Some extensions to the integral equation method for electromagnetic scattering from rough surfaces

The traditional models for wave scattering from random rough surfaces are the small perturbation method (SPM) and the Kirchhoff model (KM), which are applicable for slightly rough surfaces and surfaces with small surface curvatures, respectively [1]-[3]. In order to bridge the gap between SPM and KM, several so-called unifying methods have been developed, including the small slope approximation (SSA) [4], the phase perturbation technique (PPT) [5], the operator expansion method (OEM) [6], the unified perturbation method (UPM) [7], [8], the full wave approach (FWA)[9], [10], and the integral equation method (IEM)[11]. The ability to provide good predictions for forward and backward scattering coefficients has made IEM one of the most widely used analytical models. Yet to make the derivation of this model mathematically tractable, several assumptions were made, including [12]: 1) the removal of the spatial dependence of the local angle of incidence of the Fresnel reflection coefficient by either replacing it with the angle of incidence or the specular angle; 2) For the cross polarization, the reflection coefficient used to compute the Kirchhoff fields is approximated by 0.5(R_|| - R_⊥); 3) Edge diffraction terms are excluded; and 4) Complementary field coefficients are approximated by simplifying the surface Green´s function and its gradient in the phase terms. Concerns over the assumptions have prompted several modifications and variations of IEM in the literature. This paper represents a brief recapitulation of our work on Assumptions (1) and (4), respectively. Assumption (1) is an oversimplification of the statistical behavior of the unit normal n directed out of the surface at an arbitrary point on the surface. Specifically, n is effectively restricted to a deterministic quantity, with the direction depending on the angles of incidence and scattering, while giving up all its statistical features, which stem from the fact that the unit - - normal follows a statistical distribution, and moreover, unit normals corresponding to different points on the surface are correlated. It is such correlation that presents technical challenge if a sound compromise between model fidelity and complexity is to be made. In this regard, we proposed a statistical model for electromagnetic scattering from a Gaussian rough surface in conjunction with the IEM formulism [13], where the statistical features of the surface slopes and the effect of shadowing are included. In evaluating the Kirchhoff incoherent power, for the correlated term due to correlation between the normal vectors of two neighboring points on the surface, an approximation scheme based on the decomposition of the covariance matrix is proposed. For the cross and complementary incoherent powers, due to their subdominant nature, the cross correlations between the surface slopes at different points on the surface are neglected to reduce the computational complexity. In dealing with Assumption (4), regarding the spectral representation of the Green´s function, the simplification was discarded and full form was restored, resulting in a modification to the complementary components. The resulting model is the so called improved IEM model (I-IEM)[14]. Additional restoration of the spectral representation of the gradient of the Green´s function in its full form leads to the advanced IEM model (AIEM)[12] and the IEM2M model[15]. However, 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. 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. We demonstrated that this operation will lead to an expression containing the error function. Inclusion of the e