2 Theoretical formulas
2.1 Generation of the Gaussian rough surface
In this paper, a randomly Gaussian rough surface is employed to represent the realistic surface
(see Fig. 1) and is generated through the Monte Carlo method. It is assumed that the Gaussian
rough surfaces are discretized into N segments with spacing x over a simulated length of
L  Nx . The altitude of every point x n   L / 2  (n  0.5)x (n  1, , N ) along the rough
surface can be defined as [4]
f  ( xn ) 
1 N /2 ~
f  ( K m ) exp( iK m x n )
L m   N / 21

(1)
where, i   1 . For m  0 ,
[ N (0,1)  iN (0,1)] / 2
~

f  ( K m )  2LW  ( K m ) 

 N (0,1)
m  0, N / 2
m  0, N / 2
(2)
Fig. 1 Geometry of EM scattering from two-layered Gaussian rough surfaces at large incident angles
~
~
For m  0 , f  ( K m )  f * ( K  m ) , where the asterisk represents complex conjugate. N (0,1)
appears at each time, which indicates an independent sample taken from a zero mean, unit
variance Gaussian distribution. It is necessary to note that two-layered Gaussian rough surfaces
are independent with each other. Therefore, the sample N (0,1) for the upper rough surface is not
equal to that of the lower one. K m  2m / L denotes the spatial wave-number. W  ( K m )
indicates the Gaussian spectrum function [4]
W ( K m ) 
 2 l 
2 
exp( 
K m2 l 2
)
4
(3)
where,   and l  are the rms height and the correlation length of the two Gaussian rough
surfaces, respectively. Usually, (1) can be solved by the FFT.
2.2 Scattering Formulas for calculating EM scattering from twolayered rough surfaces
Assuming an incident wave  inc (r) impinging upon a geometry model as shown in Fig. 1,
1
where the height profile function of the two Gaussian rough surfaces are z  f  (x) and
z  f  (x) with parameters   , l  , respectively. The height profile functions satisfy
 f  ( x)  0 and  f  ( x)   d , in which d  0 is the average height between the two
rough surfaces.  i and  s represent the incident angle and the scattered angle, respectively.
The spaces are separated by the two rough surfaces into three media: the upper one  0 ( 0 ,  0 ) ,
supposed the free space, i.e. with relative permittivity  r 0  1 and relative permeability  r 0  1 ,
the intermediate one 1 (1 , 1 ) , filling the layer, and the lower one  2 ( 2 ,  2 ) .  s indicates
the scattered field in the space  0 . In this paper, to simplify the computational condition, the
relative permeability of these three media are regarded as  2  1  0 . The time dependence is
e  it and the position vector is r  xxˆ  zzˆ . Let us define the total fields in three media
{ j } j  0,1, 2 as  j , and their normal derivatives as  j / n , in which n represent the unit
normal component on the upper and the lower rough surface, respectively. Then the integral
equations of calculating EM scattering from 1-D two-layered dielectric rough surfaces in three
media are given by [19]
G (r, r )
 0 (r )
1
 0 (r)   inc (r)  [ 0 (r ) 0
 G0 (r, r )
]dS 
2

n

n


S

r  0
(4a)

G (r, r )
 1 (r )
1
 1 (r )   [ 1 (r ) 1
 G1 (r, r )
]dS 
2

n

n


S


G (r, r )
 1 (r )
 [ 1 (r ) 1
 G1 (r, r )
]dS 

n

n


S

(4b)
r  1

G (r, r )
 2 (r )
1
 2 (r)   [ 2 (r ) 2
 G2 (r, r )
]dS 
2
n
n
S

r 2
(4c)

where, {G j (r, r ) 
i (1)
H 0 (k j r  r  )} j 0,1, 2 is the Green’s function and k j is the wave4
number in the space  j , H 0(1) () is the zeroth-order Hankel function of the first kind, S 
denote the boundary of two rough surfaces.
When the point r is put on the boundary of two rough surfaces S  , the total fields  j and
2
their normal derivatives  j / n satisfy the following boundary conditions [19]
 0 (r) 1  1 (r)

n
10 n
 0 (r)   1 (r),
 1 (r) 1  2 (r)

n
21 n
 1 (r)   2 (r),
r  S
(5a)
r  S
(5b)
here, for HH polarization, 10   21  1 , for VV polarization, 10   1 /  0 ,  21   2 /  1 .
In this study, each rough surface is divided into N segments according to their location in x
axis. Substituting the boundary conditions (5a) and (5b) into the integral equations (4a)-(4c), and
applying the pulse basis functions and the point matching [18, 27] to the integral equations, one
can obtain the matrix equation for solving this problem as
A

C
G

 0
B
0
 10 D
E
 10 H
I
0
K

 ψ inc 
 V1 



 
F 
 0 
V2 

 0 
V3 
J 



 
  21 L  4 N 4 N  V4  4 N 1  0  4 N 1
0
where, V1 ( x)   0 (r ) (r  S  ) , V2 ( x) 
V4 ( x) 
(6)
 0 (r)
(r  S  ) , V3 ( x)   1 (r ) (r  S  ) ,
n
 1 (r)
(r  S ) . The elements of the impedance matrixes in (6) are provided in the
n
Appendix of [19].
In this paper, to avoid the artificial edge diffraction resulting from the finite length of the
simulated rough surface, the incident wave cannot be simply chosen as a plane wave, and is
expressed by a tapered wave in which the energy is distributed in a narrow beam about the mean
incident angle. Here, the tapered wave which satisfies the requirement and the Maxwell’s equation
in an appropriate sense is regarded as the incident wave. Consider an incident tapered wave
illuminating the geometry model shown in Fig. 1, which can be exhibited as [4]
 ( x  z tan  i ) 2
ψ inc (r )  yˆ  inc (r )  yˆ expik 0 ( x sin  i  z cos  i )[1  w(r )]exp 
g2





(7)
where, g is the tapering parameter which controls the tapering length of the incident wave. The
additional factor in phase is w(r)  [2( x  z tan  i ) 2 / g 2  1] /(k 0 g cos  i ) 2 .
3
After solving the matrix equation (6), one can obtain the unknowns V1 , V2 , V3 and V4 .
When the point r is located in the far field, and in the observational direction with
k s  k 0 (xˆ sin  s  zˆ cos  s ) , the scattered field  s in the space  0 is [27]
 s (r) 
i
2

exp(ik 0 r  i ) sN ( s ,  i )
4 k 0 r
4
(8)
where
 sN ( s ,  i )   [i(nˆ   k s )V1 ( x)  V2 ( x)] exp(ik s  r) 1  [ f  ( x)]2 dx
(9)
S
and, r  xˆ x  zˆ f  ( x) is on the upper rough surface. The BSC is defined as [27]
 sN ( s ,  i )
 ( s ) 
8k 0 g

2
cos  i (1 
2
1  2 tan 2  i
2k 02 g 2 cos 2  i
(10)
)
2.3 Additional theorem
From (4a)-(4c), it is observed that the integral kernel functions depend on the zeroth-order Hankel
function of the first kind H 0(1) () , and its normal derivatives H 0(1) () / n . Therefore, the
impedance matrixes also depend on H 0(1) () and H 0(1) () / n . Here, the additional theorem is
presented for the Hankel function to accelerate the matrix-vector product and reduce the memory
requirements of MoM. Firstly, the N segments of every rough surface are parted into G
groups based on their locations in x axis, each of which contains M  N / G segments. Then
H 0(1) () and H 0(1) () / n are expressed by [28]
H 0(1) (krmn )
1

2
2
 dβ
 ) αl 'l ( )β ln ( )
t
ml ' (
(11)
0
and

1
H 0(1) (krmn ) 
n n
2
1

2
2
 dβ
0
 ) αl 'l ( )
t
ml ' (

β ln ( )
n n
2
 d
 ) αl 'l ( )[ik (n xn cos   n zn sin  )]β ln ( )
t
β ml
'(
0
where
4
(12)
[ αl 'l ( )] 
P
H
p  P
     / 2)]
(1)
p ( krl 'l ) exp[ ip ( l 'l
[ β ml ' ( )]  exp[ ikrml ' cos(   ml ' )] ,
[ β ln ( )] exp[ ikrln cos(  ln )]
(13a)
(13b)
here,  denotes the included angle between a vector from point  to point  and x axis,
r  r  r
indicates the distance between point  and point  . The values of P depend
on parameters G and M , which have been discussed in [29]. In this paper, the number of
unknowns for each rough surface is N  8192 . Hence, M  64 , G  128 , P  16 .
It is significant to point out that the matrixes A, B , C , D depend only on the upper rough
surface, the matrixes I , J, K, L are only concerned with the lower rough surface, and the
matrixes E , F , G, H represent the interactions between two rough surfaces. Another point is
that the interaction between any two points of rough surfaces is partitioned into near (strong) and
far (weak) interactions in the FMM algorithm. Then the matrixes A, B , C , D , I , J, K, L and
E , F , G, H can be generated by MoM for the strong part and through the three-steps
(aggregation, translation, and disaggregation) for the weak part. In this study, the source points
and the field points in the three-steps of the matrixes A, B , C , D are all from the upper rough
surface, and those of the matrixes I , J, K, L are related to the lower rough surface. The source
points in the aggregation process of the matrixes E , F are from the lower surface, and the filed
points in the disaggregation process of them depend on the upper rough surface. Similarly, the
matrixes G , H represent the interactions between the field points on the lower rough surface
and the source points on the upper rough surface.
3 MPI-based parallel algorithm
To compute EM scattering from two-layered rough surfaces at large incident angles, the simulated
length of rough surfaces should be sufficiently long, which involves a large number of unknowns
and will be more time-consuming for a single-processor. Hence, it is necessary to introduce a new
technique to improve the computation efficiency. The parallel FMM algorithm is presented to
5
solve this scattering problem based on MPI of PC cluster. A typical parallel computing PC cluster
system (see Fig. 2) is composed of several PCs and is interconnected with a high speed switch.
The MPI is a language-independent communication protocol used to program parallel computers,
in which both point-to-point and collective communication are supported. The main configuration
of PC cluster in this research is as follows:
1) System composing: 12 PCs.
2) For each PC CPU: Intel Pentium 4, 2.4GHz; Memory: 1GB; Main-board: ASUS P5KSE.
3) Switch: TP-Link TL-R402M 1000M.
4) Operation System: Microsoft Windows XP 32bit.
5) Programming environment: MPICH 2.103, Compaq Visual Fortran 6.5.
Fig. 2 A typical parallel computing PC cluster system
In this paper, the BICGM [24] is applied to solve the unsymmetrical matrix equation (6) and is
parallelized according to the property of MPI. In performing the parallel algorithm, the twelve
impedance matrixes in (6) are assigned to the twelve processors involved in the parallel computing
system (see Fig. 3), where matrix A is allocated to processor 0 (the master processor),
sequentially, matrix L is distributed to processor 11.
Fig. 3 Assignment of the twelve impedance matrixes to corresponding processor
For a general nonsingular complex matrix equation
Ax  b
(14)
The BICGM is applied to solve the matrix equation, and its procedure is illustrated in the
following. Procedure [24]: BICGM (x, b, A,  , Imax)
1) Guess x .
2) Iter  0 , r  b  Ax , r1  r ,  0  0 , p  p1  0 .
3) Iterate for || r ||  || b || and Iter  Imax
(a) Iter  Iter  1 ; (b) Iter  r1T r ,    Iter /  Iter-1 ; (c) p  r   p , p1  r1   p1 ;
6
(d) v  Ap ; (e)   Iter / p1T v ; (f) x  x  p ; (g) r  r  v ; r1  r1   A T p .
Where ||  || stands for the norm number, and || x ||
N
| x
i
| 2 .  is the residual, and here
i 1
  1.0  10 3 . In the BICGM, the product Ax of the matrix A and a vector x , the product
A T x of the matrix A T and a vector x must be computed, which is a critical step to realize
the two products on the parallel system, where A T is the transposed matrix of A .
In order to parallelize the BICGM, the vector x is divided into four parts by lines and stored
in the master processor (processor 0), i.e. x  [x1 x 2 x 3 x 4 ]T . And in this paper the impedance
matrix A is segmented to the twelve matrixes (see Fig. 3).
The steps of dealing with the product of matrix and vector are presented as follows:
Step 1: The vector x1 is sent to processor 2 and 6; the vector x 2 is assigned to processor 1, 3
and 7; the vector x 3 is distributed to processor 4, 8 and 10; the vector x 4 is transmitted to
processor 5, 9 and 11.
Step 2: Compute the product of matrix and vector in each processor, and then send the products
to processor 0.
Step 3: The final product is
b  [b1 b 2 b3 b 4 ]T , in which
b1  Ax1  B x 2 ;
b 2  C x1  10 Dx 2  E x 3  F x 4 ; b3  Gx1  10 Hx 2  Ix 3  Jx 4 ; b 4  Kx 3   21 L x 4 .
Similarly, the steps for the product of the transposed matrix and the vector are given by:
Step 1: The vector x1 is sent to processor 1; the vector x 2 is assigned to processor 2-5; the
vector x 3 is distributed to processor 6-9; the vector x 4 is transmitted to processor 10 and 11.
Step 2: Compute the product of the transposed matrixes and the vector in each processor, and
then send the products to processor 0.
Step 3: The final product is b  [b1 b 2 b3 b 4 ]T , in which b1  A T x1  C T x 2  G T x 3 ,
b 2  B T x1  10 D T x 2  10 H T x 3 ,
b3  E T x 2  I T x 3  K T x 4 ,
 21 LT x 4 .
7
b4  F T x 2  J T x3 
In realizing the parallel FMM algorithm, the simulation data of rough surfaces are distributed to
every processor [30]. Performance of the proposed parallel algorithm is illustrated in Fig. 4 by
comparing the BSC, the simulation time and the parallel speedup ratio of different processors for
one surface realization, and VV polarized incident wave is considered. The simulated length of
two-layered Gaussian rough surfaces are all L  819.2 , 10 sampling points per wavelength, and
the incident angle is  i  85  . The characteristic parameters of the two Gaussian rough surfaces
are    0.2 , l    (the upper rough surface) and    0.3 , l   1.5 (the lower rough
surface), respectively. Let t p denotes the time that one simulation takes by p processors,
where p  1, 2, 4, 6, 12 , the parallel speedup ratio S p is defined as
Sp 
t1
tp
(15)
It should be noted that the method on how to allocate the relevant matrix and vectors into which
node for cases of 2, 4, 6 nodes does not present any new ideas, which is just similar as that of 12
nodes. Firstly, the twelve sub-matrixes ( A, B , C , D , E , F , G, H and I , J, K, L ) and vectors
have been allocated into the target node. Then the steps of dealing with the product of the matrix
or the transposed matrix and the vector can be obtained according to the regulation of the
matrix-blocking computation theory.
Fig. 4 Comparison of the BSC, the simulation time and the parallel speedup ratio of different
processors for one surface realization
Fig. 4 (a) presents the comparison of the serial FMM calculations with the 12-processor parallel
FMM calculations for the angular distribution of the BSC of one surface realization. It is observed
that the two scattering patterns show good coincidence with each other over the whole scattering
angular range. The simulation time and the parallel speedup ratio of different processors are given
in Fig. 4 (b). It is seen that the time consumed is reduced significantly with increasing the number
of processors. From Fig. 4 (b), one can also find that the parallel speedup ratio is almost in direct
proportion to the numbers of processors involved in the computing parallel system, and increases
linearly with processors growing in number. Another point worth noting is that the pattern of the
8
parallel speedup ration of the parallel FMM algorithm departs from that of the ideal model due to
the more communication time with the number of processors increasing.
4 Numerical simulations and discussion
Firstly, to ensure the validity of the proposed algorithm at large incident angles in this paper, the
angular distribution of bistatic scattering from two-layered Gaussian rough surfaces are calculated
in Fig. 5 through the parallel FMM and the FDTD, respectively. Both HH and VV polarizations
are considered. The simulated length of two rough surfaces are L  409.6 , N  8192 (Fig. 5
(a)) for HH polarization and L  819.2 , N  16384 (Fig. 5 (b)) for VV case. Parameters of
the incident wave are   0.3m with g  L / 4 , i  80 . The relative permittivity of the
intermediate and the lower medium are 1  (2.5, 0.18) and  2  (9.8, 1.2) , which correspond
to the soil with 3.8% moisture and 25% moisture, respectively [31]. The rms height and the
correlation length of two-layered Gaussian rough surfaces are given with k   0.6 , kl  3.0 ,
k   1.0 and kl  4.5 . The average height between two rough surfaces is d  3 . From Fig.
5, it is readily found that the angular distribution of the BSC by the parallel FMM is in good
agreement with that of FDTD technique at the most scattering angle for both HH and VV
polarizations, which demonstrates the effectiveness and the accuracy of the parallel FMM
presented in this paper.
Fig. 5 Comparisons of the angular distribution of the BSC by the parallel FMM and FDTD
To further explore the important scattering characteristics of two-layered dielectric rough
surfaces, in the next simulations, the behavior of the scattering model illustrated in Fig. 1 is taken
into account. This is done by dividing both the upper and the lower Gaussian rough surfaces into
N  8192 segments with interval x   / 10 , which represents the simulated length of the
Gaussian rough surfaces L  819 .2 . The wavelength of the tapered wave is   0.3m with
g  L / 4 . The incident angle is  i  80  for HH polarization, and  i  85  for VV case except
for Fig. 7 and Fig. 11. The relative permittivities of the intermediate and the lower medium are set
to be  1  (2.5,0.18) ,  2  (9.8,1.2) except for Fig. 10. The average height d between two-
9
layered rough surfaces is 5 for the plots in Fig. 6-Fig. 10.
The performance of bistatic scattering from two-layered Gaussian rough surfaces with different
Monte Carlo surface realizations is depicted in Fig. 6. All parameters of two rough surfaces are the
same as those in Fig. 4, and VV polarization is taken for example. From Fig. 6, it is obviously
found that 50 Monte Carlo realizations are enough for the numerical simulation to get stable and
smooth curve. Therefore in all the following numerical simulations, the BSC is computed by
averaging the results of 50 realizations.
Fig. 6 The BSC from two-layered rough surfaces
Fig. 7 The BSC from two-layered rough surfaces
with different Monte Carlo realizations
with different incident angles
In order to compare the different scattering properties between small angles and large angles of
incidence, the BSC of these two incident cases (HH, VV) are calculated by the parallel FMM and
plotted in Fig. 7. In performing the calculation, the characteristic parameters of the two Gaussian
rough surfaces are    0.2 , l    ,    0.5 , l   1.5 and    0.5 , l    ,
   0.8 , l   1.5 corresponding to HH and VV polarization, respectively. It is shown that
the angular distribution of bistatic scattering from two-layered rough surfaces mainly depends on
the incident angle and the polarization. From Fig. 7, it is also found that the specular peak of the
coherent wave is more evident under large angles of incidence no matter what the polarization of
the incident wave is. In the case of the small incidence is concerned, backscattering enhancement
is more evident for the case of  i  45  at VV polarization, which is caused by the large rms
height of the layered rough surface, and this implies a decrease of the scattered energy in the
specular direction and an increase of the incoherent scattering.
In Fig. 8 and Fig. 9, the influences of the characteristics parameters of Gaussian rough surfaces,
which include the rms height  and the correlation length l , on the BSC of two-layered
Gaussian rough surfaces at large angles of incidence are calculated and illustrated. As is apparent
in Fig. 8, the BSC for larger rms height is larger in the non-specular direction, which demonstrates
incoherent scattering depends strongly on the surface roughness. The primary reason for this is
due to the fact that the roughness of the Gaussian rough surface is determined by the rms height.
10
The larger rms height, the rougher the Gaussian rough surface, leading to more incoherent
scattering contribution to bistatic scattering in non-specular directions. Consequently, incoherent
scattering (except for the specular direction) is enhanced with surface roughness increasing. As
seen previously in Fig.7, backscattering enhancement is also observed for larger rms height in Fig.
8(b).
Fig. 8 The BSC from two-layered Gaussian rough surfaces with different rms heights
Fig. 9 The BSC from two-layered Gaussian rough surfaces with different correlation lengths
Fig. 9 presents the dependence of the BSC on the correlation length of two-layered surfaces. It
is readily seen that the BSC near the specular direction with different correlation lengths is nearly
identical for both HH and VV polarizations. It is also found that the BSC increases by keeping the
rms height constant and by decreasing the correlation length, the electromagnetic roughness is
constant, but the rms slope increases, leading to a higher angular spreading of the scattered energy,
and the BSC increases with the correlation length decreasing far from the specular direction, in
particular for VV case.
Figure 10 Comparisons of the BSC for different relative permittivities of the intermediate and the lower medium
The properties of bistatic scattering from two-layered rough surfaces with different relative
permittivities of the intermediate or the lower medium are exhibited in Fig. 10, where the
parameters of the two Gaussian rough surfaces are    0.2 , l    ,    0.5 , l   1.5 .
The relative permittivity of the lower medium is  2  (9.8,1.2) in Fig. 10(a). From Fig. 10(a), it
can be seen that the BSC of the lossless medium is larger than that of the lossy case, and the BSC
decreases as the imaginary part of  1 increases far from the specular direction. We attribute this
result to the fact that the imaginary part of a medium affects its absorption, with increasing the
imaginary part, the absorption becomes stronger, the reflectivity weakened, which leads to the
decreasing of the BSC. In Fig. 10(b), influences of  2 on the BSC are also shown, in which
1  2.5 . When both of two media are lossless, it is clear that the BSC is the smallest with
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 2   1 . It is also seen that with an increase of  2 , the BSC of the two-layered model becomes
larger in the non-specular direction as depicted in Fig. 10(b). In addition, one can also observed
that the BSC decreases rapidly when the imaginary part of  2 increases.
Fig. 11 Comparisons of the BSC for different average heights between two rough surfaces
The effect of the average height d between the two Gaussian rough surfaces on the BSC of
the scattering model of Fig. 1 is also examined in Fig. 11, in which the parameters of the two
Gaussian rough surfaces are    0.2 , l    ,    0.5 , l   1.5 . The incident angle is
i  85 under both HH and VV polarizations. The relative permittivity of the intermediate and
the lower medium are set to be  1  (2.5,0.18) and  2  (9.8,1.2) , respectively. One can notice
that the angular distribution of the BSC almost does not change at different heights in the most
scattering region for both HH and VV polarizations, in particular for large thickness d , which is
consistent with the result in [32]. That is to say, bistatic scattering is not “sensitive” to the
variation of the average height d .
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