Some Extensions to the Integral Equation
Method for Electromagnetic Scattering from
Rough Surfaces
Yang Du
zjuydu03@zju.edu.cn
Zhejiang University, Hangzhou, China
2011.7.25
Outline



The analytical models.
conventional and the unifying models
Recent advances
Statistical IEM (SIEM)
Extended AIEM (E-AIEM)
Conclusions
The Conventional Analytical Models


Small Perturbation
Model (SPM)
Kirchhoff Approximation
(KA)
The Unifying Models

There have been interests to develop a unifying model that can
bridge KA and SPM, for both theoretical compactness and
practical considerations.

A number of unifying models in the literature includes the phase
perturbation method (PPM), the small slope approximation (SSA),
the operator expansion method (OEM), the tilt invariant
approximation (TIA), the local weight approximation (LWA), the
Wiener-Hermite approach, the unified perturbation expansion
(UPE), the full wave approach (FWA), the improved Green’s
function methods, the volumetric methods, and the integral
equation method (IEM).
The IEM and AIEM Models


The IEM model, developed by A.K. Fung, Z.Q. Li, and K.S.
Chen in 1992 [1], has attracted enormous attention with its
accuracy for backscattering coefficients over large region of
validity, and has become one of the most widely used models.
It has also been recognized that IEM can be further improved.
For instance, by improving the spectral representation of the
surface Green’s function and its gradient, K.S. Chen et. al
obtain the advanced IEM (AIEM) model [2].
Ref:
[1] A. K. Fung, Z. Q. Li, and K. S. Chen, “Backscattering from a randomly
rough dielectric surface,” IEEE Trans. Geosci. Remote Sensing, vol.
GE-30, no. 2, pp. 356–369, Mar. 1992.
[2] K. S. Chen, T. D. Wu, L. Tsang, Q. Li, J. C. Shi, and A. K. Fung,
“Emission of rough surfaces calculated by the integral equation method
with comparison to three-dimensional moment method simulations,”
IEEE Trans. Geosci. Remote Sensing, vol. GE-41, no. 1, pp. 90–101,
Jan. 2003.
Assumptions Underlying IEM

According to [2], there are four assumptions underlying IEM:
 Spatial dependence of the local incidence angle of the
Fresnel reflection coefficient is removed, by either replacing
it with the incidence angle or the specular angle.
 For the cross polarization, the reflection coefficient used to
compute the Kirchhoff fields is approximated by  R //  R   2
 Edge diffraction terms are excluded.
 Complementary field coefficients are approximated by
simplifying the surface Green’s function and its gradient in
the phase terms.
Observation One

Statistical features of the surface slopes are rich and
important.
Correlation Coefficients between Slopes at Different
Points
 1

0
2 
C s
  xx

  xy
0
 xx
 xy 
1
 xy
 yy
 xy
1
 yy
0


0 

1 
2

2( x  x  )
 xx ( r , r )  1 
2
L

 xy ( r , r )  

 ( x  x ) 2  ( y  y ) 2 

 exp  
2
L



2( x  x  )( y  y  )
L
2
 ( x  x ) 2  ( y  y ) 2 
exp  

2
L


The Significance of Slope Statistics
If the conventional KA approach is
incorporated with the surface slope
statistics, the resulting model
appears almost immune to the
Brewster angle effect for vertical
polarization.
This feature is expected since the
directions of the unit normal which
lead the local angles of incidence to
approach the Brewster angle occupy
only a small portion of the directional
distribution; contributions from the
rest of the distribution become
appreciable in this new treatment.
No slope accounted
slope accounted
Smooth surface
Data: DeRoo, R.D., and F.T. Ulaby, “Bistatic Specular
Scattering from Rough Dielectric Surfaces,” IEEE
Transactions on Antennas and Propagation, Vol. 42,
No. 2, 1994, pp. 1743–1755.
Rougher surface
Added by KSC
The Statistical IEM Model
Development of the statistical IEM (SIEM) model is motivated by the
observation that slope statistics has appreciable impact on the Kirchhoff
approximation, which forms the Kirchhoff part of the IEM formalism, and
by the intuition that incorporating shadow effect directly into field rather
than scattered power may provide more physical results.
Details of the SIEM model can be found in [3].
[3] Y. Du, J. A. Kong, Z. Y. Wang, W. Z. Yan, and L. Peng, “A statistical integral equation
model for shadow-corrected EM scattering from a Gaussian rough surface,” IEEE Trans.
Antennas and Propagation, vol. 55, no. 6, pp. 1843-1855, June 2007.
Testing Cases of SIEM
SIEM – Simulation I
SIEM Simulation II
Some Concluding Remarks on SIEM




SIEM is in good agreement with MoM
SIEM has the potential to bridge the gap between KA
and SPM
IEM is a special case of SIEM
Refinement of SIEM is in need
Observation Two

There is growing interest to use the spectral
representations of the Green's function and its gradient
in complete forms, as in the advanced integral equation
model (AIEM) and the integral equation model for
second-order multiple scattering (IEM2M).

Yet there are some technical subtleties in connection
with the restoration of the full Green’s function that have
not been adequately reflected in these models.
Observation Two (Cont.)



For example, in evaluating the average scattered
complementary field over height deviation z, a split of the
domain of integration into two semi-infinite ones is
required due to the absolute phase term present in the
spectral representation of the Green's function.
This operation will lead to an expression containing the
error function. Inclusion of the error function related
terms is also encountered when one evaluates the
incoherent powers that involve the scattered
complementary field.
Thus, a complete expression for the cross scattering
coefficient or for the complementary scattering
coefficient should have two parts: one does not contain
the error function and the other includes its effect. The
latter can be regarded as a correction term and an
analysis of its effect is desirable.
Spectral form of Green’s function
v v'
gm (r , r ) = v v'
'
Ñ gm (r , r ) = -
qm =
2
j
2p
òò
1
1
qm
)
)
ˆ m
x u + y v m zq
òò
2p
2
'
'
'
ex p [ ju ( x - x ) + jv ( y - y ) - jq m | z - z | ]d u d v
qm
'
'
'
ex p [ ju ( x - x ) + jv ( y - y ) - jq m | z - z | ]d u d v
2
k m - u - v , m = 1, 2
i
F

F
s
z(x, y )
G


G

t
F , G : Propagator in upper and lower medium, respectively
 , :
Upward, downward
Added by KSC
@ksc 2003
An Illustrative Computation to Show the Inclusion of
the Error Function I
 m ,d 
IF

 ex p  i  k sz z  k iz z   q m d  z  z   
 x  is the Heavyside function,
  d  z  z 
if x  0
 1,
 x  
 0, otherw ise
Transformation of variables leads to the factorization
 m ,d 
IF
.
 m ,d 
 I F1I F 2
where
1
IF1 
 m ,d 
IF2
2 

1 
1
2 
1 




2


  k sz y1 k z y1  
y1
dy1 exp   i 

  exp   2 2 (1   ) 
2 
  2




2

k sz  k z
y2
dy 2 exp  

i
(

2
2

(1


)
2


2qm d ) y2 

 y2d 
An Illustrative Computation to Show the Inclusion of
the Error Function II
IF1
can be readily obtained as
2
 1
2 
I F 1  exp    k sz  k z  (1   ) 
 4

while
 m ,d 
IF2

2
 1
2
 exp    k sz  k z  2 q m d  (1   )   1  erf
2
 4
 
1
where erf is the error function defined as
erf ( z )
2


z
e
0
t
2
dt
  1 

 k sz  k z  2 q m d   
i
2

 
The Extended AIEM Model
Development of the extended AIEM (E-AIEM) model is motivated by the
above observations. It is found that
1. the Kirchhoff term is identical to that of IEM and AIEM,
2. The cross scattering coefficient has two parts: one free of the error
function and the other including its effect
3. The complementary scattering coefficient has two parts: one free of
the error function and the other including its effect
Details of the E-AIEM model can be found in [4].
Ref:
[4] Y. Du, “A new bistatic model for electromagnetic scattering from randomly rough surfaces,”
Waves in Random and Complex Media, vol. 18, no. 1, pp. 109-128, Feb. 2008.
Some Observations on the Cross Scattering
Coefficient



The error function free part is in agreement with the
literature (AIEM, IEM2M, I-IEM).
For the case where both media are lossless, the two
quantities involving the error function are purely
imaginary because their corresponding arguments are
purely imaginary. Moreover, all the fqp and Fqp are real.
These two facts suggest that the argument of the Re
operator is purely imaginary and thus the correction term
vanishes.
For the case where either medium is of lossy nature, the
two statements above are no longer held, nor will the
correction term vanish.
Some Observations on the Complementary Scattering
Coefficient



The error function free part is different from the literature
(AIEM, IEM2M) because the assumptions made here are
fewer and less restrictive than those in the above models.
For the case where both media are lossless, the
correction term does not vanish.
For the case where either medium is of lossy nature, the
correction term does not vanish.
EAIEM Simulation I
Macelloni, G., Nesti, G., Pampaloni, P., Sigismondi, S., Tarchi, D. and Lolli, S.,
2000, Experimental validation of surface scattering and emission models. IEEE
Transactions on Geoscience and Remote Sensing, 38, 459–469.
EAIEM Simulation II
Some Concluding Remarks on EAIEM

This new model can be regarded as an extension to the AIEM and
IEM2M models on two accounts: first it has made fewer and less
restrictive assumptions in evaluating the complementary scattering
coefficient for single scattering, and second it contains a more rigorous
analysis by the inclusion of the error function related terms for the cross
and complementary scattering coefficients. Each of these two distinctive
features bears its implication: the first suggests that our result for
complementary scattering coefficient is more accurate and more
general, even when the effect of the error function related terms is
neglected; the second suggests that for the case where both the media
above and below the rough surface are lossless, it can be shown that
the correction term vanishes for the cross scattering coefficient, but not
for the complementary scattering coefficient; for the case where either
medium is of lossy nature, the correction term due to this lossy medium
will contribute to both the cross and complementary scattering
coefficients. As a result, the proposed model is expected to have wider
applicability with a better accuracy.
Thank You !