The Small Perturbation Method (SPM) is a low frequency approximation to the electromagnetic scattering from rough surfaces. The theory involves a small height expansion in conjunction with a perturbation series expansion of the unknown scattering coefficients. Recently, an arbitrary order, iterative solution procedure has been derived for SPM: kernels at any order are expressed as a summation over lower order kernels in an iterative fashion. Such a form is very useful, because it allows evaluation of the field statistical moments in a direct manner, when considering stochastic surfaces. In this dissertation, this procedure is extended to the two layer (two rough surfaces on top of each other) problem and the complete solution is given.

Utilizing this formulation, the second and fourth order bi-static scattering coefficients for two rough surfaces characterized by two uncorrelated Gaussian Random Processes (GRP) are obtained. The effects of upper and lower roughnesses and the interaction effect in the total fourth order cross section can be identified in the theory. Studies on the ratio of the interaction effect to the total cross section are presented for example cases, investigating the relative importance of interactions among surfaces. Results show the interaction term contributes most to the cross-pol cross sections when surfaces are close to each other at near grazing incidence.

In addition, the previously developed arbitrary order SPM solution for the single layer problem is utilized to derive the fourth order term in the small slope approximation (SSA) of thermal emission from the sea surface. It is shown that this term has the form of a four-fold integration over a product of two sea spectra for a Gaussian random process sea, thereby describing emission “interaction” effects among pairs of sea waves. Interaction effects between “long” and “short” waves are considered, both through numerical and approximate evaluations of the fourth order theory. The approximation developed is a theoretical alternative to the “two-scale” model, and enables comparisons of short wave “tilting” effects between the two models in terms of spectrum independent “weighting” functions. The weighting functions obtained are found to be similar, but not identical.