## 1. Introduction

[2] Using a unified full wave approach, the scatter cross sections for rough surfaces are expressed as “weighted” sums of cross sections for the larger- and smaller-scale surfaces [*Bahar*, 1981a]. The contributions of the larger-scale surfaces (for which the radii of the curvature are larger than the electromagnetic wavelength) are the physical optics [*Beckmann and Spizzichino*, 1963] cross sections reduced by a factor equal to the characteristic function squared of the smaller-scale surface. The contributions of the smaller-scale surfaces are tilt modulated by the slopes of the larger-scale surfaces only [*Valenzula*, 1968; *Bahar et al.*, 1983a, 1983b; *Bahar and Kubik*, 1993]. The specific decomposition of the total surface height spectral density function is considered in detail in section 4. Associated with this decomposition is a variational parameter equal to the ratio of the mean square heights of the larger-scale surface and the total rough surface (*n* = 〈*h*_{l}^{2}〉/〈*h*^{2}〉). The stationarity of the solutions for the total scatter cross sections is the basis for the variational method described in this paper. For the illustrations, both Gaussian [*Brown*, 1978] spectral density functions and Pierson-Moskowitz [*Barrick*, 1974] spectral density functions are considered in detail. The decomposition is conducted in a smooth continuous manner in order to avoid artificial rapid fluctuations in the corresponding surface height autocorrelation functions [*Bahar and Kubik*, 1993]. These artificial rapid fluctuations occur when the decomposition is performed by selecting a specific spatial wave number *k*_{d} that separates the spectral density of the larger- and smaller-scale surfaces. This results in discontinuities of the individual spectral density functions. Since the surface height autocorrelation functions of the individual surfaces are the Fourier transforms of the corresponding surface height spectral density functions, it is these discontinuities that result in the artificial rapid fluctuations in the corresponding autocorrelation functions.

[3] Of special interest, in performing the decomposition, is the necessity to have finite means square slopes for the larger-scale surfaces. For the Pierson-Moskowitz spectral density function this has usually been achieved by limiting the spectrum of the larger-scale surface on selecting an artificial cutoff spectral wave number *k*_{c} [*Brown*, 1978] as the upper limit of the Pierson-Moskowitz spectrum. This results in another discontinuity for the assumed spectral density function. More recently, empirical methods have been adopted to modify the Pierson-Moskowitz spectral density function for large values of the spatial wave number *k*, such that the corresponding mean square slope remains finite. The method presented in this paper to decompose the total spectral density function does not require the selection of a cutoff spatial wave number *k*_{c}, or the adoption of any other empirical method to overcome this problem. Since we consider two-dimensional multiscale rough surfaces in this paper, both like and cross-polarized cross sections are considered.

[4] Preliminary numerical simulations of the variation technique for evaluating the scatter cross sections of multiple-scale rough surfaces have been conducted for perfectly conducting media below the interface [*Bahar and Crittenden*, 2008]. Furthermore, surfaces that are rough only in one dimension *y* = *h*(*x*) were considered. The plane of incidence was assumed to be perpendicular to the *x* axis; thus, the cross-polarized cross sections were zero. In this paper the media below the two-dimensionally rough interfaces are characterized by their complex permittivity and permeability. The tangential components of the electric and magnetic fields are continuous at the rough interface where exact boundary conditions are imposed. The scatter cross sections are shown to be stationary over a broad range of the variational parameter *n*. It has also been shown that since the horizontally polarized cross sections are much more sensitive to tilt modulations, the ratio of the vertically to the horizontally polarized backscatter cross section are near unity for near grazing angles of incidence [*Collin*, 1992, 2008]. This has been as observed in field measurements [*Cloude and Coor*, 2002].

[5] The paper is organized as follows: In section 2, a survey of the related publications is presented. An overview of the full wave method is given in section 3. In section 4, a description of the variational technique, including formulation of the problem, the analysis, and the principal results are presented. The implementation of the variational technique and the criteria for determining the stationarity of the results are discussed in section 5. References to numerical simulations are presented in section 6. In section 7, future work on numerical simulations and practical applications to remote sensing of rough sea surfaces (to determine wind speed) and rough land surfaces (to determine soil moisture content for example) are briefly discussed. Concluding remarks are given in section 8.