1
2
3
Auxiliary Material for
An Accurate and Fast Forward Model of Three Dimensional Electromagnetic Wave
Scattering in a Layered Structure with Multilayer Rough Interfaces
4
5
8
9
6
7
10
1
Chao Wu 1 , Xiaojuan Zhang
Jiancheng Shi
1 , Guangyou Fang
2
, and Shiyin Liu
3
1 ,
Key Laboratory of Electromagnetic Radiation and Sensing Technology, Chinese Academy of Sciences, Beijing, 100190, China
2
State Key laboratory of Remote Sensing Science, Beijing, China
3
Cold and Arid Regions Environmental and Engineering Research Institute, Chinese
Academy of Sciences, Lanzhou, China
11 Radio Science, 2014
12
13
14
Introduction
This auxiliary material contains higher-order perturbative analytical solutions, numerical
15 validations, full polarization numerical analysis, figures for the article “An Accurate and Fast
16 Forward Model of Three Dimensional Electromagnetic Wave Scattering in a Layered
17 Structure with Multilayer Rough Interfaces”.
18 This auxiliary material is organized as follows: In Part I, the general compact closed-form
19 explicit expressions of high-order perturbative solutions of the problem of 3D radiation and
20 propagation of EM fields inside layered structure with an arbitrary number of rough
21 interfaces is provided based on our proposed method. In addition, the kernel function
22 approach is proposed to develop a hybrid method based on the obtained essential property of
23 EM scattering from rough interfaces, which is significant and important, and can be better use
24 in practice. Part II gives the numerical validation and investigates completely the influence of

25 various parameters of layered structure and configuration of radar system on full polarization
26 radar cross sections, which indicates that the significant of the high-order contributions.
27
28
29
Part I H
IGH
-O
RDER
P
ERTURBATIVE
A
NALYTICAL
S
OLUTIONS
30 The proposed method can obtain the high-order full-polarization radar cross section of the
31 layered structure with two rough interfaces expressed by the Sarabandi and Fuks models, as
32 well as their joint mutual coupling component as below:
33
 0(2) qp

4
 k
2
0 cos
2  s

 d

 W g
0
( k
 f

(2)
0 qpF
(

 )
 f
0

(2) qpF
( k
 
 
 ) W g
0
(


 k )
 f

(2)
0 qpF
(

 )

 k )


  d

 W ( k
 
 ) W (


 k ) Re f
0

(2) qpF
(

 )

 f

(2)
0 qpS
(

 )
 

  d

 W g
1
( k f

(2)
0 qpS
(

 )

 
 ) W g
1
(


 k )
 f

(2)
0 qpS
(

 ) f

(2)
0 qpS
( k
 

 k )

 
 
(1)
34

35
 qp

4
 k
0
2  cos
2  s
 

  l

1 d

1
 d

2
 d

3
 d

 

W g
0


1

 k i



W g
0
 k
 s

 f
0
 qpF

 
 
 l

L

3
W g
0


,
2

, ,

 


( L
  f
0
 qpF



( L
 

2


 f
0
 qpF

 
1

 k i

,
2

, ,

 
,

3

 
2

 
1

, ,
 k
 s  
 
 
( l
 
  
   l

1 d

1
 d

2
 d

3
 d

 

W


1

 k i


W
 k s

,

 

  f
0 qpS

 
 
 l 
L

3
W
,
2

, ,



( L
 
 





 
( L
 


Re

  f
0
 qpF d

1
 d

2
 d

3
 l

1
 d


 

W g
1
,
2

, ,



1

 k i



W g
1
 k
 s
 


 f
0
 qpS

 
 
,
2

, ,

 l 
L

3
W g
1


 


( L
  f
0
 qpS
 
( L
 


2

 
1



 k i
 f
0

, qpS

3

 
2

 
1

, , k
 s  
 
 
( l
 
  
,
2

,
36 (2)
37 where f

( )
0 qp
represents the scattering property of high-order multiple-bounce process
38 originated from first-order to -th order on rough interface, which indicates the
39 manipulations and processes of scattering property including from single-bounce to multiple-
40 bounce. W g
0
( k  ) and W g
1
( k  ) are the power spectral densities of the rough interfaces
41 denoted by g
0
( , ) and g x y respectively. W ( k  ) is the joint spectral density
42 between the top rough interface g
0
( , ) and bottom rough interface ( , ) with correlated
43 property. The first and third terms in the RHS of the Eq. (1)-(2) represent the scattering
44 components from top rough surface and bottom rough interface, respectively, contributed to
45 the overall scattering fields. The second term in the RHS of the Eq. (1)-(2) indicates the

46 mutual coupling scattering components between correlated top rough surface and bottom
47 rough interface, contributed to the overall scattering fields in region 0.
48 The train of thought of calculating the problem of layered structure with N-rough
49 interfaces is to calculate the kernel function of high-order perturbative solutions of EM fields
50 in layered structure with one statistically rough interface inside multi-planar mediums and
51 joint mutual coupling scattering components between each correlated rough interfaces,
52 respectively. Therefore the general explicit closed-form high-order perturbative solutions of
53 the problem of 3D EM scattering from inhomogeneous layered structure with N-rough
54 interfaces can be derived as
55
 0(2) qp

4
 k
0
2  cos
2  s


N

1   m

0 d


W g m
( k
 s f
0

(2) qp g m (


)
 f
0

(2) qp g m ( k
 s
 

) W g m
(


 k i

)
 f

(2)
0 qp g m (


)
 

 k i

)


4
 k
0
2  cos
2  s
  d


W g g j
( k
 s  

)

W g g j
(


 k i

) Re

(2) f
0 qp g i (


)



(2) g f
0 qp j
(


)
 
(3)
56 with
57
58 f

(2)
0 qp g m (

 )
 f

(2) mqp g m (

 )
 k s
0 z k s mz e jk i d
T p
0, m
( k )
 e jk s d
T q m ,0
( k ) (4)
The kernel function f

(2) mqp g m (

 ) is presented in Appendix.

59
 qp

4
 k
0
2  cos
2  s

N m

1   

0 l

1 d

1
 d

2
 d

3
 d

 

W g m


1

 k i



W g m
 k
 s  

 f
0
 qp k
 s   m

 
 
 l 
L

3
W g m


,
2

, ,

 

( L
 
 f
0

 qp
( l )
 g m
( L
 


 f
0
 qp

2

 
1

 m k i


,
2

, ,

 
,

3

 
2

 
1

, ,

 
( l
 
  

4
 k
0
2  cos
2  s
   l

1 d

1
 d

2
 d

3

(5) d


 

W g g j

( l ) f
0 qp g i


1

 k i


W g g j


 
2

, ,
 k
 s  
 



 f
0
 qp
 

L l


3
W g g j


( L
 
 
( L
 
 j

,
2

, ,

 
 

 
60
61 where the superscripts ( l ) represents the order of the perturbative solutions. The coefficient
 f
0 qp m is the relative to the p -polarized incident wave impinging on the structure
62 transmission through upper half-infinite space region 0 to region m acted as equivalent
63 incident wave on rough interface g m x y and to the q -polarized l -multiple-bounce
64 scattering contribution from rough interface and transmission from above multi-planar
65
66 layered structure into the upper half-infinite space, originated from the rough interface between the layer m and m

1 . ,

{ , } denote the incident and the scattered
67 polarization states, respectively, and may stands for horizontal polarization ( h ) or vertical
68 polarization ( v ).
69 The higher-order SPM solutions of the problem of layered structure with multilayer rough
70 interfaces contains two parts: the one part is the scattering components from each rough
71 interfaces denoted by the first term in RHS of Eq. (5), which can be obtained through
72 calculating the higher-order SPM solutions of the problem of layered structure with one
73 embedded rough interface into multilayer planar interfaces and superposing these SPM
74 solutions of N problems (if there are N rough interfaces). As the procedure for deriving the
75 higher-order SPM solutions of the problem of layered structure with one rough interface is

76 much easier than the procedure for deriving the higher-order SPM solutions of the problem of
77 layered structure with multilayer rough interface, then our proposed method can be efficient
78 way to calculate the SPM solutions of complex problem of layered structure with multilayer
79 rough interfaces. In addition, as the proposed method is developed based on the theoretical
80 formula derivation in framework of SPM, therefore the obtained conclusion should be
81 guaranteed and can reveal essence property of rough interface scattering. As there were other
82 approaches for deriving the higher-order SPM solutions of the problem of layered structure
83 with restricted number of rough interfaces (less than two rough interfcaces) because of
84 awfully complex computational process and even impossible to resolve the problem of
85 layered structure with more than three rough interfaces (with per rough interface increase, the
86 complexities increase exponentially, in more rough interfaces case, the growth rate of
87 complexities increases exponentially, the growth rate even overweighs the exponential way),
88 for example the four-order SPM solution of two rough interfaces [ Demir, et al.
, 2012], the
89 authors can obtain the four-order SPM solution of problem of layered structure with
90 multilayer rough interfaces through employing the four-order kernel function of problem of
91 layered structure with one rough interfaces to obtain the fourth-order SPM solutions of
92 multilayer rough interfaces while there are some reflection/transmission coefficients should
93 be considered to correct the higher-order SPM solution provided in the physical meaning
94 section based on our proposed method. The other part is the mutual coupling scattering
95 components among each rough interface expressed by the second summation term of the
96 RHS of Eq. (5). The higher-order mutual coupling scattering components from layered
97 structure with multilayer rough interfaces can be calculated similarly based on the obtained
98 higher-order scattering term
 f
0 qp m of problem of layered structure with one rough interface
99 and the cross-correlation spectral density of each rough interface.

100 There are many methods for deriving the radar cross section of the problem of rough
101 interface scattering, such as small slope approximation (SSA), method of moments (MoM)
102 and so on. For example, when employing the MoM to calculate the radar cross section of one
103 rough interface, which MoM results are equal to the overall scattering fields obtained by
104 superposing the all SPM orders solutions, then the overall scattering fields of the problem of
105 layered structure with multilayer rough interfaces can be obtained based on the MoM results
106 of one rough interface and the reflection/transmission coefficients in multilayer planar
107 interfaces, and finally employing our proposed method to obtain the overall scattering from
108 layered structure with multilayer rough interfaces. The MoM employed to calculate the one
109 rough surface scattering can be accepted within certain computational cost and memory but
110 higher accuracy, then the proposed method can be utilized to combine the MoM method with
111 analytical method to calculate the radar overall scattering from layered structure with
112 multilayer rough interfaces. The hybrid method of numerical method and analytical method is
113 achieved based on our proposed method, the numerical method guarantees the accuracy and
114 the analytical method guarantees the efficiency. In addition, the single numerical method for
115 deriving the problem of 3D layered structure with multilayer rough interfaces is impossible,
116 the proposed method can achieve the combination of numerical method with analytical
117 method to obtain the complex problem of 3D layered structure with multilayer rough
118 interfaces in accuracy and efficiency. Certainly, the other numerical/ semi-numerical/
119 experiential/ analytical methods employed to calculate the radar cross section of rough
120 interface scattering can also be utilized to achieve the calculation of complex problem of 3D
121 layered structure with multilayer rough interfaces while some correction index of
122 reflection/transmission in layers should be considered and added based on our proposed
123 hybrid method. In a word, the proposed hybrid method can be utilized to resolve the complex
124 problem with higher accuracy and efficiency. As the conventional methods can be utilized to

125 obtain the kernel function of scattering with simple structure case (two layers with upper and
126 lower half-space), then employ our proposed method and the considered
127 reflection/transmission coefficients to achieve much more complex problem of 3D
128 inhomogeneous layered structure with an arbitrary number of rough interfaces based on the
129 kernel function.
130
131
132
133
Part II V ALIDATION AND N UMERICAL R ESULTS
To validate our proposed method of the first-order situation, we reproduce the Fig. 4 in
134 [ Imperatore et al., 2009] for the simple case of three rough interfaces in order to validate with
135 existing classical SPM in the first-order situation, which is shown as below in Fs01. The
136 comparison with the reference paper [ Imperatore et al., 2009] shows a very good agreement
137 for first-order SPM solutions when our proposed method is regressed to first-order method
138 for deriving the bistatic scattering coefficient of special cases with three rough interfaces
139 considered by Imperatore et al., 2009. As the main advantage of our proposed method
140 compared with classical SPM like [ Imperatore et al., 2009] is that the proposed method can
141 derive the high-order perturbative solutions of the problem of 3D bistatic scattering from
142 layered structure with multilayer rough interfaces, therefore we also calculate the higher-
143 order perturbative solutions of this case based on our proposed method in order to give
144 insight into the accuracy of the first-order solution, which is also shown in Fs01. From the
145 Fs01, we can see that the higher-order contributions make appreciable contributions to radar
146 cross sections especially at some observation angles, although the first-order contributions
147 dominate the overall scattering fields. Therefore, in order to obtain more accurate solutions
148 and successful inversion process, the higher-order perturbative solutions is essential,

149 meaningful and significant as the first-order solutions is inaccurate for estimating the overall
150 scattering fields.
151
Fs01. Bistatic scattering coefficients for three rough-interface layered structure, with
 
9.5
 j 0.00055
, 20.5
 j 2.55
,

1
 
1.15
,

2
 
2.80
, k
  k
  k
 k l
 k l
 k l
3.0

0.15
,
,

1.5
, f

1.0 GHz ,
 
45 o
,
 
0 o
, and
 
45 o
. Black solid line indicates the total first-order contributions
 0(1) qp
, cyan dotted-dashed line represents the first rough interface contribution f

(1)
0 qp g
0 , blue dotted line shows the second rough interface contribution f

(1)
0 qp g
1 , green dashed line denotes the third rough interface contribution f

(1)
0 qp g
2 , magenta dotted-dashed line illustrates the total second-order contributions
 0(2) qp
, red solid line illustrates the total first two orders contributions.
152
153 In the followings, we will emphasize on analysis of the double-bounce scattering
154 properties of layered structure with four rough interfaces and three intermediate layers.
155 Fs02 illustrates radar cross section of the four rough-interface layered structure as well as
156 the significance of the second-order contribution at some observation angles. The first two
157 orders bistatic scattering returns decrease when the observation angles away from the
158 specular direction.

159
Fs02. Simulation of the first two orders bistatic scattering coefficients for four rough-interface layered structure, with u
1
 u
2
 u
3
 u
4

0.025

, l
1 l l
3 l
4
0.35

, d
1

0.2

, d
2

0.8

, d
3

1.3

,
 
5.3
 i 0.005
, 11.87
 i 0.008
, 17.9
 i 0.006
, 23.8
 i 10.0
, f

1.0 GHz ,
 i
0

45
0
,

0 i 
0 and

0 s 
15
0 .
160
161 Fs03 illustrates the total radar cross section including first- and second-order SPM
162 solutions of scattering field distribution in three-dimensional space. The plot formats are a
163
164 projection of the hemisphere onto a horizontal plane, in which the horizontal and vertical axes represent sin
 s cos
 s and sin
 s sin
 s
, respectively.

165
166
167
FS03. Simulation of high-order total scattering with the same configuration as Fs02, but with an azimuth angle
 s
ranging o
[0;180] and observation angle ranging
 o o
[ 90 ;90 ] .
168
169 Fs04 illustrates the backscattered fields contributed by both first- and second-order SPM
170 solutions. The backscattering response will decrease as the correlation length increase. In
171 addition, the first-order scattering response will reduce much faster than second-order
172 scattering fields along with increasing rough correlation length in co-polarization case. In the
173 cross-polarization HV and VH scattering case, the second-order scattering fields dominate the
174 overall scattering response and the contributions from first-order scattering become zero. The
175 influence of depolarization on the global scattering response power is reflected in the cross-
176 polarization components as shown in Fs04. However, the scattering response will increase as

177 the increasing correlation length when the correlation length is less than 0.25 wavelengths.
178 The second-order backscattering fields can give more contributions to total radar cross
179 section than first-order contributions as the increasing correlation length, as well as second-
180 order effects can dominate the total scattering fields at some larger correlation length.
181
182
183
FS04. Back-scattering coefficients for four rough-interface layered structure as a function of rough interface correlation length with same configuration as Fs02.
184
185 Fs05 shows the effect of the first surface roughness on backscattering cross section. It is
186 observed that the second-order depolarized scattering cross section increases with the first
187 surface roughness. Furthermore, as the first interface roughness increase, the first surface

188 itself produces more nonspecular co-polarized scattered power than cross-polarization for
189 second-order depolarization in the incident plane.
190
191
192
FS05. Simulation of the influence of the first rough surface roughness on radar backscattering cross section with the same configuration as Fs02.
193
194
195 A PPENDIX A
196 The kernel function f

(2) g m mqp
(


)

U g m mqp
( k
 s i
, k

)

V g m mqp
(


) , the expressions of U g m mqp
and
197 V g m mqp
are presented as below:

198
199
200
201
202
203
204
205
206
207
208
209
U g m mhh


1
4
( k
2 m

1
 2 k m
) cos(
  i
)


 1

R h
,

1
( k i

) 1 cos
 i m cos
 s m
 1

R h
,

1
( k i

) 1 R h
,

1
( k s

)

U g m mhv

1
4
( k
2 m

1
 k
2 m
) sin(
  i
)


 1

R v
,

1
( k i

) 1
 cos

1  s m

1 R v
,

1
( k i

) 1

R h
,

1
( k
 s
)

R h
,

1
( k
 s
)
R h
,

1
( k s

)
 cos
 i m
(A.2)
U g m mvh

1
4
( k
2 m

1
 2 k m
) sin(
  i


1  m s
 i m

1 R h
,

1
( k i

) 1

1
R v
,

1
( k
 s
)
R h
,

1
( k i

) 1

(A.1)
R v
,

1
( k
 s
)
(A.3)
U g m mvv
 
1
4
( k
2 m

1
 k
2 m
) cos(
  i
)


 1

R v
,

1
( k i

) 1
 k k i mz s mz
 1 R v
,

1
( k i

) 1 R v
,

1
( k s

)

R v
,

1
( k s

)
V g m mhh
(


)
 
1
4
 1

R h
,

1
( k
 s
) 1
  cos(

R h
,

1
( k
 i
)

( k
2 m

1 k
2 m
 cos k
 m s m

) cos(


  i
) cos

1   m
 1

R v
,

1


  i
) sin(


) cos
  m
 1 R v
,

1
(

)

(A.4)
(A.5)
V g m mhv
(


)
 
1
4 cos
 s m
( k
2 m

1
 k
2 m



) sin
  m
 sin
 i m
1

R v
,

1
( k i

)

1
 cos(

1
R h
,

1
( k
 s
)



) sin(





 m m

1
  i
2 k m
 1

) cos

1
R h
,

1
( k
 s
) 1

R h
,

1

  m

R v
,

1 cos

 sin( i m
  
1 R v
,

1


 m

1

 m
 m


1 R v
,

1
( k
 i
)

) cos(


  i
) cos
  m
 cos
 i m

 m

1

 m
 m

1 R v
,

1
( k i

) 1

R h
,

1
( k
 s
)   1

R v
,

1

(A.6)

210
211
212
213
214
215
V g m mvh
(


)
 
1
4 cos
 s m
( k
2 m

1
 k
2 m



  i
) sin
 s m
 sin
  m
1

R h
,

1
( k i

)
R v
,

1
( k
 s
)



 k
2 m
2 k m

1
 1

R v
,

1
( )   1
 sin(


) cos(


  i
) cos cos
 
  m m

1

 m
 m

1
R v
,

1
R h
,

1
( k i

)



R v
,

1
( k s

) 1

R h
,

1
( )
 cos(


) sin(


  i
) cos
 s m
  m
 m

1

 m
 m

1 R h
,

1
( k i

) 1

R v
,

1
( k
 s
)   1

R v
,

1

(A.7)
V g m mvv
(


)

1
4
 1

R v
,

1
( k
 s
) k
2 m

1
 k m sin(
 

) cos
 i m
 sin(


  i
) cos

1

1 k
2 m
 1

R v
,

1
( k i

) 1 R h
,

1


 cos(
  m

)
   m sin
 i m
( k
2 m

1
 k
2 m
 R v m , m

1
( k i

)
 k
2 m

R v
,

1 k
2 m

1

1

R v
,

1 k
2 m

1
2 k m


 cos(
 

) cos(


  i
) cos
  m cos
 i
( k
2 m m

1
 k m
 1

R v
,

1
( k i

) 1

R v
,

1




1
4 tan
 s m sin
  m
 k
2 m

1

1
R v
,

1
( k
 s
)
R v
,

1 tan
  m sin
 i m

 k
2 m

1
 k
2 m

1 k
2 m 

 2

1



 cos(


  i
) cos
 i m

( k
2 m

1
 k
2 m




1

R v
,

1 k
2 m

1
 
R v
,

1
2 k m


R v
,

1
( k
 i
)
R v
,

1
( k i

)
(A.8) where
 i m
is the incident angle impinging on the m -th rough interface and
 s m
is the double-
216 bounce scattering angle from the m -th rough interface.
 i m
can be derived from

0 i
based on
217
218
219 the Fresnel law. R h
,

1
and R v
,

1
are the generalized reflection coefficients for H and V polarizations at the m -th rough interface. The parameters with superscript

refer to the dummy variable


.