Some Extensions to the Integral Equation Method for Electromagnetic Scattering from Rough Surfaces Yang Du Zhejiang University, Hangzhou, China

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 [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

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 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

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.

Smooth surface Rougher surface slope accounted No slope accounted 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. 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 , 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 Added by 2003 Propagator in upper and lower medium, respectively Upward, downward

An Illustrative Computation to Show the Inclusion of the Error Function I is the Heavyside function,. Transformation of variables leads to the factorization where

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

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 , Feb

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.

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