Some Extensions to the Integral Equation Method for Electromagnetic Scattering from Rough Surfaces

1
Some Extensions to the Integral Equation Method
for Electromagnetic Scattering from Rough
Surfaces

• Yang Du
• zjuydu03_at_zju.edu.cn

Zhejiang University, Hangzhou, China
2011.7.25
2
Outline

• The analytical models.
• conventional and the unifying models
• Recent advances
• Statistical IEM (SIEM)
• Extended AIEM (E-AIEM)
• Conclusions

3
The Conventional Analytical Models

• Small Perturbation Model (SPM)
• Kirchhoff Approximation (KA)

4
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 Greens
  function methods, the volumetric methods, and the
  integral equation method (IEM).

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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 Greens
  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. 356369, 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. 90101, Jan. 2003.
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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
• Edge diffraction terms are excluded.
• Complementary field coefficients are approximated
  by simplifying the surface Greens function and
  its gradient in the phase terms.

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Observation One

• Statistical features of the surface slopes are
  rich and important.

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Correlation Coefficients between Slopes at
Different Points
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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.
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No slope accounted
slope accounted
Smooth surface
Rougher 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. 17431755.
Added by KSC
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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.
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Testing Cases of SIEM
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SIEM Simulation I
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SIEM Simulation II
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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

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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
  Greens function that have not been adequately
  reflected in these models.

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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.

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Spectral form of Greens function
Propagator in upper and lower medium, respectively
_at_ksc 2003
Upward, downward
Added by KSC
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An Illustrative Computation to Show the Inclusion
of the Error Function I
is the Heavyside function,
Transformation of variables leads to the
factorization
.
where
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An Illustrative Computation to Show the Inclusion
of the Error Function II
can be readily obtained as
while
where erf is the error function defined as
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The Extended AIEM Model

• Development of the extended AIEM (E-AIEM) model
  is motivated by the above observations. It is
  found that
• the Kirchhoff term is identical to that of IEM
  and AIEM,
• 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.
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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.

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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.

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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, 459469.
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EAIEM Simulation II
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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.

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• Thank You !