................
127
The se nsitivity of the backscattering coefficient cr° to the dielectric
constant, in case o f ks = 0.22, k l = 3.8, and e2 = 15 + i3 at
9 — 44°.............................................................................................................
128
The se nsitivity of the degree of correlation a to the dielectric constant
in case o f ks = 0.22, k l = 3.8, and e2 = 15 + z‘3 at 0 = 44°. . .
129
Geometry o f a radar system illu m in a tin g a homogeneous d istrib ute d
targe t..................................................................................................................
135
Sim plified block diagram o f a polarim etric radar system ........................
142
6.3
A zim uth-over elevation and elevation-over azim uth coordinate sys­
tems ('ip,£ ) specifying a p oint onthe surface o f a sphere............................ 144
6.4
Geometry of a radar above x-y plane and transform ation to carte­
sian coordinates from (a) azimuth-over-elevation coordinate and (b)
elevation-over-azimuth c o o rd in a te .............................................................
145
P olarim etric response o f a m e tallic sphere over the entire m ainlobe
of X-band scatterometer; Normalize avv (a) corresponds to G l and
normalized crvh (b) corresponds to
Phase difference between
co-polarized (c) and cross-polarized (d) components o f the sphere
response correspond to phase variation of the co- and cross-polarized
patterns o f the antenna..................................................................................
150
Comparison between the new and old calibration techniques applied
to the X-band measured backscatter from a bare soil surface; (a), (b ),
and (c) show the difference in the co- and cross-polarized backscat­
tering coefficients and (d) demonstrates the enhancement in the ra tio
o f the cross-polarized backscattering coefficients obtained by the new
m ethod...............................................................................................................
152
6.5
6 .6
6.7
Degree o f correlation for co-polarized components o f the scattering
m a trix for L-band (a), C-band (b), and X-band (c )...................................154
xi
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6 .8
Polarized-phase-difference for co-polarized components o f the scat­
tering m a trix for L-band (a), C-band (b ), and X -band (c )....................... 157
6.9
P ro b a b ility density functions (P D F ) fo r co-polarized phase-difference
for C-band at 30°..............................................................................................
158
E xperim ental system, (a) a scatterom eter block diagram , (b) Laser
profile m eter, and (c) dielectric probe.........................................................
163
Comparison o f the measured autocorrelation functions w ith the Gaus­
sian and exponential functions (a) S i and (b ) S4...................................
170
A ng u la r response o f a°v fo r four different surface roughnesses at m od­
erately d ry co nd itio n (m „ ~ 0.15), at (a) 1.5 G Hz and (b) 9.5 GHz.
173
7.1
7.2
7.3
7.4
A ngular responses o f cr°v,
and cr°lt, for (a) a smooth surface at
1.5 GHz (L2) and (b) a very rough surface at 9.5 GHz (X 4 )..................174
7.5
A ngular dependence o f the like-polarized ra tio , <rhJcr°v, at 4.75 GHz
fo r a smooth surface and a very rough surface.........................................
175
A ngular plots of (a) tr°v and (T°hv of surface S i at X-band for two
different m oisture conditions and (b) the like-polarized ra tio , c r ^ / a ^ ,
fo r the same surface at C-band.....................................................................
176
Roughness parameters and the region o f v a lid ity of SPM , PO, and
GO m odels.........................................................................................................
178
7.6
7.7
7.8
SPM model w ith different autocorrelation functions compared to
the measured data o f L l (surface 1 at 1.5 GHz, A:s=0.13), (a) V V polarization, (b ) H H -po la riza tio n, and (c) V V -, H H -, and H V -polarizations
using an exponential autocorrelation fu n c tio n .........................................
181
7.9
PO model w ith different autocorrelation functions compared to the
measured data o f X I (surface 1 at 9.5 G Hz, k s = 0.80) for (a) V V polarization and (b) H H -p o la riza tio n ..........................................................
183
GO model compared to the measured data o f X 4 (surface 4 at 9.5
G Hz, k s = 6.0).....................................................................................................
185
The se nsitivity o f the depolarization ra tio , o-lv/a ° v, to surface rough­
ness for (a) d ry soil and (b) wet so il...........................................................
187
The se nsitivity o f the like-polarized ratio,
ness and soil m oisture at (a) 40° and (b) 50°
189
7.10
7.11
7.12
to surface rough­
x ii
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7.13
7.14
7.15
7.16
7.17
E m p irica l model compared to the measured data of surface 1 for wet
soil at (a) 1.5 G Hz, (b) 4.75 GHz, and (c) 9.5 G H z................................
192
E m p irica l model compared to the measured data of surface 4 for wet
soil at (a) 1.5 GHz, (b) 4.75 GHz, and (c) 9.5 G H z...............................
193
E m p irica l model compared w ith the data from independent data set
I I fo r a surface w ith s = 0.7 cm and I = 3.0 cm, measured at (a) 1.5
G Hz, (b ) 4.75 GHz, and (c) 9.5 GHz ......................................................
195
E m p irica l model compared w ith the data reported by Yamasaki et al.
[1991] at 60 GHz for (a) soil-1 (s=0.013 cm, 1=0.055 cm ), (b) soil-2
(s=0.051 cm, /= 0.12 cm ), and (c) soil-3 (s=0.139 cm, /=0.20 cm ).
197
Comparison between the values o f surface parameters estim ated by
the inversion technique and those measured in situ for (a) ks and (b)
the volum etric m oisture contents m v..........................................................
199
7.18
Comparison between the values o f surface parameters estim ated by
the inversion technique and those measured in situ for (a) the real
part o f er and (b) the im aginary p art o f eT.................................................... 2 0 0
8.1
S en sitivity of a°v on surface parameters at 40°............................................. 207
B .l
Inversion diagram for 1.25 GHz at 20°
217
B.2
Inversion diagram
for 1.25 GHz at 30°
218
B.3
Inversion diagram
for 1.25 GHz at 40°
219
B.4
Inversion diagram
for 1.25 GHz at 50°
220
B.5
Inversion diagram
for 1.25 GHz at 60°
221
B .6
Inversion
diagram for 5.3 GHz at 2 0 ° .......................................................
222
B.7
Inversion
diagram for 5.3 GHz at 3 0 ° .......................................................
223
B .8
Inversion
diagram for 5.3 GHz at 4 0 ° .......................................................
224
B.9
Inversion
diagram for 5.3 GHz at 5 0 ° .......................................................
225
B.10
Inversion
diagram for 5.3 GHz at 6 0 ° ......................................................
226
x iii
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LIST OF TABLES
Table
3.1
Roughness parameters used for the random surface generation
. . .
5.1
Roughness parameters corresponding to constants A and B ......................117
5.2
Constants used in the num erical com putations...............................
123
7.1
P olarim e tric scatterom eter (P O LA R S C A T ) characteristics.........
164
7.2
Sum m ary o f roughness parameters.....................................................
167
7.3
Sum m ary o f soil m oisture contents.....................................................
168
7.4
Comparison between cr° of SPM model (w ith exponential correlation)
and the measured data for wet soil surfaces....................................
182
7.5
Comparison between cr° of PO model (w ith exponential correlation)
and the measured data for wet soil surfaces..................................... 184
7.6
Measured surface parameters for the Independent. D ata Set II.
7.7
Surface parameters for Yamasaki et a l.’s measurement................
. . .
196
xiv
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43
194
LIST OF APPENDIC ES
Appendix
A.
C H A R A C T E R IS T IC F U N C T IO N FO R A G AU SSIAN R A N D O M V E C ­
T O R ..................................................................................................................... 209
B.
IN V E R S IO N D I A G R A M S ............................................................................
XV
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216
C H A PTER I
INTRODUCTION
1.1
M o tiv a tio n s and O b jectiv es
Recently, spaceborne remote sensing o f the E a rth has become an im p o rta n t source
of in fo rm a tio n in m o n ito rin g the E a rth ’s environm ent. Depending on the applica­
tion, sensors operating in different parts o f electrom agnetic spectrum have been de­
signed. The sensors in optical, therm al infrared, m illim e te r, and microwave frequen­
cies have been com m only used to retrieve in form a tion about different target types
on the E a rth ’s surface by v irtu e of th e ir spectral properties.
Propagation proper­
ties o f electrom agnetic waves at microwave frequencies offer the follow ing advantages
over o ptica l sensors: ( 1 ) microwave sensors are capable o f p enetrating clouds and
p recipitations, ( 2 ) microwave sensors may have th e ir own source o f illu m in a tio n so
th a t they are independent o f the Sun and thus the desired incidence angle, frequency,
and p olarization can be chosen, and (3) the microwave signals can penetrate to some
extent in to various types o f the surface cover such as vegetation, and thus provide
some in fo rm a tio n about the subsurface targets.
A m ong other applications, microwave rem ote sensing o f soil m oisture has been
o f p rim a ry concern to hydrologists since many atm ospheric and environm ental pro­
cesses are closely linked to the spatial d is trib u tio n o f soil m oisture. A p p lica tio n o f
1
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both passive and active sensors in microwave remote sensing o f soil m oisture have
been attem pted [Schmugge et ah, 1986 ; Dobson and Ulaby, 1986a], however only
active sensors are capable o f producing estimates o f soil m oisture w ith in fine spatial
resolution from a spaceborne p la tfo rm . Radar backscatter from te rra in is influenced
by tw o sets o f parameters:
1)
physical parameters such as com plex die lectric con­
stant o f the scatterers and the surface topography, and
2
) the radar param eters such
as frequency, incidence angle, and polarization. For bare soil surfaces, the dielectric
constant is strongly dependent upon the liq u id water content, and the effects o f other
soil parameters like soil type (p a rticle size d istrib utio n ) on the d ie le ctric constant of
the soil m edium are less im p o rta n t, p a rtic u la rly at the lower microwave frequencies
[Ulaby, 1974]. Radar backscatter, away from normal incidence, from a bare soil sur­
face is a d irect result o f the surface irreg u la rity. The strength o f the backscattered
field and its statistics are complex functions of these surface irre g u la ritie s relative to
the wavelength and the d ie lectric constant o f the soil m edium .
Researchers, for a long tim e , have been try in g to develop m a them atical models to
pre dict the backscattered characteristics o f random ly rough surfaces. A t present there
exist numerous analytical models each pertaining to specific cases [U la b y et al., 1982;
Tsang et ah, 1985; Ishim aru, 1978]. The success o f these models when applied to real
rough surfaces is very lim ite d , however, because of the over s im p lify in g approxim ate
nature of these models. There are also a large number o f experim ental data sets, all
collected in an a tte m p t to establish the relationship between the radar backscatter
and the soil moisture and surface parameters em pirically [B a rtliv a la and Ulaby, 1977;
Ulaby et ah, 1978; Jackson et ah, 1981; Dobson and Ulaby, 1986a and 1986b; Mo et
ah, 1988; Wang et ah, 1986]. However due to the lack of precise ground tru th data,
accurate calibrations, complete angular and polarization response characterization, or
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3
a sufficient span of the surface parameter, the existing data sets have not been able to
co ntribu te much to the understanding of the scattering process. Therefore, no reliable
a lg orith m has yet been developed to retrieve soil m oisture and surface roughness
parameters from radar data w ith an accuracy required for hydrologic applications.
One of the added com plexity in the soil m edia is the inhomogeneous nature of the
soil m edium which is the result of nonuniform m oisture profile.
T he problem o f scattering from random surfaces has been investigated o nly for
homogeneous surfaces. Even for this case, theoretical solutions exist only fo r lim it ­
ing cases. There are two conventional models dealing w ith rough surface scattering
[Ulaby et al., 1982; Tsang et ah, 1985]. One is the small p ertu rb a tion m ethod (SPM ),
which has been developed for surfaces whose height variations are small compared to
the wavelength and where surface slopes are much sm aller than unity. The second is
the K irc h h o ff approxim ation (K A ), which has been developed for rough surfaces w ith
large radii o f curvature. In recent years, there has been a considerable interest in de­
te rm in a tio n o f the regions o f v a lid ity of these tw o methods as well as the development
o f a more general theory th a t can bridge these two lim itin g scattering models [Brown,
1978; Bahar, 1981; W ineberner and Ishim aru, 1985; Fung and Pan, 1987]. The va­
lid ity regions of the SPM and K A have been examined previously by com paring the
model predictions to the results derived fro m exact num erical sim ulation [Chen and
Fung, 1988; Broschat et al., 1987; Thorsos, 1988].
U nfortunately, these num erical
solutions could only address one-dimensional surface roughness, thus the num erical
sim ulation does not produce depolarization and its prediction for co-polarized com­
ponents are much different from real surfaces th a t are two-dim ensional. So far, no
exact num erical solution for two-dimensional surfaces has been developed due to the
lim ite d com puter power. Therefore, carefully controlled experim ents are necessary to
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4
study the two-dimensional surface of a rb itra ry surface correlation functions.
The m a jo r goal of this thesis is to develop an alg orith m for re trieving soil m oisture
content and surface roughness parameters from radar backscatter data. To accom­
plish this task, first a p o la rim e tric radar backscattering model is developed th a t is
valid fo r n a tu ra l rough surfaces over a w ide range o f surface conditions at microwave
frequencies. Development o f the scattering model requires four m a jo r steps: (1) ac­
q uisition o f accurate p o la rim e tric radar backscatter data from bare soil surfaces over a
wide range of m oisture conditions and surface roughnesses, ( 2 ) acquisition of accurate
ground tru th data for surface roughness and soil m oisture using a laser surface profile
m eter and a dielectric probe, (3) development and enhancement o f theoretical and
num erical scattering models, (4) development of a h yb rid scattering model for bare
soil surfaces constructed based on the experim ental observations and the the o re ti­
cal and num erical models. Once the h yb rid (sem i-em pirical) model is developed and
tested, development of an inversion alg orith m capable of providing accurate estimates
of soil m oisture contents and surface roughnesses from polarim etric radar backscatter
is considered.
1.2
T h esis O verview
In this section, the structure of the thesis is explained and content o f each chapter
is briefly discussed. Figure
1 .1
shows a sim plified flow chart of the topics discussed in
this thesis. A theoretical basis for this w ork is reported in Chapters 2 and 4 which is
used to ve rify both num erical sim ulations as well as experim ental observations. The
num erical sim ulations of radar backscattering from random surfaces are introduced
in Chapters 3 and 5 and are used to examine the accuracy of the theoretical models
and also to guide the development of the sem i-em pirical scattering m odel. Chapter
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5
Theoretical
Models
Numerical
Solutions
Ch. 2 & Ch. 4
Ch.3 & Ch. 5
Measurements
Ch. 6 & Ch. 7
A Scattering
Model
Ch. 7
An Inversion
Algorithm
Ch. 7
Figure 1.1:
Investigation of backscattering from random ly rough surfaces
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6
6
is devoted to the development o f an accurate calibration technique for p olarim e tric
measurement of distributed targets. The experim ental data acquired using this cali­
b ra tio n technique were used to evaluate the existing scattering models. Development
o f the sem i-em pirical model and its inversion algorithm is described in Chapter 7. In
this chapter the scattering behavior based on the extensive experim ental observations
and the results derived from the theoretical and the num erical studies are combined
to develop the semi-empirical scattering model for the backscattering coefficients,
a lv i tfiin a hv The inversion alg orith m can provide an estim ate for soil m oisture and
surface rm s height, s, when radar parameters (frequency and incidence angle) are
known.
In Chapter 2, classical scattering models are reviewed.
In specific, the sm all
p e rtu rb a tio n method, the physical optics m odel and the geometrical optics model are
considered.
In Chapter 3, a M onte Carlo m ethod in conjunction w ith the m ethod o f moments is
introduced to solve scattering from a one-dimensional conducting surface num erically.
To make num erical sim ulation of random surfaces tractable, fin ite samples o f the
random surface must be considered. However, the edges o f the fin ite sample perturb
the scattering solution. To suppress the edge co ntribution a tapered resistive sheet
is added to each edge. Using this num erical technique, the phase difference statistics
as well as the backscattering coefficients are computed, and the existing scattering
models are examined against the num erical calculation.
In Chapter 4, an im proved high frequency solution for random surfaces is form u­
lated and evaluated num erically for a one-dimensional surface. Using th is form ulation,
the zeroth- and the first-order classical physical optics approxim ations are examined.
In Chapter 5, the effect of dielectric inhomogeneity in a soil m edium is considered
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7
by developing an efficient num erical technique for one-dimensional inhomogeneous
d ie lectric rough surfaces.
In Chapter
6
, an accurate technique for measurement o f p olarim e tric backscatter
from d istrib u te d targets is introduced. In this technique the polarization d isto rtio n
m a trix o f a radar system is com pletely characterized from the p olarim e tric response
of a sphere over the entire m ain lobe of the antenna.
In Chapter 7, the experim ental procedure and the backscattered data collected
from bare soil surfaces w ith many different roughness and m oisture conditions at
microwave frequencies are explained. These data are analyzed and compared w ith
the theoretical scattering models. Also they are used to find the dependency of the
backscattering coefficients on the radar and the surface parameters. Using the co­
polarized and the cross-polarized ratios
cr°w/cr°v), a sem i-em pirical scattering
model is developed. I t is shown th a t the semi-empirical scattering model provides a
very good agreement w ith independent experim ental observations. In this chapter an
inversion a lg orith m for the em pirical model is also developed and its performance in
e stim a ting the soil m oisture and surface roughness parameters is tested.
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CH APTER II
REVIEW OF CLASSICAL MODELS FOR
BACKSCATTERING FROM RANDOMLY
ROUGH SURFACES
2.1
In tro d u ctio n
Even though the scattering o f electromagnetic waves from a random ly rough sur­
face has been studied for many decades, no exact closed-form solutions have been
obtained because o f the com plexity o f the problem . Instead, approxim ate models are
available for a lim ite d range of random surface parameters.
The objective o f this
chapter is to study such approxim ate analytic models.
In order to study scattering models o f random surfaces, it is convenient to treat the
rough surface as a p a rticu la r realization of a random fun ctio n w ith given statistical
properties. Let z ( x , y ) be such a random function describing the height d istrib utio n
of the xy plane. Then,
2
is a random variable w ith a p ro b a b ility density function
p(z) and a correlation function given as
C ( 0 = ( z { x , y ) z{x + fx,Z/ + fv )) •
(2-1)
We assume as follows; ( 1 ) z ( x , y ) is stationary in the w ider sense. In other word,
the p ro b a b ility density function and the correlation fun ctio n are independent o f the
coordinate of x and y. (2) The surface is isotropic, which means th a t the correlation
8
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9
fun ctio n depends only on the distance regardless of its direction. (3) z ( x , y ) is meansquare differentiable w ith p a rtia l derivatives zx , z y. T h is means th a t there is a function
zx (or zy) such th a t
lim
A x—O
z{x + A x , y ) - z ( x , y )
2x(x i y )
Ax
=
0.
( 2 .2 )
Based on the p ro b a b ility density fun ctio n and the correlation fun ctio n, the surface
roughness is com m only characterized by two parameters, nam ely rm s height and cor­
re la tion length [U laby et ah, 1982]. The rms height s is the standard deviation o f the
surface height d is trib u tio n , and the correlation length / is defined as the displacement
such th a t the correlation function is equal to
1 /e
= 0.367 • ••. The rms height m
[U laby et ah, 1982] is also defined as the mean square of the slope d is trib u tio n which
is
m = {z l)k = (z l)h = ^ "(0 )1 ,
(2.3)
where C "(0 ) is the second derivative o f C (£) for £=0. I t is often convenient to use the
norm alized correlation function (or correlation coefficient fun ctio n) which is defined
as
pit) =
(2-4)
where s is the standard deviation o f the p ro b a b ility density function p(z).
T w o com m only used classical models are the small p ertu rb a tion m ethod (S P M )
[Rice, 1951] and the K irch h o ff approach (K A ) [Beckmann and Spizzichino, 1963]. The
SPM can be used fo r the random surface of which the surface rm s height is much
sm aller than the wavelength and the surface rms slope is re latively sm all. In SPM , the
surface field is expanded in a p e rtu rb a tio n series to solve for the scattered field fro m a
random surface. The SPM appears to be exact because this method is using the exact
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10
boundary conditions on the surface, but in practice the in fin ite series representing the
solution converges reasonably quickly only for very s lig h tly rough surfaces.
The K A is applicable to the random surface o f w hich the correlation length is
larger than the incident wavelength and the rms height is sm all enough so th a t the
average radius o f curvature is larger than the incident wavelength [Ulaby et ah, 1982]
where the average radius o f curvature Rc is given as
R, =
' 2 d 4c ( o y ~ 7T
(2.5)
d (*
The K A m ethod employs the so called tangent plane approxim ation to apply the
boundary conditions on the surface.
Under the tangent plane approxim ation, the
surface fields at any p oint o f the surface are approxim ated by the fields th a t would
be present on the tangent plane at th a t p oint. However, even w ith the tangent plane
approxim ation, the scattered field in the K irchhoff-approxim ated diffraction integral
is s till d iffic u lt to solve analytically. Therefore, a dd itio n al assumptions are required
to obtain an a na lytica l solution. A com m only used approxim ation is to expand the
integrand of the d iffractio n integral in terms o f the surface slope, keeping only the
lower order terms. T his additional approxim ation provides the physical optics (P O )
m odel, which is valid when the surface rms slope is small relative to the wavelength.
In the high frequency lim it as k — ►oo, the geom etrical optics (GO) m odel can be
obtained using the methods o f stationary-phase. The GO model is independent o f
the frequency, and is valid when the rms height is large relative to the wavelength.
A ltho u gh the small perturbation method and the K irch h o ff approach are the most
common models over the decades for com puting the scattering from random ly rough
surfaces, many other techniques are introduced recently to extend the v a lid ity regions
of the two classical models, including the phase p e rtu rb a tio n method (P P M ) [W inebrenner and Ishim aru, 1985a], the full-wave m ethod (F W M ) [Bahar, 1981], and the
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11
integral equation m ethod (IE M ) [Fung and Pan, 1987].
The classical models (i.e., SPM, PO, and G O ) for two-dim ensional dielectric ran­
dom surfaces are summarized and num erical examples are com puted to show the
dependency o f the backscattering coefficients on the radar and surface parameters in
section 2.2 and 2.3.
2.2
S m all P ertu rb ation M eth o d
The scattering o f electrom agnetic waves from a slightly rough surface can be
obtained by using the Rayleigh hypothesis to express the reflected and tra nsm itte d
fields in to upward and downward waves, respectively [Rice, 1951]. The surface field
amplitudes are then determined from the boundary conditions and the divergence
relations, from which the scattered fields can be obtained.
In order to illu stra te the Rayleigh hypothesis, let us assume a periodic surface
w ith period L. The scattered fields in z > B , where B = max z ( x , y ) , m ay be w ritte n
as
CO
E* =
A m„ exp[—i a ( m x + ny) — ib(m, n ) z ]
(2-6)
— CO
where
27T
a—— ,
v/fc2 — a2( M 2 + n 2)
; k2 > a 2( m 2 -f n 2)
b( m , n ) -—iyJa2( M 2 + n 2) — k 2 ; k2 < a2( m 2 -f- n 2),
and the coefficients A mn are to be determ ined. Rayleigh [1945] made the assumption
th a t the series (2.6) w ith coefficients A m„ was a valid representation for the scattered
field not only for z > B but also throughout z > z ( x , y ) . This assum ption has come
to be known as the Rayleigh hypothesis, and it was shown th a t th is hypothesis is valid
if
B < 0.448 in the case of a periodic surface [M illa r, 1973].
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In this section, the form u la tion o f the SPM is summarized by following closely the
derivation in Ch. 12 of [U laby et ah, 1986]. Considering a plane wave incident upon
a tw o-dim ensional dielectric rough surface as shown in Fig.
2.1, the orthonorm al
coordinate systems are given by (5,-, hi,k{) and (va, h 3, k 3) w ith
ki = x sin 9 cos <j) + y sin 9 sin <f> — z cos 9
h{ = x sin <f>— y cos
(2-7)
Vi = x cos 9 cos 4> + y cos 9 sin <j>+ z sin 9,
k3 = x sin 9Scos <j>3 -f y sin 03 sin <f>3 + z cos 0S
hs = x sin (f)3 — y cos <f>3
( 2 -8 )
vs = —x cos 03 cos <j)3 — y cos 0Ssin <f>3 + i sin 6S,
kT = x sin 9 cos (j> + y sin 9 sin <j>+ z cos 9
(2.9)
k t = x sin 9t cos <f>t + y sin 9t sin <j>t+ z cos 9t .
I f we consider only the backscattering direction (9S = 9 and <f>3 =
<f>=
0
7r
+ $), and set
for sim p licity, the coordinate system can be sim plified as
Ar,= x s in # — Jcos0,
hi = —y,
ks - —k{ - —x sin 9 + z cos 9,
ii =
h s = y,
x co s9 + zs l n 9,
v3 = x cos 93+ z sin 9.
(2 . 1 0 )
(2-11)
The to ta l field in m edium 0 is the sum o f the incident, reflected, and scattered
fields, where the scattered field in a homogeneous h a lf space may be represented by
superimposing plane waves w ith unknown am plitudes as follows:
E p = Ep + Ep + Ep ,
Ep = d y f ^
p = v or h,
w ith
(2 . 1 2 )
UP( ^ ’ ky) e - ' ^ ' k^ ~ ' d k xdky,
EJ, = ^ e,A'°i','r ,
(2.13)
E ; = pTR p eik° ^ r ,
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13
Z(x,y)
Medium 0
L
" 1
Medium 1
F igure 2.1:
Geometry o f the scatter problem for a two-dim ensional rough surface
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
where the subscript p stands for v or h which indicates ve rtically and horizontally
polarized incident wave, respectively, Rp is the Fresnel reflection coefficient for ppolarized wave, the U p(kx , k y) is the unknown to be determined using the boundary
conditions, and kz = y k o — k% — k%. S im ilarly, the to ta l field in m edium 1 can be
given as
E ’ = E I" + E “ ,
E pS = ( ^ / C
P = v or h,
w ith
(2.14)
D p(kx, k y) e - ik*x~ik»y' ik^ d k xdky,
(2.15)
Ep( =
PtTp eik°k,'r ,
where Tv = 1 + R v is the Fresnel transmission coefficient for p-polarized wave, and
the D p(/:x , ky) is the unknown to be determined. The boundary conditions, h x (E p —
E *) = 0 and h, x (Hp — H }J = 0, give four equations as follows,
A E y + f t -A E , = 0,
<>y
dz
A Ex + §
*A E z = 0,
dxL
9x
9A
dz
_ aA E.
dx
dAEi
3y
dAEv
dz
(2.16)
. dz_ ( d A E u _
' dy \ d x
dAEr \
dy )
. dz_ f d AE, ,
dy \ d x
dAEr )
dy
)
_
n
’
’
where A E x —Ex —Ex , A E y = E y — /?,), and A E : = E :— E\
divergence relations, V • E p =
on the surface. The
0 and V • Ep = 0, give two more independent equations,
M i
_l
«* +
aEj
dx
M i
*
iM
+
. dEl
dy +
i -
n
"
’
dE\
dz ~
n
U’
(2.17)
The six relations given by (2.16) and (2.17) p e rm it the six unknown field amplitudes
U p, U p, 6 T , D vx , D p, and D ’’ in (2 .12)-('2.15) to be determined.
Since we assume kzz to be a small quantity, we can expand a ll exponentials in­
volving kzz in Taylor series,
e± i* .*(*.») = i ± i k 2z ( x , y )
.
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(2.18)
15
T he surface field amplitudes can also be expanded in a pertu rb a tion series,
U»’(ASIfcv) = U? + U5 + -..,
(2.19)
D ^ (k I , k y) = D p
1+ D p
2 + --..
S u b s titu tin g (2.18) and (2.19) in to (2.16) and (2.17) up to the first order in m agnitude,
we can get six algebraic equations fo r the six unknown am plitudes, U Pi, Uy l , Uz l , D h ,
D y l , and D p,, where subscript
1
indicates the first order solution.
Once we find the six unknown surface field am plitudes, the (/-polarized scattered
fields can be computed in case o f the p-polarized incident fields as
E ; p = qa - E ; ,
(2.20)
where qs is h s or vs, and E* is the scattered field for a p-polarized incident wave.
T h e backscattered fields are given for backscattering direction in [Ulaby et al.,
1986],
1
K
f
f°°
= TTTi / /
( Z tt J
j
J —oo
{ ~ i2k cos 0aqvZ { k x + k sin 0, kv)} e~iklX- ik^ +ik^ d k xdky , ( 2 .2 1 )
where
h
- C O S 0 + V € r —s ill2 9
/
COS
.2.
-S tir V
i
a m = (er —
„
(2.22)
I cr cos 6 + \ / r, —sin2 9 J
&vh ~ Othv = 0.
A ssum ing kzz «
i, the ensemble average in ten sity can be approxim ated as
s
/ / / £ > « » « ) ’ | «
{ZZ-)
lt'»-»y)vdkxdkydk'xdk'y.
(2.23)
Since Z { k T, k y) is the Fourier transform o f the random function z ( x , y ) representing
the surface height d istrib utio n ,
Z(k1. , ky) =
[
J
r
z { x , y ) e ik* * - ik**dxdy,
(2.24)
J — OO
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
the ensemble average o f Z Z * can be computed as
{ Z ( k x , k y) Z' ( k' x , * ' ) ) = J J ~ j z ( x , y ) z ( x ' , y ' ) )
.eikx(x-x')+iky(y-y')d xdy dx ,dy>
= W { k x , ky)8(kx - k'x )8(ky - k'y),
(2.25)
w ith
W ( k x , k y) =
[ ! ° ° C ( u , v ) e ikxU+ik»vdudv,
J
(2.26)
J —OO
where u = x — x', v = y — y', and C ( u , v ) = (z(x, y) z( x' , y')). W ( k x , k y) is the Fourier
transform o f the correlation function and hence is the surface roughness spectrum.
S u b stitu tin g (2.25) into (2.23) and integrating w ith respect to kx and k'y, we get
(lE»f ) = (2^
{2kc° s 9)2 ^ I J Z
W{ k 1 + k s [ n 0 ' ky)dk* dky
f
A
f°°
J J
(2-2?)
hy^(lk;cCllCy
D enoting A (^EqvE qi^ as the intensity w ith in the narrow spectral bands A k xA k y
centered at kx and ky,
^
~ Jqpi^xi ky) A k xA k y = JqP(kx , k y) k2 cos 0SAQ,S.
(2.28)
The averaged in ten sity P received at a distance r from the illu m in a te d area A is
equal to the average power per u n it solid angle tim es the solid angle subtended by
the receiver,
P,ir _
^
= ^
J
k
fw {K ,
(2.29)
Since the scattering coefficient is defined in term s o f P by the product o f the angle
and the solid angle subtended by the receiver,
<2-30>
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
a° for backscattering direction is computed as
cr° = —kQcos 0 |a ,p| W ( 2 k sin 6 , 0),
(2.31)
where a qp is given in ( 2 .2 2 ).
Tw o typ ica l normalized correlation functions are the Gaussian and exponential
functions given as, respectively,
i ! 1 = v 2p a ( 0 ,
p
C g ( 0 = o-2 exp
e
C e(f) = <r2exp
l /y/ 2'
= * 2P e ( a
(2.32)
(2.33)
in which
/
OO
p(u, v)d.udv,
-CO
where I is the correlation length, 7r / 2 is the correlation area [E ftim u and Pan, 1990],
and £ is \ / u 2 + v 2 for a two-dimensional rough surface. For a Gaussian correlation
function o f (2.32), the surface roughness spectrum is
W a ( ‘2k sin 0,0) = irs2P e - lkl‘ ia8)\
(2.34)
while the surface roughness spectrum for an exponential correlation function of (2.33)
is
We(2k sin 0 , 0 ) = irs2l 2 [ l +
2
( k l sin 9 )2J
(2.35)
When we have a num erical form of correlation instead o f a functional form , e.g., a
correlat.ion measured d ire ctly from a random surface, the roughness spectrum can be
computed num erically in the follow ing form;
iy e( 2 A s in 0 ,O) = s 2 /
r
J J —oo
t oo
= 2tts2
Ja
p(Oei2hsi" 8ududv
p(£)£J 0 (2fcsinl?£)d£,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
where Jo(x) is the firs t kind Bessel fun ctio n of zeroth order, which can be evaluated
approxim ately using the polynom ials given in [A bram ow itz and Stegun, C h.9, 1972].
I f above com putation was extended to second order, the cross-polarized term
would not be zero in the backscattering d irection. The second-order backscattering
coefficient for the cross-polarization has been shown in [Valenzuela, 1967] as,
a lh = *hv =
cos2 0
“ R >^\2
r r
k
2k 2
/ -------------— — - j W ( k x — ksinO, ky) W { k x -f ksinO, ky)dkx dky
J J - oo |A?i* + er kz|
(2.36)
■
where kz = \Jk^ — k 2 — k 2, k i z = ^J(.Tk l — k* — k 2, and Rv, 7?/, are the Fresnel re­
flection coefficients for vertical and horizontal polarization, respectively. The crosspolarized backscattering coefficients can be obtained by evaluating a tw o-fold nu­
m erical integration w ith a known surface roughness spectrum W ( k x ± ksinO, ky).
W { k x ± k s i n O , k y) for Gaussian and exponential correlation functions can be com­
puted, respectively,
f
WG(kx ± k sin 0, ky) = s2 J J
f°°
u2+i»2
e—
S kl±k^
u+ikyvdudv
(kT±kim0)2l2
= irs2l 2t
ro o
r
We(kx ± k sin 0, ky) = s2
/
J
ro o
r2 ir
= s2
2
e- V ,
4
(2.37)
\ / t i 2 4tJ2
e~ ‘/ ^
e'(^ ±“
)u+a'*’Vudu
J —OO
j
e 7p5e*ik* ±k'«m9)e«i|.0 c.fc,€co«0^(/ ^
Jo
Jo
= s22tt J ™ i e ~ ^ J 0 ( ^ ( k x ± ksinO)2 + k 2
y)
= trs2l 2 [ l + { { k x ± k sin 0)2 + k2
y } l 2/ 2} ~* .
(2.38)
The va lid ity conditions associated w ith the sm all perturbation m ethod are given
by [U laby et al., 1986; Chen and Fung, 1988] as
ks < 0.3,
m < 0.3,
and , k l < 3.0.
(2.39)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
(b)
Backscattering coeff., dB
(a)
03
■o
-to.
-20 .
8o
-30.
60
c
•c
w
-40.
4-*
as
o
2
3
CQ
VV
-50.
-60.
----------------
VV,
--•
0.
10. 20. 30. 40. 50. 60. 70. 80. 90.
VV,
•o
8O
o
>
>
>
x
o
■aa
aC
i
i
i
i
i
i___i
,
Incidence Angle, Degrees
Gauss. Corr.
-
10 .
■ Expon. Corr.
-15.
%
CQ
i
(C)
CQ
2
Expon. Corr.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
dC
0
o
\
H H , Expon. Corr.
_ i _____i_i____i_i
-70.
\
H H , Gauss. Corr.
---------------- - - - -
Gauss. Corr.
-
20 .
-25.
-30.
-35.
-40.
0.
10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
F igure 2.2: Backscattering coefficients of a rough surface w ith ks = 0.2,k l = 2.0, and
t r = (10,1) using the SPM; (a) polarization response of the surface of an
exponential correlation, (b) a°hh and a°v, and (c) the ra tio o°hJ o ° v o f the
surface w ith two different correlation functions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
\ \
20
(a)
Backscattering
coeff. ( W ) ,
dB
(b)
CQ
■o
- 10 .
-
-
20 .
10.
8
cj
V'
M
-30.
1cQ
-40.
k l= l
o
i2
■ ks=0.2
8
CQ
-50.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
—
k l= 3
-50.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
Incidence Angle, Degrees
(c)
CQ
’O
- 10.
its
- 20 .
a
‘C
-30.
8
0
-
1
Io
-40.
«!
CQ
e=(1 6,3.2)
• e =(8,1.6)
• £,=(4,0.8)
-50.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
Figure 2.3: Backscattering coefficients at uw-polarization using the SPM for the sur­
face o f an exponential correlation, (a) k l = 2 . 0 , er = ( 1 0 , 2 ), and the
various values o f k s , (b) ks = 0 .2 , er = ( 1 0 , 2 ), and the various values o f
kl, and (a) ks = 0 .2 , k l = 2 .0 , and the various values of eT.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
(a)
ca
■o
-5.
0>
o
u
cd
O
J2
3
- 10.
<*-.
o
>
>
-15.
- 20.
52
33
■.—
J
o
•a
03
Pi
8
O
-10.
n
a
8
-15.
a
<4-1
ca
/-“■S
(b)
CQ
-25.
-
e = (4 ,0 .8 )
•
e = (8 , 1.6)
•
er= (1 6 ,3.2)
-30.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
O
N
>
2:
a ■*
'—
o
a
03
a
(c)
0)
o
u
- 10 .
cd
-15.
O
J2
3
a
•
ks=0.2
•
ks=0.1
-20.
-25.
-30.
-35.
>
-30.
SC
O
-35.
cd
u
o
o
cd
(d)
kl=3
-10
• kl=2
-15
o
a
s
-20
0
-25
-30
1DC
>
C4
■ I__I__L-
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
a
a
aU-t
-25.
'
-40.
Incidence Angle, Degrees
■*-J
- 20 .
c = (8 , 1.6)
• £ = (1 6 ,3 .2 )
Incidence Angle, Degrees
a•o
Er= ( 4 ,0.8)
'w'
-40.
0.
10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
o
•x
s
cd
PC
-35
-40
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
Figure 2.4: The ratios of the backscattering coefficients at uu-polarization using the
SPM fo r the surface o f an exponential correlation; (a)
and (b)
<jhvl°hh f ° r ks = 0 -2 ) k l = 2 .0 , and different er , and the ra tio <?°hvfo°hh for
(c) k l = 2 .0 , er = ( 1 0 , 2 ), and different ks, and (d) ks = 0 .2 , er = ( 1 0 , 2 ),
and different kl.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
The backscattering coefficients for various rough surfaces computed using the SPM
are illu s tra te d in Figs. 2.2-2.4. The polarization responses of a rough surface w ith
ks =
0 .2
, kl =
2 .0
and er = ( 1 0 , 2 ) are p lo tte d as a fu n ctio n of the incidence angle in
Figs. 2.2(a)-(c). cr%v is higher than cr°hh and the ra tio cr°v/ a l h increases as 0 increases.
The cross polarized response <r%v is much lower than co-polarized response and the
ra tio of cr^u/o-°y is constant in a wide range o f incidence angle as shown in Figs. 2.2(a)(c). The angular p a tte rn of the backscattering coefficients depends on the type of
correlation functions as shown in Fig. 2.2(b).
The sensitivities o f <7 °^ to surface parameters, ks, kl, and er , are illustra ted in
Figs. 2.3(a), (b), and (c), respectively, for a surface w ith an exponential correlation
function. Figure 2.3(a) shows th a t <7 °^ is very sensitive to ks where k = 2 i r / \ and
s is the rm s height.
W hen kl increases, only the slope of the angular pattern of
<7 °^ increases especially in the range o f sm all incidence angles, b u t the level of <t° v
does not change as shown in Fig. 2.3(b). Increasing the dielectric constant eT also
increases cr°v by a constant value for all incidence angles as shown in Fig.
2.3(c)
w ith less se n sitivity compared w ith the se nsitivity to ks. Figures 2.4(a)-(c) show tha t
sensitivities o f the ratios, <7jjA/<7°„ and crfLV/ a ° v, as functions o f surface parameters, ks,
kl, and er , o f a rough surface having an exponential correlation. The ra tio crf^Jcr^,,
is ju s t the ra tio of \ahh\2/\<Xw\2 (2.22) w hich is independent of ks and kl. The ratio
however, depends on er and 0 as shown in Fig. 2.4(a), i.e., <7/°/i/<7°„ decreases
as 0 increases and the rate of change increases as eT increases. The ra tio ° h v K , ^ a
weak fun ctio n o f eT and k l b u t a strong function o f ks as shown in Figs. 2.4(b)-(d).
^ h v l a lv decreases by 10 dB as ks decreases from 0.3 to 0.1 for a rough surface w ith
kl =
2 .0
and er = ( 1 0 , 2 ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
2.3
K irchhoff A pp roach
The vector fo rm u la tio n o f the K irch h off m ethod has been often form ulated by
S tratto n-C h u representation [S tratton , 1941]. The scattered field outside or on the
surface can be represented by the H ertz vectors w hich are sim ple and valid fo r open
and closed surfaces [Senior, 1992].
For a plane wave in cid en t upon a random surface as shown in Fig.
2.1, the
o rtho n orm a l coordinate systems (u,-, hi, k{) and ( vs, h s, k s) are given in (2.7)-(2.7).
T he scattered field
Es(r)
above or on the surface z ( x ' , y ' ) can be w ritte n in terms o f
the H e rtz vectors,
E 5 (r ) = V x V x n e(r) + i k 0Z 0V x n m( r )
(2.40)
The e le ctric and m agnetic H ertz vectors, n * and n ^ ,, are represented using the surface
current,
Je(r') =
h' x
H(r')
and
Jm(r')
= —n' x
E(r'),
which are equivalent sources,
respectively,
n :(r ) = I T
K0
I l h ' x H ( r ') G 0 ( r , r > '
j
n ;(r) =
(2.41)
Js
/ fn ' x
Kq J
E(r')G0(r, r >
'
(2.42)
J3
where G0{ r, r;) is the free-space scalar Green’s fu n ctio n given by
_»fc0|r—r'|
G“ ( r - r ') = 5
^
M
assuming the tim e dependence o f e~*wt. In the fa r field ( r > 2 D 2/ X 0), the vector
operator V x and the Green’s fun ctio n can be approxim ated as follows;
V
X
(• • •) *
ik o t, X
(• • •)
G o ( r , r ') « ^ le x p ( - a a , T ') .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
S u b s titu tin g (2.44) in to (2.41) and (2.42), the scattered fields in (2.40) can be obtained
in term s o f the surface fields,
E '( r ) = ^
ikork x J J
[(«' x
x
E ( r ') ) - Z 0k
( h‘
x
H ( r ') ) ]
ds', (2.45)
which is the same form as what has been derived from the S tratto n-C h u representation
through q uite com plicated co m p uta tion [U laby et al., 1982].
In order to find the tangential surface fields
n' X
E ( r ') and
h' x
H ( r ') , we assume
the surface fields at any p o in t o f the surface can be represented by the fields on the
tangent plane at th a t p oint (tangent plane approxim ation). Let the incident field be
E 1' =
(2.46)
a e ikoki'T ,
where a is a u n it p olarization vector. We can define a local coordinate system (£, d, &,)
for the lo ca lly fla t tangent plane such th a t
t = pr———- 7 ,
\k{
where
hi
d= k
x n/
x
t,
k = i
x
d,
(2.47)
is a u n it norm al vector o f the tangent plane, w hich is given by
=
- Z i - Z , y + z
Jzi + zi + i
and Z x and Z y are the local slopes in the x and y directions, respectively,
r? ( t
i\
dz(x',y')
Zx[x , y ) = —
— ,
and
,
dz(x',y')
Zy{x,y) = — — — .
(2.49)
The e le ctric and m agnetic incident fields can be decomposed in to the locally perpen­
d icu lar and parallel polarizations using the local coordinate system,
E* = f(a • i ) t + (a ■d)d] e,ko^''T'
1
J
and
H - = ijfc,- x [(a • i ) i + (a • d)d\ eik°k' r ' ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.50)
25
respectively. Then, the electric and magnetic local reflected fields are, respectively,
E T = [.ft/zda • i) t -f- Riv(a • d)d] e,k° ki'r ' and
.
.
Hr=
ki X JRth(a • t)t + Riv(a • d)dj e' k°ki'r ' ,
(2-51)
where Rih and R[v are the horizontal and vertical Fresnel reflection coefficients for
the local angle
0 /,-,
respectively.
Using the boundary conditions of
E (r') = E ‘ ( r ') + E U r')
and
(2.52)
H ( r ') = H ’ ( r ') + H r (r')
(on the surface),
we can o b ta in the follow ing relations,
n/ x E ( r ') =
[(1
n ; x H ( r ') = g Then, the
+ Rih)(a ■t ) ( h i x i ) — ( 1 — R i v)(a • d)(hi • &,-)£] e,k°ki'T'
[(-1
6 -polarized
+ Rih){a ■t ) { h i ■k ) t -
(1
and
+ R h ) { a • d)(fii x {)] e,ko'ki'r ' .
scattered field from a lo ca lly fla t plane fo r the n-polarized
incident fie ld can be computed by su bstitu ting (2.53) in to (2.45),
EL =
b ■E : ( r ) = D 0 j j s f baeik°k*-r ' dxdy
(2.54)
w ith
Do =
kd = h — ks
(2.55)
h a = b • { * , x [n, x E a(r ')] + Z 0 [n, x H 0 ( r ') ] } J z f + z f + l ,
where the subscripts a and b can be v and h which indicate the ve rtica l and horizontal
p olarization, respectively.
For the backscattering d irection, the scattering am plitudes f ba areobtained after
algebraic com putation using the coordinate system given in (2.10), (2.11), (2.47), and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
(2.48),
2
(cos 9 + sin 9Zx)[ Rih(Zx cos 9 - sin 9)2 - R i vZ 2]
Jhh
/ r/
r\/j\9 i *72
'
\ Z X cos 0 — sin v ) z - f Zy
.5 6 }
2(cos0 + s\ n0 Zx)[ Riv( Zx cos9 - s in # ) 2 - R i h Z 2]
}vv ~
t
(zxcose - sine y
_ t
+
z2
{
_ 2 ( c o s 0 + sm 6Zx ){sm 6 - Zx cos 0)(Riv + R i h)Zy
- J«h ~
}
/n ecn
(Zx cos 0 - sin e y + Z l
{
’
w ith
Rih —
Z x sin 0 - f cos 0 — J e r { 1 + Z 2 + Z 2) — Zy — ( Z x cosO — s in 0 ) 2
/ ——
=,
Z x sin 9 + cos 0 + s j t r { 1 + Z 2 + Z 2) — Z 2 - ( Z x cos 9 — sin 0)2
(2.59)
er ( Zx sin 9 + cos 0) — ^£,.(1 + Z 2 + Z 2) — Z 2 — ( Zx cos 0 — sin 0) 2
Riv —
/•
~•
— •
~~~
—,
(2.60)
eT( Zx sin0 + cos 0) + ^£,-(1 + Z x + Z 2) — Z 2 — ( Z x cos0 — sin 0)2
where the local Fresnel reflection coefficients are obtained using
cos 0n = —hi ■ki =
Z x sin 9 + cos 9
s jz i+ z i + r
Consequently, the backscattered mean in te n s ity is given as the ensemble average of
the p ro du ct of the backscattered field and its complex conjugate,
( E X )
= p o |2/
j^ d x .d y , j
■{ f ba(ZXl, Z yJ f : a( Z X2 , Z y2)eikdAsi- 22)) ,
where z j, ZXl, Z X2, Z y i, and Zy2 form a random vector, | A
z(x \iH\)i
z2
)|2
(2-61)
= ^ o /(4 7rr)2) z\ —
= z ( x 2 i 2/ 2 )) and z j = z\ — z2. Since the integrals o f the mean in ten sity
are s till very d iffic u lt to evaluate analytically, we need additional approxim ations to
get a closed form o f the mean intensity.
The v a lid ity conditions for this kirch h off approach are given [U laby et al., 1982]
as
kl >
6
and
R c > A ( l 2 > 2.76sA
fo r a Gaussian correlation)
where R c is the average radius of curvature given in (2.5).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.62)
27
2.3.1
Physical Optics Model
A com m only used approxim ation is to expand the integrand in (2.45), fba{ Z Xi Z y),
about zero slopes and keep only the first few term s as follows;
fba{Zx, Zy ) — /6a(0 ,0) + Z3
dfba
dfbc
dZx
+ Z y d Z v Zx Zy 0
+
(2.63)
For surfaces w ith a sm all rm s slope, the scattering a m p litu d e fba( Z x , Z y) can be
approxim ated by the firs t te rm o f the series (2.63) [Tsang et al., 1985] where
fh h M
= 2 cos 0Rh
f w { 0,0) = 2 cos 0RV
(2.64)
fv h M
Rh, R v =
= f hv(0,0) = 0
Fresnel reflection coefficients.
T h is approxim ation m ay be called zeroth order approximation because the slope terms
o f Z x and Z y are ignored. W ith this approxim ation, the cross-polarized backscattered
fields are zero as seen in the above equations. Since /j,a( 0 ,0 ) / 6*a(0 ,0) is not dependent
on the random variables anymore, the ensemble average te rm is given as
(fba{ZXl, Z y i ) f ; a( Z x„ Z y2)eik^ - ^ )
= | / 6a(0, 0) | 2
(2.65)
W hen the random ly rough surface is assumed to have a stationary Gaussian height
d is trib u tio n ,
1
Pz ( z ) =
\f2rrs
e
-ZL
(2.66)
where s is the standard deviation of the surface height d is trib u tio n , the characteristic
fu n ctio n fo r a Gaussian random vector [Stark and Woods, 1986] is
$^(o;) =
= exp
1
M
N
L m= 1 n = l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.67)
28
in case o f zero mean (Ji =
0
), where £ is a random vector and u> is a param eter vector.
Therefore, the ensemble average in (2.65) can be represented as
_ e—(*d*)2s2[l—p(«,v)]
(2.68)
where p(u, v ) is the norm alized correlation function given in (2.4). Changing variables
in a double integral, u = x i — X2 , v = y i ~ y2i and using the characteristic fun ctio n of
( 2 .6 8 ), the backscattered mean in te n sity w ith zeroth order approxim ation is obtained
in the form o f
k2
(1^.1’L
= (f47i rfr )e2 (2cos
lft|2 1
I I dudv(-2 L ~
|u|)(2i - M>
• eikdxU e~(kd*)2s2[1~p(u'v^.
(2.69)
Instead o f a series expansion of fba { Z x , Z y) about zero slopes, fba ( Z x , Z y ) can
be approxim ated app ro p ria tely for surfaces w ith sm all rm s slope [Fung et al., 1992].
A ssum ing sm all rms slope ( Z 2, Z 2 <C 1), f b a { Z x , Z y) can be approxim ated fro m (2.56)(2.60) as follows;
f h h ( Z x , Z y ) « 2(cos 6 + Z x sm9 )Rh
f vv{ Z x , Z y) « 2(cos 9 +
sin 9 ) R V
(2.70)
fvh ( Z x , Z y ) = f h v { Z X, Z y ) W 0
where Rh and R v are the Fresnel reflection coefficients. T his approxim ation may be
called f i rs t order
approximation since the co-polarized scattering am plitudes include
the firs t order of the slope terms. The cross-polarized backscattering fields are zero
w ith th is approxim ation too.
S u b stitu tin g (2.70) in to (2.54) and in teg ra tin g by p a rt [Beckmann and Spizzichino,
1963], the backscattered fields can be computed as
E'aa = D 02R a— tt [ f eikokd'r> dxdy + 0 { L )
cos U J J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.71)
29
w ith
* fi
~\
~ Sm0 cikpkd.T
2iko cos 0
-L
0(L) =
where the term o f 0 ( L ) is an edge effect [Beckmann, 1968] which is in significant for
a surface of sm all rms slope and small incidence angles near norm al. Therefore, the
backscattered mean in te n sity is given, ignoring the edge effect term ,
ko
2
(
2
r r 2L
„
(4 7 r r ) 2
•
eikdlU e
-
( * rd * ) 2 s 2 [ w
(2
( u ’v ) ].
7 2 )
Comparing (2.72) w ith (2.69), the backscattered mean in ten sity w ith firs t order ap­
proxim ation is th a t w ith zeroth order approxim ation divided by cos4 #, i.e.,
0
^ ° “ ! ) u t = cos4 9
)oth '
(2‘ 73)
The first and zeroth order approxim ated PO solutions are examined in m ore detail
in Section 2.5.
In order to get a closed form o f the backscattered intensity, we may set L — ►oo
assuming the illu m in a te d rough surface contains m any correlation lengths L
/.
Using a series expansion fo r the exponential term ,
= £
( k dz^2p { u ,v )T ^
n=0
n '
the backscattered mean in te n sity can be rearranged as
=
j&
y
(W )J
^
(2i)!
•J J
£
{3 ^
-
e,kdxU pn(u,v)dudv.
(2.75)
The n = 0 term corresponds to coherent scattering [U laby et al., 1982] since
[(£,*„) L
=
(2
cos e f
(2 £
)2
J
dxtdv,
(2.76)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
and the rest o f the series in (2.75) represents incoherent scattering.
W hen we consider a Gaussian correlation fun ctio n given in (2.32), the integral in
(2.75) can be com puted [Gradshteyn and R yzhik, Ch6, 1980] to be
Ig =
f
.-I.
J J —oo
..
e dx e
u2 + «2 ,
IT I 2
,
i2 dudv = — e
( k l sm» ) 2
»
Tl
y
(2.77)
w hile the in teg ra l for an exponential correlation fu n ctio n given in (2.33) is computed
to be
r
Ie =
r oo
7 u 2 + ti2
e,kdxUe
I
7t t j / 2
dudv =
J J ~°°
-------- — -------- r ,
(2.7S)
[n2 + 2 { k l s in 0)2\ 2
where kdx = 2k sin 6 and I is the correlation length.
Since the backscattering coefficient corresponding to the backscattered mean in ­
te n s ity can be given as
in r'‘ R c { [ \ E l S ' ) h ' , \
^ - A a —
where the in trin s ic impedance
77*
=
v ,.}
77*
=
770
■
and |£ 0| =
<2-79)
1 .0
in this problem , the
incoherent backscattering coefficients fo r PO m odel are given by
R tf
t
{2ksC0° ° r
-l
(2.80)
n=l
where R a is the Fresnel reflection coefficient fo r a-polarization and I is Ig in (2.77)
or I e in (2.78) fo r a Gaussian or an exponential correlation function, respectively.
The a d d itio n a l approxim ation o f the PO m odel ( zeroth or firs t order approxim a­
tio n ) lim its the v a lid ity conditions to the small slope o f the rough surface. Therefore,
in a d d itio n to the v a lid ity conditions in (2.62), another condition th a t lim its the rms
slope is usually given [U laby et al., 1982] as
m < 0.25.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.81)
31
(a)
t
1 i
1 i
-
n■o
1 r
VV, Gauss. Corr.
HH, Gauss. Corr.
- VV, Expon. Corr.
-
HH, Expon. Corr.
60
S
•c
u
ts
G)
U
2
3
«
v
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle, Degrees
30.
u
ts
cd
O
2
3
a
e=(4, 0.8)
- 10.
o
•a
R)
ft!
e =(16,
3.2)
- 20 .
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle, Degrees
F igure 2.5: Backscattering coefficients o f a rough surface w ith ks = 1 and k l = 8 using
the PO model; (a) o^h and a°v fo r two different correlation functions and
er = (10,2) and (b) the ra tio c r ^ / c r ^ for the various values o f er w ith an
exponential correlation function.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
(a)
(b)
03
03
■o
4>
6
§?
e
u
J2
&
03
■o
-
o
o
U
60
a
10.
- 20 .
ks=2.0
-30.
-
10.
- 20.
'5
\
su
• ks=1.0
is
g
ks=0.5
• kl=12
-30.
kl=24
03
-40.
kl=6
-40.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
Incidence Angle, Degrees
(c)
10.
■u
■
>
—'
-
10 .
-
20 .
60
-30.
-40.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle, Degrees
Figure 2.6: Backscattering coefficients at M -p o la riz a tio n using the PO m odel for a
surface o f an exponential correlation, (a) k l = 8, er = (1 0 ,2 ), and the
various values o f k s , (b) ks = 1, eT = (10,2), and the various values o f kl,
and (a) ks = 1, k l = 8, and the various values o f er .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
T he backscattering coefficients fo r various rough surfaces computed using the
zeroth-order PO model are illustra ted in Figs. 2.5 and 2.6. Since the polarization
response depends only on the Fresnel reflection coefficients,
is higher than a°v as
shown in Figs. 2.5(a) and (b) in contrary to the polarization response in the SPM.
The ra tio n
has a peak at the corresponding brewster angle as shown in Figs.
2.5(a) and (b). Figures 2.6 show th a t the backscattering coefficients using PO mode)
are very sensitive to the variation of ks and less sensitive to the variations o f k l and
eT. W hen ks increases (a n d/o r k l decreases) a°lk decreases at lower incidence angles
and a°lh increases at higher incidence angles as shown in Figs. 2.6(a) and (b). The
dependence of <r° versus 9 on ks using PO m odel is q uite different w ith th a t using
the SPM which shows a constant increase in cr° versus 0 fo r increasing ks as shown in
Fig. 2.3(a). The dependence of cr° on t T using PO model is s im ila r to that obtained
using the SPM, i.e., increasing er produces an approxim ately constant increase in <r°.
2 .3 .2
G e o m e tric a l O p tic s M o d e l
The asym ptotic solution to the K irch h off-d iffractio n integral (2.54) can be derived
using the stationary-phase approxim ation in the geom etric lim it as k — ►oo. Under
this approxim ation, the scattering coefficient w ill be p ro po rtio n al to the p ro b a b ility
of the occurrence o f the slopes which w ill specularly reflect the incident wave to the
observation direction. Hence, local diffractio n effects are excluded in this approxim a­
tion.
T he phase tj) is said to be stationary at a p o in t if its rate o f change is zero at the
p oint where
^ = k0kd ■r ' = kdxx' + kdyy' + kdzz(x', i/'),
(2.82)
w ith kdx = 2&osin0, k dy = 0, and kdz = —2kocos0 for the backscattering direction.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
The slopes Z x and Z y can be found for the backscattering d irection as
= 0 = kdx + kdzZ x0 = > Z xo = =^ - = tan 6,
= 0 = kjy + kdzZy0 =$■ ZyO = 0.
Using these slopes w hich are not functions of the random position vectors any more,
the mean in te n s ity of (2.61) can be reduced in the form o f
(E L E u ) = \Do\2 \ f ba{ZxQM
2- ( i n
(2-83)
where
(II*) = J
J ™ ( 2 L - \u\)(2L - |w|)e<fc* “ e - ^ ^ - ^ ' ^ d u d v .
(2.84)
From (2.56)-(2.58) we can find the scattering am plitudes for the backscattering d i­
rection as follows;
fhh(Zxo ,0 ) = ^ 7 ^ ( 0 )
f vv( Z xO, 0 ) = ^ R v (0)
(2-85)
fv h( Zx0 , 0 ) = fhv( Zxo , 0 ) = 0,
where i?o(0) is the Fresnel reflection coefficient evaluated at norm al incidence for
a -polarization.
W hen we assume k%zs2 is large (^> 1) so th a t the co n trib u tio n to
the integral in (2.84) is significant o n ly fo r sm all values o f u and v , the norm alized
correlation fu n c tio n p(£) can be approxim ated by the firs t two term s of its Taylor
series expansion about the origin,
Changing the variables as £ = y/u2 + v2 and integration lim it as 2L —> oo, ( I I ' )
reduces to
(II*) = A r
f 2* eikd*isin,p e - l kd‘ ? pi! ^ P { d ( { h / >
Jo Jo
27xA
(2fccos Q)2s2 |p"(0)|
tan2 9
e 2s2Ip"<°)I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.86)
35
where A = (2L ) 2. S ub stitu ting (2.85) and (2.86) into (2.83), the backscattered mean
in te n s ity is computed as
■ A
<2 -8 7 >
From (2.79) the backscattering coefficients for the geometrical optics (G O ) model is
2
o
l ^ ( 0)1
ff“ W = 2 ^ W 9 exp
tan 2 0
2m 2
(2 .88)
where the rm s slope m is sy|/3 "(0 )|, aa is vv or hh, and cr°h = crflv = 0.
In a dd itio n to the v a lid ity conditions of (2.62) for the tangent plane approxim ation,
an a dd itio n al condition is required fo r the GO model [Ulaby et ah, 1982] as
k j zs2 > 10
or
v/2^5
ks > —
cos a
(2. 89)
W hen the angle of incidence is large, some points on the rough surface may not
be illu m in a te d d ire ctly and shadowed by other parts of the surface in te rru p tin g the
in cid en t wave. Since the PO and the GO models do not include the effect o f shadow­
ing, the scattering coefficients for these models should be m odified for the shadowing
effect. The shadowing function which is the p ro b a b ility th a t a p oint on a rough sur­
face w ill not be illu m in a te d by an incident wave, is given by [S m ith, 1967 and Sancer,
1969],
m
-
i+ r b o
( 2 -9 0 )
w ith
nr/ / i
\
/2 m
. « <2»
/'cot#
2f(6,m ) = \
-e
—erfc
7r cot 6
V \/2 m ,
where e r f c is the com plem entary error function, m is the rm s slope, and the corrected
backscattering coefficients a°' = o° • R(0).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
(a)
20.
CQ
•o
m=0.2
m=0.4
X
m
(D
o
U
-
10.
V
o
J2
o
ta
- 20 .
CQ
-30.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle, Degrees
(b)
CQ
•o
£
-
U8
10.
on
-20.
-
CQ
O
C/3
er= (16,3.2)
• e,=(8,1.6)
O
CQ
■ £,=(4,0.8)
-40.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle, Degrees
Figure 2.7: Backscattering coefficients of a rough surface at v v ( = h h )-polarization
using the GO model; (a) for eT = (10,2) and the various values of m and
(b) for m = 0.4 and the various values o f er .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
3
GO
2
1
PO
SPM
2
6
4
8
kl
Figure 2.8: The v a lid ity regions of the classical models which are the SPM, the PO
model and GO model.
Figure 2.7(a) shows the backscattering coefficient using GO model as a function
o f 6 fo r the various values o f the rms slope m. The curve o f a° versus 6 drops more
slowly as the surface slope m increases, where the backscattering coefficient o° does
not depend on polarization, i.e., <Tlh= a ° v. Similar to the SPM and PO model, the
GO m odel predicts a constant increase of a° versus 9 for increasing eT as shown in
Fig. 2.7(b).
The v a lid ity conditions o f the SPM, PO, and GO models for backscatter given in
(2.39), (2.62), (2.81), and (2.89) are illustrated in Fig. 2.8 where (2.89) is applied for
the case o f 0 < 50°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H APTER III
NUM ERICAL SOLUTION FOR SCATTERING
FROM ONE-DIMENSIONAL CONDUCTING
RANDO M SURFACES
3.1
In tro d u ctio n
Since the m ethod o f moments [H arrington, 1968] was applied to the estim ation
o f the scattering coefficient of conducting random surfaces [Lentz, 1974], many other
num erical methods w ith some m odifications have been introduced to solve the scat­
te rin g problem o f random surfaces [A xlin e and Fung, 1978; Fung and Chen, 1985;
Nieto-vesperinas and Soto-crespo, 1987; Durden and Vesecky, 1990; and Rodriguez et
ah, 1992], The incident field in a ll o f these methods was a tapered wave to elim inate
the edge-effect co ntribu tio n due to the fin ite length o f the sample surface. However,
at large incidence angles (0 > 60°), the w id th o f the sample surface m ust be very
large to e lim in a te the edge-effect, which results in excessive co m p uta tion tim e. O th ­
erwise, the w indow should be very narrow to elim inate the edge c o n trib u tio n at a
large incidence angle, which results in an incorrect o u tp u t by excessive smoothing.
In this chapter, a new technique, adding a resistive card at each end o f an illu m i­
nated surface, is introduced to e lim ina te the edge co ntribu tio n even at large incidence
angles. To illu s tra te this technique, one-dimensional random surfaces are generated
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
in Section 3.2. Integral equations are form ulated for one-dimensional conducting ran­
dom surfaces and solved by the m ethod of moments fo r vv- and M -p o la riza tio n s in
Section 3.3, and num erical results are analyzed in Section 3.4. Using this num eri­
cal technique, the existing models are evaluated for scattering from one-dimensional
conducting random surfaces in Section 3.5.
3.2
R and om Surface G eneration
A sequence o f independent Gaussian deviates w ith zero mean and u n it variance
(N [0 ,1]) can be obtained from a standard routine [Presset al.,
1986]. Then
these
independent Gaussian deviates can be correlated to a specific correlation function
using the concept of d ig ita l filte rin g [Fung and Chen, 1985].
A t first, the desired
surface height profile { Z ( k )} can be w ritte n as the sum m ation o f the product o f the
independent Gaussian deviates { X ( j - f &)} and a discrete weighting factor {V U (j)}
w hich is to be determ ined,
Z(k)=
M
Y,
W ( j ) Z ( j + k),
(3.1)
j= -M
where Z(k) is the &th p o in t of a discrete height profile and M is the to ta l num ber of
sample points o f the weighting factor W ( j ) .
The correlation coefficient function C { i ) o f the desired surface profile is given by
the definition as follows;
C ( i ) = E [ Z ( k ) Z ( k + *)] = Y Y W ( j ) W ( m ) E [ X ( j + k) X ( m + k + *)]■
j
(3.2)
"i
Since the Gaussian deviates are m u tu a lly independent, i.e.,
{
0
,
j
7^
m + i
(3.3)
1, j = m + i,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
the correlation function can be sim plified as the self convolution of the weighting
factor,
C(i) = £
W (j)W (j - i ) = W *W .
(3.4)
Using the Fourier transform theorem, the unknown w eighting factors can be found
analytically,
(3.5)
W hen we choose the Gaussian correlation fun ctio n of the form
C ( i ) = s2 exp
(3.6)
its spectrum is given by
•F[p{i)\ = s2y/TrLexp
L 2f 2
(3.7)
T he corresponding weight factor can be obtained as
W {j)
; ■,— exp
(3.8)
where s is the standard deviation o f the height d is trib u tio n (rm s height), L is the
discrete num ber given as I / A x , I is the correlation length o f a random surface, and A x
is the sam pling interval. Then, the surface height profile can be com puted by (3.1)
and (3.8) for given surface statistics such as the rm s height, the correlation function.
The proper value for the w id th o f an independent surface D and the sam pling interval
A x is a fu n ctio n of the surface correlation length and frequency o f the incident wave.
In order to get m eaningful statistics o f the backscattering coefficient, the num ber
o f sample surface, N , should be large enough. For example, N > 60 to suppress the
speckle noise in the estim ated backscattering coefficient to w ith in ± 1 dB from the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
mean value for the 5% and 95% cum ulative d is trib u tio n levels [Ulaby and Dobson,
1989].
Considering fin ite com puter storage and p ractical restrictions on com puter exe­
cu tion tim e , each surface must have fin ite length, D. W h ile the upper lim it of the
surface w id th is decided by the size o f com puter storage, the lower lim it may be con­
sidered as D > 10A and D > 10/ where A is the wavelength and I is the correlation
length.
The sample in terval, A x , is chosen as A x < A/12, w hich is comparable w ith
others [Fung and Chen, 1985; Rodriquez et ah, 1992]. The backscattering coefficient
is a pp roxim ately proportional to the surface roughness spectrum , W (2fcsin0), where
the roughness spectrum fo r a Gaussian correlation fun ctio n is given as:
W { 2ks\n9) = y / r s 2l e - l h‘ s'"x6)\
(3.9)
I f we choose the w id th o f the spectrum as tw ice o f the frequency where W ( 2 k s \ n 0 )
has dropped to e-1 W (0 ), the spatial frequency kx = 2 • 2 k s i n 0 = 4 //. A p p ly in g the
sam pling theorem, A x <
= //8 . Therefore, the sam pling interval A x is chosen
to satisfy b o th conditions of A x < A/12 and A x < 1/8.
Three different surfaces w ith given roughness parameters (Table 3.1) are generated
according to above c rite ria w ith M = 45000 and the to ta l length o f N x D. Figures
3.1 (a)-(c) show typ ica l sections o f the surface height profiles o f S -l, S-2, and S-3,
respectively.
The height d istrib u tio n of the generated surfaces are compared w ith
Gaussian p ro b a b ility density functions
fx{x) =
as shown in Fig.
3.2.
V^7TS
exp
x
'2 ?
(3.10)
The autocorrelation functions of the com puter generated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
(a )
0.3
0.2
'u'
0)
0)
0.1
N
0.0
^
!£
-o.i
-
0.2
-0.3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
7.0
8.0
9.0
10.0
7.0
8.0
9.0
10.0
Distance, x (meter)
(b)
0.3
0.2
0.1
g
N
0.0
oi
i i 1
-o .i
S3
-
0.2
-0.3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Distance, x (meter)
(c )
0.3
0.2
<u
<3
0.1
0.0
-
0.1
-
0.2
-0.3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Distance, x (meter)
F igure 3.1: T yp ica l sections o f height profiles for (a) S -l, (b ) S-2, and (c) S-3 surfaces.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
Table 3.1:
Surf.
Roughness parameters used for the random surface generation
ks
kl
A
s
Name
1
Ax
D
N
meters
A p p l.
M odel
S -l
0.21
2.2
0.24
0.0079
0.082
0.01
2.4
60
S-2
0.62
4.6
0.24
0.0237
0.175
0.02
4.8
60
S-3
1.04
7.4
0.24
0.0396
0.281
0.02
4.8
60
SPM
PO
50.
S-l (s=0.0079)
S'
a
o
S-2 (s=0.0237)
40.
S-3 (s=0.0396)
•a
o
s
30.
20.
■§
a
s
10.
-
0.10
-0.05
0.00
0.05
0.10
Normalized H e ig h t (m eters)
Figure 3.2: The height d is trib u tio n o f the generated surfaces (dots) as compared w ith
Gaussian p ro b a b ility density functions (solid lines).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
1.0
S-l (1=0.082)
0.8
a
0
1
8
o
0
2
S-2 (1=0.175)
0.6
0 .4
1
73
0.2
'Z
0.0
-
0.2
0.0
0.1
0.2
0 .4
0 .3
0.6
0 .5
Displacem ent (meters)
Figure 3.3: The autocorrelations of the generated surfaces (dots) as compared w ith
Gaussian functions (solid lines).
3 .0
S-l (Snl=0.14)
-
S-2,3 (sm=0.19) -
2 .5
2.0
't/a
a
4>
Q
&
o
55
0 .5
0 0
' -1 .0
It n r l , r~, n flU T m
- 0 .8
- 0 .6
I------1------1------1------1------1___ i
- 0 .4
- 0 .2
0 .0
0 .2
m
0 .4
0 .6
k
0 .8
r -i__
1 .0
Slope, tan0
F igure 3.4: The slope d is trib u tio n of the generated surfaces (dots) as compared w ith
Gaussian p ro b a b ility density functions (solid lines).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
surfaces agree well w ith Gaussian functions o f (3.6) as shown in Fig.
3.3. Figure
3.4 shows the slope d istrib utio n s o f the generated surfaces compared w ith a Gaussian
density fu n ctio n given as
m2
/ M (m ) = —? L — exp
v27rsm
(3.11)
2s5L
where m is the slope (tan 9) and sm is the standard deviation o f slope d is trib u tio n (rms
slope) w hich can be obtained as sm = s2p//(0) (= y / 2 s / l for Gaussian correlation).
3.3
S o lu tio n by th e M e th o d o f M om ents
T he backscattering coefficient o f a computer-generated one-dimensional conduct­
ing surface can be obtained by N repeated com putation o f the electric field scattered
from each independent segment of a random surface as:
N
n
n
2
Y E n,pp
a
2'
,
pp = vv or hh,
(3.12)
n=l
where D is the w id th o f each segment o f the random surface, vv and hh denote tha t
both the in cid en t and scattered waves are V - and H-polarized, respectively. The
scattered field can be represented by the convolution o f the surface current density
J e and the Green’s function as follows:
E '(7 ) =
H ^ m p -r\)d r,
(3.13)
where k0 and Z 0 are the wave num ber and the in trin sic impedance o f free space,
respectively.
is the zeroth order Hankel function o f the firs t kin d , and "p and p 1
are the position vectors of observation and source points, respectively. The surface
current density J e(p*) due to the incident plane wave in (3.13) is to be determined
n um erically by the method of m om ent (M o M ) [Harrington, 1968].
hh-Polarization
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
z
HH-pol.
VV-pol.
Air
Conductor
Figure 3.5:
G eom etry of the scatter problem .
The electric fie ld integral equation (E F IE ) fo r one-dimensional conducting surface
can be w ritte n as
E ‘ (p) =
f Je(p') Ho^^kolp — p'\)dl',
4:
~p on interface
(3-14)
t/|
where the incident wave E *(p) is given by
E ’ (p) = Tjexp[ik0ki -p]
=
y E y%(p),
(3.15)
and an o rtho n orm a l coordinate system o f (hi, v,, k{) is defined by &,• = sin 0{X—cos 0,z,
hi = y, and t),- = cos 0,-x + sin0,£ as shown in Fig. 3.5.
The sim plest M o M solution of (3.14) consists o f using the pulse basis and p oint
m atching. A fte r discretizing a sample surface in to M ( = D / A x )cells, the pulse basis
fu n ctio n can be applied as:
M
J .M = E U P )j
71=1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3-16)
47
where
1,
on A x n
0,
on a ll other A x m.
(3.17)
In ( ? ) =
Then, (3.14) can be rearranged using dl' = y j l + (d z ( x ' ) / d x ' ) 2d x ',
M
[
4
71= 1
H ^ \ k o y f (X -
1
X n ) 2 + { z - 2 u ) 2) \
JAx„
\
+ I p - ) dx„
dxn
= exp[z&o(sin 0,x — cos 0,z)]
(3.18)
Since (3.18) can be matched every p o in t on the surface (x, z) = ( x i, z i), • ■•, ( x m, z m), •
•, ( x m , z m ), (3.18) can be casted in to a m a trix equation,
(3.19)
[ Z mn] [In] = [Vm],
where each element o f the impedance m a trix [ Z mn\ is given by
Zmn —
koZ0
■[
H q \ kQJ { x m - x n)2 + (zm - zn) 2)
J Ax„
dzn
1+
dxn
dxn,
(3.20)
the elements o f the excitation vector [Vm] are given by
vm = exp[f& o(sin0,xm - cos0,zm)],
(3.21)
and [Tn] is the surface current vector which is to be determ ined.
Since H o \ k 0p) has an integrable sin g u la rity for diagonal elements [ m = n ) of
the 2 -m a trix , sm all argument expansion of the Hankel fu n ctio n [H arrington, 1961] is
used to o bta in z„n,
X 2\
« a -
X2
-T
4 ) + 7
t
7 = 1.781 • • •.
(3.22)
Equation (3.20) can be evaluated a n a ly tic a lly using (3.22) for the diagonal element
as.
koZo A d
l + i 2 ln / fc07 A d \
7T
4e
kp(Ad)2 f 1 — 7
24
i
7r
In
&o7 A d
4 e1/ 3
(3.23)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
where A d = A x ny l + (dzn/ d x n)2 and e = 2.718 * • •. The non-diagonal elements, zmn,
can be obtained by evaluating the integral num erically, e.g., using the four-points
G aussian-Q uadrature integration technique.
Once the surface current vector [.Tn] is found, the hh-polarized scattered field, Eflh,
in the far-fie ld can be computed using the far-field approxim ation o f H o \ k 0p),
H<1)(k0\ p - p ' \ )
fo r large argum ent — >■\ —y —
V TTtfO/9
( 3. 24)
in to (3.13), w hich results in
s
^
( 3 -2 5 )
where ks = s in t^ x -f- cos 9sz, hs = y, and vs = — cos 6sx + sin 9az.
v v - P o la r iz a tio n
For the solution o f V V -p o la riza tio n case, the magnetic fie ld in teg ra l equation
(H F IE ) can be used to compute the surface current as follows:
- n x H*'(p) = - l j e( p ) + -l J n x { j e( ^ ) x V H ^ ( k 0\p - ? | ) } d l \
where p1 is on interface, and the incident m agnetic field is given as
n i { - p ) = y e ik^
= y H y.
Since
V H ^ { k a\ p - r \ ) = k0H ? \ k 0\ p - - f f \ ) R,
h x J e(p') x R = J e(p/) ( n ■R^j ,
fi
~P~~P'
\P ~ P \
n
i— —/i >
f + W
- h X H’Q?) = _■g + dxZ
f + i
*
—
)
Je**o(sm
h
Bix-cosOiz)^
if
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.26)
49
(3.26) can be rearranged as,
(-ft x H ' W ) p =
- \j.M
+^
(ft •R )
J j.A ? )
dz1
' \ 1 + I dx1 ) dx>'
P = * ’ y'
^3' 27^
W hen the pulse basis and p o in t m atching technique is used, (3.27) can be casted
in to m a trix equations as,
(3.28)
r = x,z.
[•Z»»l K l = M .
The element o f the impedance m a trix [ 2 mn] is computed by
f*mn + ^ 0 ^
■2’Tnn —
( j i - R j if i (1)(fc o \/(x m - x n)2 + (zm - zn)2)
(3.29)
\ + [p-\ix,
dxn
where 8mn is the Kronecker delta function. The excitatio n vector elements are given
as
1
X
. p ( s i n 9,"®m —cos
vl =
f +
M
«m)
z _
dr
m
i
'
and [Z*] and [J*] are the current vectors to be determ ined where i* = (dzm/ d x m) i * .
Once the surface currents are found sim ila rly as described in the H H -polarization
case, the scattered field can be com puted as
M
_
EV
SV(0S) = -
A i k n o —i t l
/ q I t :; eV(W- 7r/4) £
y/8irk0p
n=l
— cos
n
.
•
/]
d zn
+ sin 0S- —
u X jj
L
.e- ' kok,-pn£ Xn
1+
dxi^ni ^n)
dzn
d x nj
(3.30)
Suppression of Edge Contribution
In order to dem onstrate the edge effect, the surface current J e(p) is computed
for a fla t conducting surface o f the w id th o f 12 A at 0° incidence. W hile the surface
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
current induced by v-polarized incidence wave does n o t show any edge effect and
agrees w ith the PO (physical optics) current J e, =
2h x H ', the surface current
induced by h-polarized incidence wave shows the peak currents at the edges of the
surface as shown in Fig. 3.6 (a). Corresponding backscattered radar cross section
(RCS) of the conducting s trip is com puted fo r b o th hh- and vv-polarizations as shown
in Fig. 3.6 (b). For the sim u la tio n o f the random surface scattering, the surface is
assumed to have in fin ite w id th and to give no backscatter at 90° incidence.
To suppress the edge co n trib u tio n to the backscatter o f the fin ite conducting
surface, each end o f the surface is extended w ith a resistive sheet in the hh-polarization
case. Using the tra n s itio n conditions on the resistive sheet given by [Senior et al.,
1987]
[:h x E ] i = 0,
h x (n x E ) = —RJ,
(3.31)
where R is the re sis tiv ity o f the resistive sheet, eq. (3.14) can be revised fo r a resistive
sheet as
E ’ (p) = R ( p ) 3 e(p) +
J
H o \ k 0\p — pi\)dl',
p1 on interface.
(3.32)
Consequently, the element o f impedance m a trix in (3.20) is revised as
Zrnn = R ( x n , Zn ) 8 mn +
4
f
H ^ \ k 0\/(x J A xn
Xn) 2 +
(z
-
1+ (
Zn ) 2)
\ 8.xn-
(3.33)
\
T h e re sistivity profile, R ( x ) , is chosen s im ila rly as the profile used in [Leo et al.,
1993] as
0,
|x| < D l 2 - f d
0.005Zo ( ^ 2 l ) 4 ,
D / 2 - d > \ x \ < D/ 2
R(x) =
(3.34)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
(a)
3.
2.
-
h-pol.
• v-pol.
’ -8.
-6.
-4.
-2.
0.
2.
4.
6.
8.
Position (x), [ A ]
(b)
20.
B
m
S
10.
hh-pol.
■s
0.
vv-pol.
T3
O
X!
O
W
s
c5
a
M
%
m
-
20.
-30.
0.
10. 20.
30.
40.
50.
60.
70.
80. 90.
Incidence Angle (Degrees)
Figure 3.6: Backscattering from a fla t conducting s trip o f the w id th o f 14A, (a) the
current d is trib u tio n at 0° incidence, and (b) the backscatter echo w id th .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
where D is the to ta l w id th o f the surface and d is the w id th o f the resistive card. The
re s is tiv ity and the surface currents induced by h- and v-polarized incidence waves
are shown in Fig. 3.7 (a) in case o f D = 12A and d = 1A. Since the current d is tri­
b u tio n near resistive cards shows ripples, the backscatter RCS is computed ignoring
the surface current at each end (1A) o f the surface. In this case, the hh-polarized
backscatter RCS also decreases as the incidence angle increases, follow ing the vvpolarized backscatter RCS as shown in Fig. 3.7 (b).
3.4
N u m erica l R esu lts
The backscattering coefficient of a random surface w ith given roughness param e­
ters can be computed by a M onte Carlo m ethod as described above, i. t., the scattered
fields from N random ly generated segments o f the surface are com puted num erically,
and the backscattering coefficient o f the random surface is obtained from the statis­
tics o f the scattered fields. Each segment of the surface has the w id th o f D w ith an
extended region of D r and a resistive sheet o f the length
in Fig. 3.8. B oth
D
r
and
D
r
D
r
a t each end as shown
are chosen to be 1A considering the trade-off between
co m p uta tion tim e and edge efFect reduction. The re sistivity o f the resistive cards is
given in (3.34) and shown in Fig. 3.8. Even though currents on the whole regions are
com puted by the m ethod o f moments, the currents only on the region o f consideration
are used to compute the scattered field, ignoring those on the extended regions and
the resistive regions.
In order to test the num erical technique described above, the backscattering co­
efficients of the surface S -l and S-3 in Table 3.1 were com puted and compared w ith
the sm all pertu rb a tion m ethod (S P M ) solution and the physical optics (P O ) solution,
respectively. The roughness parameters of the surface S -l given in Table 3.1 satisfy
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
(a)
3.
>
•a
Vi
•
Vi
2,
<0
Pi
-
O
"O
3
• r*1
1.
current, h-pol. inc.
• current, v-pol. inc.
cbD
• Resistivity
C3
s
0.
*■*
c
£
3
U
1.
8.
-6.
-4.
-2.
0.
2.
4.
6.
8.
P osition(x), [A ,]
(b)
20.
-
hh-pol.
■ vv-pol.
o
-
10.
tS
Ui
<D
-30.
-40.
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
Figure 3.7: B ackscattering fro m a flat conducting s trip w ith resistive cards, (a) the
current d is trib u tio n at 0° incidence and re sistivity d is trib u tio n , and (b)
the backscatter echo w idth.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
z
-D /2-2X
-D /2- X
D /2
D /2 D/2+2. D/2+22.
Region o f consideration
— Extended regions
—
Regions with resistive cards
R(Resistivity)
I
I
I
V i
I
I
I
-I^ L
-D/2-2 X
Figure 3.8:
-D/2-X
D/2
D /2
D/2+2.
D /2 + 2 X
The extension o f the random surface w ith resistive cards.
10.
i
i
| i
j i j
i | i
i
i j
i
r
0.
02
P.
-
10.
- 20.
U
CQ
U
J3
M
02
SPM, vv-pol.
-30.
“-J3
SPM, hh-pol.
-40.
-50.
o
M oM , vv-pol.
0
M oM , hh-pol.
j
-60.
0.
10.
i
i
. i i i
20.
30.
40.
'\0
\0
i
i
50.
. i
60.
. i i
70.
lJj
80.
90.
Incidence Angle (Degrees)
Figure 3.9: The solution by the m ethod o f moments compared w ith the sm all per­
tu rb a tio n m ethod fo r the random surface, S -l, o f ks=0.21 and kl= 2 .2 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
20.
ca
2°e>
SBr
o
o
U
cS
o
8
8
CQ
1 i
1 i
1
i
1
i
1
,
i
-
10.
,
PO, vv- & hh-pol.
©
M o M , vv-pol.
0
M oM , hh-pol.
0.
-10. -20.
-30. -
i
°
© 'N
□
, i ,
70.
80.
-40. , i , i
-50.
0.
10.
............................
20.
30.
40.
50.
,
i
,
60.
(
T
1
1
90.
Incidence A n g le (Degrees)
Figure 3.10: The solution by the m ethod o f moments compared w ith the physical
optics solution fo r the random surface, S-3, o f ks=1.04 and kl=7.4.
10.
«
o.
°e
-10.
Vw'
SB
8
io0
- 20.
u
i
ks=0.6, kl=4.5
-30.
' 00© ,
J2
I
M o M , vv-pol.
-40.
M o M , hh-pol.
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence A n g le (Degrees)
Figure 3.11: The solution by the m ethod of moments for the random surface, S-2, of
ks=0.62 and kl=4.6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
the v a lid ity condition of the SPM which is ks < 0.3, k l < 3.0, and \ f 2 s / l < 0.3 fo r a
Gaussian correlation. The backscattering coefficients by the num erical sim ulation for
vv- and hA-polarizations show an excellent agreement w ith the solution of the SPM
as shown in Fig. 3.9, where the SPM model for a one-dimensional conducting random
surface is given in A ppendix A . The roughness of the surface S-3 in Table 3.1 is valid
for the PO model {ks < 6, \ / 2 s / l < 0.25 [U laby et al., 1986]). The num erical solution
fo r S-3 shows an excellent agreement w ith the PO solution at 9 < 85° as shown in Fig.
3.10, where the PO model is form ulated and evaluated e xactly for a one-dimensional
conducting surface as given in Chapter 2. Since the num erical solutions agree very
well w ith the theoretical models at tw o extrem e roughness conditions, we can apply
this num erical technique w ith confidence to the in term e dia te roughness conditions
which cannot be solved by existing classical theoretical models. The backscattering
coefficient fo r one o f such surfaces, S-2 in Table 3.1, is com puted and shown in Fig.
3.11.
f t is well known th a t the phase-difference statistics has valuable in fo rm a tio n in
add itio n to the m agnitude as shown in Chapter 5. T h is num erical technique is used to
com pute the co-polarized phase-difference statistics o f a random surface S-2 ( ks = 0.6,
k l = 4.5).
The co-polarized phase-difference angle <f)c = <j>hh — <f>w for a sm aller
incidence angle (20°) shows narrower and higher shape o f d is trib u tio n curve than
th a t fo r a larger incidence angle (50°) as shown in Figs.
3.12 (a) and (b).
The
standard deviation of the <j>c d is trib u tio n increases as the incidence angle increases
while the mean of the d istrib u tio n stays at zero as shown in Figs. 3.13 (a) and (b).
The degree o f correlation a and the coherent phase-difference ( are parameters o f the
phase-difference statistics defined in [Sarabandi, 1993]. As described in Chapter 5,
the degree of correlation a and the coherent phase-difference £ are the measures of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
(a)
40.
0 = 20'
t/5
O
o
c
Mean = -2.1
30.
§3
o
o
O
Dev. = 10.0‘
20.
o
o
.O
S
3
2
10.
0.
-80.-60.-40.-20. 0. 20. 40. 60. 80.
Phase Difference (Degrees)
(b)
0 = 50°
t/J
U
o
a
<u
o
u
O
lH
<
U
X5
e
3
Mean = -5.6°
30. -
Dev. = 27.1°
20. -
10.
2
0.
M
\
h n
-80.-60.-40.-20. 0. 20. 40. 60. 80.
Phase Difference (Degrees)
F igure 3.12: The d is trib u tio n o f the phase difference between a°hh and cr°„ o f the
surface, S-2, (ks=0.62, kl= 4 .6 ), (a) at 20° and (b) at 50° incidences.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
(a)
120.
CO
0)
£
b0
u
100.
kl = 4.5
G©'
80.
s
o
2
4)
Q
60.
40.
•a
20.
C/5
0. 10. 20. 30. 40. 50. 60. 70. 80. 90,
Incidence Angle (Degrees)
(b)
90.
ks = 0.6
60.
t/3
o
a
60
U
Q
%
a
s
kl = 4.5
30.
Po
-30.
-60.
-90.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
F igure 3.13: The d is trib u tio n o f the phase difference between a°hh and a°v o f the
surface, S-2, (ks=0.62, k l= 4 .6 ), (a) standard deviation and (b) mean
values.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
(a)
1.0
c
o
0.8
'M
8
o
U
0.6
0.4
a
0.2
0.0
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
(b)
90.
ks = 0.6
00
<0
Q
60.
u
o
c
30.
kl =4.5
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
Figure3.14: The statistics of the phase difference between <7l h and <7°^ of the surface, S-2, (ks=0.62, k l= 4 .6 ), (a) the degree o f correlation, a and (b) the
coherent phase-difference, (.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
(a)
1.0
c
0
1
8o
U
0.6
o
0.4
For ks=0.6,
kl=7.5
0.2
kl=4.5
kl=2.1
0.0
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
(b)
a
0
0.9
1
8
o
U
0.8
o
0.7
<D
00
<o
For kl=4.5,
0.6
0.5
k s = 0 .1 5
ks=0.6
ks=l .2
0.4
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
Figure 3.15: The degree of correlation, a, (a) for three different k l values at a fixed
value o f ks = 0.6, and (b) for three different ks values at a fixed value
o f k l = 4.5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the w id th (standard deviation) and the mean o f <j>c d is trib u tio n , respectively. The
degree of correlation a and the coherent phase-difference £ computed by the num erical
sim u la tio n fo r the surface S-2 are shown in Figs. 3.14 (a) and (b), respectively. The
degree o f correlation a depends on the roughness o f random surfaces as shown in
Figs. 3.15 (a) and (b). Figures 3.15 (a) and (b) show the k l and ks dependencies
o f the degree o f correlation for fixed ks and kl, respectively. Based on the num erical
results, the degree of correlation a seems to be a strong function o f the rms slope for
one-dim ensional conducting random surfaces.
3.5
E valu ation o f T h eo retica l S ca tterin g M od els
A t first, the form ulations o f the theoretical models are summarized for scattering
fro m a one-dimensional conducting random surface having a Gaussian correlation.
T he sm all p e rtu rb a tio n m ethod (SPM ), the physical optics (P O ) model, and the
geom etrical optics (G O ) m odel have been presented in the previous chapter for twodim ensional dielectric surfaces, and those w ill be m odified for one-dimensional con­
d u ctin g surfaces in this chapter. M any other scattering models have been presented
recently, in clu d in g phase p e rtu rb a tio n m ethod (P P M ) [W inebernner and Ishim aru,
1985a], full-w ave m ethod (F W M ) [Bahar, 1981], and integral equation m ethod (IE M )
[Fung and Pan, 1987].
Those models are also evaluated in this section.
Since a
one-dimensional random surface does not produce depolarization, the cross-polarized
backscattering coefficients are zero, a°w = vlh — o.
3.5.1
Small Perturbation Method
T he like-polarization backscattering coefficients by the first-order sm all p e rtu r­
b ation model (SPM ) have the form given in (2.31) for two-dim ensional surfaces.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
The backscattering coefficients crpp fo r the SPM are p ro portional to the roughness
spectrum w hich is the Fourier transform o f the surface correlation fu n ctio n and also
pro portional to the m agnitude square o f the far-field Green’s fu n ctio n given in (3.24)
and (2.44) fo r one- and two-dim ensional surfaces, respectively. Since the one- and
two-dim ensional surfaces correspond to the two- and three-dim ensional scattering
problems, respectively;
<x\d oc 2irp • —
• W u { 2 k sin 0)
oirkp
(3.35)
a l d oc 47rr2 • • W 2i{2ksm 0 , 0).
yi'Kry
Com paring W id(2 & sin 0) given in (3.9) and W 2d{2k sin 6 , 0) given in (2.34) fo r a Gaus­
sian correlation function,
c\d
7T
W i d( 2 ks m9 )
a\d
k
W u { 2 k s in 0 ,0)
^ y^
.
.
kl
Therefore, the backscattering coefficient for the one-dimensional conducting surface
can be given fro m (3.36) and (2.31) fo r a Gaussian correlation;
a0
pp = kypK (k s f k l cos4 6 \acpp\2 e- ^ ,s'm0^
(3.37)
where a pp is m odified for a conducting surface as,
« « (*) = - i
(3-38)
3.5.2
Physical Optics Model
The incoherent like-polarized backscattering coefficient o f the physical optics (P O )
m odel for a one-dimensional random surface can be obtained by m o d ifyin g the m odel
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
for a two-dim ensional surface given in (2.80). S im ila rly to (3.36), the ra tio between
cr\d and a^d for the PO model is
Z li- 1 .
"u
^ n W j d)( 2 k s m 6 )
k ' E nW ^\2ksm e,oy
where W^d\ 2 k sin 6 , 0) is the Fourier transform o f the n th powered norm alized corre­
la tio n fun ctio n for a two-dim ensional surface and given as Ig in (2.77).
sin 0)
is the Fourier tra nsfo rm o f pn(£) f ° r a one-dimensional surface,
Wi?{2ksm6) =
(3.40)
Vn
Therefore, the backscattering coefficients fo r a one-dimensional conducting surface
w ith the Of/i-order approxim ation can be obtained as
co,’ * «-<“ ■“ «’ £
P -« )
when a Gaussian correlation function is assumed.
The first-o rd e r approxim ated
w
backscattering coefficient for the PO model is given by (2.73)
a°
(0) = <TpPu,} ° J ■
PPoth' >
cos 0
3.5.3
(3.42)
Phase Perturbation Method
Using a p e rtu rb a tio n expansion of the surface field phase fro m rough surfaces,
the phase p ertu rb a tion method (P P M ) was developed in the case o f scalar wave
scattering from surfaces for which D irich let boundary conditions hold [W inebernner
and Ishim aru, 1985a and 1985b]. The backscattering coefficient fo r a one-dimensional
surface is expressed as [Broschat et al, 1987],
cr° = kcos29 exp[—2Re[lV2]]
f
exp[i2k x sin0](exp[A rn(a;)] — 1) dx
(3.43)
J —OO
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64
w ith
JV2 = 2k2 cos 6 J ° ° W { U ) P
/
oo
--~k— — j dU,
. U + k sin 0 ' 2
W { U ) e x p [ i U x ] cos 0 + /3 ( — — -------dU,
•C O
where W ( U ) is a roughness spectrum for one-dimensional random surfaces given in
[Broschat et al, 1987] as
U 2l 2'
w(u)= ^
The P P M does not include polarization dependence, and above fo rm u la tio n is
equivalent to electrom agnetic wave scattering from perfect conducting rough surfaces
fo r /i/i-p o lariza tion . Calculations o f the backscattering coefficient using (3.43) involve
a double integral having in fin ite lim its o f integration, where the integrand fluctuates
by being non-oscillatory and h ig h ly oscillatory [Broschat et al, 1987].
It has been claimed th a t the P P M reduces to the two classical models, nam ely
the S PM and the PO in the appropriate lim its and sm oothly interpolates between
the SPM and the PO [Ivanova et al, 1990; Broschat, 1987].
3.5.4
Full Wave Method
The “ full-wave” m ethod (F W M ) was developed fo r random surface scattering by
Bahar [Bahar, 1981; Bahar, 1991a; and Bahar,1991b].
Even though the F W M is
fo rm a lly exact, approxim ations are necessary to o btain results for rough surface scat­
terin g since the general form includes ten-fold num erical integrals fo r tw o dimensional
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
surfaces. The scattering coefficient for a one-dimensional perfect conducting Gaussian
surface is given as [Bahar, 1991b],
a°pp = 2kG%2 j f
[l - ££
{ x z K i - v y) [ l + G f 2(m 2A - u2
y B 2)]
w ith
(3.45)
X iy v) = exp
(3.46)
where s is the rm s height, m is the rms slope, vx = —2k sin 0, vy = 2k cos 0, and
Go" = cos 9, G ? = tan 6 for the h orizon ta lly polarized backscatter.
Bahar has shown a n a lytica lly th a t the F W M can reduce to both of the SPM
and the PO when appropriate conditions are imposed [Bahar, 1991a; Bahar,1991b].
Thoros and W inebernner, however, claim ed th a t the F W M does not reduce to the
SPM in th e ir exam inations [Thoros and W inebernner, 1991].
3.5.5
Integral Equation Method
T he integral equation method (IE M ) was developed based on an approxim ate so­
lu tio n o f a p a ir o f integral equations for the tangential surface fields [Fung et al., 1992].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
Backscattering coefficients for a rough dielectric surface are given in a quite lengthy
form for each case o f ks < 3 and ks > 3 in [Fung et al., 1992]. The backscattering
coefficients for a p erfe ctly conducting rough surface are given in [Fung and Pan, 1987]
for a two-dim ensional random surface. For a one-dimensional surface, a°v and cr°lh
can be m odified as
hh
=
«
cos2 8
n ly /n
" V
c o s tT
(3.47)
| 2 2ti-2 e xp(—2k 2s2 cos2 8) ± 2n sin2 8 e x p (—k2s2 cos2 8) + sin4
.
I f a Gaussian correlation function is assumed, W i^ (2 & s in 0 ) can be replaced by (3.40)
as
(348)
hh
COS20
n ly /n
^2n e xp(—k?s2 cos2 8) ± 2 sin4
.
I t was claimed th a t the IE M reduces to the SPM and the PO fo r the low-frequency
lim it and the high-frequency lim it, respectively [Fung and Pan, 1987 and Fung et al.,
1992].
3.5.6
Numerical Results
T he v a lid ity regions o f the scattering models and three roughness conditions given
in Table 3.1 are illu s tra te d in Fig. 3.16. The v a lid ity regions of the phase p ertu rb a tion
m ethod (P P M ), the full-w ave method (F W M ), and the in teg ra l equation method
(IE M ) need to be examined in detail; however, each o f these models is assumed
to meet the v a lid ity conditions o f the small p e rtu rb a tio n m ethod (S P M ) and the
physical optics approxim ation (P O ), as well as the interm ediate roughnesses between
the v a lid ity regions of the SPM and the PO m odel [Bahar, 1991; Broschat et al., 1987;
and Fung and Pan, 1987].
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67
O : Model Testing Points
IEM
2
PPM
ks
FWM
l
SPM
o
o
PO
2
6
4
8
kl
Figure 3.16:
The v a lid ity regions of the scattering models.
T he backscattering coefficients o f vv- and M -p o la riza tio n s fo r a p erfe ctly con­
d uctin g rough surface are computed by the SPM , the PO, and the IE M at three
roughness conditions as indicated (o) in Figure 3.16 and those values are compared
w ith the solution fro m the m ethod o f moments as shown in Figs. 3.17(a)-(c). Since
the P P M provides only the A/i-polarized solution [W inebernner and Ishim a ru , 1985]
and the only hh-polarized scattering am plitude o f the F W M is given e x p lic itly in
B ahar [1991], the M -p o la rize d backscattering coefficient for a p e rfe ctly conducting
rough surface is com puted by the P P M and the F W M and compared w ith those of
the SPM , the PO, and the IE M as in Figs. 3.18(a)-(c).
3.6
C onclu sion s
A M onte Carlo m ethod in conjunction w ith the m ethod o f m om ents (M o M ) is
applied to obtain an exact solution o f scattering fro m a one-dimensional conducting
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68
random surface. A new approach is introduced in this m ethod to elim inate the edge
effect which results from the num erical sim ulation o f scattering from a random surface
of fin ite w id th . B y adding a resistive sheet at each end of an illum ina te d random
surface, the edge effect can be elim inated even at large incidence angles less than 85°.
The num erical solution w ith th is technique agrees very well w ith existing theoretical
models, SPM and PO, at th e ir v a lid ity regions. In a dd itio n to the m agnitude, the
phase-difference statistics was computed by this technique, and it is shown th a t the
degree o f correlation a shows strong dependency not only on the incidence angle
b ut also on the roughness of the surface. This num erical technique has been used
to evaluate existing models o f SPM , PO, P PM , F W M , and IE M for scattering from
one-dimensional conducting random surfaces.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
Backscattering Coeff. (dB)
(a)
ks=0.21
kl=2.15
M o M , VV -pol.
M o M , HH-pol.
SPM, W - p o l.
SPM, HH-pol.
PO, Oth-order
PO, lst-order
IE M , V V -p ol
---------------
0.
10.
IE M , HH-pol.
20.
30.
40.
50.
60.
Incidence Angle (Degrees)
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70
Backscattering Coeff. (dB)
(b)
MoM, VV-pol.
MoM, HH-pol.
SPM, VV-pol.
SPM, HH-pol.
PO, Oth-order
PO, 1st-order
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
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71
20.
(c)
i— |— i— |— i— |— i— |— .— |— i— |— .— |ks=1.04
kl=7.4
Backscattering Coeff. (dB)
10.
•
MoM, VV-pol.
■
MoM, HH-pol.
------------
SPM, VV-pol.
................
SPM, HH-pol.
--------------
PO, Oth-ordcr
-------
po, lst-ordcr
0.
\
-
''V
\
10.
\ *
\ \
-
------------- IEM, VV-pol.
\ \\ TA
20 .
\
--------------
.
V
\ \N\tv *
-30.
\
-40.
■
IEM, HH-pol.
*
_
s
\ ^
\ \
■ I
.
i
i
70.
80.
■ •
\ \
LLLi
-50.
20.
30.
I
40.
i
I
'
■\ i
50.
%
60.
v
Incidence Angle (Degrees)
F igure 3.17: Comparison o f models w ith an exact num erical solution fo r (a) ks = 0.21
and k l = 2.2, (b ) ks = 0.62 and k l = 4.6, and (c) ks = 1.04 and k l = 7.4
for b oth o f vv- and /i/i-polarizations.
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72
(a)
Backscattering Coeff. (dB)
- 10.
HH-pol.
- 20 .
ks=0.21
kl=2.2
M oM
-30.
SPM
• PO, Oth-order
PO, lst-order
-40.
»_
\
*» \
IE M
• «s
PPM
FW M
-50.
0.
10.
20 .
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
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73
(b)
- 10.
H H-pol.
ks=0.62
kl=4.6
- 20 .
M oM
SPM
-30.
PO, Oth-order
PO, lst-order
IE M
-40.
PPM
FW M
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
(c)
20.
HH -pol.
ks=1.04
10.
k l= 7 .4
Backscattering Coeff. (dB)
MoM
SPM
0.
• PO, 0th-order
■ PO, lst-order
- 10.
IE M
PPM
- 20 .
FW M
-30.
-40.
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 3.18: Comparison of models w ith an exact num erical solution for (a) ks = 0.21
and k l — 2.2, (b) ks = 0.62 and k l — 4.6, and (c) ks = 1.04 and k l = 7.4
for /i/t-polarization.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A P T E R IV
A N IM PROVEM ENT OF PHYSICAL OPTICS
MODEL
4.1
In tro d u ctio n
In Chapter 2 the KirchhofF approxim ation in the com putation o f the radar backscat­
te rin g coefficient was introduced briefly. The scattered fields fro m a random ly rough
surface can be form ulated exactly in terms o f surface currents (or equivalent tangential
surface fields). Since i t is very d iffic u lt i f not im possible to com pute the exact form of
tangential fields on the random surface, we have used the tangent plane approxim a­
tio n (o r the KirchhofF approxim ation) i.e., the surface fields at any p o in t o f the surface
are represented by the fields computed by approxim ating the boundary interface as
a tangent plane at th a t p oint. Then, the backscattered mean in tensity, from which
the backscattering coefficients can be com puted, can be form ulated in term s o f the
ensemble average of the conjugate-m ultiplied scattering am plitudes. The scattering
am p litud e is a fu n ctio n o f the local surface slopes and the local reflection coefficients
where the local reflection coefficients are also the function of the local surface slopes.
In order to avoid the co m p lexity o f the a nalytical com putation o f the ensemble average
te rm , the scattering a m p litud e has been expanded in a series as a fun ctio n of surface
slope, and only the firs t few term s have been kept to produce the physical optics (PO)
75
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model. W hen only the firs t term is kept and all other slope term s are ignored, the
zeroth-order approxim ated PO model is obtained as shown in the last chapter. On
the other hand, when one more a dditional te rm (first-order slope te rm ) is kept, and
the edge effect term [Beckmann and Spizzichino, 1963; Beckm ann, 1968] produced
by the integration by parts is ignored, the first-o rd e r approxim ated PO model is ob­
tained. For both o f these PO models, the integration lim its fo r the tangent plane
have been approxim ated by in fin ity to obtained closed form o f the models. Instead
o f using the series expansion o f the scattering am plitudes, the stationary-phase ap­
p ro xim a tio n can be used in the geom etric lim it, ignoring th e local d iffractio n effect.
T his approxim ation w ith an assumption o f large value of ks cos 0 (where k = 27t/A,
s is rm s height) leads to the geometrical optics (G O ) model. The mean in ten sity has
been form ulated exactly, w ith o u t any fu rth e r approxim ation, using a spectral rep­
resentation o f the d elta fun ctio n and assuming a Gaussian height d istrib u tio n , and
evaluated approxim ately in case o f a very rough surface [Stogryn, 1967; Holzer and
Sung, 1978].
T he goal o f this chapter is to evaluate e xactly the backseattered mean intensity
w ith o u t using any fu rth e r approxim ation except the tangent plane approxim ation so
th a t the exact K irch h o ff solution can be used to examine the zeroth- and first-order
approxim ated PO models. In other words, the slope term effect and the edge term ef­
fect (Ch. 2) can be exam ined using the exact K irc h h o ff solution. The exact K irchhoff
solution m ay also be used to examine other theoretical models and num erical solu­
tions. In th is chapter, the backscattered mean in ten sity is form ulated exactly for a
tw o-dim ensional dielectric random surface and evaluated e xactly for one-dimensional
d ie lectric and conducting surfaces. The exact KirchhofF solution is compared w ith a
m ethod of moments solution for scattering fro m a one-dimensional conducting surface,
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77
Exact Scattered Field Formulation
Tangent Plane Approximation
(Kirchhoff Approximation)
Ensemble Average of Backscattered Intensity
Approximation by
First Few Terms of
Series Represented
Scattering Amplitude
Zeroth-order
Approximated
Physical Optics
(PO) Model
First-order
Approximated
Physical Optics
(PO) model
Comparison
Spectral
Represent.
Assuming
very rough
First-order
Slope Term,
Ignore
Edge Term,
No Slope
Term
Figure 4.1:
StationaryPhase
Approx.
Geometrical
Optics (GO)
Model
Assuming
Gaussian
Height
Distribution
Exact Formulation
Exact
Evaluation
Approximate
Evaluation
Illu s tra tio n o f the development of an exact K irc h h o ff solution.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
as well as w ith the zeroth- and first-order approxim ated PO models. The K irchhoff
approach previously studied is illu stra te d by solid boxes and lines, and the work to
be studied in this chapter is indicated in dotted box and lines in Fig. 4.1.
4.2
Form ulation for a T w o-d im en sion al D ie le c tr ic Surface
The backscattered mean in te n sity for the K irchhoff a pp roxim ation has been given
in (2.61) and w ill be presented here again for convenience;
(F 6sai C > = IA ) |2/ J^l dxidyi
J f_ h dx2dy2 e * * <«.-«*)
• { f ba(Z x l , Z J K a{Z X2, Z y2) eik^ “ ) ,
(4.1)
w ith f ba(Z x , Z y) given in (2.56-2.58) where
d z(x u yi)
d z ( x 2, y 2)
’ 12“
8x
dx
d z { x l , y 1)
’
3/1”
dy
d z { x 2, y 2)
’
Vi ~
dy
’
Zd = Z\ — Z2, Z\ = z { x i , y i ) , z2 = z ( x 2, y 2), kdx = 2&o sin0, kdz = —2k0 cosO, and
\D q\2 = fcg/(47rr)2. Since the integral o f the ensemble average te rm in (4.1) is very
d iffic u lt to evaluate, additional approxim ations were used to get the PO and GO
models as summarized in Ch.
2. The ensemble average te rm o f the integrand in
(4.1), however, can be evaluated exactly by using the spectral representation of the
delta fun ctio n and the siftin g p ro pe rty o f the delta fu n ctio n in teg ra tio n. Since the
delta function can be represented by
* (x ’ y) = ( 2 hj> I
J1
e<ax eWy dad/3’
(4,2)
the scattering am plitude can be w ritte n in terms of the dum m y variables a and /? as
M Z „Z ,) =
J J
fba(Px , P ,)6 ( f t - f t , f t - Z„) I i f t f t f t
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79
In the above equation, fba(Zx, Z y) is replaced by fba(Px,Py) w hich is not a fun ctio n of
random variables anymore. A p p lyin g above relations to (4.1), the ensemble average
te rm can be represented by
=
(2 jr)4 /
■[J Jf—
J J
/
dceXld a X2
00
[J J[
, fly,
d a y iday2eic,x^
Py-j)
+ia^ p^ ■
~00
.
^ * 1 *~, a f S/l ^1 /1
. g * Qr* 2 ^ x 2 ^ ”,0 fS'2 ^ 2
. ^ ^ d x z d^j
^4.4)
W hen we assume the random surface o f this problem has a Gaussian height d is tri­
b u tio n , the ensemble average term , ( • • • ) , is a characteristic fun ctio n for a random
vector, x = [ZXJ, ZX2, Z y i, Z y2, z j] T, and a param eter vector, uJ = [—a Sl, a X2, —a y i,
a y2, kdz]T . In fact, the most n a tu ra l rough surfaces have Gaussian height d is trib u ­
tio n .
A ty p ic a l exam ple o f surface height d istrib utio n s w hich were measured fro m
bare soil surfaces is shown in Fig. 4.2. Figure 4.2 includes a large num ber of points
( > 8000) measured by a laser profile m eter and shows an excellent agreement between
the measured height d is trib u tio n and the Gaussian p ro b a b ility fun ctio n for the same
standard deviation o f 1.12cm.
T h e characteristic fun ctio n o f the ensemble average form , (• • •), fo r the Gaussian
random vector can be com puted using the param eter vector and the correlation m a­
tr ix , where the correlation m a trix components are functions o f a correlation coefficient
and its derivatives as shown in A p p e n d ix A . B y changing in teg ra l variables fro m d x i
dyi to du dv (where u = x i — x 2, v = ?/i — j/ 2 ) and in te g ra tin g over dx2 dy2, the
follo w in g in te g ra l id e n tity can be obtained;
JJ
d x \d x 2
j J
dy\dy2 = $ -
JJ
(2L — |u|) (2L — |u|) dudv,
(4.5)
Using the characteristic fun ctio n and the in teg ra l id e ntity, the backscattered mean
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80
0.6
Measured Dist.
0.5
E
Gaussian PDF
e
0.4
6
.■
'wHr
0.3
G
&
&
0.2
S
£>
O
Ui
Dh
0.1
c£
-3.
0.0
-4.
-2.
-1.
0.
1.
2.
3.
4.
Height Distribution (cm)
F igure 4.2: A typ ica l example o f surface height d istrib u tio n s measured fro m natural
rough surfaces.
intensity, (|.££a|2), in (4.1) can be w ritte n in a ten-fold integral equation o f w hich the
integrand is an algebraic equation w ith o u t ensemble averaging term s given as
< \K .\2 > =
•
/ f ^ d u d v ( 2 L - |u |)(2 £ - | „ | ) e * -
[r
J
lr
J —00
■s
J J —oo
(4-6)
w ith
S=/r
daXlda X2 j [° ° day id a y2eiax^
J J—OO
■eiax^ z' +ia^
+io^
j J —oo
•exp [ y / > Uu(0,0) { <
+ <
+ a 2, + a 2J - <r2A£ {1 - /> (u ,v)}
•exp [—a 2 { a Xia X2pnu( u , v ) + OiyiOLy2puv{ u , v ) + (a Xla y2 + a X2a yi) p „ „ ( u ,u ) } j
• e x p
\ - a r 2kdz {
( q
, j
-
a X2) pu(u,v)
+
( « y ,
~
«
»
)
/> * ,( « > u
) } ]
•
( 4 -7 )
T he term s involving th e cross correlation fu n ctio n pUv{u, v) have been ignored
in [Holzer and Sung, 1978]; however, puv( u , v ) is not sm all enough to be ignored as
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81
8.
P (u .v )
7.
s>
•s
>
•c
<u
Q
a
•a
s
ra
CL)
o
6,
- Puu ( u -v )
5.
Puv ( u ' v >
4.
3.
2.
a)
1.
U
0.
ofc
1
■2 .
0.0
0.2
0.4
0.6
0 .8
1.0
1.2
1.4
1.6
1.8
2.0
Normalized Displacement in Corr. Length
F igure 4.3: Comparison between a Gaussian correlation coefficient and its derivatives
in case o f / = 0.5m.
illu s tra te d in Fig. 4.3.
In Fig.
4.3, a Gaussian correlation coefficient function is
considered as an exam ple w ith the correlation length o f I = 0.5m and the magnitudes
o f p ( u , v ) and its derivatives, pu(u ,v ), puu( u , v ) , and pUv {u ,v ), are compared among
one another.
In order to integrate the last term S a na lytica lly, it w ill be b e tte r to rearrange it
as follows;
S = exp [ - c r 2k l t {1 - p(u, v)}J • I t l I i ,
(4.8)
w ith
I ai = r
da XteiPxi “ xi . e ^ p““ (0’0)“ ' i • e~a2k^Pu(u,v)axi
1
J-oo
^
?
(4.9)
2
where
•-aj.
—
[
e - i'0x2<»x2 .
e * T -/> u u (
0,0)
a l
2 .
e - a
2p„„(u.vjor*,a I2
J—oo
„<r2kdzpu('u,v)aX2 j
e
X " ! /l
>
(4.10)
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82
where
^
J—00
d a y iei0»*ayi • e^ uu(0'° K i •
Iq ^ 5
(4 1 1 )
and where
_
/°° Jq, e-'0V2aV2 . gir / ’<iu(0.0)“ y2 . g-^fpiiuCu.vjQyj+PuvKwJojJajjj
/
^2
J —oo
. e»2*-drPu(u,w)oy2_
(4-12)
Since a ll o f the above integrals in (4.9)-(4.12) have the same fo rm o f
f°°
A
AJ
iB x
- C x 2- D x
e,axe Ux
7
A
F*
d2
— b~ .
dx = A y — e *c
-{M S .
.e
the integrals I axi, I Qj2, I ayi, and I 0j/2 can be computed consequently by backward
su bstitu tion s. A fte r a q uite com plicated algebraic com putation, th e final form o f S
in (4.8) can be obtained as
S = _____
sY o^
(2?r) 2
7
b
^
e-P2(*dr)2(l-p) . e[P2(2*ir)V„]/[po(l+Pfl)J
7
e
.eb2( ^ ) 2(p2
u- f ^ ) 2]/[Po(l+Pfl)] , e-[/321 +/922 -2pB/3y1 /?a2 ]/[2po*2 (l-P 2
B)l
2
Px2-
-PBPy2)
/[2P°j2] .
e - i [ 2 k U P«(Py i +l3y2) ] / [ p0( l + p B )]
*12
•e
e
Pxl - P E P x 2PDPy 2+ ' ^ F { P y i - PBPy2 )( P B + P E )
/[2pos2 ( l - P B)l
a
—i2kj Z(Pu~i2pg) \P*\ +0*2 - P d Pv2 +
L
■{Pyj ~PBpy2) ( l —PB) / [ p o ( i
B
•
+
p b
)1
(4.13)
where m ore sim plified notations have been defined as pu = d p ( u , v ) / d u , puu =
"ail*’”* »
= -P«tt(0,0) PB = P n n ( u , v ) / p uu( 0 ,0 ), pD = /»««(«, v ) / p uu( 0 , 0), PE =
P w ( u , v ) / p uu( 0 , 0), and s = er. Above fo rm u la tio n is good fo r any typ e o f correlation
fu n ctio n w hich has derivatives throughout the coordinate system.
Once we found the mean in te n sity < |-Ej0|2 > in (4.6) w ith S given in (4.13),
the backscattering coefficient cr£a can be computed using (2.79) for a two-dim ensional
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83
die lectric random surface. The exact fo rm u la tio n o f the K irch h o ff approxim ated mean
in te n s ity in (4.6), however, is in the form o f six-fold integrals and should be solved
num erically. Since the six-fold num erical in te g ra tio n is a challenging problem w ith the
present com puter powers, we can reduce the co m p uta tion either by approxim ating
the integrand in (4.6) and (4.13) o r by s im p lify in g the problem itself.
Since the
goal o f this chapter is to get an exact solution o f the backscattering coefficient for
e xam ination o f the zeroth- and first-order PO models (o r the effects o f the slope term
and the edge te rm ), a sim ple problem o f a one-dim ensional random surface w ill be
considered in th e n ext section.
4.3
E v a lu a tio n for a O n e-d im en sio n a l D ielectric Surface
A n exact fo rm u la tio n fo r the K irch h o ff approxim ated backscattering coefficient
fo r a one-dimensional d ie lectric random surface w ill be derived and also solved nu­
m e ric a lly in this section. For a one-dimensional random surface (no surface height
va ria tion in y d ire ction , i.e. Z y = 0), the scattered fie ld can be obtained sim ila rly
as the problem o f a two-dim ensional surface given in (2.40) except th a t the Green’s
fu n c tio n for a tw o-dim ension scattering problem (corresponding to a one-dimensional
random surface) is given by
G o i r y ) = l- H ^ ( k r ) ,
(4.14)
where the position vector r is employed instead o f p fo r a one-dimensional surface
(tw o-dim ensional scattering problem ) in order to avoid confusion between the position
vector and the co rre la tion coefficient fu n ctio n p(u). The firs t-k in d Hankel function of
order zero, H q \ can be approxim ated in the fa r field ( r > 2 D 2/ X q) as
H ^(k r) w
e- iko‘k’ -?'.
y 7C/CTq
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(4.15)
Therefore, the corresponding backscattered mean in te n sity can be computed as
( |£ ? ,|2) = P o l2 / “
(2L - H i e ' * " ” ( h . ( Z n )
e * * - ) in ,
(4.16)
J—
where |.Do|2 = ko/(8irro). The scattering am plitude fba( Z x) can be obtained by sub­
s titu tin g Z y = 0 in to (2.56)-(2.58) as
f hh = 2R ih ( Z x , 0)(cos 9 - f sin 9ZX)
f vv = 2Riv ( Zx , 0)(cos 6 + sin 6ZX)
U
= fhv = 0
(4.17)
w ith
Zx sin 9 + cos 9 — \Jer { \ + Z£) — (Z x cos0 — s in 0 )2
=
R lv
Zx sin 9 + cos 9 + ^ e r ( l -f Z 2) — (Z x cos0 — sin Q)2
er (Z x sin 9 + cos 9) — J c T{ 1 + ZJ) — (Z x cos 9 — s in 0 )2
.......
........ —
— ---------------------------------- .
eT(Z x sin 9 + cos 9) + \J er ( l + Z 2) — (Z x cos 9 — sin 9)2
(4. 18)
No cross-polarized backscattering coefficient can be obtained fo r a one-dimensional
surface as indicated in (4.17). W hen the spectral representation o f the delta fun ctio n
is used, the scattering a m plitude and its conjugate can be pulled out o f the ensemble
average te rm as
= ( j I 5 5 / /_ “
•J
j° °
d a xd a 2
/..(A )/:.(A )
• e‘ “ 2^
• e ik i' Z i) .
(4.19)
T h e te rm (• • •) at the rig h t hand side o f (4.19) is the characteristic fun ctio n o f the
Gaussian random vector x = [ZXl 1 Zx21Zd]T and the param eter vector a; = [—au, a 2,
kdz] T- The characteristic fun ctio n for a one-dimensional surface is a q uite sim ple form
compared w ith th a t for a two-dim ensional surface and is represented as a fun ctio n
o f the param eter vector components, the correlation coefficient and its derivatives
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85
as shown in A ppendix A. The backscattering mean in te n sity for a one-dimensional
die lectric random surface is
< \K .? > =
ID P
tA ;
..
t
t° °
- M K ‘ " “ ■J J _ J P
(2ir)
/ - ( A ) / : . ( * ) • S (4.20)
w ith
(4.21)
where
/
OO
dot\ exp [i/?iO i] • exp
a
/n \
2
■y/JuuCO)^
exp [- c r 2^ s/)u( u ) a i] • I Q2, (4.22)
•OO
and
-r
102 =
I
d a 2 exp [ - i/ ? 20 !2 ] • exp
0 Puu(0)tt2
J —c
exp
exp [-o -2puu( u ) a ia 2]
.
(4.23)
A fte r in teg ra tin g I Ql and I Q2 a nalytically, the backscattered mean in te n sity is obtained
as,
I2
/-2L
(l^ a a l2) = 2 ns*poj - 2 L dU (2jL _ lWD ‘ eXP
‘ exP
° 2kizP l
.Po(l + Pb ).
• exp
1
•
-f= —
Ji -
/?1 "b @2 ~ 2PB 0102
2p0s2( l - p j )
J
* eXP [ ~
r oo
/
•/-°°
~
/J(W) } ]
roo
dP1 f aa(0l)
/
d ft £ „ ( & )
■kdzPu(01 + 02)
GXP [
p0( l +
(4.24)
Pb )
where pu = d p ( u ) /d u , p0 = - p uu{0), pB = pu«(n)//9uu(0), and s = a .
T he effect of shadowing can be p roperly accounted fo r the mean in te n sity in (4.24)
ju s t by re strictin g the lim its o f in teg ra tio n w ith respect to d0j and d02 corresponding
to th e surface slopes.
In the backscattering case, m a xim u m slope can be in fin ity
0max = oo; however, the m in im u m slope should be restricted by the incidence angle
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86
M ax. Slope
M in . Slope
Slope =
Figure 4.4:
Surface Facet
Illu s tra tio n o f the shadowing correction in backscattering direction.
Oi as /3m{n = — cot Qi to get proper orientations o f surface facets as illu stra te d in
Fig. 4.4. Therefore, the in tegration in (4.24) w ith respect to d/3i and d/32 should be
restricted for the shadowing correction as
/
oo
roo
/
r oo
F tfu P iW fr d P i = » /
-oo J —oo
roo
/
^ ( A i f t ) M .
J —cot QiJ —cot $%
Since the incoherent backscattering coefficient fo r a one-dimensional random sur­
face is defined by
( m 2)
„
<r‘ " = ,* !£ l L
'
-
---------------------
’
(4 ' 25)
where |(i? ia)|2 is the coherent in ten sity and \E'af is the incident in te n sity given as 1.
The coherent scattered field (E%a) can be obtained from (2.54) as
[L
(E "a) = D 0
J —L
dx exp [ikdxx\ • {f aa{Z x) exp [ikdzz])
(4.26)
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87
Using the spectral representation o f the delta fun ctio n , the ensemble average te rm at
the rig h t-h an d side o f (4.25) becomes
1
1
(faa(Zx) exp [ik dzz]) = —
fr°O°O
A lt J —OO
r co
d/3 I
da exp [m/3]
J —C O
• (e x p [—i a Z x] exp [ikdzz\ ) .
(4.27)
For a Gaussian height d is trib u tio n , the term (exp [—i a Z x] exp [ikdzz]) is the charac­
te ris tic function fo r the random vector x = [Zx , z]T and the param eter vector uJ =
[—a , kdz]T. Since the components o f the correlation m a trix are computed as
(• Z j) = ~cr2puu(0),
(Z x z) = 0,
and
( z 2} = cr2),
(4.28)
the characteristic fu n ctio n can be com puted as follow ;
/
-P u u { 0)
(exp [—i a Z x] exp [ikdzz]) = exp
Zd
—a
\
( a , kdz)
V
= exp
0
y
0
^ ^dz j
( - a 2Ptt«(0) + k d
2z)
(4.29)
In te g ra tin g w ith respect to da, the coherent in te n s ity can be rearranged as
— 2 ~kdz J L dx exp [ik dxx]
/
oo
d/3 faa(P)exp
• cot 9
2s2puu(0)
(4.30)
T h e backscattering coefficient o f the K irc h h o ff m ethod has been evaluated exactly
fo r the one-dimensional die lectric random surface having a Gaussian height d is trib u ­
tio n and an a rb itra ry correlation coefficient function. The backscattering coefficient
can be evaluated by in te g ra tin g three-fold integral given in (4.24), and the num erical
results w ill be given in Sec. (3.5).
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88
4.4
E valuation for a O n e-d im en sion al C o n d u ctin g Surface
In the last chapter a num erical solution fo r scattering fro m a one-dimensional
conducting random surface has been developed. In order to compare the exact PO
solution described in this chapter w ith the num erical solution, we w ould need the
evaluation of the exact PO solution fo r a one-dimensional conducting surface. For
a conducting surface, the mean intensity can be fu rth e r sim p lified fro m (4.24) since
the local reflection coefficients are constant (R h = —1, Rv =
1).
T he scattering
am plitudes can be sim plified from (4.17) fo r a one-dimensional conducting surface,
and the backscattered mean intensity can be re w ritte n as
|A>|2 r2L
(l^ a a l2) = 2— °T ^ J_2L du ( 21 - M ) • eXP \i k d M ■eXP [ ~ s2kdz U - p (u ) } ]
exp
° 2k l p l
po(l +
Pb
).
I,■ft
(4.31)
w ith
LPl
/
= ~7===f
y /i - p i
oo
r
*pi u m
■ L02
PI + P I - 2 P B P M
faa ( 1^ 2 ) - e x p
2p0s2( l — Pb )
•00
(4.32)
and
• 6Xp
;
kdzPu{Pl
+
P 2)
P o {^ + P b )
.
,(4.33)
where
fhh(Pi) = -2 (c o s 9 + sin 9Pi)
f w ( P i) = 2(cos 0 + sin 0(5i)
U (P i) = M P i ) =
(4.34)
0.
The integrals, I p2 and 1 ^ , can be integrated a na lytically, and the results are, respec­
tive ly,
I/?2«. = ± 2 ^ 2 i r p 0\ J l - p%s exp
.
exp —i
kdzpupB$ 1
P lft
2p0s2( l - p%)
s2( l - pB ) { k dz puf
2/5q(1 + P b )
[sin0 { p b P i - is 2( 1 - pB)kdzPu} + cosfl] ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.35)
89
and
Ip, «v = ( ± 2 ) 227T/?0s2 exp
hh
s 2( l - pB )( kdz Pn) 2
s2po sin2 dps | l —
2po(\
+
pB )
• exp
S2( k dz p n ) 2
2po
| -). cos2 9 — i sin 9 cos 0s2( l — pB)kdzpu
- s 2kdzpu { i sin 9 cos 9(1 -f pB ) + s2kdzpu sm2 9(1 - ps ) } ] .
(4.36)
S ub stitu ting (4.36) in to (4.31), the exact evaluation o f the K irc h h o ff mean in tensity
for a one-dimensional conducting surface can be obtained as
(l-Ewl2) = (P w ,P ) =
/ “ .ft. (2L - |»|) • exp [ikd, u\
• exp [—s2k%z {1 — />(u)}j • | ^cos 9 - i sin 9s2kdzpu)
+ sin2 9s2p0pB^ ,
(4-37)
where |D 0|2 = k0/ { 8 i r r 0), pu = d p (u )/d u , p0 = - p « u(0), pB = pUu{u)/Puu(0), s = cr,
kdx = 2ko sin 9, and kdz = —2ko cos 9. The coherent in te n sity fo r a conducting surface
can be com puted fro m (4.30) using the scattering am plitudes given in (4.34) as
sin2(kdz L )
! ( £ ; „ ) I2 = I M / J I 2 = 4 P o l2exp [ - e 2J & ]
(4.38)
The backscattering coefficient o f a one-dimensional conducting surface using the
K irc h h o ff a pp roxim ation can be evaluated exactly using (4.25), (4.37), and (4.38).
Since the integrals, I p2 and 1 ^ , are integrated over —oo to oo, the shadowing effect
should be accounted fo r by the shadowing fun ctio n given in (2.90).
4.5
N u m er ica l R esu lts
A num erical technique introduced in the previous chapter for scattering from
a one-dimensional conducting random surface w ill be used to examine the exact PO
model evaluated in this chapter. Since the generated random surfaces given in the last
chapter have a Gaussian correlation fun ctio n, the same form o f correlation function
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90
w ill be used fo r num erical examples as
p (u ) = e-u2/ ,J
* ( « ) = - y e - u2/'2,
__2
puu(w) =
_
/2 +
/4
e -u2/ '2,
(4.39)
2
PO =
—/Jau(0) =
The exact PO solution is tested against the exact num erical solution (M onte Carlo
m ethod w ith the m ethod of moments technique) for conducting random surfaces
at five roughness conditions.
Figure 4.5(a) illustrates the five roughnesses, ks\ =
0.62, k li = 4.6; ks2 = 0.6, k l 2 =
ks5 =
1,
k l5 =
10
6
; ks 3 = 1, kls =
6
; &S4 = 1, kU =
8
; and
where k = 2 i r / \ , s is the rms height, and I is the correlation
length. Figures 4 .5 (b )-(f) show good agreements between the exact PO m odel and
the m ethod o f m om ents(M oM ) solution. There is no d istin c tio n between the hh- and
uu-polarizations fo r the exact PO solution fo r a one-dimensional conducting surface.
The uu-polarized backscattering coefficient o f the m ethod o f moments solution is
higher tha n the /1/1-polarized one; however, the difference becomes negligible when
the roughness condition satisfies the v a lid ity region o f the PO solution as shown in
Figs.
4.5(e) and 4.5(f).
The difference between the tw o polarizations is large for
smooth surfaces and the a° of the exact PO solution has the values between the hhand wv-polarized a° o f the M oM solution as shown in Fig. 4.5(b).
Since the exact PO solution shows an excellent agreement w ith the M o M solution,
the exact PO solution can be used to examine the zeroth- and first-order approxi­
mated PO models (see Ch.
2).
In order to examine the zeroth- and first-order
approxim ated PO models using the exact PO solution derived in the previous sec­
tions, the approxim ated models for a one-dimensional die lectric random surface are
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91
(a)
X : Roughness for Testing
ks
0.5
PO
0.0
kl
(b)
10.
-
10.
oo
ks=0.62
- 20 .
kl=4.6
-30.
Exact PO
MoM-w
-40.
MoM-hh
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
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92
(c)
- 10.
M
ks=0.6
- 20 .
-30.
Exact PO
MoM-w
-40.
MoM-hh
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(d)
10.
-
10.
-
20 .
00
ks=l
kl=6
-30.
Exact PO
MoM-vv
-40.
B-
MoM-hh
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
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93
(e)
10.
m
•o
<u
o
0.
-
10.
U
00
a
'S
+-»
03
U
J3
«
- 20 .
kl=8
-30.
Exact PO
M o M -w
-40.
MoM-hh
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
70.
80.
Incidence Angle (Degrees)
(f)
PQ
^3
u
U
W)
o
- 10.
s
-20.
•c
u
cfl
ks=l
kl=10
O
J2
-30.
CQ
-40.
Exact PO
M o M -w
MoM-hh
-50.
0.
10.
20.
30.
40.
50.
60.
Incidence Angle (Degrees)
Figure 4.5: Comparison between the exact physical optics model and th e m ethod of
moments solution; (a) illu s tra tio n o f the roughness conditions and the
backscattering coefficients fo r (b) ks = 0.62, k l = 4.6, (c) ks = 0.6,
k l — 6 , (d ) ks = 1 , k l = 6 , (e) ks = 1 , k l = 8 , and (f) ks = 1 , k l = 1 0 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
summarized as;
( \ E L \ 2) oth = U > . | W * | J
I( K X a
? .|2
f
j
2
L - |u\)eikd*u • e -*2^
I2 = | A , | 24 c o s 2 « |J!,i | V
i ‘ 3. .
( I ^ J 2),., =
(I3 .-.P W
211- ^ 1^ ,
(4 .4 1 )
1
K - e j.W I2
(4.40)
,4
« » 4o'
'
The backscattering coefficients o f the zeroth- and first-order approxim ated PO models
for a one-dimensional conducting random surface can be obtained by substitu ting
|f ? „ | 2 = 1 in to (4.40)-(4.42).
I t should be noted th a t if the edge te rm is included
in the evaluation o f the coherent intensity o f the first-order PO m odel, the coherent
in ten sity | ( ^ a)lsJ 2 becomes |( £ * a) 0 J 2.
Figures 4.6(a) and (b) show the comparison between the exact PO solution and
the zeroth-order approxim ated PO model (no slope term in the series of scattering
a m plitude) fo r a conducting and a dielectric surface, respectively. The zeroth-order
approxim ated PO m odel underestimates the backscattering coefficients. For example,
for a re la tiv e ly smooth surface (ks =
1,
2 ~ 5 dB at 20° — 70° as shown in Figs.
kl =
8
) the slope te rm effect is about
4.6(a) and (b).
Figures 4.7(a) and (b)
show the comparison between the exact PO solution and the first-o rd e r approxim ated
PO m odel fo r a conducting and a dielectric surface, respectively.
The first-order
approxim ated PO m odel was obtained by in clu d in g the first-order slope term in the
series o f scattering a m p litude, integrating by parts, and discarding the edge term
(see Ch. 2). The first-order approxim ated PO m odel shows a good agreement w ith
the exact PO model at lower incidence angles (9 < 40°), b ut, overestimates at large
incidence angles
(6
> 40°). For example, the edge term effect is about 5 dB at 45°
and 15 dB at 70° for a re la tively smooth surface (ks = 1, k l =
8
) as shown in Figs.
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95
(a)
20.
Conducting
03
10.
ks=l, kl=8
•o
■
Exact
<D
O
u
00
a
- 10.
‘S
-4-»
- 20 .
c3
a
J2
-30.
03
-40.
• Approx-0
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(b)
20.
HH-pol.
PQ
•X3
£=(10,2)
£3
<0
O
U
ks=l, kl=8
Exact
- 10.
00
• Approx-0
c
•a
<D
4 -1
•4— 1
C3
O
J2
-30.
PQ
-40.
- 20 .
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 4.6: Comparison between the exact physical optics m odel and the m ethod of
moments solution fo r the roughness o f ks = 1, k l = 8 fo r (a) a conducting
surface and (b) for a d ie le ctric surface o f e = (10,2) fo r /i/i-p o lariza tion .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
(a)
20.
Conducting •
10.
PQ
ks=l, kl=8 ‘
•tJ
Exact
<u
o
U
eo
-
10.
s
*44—->s1
- 20 .
cd
o
ja
-30.
-40.
pq
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(b)
20.
HH-pol.
10.
PQ
er=(10,2)
•o
ks=l, kl=8
<u
o
U
Exact
-
10.
• Approx-1
(so
c
•c
<D
4-^
4-»
-
20 .
C3
o
J2
-30.
PQ
-40.
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 4.7: Comparison between the exact physical optics m odel and the m ethod of
moments solution for the roughness of ks = 1, k l = 8 fo r (a) a conducting
surface and (b) fo r a dielectric surface of e = (10,2) for /i/i-polarization.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
(a)
CQ
3
- 10.
ks=l
- 20 .
kl=6
J
-30-
PQ
-40.
Exact
■ Approx-0
• Approx-1
-50.
0.
10.
20.
30.
40.
50.
60.
70.
80.
90,
Incidence Angle (Degrees)
PQ
■o
<D
o
U
M
a
•G
<
Wu
cS
o
V)
M
O
«
Approx-0
Approx-1
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
Figure
4.8:
Comparison between the exact PO solution and the approxim ated PO
solutions fo r a one-dimensional conducting surface o f (a) ks = 1, k l = 6
and (b) ks = 1, k l = 10.
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98
30.
✓—\
CQ
"O.
>
e= (10,2 )
25.
ks=l, kl=8
20.
Exact PO
Approx. PO
8
u
d
J2
3
m
-
10.
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence A n g le (Degrees)
F igure 4.9: The ra tio a^h/^vv
^he exac^ PO solution and the approxim ated PO
solutions for a one-dimensional dielectric random surface o f ks = 1, k l =
8, and eT = (10,2).
10.
•o
^
-
10-
0)
O
U
M
C
-20.
'§
-3 0 .
ks=1.2
L =51
L=101
0
1
L=201
-4 0 .
£
«
L=40l
-5 0 .
-6 0 .
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence A n g le (Degrees)
Figure 4.10: The backscattering coefficients o f the exact PO solution fo r a various
values of the in teg ra tio n lim its for a one-dimensional conducting random
surface o f ks = 1.2, k l = 6.1.
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99
4.7(a) and (b). Figures 4.8(a) and (b) show the comparison between the exact PO
solution and the approxim ated PO solutions fo r a re la tiv e ly rough surface (ks = 1,
k l = 6) and a re la tively smooth surface (ks = 1, k l = 10), respectively. The slope
term effect for a rough surface is larger tha n for a sm ooth surface, and the edge term
effect for a rough surface is sm aller than fo r a smooth surface as shown in Figs. 4.8(a)
and (b). The difference between hh- and im -polarized backscattering coefficients is
the ra tio o f the Fresnel re fle c tiv ity
and |7?t,|2. For the exact PO solution, the
local reflection coefficient w hich is a fun ctio n of the surface local slope affects the
polarization differences. Figure 4.9 shows the comparison between the ra tio crlh/ a ° v
of the approxim ated PO model and the ra tio o f the exact PO model fo r the roughness
o f ks = 1, k l = 8 and the dielectric constant o f eT = (10,2). The ra tio <r^/or“„ for
the approxim ated PO model has a peak a t 9
66° w hich is the brewster angle for
er = (10,2), w hile the ra tio fo r the exact PO solution is < 0 dB a t low incidence
angles and > 0 dB at large incidence angles as shown in Fig.
4.9.
In order to
get a closed form of the backscattering coefficient, the in teg ra tio n lim it in (4.40) is
changed as L —►oo assuming L
I where / is the correlation length. Figure 4.10
shows the backscattering coefficients of the exact PO solution fo r the one-dimensional
conducting random surface o f ks = 1.2, k l = 6.1 for the values o f L \ = 51, L 2 = 10/,
Lz = 201, Ln = 40/, and L 5 = oo. A t low incidence angle
(6
< 45°) the backscattering
coefficient is independent o f the integration lim it L. The backscattering coefficient
w ith L = 51, however, is much higher tha n cr° w ith L = oo as shown in Fig. 4.10.
The exact PO solutions presented in Figs. 4.5, 4.6, 4.7, and 4.8 have been com puted
for the in teg ra tio n lim it o f L = 10/.
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100
4.6
C onclusions
The backscattering coefficient o f a two-dim ensional dielectric random surface has
been form ulated exactly w ith o u t fu rth e r a pp roxim ation other than the tangent plane
approxim ation (or K irch h o ff approxim ation).
The exact PO solutions fo r a one­
dim ensional random surface have been com puted num erically and compared w ith
the exact num erical solution (M onte Carlo m ethod w ith the m ethod o f m om ents) in
the case o f conducting surfaces.
The exact PO solution was used to exam ine the
zeroth-order and the first-order approxim ated PO models, which is equivalent to the
exam ination o f the effects o f the slope te rm in the series of scattering a m p litud e
and the edge te rm in the form u la tion of scattered field, respectively. T h e zerothorder approxim ated PO m odel (no slope te rm ) underestimates the backscattering
coefficient in the range o f 2 ~ 5 dB at 20° — 70°. The first-order approxim ated PO
m odel shows an excellent agreement w ith the exact PO solution at sm all incidence
angles
6
< 40° and overestimates at large incidence angles 0 > 40° in the range of
5 ~ 20 dB depending on the incidence angle.
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CHAPTER V
A NUMERICAL SOLUTION FOR
SCATTERING FROM INHOMOGENEOUS
DIELECTRIC RANDOM SURFACES
5.1
In tro d u ctio n
A n exact solution for scattering by inhomogeneous, dielectric, random surfaces
does not exist a t the present time. This chapter presents an efficient num erical tech­
nique for co m puting the scattering by inhomogeneous dielectric rough surfaces. The
inhomogeneous dielectric random surface, w hich is intended to represents a bare soil
surface, is considered to be comprised o f a large num ber o f random ly positioned dielec­
tric humps o f different sizes, shapes, and dielectric constants, lyin g on an impedance
surface. Clods w ith non-uniform m oisture content and rocks are modeled as inho­
mogeneous d ie le ctric humps and the underlying sm ooth wet soil surface is modeled
as an impedance surface.
In this technique an efficient num erical solution for the
constituent d ie le ctric humps is obtained using the m ethod o f m om ents in conjunction
w ith a new G reen’s fun ctio n representation based on the exact image theory. The
scattered field fro m a sample of the rough surface is obtained by sum m ing the scat­
tered fields from a ll the in d ivid ua l humps of the surface coherently, ignoring the effects
o f m u ltip le scattering between the humps. The behavior of the scattering coefficient
101
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102
a° and the phase difference statistics are obtained from calculations o f the scattered
fields for m any different surface samples o f the same process. N um erical results are
presented fo r several different roughnesses and dielectric constants o f the random sur­
face. The num erical technique is verified by com paring the num erical solution w ith
the solution based on the sm all-perturbation m ethod and the physical-optics model
fo r homogeneous rough surfaces. This technique can be used to stud y the behavior
o f the scattering coefficient and phase difference statistics o f rough soil surfaces.
Investigation of the radar scattering response o f n atu ra l surfaces is an im p o rta nt
problem in rem ote sensing because of its p ote n tia l in re trie vin g desired physical pa­
rameters o f the surface, nam ely its soil m oisture content and surface roughness. Soil
m oisture is a key ingredient of the biochemical cycle and an im p o rta n t variable in
hydrology and land processes. A lthough the problem of electrom agnetic wave scat­
te rin g from random surfaces has been investigated fo r m any years, because of its
com plexity, theoretical solutions exist only fo r sim ple lim itin g cases. These include
the sm all p e rtu rb a tio n m ethod (SPM ) [Rice, 1951] and the K irc h h o ff approxim ation
(K A ) [Beckmann and Spizzichino, 1987], b oth o f which are applicable fo r homoge­
neous surfaces over restricted regions of va lid ity. Numerous techniques based on the
basic assumptions o f the SPM and K A have been developed in the past in an attem pt
to extend the regions o f v a lid ity o f these models; however, they a ll have the basic lim ­
ita tio n s of the original models [Brown, 1978]. O ther theoretical models are available
also, such as the fu ll wave analysis technique [Bahar, 1981], the phase p ertu rb a tion
technique [W ineberner and Ishim aru, 1985], and the integral equation m ethod [Fung
and Pan, 1987], b ut they are not applicable for inhomogeneous surfaces and th e ir
regions o f v a lid ity have not been fu lly determined yet. Several num erical solutions of
the scattering problem have been proposed to id e n tify the regions o f v a lid ity and ac­
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103
curacies o f these theoretical models. A scattering solution fo r a p erfe ctly conducting
random surface using the method of mom ents has been suggested by A x lin e and Fung
[1987] who used a tapered incident field as the excitation to elim ina te the edge-effect
c o n trib u tio n due to the boundaries o f the illu m in a te d area. Since then, m any other
n um erical solutions w ith some m odifications have been introduced [Fung and Chen,
1985; Nieto-Vesperinas and Soto-Crespo, 1987; Thorsos, 1988; D urden and Vesecky,
1990; Lou et al., 1991; Rodriguez et al., 1992], a ll for scattering fro m p erfe ctly con­
d uctin g random surfaces. A num erical solution fo r homogeneous d ie le ctric random
surfaces has recently been reported [Sanchez-Gil and Nieto-Vesperinas, 1991] where
again a tapered illu m in a tio n is used to lim it the size of the scattering area. The ac­
curacy o f the num erical solution w ith tapered illu m in a tio n decreases w ith increasing
incidence angle. To our knowledge, a solution fo r scattering fro m an inhomogeneous
rough surfaces does n ot yet exist.
A nalysis o f microwave backscatter observations by O h et al. [1992] reveals th a t
the existing theoretical models cannot adequately explain the scattering behavior of
soil surfaces. The deviation between theoretical predictions and experim ental data
is a ttrib u te d to three factors. F irst, the roughness parameters o f some surfaces are
often outside the region of va lid ity of the theoretical models. Second, the autocorre­
la tio n functions associated w ith the measured height profiles o f n a tu ra l surfaces are
very com plicated and are not Gaussian or exponential functions. F in a lly, the most
im p o rta n t reason is th a t in most cases n a tu ra l surfaces are not homogeneous dielec­
t r ic surfaces, i.e., the m oisture content is not u n ifo rm in depth. The top rough layer,
w hich includes clods and rocks, is usually d ry and the u nderlying soil layer is m oist
and smooth.
In this chapter we model a soil surface as an inhomogeneous die lectric random
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104
y
F igure 5.1:
G eom etry o f the scatter problem for a two-dim ensional rough surface.
surface comprised o f a large number o f random ly positioned two-dim ensional dielectric
hum ps of different sizes, shapes, and dielectric constants, all ly in g over an impedance
surface as shown in Fig.
5.1.
A t microwave frequencies, the m oist and smooth
u nd erlying soil layer can be modeled as an impedance surface, and the irregularities
above i t can be treated as dielectric humps o f different dielectric constants and shapes.
For th e field scattered by a single dielectric hum p over an impedance surface, we have
an available efficient num erical solution th a t uses the exact image theory for the
G reen’s fu n ctio n in conjunction w ith the m ethod o f moments [Sarabandi, 1992]. In
the solution o f a single hum p, it has been shown th a t the b ista tic scattered field is
very weak a t points in close p p ro xim ity to the impedance surface; thus, the effects
o f m u ltip le scattering between humps can be ignored. In this case the scattered field
fro m a collection o f random ly positioned die lectric humps can easily be obtained by
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105
sum m ing the scattered field o f all the constituent humps coherently. The scattering
coefficients (o-0) and the phase difference statistics are obtained by a M onte Carlo
sim ulation.
In Section 2 we sum m arize the procedure fo r the num erical solution o f a single
hum p above an impedance surface. Section 3 outlines the procedures used fo r gen­
erating the random surfaces and for evaluating the statistics o f the scattered field.
N um erical results and th e ir comparison w ith theoretical models are presented in Sec­
tio n 4.
5.2
S ca tterin g From Individual H u m p s
In th is section we brie fly review the procedure used for the num erical solution
of scattering fro m a two-dim ensional dielectric object above a u n ifo rm impedance
surface [Sarabandi, 1992]. The radiated field for a dipole source above a dissipative
half-space m edium (Green’s fu n ctio n ) is usually evaluated using the Sommerfeld in ­
tegral [S tratton , 1941].
This in fin ite integral, in general, is h ig h ly o scilla tory and
co m p u ta tio n a lly ra the r inefficient. Recently, the Green’s fu n ctio n o f an impedance
surface was derived in term s o f ra p id ly converging integrals using appropriate integral
transform s s im ila r to those employed by L in d e ll and Alanen [1984] in th e ir derivation
o f the exact image theory. The scattering problem was then form ulated by integral
equations w hich were solved n um erically using the m ethod of moments.
Suppose a dielectric object, possibly inhomogeneous, is located above an impedance
surface and is illu m in a te d by a plane wave. The incident field E ' induces conduc­
tio n and displacement currents in the dielectric object which together are known as
the p o la riza tio n current J e. The polarization current can be represented in terms of
the to ta l electric field inside the dielectric object, which is comprised o f the incident,
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106
reflected, and scattered fields denoted by E ’ , E r , and E s, respectively. Thus
U
p)
(5.1)
= - i k o Y M ? ) - 1 )(E ’ (/>) + E r (p) + E *(p )),
where k 0 = ui^po^o, Vo = yjto jp o, and e(p) is the relative d ie le ctric constant of the
o bject at the p o in t ~p{= x x -j- yy). T he fields E ‘ , E r , and E * are, respectively, given
by
E ’ (p) = (E'hhi + E'vVi) e xp[ik0ki ■p],
(5.2)
E r (p) = ( RhE'hhr -I- R vE'vvt ) exp[ik 0 kr • p],
(5.3)
E *(p) = i k QZo I ^ ( p , 7 ) • J ( 7 W ,
Js
(5.4)
where E^ and E'v are the horizontal (E -polarized) and vertical (H -polarized) compo­
nents o f the in cident field, respectively. Rh. and R v are the h orizon ta l and vertical
Fresnel reflection coefficients and G (/j, p') is the dyadic Green’s fu n ctio n o f the prob­
lem. The dyadic Green’s fun ctio n can be decomposed into tw o components: (1) the
dyadic Green’s fu n ctio n o f the free space Go(p, p1) and (2) the dyadic Green’s function
due to the presence o f the impedance surface G r (p ,p '); th a t is:
(5.5)
G ( P i 7 ) — G o (p ,7 ) + G r (p ,p /)
Since ^ = 0 in a tw o-dim ensional scattering problem , the dyadic Green’s fun ctio n o f
free space Go(p, 7 ) takes the follow ing form
(l + fcgSr?) 9o{p, P')
Go ( p ,P ) =
where
V>&aj9o{7 7)
0
k%dy'dx’ 9 ° ( P ’ P')
+ fcpp7) 9 o (p ,7 )
0
0
o
go(p,7)
go(p,7) is the scalar free-space Green’s fu n ctio n
3
o(p, 7>') = - //n 'V ^ o
)
(5-6)
given by
/>o = \ / ( x - x ' ) 2 + (y - y ' ) 2 ,
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(5.7)
107
and H q^ is the Hankel fu n ctio n of the firs t k in d and zeroth order. The co m p uta tion ­
a lly efficient dyadic Green’s fun ctio n G r ( p , P ) is given by [16]:
"“
(****
( P i P 1)
kld x'd y'S r
0
+
o
0
0
.
(5.8)
9 ? {p ,p ')
where
9?
(p rf) =
9
i(p i) -
2a
f
Jo
e avg 2 {p 2 )dv
9r (p, f ) =
9
i { p i ) - 2/3 /
Jo
e - pug2 {p 2 )dv
(5.9)
w ith
9i{Pi) =
pi), i = 1,2,
pi = \J{x - x ' ) 2 + {y + y ' ) 2 ,
(5.10)
P2 = \ ] { x ~ x ' ) 2 + (y + y ' + i v f ,
and a = k0 /t], j3 = k 0 y. Here y is the norm alized impedance o f the impedance surface
defined by y = Z / Z 0.
T here is no known exact solution for the integral equation given by (5.1). Hence,
an approxim ate num erical solution of this equation m ust be obtained using the
m ethod o f moments.
This is done by d iv id in g the cross section of the dielectric
s tru ctu re in to N c sufficiently sm all rectangular cells such th a t the dielectric constant
and the p olarizatio n current over each cell can be approxim ated by constant values.
Using the p oint-m atch in g technique, the in teg ra l equation can be cast into a m a trix
equation o f the follow ing form :
[Z XX\
[ZXy\
0
lz „ ]
[Z „]
0
0
0
[Z „]
[V,]
f[2»]
o]
. [lz ].
—
=
[V„]
[Vy]
[v j _
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(5.11)
108
where [-Hpg] is the impedance m a trix , [Xp] is the unknown vector whose entries are the
values o f the p olarizatio n current at the center o f each cell, and [Vp] is the excitatio n
vector w ith p = x, y, or z. The entries o f [Vp] are sim ply given by
vP,n = i k 0 Y0 [e(xn, y n) - 1] ([E p(® „,y „) + E p(® „,j/„)]) • p ,
(5.12)
and the entries o f [Z pq] can be evaluated from
'p q m n
where Spq and 5mn are the Kronecker delta functions, and p,q = x , y , or
2
.
E x p lic it expressions fo r the elements o f the impedance m a trix are given in [Sarabandi, 1992] where off-diagonal elements are obtained by approxim ating the Green’s
fun ctio n via its Taylor series expansion around the m id p oint o f each cell and then
the in tegration over the cell surface is perform ed analytically. For diagonal elements
the free-space G reen’s fun ctio n is approxim ated by its sm all argum ent expansion and
then in teg ra tio n is perform ed analytically.
Once the impedance m a trix for a given dielectric hum p is calculated and inverted,
the scattered far-fie ld can be computed fro m (5.4) for any desired com binations of
incident and scattered directions.
5.3
M o n te Carlo S im u lation o f R ou gh Surface S ca tterin g
M onte Carlo sim u la tio n of scattering by a rough surface com prised o f a fin ite
collection o f d ie le ctric humps involves the execution of five m ajor steps, as shown in
Fig. 5.2. The firs t step is to choose the type (size, shape, and d ie lectric constant) and
num ber o f co nstituent humps. The second step deals w ith generating a surface sample
by positioning a large num ber of humps w ith a prescribed p ro b a b ility d istrib u tio n
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109
K Constituent Humps
Random Numbers
MoM/
rms Height /
Corn Length
Green’s
Function
Method
A Random Surface with
M Humps ( M » K )
N Samples
Scattering Coefficient CJ° and
Phase-Difference Statistics
F igure 5.2: Flow chart of the M onte Carlo sim ulation for the rough surface scattering
problem .
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110
fun ctio n . The num ber o f humps in the surface sample m ust be chosen large enough
so th a t the surface length is longer than fifty correlation length. T h e th ird step in
this a lg o rith m is to compute the inverse impedance m atrices for a ll co nstituent humps
using th e num erical m ethod explained in the previous section. N e xt, the scattered
fie ld fro m the surface is com puted by coherent sum m ation of the scattered fields from
a ll o f th e humps in the surface sample.
F in a lly, the scattering coefficient <r° and
the phase-difference statistics are obtained by repeating the fo u rth step for a large
num ber o f independently generated surface samples. For example, N is chosen to
be around 100 to reduce the standard deviation associated w ith th e estim ation of
mean backscattered power (cr°). The standard deviation o f estim ated a° is inversely
p ro p o rtio n a l to y / N [U laby and Dobson, 1989].
The types of constituent humps, in a dd itio n to th e ir p ro b a b ility o f occurrence,
fu lly characterize the statistics o f the random surface. Figure 5.3 shows the geometry
and d ie le ctric profiles o f different types of d ie le ctric humps th a t can be handled by
th is a lg o rith m . For example, Figure 5.3(a) shows a ty p ic a l hump arrangem ent for a
d ry clod above a m o ist and smooth underlying soil layer (e0 < ei < e2), and Figure
5.3(b) shows the same hum p when the clod and und e rlyin g layer are b oth m oist (a
homogeneous surface). The hum p its e lf may be considered to be inhomogeneous as
shown in Fig. 5.3(c). Isolated irregularities such as rocks above a fla t surface can
be represented by the hum p example shown in Fig. 5.3(d) where th e bum p occupies
only a p a rt o f the to ta l w id th allocated to an in d iv id u a l hump. W hen the surface
is very rough w ith a short correlation length, the geom etry of the hum ps are more
com plicated. Two examples o f such humps are shown in Figs. 5.3(e) and (f). The
profiles o f Figs. 5.3(a)-(e) used in this chapter are given by the follo w in g functionals;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
e2
(a )
(b )
W
F igure 5.3:
(c )
(d )
(e)
(f)
Hump types fo r the rough surface considered in this chapter.
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112
for (a)-(c)
,
y (*) =
W
2 /w x \
- J cos ( j j f J »
W ^
^ ,W
- - j- < X < Y ,
_
(5.14)
for (d )
a;2
y(a:) = ,4 ^1 - — J ,
- B < x < B,
B <W ,
(5.15)
and fo r (e)
y ( i ) = A F i ( x ) + £ F 2( x ) ,
w ith
F \(:r) = cos" f e - f ) f e ' T
U '
VlV
2)W)
F 2(
x
)
=
COSn
—
2)
( l
| , 0 < a : < W ,
(5.16)
— jfr )
where A and B are constants, n and m are integers, and W is the w id th o f a hum p.
The set o f constituent humps fo r a surface can be constructed by choosing a fin ite
num ber o f param eters and desired dielectric constants in the desired functionals. The
profile of Fig. 5 .3 (f) is very com plicated and should be obtained num erically by the
procedure o utline d in [Fung and Chen, 1985]. In th is procedure the hum p profile is
obtained fro m a sequence o f independent Gaussian deviates w ith zero mean and u n it
variance w hich are correlated by a set of w eighting factors derived from the desired
correlation fun ctio n.
Suppose the set o f in d ivid u a l humps includes K different humps (in clu d in g size,
shape, and d ie le ctric constant) and the profiles o f the humps in the set are repre­
sented by f i ( x ) , i = 1, • * -, K . Then a sequence o f random numbers ranging fro m 1
to K , w hich is generated by a random num ber generator w ith the prescribed proba­
b ility d is trib u tio n function, is used to position a large num ber of humps random ly to
construct a surface sample. If the to ta l num ber o f humps (M ) in the surface sample
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113
is much larger than the num ber of constituent humps [ K ) and the random number
generator has a u nifo rm d istrib u tio n , the p ro b a b ility of occurrence of each hump in
the surface w ill be about M / K . A fu n ctio n a l form of the generated surface profile
can be represented by
M
/
m —1
\
y(x) = J 2 f i m U ~ Y l wj,
m =l
\
;=1
(5.17)
/
where i m, i i 6 {1, • • - , K } and Wit represents the w id th o f the hum p of the z'/th type.
The roughness parameters, rms height s, correlation length /, and rm s slope m [Ulaby
et al, 1982], can be com puted either num erically or a nalytically fro m the surface profile
given in (5.17). The a nalytical com putation is possible fo r sim ple fun ctio na l forms.
The average height o f the surface can be com puted from
K
1 A
yW,
y (x ) = T ^ P i
fi(x )d x ,
L t=1 Jo
_
where L =
(5.18)
PiW( and p,- is the p ro b a b ility o f occurrence o f the hum p o f type z.
The rm s height s and the rms slope m , respectively, can be evaluated from
I
' 1 A rWi
= ( b W - F W ) 2} 1 =
J0
( p i f ‘ (x ) - v ( x ) f dx
(5.19)
and
m =
'd y ( x )
dx
/ d y(x)
2\ 2
’1 K
t£ p <
dx
L .= 1
W
„
/
( / / ( * ) ) dx
(5.20)
Jo
Assum ing the surface has a Gaussian correlation function, the correlation length I is
related to rms height and rms slope by,
l = V2 —
m
.
(5.21)
I t is often required to generate a random surface of a specified rm s height s and
correlation length I. In th a t case, the required surface can be obtained by an iterative
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114
process where some in itia l values for the hum p param eters are chosen.
Then the
roughness param eters are calculated and compared w ith the desired ones. Depending
on the difference between the calculated s and I and the desired ones the hum p
parameters are m odified and this process is repeated u n til the difference becomes
sm aller than a tolerable error.
Once the set of in d ivid u a l hum ps fo r a random surface w ith given s and I is formed,
the impedance m atrices, [Z pq]i, i = 1,- • •, K , can be com puted using the method o f
moments described in the previous section. Since the scattered field o f a hum p near
the impedance surface is very weak [16], the effect o f m u ltip le in te ra ctio n between
hum ps in a surface sample can be ignored. Therefore by in v e rtin g and storing the
impedance m atrices o f the constituent humps, the scattered field o f any surface sample
comprised o f M humps (M
K ) can be computed very e fficie n tly fo r any incidence
and observation directions. For a given direction o f incidence the p olarizatio n currents
in the j t h hum p fo r the ve rtical and horizontal polarizations, respectively, are given
by
-1
[zxx] \Zxy\
K J .
i
.
IZyr] [Zyy] .
" [Vx] ’
. M
and
(5.22)
.
(5.23)
where j € {1, • • - , M } and i j € {1 ,- • •, K } representing the hum p o f the zth type.
The e xcitation vector [V ]j is com puted fro m (5.12) where the position vector p is
specified by the discretization procedure and the profile fu n ctio n (5.17). The electric
polarization cu rre n t induced inside the surface sample can be represented by
(5.24)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
where [Xp],- is the p-polarized current inside the zth hump. The radiated fa r field can
be evaluated from
E*pp = ] j ^ p ei{hoP~1',4)
.
VP = hh or uu
(5.25)
where Spp is the far-field am plitude given by
k 7 Nt
Shh = — 2 - l ' £ i J2(xn, yn) A x nA y ne~'k° sm6iXn L - ^ c a s O ^ + R ^ y h c o s e ^ (5 26)
4
n=l
I. 7 -Nt
° ° £ A
4 „ =1
x„ A
ynC-«0 “ “ «■*»
{ j* ( z „ , j/ „ ) c o s 0 3 {[ e - ik^ os0^
- R H {Os)eihoCOse^ )
- J y ( x n,y n) s m 6 s ( e- ifc° c°se**" + ^ ( ^ ) e ,fc° cos9* ^ ) } . (5.27)
Here N t is the total num ber of cells in the surface sample.
The statistical behavior o f the scattered field is obtained fro m evaluation o f E pp
fo r m any independent surface samples.
For a sufficiently large num ber o f surface
samples ( N a), the incoherent scattering coefficient is computed fro m
2k p
<r° = lim
pp
P-oo N , L a
where L av =
**• l
y4 - /
j =1
Es .
PP.J
12
-
1
—
AT
N,
f4— 1 EsPP,J
pp = hh, vv,
(5.28)
i= i
and Z j is the to ta l length o f the j t h random surface.
In the past, the study o f scattering by random surfaces was confined to exam ina­
tio n of the incoherent scattering coefficients, cr°„, a^h, and cr£„. W ith the in tro d u c­
tio n o f radar polarim etry, it was recently shown th a t the co-polarized phase angle <j)c,
defined as the phase difference between the H H - and V V -polarized scattering a m p li­
tudes: <f>c =
4>hh
— <f>vv, also depends on the roughness and die lectric constant o f the
surfaces [Sarabandi et a]., 1992; Oh et al., 1993]. In this chapter, the statistics of
the phase difference, in a dd itio n to the scattering coefficients, are used to study the
radar response of rough surfaces. I t is shown th a t the PD F o f <j>c, f(<t>c) > can be ob­
tained fro m the Mueller m a trix o f the d istrib u te d target and characterized com pletely
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
116
£=30'
a=0.9
TJ
<2
a=0.7
0.8
a=0.4
C/D
a
<D
Q
•S
o
0.6
a=0
0.4
0.2
0.0
180.
-120.
-60.
0.
<f>c = ‘t’hh -
60.
120.
180.
(Degrees)
F igure 5.4: P ro b a b ility density function o f the co-polarized phase angle (j>c =
fo r a fixed value o f £ and four different values o f a.
by two param eters; the degree of correlation a and the polarized-phase difference £
[Sarabandi, 1992]. T he degree of correlation is a measure o f the w id th o f the P D F
and the polarized-phase difference is the value of <j>c at w hich the P D F is m a xim u m
as shown in Fig. 5.4. T h e f{<f>c) is given by
1 —a 2
I.
2tt[1 - a 2 cos2 (<t>c - 0 ] \
- + tan
2
-i
1+
a cos (<^c — C)
y / l - a 2 cos2 {<j>c - £)
a cos{ ^ c ~ 0
,---------------------------y j 1 — a 2 cos2 (<f>c — £)
(5.29)
w ith
1
( M 3 3 -(- M
a ~ 2V
£ = tan - 1
M 34
4 4 ) 2
+ (M 34 — M 4 3 ) 2
M n M 22
— M i43
.M33 -f M
4 4
.
where M { j are the elements o f the ensembled M ueller m a trix .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.30)
(5.31)
117
Table 5.1:
Roughness parameters corresponding to constants A and B .
A pprox. f
Case
A
B
s
/
in cm
E xact J
A t 5 GHz
s
I
ks
kl
Remarks
SPM region
in cm
1
15
0.20
0.115
2.21
0.115
2.03
0.12
2.13
2
15
0.36
0.208
3.98
0.207
3.63
0.22
3.80
3
15
0.70
0.405
7.74
0.405
7.15
0.42
7.49
PO region
f A p p ro xim a tio n by equations (32-33) and (21),
J N um erical evaluation w ith 4000 humps,
s : rm s surface height,
1
: co rre la tion length.
5.4
N u m erica l R esu lts
To dem onstrate the performance o f the technique proposed in th is chapter, we
shall use i t to com pute the scattering fo r some sample surfaces and then compare
the results w ith those predicted by the available the o re tical scattering models, when
conditions apply. F irs t, we consider a surface w ith homogeneous d ie lectric humps as
shown in Fig. 5.3(a). The functional form o f the humps are given by (5.14) where
the param eters A and W are varied to generate the set o f the constituent humps.
Keeping A as a constant co ntro lling the height and varying W , a set o f sim ilar humps
can be generated. A random num ber generator w ith o u tp u t i E {1, • • •, K ) selects the
param eter W{ = B M , where B is a constant co n tro llin g the w id th o f the humps and
A is th e wavelength. In this example the hum p param eters were chosen according to
Table 5.1 and the random num ber generator was given a u nifo rm d is trib u tio n w ith
K = 10. Before presenting the statistica l scattering behavior of the surface, it is useful
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
118
to dem onstrate the v a lid ity o f the assumption regarding the significance o f the effects
o f m u ltip le scattering among the humps. Figures 5.5 (a) and (b) show the b istatic
echo w id th o f a squared-cosine hump w ith W = 0.72A, H = 0.07A, ei = 15 + z'3
above a surface w ith rj = 0.254 — i0.025 (w hich corresponds to e2 = 15 + iZ) at 5
GHz when the incidence angle 0,- = 0° and
6{
= 45°, respectively. I t is shown th a t
the b is ta tic echo widths at the large scatter angles (near the surface) are very weak
which im plies th a t the effect o f m u ltip le scattering between humps can be ignored.
In order to illu stra te the effect o f m u ltip le scattering, a surface segment comprised
o f three squared-cosine hum ps w ith
— 15 4- iZ above an impedance surface w ith
rj = 0.254 — z’0.025 was considered (see Fig. 5.6). Dimensions o f the three humps
are, respectively, given by: W \ = 0.8A, H i = 0.08A; W 2 = 1.0A, H 2 = 0.1A; and
W 3 = 0.6A, H 3 = 0.06A. The backscatter echo w id th o f the surface segment was
com puted twice. In one case the scattered field was computed from the polarization
current of isolated humps (ignoring the effect o f m u tu a l coupling) and in the other case
the p olarizatio n current o f the three-hum p stru cture was obtained d ire c tly fro m the
m ethod o f moments solution (including the effect o f m u tu a l coupling). Figures 5.6(a)
and (b) show th a t the effect o f m u ltip le scattering is negligible fo r b o th polarizations.
As long as the ra tio of rm s height to correlation length o f the surface (s / l ) is small,
this approxim ation provides accurate results. For most n a tu ra l surfaces s / l < 0.3
which satisfies this condition [Oh et al., 1992]. However i f the ra tio ( s / l ) is re latively
large, the hum p type o f Fig. 5.3(f) m ust be used to include the effect o f m u ltip le
scattering at the expense o f com putation tim e.
The rm s surface height s and the rm s surface slope m fo r this surface can be
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119
(a)
-15.
E
m
"C
■.—J'
•3
•a
-
20 .
-25.
o
-
-30.
a
o
3»
■*c—
E-pol.
• H-pol.
-35,
0j=O°
00
•H
pa
-40.
■90.
-60.
-30.
0.
30.
60.
90.
Scatter Angle (Degrees)
(b)
-15.
- 20 .
-25.
-
-30.
E-pol.
• H-pol.
-35.
0(=45°
-40.
■90.
-60.
-30.
0.
30.
60.
90.
Scatter Angle (Degrees)
Figure 5.5: B istatic echo w id th of a squared-cosine hum p o f t \ = 15 -f z’3, W =
0.72A, H = 0.07A over an impedance surface o f rj = 0.254 — iO.025 at (a)
= 0° and (b) 0,- = 45° at / = 5 GHz for E- and H-polarizations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
(a)
3
-
10.
- 20.
-30.
-40.
-50.
-60.
CQ
hh-pol.
W ithout multi, scatt.
• W ith multi, scatt.
-70.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
(b)
-
10.
- 20 .
-30.
-40.
-50.
-60.
m
vv-pol.
W ithout multi, scatt.
• W ith multi, scatt.
-70.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
Incidence Angle (Degrees)
F igure 5.6: M u ltip le scattering effect on the backscatter echo w id th of a surface
segment consisting o f hum p-4 , hump-5, and hum p-3, corresponding to
the roughness o f ks = 0.36, k l = 2.2, w ith ej
=
15 + i3 over an
impedance surface o f rj = 0.254 — t'0.025 at (a) h h-polarization and (b)
v v-p o la riza tio n at / = 5 GHz .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
121
com puted from (5.19) and (5.20) respectively and are given by
A
s=
/ 3 Wf
A2
Lw
W?_
(5.32)
A 1
7T
m =
(5.33)
y /2 A
where
1
V = nAT
2ALw
K
,= 1
K
'
“ <l
L w = 'Z .W i,
„ i
I t should be noted th a t the rm s surface slope m o f this surface depends o n ly on the
constant A. Therefore fo r a fixed value of A, both th e rm s height and the correlation
length increase at the same rate w ith increasing B . Table 5.1 shows several values o f
roughness parameters, s and I, corresponding to different values of A and B .
A random num ber generator was used to select and position 4000 squared-cosine
humps over the impedance surface (rj = 0.254—iO.025). Then this surface was divided
in to 100 segments to o btain 100 independent surface samples each having 40 humps.
The length o f the surface segment was chosen to be in the range o f 44A to 154A
depending on the correlation length o f the surface w hich corresponds to the size of
in d iv id u a l humps. In the m ethod o f moments solution o f in d iv id u a l hum ps, the size
of a discretized cell was chosen such th a t A x = A y = A/15 (where A = Ao
Table 5.2 summarizes the characteristics of the surfaces and th e ir co nstituent humps
used in the examples considered in th is study. F igure 5.7(a) shows a surface sample
w ith A = 30 and B = 0.2 (Case 1 in Tables 5.1 and 5.2). In Fig. 5.7(b) the correlation
fu n ctio n o f the surface is shown.
The correlation fu n ctio n , as com puted from the
surface samples, is compared w ith Gaussian and exponential correlation functions
w ith the same correlation length. W ith in the m ainlobe of the correlation fun ctio n
(sm all displacements), the actual correlation fun ctio n is very sim ila r to the Gaussian
fu n ctio n ; however, the ta il o f the correlation fu n ctio n is very much different from
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122
(a)
1.0
0.6
e
o
>>
*s
M
0.2
DO
-
0.2
-
0.6
-
1.0
80.
90.
110.
100.
120.
130.
Distance, X (cm )
(b )
1.0
C
0
0.5
1
£
0.0
o
Computer generated
- Gaussian
-0.5
Exponential
-
1.0
0.
5.
10.
15,
20.
25.
D isplacem ent, X (cm )
Figure 5.7: A random surface generated using squared-cosine humps, (a) a sample
surface profile, (b) the autocorrelation fun ctio n o f the surface as compared
w ith a Gaussian and an exponential fun ctio n w ith identical correlation
length.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
Table 5.2:
Constants used in the num erical com putations.
Individuc il hum p size
Case
No.
W i d th
Hei ght
No. o f humps
Length of
No. of
fo r each
surface
segments
min.
m ax.
m in.
max.
surface
segment
fo r a
(A)
(A)
(A )
(A)
segment
(A)
surface
1
0 .2
2 .0
0.0066
0.066
40
44
100
2
0.36
3.6
0 .0 1 2
0 .1 2 0
40
79
100
3
0.7
7.0
0.023
0.233
40
154
100
the Gaussian function.
The correlation fun ctio n o f this surface is very sim ila r to
the correlation functions determ ined from measured height profiles o f n a tu ra l rough
surfaces [Oh et al., 1992].
The backscattering coefficients for the surface at 5 G Hz w ith ks = 0.12 and
k l = 2.13 (Case
1
in Table 5.1) are computed by the M onte Carlo sim u la tio n tech­
nique fo r a homogeneous surface w ith e\ = e2 = 15 + f3 (Fig. 5.3(b)), and compared
w ith the analytical results based on the SPM as shown in Figs. 5.8(a) and (b ). For the
SPM solution, the scattering coefficient <r° is p ro po rtio n al to the roughness spectrum
(Fourier transform o f the correlation function). B oth the actual and Gaussian cor­
re la tion functions are used in the calculation o f the backscattering coefficients using
the SPM. I t is shown th a t the M onte Carlo sim ulation agrees very well w ith the SPM
p re dictio n when the actual correlation function is used. The discrepancies between
the M onte Carlo sim ulation and the SPM w ith Gaussian correlation fu n ctio n indicate
the im portance of the ta il section of the correlation function in the estim a tion o f a 0.
Using the first-order SPM solution [Ulaby et al., 1982; Tsang et al., 1982], it
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
(a)
©
This Technique
SPM, (numer. corn)
SPM, (Gauss, corr.)
' 0.
10.
20.
30 .
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
(b)
0. ---■
---1---'---1---'---1---'---1---1---1--->---1---'---1---■
---1---r
G
This Technique
SPM, (numer. corr.)
SPM, (Gauss, corr.)
’ 0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
Figure 5.8: B ackscattering coefficient o° o f the random surface w ith ks = 0 . 1 2 ,
k l = 2.13, and
= e2 = 15 + *3 as com puted by the SPM and the
num erical technique; (a) H H -polarization and (b) V V -p o la riza tio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
1.0
a
0.8
c
o
£
o
U
0.6
0 .4
0.2
0.0
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
F igure 5.9: Degree o f correlation a o f the random surface w ith ks =
and £i = e2 = 15 + iS.
0 .1 2 ,
k l = 2.13,
can easily been shown th a t the degree o f correlation (a ) fo r the phase difference
(4>w — <f>hh) is equal to u n ity ( a = l) . For the surface under consideration (case 1), Fig.
5.9 compares the values o f co-polarized a computed using the num erical sim ulation
w ith those derived from the SPM. The SPM is a first-order solution; hence, it predicts
th a t the degree o f correlation a between the H H - and V V - polarized scattering fields
is always equal to u n ity ( a = l) . A plot o f a , computed using the num erical sim ulation
technique, is shown in Fig. 5.9. A t sm all incidence angles (0; < 20°), the degree o f
correlation a w l , and then it decreases slowly as 0,- increases. I t should be noted
th a t the measured angular response o f a fo r smooth bare soil surfaces at Z-band
frequencies shows a sim ila r trend [Sarabandi et al., 1992; O h et al., 1993].
T he num erical sim ulation was also perform ed for a surface at 5 G Hz w ith ks =
0.42, k l =
7.49 (Case 3 in Table 5.1), and t\ = e2 = 15 + iS.
The roughness
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
-i 1-- .-- 1-- ■-- 1-- 1---- 1-1---1--■-- 1-- 1-- 1-- 1-- 1-- rO
This Technique
-------------------- PO,(numeric, corr.)
09
T>
PO, (Gaussian corr.)
8
u
u
J2
8
09
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
F igure 5.10: Backscattering coefficient a° o f the random surface w ith ks = 0.42,
k l = 7.49, and £i = e2 = 15 + i 3 as com puted by the PO m odel and
the num erical technique fo r H H -polarization.
parameters of this surface fa ll w ith in the v a lid ity region o f the physical optics (PO )
model; therefore the num erical solution can be compared w ith the PO solution. The
scattering coefficient a \ h predicted by the PO m odel using the actual correlation
fun ctio n agrees very well w ith the results computed by the num erical technique, as
shown in Fig.
5.10.
In this figure the PO solution using a Gaussian correlation
function w ith the same correlation length as the actual correlation fu n ctio n is also
compared w ith the num erical sim ulation. I t is shown th a t the agreement is good only
for low incidence angles ( 0 ,- <
20
°) and the discrepancy between the tw o solutions
becomes very significant fo r higher incidence angles.
In this case, s im ila r to the
previous case (S P M ), it is shown th a t the ta il of the correlation function plays an
im p o rta n t role in determ ining the angular patterns o f the backscattering coefficients.
W ith the success o f the M onte Carlo sim ulation in p re d ictin g the scattering be-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
(a)
/—
N
CQ
T3
-10.
°t>
-20.
Jtf
8
u
60
c
•c
§03
o
2
-30.
-40.
V V -p ol.
-50.
&
CQ
HH -poI.
-60.
e ,= (3 ,0.6)
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
(b)
ra
■o
8a
8
u
eo
a
•»*
1o
V V -p ol.
2
8
a
HH-pol.
e.
0.
10.
20.
= (12, 2.4)
30.
40.
50.
60.
70.
80.
90.
Incidence Angle (Degrees)
Figure 5.11: Backscattering coefficient a 0 o f the random surface w ith ks = 0.22,
k l = 3.8, (a) ei = 6 + i0.6 and e2 = 1 5 -H 3 , (b) ei = 12 + i2 .4 a n d
e2 = 15 4 - *3 fo r V V - and H H -polarizations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
-10.
e = 44(
Ok
— -0-----
W -p o l
— - 0— -
HH-pol
- 20.
0
8
u
60
a
•n
53
ts
u
J2
-30.
Gi­
-40.
«
-50.
0.0
3.0
6.0
9.0
12.0
15.0
18.0
Dielectric Constant, e,’ ( £j” = 0.2 £ ,’ )
F igure 5.12: The se nsitivity o f the backscattering coefficient cr° to the d ie le ctric con­
stant, in case o f ks = 0.22, k l = 3.8, and
— 15 -(- *3 at 9 = 44°.
havior o f rough surfaces in the sm all p e rtu rb a tio n and physical optics regions, the
num erical m odel can be used to stud y complex surfaces w ith interm ediate roughness
parameters and inhomogeneous d ie lectric profiles. For exam ple, consider an inhomogeneous surface at 5 GHz w ith ks = 0.22 and k l = 3.80 (Case 2 in Table 5.1). Figures
5.11(a) and 10 (b ) show the backscattering coefficients o f the surface fo r b o th polar­
izations w ith
€-2
= 15 + *3 and tw o values of ei, nam ely
= 3 + *0.6 and ei = 12-H2.4.
To dem onstrate the se nsitivity o f radar backscatter to the m oisture content o f the top
layer, th e scattered fields for three other surfaces w ith
= 6 + *1.2, £i = 9 + *1.8, and
€\ = 15 + *3 were also computed. The backscattering coefficients a° and the degree
o f correlation a are shown in Figs. 5.12 and 5.13, respectively, as functions o f the
d ie le ctric constants
at 9 = 44°. We note th a t the scattering coefficients, a°v and
Ojhj, as well as the ra tio o f a°v/ a ^ , increase as the d ie le ctric constant increases. The
degree o f correlation a also shows se nsitivity to the d ie le ctric constant o f the surface;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
1.0
0.9
0.8
0.7
0.6
0 = 44'
0.5
0.0
3.0
6.0
9.0
12.0
15.0
18.0
Dielectric Constant, e ,’ ( e ,” = 0.2 e / )
F igure 5.13: The se n sitivity o f the degree o f correlation a to the die lectric constant
in case o f ks = 0.22, k l = 3.8, and €2 = 15 + i3 a t 0 = 44°.
i.e., a decreases as £i increases. A ll of these trends are in lin e w ith experim ental
observations [Oh et al., 1992; Oh et al., 1993].
5.5
C onclu sion s
In this chapter an efficient M onte Carlo sim ulation technique is proposed for com­
p u tin g electrom agnetic scattering by inhomogeneous one-dimensional rough surfaces.
T he surface irreg u la ritie s are represented by inhomogeneous d ie le ctric humps o f d if­
ferent shapes and the u nderlying layer is represented by an impedance surface. A
m om ent-m ethod procedure, in conjunction w ith the exact image theory, is used for
calculation of the fie ld scattered by the die lectric humps.
I t was shown th a t the
scattered field near the impedance surface is weak , and hence th e effect o f m u ltip le
scattering between hum ps can be ignored.
To check the v a lid ity o f the Monte Carlo sim ulation, the num erical results were
compared w ith the existing analytical solutions for surfaces at extrem e roughness con-
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130
ditions. A sm ooth surface th a t satisfies the v a lid ity region of the SPM and a surface
th a t satisfies the v a lid ity region of the PO m odel were considered, and in b oth cases
excellent agreement was obtained between the analytical results and those com puted
using the proposed technique. I t was found th a t away fro m norm al incidence the
ta il o f the correlation function plays an im p o rta n t role in the determ ination o f the
backscattering coefficients.
T he analysis presented in this chapter is only fo r one-dimensional surfaces and
therefore is incapable o f predicting the cross-polarized scattering coefficients. A nu­
m erical sim u la tio n fo r a two-dim ensional rough surface using a sim ila r m ethod is
co m p u ta tio n a lly tractable.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER VI
M EASUREM ENT PROCEDURE - RADAR
CALIBRATION FOR DISTRIBUTED
TARGETS
6.1
In trod u ction
The recent interest in radar p o la rim e try has led to the development of several
ca lib ra tion techniques to retrieve the M u e lle r m a trix o f a d istrib ute d targe t from the
m u lti-p o la riz a tio n backscatter measurements recorded by the radar system. Because
a d is trib u te d target is regarded as a s ta tistica lly u nifo rm random m edium , the mea­
surements usually are conducted fo r a large num ber o f independent samples (usually
sp atia lly independent locations), from which the appropriate statistics characterizing
the elements o f the M ueller m a trix can be derived. E xistin g calibration methods rely
on tw o m a jo r assumptions. The firs t is th a t the illu m in a te d area of the d istrib ute d
target is regarded as a single equivalent p oint target located along the antenna’s
boresight direction, and th a t the statistics o f the scattering from all o f the measured
equivalent p o in t targets (representing the spatially independent samples observed by
the radar) are indeed the same as the actual scattering statistics of the d istrib ute d
target.
The second assumption pertains to the process by which the actual mea­
surements made by the radar fo r a given illu m in a te d area are transform ed in to the
131
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scattering m a trix of th a t area. The process involves measuring the ra d ar response
o f a p o in t ca lib ra tio n ta rg e t o f know n scattering m a trix , located along the boresight
dire ctio n o f the antenna, and then m o d ifying the measured response by a constant,
known as the illu m in a tio n integral, when observing the d istrib u te d target. The illu ­
m in a tio n in te g ra l accounts for only m agnitude variations o f the illu m in a tin g fields.
Thus, possible phase variations or antenna cross-talk variations (between orthogonal
pola riza tio n channels) across the beam are to ta lly ignored, which m ay compromise
the c a lib ra tio n accuracy. To re ctify th is deficiency o f e xisting ca lib ra tio n techniques,
a new technique is proposed w ith w hich the radar polarizatio n d is to rtio n m a trix is
characterized com pletely by measuring the p o la rim e tric response o f a sphere over the
e n tire m ain lobe o f the antenna, ra the r than along only the boresight d ire ction . A d d i­
tio n a lly , the concept of a “ d iffe re n tial M ueller m a trix ” is introduced, and by defining
and using a co rre lation-calibration m a trix derived fro m the measured ra d a r d istor­
tio n m atrices, the diffe re n tial M u e lle r m a trix is accurately calibrated. Comparison
o f data based on the previous and the new techniques shows significant im provem ent
in the measurement accuracy o f the co-polarized and cross-polarized phase difference
statistics.
The lite ra tu re contains a variety o f different methods fo r measuring the backscatte rin g cross section o f p o in t targets.
In a ll cases, however, the ca lib ra tio n p a rt of
the measurement process involves a comparison o f the measured radar response due
to the unknow n target w ith the measured response due to a ca lib ra tion target of
know n radar cross section. Under ideal conditions, b o th the unknown and calibra­
tio n targets are placed along the antenna boresight d ire ctio n , thereby in su rin g th a t
b o th targets are subjected to the same illu m in a tio n by the radar antenna. The s it­
u a tio n is m a rke d ly different fo r d istrib ute d targets; the unknown d is trib u te d target
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133
is illu m in a te d by the fu ll antenna beam, whereas the ca libration ta rg e t - being o f
necessity a p oint target - is illu m in a te d by o nly a narrow segment o f the beam cen­
tered around the boresight direction. Consequently, both the m agnitude and phase
variations across the antenna p atte rn become p a rt o f the measurement process.
A ccurate calibration o f p o la rim e tric radar systems is essential fo r e xtra ctin g ac­
curate biophysical in fo rm a tio n o f earth terrain.
The concept and fo rm u la tio n o f
p o la rim e tric calibration were developed by Barnes [Barnes, 1986], who used three
in-scene calibration targets. M any other p o la rim e tric ca lib ra tio n techniques were re­
ported, including the generalized ca lib ra tion technique (G C T ) [W h itt et al, 1991],
the isolated-antenna ca lib ra tion technique (IA C T ) [Sarabandi et al, 1990], and the
single-target calibration technique (S T C T ) [Sarabandi and Ulaby, 1991]. T h e phase
va ria tio n across the antenna p atte rn , however, has been ignored among those calibra­
tio n techniques w h ile the m agnitude va ria tion usually is taken into account through
a calculation of the illu m in a tio n integral [U laby et al., 1982; Sarabandi et al., 1991;
Tassoudji et al., 1989; and U laby and E lachi, 1990]. The role o f this phase variation
across the beam w ith regard to p o la rim e tric radar measurements and th e means for
ta k in g it into account in the measurement process are the subject of this paper.
Terrain Surfaces, in clu ding vegetation-covered and snow-covered ground, are treated
as random media w ith s ta tis tic a lly u n ifo rm properties. In radar measurements, the
qua n tities of interest are the sta tistica l properties o f the scattered field p e r u n it area.
One such q u a n tity is the scattering coefficient <7°, w hich is defined in term s o f the
second m om ent o f the scattered field:
W
< | E ■ |»>
where E ' and E s are the in cident and scattered fields, A is the illu m in a te d area, and r
is the range between the target area and the observation point. The above d e fin itio n of
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134
a ° is based on the assum ption th a t the target is illu m in a te d by a plane wave. Although
in practice such a condition cannot be absolutely satisfied, it can be approxim ately
satisfied under certain circumstances. The correlation length I o f a d istrib u te d target
represents the distance over which two points are lik e ly to be correlated, im p lyin g th a t
the currents induced at the two points due to an incident wave w ill lik e ly be correlated
as well. Thus, the correlation length may serve as the effective dimension o f in d ivid ua l
scatterers com prising the d istrib u te d target. The plane-wave a pp roxim ation may be
considered valid so long as the m agnitude and phase variations o f the incident wave
are very sm all across a distance o f several correlation lengths.
In m ost practical
situations, this “local” plane-wave approxim ation is almost always satisfied. When
this is n ot the case, the measured radar response w ill depend on b o th the illu m in a tio n
p a tte rn and the statistics o f the d istrib ute d target [Eom and Boerner, 1991; Fung and
Eom , 1983].
A n im p lie d assum ption in the preceding discussion is th a t the phase variation
across the antenna beam is the same for both the tra n sm it and receive antennas.
W hen m aking p o la rim e tric measurements w ith dual-polarized tra n s m it and receive
antennas, the phase va ria tion o f the tra n sm it and receive patterns m ay be different,
w hich m ay lead to errors in the measurement o f the scattering m a trix o f the target,
unless the variations are known fo r a ll of the p olarizatio n com binations used in the
measurement process and they are properly accounted fo r in the ca lib ra tio n process.
In this paper, we introduce a calibration procedure th a t accounts for m agnitude
and phase imbalances and antenna cross-talk across the entire m ain beam o f the
antenna.
B y applying th is procedure, we can make accurate measurements o f the
diffe re n tial M ueller m a trix o f a d istrib ute d target using the local plane-wave approx­
im a tio n . The differential M ueller m a trix can then be used to com pute the scattering
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135
Radar
Boresight direction
A S (x,y)
Random surface
Illuminated area
F igure 6.1: Geom etry o f a radar system illu m in a tin g a homogeneous distributed ta r­
get.
coefficient for any desired com bination o f receive and tra n s m it antenna polarizations,
and by em ploying a recently developed technique [Sarabandi, 1992], the statistics
o f the polarization phase differences can also be obtained.
B y way of illu s tra tin g
the u tilit y o f the proposed measurement technique, we w ill compare the results of
backscatter measurements acquired by a p o la rim e tric scatterom eter system fo r bare
soil surfaces using the new technique w ith those based on ca lib ra tin g the system w ith
the tra d itio n a l approach which relies on measuring the response due to a calibration
target placed along only the boresight direction o f the antenna beam.
6.2
T h eo ry
Consider a planar d istrib ute d target illu m in a te d by a p o la rim e tric radar system
as shown in Figure 6.1. Suppose the d istrib u te d target is s ta tis tic a lly homogeneous
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136
and the antenna beam is narrow enough so th a t the backscattering statistics of the
target can be assumed constant over the illu m in a te d area. Let us subdivide the illu ­
m in a tio n area in to a fin ite num ber o f pixels, each in cluding m any scatterers (or m any
correlation lengths) and denote the scattering m a trix of the i j t h p ixe l by A S [ x { , y j) .
The scattering m a trix o f each pixel can be considered as a com plex random vector.
I f the radar system and its antenna are ideal, the scattered field associated w ith the
i j t h p ixe l is related to the in cid en t fie ld by
1
p 2 i k Qr ( x i ty j )
A S vv(x{, y j )
A S Vh(xi i y j )
ei
K
AShv{xii
V j
)
AShhfoii
Vj)
■sf
E{
i
where E v and Eh are the components o f the e le ctric field along tw o orthogonal direc­
tions in a plane perpendicular to the d irection o f propagation, and K is a constant. In
reality, radar systems are not ideal in the sense th a t the ve rtica l and horizontal chan­
nels o f the tra n s m itte r and receiver are not id e n tica l and the radar antenna introduces
some coupling between the v e rtica l and horizontal signals a t b o th transm ission and
reception. Consequently, the measured scattering m a trix U is related to the actual
scattering m a trix o f a p o in t targe t S by [Sarabandi and Ulaby, 1990]
p 2 i h 0r __________
U = —
R S T
(6.2)
where R and T are known as the receive and tra n s m it d isto rtio n m atrices. For sm all
point targets where the illu m in a tio n p atte rn o f the incident field can be approxim ated
by a u n ifo rm plane wave, measurement o f S is ra th e r straightforw ard and in recent
years this problem has been investigated tho ro u gh ly by m any investigators [Barnes,
1986; W h itt et al., 1991; Sarabandi et al., 1990].
The d is to rtio n m atrices are ob­
tained by measuring one or m ore targets of known scattering m atrices, and then by
in ve rting (6.2) the scattering m a trix o f the unknow n target is obtained. In the case o f
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137
d is trib u te d targets, however, d istrib u te d ca lib ra tion targets do not exist. Moreover,
the d is to rtio n matrices and the distance to the scattering points are all functions of
position. T h a t is for the i j t h p ixel the measured d iffe re n tial scattering m a trix A U
can be expressed by
g 2»'fcor(a:,' ,Vj) __
A S r„ „ ( x , ', y j )
A S vh ( x i , y j )
T (x i,y j)
(6.3)
A Shv(xi,yj) A Shh(x{,yj)
The radar measures the sum o f fields backscattered fro m all pixels w ith in the illu m i­
nated area coherently; i.e.,
e2ikoT(xi,yj)_
_
—
u = J 2 Y , ~ 2 (a..
R (x n Vi) A S (® i, y j) T ( x ;, y j)
(6.4)
Thus, the measured scattering m a trix is a linear fu n c tio n o f the random scattering
m atrices o f the pixels. For u nifo rm d istrib u te d targets, we are interested in deriving
in fo rm a tio n about the statistics o f the differential scattering m a trix from statistics of
the measured scattering m a trix U . One step in re la ting the desired quantities to the
measured ones is to perform a ca lib ra tion procedure to remove the distortions caused
by the radar and the antenna systems. The tra d itio n a l approach used for ca lib ra ting
p o la rim e tric measurements of extended-area targets relies on two approxim ations.
F irst, it is assumed th a t for each measured sample, the differential scattering m a trix
o f the illu m in a te d area is equal to some equivalent scattering m a trix at boresight.
Using this approxim ation it is hoped th a t the equivalent scattering m a trix has the
same statistics as the original differential scattering m a trix . This approxim ation is
p urely h eu ristic and cannot be ju s tifie d m athem atically. Second, the measured data
for each sample is calibrated as i f i t were a p oint ta rg e t and the result is m odified
by a constant known as the illu m in a tio n integral to account for the non-uniform
illu m in a tio n [Tassoudji et al, 1989; U laby and Elachi, 1990]; thus, the cross-talk
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138
variations away from the antenna’s boresight direction over the illu m in a te d area are
ignored.
The illu m in a tio n integral accounts fo r only m agnitude variations of the
gain patterns o f the tra n s m itte r and receiver antennas, and no provision is made for
accounting for any possible phase variations in the ra d iatio n patterns.
In this paper we a tte m p t to derive the second moments o f the diffe re n tial scattering
m a trix fro m the statistics o f the measured m a trix w ith o u t m aking any approxim ation
in the radar d isto rtio n m atrices or using the equivalent differential scattering m a trix
representation. In random polarim etry, the scattering characteristics o f a d istrib u te d
targe t usually are represented by its M ueller m a trix , w hich is the averaged Stokes
m a trix [Ulaby and Elachi, 1990]. T he M u e lle r m a trix contains the second moments
o f the the scattering m a trix elements. B y the central lim it theorem, i f the scatterers
in the illu m in a te d area are numerous and are o f the same type, then the statistics
describing the scattering is Gaussian (Rayleigh statistics). In such cases, knowledge
o f the M ueller m a trix is sufficient to describe the scattering statistics of the target
[Sarabandi, 1992].
In a manner analogous w ith the d efin itio n o f the scattering coefficient as the
scattering cross section per u n it area, let us define the differential M u e lle r m a trix M
as the ra tio of the the M u e lle r m a trix ( A M ) derived from the d iffe re n tial scattering
m a trix (A S ) to the d iffe re n tial area; i.e.,
M
=
lim
A A -fO
AM
. ,
AA
To compute the d iffe re n tial M ueller m a trix , the ensemble average o f the cross products
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139
o f the differential m a trix components are needed. Let us define
W
Co* c °
Co* c °
<
°v v °v v
>
<
^vh^vh
<
CO* QO
^hv
hv
>
<
° h h D hh
CO* C°
>
<
^hh^vh
>
<
°vk°hh
=
<
J h v J vv
<
^ v v '-'h v
Co* c °
Co* c °
CO* Co
Co* C°
Co* c °
>
<
°v h °v v
>
<
dhh^hv
>
<
dhh^vv
>
<
dvh^hv
Co* c °
Co* c °
Co* Co
>
<
> <
Co* c °
>
Co* c °
>
J h v ‘-) h.h.
Co* c °
(6.5)
>
<
^hv^vh
>
>
<
Co* c °
‘- ^ v v ^ h h
>
where
< s;;s;t >= &H
m
A -+ 0
< A S ; qA S 3t >
AA
:o
In term s of the correlation m a trix W , the differential M ueller m a trix can be com­
puted from
M
= 4 7tj/W
v~ l
(6.6)
where [Ulaby and E lachi, 1990]
/
1 0
0
0
01 0
0
0
01
1
0
0 —i
i
\
v =
\
/
In order to calibrate a radar system so as to measure the d iffe re n tial M u e lle r m a trix,
let us represent each 2 x 2 m a trix in (6.4) by a corresponding four-com ponent vector,
in w hich case (6.4) sim plifies to
_
e2ikor(Xi'yi)-—
u =
«
___
...\ D (x «;. > y j)A S (x u y j)
’ (*<> Vi)
j
(6.7)
where
uvv
uvh
uhv
. Uhh.
A S vv(xi, y j)
A S vh (x{, y j)
,
A S (x i,y j) =
(6.8)
AShv(x{, y j)
A Shh(xi,yj)
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140
1
i
a
sS
jr
Ryh ( x i ) V j )
, T ( x i , y j)
'R-ixi, y j) —
Rhy ( ^
yj)
R v h ^ ii
yj)
—
Thv(x i) y j )
i 7y j )
Rhh(x ii
Tyy(x j ,
jyj)
yj)
and it can be easily shown th a t
RvvTvv
RyyTyh
RvhTyv
RvhTvh
RwThv RwThh RvhThv RvhThh
(6.9)
D (*i» Vj) =
RhvTw
RhvTvh
RhhTyv
RhhTyh
RhyThy
RkyThh
RhhThy
RhhThh
Jl________________________________________ _
The m
component o f the measured targe t vector (Um) defined by (6.8) can be
obtained fro m (6.7) and is given by
_2tfcor(xj,yj)
i j
D me( x i , y j) A S e(xi, y j)
r 2( x i , y j )
Thus, the averaged cross products of these components are
<
>
=
E E
W)
< ASe(xi,yj)AS*(xi>,yj>) > ( 6 .10)
I f the num ber o f scatterers in each pixel is assumed to be large, or the correlation
le n gth o f the surface is much smaller than the p ixe l dimensions, then
0
i ^ V and j ^ j 1
< S (S °* > A A i j
i = i ' and j = j '
< A S e ( x i , y j ) & S * ( x i', y j') > =
I t should be m entioned here again th a t the target is assumed to be sta tistica lly
homogeneous and the antenna beam is assumed to have a narrow beam.
<
Hence
> is n ot a fun ctio n of position w ith in the illu m in a te d area. In the lim it as
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141
A A approaches zero, (6.10) takes the follow ing form
4
4
< l« C > = £ £
t= 1 p = 1
//^
- D me( x , y ) D * ( x , y ) d x d y
, V)
<S?S?m>
(6.11)
npv
E quation (6.11) is valid fo r all com binations o f m and n and, therefore, i t constitutes
16 equations for the 16 correlation unknowns. Let us denote the measured correlations
by a 16-component vector
y
and the actual correlations by another 16-component
vector X so th a t
X{ = < S \S a
; >
; i = 4 ( * - l) + j>
y j = < Umli* >
; j = 4(m - 1) + n
In th is form , (6.11) reduces to the follow ing m a trix equation
y
= Il
X
(6.12)
where the i j element o f B is given by
bii = J L
h (x
y j D m t ( x ' y ) D np(x ’ y ) d x d y
(6-13)
and as before,
i = 4(£ — 1) -f p ; j = 4(m — 1) + n
Once the elements o f the correlation calibration m a trix B are found fro m
(6.13),
equation (6.12) can be inverted to o bta in the correlation vector X . T he elements of
the correlation vector are not a rb itra ry complex numbers; fo r example, X 2 and X5 are
com plex conjugate o f each other and X \ is a real num ber; thus, these relationships
can be used as a crite rio n for ca lib ra tion accuracy. The d iffe re n tia l M u e lle r m a trix
can be obtained fro m the correlation m a trix W
whose entries in term s o f the vector
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142
V-Transmitter
Circulator
Direction
Circulator
H-Transmitter
(V &
O-D
Antennas
V-Receiver
F igure 6.2:
H-Receiver
Sim plified block diagram o f a p o la rim e tric radar system.
X are given by
*i
*e
X2
X5
X \\
*16
x12
*15
*3
*s
X4
*7
* 9
*14
* io
*13
E valuation o f the elements of B requires knowledge o f the radar d isto rtio n m atrices
over the m ain lobe o f the antenna system. The d isto rtio n m atrices of the radar can
be found by app lyin g the ca lib ra tion m ethod presented in the n ext section.
6.3
C alib ration P roced ure
As was shown in the previous section, the correlation vector X can be obtained if
the ca lib ra tio n m a trix D(a;, y ) given by (6.9) is known. A sim p lified block diagram o f
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143
a radar system is shown in Fig. 6.2. The quantities 4 , 4 , 4 , 4 represent flu ctu a tin g
factors o f the channel imbalances caused by the active devices in the radar system.
W ith o u t loss o f generality, i t is assumed th a t the n om inal value of these factors is
one and th e ir rate o f change determines how often the radar m ust be calibrated.
T he antenna system also causes some channel d is to rtio n due to variations in the
antenna p a tte rn and path length differences. The cross-talk contam ination occurs in
the antenna stru cture which is also a function o f the direction o f ra d ia tio n . I t has
been shown th a t the antenna system , together w ith tw o orthogonal directions in a
plane perpendicular to the dire ction o f propagation, can be represented as a fourp o rt passive netw ork [Sarabandi ana Ulaby, 1990]. Using the re cip ro city properties
o f passive networks, the d isto rtio n m atrices o f the antenna system were shown to be
[Sarabandi ana Ulaby, 1990]
ft»(0,O =
r„(0,£)
0
1
(6.14)
0
T.(0,0 =
< 7 (0 ,0
1
< 7 (0 ,0
(7 (0 , 0
4 ( 0 ,0
1
0
(6.15)
< 7(0,0
1
J [
°
4 (0 ,0
where 0 ,£ are some coordinate angles defined w ith respect to the boresight d irection
o f propagation. The q ua n tity < 7 (0 ,0 is the antenna cross-talk factor and r „ ( 0 , 0 ,
0 ) 4 ( 0 , 0 ) 4 ( 0 ) 0 are the channel imbalances caused by the antenna system.
These quantities are not subject to change due to variations in active devices and
once they are determ ined, they can be used repeatedly.
In order to fin d the radar d isto rtio n parameters at a given p o in t (x, y ) on the
surface, we firs t need to specify a convenient coordinate system w ith respect to the
antenna’s boresight direction so th a t the distortions become independent o f incidence
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144
^
Boresight
F igure 6.3: A zim u th-ove r elevation and elevation-over a zim u th coordinate systems
(ip, £) specifying a p o in t on the surface of a sphere.
angle and range to the target. The azim uth-over-elevation and elevation-over-azim uth
coordinate angles (?/>,£) provide a coordinate system th a t is appropriate fo r antenna
p a tte rn measurements. The angle f specifies the elevation angle and ip specifies the
azim u th angle in a plane w ith elevation f as shown in Fig. 6.3. The m apping from
( ^ , £ ) coordinates to (x , y ) coordinates can be obtained by considering a radar at
height h w ith incidence angle do and the boresight d irection in the y —z plane as
shown in Figs. 6.4(a) and (b ) fo r azim uth-over-elevation and elevation-over-azim uth
coordinate systems, respectively. I t is easy to show th a t co n sta n t-f curves on surface
o f a sphere map to constant-j/ lines and co n sta n t-^ curves m ap to hyperbolic curves in
case of azim uth-over-elevation coordinate system. The m apping functions are given
by
h ta n ip
cos(Q0 + 0
’
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
constant
constant
x
(b)
z
constant \|/
constant ^
F igure 6.4: G eom etry o f a radar above x-y plane and tra nsform ation to cartesian
coordinates from (a) azim uth-over-elevation coordinate and (b) elevationover-azim uth coordinate .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
y = h tan (0 o + f )
where ip = f = 0 represents the boresight direction. For the elevation-over-azim uth
coordinate system, constant-^ curves on surface of a sphere m ap to e llip tic curves
and constant-^ curves m ap to straight lines e m ittin g fro m one o f focus points o f the
ellipsoids, and the m apping functions are given by by
x =
V=
hsim p
cos9o cosip — sin0o t a n f ) ’
/i(sin0o cosip + cos0o t a n f )
cos 0o cos ip — sin 0o tan if)
T h e entries of the ca lib ra tion m a trix D (ip, f ) as defined by (6.9) should be obtained
through a ca lib ra tion procedure. Following the single target ca lib ra tion technique
given in [Sarabandi and Ulaby, 1990], a single sphere is sufficient to determ ine the
channel imbalances as well as the antenna cross-talk fa cto r fo r a given direction.
Hence, by placing a sphere w ith radar cross section a 3 at a distance r 0 and a direction
( i p , 0 w ith respect to the radar, the receive and tra n sm it d is to rtio n parameters can
be obtained as follows:
p
nr
-itt/v * vv
—
*“
U3
r 2 „ - 2ik o r0
*o
(1 + C 2) y j crs/ Air
± R
2C
hh
R vv
a
(1 + C 2) U ; h
a Thh
1+ c 2U 3h
Tvv
C
U shh _ 1 + C 2 U h
3v
2C
=
U 3V
2C
U 3V
(6.16)
± - ^ ( 1 - \ / l - a)
V a,
where
A UyhUhv
Uvvu
L U h.h
i
and U
is the measured (uncalibrated) response of the sphere at a specific direction
(V>,f). In terms o f the known quantities given by (6.16), the ca lib ra tio n m a trix D
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
can be w ritte n as
1
Ca
C
C 2a
C
a
C 2 Ca
C/3
C 2a/3
/3
(6.17)
D(V>,£) = RvvTv,
C 2/3 Ca/3
Ca/3
C/3 a/3
where the dependencies on r/> and £ o f a ll parameters is understood.
In practice it is impossible to measure the sphere for a ll values o f rp and £ w ith in
the desired dom ain; however, by discretizing the dom ain o f rp and £ (m ain lobe)
in to sufficiently sm all subdomains over which the antenna characteristics are almost
constant, the in teg ra l given by (6.13) can be evaluated w ith good accuracy.
P olarim e tric measurement of a sphere over the e ntire range o f rp and £ is very
tim e consuming, and under field conditions perform ing these measurements seems
impossible.
However this measurement can be perform ed in an anechoic chamber
w ith the desired resolution Arp and A f only once, and then under field conditions we
need to measure the sphere response only at boresight to keep tra ck of variations in
the active devices. W ith o u t loss o f generality, let us assume th a t f v = f/, = t v = £/t = 1
fo r the sphere measurements when perform ed in the anechoic chamber, and th a t these
quantities can assume other values fo r the measurements made under field conditions.
I f the measured d istortion parameters at boresight (field co nd itio n ) are denoted by
prim e and calculated from (6.16), then the channel imbalances corresponding to the
field measurements are
f f
= ( ^*0 \2 —2ifcnfrl—rn)
h
=
iv
ffc
rv
7 1 (0 ,0 ) '7 k ( 0 , 0 )
_ ^ ( 0 , 0 ) , -fiy„(0, 0)
^ C ( 0 ,0 )
^ (0 ,0 )
V
T U M
t vv(
Q)^vv(Q> 0)
Rw(o, o),rw(o,o)
0,0)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(6.18)
148
Now the ca lib ra tio n m a trix at any d ire ction (D (0 , £)) can be obtained fro m (6.17)
by replacing RyVTvv, a , and /? by R'VVT^V, a 1, and j5' where
R yyTvV
=
rJ v R w T y y
a' =
&-OC
(6.19)
*v
(?= £/?.
Having found the calibration m atrices fo r all subdomains, the element i j o f the
co rre lation-calibration m a trix (B ), as given by (6.13) takes the follow ing form
where f l is the solid angle subtented by the illu m in a te d area (m ain lobe o f the an­
tenna) and
is the Jacobian fo r the transform ation o f integral variables.
Since r(^>, £) and |J ( x , y, Vs£)| in the azim uth-over-elevation coordinate system are
computed as, respectively,
= cos^cos(0o + O
COS2
COS3 ( ^ 0 +
0
’
the i j t h element o f the m a trix B in (6.20) in the azim uth-over-elevation coordinate
system is given by
b ij= J L
(6.21)
In the elevation-over-azim uth coordinate system, r ( 0 , f ) and \J(x ,y,iJ;,£ )\ are com­
puted as, respectively,
cos £ (cos 9q cos %j) — sin 0Ota n f )
01 = — r r : —
:—
- —
r-
cos2f (cos 6q cos xp — sin 60 tan £)"
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
the ijfth element o f the m a trix B in (6.20) in the elevation-over-azim uth coordinate
system is given by
i
bn
6.4
f f
n /-\ cos2 f | cos 60 cos xb — sin 80 ta n £ )| ..
D me{xJ>, ()D*np{il>, 0
i!
2
r ------------ 2-------i l l dlpd{ '
( 6 >2 2 )
E x p erim en ta l P ro ced u re and C om parison
To dem onstrate the performance o f the new ca lib ra tion technique, the p o la rim e t­
ric response o f a random rough surface was measured by a tru ck-m o u nte d L-, C-,
and X -b an d p o la rim e tric scatterom eter w ith center frequencies a t 1.25, 5.3 and 9.5
GHz.. P rio r to these measurements, each scatterometer was calibrated in an anechoic
chamber. The scatterom eter was m ounted on an azim uth-over-elevation positioner
at one end o f the chamber and a 36cm m e tallic sphere was positioned at the antenna
boresight at a distance of 12m. Then the p o la rim e tric response of the sphere was mea­
sured over the m ainlobe of the antenna. The sphere measurements a t L-band, which
has the w idest beam o f the three systems, was perform ed over ( 0 ,C) £ [—2 1 °, + 2 1 °]
in steps o f 3° and the ranges o f (%/>, ( ) fo r C- and X-band were ±10.5° and ± 7 ° w ith
steps o f 1.5° and 1°, respectively. To im prove the signal to noise ra tio by rem oving the
background co n trib u tio n , the chamber in the absence o f the sphere was also measured
for a ll values o f
0
and (.
Figures 6.5(a) and (b) show the co- and cross-polarized responses o f the sphere at
X -band, and Figs. 6.5(c) and (d) show the co- and cross-polarized phase differences
{(f>hh ~ $vv, <i>hv — <t>w)- S im ilar patterns were obtained for L- and C-band. Using the
sphere responses the correlation-calibration matrices were determ ined as o utlined in
the previous section.
To evaluate the im provem ent provided by the new calibration technique, we shall
compare results o f p o la rim e tric observations o f a bare soil surface processed using the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
(a)
(b)
SigmafVV)10!
Sigma(VH)^T
-201
-40!
(d)
(c)
P h i(H H /V \$ 0'
P h i(V H /V \$ 0]
Figure 6.5: P ola rim e tric response o f a m e ta llic sphere over the entire m ainlobe o f X band scatterometer; N orm alize crvv (a) corresponds to G l and norm alized
crvh (b ) corresponds to GvGh\ Phase difference between co-polarized (c)
and cross-polarized (d) components o f the sphere response correspond to
phase va ria tion o f the co- and cross-polarized patterns o f the antenna.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
new technique w ith those obtained previously on the basis o f the boresight-only ca li­
b ra tio n technique. The data were acquired fro m a truck-m ounted 17-m high p la tfo rm
for a rough surface w ith measured rm s height o f 0.56cm and correlation length of
8
cm. The p o la rim e tric backscatter response was measured as a fu n ctio n o f incidence
angle over the range 20° — 70°. To reduce the effect of speckle on the measured data,
100
sp a tia lly independent samples were measured at each frequency and incidence
angle. Also, the response of the sphere a t boresight was measured to account fo r any
possible changes in the active devices. T he collected backscatter d ata was calibrated
by the new and old methods. The firs t test o f the new accuracy o f the calibration
a lg o rith m was to make sure th a t the components o f the correlation vector X satisfy
th e ir m u tua l relationships as explained in the previous sections. For a ll cases, these
relationships were found to be valid w ith in ±0.05% .
T h e second step in the evaluation process is the relative comparison o f the backscatte rin g coefficients and phase statistics derived fro m the tw o techniques.
6 .6
Figures
(a )-(c) show the co- and cross-polarized backscattering coefficients as a fun ctio n
o f incidence angle, calibrated by the old and the new methods. The differences in
backscattering coefficients, as shown in these figures, are less tha n 0.75dB. I t was
found th a t the difference in backscattering coefficients is less tha n 1 dB for all fre ­
quencies and incidence angles. A lthough ld B error in <r° m ay seem negligible, in some
cases, such as the va ria tion w ith soil m oisture content for w hich the to ta l dynam ic
range o f <r° is about 5dB, the ld B error becomes significant. F igure
6 .6
(d) shows the
ra tio o f tw o cross-polarized scattering coefficients after ca lib ra tion by each of the two
m ethods. T h eo re tica lly this ra tio m ust be one and independent o f incidence angle.
In th is figure it is shown th a t the new ca lib ra tio n m ethod m ore closely agrees w ith
the o re tical expectations than the old m ethod.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
(a)
-
(b)
10.
-
- 12.
10.
- 12.
-14.
-14.
3
>
>
-16.
JS
-18.
-18.
—
ol d Tech.
O ld Tech.
- 20.
-20.
—
0
.— . New Tech.
— E>—
- 22 .
10.
-16.
-
20.
30.
40.
50.
60.
70.
New Tech.
22 .
80.
10.
20.
Incidence Angle (Degrees)
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(d)
-i—>— i—'—r
1.30
(rms error)
1.25
- 22 .
1.20
w
\\
-24.
(ap)
\ \\ >
\o .
-26.
o—-
Old Tech. (0.076)
□
New Tech. (0.020)
1.15
1.10
1.05
\\
-28.
-30.
GO
o
Old Tech.
— 0 — . N ew Tech.
-32. - i I __■ ■ ■__ i__I__ ___ L.
10. 20. 30. 40. 50. 60.
1.00
'S
O
©
0
0
O
©
0
-H ................... 0..............
0
Q
0.95
70.
Incidence Angle (Degrees)
80.
0.90
10.
-I
20.
I
I
30.
40.
■
i
50.
■
L.
60.
70.
80.
Incidence Angle (Degrees)
Figure 6 .6 : Comparison between the new and old ca lib ra tio n techniques applied to
the X -band measured backscatter from a bare soil surface; (a), (b ), and
(c) show the difference in the co- and cross-polarized backscattering coef­
ficients and (d) demonstrates the enhancement in the ra tio of the cross­
polarized backscattering coefficients obtained b y the new method.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
153
(a)
1.0
e
0
1
8o
U
u
a
00
o
Q
T
0.8
T
0
-
0
------- -Q
"0-
_
'©
0.6
0.4
0.2
'q
- 0 —-
New Tech.
~G— • Old Tech.
at 1.25 GHz
I__
0.0
20.
I
30.
i
L
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(b)
1.0
o........
c
o
-4— »
a
13
a
o
U
<o
u.
oo
0>
........
0.8
©
0
—
_
-0
-O
-
0.6
0.4
0.2
-
0
—
--0-—
N ew Tech.
Old Tech.
at 5.3 GHz
_i
0.0
10.
20.
30.
i
i__
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
154
(C)
1.0
c
0
1
8o
U
<0
00
<0
Q
n— 1— r
0.8
Q-
0.6
......
-a .
'*■©
13-— - B-...
0.4
N ew Tech.
- - Q
0.2
- -
I
0.0
10.
20.
Old Tech.
at 9.5 GHz
_j
.
30.
i
■ '__ i
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 6.7: Degree o f correlation fo r co-polarized components o f the scattering m a trix
fo r L-band (a), C-band (b ), and X -band (c).
The th ird step involves a comparison o f the phase difference statistics o f the dis­
trib u te d target. I t has been shown th a t when the dimensions o f the antenna fo o tp rin t
are much larger tha n the correlation length, the p ro b a b ility density fun ctio n (p d f)
of the phase differences can be expressed in term s o f tw o parameters: the degree of
correlation (a ) and the polarized-phase-difference (£) [Sarabandi, 1992]. The degree
o f correlation is a measure o f the w id th o f the p d f and the polarized-phase-difference
represents the phase difference at w hich the p d f is m axim um . These parameters can
be computed d ire c tly from the components o f the M u e lle r m a trix and are given by
[Sarabandi, 1992]
_
1_
/ (A 433 + M 4 4 ) 2 -f- ( M 3 4 — A d43)^
2y
A/i i i A /i22
f
,
C = ta n
_1
/ M -Z i — M - iZ
~rz
n —
V Adzz T M 4 4
Param eter a varies from zero to one, where zero corresponds to a uniform d istrib u tio n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
and one corresponds to a delta-function d is trib u tio n (fu lly polarized wave). Parameter
£ varies between —180° and 180°.
Figures 6.7(a)-(c) show the degree of correlation calculated by the new and old
m ethods fo r the co-polarized phase difference ((f>hh — <f>w) at L -, C-, and X-band, re­
spectively. There is a significant difference between the tw o methods in a ll cases. The
p a rtia lly polarized backscattered Stokes vector obtained by the old ca lib ra tio n m ethod
appears more unpolarized tha n the Stokes vector obtained by the new m ethod. The
v irtu e o f this result can be checked in the lim itin g case if an a na lytica l solution is
available. A first-order solution o f the small p e rtu rb a tio n m ethod for s lig h tly rough
surfaces shows th a t the backscatter signal is fu lly polarized and, therefore, the p d f
of the co-polarized phase difference is a d elta fun ctio n, corresponding to a = 1. The
roughness parameters of the surface under in ve stiga tio n falls w ith in the v a lid ity re­
gion o f the sm all p e rtu rb a tio n m ethod at L-band. T he value o f a at L-band derived
fro m the new ca lib ra tion m ethod is in m uch closer agreement w ith theoretical ex­
pectations tha n the value obtained by the o ld m ethod. Figures 6.8(a)-(c) show plots
o f the co-polarized phase difference at L-, C-, and X -band, respectively. A t L- and
X -band the value of ( obtained by the two m ethods are positive and not very differ­
ent fro m each other. Also, i t noted th a t £ has a positive slope w ith incidence angle.
However, this is not the case fo r C-band; the value o f ( obtained by the old m ethod
is negative has a negative slope w hile the behavior o f £ obtained by the new m ethod
is very s im ila r to th a t at the other tw o frequencies. Figure 6.9 shows the deviation
between the PDFs (p ro b a b ility density functions) o f co-polarized phase difference o f
the tw o methods for C-band a t 30°. T his deviation shown in Fig. 6.8(b) and Fig. 6.9
is due to th e large variation o f phase difference between the V - and H-channels of the
C-band radar over the illu m in a tio n area, and since the old m ethod does not account
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
156
(a )
U
OQ
a
— <•>---
New Tech.
— E>—
Old Tech.
40.
30.
cd
£i
T3
20.
/ /
V
...........
o.
20.
\©
0
y -o
10.
'S
(2
at 1.25 GHz
_!
i
—j-,'"
q
I__
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(b)
U
00
&
w
V
c
21
{§
U-i
sI
0
s
£
1
i
c2
20.
— ©■— ■ New Tech.
ej.—
-10.
-20.
o ld Tech.
at 5.3 GHz
■T3-—
-30.
-a .
TD-
-50.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
(C)
&
&
50.
T
1— ■— r
New Tech.
40.
--Q- —
88
u
3
£i
•a
N
I
£
30.
Old Tech.
at 9.5 GHz
J9
20.
10.
a --
_L
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 6.8: Polarized-phase-difference for co-polarized components o f the scattering
m a trix fo r L-band (a), C-band (b), and X -band (c).
fo r phase variations, it is incapable o f correcting the resulting errors. S im ila r results
were observed fo r the statistics of the cross-polarized phase difference (<j>hv — <f>vv)-
6.5
C on clu sion s
A rigorous m ethod is presented fo r ca lib ra tin g p o la rim e tric backscatter measure­
ments o f d istrib u te d targets. B y characterizing the radar distortion s over the entire
m ainlobe o f the antenna, the differential M u eller m a trix is derived fro m the mea­
sured scattering m atrices w ith a high degree o f accuracy. I t is shown th a t the radar
d istortion s can be determ ined by measuring the p o la rim e tric response o f a m etallic
sphere over the m ainlobe o f the antenna. The radar distortions are categorized into
tw o groups, namely, distortions caused by the active devices and d istortions caused
by the antenna stru ctu re (passive). Since passive distortions are im m une to changes
once th e y are determ ined, they can be used repeatedly. The active distortions can
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
158
1.00
0.90
New Tech.
0.80
• Old Tech.
0.70
0.60
Q
»S'
w
•H
•s
i
0.50
0.40
0.30
0.20
0.10
0.00
-180. -135.
-90.
-45.
0.
45.
90.
135.
180.
^hh-^w (Degrees)
F igure 6.9: P ro b a b ility density functions (P D F ) for co-polarized phase-difference for
C-band at 30°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
be obtained by m easuring the sphere response only at boresight, thereby reducing
the tim e required fo r calibration under field conditions. The ca lib ra tio n a lg orith m
was applied to backscatter data collected fro m a rough surface by L-, C-, and X band scatterometers. Comparison of results obtained w ith the new a lg orith m w ith
the results derived fro m old ca libration m ethod show th a t the discrepancy between
the tw o methods is less ld B fo r the backscattering coefficients.
T he discrepancy,
however, is more drastic for the phase-difference statistics, in d ic a tin g th a t removal
o f the radar distortions from the cross products of the scattering m a trix elements
(d ifferen tia l M u e lle r m a trix elements) cannot be accomplished w ith the tra d itio n a l
ca lib ra tion methods.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A P T E R V II
MICROWAVE POLARIMETRIC RADAR
MEASUREMENTS OF BARE SOIL
SURFACES, A SEMI-EMPIRICAL MODEL
AND A N INVERSION TECHNIQUE
7.1
IN T R O D U C T IO N
P ola rim e tric radar measurements were conducted fo r bare soil surfaces under a
variety o f roughness and m oisture conditions at L-, C-, and X-band frequencies at
incidence angles ranging fro m 10° to 70°. Using a laser p rofile m eter and dielectric
probes, a complete and accurate set of ground tru th data were collected fo r each
surface condition, fro m which accurate measurements were made o f the rms height,
correlation length, and dielectric constant. The angular and spectral dependencies
of co-polarized ra tio (<T°v/er^h) and cross-polarized ra tio
( c r ° ^ / c r ° „ )
fo r a wide range
of roughnesses and m oisture conditions are examined. Based on knowledge of the
scattering behavior in lim itin g cases and the experim ental observations, an em pirical
model was developed for cr°v, a°lh, and <r£w in term s of ks (where k = 27t/A is the
wave num ber and s is the rm s height) and the relative dielectric constant of the
soil surface. The model, which was found to yie ld very good agreement w ith the
backscattering measurements o f this study as well as w ith measurements reported in
other investigations, was used to develop an inversion technique for pre dictin g the rms
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
height o f the surface and its m oisture content fro m m u ltip olarize d radar observations.
Investigation o f the radar backscattering response o f n atural surfaces is of great
im portance because o f its p o te n tia l in re trie vin g the desired physical surface param­
eters. Soil m o isture content and surface roughness are two such param eters.
The pro ble m o f electrom agnetic wave scattering w ith random surfaces have long
been studied and because o f its com plexity theoretical solution exist o nly for lim it ­
ing cases. W hen deviation o f the surface profile is s lig h tly different fro m th a t o f a
smooth surface, p e rtu rb a tio n solutions can be used. In the classic tre a tm e n t of small
p e rtu rb a tio n m e thod (SPM ) [Rice, 1951; Tsang et ah, 1985] i t is required th a t the
rm s height be m uch smaller than the wavelength and the rms slope be same order
o f m agnitude as the wavenumber tim es the rms height.
Recently, a p ertu rb a tion
m ethod based on p e rtu rb a tio n expansion o f phase o f the surface fie ld (P P M ) was
developed w hich extends the region of v a lid ity of SPM to higher rm s height but w ith
modest slope and curvature [W ineberner and Ishim aru, 1985]. The oth e r lim itin g
case is when surface irreg u la ritie s are large compared to the wavelength, nam ely the
radius o f curvature a t each p o in t on the surface is large. In th is lim it the solution
is known as K irc h h o ff approxim ation (K A ) [Beckmann and Spizzichino, 1963; Ulaby
et ah, 1982]. Various types o f m odifications and im provem ents to this m odel can be
found in lite ra tu re . In these papers the effects of shadowing and m u ltip le scattering
are discussed w hich basically extends the region o f K A slig h tly [Fung and Eom , 1981].
Com bined so lu tion o f K A and SPM w hich is applicable for composite surfaces has
basically the same regions of v a lid ity as the in d iv id u a l models [B row n, 1978].
A t m icrowave frequencies most of n atu ra l surfaces do not fa ll in to the va lid ity
regions o f the the o re tical models. Also a complete set o f measured data does not exist
to characterize the role of in flu e n tia l parameters in the scattering mechanism. Thus
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
the m a jo r goal in this investigation is to find the dependency o f the radar backscatter
to the roughness parameters and soil moisture condition through extensive backscatter
measurement for variety o f m oisture and roughness conditions and over a wide range
o f incidence angles and frequencies. Once the dependency of the radar backscatter
to these parameters are obtained, the em pirical m odel can be used to retrieve the
surface roughness and soil m oisture content fro m measured data.
The radar backscatter of bare soil surfaces under va rie ty o f conditions are mea­
sured using a tru c k m ounted netw ork analyzer based scatterom eter (L C X P O LA R S C A TS )
[Tassoudji et al., 1989]. The data are collected p o la rim e tric a lly at three L-, C-, and
X-band frequencies at incidence angles ranging fro m 10° to 70°. A n e m pirical model
is form ulated based on a set o f measured data and another set o f data is used to verify
the em pirical m odel. Excellent q u a lita tive and reasonable q u a n tita tiv e agreement is
obtained. The p olarim e tric measurements included recordings of th e phase statistics
o f the backscattered signal, b u t these w ill be discusseded in the n e xt chapter.
7.2
Fig.
E X P E R IM E N T A L P R O C E D U R E
7.1 shows simple sketches o f experim ental set-ups for a scatterom eter, a
profile m eter, and a dielectric probe. A description of each of the set-ups is given
below briefly.
7.2.1 Scatterometer
The U n ive rsity o f M ich ig a n ’s L C X P O LA R S C A T [Tassoudji et ah, 1989] is de­
signed w ith the ca pa b ility to measure the scattering m a trix of p o in t or d istrib u te d
targets a t L -, C- and X-band frequencies (1.25, 4.75, and 9.5 GHz o f center frequen­
cies, respectively) as shown in Table 7.2.1. The scatterom eter consists o f an autom atic
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
Platform on the Top
o f the Truck Boom
I
RF Circuitries
RF Cables
Contol Room on Truck Bed
Relay
Actuator
HP-IB
Disk
Drive
f
Pulsing
C irc u itry
Computer
Unit
Network
Analyzer
Antennas
Control
Cables
Dielectric Probe
Reflectometer
Assembly
X^L
(a) Scatterometer
Signal Processing
and Computer
Assembly
Laser Distance Meter
(c) Dielectric
Probe
Control
Unit
Laptop
Computer
(b) Laser Profile Meter
F igure 7.1: E xpe rim en ta l system, (a) a scatterom eter block diagram , (b) Laser profile
m eter, and (c) dielectric probe.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
L
C
X
Center frequency
1.50 GHz
4.75 GHz
9.50 GHz
Frequency bandw idth
0.3 GHz
0.5 GHz
0.5 GHz
A ntenna type
D ual polarized p yra m id a l horn
A ntenna gain
22.1 dB
25.3 dB
29.5 dB
Beam w id th
12.0°
00
oo
Table 7.1: P olarim etric scatterom eter (P O L A E S C A T ) characteristics.
5.4°
Far field (2d2/A )
8.5 m
5.8 m
10.5 m
P la tfo rm height
18 m
18 m
18 m
Cross-pol Isolatio n !
45 dB
45 dB
45 dB
C a libration accuracy
± 0 .3 dB
±0.3 dB
± 0 .3 dB
precision(N>100)
± 0 .4 dB
±0.4 dB
± 0 .4 dB
Phase accuracy!
±3°
±3°
±3°
Measurement
f A fte r p o la rim e tric calibration using S T C T [Sarabandi and Ulaby, 1990]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
vector netw ork analyzer (H P 8753A), a com puter u n it, a disk drive fo r d a ta storage,
an a m p lify in g and pulsing c irc u itry fo r hardware range gating, a relay actuator, and
L-, C-, X -b an d R F circu itries and antennas as shown in Fig. 7.1. Antennas are dual­
polarized w ith orthogonal mode transducers (O M T ) to tra n s m it and receive a set o f
orthogonal polarizations.
A co m puter is used to control the netw ork analyzer through H P -IB (interface bus)
to acquire the desired data a uto m a tically. The com puter also controls a relay actuator
which energizes the desired frequency and p olarization switches. Table 7.2.1 shows
a basic characteristics o f the scatterom eters in clu ding specifications o f frequencies,
antennas, and overall performances.
To achieve good sta tistica l representation o f the measured backscatter for dis­
trib u te d targets, a large num ber o f s p a tia lly independent samples are required. In
this experim ent 90 and 60 independent samples were taken a t incidence angles of
10°, 20°, 30° and 40°, 50°, 60°, 70°, respectively. To achieve tem poral resolution and
also to increase the num ber of independent samples, measurements were perform ed
over 0.3 G Hz fo r L-band and 0.5 G Hz fo r C- and X -band, assuming backscattering
coefficient is constant over the m entioned bandwidths. Thus the tota l num ber of in ­
dependent samples in clu ding those achieved by these bandw idths, for each incidence
angle, is m ore th a n 1000.
In a d d itio n to the soil backscatter data, the noise background level was measured
by p o in tin g the antennas towards the sky. The noise background level was subtracted
fro m the soil backscatter data coherently to im prove the signal to noise ra tio . The
p o la rim e tric response o f a conducting sphere was measured to achieve absolute cal­
ib ra tio n o f the radar system [Sarabandi and Ulaby, 1990].
To m inim ize the tim e
elapsed between the four p ola riza tio n measurements th a t completes a p o la rim e tric
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
166
set, the soil backscatter data were collected in a raw -data fo rm a t. The radar data
was post-processed to separate the unwanted short-range returns fro m the target re­
tu rn using the tim e dom ain gating capability. The gated target response was then
calibrated using the sphere data.
7.2.2
Laser Profile Meter
The height profiles o f soil surfaces are measured by the Laser p rofile m eter mounted
on a stepper-m otor driven X Y -ta b le as shown in Fig. 7.1. The Laser profile meter can
measure a surface profile w ith 1 m m horizontal resolution and 2 m m ve rtica l accuracy.
A la ptop com puter is connected to the stepper-m otor controllers to position the Laser
distance m eter w ith the desired steps in X and Y directions. The heights measured
by the Laser distance m eter are also collected and stored by the same computer. A
m in im u m o f ten one-meter profiles are collected fo r each surface w ith steps of 0.25 cm
in the h orizontal direction. In a ddition to the surface profiles acquired by the Laser
p rofile m eter, a couple of three-m eter-profiles were collected using chart paper and
spray p a in t to m o n itor large scale roughnesses variations. Radar measurements were
conducted fo r four surface-roughness conditions, covering the range fro m 0.32 cm to
3.02 cm in rm s height as shown in Table 7.2.2.
7.2.3
Dielectric Probe
D ie le ctric constants o f the soil fields were measured by a C-band field-portable
die lectric probe [B run fe ldt, 1987].
The probe consists o f a reflectom eter assembly
w ith a coaxial probe tip and a signal processing assembly w ith a calculator. D ielectric
constants were measured a t the top and a t the depth o f 4 cm fo r more than fifty
spots ra n do m ly chosen over each surface. T h e d ie lectric constants (er ) were used to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
167
Table 7.2: Sum m ary of roughness parameters.
Surface
S -l
S-2
S-3
S-4
s (cm )
0.40
0.32
1.12
3.02
I (cm )
8.4
9.9
8.4
8.8
m
0.048f
0.032f
0.133f
0.485J
Freq(G Hz)
ks
kl
D enotation
1.50
0.13
2.6
LI
4.75
0.40
8.4
Cl
9.50
0.80
16.7
XI
1.50
0.10
3.1
L2
4.75
0.32
9.8
C2
9.50
0.64
19.7
X2
1.50
0.35
2.6
L3
4.75
1.11
8.4
C3
9.50
2.23
16.7
X3
1.50
0.95
2.8
L4
4.75
3.00
8.8
C4
9.50
6.01
17.5
X4
f m = s/1 assuming exponential autocorrelation function,
| m = y/2s/ 1 assuming Gaussian autocorrelation function.
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168
Table 7.3: S um m ary o f soil m oisture contents.
No.
Meas. er (4.8 GHz)
E stim . m v
calculated eT (er
Top soil
4cm depth
Top
4cm
1.5 GHz
4.75 GHz
9.5 GHz
1 -wet
14.15, 4.62
16.74, 5.94
0.29
0.33
15.57 3.71
15.42 2.15
12.31 3.55
1 -d ry
6.58, 1.54
11.05, 3.27
0.14
0.24
7.99, 2.02
8.77, 1.04
5.70, 1.32
2 -wet
14.66, 4.18
14.30, 4.08
0.30
0.29
14.43 3.47
14.47 1.99
12.64 3.69
2 -d ry
4.87, 0.83
8.50, 2.15
0.09
0.19
Oi
bo
S*1
Surf.
1.46
6.66, 0.68
4.26, 0.76
3 -wet
15.20, 5.84
15.10, 5.71
0.31
0.31
15.34 3.66
15.23, 2.12
13.14 3.85
3 -d ry
7.04, 1.85
10.02, 3.15
0.15
0.22
7.70, 1.95
8.50, 1.00
6.07, 1.46
4 -wet
8.80, 2.38
10.57, 3.27
0.19
0.23
8.92, 2.24
9.64, 1.19
7.57, 1.99
4 -d ry
7.28, 1.93
8.84, 2.58
0.16
0.19
7.23, 1.83
8.04, 0.92
6.28, 1.53
estim ate the m oisture contents (m „) by in ve rtin g a sem iem pirical model [H allikainen
et al., 1985] w hich gives er in terms o f m v. The real part of er is chosen since the error
in measuring the im a ginary part o f eT by dielectric probe is re la tive ly higher [Jackson,
1990]. The mean value o f m v then was used in the same sem iem pirical m odel to obtain
an estim ate fo r eT at L-, C- and X-band frequencies. Table 7.2.3 gives the measured
er at 4.8 GHz and the estimated values o f m v fo r the top surface and 4-cm deep
layers, from w hich the 0-4 cm average d ie lectric constant was calculated at L-, C-,
and X-band. Soil density was determ ined fro m soil samples w hich known volume.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
169
7.3
E X P E R IM E N T A L O B SE R V A T IO N S A N D C O M P A R ­
IS O N W IT H C L A SSIC A L S O L U T IO N S
In this section we present samples o f the measured radar backscatter to demon­
stra te the spectral, angular, and p o la rim e tric behavior o f rough surface backscattering
coefficients. N ext the theoretical solutions fo r rough surface backscattering w ill be
compared w ith experim ental data where applicable.
7.3.1
Experimental Observations
Four different fields (S i, S2, S3, and S4) were considered. Each one of fo u r surfaces
was measured under tw o different m oisture conditions, re la tive ly wet and re la tive ly
dry. The roughness param eters o f the surfaces such as rm s height s, autocorrelation
fu n c tio n /?(£), correlation length I, and rm s slope m , are calculated fro m the measured
surface height d istrib u tio n s as given in Table 7.2.2.
T he surface height d istrib u tio n s o f a ll fo u r surfaces f it well to Gaussians d is trib u ­
tio n , where
K 2) = - ^ exP
V27TS
(7.1)
2s2
T h e autocorrelation functions for surfaces 1, 2, and 3, f it b ette r to exponential func­
tio ns (p(£) =
exp [—1£|//]) than to Gaussian functions (/>(£) =
exp [—£2/ / 2]). The
Gaussian form provided a b ette r f it fo r the roughest field, S4.
T his is illu s tra te d
in Fig.
7.2 fo r fields S i and S4.
The surface rm s slopes can be calculated from
m = s y V '( 0 ) | , where ^"(0 ) is the second derivative o f />(£) evaluated at £=0.
Measured surface roughness param eters are summerized in Table 7.2.2 and Fig.
7.7. A m ong the fou r surfaces, surface S2 is the smoothest (s = 0.32 cm ), surface S i
(s = 0.4 cm ) is s lig h tly rougher, surface S3 (s = 1.12 cm) represents an interm e dia te ­
roughness condition, and surface S4 (s = 3.02 cm) is a very rough surface th a t was
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
170
(a)
1.00
Measured
0.75
• Gaussian
0.50
Exponen.
1 = 8.4 cm
0.25
0.00
-0.25
0.
10.
20.
30.
40.
50.
Displacement, %(cm)
(b)
1.00
Measured
0.75
Cl
c
o
>
• Gaussian
0.50
Exponen.
1= 8.4 cm
0.25
3
<
0.00
-0.25
0.
10.
20.
30.
40.
50.
Displacement, £ (cm)
Figure 7.2: Com parison o f the measured autocorrelation functions w ith th e Gaussian
and exponential functions (a) SI and (b) S4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
171
generated by plow ing the top 15-cm surface layer. Electrom agnetically, these surfaces
cover a wide range o f roughness conditions, extending fro m ks = 0.1 to ks — 6.01
(where k = 2 ir / \ is the wave num ber) and from k l = 2.6 to k l = 19.7. Surface 1, for
exam ple, may be considered smooth at 1.5 GHz (ks = 0.13, where k is a wave num ber),
and m edium rough a t 4.75 GHZ and 9.5 G Hz (ks = 0.42 and 0.80, respectively). The
12 roughness conditions corresponding to the four surfaces and three wavelengths are
id e ntifie d in ks - k l space in Fig. 7.7, together w ith the boundaries for the regions
o f v a lid ity of the sm all pertu rb a tion m odel (SPM ) and the physical optics (P O ) and
geom etrical optics (G O ) solutions o f the K irc h h o ff approxim ation.
Fig.
7.3 shows angular responses o f the uu-polarized backscattering coefficient
(cr°v) fo r four different bare soil surfaces w ith rms heights ranging fro m 0.3 cm to 3.0
cm a ll at a m oderately d ry condition (ro„ Ci 0.15). T h e se nsitivity o f v°v to surface
roughness is clearly evident at both 1.5GHz ( Fig. 7.3a) and 9.5 GHz (F ig. 7.3b); over
the 30° - 70° angular range. A t L-band (1.5 GHz) there is a to ta l of 16 d B dynam ic
range in a°v when surface roughness (s) changes fro m 0.3 cm to 3 cm at the angles
ranging fro m 30° to 70°. Fig. 7.3(b) shows the angular p a tte rn o f the same surfaces
at X -band (9.5 G H z). In this case the to ta l dynam ic range is reduced to 10 dB and
the changes in a °v fo r s = 1.1 cm and s = 3.0 cm is negligible. The radar backscatter
is very sensitive to surface roughness at lower values o f ks (ks < 2.0), b u t insensitive
at higher values of ks (ks > 2.0).
F u rth e r illu s tra tio n o f the effect o f surface roughness on the angular response
o f <7° is shown in Fig. 7.4 which contains plots o f the three p rin cip a l p olarization
components for the smoothest case (Fig. 7.4a), corresponding to surface S2 at 1.5
GHz, and fo r the roughest-surface condition (Fig. 7.4b), corresponding to surface S4
at 9.5 GHz. Based on these and on the data measured fo r the other surfaces, we
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
172
(a)
30.
1.5 GHz
Backscattering Coefficient, a°vv (dB)
20.
rms hgt. (ks)
3.02 cm (0.95)
10.
1.12 cm (0.35)
0.40 cm (0.13)
0.32 cm (0.10)
0.
-
10.
-20.
-30.
V.
"A
•V
60.
70.
-40.
0.
10.
20.
30.
40.
50.
80.
Incidence Angle (Degrees)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
(b)
~ r~
30.
9.5 GHz
Backscattering Coefficient, a°
(dB)
20.
rms hgt. (ks)
3.02 cm (6.01)
10.
-a -
1.12 cm (2.23)
-A -
0.40 cm (0.80)
0.32 cm (0.64)
0.
- 10.
- 20.
-30.
-40.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
F igure 7.3: A ng u la r response o f
fo r four different surface roughnesses a t m oder­
ately d ry co nd itio n (m v ~ 0.15), a t (a) 1.5 G Hz and (b) 9.5 GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
174
(a)
20. ----
PQ
...... „ !
, ....."i
i
i
TJ
10.
C
S
o
ss
u
o
U
to
c
•c
u
+-»
-30.
o
w
-40.
c3
«
-50
|
-
1.5 GHz
0
0.
\
- 10 .
----- 0-----
VV-pol. (L2)
— ■-0......
HH-pol. (L2)
----- A----- VH-pol. (L2)
■
'
-
___
- 20 .
\
:
0.
V
—0
.
i
i
i
10.
20.
30.
1 —A
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(b)
30.
PQ
"1--------- 1--------- 1----------r
•o
9.5 GHz
20.
VV-pol. (X4)
c
<L>
• iH
o
s
<0
o
U
DO
e
•c
%
rt
a
w
&
PQ
10.
0.
-
-0-----
HH-pol. (X4)
A
VH-pol. (X4)
10.
A — -A
A -----
- 20 .
-A -----
-30.140.
0.
10.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Figure 7.4: A ngular responses o f <r°v, a%h, and
for (a) a sm ooth surface at 1.5
GHz (L2) and (b) a very rough surface a t 9.5 G H z (X 4).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
10.
1
r
4.75 GHz
— ■©-----
C4 (s=3.0 cm, mv=0.16)
— a
C l (s=0.4 cm, m =0.14)
5. -
CQ
■a
>
o
'•0.......
-5.
-
73— - .
■0
60.
70.
10.
10.
20.
30.
40.
50.
80.
Incidence Angle (Degrees)
Figure 7.5: A ngular dependence of th e like-polarized ra tio , cr°hhla ° v, at 4.75 GHz fo r
a sm ooth surface and a very rough surface.
note th a t the ra tio o f erj^ to a°v, w hich w ill be referred to as the co-polarized ra tio ,
is always sm aller th a n or equal to 1, and it approaches 1 as ks becomes large. Very
rough surfaces such as C4 (surface 4 a t C-band) and X4 do n o t show any noticeable
differences between a°v and cr%h, while sm ooth surfaces show values o f ^ h /^ v v sm aller
th a n 1. I t is also observed th a t co-polarized ra tio is a fu n ctio n o f incidence angle fo r
sm ooth surfaces and increases as the incidence angle increases. The se n sitivity o f the
co-polarized ra tio a \ h/(.r°„ to surface roughness and the incidence angle is shown in
Fig. 7.5. For very rough surfaces (ks > 2) cr°hhlcr0
vv
~
1 independent of incidence
angle. A nother p o in t w o rth noting is th a t the shape of the angular p atte rn o f the
cross-polarized backscattering coefficient
is sim ila r to th a t o f a°v, b ut the ra tio
o°hJ a 0
Vv, w hich w ill be referred to as cross-polarized ratio, increases w ith ks as shown
in Figs. 7.4 (a), (b ) (and more e x p lic itly in Fig. 7.11).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
176
(a)
15.
CQ
"O
i
1------- 1------- 1
r—
VV-pol, X I (mv=0.29)
5.
c
<L>
o
S3
■a
*-1
>
-5.
-Q
VV-pol, X I (mv=0.14)
A— -
HV-pol, X I (mv=0.29)
- - -V - -
HV-pol, X I (mv=0.14)
o
U
-15.
(30
e
‘S
S
-25.
11
«
-35.
0.
10.
20.
30.
40.
50.
j
60.
i_
70.
80.
Incidence Angle (Degrees)
(b)
5.
~i
1------------1
r
4.75 GHz
c i( m v=0.14)
PQ
•o
0.
.— a .....
□.....
>
C l (mv=0.29)
*0.
£
-5.
-
T3.
J
10.
0.
10.
20.
L
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
F igure 7.6: A ngular plots o f (a) <r“„ and <r£„ o f surface SI at X -band fo r tw o different
m oisture conditions and (b) the like-polarized ratio, cr^h/< rlhi for the same
surface at C-band.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
177
T he backscattering coefficients o f a surface is a fun ctio n of m oisture content. F ig ­
ure 7.6(a) shows the backscattering coefficient o f surface 1 fo r tw o m oisture conditions,
m „= 0 .2 9 and m „= 0 .1 4 . The ra tio o f a°v (or c r£ j of wet soil to cr°v (or a^v) o f d ry
soil is about 3 dB at the angles ranging from 20° to 70°. The se n sitivity of <7° to
m oisture contents (« 3 dB in Fig. 7.5 (a)) is m uch lower than the se n sitivity to sur­
face roughness (« 1 6 dB in Fig. 7.3 (a)) in this experim ent. F igure 7.6 (b ) shows
the angular response o f the co-polarized ra tio
fo r a fixe d roughness at two
different m oisture contents. The m agnitude o f the co-polarized ra tio is larger for the
wet surface (6 dB at 50°) than for the d ry surface (3 dB at 50°).
7.3.2
C om parison w ith C lassical Solutions
T h is section evaluates the a p p lic a b ility o f the sm all p e rtu rb a tio n m ethod (SPM ),
the physical optics (P O ) m odel, and the geom etrical optics (G O ) m odel to the mea­
sured radar data. Expressions fo r the backscattering coefficient <7° and the regions
o f v a lid ity o f these models are given in Chapter 2. Measured roughnesses fo r all four
surfaces at three frequencies are shown in Fig. 7.7 in term s o f ks and kl. Also shown
are the regions o f v a lid ity of the models; SPM , PO , GO models. The lower lim it of
the ks value of the v a lid ity region fo r GO m odel given by ks > \ /2 .5 / cos 0 is chosen
at the incidence angle 6 o f 30°, where the lower lim it varies fro m ks = 1.62 a t 0 = 10°
to ks = 6.32 at 6 = 60°
Sm all Perturbation M odel
SPM is applicable only on L I (surface 1 at L-band frequencies) according to Fig.
7.7, and the backscattering coefficients com puted by SPM are compared w ith those
measured fo r L I in Figs.
7.8 (a)-(c).
The tm -polarized backscattering coefficients
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
178
X4
GO
C4
X3
SPM
L4
L3
PO
C1 C2
X1
X2
kl
F igure 7.7: Roughness parameters and the region o f v a lid ity o f SPM , PO , and GO
models.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
179
(T°„ computed using SPM are compared w ith measured a°v as shown in Fig. 7.7 (a)
for three types o f autocorrelation functions, namely, a Gaussian, an exponential, and
a num erical form o f autocorrelation functions. These autocorrelation functions and
their Fourier transform s (w hich are the norm alized roughness spectrum s) are given
in Chapter 2.
The backscattering coefficient a°v o f SPM w ith exponential a utocorrelation func­
tion fits b e tte r to the measured a°v than those w ith the other autocorrelation func­
tions, even though there is about 5 dB discrepancies at higher incidence angles. The
measured a \ h agrees quite well w ith SPM model (w ith in about 1 dB tolerance) when
exponential autocorrelation is assumed as shown in Fig. 7.7(b). As m entioned earlier,
the measured autocorrelation fu n ctio n of surface 1 fits better to exponential than to
Gaussian autocorrelation fu n ctio n , w hich makes sense to the results in Figs. 7.8 (a)
and (b).
The cross-polarized backscattering coefficient fo r L I computed using the secondorder SPM w ith exponential autocorrelation fu n ctio n is shown in Fig. 8 (c), and the
angular tre n d o f the calculated &°hv fits well to th a t o f the measured <x£„ w hile the
values of the measured a%v are higher than those o f the SPM. F ig . 7.7 (c) also shows
th a t the angular p attern of crj^ calculated by SPM is q uite s im ila r to th a t o f a^v.
SPM m odel is applicable to the case o f L2 (ks = 0.10, k l = 3.1) and L3 (ks =
0.35, k l = 2.6) also. B u t SPM m odel failed on the case o f L4 (ks = 0.95, k l = 2.8)
as indicated in Table 7.3.2.
Physical O ptics M odel
Several roughnesses o f this experim ent, C l, C2, C3, X I , X2, and X 3, are in the
region o f v a lid ity o f the PO M odel.
One case o f those roughnesses, X I , is closely
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
180
(a)
PQ
0.
>
-10.
°e>
-20.
*-T
o
i
'o
iS
o
C
f=1.5GHz, s=0.4cm
-40.
1
PQ
SPM (Gaus. Corr.)
SPM (Expo. Corr.)
p
U
00
c
0
>»
•*-»
4—
-60.
a
o
&
M
O -70.
a
SPM (Meas. Corr.)
©
_i_
0.
10.
Measured
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
Measured
Incidence Angle (Degrees)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
-
PQ
.’O
c
u
o
s
lU
o
10.
- 20 .
-30.
A
-40.
A
A
U
A
A
taO
a
•c
o
<4—*
4-»
cd
O
■s
ca
\
VV-pol., SPM
-50.
HH-pol., SPM
-60.
HV-pol., SPM
PQ
-70.
O
VV-pol., Measured
f=1.5 GHz
□
HH-pol., Measured
s=0.4 cm
HV-pol., Measured
mv=0.29
1______ L
A
_L
-80.
0.
10.
_l_____ i
20.
30.
I
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
F igure 7.8: SPM m odel w ith different a utocorrelation functions compared to the mea­
sured data o f L I (surface 1 a t 1.5 GHz, ks= 0.13), (a) V V -p o la riza tio n ,
(b) H H -polarization, and (c) V V -, H H -, and H V -p o lariza tion s using an
exponential autocorrelation function.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
182
Table 7.4: Comparison between a 0 o f SPM m odel (w ith exponential correlation) and
the measured d ata for wet soil surfaces.
Surface
^spm /^m eas (d B ) at 40°
V V -p o l
H H -p o l
H V -pol
LI
2.9
0.7
-6.7
L2
0.5
0.8
-8.7
L3
-0.4
O
1
-3.8
L4
5.6
1.6
3.4
examined and the results are shown in Figs.
9 (a)-(c).
a°v modeled by PO w ith
Gaussian autocorrelation fun ctio n deviates from the measured a°v except at sm all
incidence angles, but <r°v modeled by PO w ith exponential auto corre la tion function
agrees to the measured <
j °v q uite well over a wide range of angles (6 < 40°) as shown
in Fig. 7.9 (a). The comparison between a°hh o f PO model and a°hh measured for the
surface of X I is shown in Fig. 7.9 (b). The ratios o f the backscattering coefficients
computed by PO model c°0 to those measured in this experim ent (cr^jens) are tabulated
in Table 7.3.2 fo r incidence angles o f 20°, 40° and 60° for wet soil surfaces. The hhpolarized backscattering coefficients cr^h o f the PO m odel agree, w ith in about 3 dB,
to the measured values for the surfaces of C2, c3, and X I at the angles less than 40°.
The vu-polarized backscattering coefficients <r°v o f the PO m odel, however, deviate
much from the measured values for all surfaces. The deviation is very large at large
incidence angles (6 > 50°) as shown in Fig. 7.9 (a) and Table 7.3.2.
PO model failed
on the p re d ictio n of a° for the most surfaces w hich roughness param eters are in the
region o f v a lid ity of the PO m odel. The cross p olarization backscattering coefficient
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
183
(a)
m
•o
T
1------- 1------- 1------- 1---
Measured, f=9.5GHz, s=0.4cm
>
>
PO (Gauss. Corr.)
PO (Expon. Corr.)
M
<D
O
U
60
a
•c
0)
cd
o
J2
c3
CQ
0.
J
I
10.
20.
LI
30.
I
I
I
40.
50.
60.
L.
70.
80.
Incidence Angle (Degrees)
(b)
CQ
•o
i
1------------1------------r
Measured, f=9.5GHz, s=0.4cm
E
E
PO (Gauss. Corr.)
PO (Expon. Corr.)
s
•u
*■4
o
O
U
■O
©
40.
50.
60
C
'S
w
03
O
C/3
CQ
0.
10.
20.
30.
60.
70.
80.
Incidence Angle (Degrees)
F igure 7.9: PO model w ith different autocorrelation functions compared to the mea­
sured data o f X I (surface 1 at 9.5 G Hz, &s=0.80) for (a) V V -p o la riza tio n
and (b) H H -polarization.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
184
Table 7.5: Comparison between a° o f PO model (w ith exponential co rre la tion ) and
the measured data for wet soil surfaces.
0rp o / crmeas (A®)
Surface
H H -p o l
V V -p o l
o
O
O
o
O
O
Cl
2.6
-5.8
-17.6
3.7
1.0
-3.0
C2
0.5
-6.7
-20.1
2.2
-0.2
-6.9
C3
-4.7
-5.6
-15.3
-2.8
-1.5
-5.1
XI
1.6
-6.3
-17.5
2.3
-1.8
-5.7
X2
0.8
-5.8
-20.0
3.6
-0.2
-8.5
X3
-7.6
-5.6
-12.8
-6.5
-2.4
-5.4
60°
O
O
20°
is n ot available fo r PO model. The measured cr°„ is higher tha n or equal to a°hh in all
cases o f roughnesses, m oisture contents, and incidence angles, which is co ntra ry to
the PO model. T h is is due to the fact th a t <7°„ and c r^ o f the PO m odel are d ire c tly
p ro p o rtio n a l to the Fresnel reflectivities.
G eom etrical O ptics M odel
The GO m odel agrees w ith the measured a°v and o^h w ith in 4 dB tolerance for
C4 case and w ith in about 2 dB tolerance for X4 fo r the incidence angles less than
or equal to 50°.
Figure 7.10 shows the measured backscattering coefficients o f X4
case compared w ith the GO model. The coherent com ponent o f the backscattering
coefficient is negligibly small fo r the very rough surfaces lik e C4 and X 4, and the
noncoherent component dominates a t a ll angles including norm a l incidence.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
185
25.
1
i
PQ
"O
15. -
e
<d
'3
£
(U
o
U
00
c
■S
so
J3
5. -
-5.
____J i}_
-15.
$
(3
A
1
i
i
i
Measured, VV-pol.
(= 9.5 GH;
□
Measured, HH-pol.
s=3.0 cm-
V
Measured, VH-pol.
11^=0.16 ■
A
Measured, HV-pol.
0
■
0
$
*
-25.
-35.
A \
* A
o
i
i
i
i
i
i
o
PQ
i
0
20.
30.
40.
50.
60.
S
■
\ 1
70.
80
Incidence Angle (Degrees)
Figure 7.10: GO m odel compared to the measured d ata o f X 4 (surface 4 at 9.5 GHz,
fcs=6.0).
The GO m odel like the PO m odel is incapable of p re d ictin g the cross-polarized
term s. This m odel also failed to p re dict <r°„ (or e r^) at larger incidence angle (cr > 60°)
as shown in Fig. reffig:5-10.
The m a jo r conclusions we drew fro m our analysis o f th e measured ra d ar data
when compared w ith the predictions o f the SPM , PO, and GO models are:
1. Some n a tu ra l surface conditions fa ll outside the regions o f v a lid ity of a ll three
models.
2. None o f the models provides consistently good agreement w ith the measured
data, p a rtic u la rly at incidence angles greater th a n 40°.
3. The PO model predicts th a t cr°v < c r^ , contrary to a ll observations.
A d d itio n a lly, being first-order solutions, b o th the PO and GO models cannot be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
186
used fo r a^v. Faced w ith these inadequacies o f the available theoretical scattering
models, we decided to develop an em pirical m odel th a t relates
and crlv to
the roughness (As) and dielectric constant (er ) o f the surface. T h is is the subject of
the next section.
7.4
S E M I-E M P IR IC A L M O D E L (SE M )
The fa ilu re o f existing models to cover roughness conditions th a t occur more often
in nature as described in the previous section, prom pts development o f sem i-em pirical
backscattering models for random surfaces. A no th er reason fo r developing the semiem pirical m odel is to generate an inversion a lg o rith m to retrieve soil m oisture and
surface roughness from the measured radar backscatter.
Sem i-em pirical models o f backscattering coefficients, cr°v, aflh and a%v, are devel­
oped based on the measured radar backscatter o f the eight soil surfaces and knowledge
o f scattering solutions in the lim itin g cases.
For th is set o f data, surface rough­
nesses and the volum etric m oisture contents are in the range of 0.1 < ks < 6.0,
2.6 < kl < 19.7, and 0.09 < m v < 0.31, which is the region of interest at microwave
frequencies.
7.4.1
D evelopm ent
we begin w ith an exam ination o f the cross-polarized ra tio q = a h.vla l v
The
angular p a tte rn o f a \ v follows the p atte rn o f cr°v and thus the cross-polarized ra tio
is only a fu n ctio n o f roughness and frequency as shown previously. Cross-polarized
ra tio increases when frequency a n d /o r surface rm s height s increases.
Figs. 7.11 (a) and (b) show the variation o f crlv/cr°v as a fu n ctio n o f ks which
includes both frequency and rm s height, for incidence angles of 30°,40° and 50°. The
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
187
- 10.
>
>
D
q=0.23*ro
-15.
(l-exp(-ks))
dry ( er = 6.5 8-j 1.55)
-25.
0.
1.
2.
3.
4.
q=0.23*ro
5.
6.
7.
(l-exp(-ks))
wet (e r = 15.34 -j 3.66)
Figure 7.11: The se nsitivity of the depolarization ra tio , crlv/cr°„, to surface roughness
fo r (a) dry soil and (b) wet soil.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
188
cross-polarized ra tio starts from values of about -20 dB for ks « 0.1 and increases
lin e arly w ith ks u n til v%v/c rlv saturates at values about -10 dB (ks > 2) as shown
in Fig.
7.11 (a).
Cross-polarized ratios for the re la tive ly wet surfaces start from
smaller values than those fo r the re la tively d ry surfaces and increases more ra p id ly
as ks increases, and the ra tio saturates at higher values ( « -9 d B ) as shown in Fig.
7.11
(b). A n e m pirical form fo r the cross-polarized ra tio °h v / cr°v is obtained by curve
fittin g of the measured data as follows:
q £
= 0 .2 3 i/r\4 l - exp(-Jfcs)]
(7.2)
a vv
where T0 is the Fresnel re fle c tiv ity o f the surface at nadir,
2
(7.3)
ro=
1 + y/^T
The e m p irica l function fo r the cross-polarized ra tio is compared w ith the measured
data as shown in Figs. 7.11 (a) and (b).
N e xt, we shall examine the co-polarized ra tio p = (T°hhla%v. The measured values of
the co-polarized ra tio crj^/cr^ is a fun ctio n o f surface roughness, soil m oisture content,
and incidence angle. Figs. 7.12 (a) and (b) show the variation o f the ra tio crj^/cr^
w ith respect to the roughness param eter ks for b oth o f wet and d ry conditions a t the
incidence angles o f 40° and 50°, respectively. The co-polarized ratios sta rt from —7.5
dB for wet soil and —4.5 dB fo r d ry soil at 50° incidence, which are much lower than
those fo r 40° incidence. B oth Fig. 7.12 (a) and (b) show th a t the co-polarized ra tio
a hhla lv ° f wet soil are lower than those of the d ry soil. The curves shown in Fig.
7.12
are based on the e m pirical expression
rZo
^
= u
/ n o \ [ i / ( 3 .or„)]
t = 1 ~ f c )
'exp(' fa°
where 6 is the incidence angle in radians.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( ?
' 4 )
189
Wet(e=15.0-j3.0), Model
Dry(e= 7.5-j 1.5), Model
Wet, Measured
Dry, Measured
PQ
•o
Wet(e=15.0-j3.0), Model
Dry(e= 7.5-j 1.5), Model
O
II
Cu
Wet, Measured
Dry, Measured
F igure 7.12: The se nsitivity o f the like-polarized ra tio , (r lh/ a °v, to surface roughness
and soil m oisture at (a) 40° and (b) 50°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
190
H aving established em pirical formulas for q = cr°lv/<J°v and p =
th a t
provide reasonable agreement w ith the measured data, the rem aining task is to relate
the absolute level o f any one of the three lin e arly polarized backscattering coefficients
to the surface param eters. Upon exam ining the measured data, an e m p irica l form ula
fo r the m agnitude o f the backscattering coefficient cr°v is found to have the following
form :
n rn s 3 0
<r°vv(0, eT, ks) = —
9 — ■[Tv(6) + r fc(0)]
VP
(7.5)
g = 0.7 [ l — exp(—0.65(A:s)1‘8)j ,
(7.6)
where
and p is given in (7.4).
The m u ltip ly in g te rm o f I\,(0 ) + ^ ( 0 ) is included in (7.5) to provide a m oisture
dependency o f <7°^ as shown in Fig. 7.6 (a), w hich m ay be a reasonable assumption
since a ll three the oretical models described in previous sections include the Fresnel
re fle c tiv ity or equivalent forms o f the Fresnel re fle c tiv ity in th e ir form ulae of the
backscattering coefficients.
The measured angular patterns o f the backscattering
coefficient a°v follo w the form o f (cos 0 )m, where m is ranging fro m 2.0 to 4.0. The
angular patterns o f backscattering coefficients a °v fo r a very rough surfaces approach
the fo rm o f cos2 0 w hich is the scattering p attern o f a Lam bertian surface [Ulaby et
al, 1982], and those for the smooth and dry surfaces follow the fo rm o f cos4/?. To
avoid com plication o f the em pirical form ula, the te rm cos3 0 was chosen, which can be
applicable generally to most surfaces. The roughness dependency of <7°^ is represented
by fa cto r g in (7.6) w hich is found by cu rve -fittin g o f the measured data.
Consequently, the backscattering coefficient
can be obtained fro m (7.4)-(7.6)
and is given by
°h h (9 ,eT,ks) = gy/pcos3 0 [r „ ( 0 ) + Tfc(fl)]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7.7)
191
and the backscattering coefficient <t£„(= cr°h) is sim ply given by
ffhv{9> €r, ks) = q a lv(0, er , ks).
(7.8)
As we w ill see next, the sem i-em pirical model was found to provide a good represen­
ta tio n of the measured data at a ll frequencies and over a wide angular range. The
m odel was evaluated against three data sets: (a) the data measured in this study, (b)
another independently measured d a ta set th a t was n ot used in the developm ent of
this model, which shall be referred to as Independent D ata Set I I , and (c) a data set
th a t was recently reported by Yamasaki et al. [1991] at 60 GHz.
7.4.2
Com parison W ith M easured D ata
Because o f the space lim ita tio n s, we w ill present only two ty p ic a l examples i l ­
lu s tra tin g the behavior o f the sem i-em pirical model, in comparison w ith the data
measured in this study. This is shown in Fig. 7.13 and 7.14 for surface SI (represent­
ing a very sm ooth surface w ith s = 0.40) and surface S4 (representing a very rough
surface w ith s = 3.02). In both cases, very good agreement is observed between the
m odel and the measured data at a ll three frequencies and across the e ntire angular
range between 20° and 70°. The levels o f the measured values o f cr°„ and <r^h at
9 = 10° for surface SI include a strong co ntribution due to the coherent backscat­
te rin g component th a t exists at angles close to norm al incidence. No a tte m p t has
been made at th is stage to add a coherent component to the em pirical, and therefore
its range o f a p p lic a b ility does not include the angular range below 20° fo r smooth
surface. I f the surface is rough, as is the cse for surface S4 (Fig. 7.14), the coherent
backscattering coefficient is negligibly sm all, in which case the e m p irica l m odel may
be used at a ll angles between 0° and 70°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
192
(a)
0.
PQ
1.5 GHz
*T3
s=0.4cm, mv=0.29
10.
d
v
• pH
o
s<4-H
o
o
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•d
«y
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J2
3
VV-pol., Model
20 ,
HH-pol., Model
•
VH-pol., Model
•30,
-40.
-50.
0. 10. 20. 30. 40. 50. 60. 70. 80.
©
W -p o l., Measured
b
HH-pol., Measured
A
VH-pol., Measured
V
HV-pol., Measured
Incidence Angle (Degrees)
(b)
PQ
0.
PQ
T3
4.75 GHz
T3
9.5 GHz
- 10.
d
C
o
o
<D
• pH
• pH
o
£
t4O
<H
- 20.
£
<4-H
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*C
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4-4
-30.
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d
a
o
CQ
M
3
PQ
-40.
o3
O
c/3
-50.
PQ
3
0.
10. 20. 30. 40. 50.
60. 70. 80.
Incidence Angle (Degrees)
J
0.
10.
i
I
i
I
20. 30.
i
I
40.
i
I
50.
i
I
i
L.
60. 70.
80.
Incidence Angle (Degrees)
F igure 7.13: E m p irica l m odel compared to the measured data of surface 1 fo r wet soil
at (a) 1.5 GHz, (b ) 4.75 GHz, and (c) 9.5 GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
193
PQ
1.5 GHz
•o
s=3.0cm, mv=0.19
c
u
o
W -pol., Model
HH-pol., Model
CD
O
U
W)
a
’C
(D
4-^
fS*
VH-pol., Model
#
4—»
cd
O
J2
o
W -pol., Measured
h
HH-pol., Measured
A
VH-pol., Measured
V
HV-pol., Measured
£
CQ
0. 10. 20. 30. 40. 50. 60. 70. 80.
Incidence Angle (Degrees)
(c)
(b)
10.
CQ
4.75 GHz
D
0.
a
(D
O
s
-
9.5 GHz
- 10.
10.
<D
d3
O
U
oo
a
•G
O
U
W)
c
*C
<D
w
- 20 .
cd
-30.
cd
U
-30.
-40.
iPQ
-40.
r '~ l"
2
4->
o
00
%
PQ
-20.
0.
10. 20. 30. 40. 50. 60. 70. 80.
Incidence Angle (Degrees)
0.
10. 20. 30. 40. 50. 60. 70. 80.
Incidence Angle (Degrees)
Figure 7.14: E m p irica l model compared to the measured data o f surface 4 fo r wet soil
at (a) 1.5 GHz, (b ) 4.75 GHz, and (c) 9.5 GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
194
Table 7.6: Measured surface parameters fo r the Independent D a ta Set II.
Surface
7.4.3
M oisture contents, m v
rms height,
corr el.
No
Top soil
3cm depth
6cm depth
s (cm )
length, I (cm )
1
0.11
0.20
0.27
0.46
3.0
2
0.11
0.18
-
1.10
2.9
3
0.08
0.18
-
2.88
4.0
Com parison W ith Independent D ata Set
P rio r to conducting the measurements reported in this study, another data set was
acquired by the same radar system fo r three surface roughnesses. The surface profiles
were measured by inserting a p la te in to the surface and spraying i t w ith p aint. Such
a technique provides an approxim ate representation o f the surface, b u t i t is not as
accurate as th a t obtained using the laser profiler. Hence, our estim ate o f the values
of ks and k l fo r Independent D ata Set are not as accurate as those we obtained w ith
the laser p ro file r fo r the surfaces discussed in the preceding sections o f this section.
Nevertheless, we conducted an evaluation o f the sem i-em pirical m odel by com paring
its p re d ictio n w ith the backscatter data of Independent D ata Set I I and found the
agreement to be very good a t all three frequencies, provided we are allowed to m o d ify
the values o f s measured w ith the m etal plate technique. A n example is given in Fig.
7.15 in w hich the curves were calculated using the e m pirical model w ith s = 0.7; the
value of s estim ated from the m etal-plate record was 0.46 cm (Table 7.4.3).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
195
(a)
1.5 GHz
s=0.7 cm, 1=3.0 cm
- 20 .
VV-pol., Model
HH-pol., M odel
ao
-30.
VH-pol., Model
o
VV-pol., Measured
0
HH-pol., Measured
A
VH-pol., Measured
CO
M
-50.
20.
30.
40.
50.
60.
70.
80.
Incidence Angle (Degrees)
(C)
m
•o
4.75 GHz
9.5 GHz
c
•w
o
s<4-<
o
o
'A
.
A
u
DO
c
•c
u
.*— »
c3
o
A
-30. -40. -
PQ
50.
60.
70.
Incidence Angle (Degrees)
-50.
20.
_i
30.
i
i
40.
i
i
50.
i
i
i
60.
70.
80.
Incidence Angle (Degrees)
Figure 7.15: E m p iric a l m odel compared w ith the d ata fro m independent d ata set II
fo r a surface w ith s = 0.7 cm and / = 3.0 cm, measured a t (a) 1.5 GHz,
(b) 4.75 GHz, and (c) 9.5 GHz
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
196
Table 7.7: Surface parameters fo r Yamasaki et a l.’s measurement.
Soil-1
Frequency (G H z)
Soil-2
Soil-3
57-58
Soil moisture
30%
1.9 -j0.4
7.4.4
s (cm )
0.013
0.051
0.139
I (cm )
0.055
0.12
0.20
ks
0.16
0.64
1.75
kl
0.69
1.51
2.51
Com parison W ith 60 GH z D a ta
O ur fin a l comparison is w ith a 60 GHz data set th a t was recently reported by
Yamasaki et al. 1991. Even though k = 2tt/ \ = 1260 at 60 G Hz, the three surfaces
exam ined in this stud y were extrem ely sm ooth, w ith rm s heights o f 0.055 cm, 0.12 cm,
and 0.20 cm. The corresponding values o f ks are 0.16, 0.64, and 1.75 as summarized
in Table 7.4.4. Good overall agreement is observed (F ig.
7.16) between this data
and the e m pirical m odel, inspite of the fact th a t the correlation lengths fo r a ll three
surfaces are smaller th a n the smallest correlation length o f the surfaces on the basis
o f w hich the em pirical m odel was developed.
7.5
IN V E R S IO N M O D EL
H aving established in the preceding section th a t the sem i-em pirical model (S E M )
is a good estim ator o f cr°v, a lh, and
over a wide range o f ks (0.1 to 6), we shall
now in v e rt the model to obtain estimates o f s and the m oisture content m v from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
(a)
CO
- 10.
s=0.013 cm
•o
60 GHz
- 20 .
c
<D
O
s
o
o
U
V V-pol., Model
-30.
HH-pol., Model
-40.
VH-pol., Model
OD
C
‘S
'W
©
VV-pol., Measured
0
HH-pol., Measured
A
VH-pol., Measured
V
H V-pol., Measured
-50.
4_»
a
o
-60.
3
o
n
-I
-70.
0.
10.
20.
30.
i
40.
1___
50.
L.
60.
70.
Incidence Angle pegrees)
(b)
o.
m
•o
r
1
1---
(c )
. r . .,
|—
, ,_1-----,— , — ,—
cq
G
u
o
£
O
0
a
®
|
9
0
c
0
4)
O
-30.
o
60
C
■c
o
s
<D
o
U
7
♦
❖
8
m
G
■a
o
c3
-50.
•
-60.
0.
r>» |
i
|
— * —i— i— j— i—
s=0.139cm
1
t
10.
20.
- 10.
- 20 .
-
i
0
0
§
a
'
00
-40. "
4—>
C3
i
0. •
- 20.
U
'
•a
s=0.051 cm
- 10.
10.
—
i
30.
—i
40.
. i
50.
a
■8
. i. .
60.
Incidence Angle Pegrees)
PQ
70.
-30.
-40.
.
-50.
0.
i
10.
.
i. .
20.
i
30.
.
i
40.
.
i
50.
.
i
60.
.
70.
Incidence Angle (Degrees)
F igure 7.16: E m p irica l model compared w ith the data reported by Yamasaki et al.
[1991] at 60 GHz for (a) soil-1 (s=0.013 cm, /=0.055 cm ), (b) soil-2
(s=0.051 cm, (=0.12 cm ), and (c) soil-3 (s=0.139 cm, (=0.20 cm).
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198
observation o f cr°v, <T%h, and a°h. Because the sem i-em pirical model was developed on
the basis o f data for surfaces w ith k l in the range 2.6 < k l < 19.7, we cannot ascertain
its a p p lic a b ility o f the inversion model for surfaces w ith k l outside this range. Suppose
we have measurements o f a°v, cr°lh, and a°h for a given surface at a given incidence
angle 0 and wavelength A. From these measurements, we compute the co-polarized
and cross-polarized ratios p = v l h K v and ? = a L K v
B y e lim in a tin g ks from (7.2)
and (7.4), we o b ta in the follo w in g nonlinear equation fo r r o:
where 9 is the incidence angles in radians.
A fte r solving for T0 using an iterative
technique, we can calculate the real p a rt o f the d ie le ctric constant t'T fro m (7.3) by
ignoring the im a g in ary p a rt e", which is a va lid approxim ation for a soil m aterial.
N ext, the m oisture content m v and the im aginary p a rt of the dielectric constant e"
can be determ ined from the m odel given in [H allikainen et al., 1985]. F in a lly, w ith
r „ known, the roughness param eter ks can be determ ined from (7.4).
Because the co-polarized and cross polarized ratios p and q are not sensitive to
surface roughness for very rough surfaces (ks > 3), this technique cannot estimate
ks for such surfaces. Hence, i t is preferable to use radar observations at the lowest
available frequency fo r estim ating the m oisture content and rms height o f a bare soil
surface. B y way o f illu s tra tin g the ca pa b ility of the inversion technique, we present
in Fig. 7.17(a) the values o f ks estim ated by the inversion technique p lo tte d against
the values measured in situ. The data points include the data measured in support
o f this s tu d y fo r a ll surface conditions (1990 data) in a dd itio n to the data measured
independently one year la ter (1991 data), b ut exclude the surfaces for w hich ks > 3.
Each data p o in t in Fig. 7.17(a) represents the average values for four incidence angles
(30° to 60°) and tw o different m oisture conditions (w et and d ry). Figure 7.17(b) shows
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199
(a)
4.0
correl. coeff. = 0.98
1990 data
1991 data
3.0
C /3
•o
O
3
2.0
B
*Z3
Vi
w
1.0
00
0.0
0.0
1.0
2.0
3.0
4.0
Measured ks
(b)
0.4
correl. cocff. = 0.97
0
O
1990 data
G0 /
0
1991 data
*'
/
0.3
/
/ /'0 0
✓0
• V'
•a
I
0.2
n /® '
0
E
•a
c/3
w
0.1
/
/
/
/
0.0
0.0
0.1
0.2
0.3
0.4
Measured mv
F igure 7.17: Comparison between the values o f surface parameters estimated by the
inversion technique and those measured in situ fo r (a) ks and (b) the
vo lu m e tric m oisture contents m v.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
(c )
25.
1
1
1
r
1
.
\
-------[_
_
06
•o
I
E
U3
W
*
©
@
15.
1
•03
73
1
1
20.
'
10.
0
©
0.
5.
rms error = 3.34
_
_
_1
____1
0. 1----.-------1
5.
...
.
1
10.
1
15.
20.
25.
Measured R e a l ( e r )
(d)
6.
1
,
.
i
1
1
'
1
1
1
1
.
5. 00
W
'w'
*
0
©©.'
1
0
l
0
cd
© ,'
O
2.
l
B
-
©
3.
0
**3
00
u
/
°0
"O
4. ,
&D
cd
B
►
-H
• y
1. -
© <3
rms error = 0.81 ■
/
0.
1.
2.
1
1
3.
4.
.
1
5.
Measured I m a g (e r)
F igure 7.18: Comparison between the values o f surface parameters estim ated by the
inversion technique and those measured in situ for (a) the real part of
er and (b ) the im aginary p a rt o f er .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
201
the results for m v where each data p o in t presents the average values for four incidence
angles (30° to 60°) and three different frequencies (L, C, and X -band). Figures 7.1S
(a) and (b) show the results for e' and e", respectively, fo r all surfaces measured in
this stud y (the inversion technique is capable o f estim ating e ', e", and m v fo r any ks,
b u t it is incapable o f estim ating ks i f ks > 3). Note tha t fo r each value of m v, we have
three sets of values fo r e' and e” , corresponding to the three frequencies used in this
study. Inversion diagrams could be generated (see A pp e nd ix B ) fo r the frequencies
o f 1.25 and 5.3 GHz at 20°, 30°, 40°, 50°, and 60° using the sem i-em pirical scattering
m odel developed in this study. The approxim ated values o f the m oisture content m v
and the rm s height s can be estimated q u ickly by looking up the inversion diagrams
w ith known values o f co- and cross-polarized ratios.
The results displyed in Figs. 7.17 and 7.18 represent the firs t dem onstration ever
reported o f a practical alg orith m for e stim ating the roughness, d ie lectric constant, and
m oisture content o f a bare soil surface fro m m u lti-p o la rize d radar observations. Before
this technique can be w id e ly applied, however, it is prudent to conduct a dditional
experim ents over a wide range of roughness and m oisture conditions.
7.6
C O N C L U SIO N S
T h e m a jo r results o f this study are sum m arized as follows:
1. A t microwave frequencies, the available rough-surface scattering models are
incapable o f p re dictin g the scattering behavior observed fo r bare-soil surface.
2. The co-polarized ra tio p = (rlh/cr°v < 1 fo r all angles, roughness conditions, and
m oisture contents; for a ll values o f incidence angle, roughness, and m oisture
contents. The ra tio p increases ra p id ly w ith increasing ks up to ks ~ 1, then
i t increases at a slower rate, reaching the value 1 fo r ks > 3. For ks < 3, p
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202
decreases w ith increasing incidence angle and w ith increasing m oisture content.
3. The cross-polarized ra tio q = crlh/ a l v exhibits a strong dependence on ks and a
re la tive ly weak dependence on m oisture content. T h e ra tio q increases ra p idly
w ith increasing ks up to ks ~ 1, then it increases at a slower rate, reaching the
value (th a t depends on the m oisture content) fo r ks > 3.
4. The proposed scattering m odel (SEM ) provides very good agreement w ith ex­
perim ental observations made over the ranges 0.1 < ks < 6.0, 2.5 < k l < 20.0,
and 0.09 < m v < 0.31. The m odel was found to be equally applicable when
tested against radar data measured for surfaces w ith parameters outside the
above ranges, re outside o f the above ranges.
5. Soil m oisture content (m v) and surface roughness (ks) can be retrieved from
m u ltip olarize d radar observations by applying the m odel-driven inversion tech­
nique developed in this chapter.
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CHAPTER V III
CONCLUSIONS AND RECOMMENDATIONS
8.1
S u m m a ry
The m a jo r contributions o f this thesis have been the developments o f a semiem pirical m odel (S E M ) fo r microwave backscattering from bare soil surfaces and
an inversion a lg o rith m for re trie vin g soil m oisture and surface roughness from the
p olarim e tric radar backscatter. Besides these m a jo r accom plishm ents, the classical
scattering models were evaluated using exact num erical solutions and extensive ex­
perim ental observations. Also, an accurate ca lib ra tio n technique fo r the measurement
o f p o la rim e tric backscatter fro m d istrib ute d targets was developed.
In C hapter 2, classical scattering models fo r radar backscattering fro m dielectric
random surfaces were reviewed. In specific, the sm all p e rtu rb a tio n m ethod and the
K irch h off a pp ro xim a tio n (the physical optics and geom etrical optics models) were
considered.
In C hapter 3, a M onte C arlo m ethod in conjunction w ith the m ethod o f moments
was developed to solve scattering fro m a one-dimensional conducting random surface.
In order to make the num erical sim ulation o f the random surface scattering tractable,
the random surface sample o f fin ite length m ust be considered. Since the edges o f
the fin ite sample p e rtu rb the scattering solution, the edge c o n trib u tio n should be
203
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204
suppressed. This is done by adding a resistive sheet at each end o f the illu m in a te d
surface sample. This num erical technique was used to exam ine existing scattering
models and to compute the phase-difference statistics as w ell as the backscattering
coefficients.
In C hapter 4, an im proved high frequency scattering solution was form ulated and
evaluated num erically fo r a one-dimensional random surface. T h is solution showed an
excellent agreement w ith the num erical solution developed in chapter 3. Using th is
fo rm u la tio n , the zeroth- and the first-o rd e r classical physical optics approxim ations
were examined.
In C hapter 5, the effect o f dielectric inhom ogeneity in a soil m edium was considered
by developing an efficient num erical technique for one-dimensional inhomogeneous
die lectric rough surfaces.
In C hapter 6, an accurate technique fo r measurement o f p o la rim e tric backscatter
fro m d is trib u te d targets was introduced. In th is technique the polarizatio n d isto rtio n
m a trix o f a radar system was com pletely characterized from the p o la rim e tric response
o f a sphere over the entire m ain lobe o f the antenna.
In C hapter 7, the experim ental procedure and the backscattered data collected
fro m bare soil surfaces w ith m any different roughness and m oisture conditions at m i­
crowave frequencies were explained. These data were analyzed and compared w ith
the results fro m the theoretical scattering models. Also they were used to fin d the
dependency o f the backscattering coefficients on the radar and the surface param ­
eters.
Using the co-polarized and the cross-polarized ratios (o-^/(T°u, cr^/cr®,,), a
sem i-em pirical scattering m odel was developed. I t was shown th a t the sem i-em pirical
scattering m odel provided a very good agreement w ith independent experim ental ob­
servations. In this chapter an inversion a lg o rith m for the e m p irica l model was also
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205
developed, and its performance in estim ation o f the soil m oisture and surface rough­
ness param eters was tested.
8.2
R eco m m en d a tio n s for Future W ork
N um erical solution of electromagnetic wave scattering from ra n d o m ly rough sur­
face exists at present for only one-dimensional surfaces. Since the num erical solution
o f a one-dimensional random surface doesn’t p re dict the cross-polarized scattering
coefficient and moreover gives anisotropic results, development o f a num erical code
fo r scattering fro m a two-dim ensional random surface would be e xtrem ely useful be­
cause i t can be used for ve rifica tion of not only the existing scattering models b ut
also experim ental observations for the co- and cross-polarized backscattering coef­
ficients and the phase difference statistics. One way m ay be to s ta rt by solving a
scatter problem o f a three-dim ensional dielectric hum p above an im pedance surface.
Then, scattering from two-dim ensional random surface could be solved s im ila rly as
the M onte Carlo m ethod presented in Chapter 5. In order to achieve this work, either
a powerful com puter may be used, or the problem its e lf m ight be sim p lified w ith an
efficient num erical technique.
E xisting scattering models fo r rough surfaces, in clu d in g the sem i-em pirical model
developed in th is study, have ignored the in form a tion contained in the phase d if­
ference statistics. However, i t have been shown th a t the phase difference statistics
measured fro m bare soil surfaces by a polarim etric radar depends on radar parameters
(frequency, incidence angle, and polarization) and target parameters (soil m oisture
and surface roughness). The p re lim ina ry results show th a t the co-polarized phase
difference d is trib u tio n has a strong dependency on the target and radar parameters,
w hile the cross-polarized phase difference d is trib u tio n doesn’t have any dependency
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206
on the target and radar parameters. In order to u tiliz e the phase difference in for­
m a tio n, i t seems necessary to extend the scattering models to relate the co-poIarized
phase difference d is trib u tio n to surface roughness and soil m oisture.
The sem i-em pirical model is based on radar measurements of a lim ite d num ber of
soil surfaces. In order to increase the v a lid ity of the m odel, i t would be recommended
to conduct a d d itio n a l experim ents over a wide range o f roughnesses and soil types.
Since the backscattering coefficients o f soil surface show strong dependencies on soil
m oisture and rm s height, the sem i-em pirical model was given as a fu n ctio n o f those
parameters. However, the backscattering coefficients was also affected by other pa­
rameters such as the correlation length and soil type o f the surface even though the
dependencies are re la tive ly weak as shown in Fig. 8.1. I believe the v a lid ity o f the
sem i-em pirical m odel can be extended by fin ding m ore fun ctio na l relationships and
using m ore target parameters w ith a d d itio n a l extensive experim ents.
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207
(d B )
Parameters
( o ;v/
M
ia )
Surface Roughness
(ks and Id)
ks with fixed kl
Sensitivity
Soil Moisture
( 0.0 4 - 0. 3 5 )
kl with fixed ks
Soil Type
(Sand-C Iay)
Temperature
Maximum
Minimum
(5°- 25°c)
Range o f Parameters
F igure 8.1:
S e n sitivity o f a°v on surface parameters at 40°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIC ES
208
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209
A P P E N D IX A
CHARACTERISTIC FUNCTION FOR A
GAUSSIAN RANDO M VECTOR
The characteristic fun ctio n is defined as;
*3 r(")
= (e'"rs)
= /°° -Pr(z) e” ' 1 dx
'
‘/-00
( A . l)
where x = ( s j, • • •, x n)T is a real n-component random vector, uJ = (o>i, • • ■,0Jn)T is a
real n-com ponent parameter vector. The p ro b a b ility density fu n ctio n (P D F ) P y ( x )
for a Gaussian random vector can be w ritte n com pactly as;
P x (x ) =
-=
exp
(2ir)n/ 2[d e t{K )]V 2
(A .2)
where Ji = ( / ij, •••, fin)T is a n-component mean vector and K is the covariance m a trix
defined as the average value o f the outer vector product (x — Ji) • (x — Ji)T, i.e.,
K = ( U - ; i ) ( i - (i)1} .
(A .3 )
S u b stitu tin g (A .3 ) into ( A . l) , the characteristic fu n ctio n for a Gaussian random vector
can be obtained. When we assume a Gaussian random vector has zero mean, the
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210
covariance m a trix becomes a correlation m a trix R,
<
Xi
Xi
>
<
Xi
< Xi x n >
x 2 >
R = (^x x T^ =
(A .4 )
<
Xn Xi
>
< xn x 2 >
•••
< xn xn >
w hich is a real sym m etric m a trix , and the corresponding characteristic fun ctio n is
given [S tark and Woods; 1986] as;
r i
— i
r
= eXP —ZUTRw \ = exp
*
J
i
n
"
^rn^n (x mXn)
(A .5)
1 m =l n=l
In order to get the characteristic function fo r a tw o-dim ensional random surface
h aving a Gaussian height d is trib u tio n w ith the random vector x — [ZXl, ZX2, Z y i,
Zy2) Zd]T and the param eter vector a; = [—a Xl, a X2, —a y i, a y2,kdz]T (Ch. 3), a t firs t
the correlation m a trix is w ritte n e x p lic itly as;
R = ( x x T^
< Z *i >
< Z X1Z X2>
< Z X1Z X2
> < Z l2 >
< Z xJZyJ
> < Z X2 Z yi
>
< Z XlZ y i >
< Z X lZy2 >
< Z Xizd >
< ZX2Z yx >
< Z X2Z y2 >
< ZX2zd >
< ZyJZy2 >
< Z y xZd >
< Z y 2 Zd >
< Z yi >
< Z x J Zy2 >
< Z X2 Z y 2 >
< Zy j Z y2 >
< Z
>
< Z Xlzd >
< Z X2Z d >
< Z y iZ d >
< Z V2zd >
(A .6)
< z \>
When, we consider the random process of zero mean, /i = (z (x, y )) = 0, the variance is
defined as cr2 = (z 2( x , y)). Since we assume th e random process { z ( x , y ) } is stationary
in the w ider sense, the correlation o f z { x i,y { ) and z (x 2, y 2) can be com puted as;
C ( u ,v ) = C ( x i - x 2, y x - y2) = ( z ( x i, y i) z ( x 2, y 2)) = a 2p (u ,v )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A .7)
211
where the correlation coefficient
/
v
p{u, v
) is defined by
(z {x u yx )z {x 2, y i) )
a
p(U ,v) =
(A .8)
\J {z 2{ x i , y 1))y /{z 2( x 2, y 2))
and (z2( x i, y i) ) = (z 2(x 2 , y 2 )) = cr2- Quadratic-m ean derivative of a random process
{ z ( x ) } at a p o in t x is defined [Wong and H ajek, 1985] as
„
A d z (x ) A
z (x H- A x ) - z (x )
Z x = —7T— = h m
d x
A x — O
Ax
(A .9)
when the follo w in g conditions are satisfied;
lim
A x ,A x
z(x + A x ) — z (x )
z (x + A x ') — z (x ) 2
=
Ax'
Ax
1—*0
0.
One element < Z 2} > of the correlation m a trix R can be computed using (A .9 ) and
the lin e a rity o f expectation as
= Alim o ^ - ^ ( z 2(x 1 - |- A u , 2/i) +
=
2 2( x ! , j/ i)
Alim
T A
u -*0„ 7( A
u )\2
2 2or2
t 1
W hen we expand the correlation coefficient
“
- 2 z {x 1 + A u ,y x) z ( x u y i) )
P(Au>
° ) ]
p(u ,v)
(A .10)
■
in a T a y lo r’s series fo r a function
of two variables,
ti= a
(A .X l)
4*
t/=6
Since the correlation coefficient p{u, v) is an even fu n c tio n having a m a xim u m of 1 at
u — v = 0, ( A .11) can be re w ritte n su b stitu tin g h = A u, k = 0, and a = b = 0 as,
.
.
(A u )2 d 2p (u ,v )
u=o
2!
du2
v=0
p(Au, 0) = 1 + -— ------ — n
’ '
+
( A u )4 d4p (u ,v )
4|
du4
u=0
+
( A .12)
u=0
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212
Therefore, su b stitu tin g ( A .12) in to (A .10, < Z 2X > can be computed as
1 d 2p (u ,v )
i z 2 ') — 2cr2 lim
\
11
/
Au—
*o
2
(A u )2 <9V(u,u)
u= 0
du2
41
u=0
duA
+
u=0
u=0
d 2p ( u ,v )
= -<r
5u2
(A.13)
u= 0
u=0
In order to avoid co m p licity in w ritin g , new notations are defined fo r convenience as
follows;
d p (u ,v )
ou
u=:a — Pu{aj b),
d p ( u ,v )
dv2
. /.. m
d 2p {u ,v )
A .
/
..
&)
usa
dv
v=b
d 2p (u ,v )
a
v=b
l \
0 2/>(u,u)
v=o
= puv(a ,b),
etc.
(A -14)
v=b
Using statio n ary characteristics and new notations, the diagonal elements o f the cor­
re la tion m a trix can be computed as;
(Zj,) = ( 2 i) = -ffV..( 0,0),
( zl ) =
+ z2( z 2,!/2) - 2 3 ( 1 !, ^ / ! )
2
(A.15)
( ^ 2 , J/2) ) = 2cr2 [ l - / j ( u , u ) ] .
(A .17)
The off-diagonal elements o f the correlation m a trix can also be computed using the
quadratic-m ean derivatives (A .9) and the the T aylor’s series expansion ( A .11).
(Z X1 Z X2) = j i m
+ A u , y i)
-z (x
= Alim Q
1
2 (1 2
+ A u ,y 2) + z (x x, y i) z {x 2, y 2)
+ A u , 7/ 1 ) z (x 2, y2)- z {x !, r/i) z (x 2 + A u , 7/ 2 ))
u ) “ P (u +
“
P (u ~ A u ><01 •
( A .18)
( A -1 9 )
The correlation coefficients, p (u + A u ,7 ;) and p(u —A u ,v ) can be obtained fro m ( A .11)
w ith h = A u , k = 0, a = u, and b = v as,
p(u + A u , v ) = p(u, v) + A upu(u, v ) + ^ j ^ ~ p uu(u, v) -j
,
p (u - A u , v) = p[u, v ) - A upu(u, v ) + ^ p - / 9 „ u(u, v ) ---------.
(A .20)
(A .21)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
213
S ub stitu ting (A .20) and (A.21) in to (A .19),
(Z XJ Z X2) = - a 2puu(u ,v ).
(A.22)
(z vi Z V2 ) = -< r2P w {u ,v ).
(A .23)
S im ilarly,
O ther components of the correlation m a trix can be computed in s im ila r manners as
follows;
(z * i Z v i)
=
( z { x i + A u , yx) z ( x 1, y 1 + A i / ) + z2( i i , t / i )
-z (x i
2
=
+
A u , yx) z ( x i, y i) - z ( x u yx + A u ) z (x x, y x))
( K ^ j* ^ A u > A u ) + 1 ~ ^ A u >° ) ~ ^ (°> A u )] •
(A -24)
where
p (A u , A u ) = p (0,0) + A u [/ju(0, 0) + pv(0,0)]
+
(A u )2
2!
[/3uu(0,0) + 2M M ) + M M ) ] + ■• ’
= 1+
p (A u , 0)
=
p(0, A u )
=
( A u l2
k u ( 0 , 0) + pw (0 ,0)] + • • •
1 + ^—^ —puu{®, 0) + • • •
1 + ^ ^ ^ -/9 t,u (0 ,0) H
.
(A.25)
(A .26)
(A .27)
S ub stitu ting (A .25)-(A .27) in to (A .24),
(Z xj Z y i) = (ZX2 Z y2) = 0.
(A .28)
In a s im ila r m anner, other components are obtained as;
(Z Xl Zy2) = (Z X2 Zy,) = - a 2puv{ u ,v ),
(A .29)
[Z xj zd) = (Z X2 zd) = - c r 2pu(u ,v ),
(A .30)
{Zvi Zd) = (Zy2 zd) = - a 2pv(u ,v ).
(A .31)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
214
F in a lly , th e c o rre la tio n m a tr ix can be w r itte n w ith the c o rre la tio n co efficient and its
d e riv a tiv e s , assum ing an is o tro p ic ra n d o m surface such th a t puu( 0 ,0) = Pur(0, 0),
/» „„(0 ,0 )
Puu(u,v) 0
Puu(u,v)
p u u ( 0 ,0 )
0
puv( u , v ) puv( 0 ,0 )
Puu(u,v)
pn(u,v)
Puv(u,v)
0
Puu(u,v)
/> „„(0 ,0 )
P u( u , v )
P u( u , v )
p u{ u , v )
pu( u , v )
P u( u , v)
—2 [1 — p(u, u)]
R = -a 2
Puv{u,v)
Puv(u,v)
0
p u( u , v )
p u( u ,V )
■ ( A .32)
T h e re fo re , th e c h a ra c te ris tic fu n c tio n fo r a th e Gaussian ra n d o m ve cto r, x = [Z X l, Z X2,
Z Vl, Zy2, Zd]T , can be co m p u te d using th e com ponents o f th e c o rre la tio n m a tr ix , R,
and th e p a ra m e te r v e c to r, u> = [—a X l , a X2, —a m , a y2, kdz]T . A fte r algebraic co m p u ­
ta tio n s , th e fin a l fo rm o f th e c h a ra c te ris tic fu n c tio n is given by
( e ' wTx^ = exp
=
+ Q & } - cr2k2
dz {1 - p { u , v ) }
= exp | y p uu(0 ,0 ) { a 2
xi + q £ +
&
^)
“f "
®-y\ ^ 3 /2
“f~ ( < * x i ^ 3 /2
) Pvvi^ j
^ }
- < r 2kdz { ( a * ! - a X2) pu( u , v ) + ( a yi - a y2) pv(u, v ) } ] .
(A .3 3 )
F or a o n e -dim e n sio na l ra n d o m surface, th e c o rre la tio n m a tr ix fo r a ra n d o m v e cto r
o f x = [ Z xj , Z X2,Zd]T can be co m p u te d s im ila rly w ith th e tw o -d im e n sio n a l surface
case.
U s in g th e q u a d ra tic -m e a n d e riv a tiv e and th e T a y lo r’s series expansion, th e
c o rre la tio n m a tr ix R \ is o b ta in e d as
R \ = - O '2
/,uu(0)
Puu(u )
Pu(u)
puu(u)
puu( 0)
p u( u )
Pu{u)
pn( u)
~ 2 [1 ~ p(u)\
•
(A .3 4 )
T h e c h a ra c te ris tic fu n c tio n fo r a one-dim en sional surface h a v in g a G aussian h eight
d is tr ib u tio n can be co m p u te d using th e com ponents o f th e c o rre la tio n m a tr ix R \ and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
215
the p a ra m e te r vector uJi = [• -£ *1, a 2, kdz]T as
^ e ~ i a i Z Xl + i a 2 Z X2 + i k dcz d ^
= exp
=
= exp
=
y /» ™ *(0) { ol\ + a j } - a 2k 2
dz {1 - p { u ) }
- a 2 puu{ u ) a ^ a 2 - <r2pu {u)kdz (c*i - a 2)] .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.35)
216
A P P E N D IX B
INVERSION DIAGRAMS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
hh
/<
©
Vy (dB)
K>
©
©
£
©
0>i
$
s.
§
??
£?
#
&
» o
I
O,
.1
o
§
218
^S U re
2^ e r sion
fo r , , , „
■J'5 G f li
z
00Pyright
°Wner
Furtherr«n
reProdLuotion
pr°hibited wit
3 qo
Inversion Diagram (1.25 GHz, 40°)
M v=0.l0
(ap)
Mv=0.15
Mv=0.25
Figure B.3:
Inversion diagram for 1.25 G Hz a t 40°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
220
Inversion Diagram (1.25 GHz, 50°)
Mv=0.10
Mv=0.15
■Mv=0.20
■Mv=0.25
Mv=0.30
-------
-24.
-22.
-20.
-18.
q = o0hv/o 0 w
Figure B.4:
-16.
-14.
-12.
-10.
m
Inversion diagram fo r 1.25 GHz at 50°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
221
Inversion Diagram (1.25 GHz, 60°)
(ep)
Mv=0.10
Mv=0.15
Mv=0.20
Mv=0.25
Mv=0.30
-24.
-22.
-20.
Figure B.5:
-18.
-16.
(3 = 0 ° h v ^ ° vv
(d B )
-14.
-12.
-10.
Inversion diagram fo r 1.25 GHz at 60°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
222
Inversion Diagram (5.3 GHz, 20°)
s=3.5 cm
s=2.5 cm
0.05
[v=0.10
s=1.9 cm
s=1.4cm
FM v=0.15
-
1.0
PQ
PMv=0.20
'w'
>
>
e
O
II
Cu
-1.5
Mv=0.25
-2.0 r-Mv=0.30
s=0.7
0.35
-2.5
s=0.45
•0.40
-3.0
-16.
s=0.3 cm
-15.
-14.
-13.
-
12.
-
11.
-
10.
■9
q = a °h v /o 0w (dB)
Figure B.6:
Inversion diagram fo r 5.3 GHz at 20°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
223
Inversion Diagram (5.3 GHz, 30°)
s=3.5 cm
s=2.5 cm
s=1.9 cm
Mv=0.10
s=I.4cm
Mv=0.15
(ap)
s=1.0cm
Mv=0.20
o
ii
-Mv=0.25
Q,
-3.0
Mv=0.30
s=0.45
s=0.3 cm
-16.
-15.
-14.
-13.
-12.
-11.
-
10.
-9.
q = ° 0hv/ o Ovv (dB)
Figure B.7:
Inversion diagram for 5.3 G Hz at 30°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
224
Inversion Diagram (5.3 GHz, 40°)
s=3.5 cm
s=2.5 cm
s=1.9cm
-
Mv=0.10
s=1.4cm
s=1.0cm
Mv=0.20
'Mv=0.25
Mv=0.30
s=0.45
13.
-12.
-11.
-10.
q = ° V ° 0Vv m
F igure B.8:
Inversion diagram for 5.3 GHz at 40°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
225
Inversion Diagram (5.3 GHz, 50°)
s=3.5 cm
s=2.5 cm
s=l .9 cm
-
1.0
0.05
s=1.4cm
Mv=0.10
-
2.0
Mv=0.15
(ap)
-3.0
>
>
e>
-4.0
s=0.7
Mv=0.20
Mv=0.25
-5.0
Mv=0.30
-
6.0
s=0.45
0.35
,s=0.3 cm
0.40
-7.0
-16.
-15.
-14.
-13.
q = °°hv/ < v
Figure B.9:
-
12 .
-
11 .
-
10.
■9,
(dB)
Inversion diagram for 5.3 G Hz at 50°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
226
Inversion Diagram (5.3 GHz, 60°)
s=3.5 cm
s=2.5 cm
s=1.9 cm
-
1.0
0.05
s=1.4 cm
-
2.0
Mv=0.10
(ap)
-3.0
Mv=0.15
>
>
-4.0
s=0.7
O
II
-5.0
Q.
Mv=0.20
-
6.0
-7.0
Mv=0.25
s=0.45
Mv=0.30.
s=0.3 cm
0.35
-
1.40
8.0
-16.
-15.
-14.
Figure B.10:
-13.
- 12.
q = °V °°w
m
-
11 .
- 10.
9
Inversion diagram fo r 5.3 G H z at 60°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
227
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
228
B IB L IO G R A P H Y
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