4.2. Principal Advantages of the Full Wave Variational Technique
 The major advantages of the novel unified full wave variation techniques over the standard hybrid two-scale solutions are summarized below.
 1. The decompositions of the surface height spectral density functions are performed in a smooth continuous manner (rather than a discontinuous manner [Brown, 1978]) such that there are no non physical rapid fluctuations in the corresponding surface height autocorrelation functions [Bahar, 1969; Bahar and Kubik, 1993].
 2. The slopes of the larger-scale surfaces which modulate the scatter cross sections of the smaller-scale surfaces are finite, without introducing an artificial cutoff spatial wave number to the Pearson Moskowitz spectral density function [Brown, 1978].
 3. The total scatter cross sections are expressed as weighted sums of the scatter cross sections for the larger- and smaller-scale rough surfaces [Bahar, 1981a]. They are not summed up in an ad hoc manner [Brown, 1978]. This weighted sum is consistent with energy conservation. It also makes the total scatter cross sections stationary over a wide range of the variational parameter (the ratio of the mean square heights of the larger-scale surface and the total surface) [Bahar and Lee, 1994; Bahar and Crittenden, 2008].
 4. The solutions for the scatter cross sections for the smaller-scale surfaces are not limited to surfaces with small mean square heights compared with the electromagnetic wavelength, as is the case when small-perturbation solutions are used [Rice, 1951].
 5. The scatter cross sections are polarization dependent [Beckmann and Spizzichino, 1963; Bahar, 1981b; Bahar and Lee, 1994].
 6. These full wave solutions explain why the cross sections for the horizontally polarized waves are much more sensitive to tilt modulation than the cross sections for vertically polarized waves [Bahar and Kubik, 1993; Bahar et al., 1983a, 1983b; Bahar and Crittenden, 2008].
 7. These solutions explain why the ratios of the horizontally polarized to vertically polarized cross sections can approach unity even at near grazing angles, contrary to prediction based on the small-perturbation approach [Collin, 1992; Cloude and Coor, 2002].
 8. Numerical simulations based on these Full Wave analytical solutions can be conducted much more rapidly than those based on averaging over many Monte Carlo simulations of rough surfaces.
4.3. Full Wave Analysis
 For plane wave, linearly polarized excitations the second-order linearly polarized scattered fields are expressed in matrix form as [Bahar and Lee, 1994]
where GSf and Gi are 2 × l matrices, whose elements are the vertically, V, and horizontally, H, polarized components of the scattered and incident fields ESPf and EPi(P = V, H), respectively. The integration is over the surface variables xs and zs as well as the wave vector variables k′0y, k′0z for the radiation fields. It is assumed here that the mean plane of the rough surface is y = 〈h(xs, zs)〉 = 0.
 The 2 × 2 surface element scattering matrix Su is given by [Bahar and Lee, 1994]
The elements of the matrix Ru in (3) are
The wave vectors in the scatter and incident directions are
In (9), θ0 is the elevation angle (measured from y axis) and ϕ is the azimuth angle (measured from x axis toward the z axis).
In (4)–(7), ɛr = ɛ1/ɛ0, μr = μ1/μ0, nr = (ɛrμr)1/2 and ηr = (μr/ɛr)1/2 are the relative permittivity, permeability, refractive index and intrinsic impedance of the medium below the rough interface (1). The sines and cosines of the angles θ0 and θ1 above and below the rough interface (denoted by S0, S1 and C0, C1) are related through Snell's law.
 The vectors ′ and ′ are given in terms of the incident 0i and scatter 0i wave vectors. The subscripts 0, 1 are for the medium 0(y > h (xs, zs)) and medium 1(y < h (xs, zs)), respectively:
and s and are position vectors from the origin to points on the rough surfaces and to the observation point, respectively:
On integrating (2) by parts, it is expressed as [Bahar and Lee, 1994]
In (15) it is assumed that h(xs, zs) vanishes for ∣xs∣ > L and ∣zs∣ > l, (1), and
Since ′ · , it follows that
The expression for the surface element scattering matrix S(′, ′) is given by (4)–(7) with exception that Cu and Du are replaced by cos(ϕ′ − ϕi) and sin(ϕ′ − ϕi), respectively.
 The solutions presented in (15) are referred to by Collin  as the original full wave solutions. Collin  uses a different full wave approach to the problem of scattering of plane waves from perfectly conducting surfaces based on an inversion of an integral equation. Collin's results are also shown to be in complete agreement with the original full wave results for the perfectly conducing case (∣ɛr∣) → ∞, μr = 1).
 The second term GDf in (15) can be integrated with respect to xs and zs. For L → ∞ and l → ∞, the integrations yield the Dirac delta functions δ(v′x)δ(v′z). Thus, this term GDf reduces to the specularly reflected plane wave. The full wave solution Gf(15) includes the diffusely scattered field GSf, as well as the specularly reflected field GDf.
 When the observation point is at a very large distance from the rough surface (k0r ≫ k0L ≫ 1 and k0r ≫ k0l ≫ 1), the integration with respect to the scatter wave vector variables k′0y, k′0z can be performed analytically using the stationary phase method. Thus, if the observation point is in the direction
the diffuse far fields scattered from the rough surface are [Bahar and Lee, 1994]
The expressions for S(f, i) in (19) are the same as the expressions S(′, i) in (15) except that the scatter wave vector ′ is replaced in f. Thus, 0f = k0f and 1f for y < h (xs, zs), are related to 0f through Snell's law. Furthermore,
In (20), vy = k0(C0i + C0f) = k0 (cosθ0i + cosθ0f) When the integrations with respect to xs and zs are performed, the term GDf is shown to be the flat surface quasi-specular (zero-order) scattered field which is proportional to (4Ll/vxLvzl)sin vxL sin vzl. The expression for the quasi-specular scatter term GDf is the same as the expression for the total field Gf except that s in Gf is replaced by t in GDf (equation (16)). Thus, for h(xs, zs) = 0, they are identical and Gsf = 0.
 For surfaces with small Rayleigh roughness parameters β = 4k02 〈h2〉 ≪ 1, the exponent exp(ivyh) appearing in Gf is expanded in a Taylor series. In this case the first term in the integrand of (19), which is proportional to ivyh, is precisely equal to the first-order small-perturbation solution [Rice, 1951].
 To remove the small-slope assumption used to derive the iterative original full wave solution (19) (from the generalized telegraphists' equation), the surface element (differential) scattering matrix S(f, i) (which accounts for the sources induced on the rough surface by the incident field) is replaced by the following scattering matrix associated with the local coordinate system [Bahar, 1987]:
The quantity Sn is obtained from S, in (3.2) by replacing the unit vector normal to the mean surface, y by the unit vector normal to the actual surface [Bahar and Lee, 1994]
Thus, the angles of incidence 0i and scatter 0f with respect to the fixed, reference coordinate system (x, y, z) are replaced by the angles of incidence 0in and scatter 0fn in the local coordinate system (1, 2 = , 3) (see Figures 1a and 1b). Furthermore, the fixed planes of incidence and scatter (normal to and ) are replaced by the local planes of incidence and scatter (normal to and ). The surface element scattering matrix Sn is invariant to coordinate transformations. The scattering coefficients SnPQ where (as in (4) the first superscript P denotes the polarization of the scattered wave and the second superscript Q denotes the polarization of the incident wave) vanish for −i · ≤ 0 and −f · ≤ 0, corresponding to self-shadow. The transformation matrix Ti transforms the vertically and horizontally polarized waves in the fixed (reference) coordinate systems to the corresponding vertically and horizontally polarized waves in the local coordinate system, while the transformation matrix Tf transforms the vertically and horizontally polarized waves of the local coordinate system back to the reference coordinate system.
 Note that for the specular direction with respect to the reference coordinate system.
and the matrix R in (3) reduces to
where RV and RH are the Fresnel reflection coefficients for the vertically and horizontally polarized waves at the specular, stationary phase points, where
At these specular points, the local angles of incidence θ0is and scatter θ0fs are given by [Bahar et al., 1983a, 1983b]
Thus, on applying the stationary phase method to evaluate the integral in the high-frequency limit in closed form, the full wave solutions reduce to the physical optics stationary phase solution in the high-frequency limit [Beckmann and Spizzichino, 1963]. The corresponding expression for RnPQ(P ≠ Q) vanish at the specular points.
 Unlike the solution that employs the Kirchhoff approximations for the surface fields induced by the incident wave, the generalized telegraphists' equations that satisfy exact boundary conditions are intrinsically in agreement with reciprocity.
 On retaining terms in first order of smallness, the full wave solution reduces to
Equation (29) is precisely equal to the first- and second-order small-perturbation solution [Rice, 1951]. Thus, the same full wave expression for the scattered fields presented here accounts for, in a uniform, self-consistent manner, (high-frequency) specular point scattering [Beckmann and Spizzichino, 1963] as well as (low-frequency) polarization-dependent Bragg scattering predicted by using a small-perturbation approach.
 The normalized scattering cross sections for two-dimensional surfaces are defined as follows:
where Ay is the projection of the surface area (radar footprint) onto the mean (reference) plane y = 0.
 The coherent scattering cross sections are defined as
and the incoherent scatter cross sections are defined as
For homogeneous rough surfaces, the surface height autocorrelation is only a function of xd(≡ xs1 − xs2) and zd(≡ zs1 − zs2). Furthermore, if the radii of curvature of the larger-scale surfaces are assumed to be large compared with the free-space wavelength the slopes at point 2 may be approximated by the value of the slopes at point 1 hx2 ≈ hx1,hz2 ≈ hz1.
 Full wave solutions for rough surface scattering, with height-slope correlations included, has been considered [Bahar, 1991]. However, when the surface slopes are small, the surface heights and slopes can be assumed to be uncorrelated and the full wave expression reduces to [Bahar and Lee, 1994]
where the angle brackets denote statistical average over the slopes hx, hz and
where U(·)P2(·) is Sancer's  shadow function. Furthermore,
where vxz2 = v2 − vy2. When the surface slopes (〈hx2〉 ≪ 1, 〈hz2〉 ≪ 1) and the Raleigh roughness parameter β = 4k02〈h2〉 ≪ 1 are of the same order of smallness,
where C is the normalized surface height autocorrelation function. Thus,
where W(vx, vz) is the two-dimensional surface height spectral density function. For Gaussian autocorrelation functions,
When the surface slopes are assumed to be small, IPQ can be expressed as ∣DPQ(hx, hz)∣2/π. Thus, on retaining first-order terms in (34),
where D0 is the zero slope value of D (equation (22)). Thus,
 In the small height/slope limit the full wave solution reduces to the small-perturbation solution of Rice . Using a stationary phase approximation of the full wave solution, the most significant contributions to the scattered field come from the neighborhood of the specular points on the rough surface where
The product of IPQ in (42) and Q in (35) yields the physical optics solution of Beckmann and Spizzichino :
In the high-frequency limit, it can be shown that
where δ(·) is the Dirac delta function of the slopes at the specular points. Thus, in this limit, (40) reduces to the closed form geometrical optics solution [Beckmann and Spizzichino, 1963]
where p(hxs, hzs) is the slope probability density function evaluated at the specular points, where = s (equation (41)).
4.4. Unified Two-Scale Solutions for Composite Random Rough Surfaces
 For composite rough surfaces with very large to very small scales of roughness (associated with the correlation lengths or radii of curvature), neither the physical optics solutions nor the small-perturbation solutions are suitable for the evaluation of the like and cross-polarized scatter cross section. The polarization-dependent uniform full wave solutions which are not restricted by the Rayleigh roughness parameter β = k02〈h2〉 have been shown to merge uniformly with the high-frequency physical optics solutions and the low-frequency small-perturbation solutions. Introduction of the local coordinate systems was used to remove the small slope restriction of the original full wave solution. However, only the slopes of the larger-scale surface need to be accounted for in determining the normal to the rough surface since, the smaller-scale surface does not satisfy the large radii of curvature criterion.
 In this section a two-scale unified full wave solution is presented to account for the contributions to the scatter cross section from the larger- and smaller-scale surfaces [Bahar, 1981a].
 Consider a composite random rough surface with surface height
where the subscripts l and s, denote larger and smaller. There is no restriction to the mean square heights 〈hl2〉 and 〈hs2〉; however, it is assumed that lcl of the larger-scale surface is larger than the correlation length lcs of the smaller-scale surface hs:
Denote χl and χs as the characteristic functions of the larger- and smaller-scale surfaces and denote χ2l and χ2s as the joint characteristic functions for the larger- and smaller-scale surfaces. Since the characteristic functions are related to the Fourier transforms of the surface height probability density functions,
where χ2 and χ are the joint characteristic function and characteristic function of the total surface (equation (46)).
 Equation (48) can be expressed as
For distances rd ≈ lcs ≪ lcl,
Thus, (49) can be approximated by (see Appendix A)
The physical interpretation of (51) is the larger-scale surface contribution to the scattering cross section is diminished (due to superposition of the smaller-scale surface upon the larger-scale surface) by the factor ∣χs∣2 < 1. This feature of the unified full wave solution is consistent with conservation of energy; total power scattered by the rough surface cannot increase by superimposing a smaller-scale surface upon the larger-scale surface. Since the larger-scale surface meets the large radii of curvature criterion, the contribution of the larger-scale surface is given by the physical optics solution reduced by the characteristic function squared of the smaller-scale surface, while the contribution of the smaller-scale surface to the cross section is tilt modulated by the slopes of the larger-scale surface only.
 In section 5, the decomposition of the spectral density function into spectral density functions for the larger- and smaller-scale surface is considered in detail. This decomposition results in the corresponding expression for the total cross section as a weighted sum of the physical optics contribution of the larger-scale surface 〈σlPQ〉 and the tilt-modulated contribution of the smaller-scale surface 〈σsPQ〉 [Bahar, 1981a],
The above expression for the scatter cross sections, in terms of weighted sums of cross sections for the larger- and smaller-scale surfaces, was introduced in 1981 [Bahar, 1981a]. This feature of the unified full wave solutions significantly contributes to the stationarity of the variational approach [Bahar and Crittenden, 2008].