Radio Science

Like and cross-polarized scatter cross sections for two-dimensional, multiscale rough surfaces based on a unified full wave variational technique

Authors


Abstract

[1] A variational method is used to select the specific, smooth decomposition of the total surface height spectral density function into surface height spectral density functions for the larger- and smaller-scale surfaces. Using this decomposition, the total like and cross-polarized scatter cross sections are expressed as weighted sums of physical optics scatter cross sections associated with the larger-scale surfaces and the tilt-modulated scatter cross sections for the smaller-scale surfaces. This variational technique has been shown to be stationary over a wide range of the variational parameter. Since only the slopes of the larger-scale surfaces tilt modulate the cross sections of the smaller-scale surfaces, it is necessary to select surface height spectral density functions for the larger-scale surfaces that do not require the introduction of artificial spatial cutoff wave numbers for the spectral density functions. The methods used to smoothly decompose the surface height spectral density functions result in no artificial rapid fluctuations in the corresponding surface height autocorrelation functions for the smaller- and larger-scale surfaces. This method can be applied to the remote sensing of rough sea or land surfaces.

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 = 〈hl2〉/〈h2〉). 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 kd 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 kc [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 kc, 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.

2. Comparisons Between Solutions for Rough Surface Scatter Cross Sections Based on Geometric Optics, Physical Optics, the Standard Hybrid Two-Scale Solutions, the Inversion of an Integral Equation, and the Unified Full Wave Technique

[6] The study of rough surface scattering has several applications in remote sensing [Barrick, 1970, 1972]. Solutions for the scatter cross sections of rough surfaces with multiple scales of roughness have been published [Hagfors, 1966; Burrows, 1973; Barrick and Peake, 1968; Valenzuela, 1968; Wright, 1968]. Several of them are based on a hybrid physical optics-perturbation approach [Brown, 1978]. Using this approach, the smaller-scale surfaces are depicted as if they were superimposed upon the larger-scale surfaces (see Figures 1a and 1b). To this end the surface height spectral density functions for the entire surfaces are decomposed into spectral density functions for the smaller-scale and the larger-scale surfaces. The perturbation approach [Rice, 1951] has been applied to obtain the solutions associated with the smaller-scale surfaces while the physical optics approach [Tyler, 1976; Beckmann and Spizzichino, 1963; Beckmann, 1968] has been applied to obtain the solutions associated with the larger-scale surfaces. These two solutions are combined in an ad hoc manner to obtain the hybrid two-scale solutions for the scatter cross sections of the entire rough surface [Brown, 1978]. Detailed comparisons between these different methods have been published by Barrick and Peake [1968] and Bahar [1987]. The principal problem with the hybrid, ad hoc approach is that the solution critically depends upon the manner in which the spectral splitting is accomplished [Bahar and Barrick, 1983]. Furthermore, this approach apparently contradicts conservation of energy, since the larger-scale rough surfaces cannot scatter more power by simply superimposing smaller-scale rough surfaces upon them. A comparison between the full wave solution presented here and a solution based on the inversion of an integral equation has been conducted [Collin, 1992]. The unified full wave approach described in section 4, on the other hand, is given in terms of a “weighted” sum of two cross sections [Bahar, 1981a; Bahar and Fitzwater, 1984, 1985; Bahar et al., 1983a, 1983b]. The contribution associated with the larger-scale surface is reduced by a factor equal to the magnitude squared of the characteristic function for the smaller-scale surface, while the cross section, associated with the smaller-scale surface is tilt modulated by the slopes of the larger-scale surface. Thus, depending on the specific spectral splitting assumed, as the contribution associated with the smaller-scale surface is increased, the contribution associated with the larger-scale surface is decreased and visa versa. This feature of the unified full wave approach makes the solution stationary over a very wide range of the variational parameter associated with this technique. The details of this variational technique are described section 4.

Figure 1.

(a) Rough surface y = h(x, z), wave vectors and angles of incidence and scatter in the fixed coordinate system. (b) Rough surface y = h(x, z), wave vectors and angles of incidence and scatter in the local coordinate system.

3. Overview of the Full Wave Solutions and Formulation of the Problem

[7] The overview of the full wave solutions starts with the formal solutions to the generalized telegraphists' equations [Schelkunoff, 1955] for electromagnetic wave scattering from two-dimensionally rough surfaces. These generalized telegraphists' equations are sets of coupled differential equations for the forward and backward propagating wave amplitudes. Schelkunoff's method leading to coupled differential equations for the wave amplitudes is distinct from integral equation methods. The generalized telegraphists' equations are derived from Maxwell's equations upon substituting complete expansions for the electromagnetic fields. They consist of the radiation term, the lateral waves and the guided surface waves [Bahar, 1973a, 1973b]. In this paper the scattered far fields are of particular interest; thus, only the radiation term is presented in detail. Since the expansions of the fields are in the vertical planes (x = const), exact boundary conditions are imposed at the rough interface between free space (with permittivity ɛ0 and permeability μ0 and the medium below the interface with permittivity ɛ and permeability μ) (see Figures 1a and 1b).

[8] For waveguides with finite cross sections and perfectly conducting rough boundaries considered by Schelkunoff, the complete expansion of the fields consists of a discrete set of propagating and evanescent waveguide modes. For the problem considered here the complete expansions consists of a branch cut integral associated with the radiation fields and a branch cut integral associated with lateral waves and guided surface waves associated with the poles of the Fresnel reflection coefficients. Coupling between these three species of the full wave expansions have been considered in detail [Bahar, 1977]. They are of particular interest for receivers near the interface. Lateral waves and surface waves are not excited when the surface is perfectly conducting.

[9] In Figure 1a the two-dimensionally rough surface

equation image

is represented in the fixed coordinate system with unit vectors equation imagex, equation imagey and equation imagez, and equation imagei, equation imagef are unit vectors in the directions of the incident wave and scattered waves above the interface. In Figure 1b, the rough surface is represented in the local coordinate system, with equation image normal to the larger-scale surface. For the smaller-scale surface, the correlation length (or the radius of curvature) is smaller than the wavelength λ of the electromagnetic excitation. The two other orthogonal unit vectors of the local coordinate system lie in the local tangent plane (see Figures 1a and 1b).

[10] The first-order iterative solutions to the telegraphists' equations are the fields impressed on the surface, with wave coupling neglected. The second-order iterative solutions take into account wave coupling [Bahar, 1981b, 1987; Bahar and Rajan, 1979]. Collin [1992] refers to these solutions as Bahar's “original” full wave solutions. Higher-order solutions, which take into account multiple scatter, are shown to be associated with enhanced backscatter [Bahar and El-Shenawee, 2001]. Solutions, based on the inversion of integral equations, are show to be in total agreement with the original full wave solutions [Collin, 1992]. The original full wave solutions are applicable to rough surfaces with small mean square slopes. However, unlike Rice's [1951] small-perturbation solutions, applicable to rough surfaces with RMS heights and slopes of the same order of smallness, the original full wave solutions are not restricted to rough surfaces with small mean square heights. On applying a unified full wave, two-scale model of the rough surface, the scatter cross sections are expressed as weighted sums of physical optics cross sections for the larger-scale surfaces and cross sections for the smaller-scale surfaces that are tilt modulated by the slopes of the larger-scale surfaces only, since the slopes of the smaller-scale surfaces do not contribute to tilt modulation. The slopes of the larger-scale surfaces need not be small as with the original full wave solution.

4. Description of the Full Wave Variational Technique

4.1. Formulation of the Problem

[11] Traditional analytical techniques to determine the like and cross-polarized scatter cross sections for rough surfaces are classified as physical optics solutions [Beckmann and Spizzichino, 1963] and perturbation solutions [Rice, 1951]. The physical optics solutions are generally restricted to rough surfaces with large radii of curvature compared with the electromagnetic wavelength. Physical optics scattering occurs primarily in the neighborhoods of the stationary phase, specular points on the surface where the normal to the surface bisects the wave vectors associated with the incident and scattered waves (see Figures 1a and 1b). The perturbation solutions are generally restricted to smaller-scale surfaces with mean square heights of the order of the electromagnetic wavelength or less.

[12] For good conducting surfaces, the physical optics solutions are practically independent of the polarization of the waves. However, the perturbation solutions are strongly dependent on polarization, particularly for backscatter at near grazing incidence. Very often natural rough surfaces do not satisfy the restrictions associated with the physical optics solutions or the perturbation solutions. It is therefore necessary to consider two-scale models of the rough surfaces [Hagfors, 1966; Burrows, 1973; Valenzuela, 1968; Wright, 1968; Barrick and Peake, 1968; Brown, 1978] for problems of scattering from composite rough surfaces. These solutions require the decomposition of the surface height spectral density functions into individual spectral density functions for the larger- and smaller-scale rough surfaces. Solutions based upon an ad hoc, hybrid, physical optics-perturbation approach, are shown to be very sensitive to the specific decomposition of the surface height spectral density functions for the entire surface into spectral density functions for the larger- and smaller-scale surfaces [Brown, 1978].The unified full wave solutions presented here have been shown to reduce to the physical optics solutions [Beckmann and Spizzichino, 1963] in the high-frequency limit and to the small-perturbation solution [Rice, 1951] in the low-frequency limit. Moreover the full wave solutions based on a two-scale model of the rough surface [Bahar, 1981a] are expressed as weighted sums of larger- and smaller-scale surfaces. This feature of the unified full wave solutions contributes significantly to the stationarity of the variational technique presented here.

4.2. Principal Advantages of the Full Wave Variational Technique

[13] The major advantages of the novel unified full wave variation techniques over the standard hybrid two-scale solutions are summarized below.

[14] 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].

[15] 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].

[16] 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].

[17] 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].

[18] 5. The scatter cross sections are polarization dependent [Beckmann and Spizzichino, 1963; Bahar, 1981b; Bahar and Lee, 1994].

[19] 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].

[20] 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].

[21] 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

[22] For plane wave, linearly polarized excitations the second-order linearly polarized scattered fields are expressed in matrix form as [Bahar and Lee, 1994]

equation image

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 k0y, k0z for the radiation fields. It is assumed here that the mean plane of the rough surface is y = 〈h(xs, zs)〉 = 0.

[23] The 2 × 2 surface element scattering matrix Su is given by [Bahar and Lee, 1994]

equation image

The elements of the matrix Ru in (3) are

equation image
equation image
equation image
equation image

The wave vectors in the scatter and incident directions are

equation image

and

equation image

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).

equation image
equation image
equation image

In (4)(7), ɛr = ɛ10, μr = μ1/μ0, nr = (ɛrμr)1/2 and ηr = (μrr)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.

[24] The vectors equation image′ and equation image′ are given in terms of the incident equation image0i and scatter equation image0i wave vectors. The subscripts 0, 1 are for the medium 0(y > h (xs, zs)) and medium 1(y < h (xs, zs)), respectively:

equation image

and equation images and equation image are position vectors from the origin to points on the rough surfaces and to the observation point, respectively:

equation image

On integrating (2) by parts, it is expressed as [Bahar and Lee, 1994]

equation image

In (15) it is assumed that h(xs, zs) vanishes for ∣xs∣ > L and ∣zs∣ > l, (1), and

equation image

Since equation image′ · equation image, it follows that

equation image

The expression for the surface element scattering matrix S(equation image′, equation image′) is given by (4)(7) with exception that Cu and Du are replaced by cos(ϕ′ − ϕi) and sin(ϕ′ − ϕi), respectively.

[25] The solutions presented in (15) are referred to by Collin [1992] as the original full wave solutions. Collin [1992] 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).

[26] 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(vx)δ(vz). 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.

[27] When the observation point is at a very large distance from the rough surface (k0rk0L ≫ 1 and k0rk0l ≫ 1), the integration with respect to the scatter wave vector variables k0y, k0z can be performed analytically using the stationary phase method. Thus, if the observation point is in the direction

equation image

the diffuse far fields scattered from the rough surface are [Bahar and Lee, 1994]

equation image

The expressions for S(equation imagef, equation imagei) in (19) are the same as the expressions S(equation image′, equation imagei) in (15) except that the scatter wave vector equation image′ is replaced in equation imagef. Thus, equation image0f = k0equation imagef and equation image1f for y < h (xs, zs), are related to equation image0f through Snell's law. Furthermore,

equation image

and

equation image

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 equation images in Gf is replaced by equation imaget in GDf (equation (16)). Thus, for h(xs, zs) = 0, they are identical and Gsf = 0.

[28] For surfaces with small Rayleigh roughness parameters β = 4k02h2〉 ≪ 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].

[29] 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(equation imagef, equation imagei) (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]:

equation image

The quantity Sn is obtained from S, in (3.2) by replacing the unit vector normal to the mean surface, equation imagey by the unit vector normal to the actual surface [Bahar and Lee, 1994]

equation image

Thus, the angles of incidence equation image0i and scatter equation image0f with respect to the fixed, reference coordinate system (equation imagex, equation imagey, equation imagez) are replaced by the angles of incidence equation image0in and scatter equation image0fn in the local coordinate system (equation image1, equation image2 = equation image, equation image3) (see Figures 1a and 1b). Furthermore, the fixed planes of incidence and scatter (normal to equation image and equation image) are replaced by the local planes of incidence and scatter (normal to equation image and equation image). 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 −equation imagei · equation image ≤ 0 and equation imagef · equation image ≤ 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.

[30] Note that for the specular direction with respect to the reference coordinate system.

equation image

and the matrix R in (3) reduces to

equation image

where RV and RH are the Fresnel reflection coefficients for the vertically and horizontally polarized waves at the specular, stationary phase points, where

equation image

At these specular points, the local angles of incidence θ0is and scatter θ0fs are given by [Bahar et al., 1983a, 1983b]

equation image

and

equation image

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(PQ) vanish at the specular points.

[31] 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.

[32] On retaining terms in first order of smallness, the full wave solution reduces to

equation image

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.

[33] The normalized scattering cross sections for two-dimensional surfaces are defined as follows:

equation image

where Ay is the projection of the surface area (radar footprint) onto the mean (reference) plane y = 0.

[34] The coherent scattering cross sections are defined as

equation image

and the incoherent scatter cross sections are defined as

equation image

For homogeneous rough surfaces, the surface height autocorrelation is only a function of xd(≡ xs1xs2) and zd(≡ zs1zs2). 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 hx2hx1,hz2hz1.

[35] 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]

equation image

where the angle brackets denote statistical average over the slopes hx, hz and

equation image

where U(·)P2(·) is Sancer's [1969] shadow function. Furthermore,

equation image

where vxz2 = v2vy2. When the surface slopes (〈hx2〉 ≪ 1, 〈hz2〉 ≪ 1) and the Raleigh roughness parameter β = 4k02h2〉 ≪ 1 are of the same order of smallness,

equation image

where C is the normalized surface height autocorrelation function. Thus,

equation image

where W(vx, vz) is the two-dimensional surface height spectral density function. For Gaussian autocorrelation functions,

equation image

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),

equation image

where D0 is the zero slope value of D (equation (22)). Thus,

equation image

[36] In the small height/slope limit the full wave solution reduces to the small-perturbation solution of Rice [1951]. 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

equation image

and

equation image

The product of IPQ in (42) and Q in (35) yields the physical optics solution of Beckmann and Spizzichino [1963]:

equation image

In the high-frequency limit, it can be shown that

equation image

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]

equation image

where p(hxs, hzs) is the slope probability density function evaluated at the specular points, where equation image = equation images (equation (41)).

4.4. Unified Two-Scale Solutions for Composite Random Rough Surfaces

[37] 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 = k02h2〉 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 equation image to the rough surface since, the smaller-scale surface does not satisfy the large radii of curvature criterion.

[38] 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].

[39] Consider a composite random rough surface with surface height

equation image

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:

equation image

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,

equation image

where χ2 and χ are the joint characteristic function and characteristic function of the total surface (equation (46)).

[40] Equation (48) can be expressed as

equation image

For distances rdlcslcl,

equation image

Thus, (49) can be approximated by (see Appendix A)

equation image

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 ∣χs2 < 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.

[41] 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],

equation image

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].

5. Implementation of the Procedure to Decompose the Surface Height Spectral Density Functions and the Criteria Used to Determine the Stationarity of the Solutions

[42] Two different surface height spectral density functions (Gaussian and Pierson-Moskowitz) are considered here since the spectral decomposition into spectral density functions for the larger- and smaller-scale surfaces are achieved into two different ways. In both cases a smooth and continuous decomposition of the spectral density functions is performed. This is done in order to avoid non physical fluctuations in the corresponding surface height autocorrelation functions, which are the Fourier transforms of the spectral density functions.

[43] Initially, a two-dimensional Gaussian isotropic surface height autocorrelation function R(rd) is assumed for the entire surface:

equation image

where rd2 = xd2 + zd2, 〈h2〉 is the mean square height, and lc is the correlation length. The corresponding surface height spectral density function is

equation image

where k2 = kx2 + kz2 and the inverse Fourier transform of W(k) is the two-dimensional autocorrelation function,

equation image

For the Gaussian spectral density function (54), the larger-scale spectral density function Wl, is defined first. It has the same form as (54) except that 〈h2〉 is replaced by 〈hl2〉, the mean square height and lc is replaced by lcl the correlation length. Furthermore, Wl(0) = W(0) such that

equation image

Therefore, the spectral density for the smaller-scale surface is

equation image

As a result of the choice (55), Ws(0) = 0. Note that all these spectral density functions are positive real for all real values of the variable k. Furthermore, the surface height autocorrelation function for the smaller-scale surface is

equation image

The corresponding mean square slopes for the total and the larger-scale surfaces are

equation image
equation image

The slopes of the smaller-scale surface do not tilt modulate the scatter cross section of the smaller-scale surface. The probability density function for the larger-scale slopes that tilt modulates the cross sections for the smaller-scale surface is assumed to be Gaussian

equation image

where hx and hz are derivates of the surface height of the larger-scale surface with respect to x and z. Likewise, the shadow function depends on the normal of the large-scale surface equation imagel as

equation image

Thus, the local angles of incidence and scatter cannot be larger than π/2.

[44] For the Pierson-Moskowitz surface height spectral density function, it is the surface height spectral density function for the smaller-scale surface that is initially chosen. It differs from the surface height spectral density function of the entire surface only in its mean square height and the value of k for which the wind generated surface height spectral density function is maximum, WMX. Thus, an isotropic Pierson-Moskowitz surface height spectral density function is given explicitly in terms of the mean square height by

equation image

The corresponding surface height normalized autocorrelation function is,

equation image

In (62), 〈h2〉 is the mean square height. In (63) the K are modified Bessel functions of orders 0 and 1 and the argument ζ = κrd, where

equation image

and V is the wind speed in m/s. The maximum value of W(k) is for k = κ:

equation image

The surface height spectral density for the smaller-scale surface is

equation image
equation image

where 〈hs2〉 is the mean square height of the smaller-scale surface and Ws peaks for k = κ. We select 〈hs2〉 ≤ 〈h2〉 and 〈hs2〉κs2 = 〈hs2〉κ2 such that W(k ≫ κ) = Ws(k ≫ κs).

[45] Thus, Ws(k) ≤ W(k) for all values of k. The corresponding expression for the normalized surface height autocorrelation function for the smaller-scale surface is the same as (11) except that ζ = κsrd.

[46] The surface height spectral density function for the larger-scale surface is

equation image

for all values of k. The corresponding mean square height for the larger-scale surface is

equation image

The corresponding normalized surface height autocorrelation function is

equation image

Unlike the mean square slopes for Gaussian spectral density functions (8), for the Pierson-Moskowitz spectral density functions the integral in (4.7a) with the limits (−∞, ∞) is not finite. Therefore it is common practice [Brown, 1978] to truncate the limits of integration to (−kc, kc) where kc, the cutoff spatial wave number is usually chosen to be less than π/λ and λ is the wavelength of the electromagnetic excitation. However, since Wl(k) defined in (68) varies as k−6 for k ≫ κs the integration in (59a) can be performed with the limits (−∞, ∞) without choosing an artificial cutoff spatial wave number, kc.

[47] The mean square slope of the larger-scale surface is

equation image

This is a very desirable feature of Wl(l) since the mean square slope with the limits (−kc, kc) is sensitive to the choice of kc. For the purpose of the analysis in this paper, the mean square slopes of the total surface or the smaller-scale surface are not needed since only the slopes of the larger-scale surface tilt modulate the scatter cross sections of the smaller-scale surface, not the mean square slope of the smaller-scale surface.

[48] The variational parameters n that is suitable to examine the stationarity of values of the scatter cross sections is the ratio of the mean square height of the larger-scale surface to the total mean square height of the entire rough surface:

equation image

Thus, for n = 1, the entire rough surface is regarded as larger-scale and the total scatter cross sections are given by the high-frequency physical optics solution which is obtained from the full wave solution through stationary phase integrations. This integration results in the evaluation of the surface element scatter coefficients at the specular points on the surface, where the normal to the surface is along the vector equation image = equation imagefequation imagei which bisects the scatter and incident wave vectors. For n = 0, the entire rough surface is regarded as smaller scale and the scatter cross sections are given by the original full wave solution with no slope modulation [Bahar and Rajan, 1979].

[49] For the values of 0 < n < 1, the scatter cross sections are given by the weighted sum of the larger-scale (physical optics) cross section (multiplied by the factor equal to the absolute value of the characteristic function squared ∣χs2 of the smaller-scale surface) plus the scatter cross section for the smaller-scale surface, tilt modulated by the slopes of the larger-scale surface. Thus, as n decreases from n = 1 to n = 0, the contribution of the smaller-scale cross sections increases and the contribution of the larger-scale surface decreases due to the weighting factor that multiplies the physical optics cross section (since ∣χs2 decreases as n decreases). This built-in feature of the unified full wave solution results in the stationarity of the cross section over a broad range of the variational parameter n.

[50] The criterion used to determine the stationarity of the solutions for the backscatter cross section is the following norm of the error over, the full range of incident angles, from normal to near grazing [Bahar and Crittenden, 2008]:

equation image

where 〈σnnPP〉 and 〈σnPP〉 are values for the backscatter cross sections corresponding to two consecutive values of mean square heights of the larger-scale surface (hii)n+ Δn and (hli)n, respectively. The norm of the error defined in equation (72) is a discretized form of the following integral:

equation image

In (72) the evaluated normalized scatter cross sections σPP(n, θ0i) depend on the variational parameter n and the angle of incidence and K is constant. The stationarity of the results over a broad range of the variational parameter n, associated with the decomposition of the surface height spectral density function W(k), manifests itself as a broad minimum of norm of the error E(n) [Bahar and Crittenden, 2008].

6. Numerical Simulations to Substantiate the Variational Technique

[51] The variational technique described in this paper has been substantiated by conducting a series of numerical simulations for surfaces that are rough in one dimension [Bahar and Crittenden, 2008]. In these simulations the scatter cross sections for vertically and horizontally polarized waves were evaluated for values of the variational parameter n (equation (71)) ranging from 0 to 1 at angles of incidence from normal to near grazing. Rough surfaces with a wide range of mean square heights were considered. The criteria used to determine the stationarity of the solutions described in section 5 (equation (72)) were applied to these numerical simulations. It was shown that the results were stationary over a very wide range of the variational parameter n ranging from 3 to 7. The norm of the error (72) was shown to be largest for values of n around zero and one, where n = 0 corresponds to regarding the entire surface as the smaller-scale surface and n = 1 corresponds to regarding the entire surface as the larger-scale surface.

[52] These numerical simulations also clearly demonstrate that for horizontally polarized waves the cross sections are strongly dependent on slope modulation by the tilt of the larger-scale surface for near grazing incidence. However, for vertically polarized waves the dependence on slope modulation is practically insignificant [Collin, 1992; Bahar and Crittenden, 2008]. These results are consistent with experimental observations that the ratio of the two cross sections is near one for grazing incidence [Cloude and Coor, 2002]. This is contrary to results based on the standard perturbation solution.

7. Future Work on the Proposed Variational Technique Applied to Two-Dimensionally Rough Surfaces

[53] Additional series of numerical simulations and experimental observations are to be performed for surfaces that are rough in two dimensions pursuant to the analysis given in this paper. Surfaces that are characterized by finite conductivity, as described in this paper, also need to be considered.

[54] Applications to remote sensing of sea surfaces (to determine wind speed for instance) and land surfaces (to determine soil moisture content for instance) are some of the practical applications of this analysis.

8. Concluding Remarks

[55] The procedures leading to variational solutions for the like and cross linearly polarized scatter cross sections for two-dimensionally rough surfaces are presented in detail in this paper. It is based on the use of unified full wave solutions to express the total cross sections as weighted sums of two cross sections [Bahar, 1981a]. The first, associated with the larger-scale surface is the physical optics cross section reduced by absolute value of the square of the characteristic function for the smaller-scale surface. The second is the scatter cross section for the smaller-scale surface that is tilt modulated by the slopes of the larger-scale surface alone. This feature of the unified full wave solutions [Bahar, 1981a] contributes significantly to the stationarity of the variational approach presented here.

[56] The method for decomposing the scatter cross sections for the entire surface height spectral density function into spectral density functions for the larger- and smaller-scale surfaces is given separately in detail in section 5 for Gaussian and Pierson-Moskowitz spectral density functions, since the procedures for the decomposition of these spectral density functions are quite distinct.

[57] For entire surfaces with Gaussian spectral density functions, we first choose Gaussian spectral density functions for the larger-scale surface, such that the spectral density function for the smaller-scale surface vanishes for the spatial wave number k = 0. Thus, when the total surface has a Gaussian spectral density function, 〈h2lc2 = 〈hl2lcl2. Therefore for the variational parameter, n = 〈hl2〉/〈h2〉 ≤ 1, lcl2lc2.

[58] For the total surfaces with Pierson-Moskowitz spectral density functions we first choose a Pierson-Moskowitz spectral density function for the smaller-scale surface such that Ws(k → ∞) = W(k → ∞). In this case therefore the variational parameter 1 − n = 〈hs2〉/〈h2〉 = κ2s2 in which the parameters κs corresponds to κ (related to the surface wind speed). In this case the spectral density function for the larger-scale surface has a finite mean square slope, without introducing an artificial cutoff spectral wave number. The corresponding surface height autocorrelation functions for the larger- and smaller-scale surfaces (the Fourier transforms of the surface height spectral density functions) do not possess artificial rapid fluctuations, since the decompositions are achieved in a smooth, continuous manner, rather than a discontinuous manner.

[59] For the preliminary numerical simulations, the scatter cross sections are shown to be stationary over a wide range of the variational parameter n [Bahar and Crittenden, 2008]. These results are also consistent with observations that the ratio of the backscatter cross sections for the horizontally and vertically polarized waves could be near unity for near grazing incidence [Cloude and Coor, 2002]. This is because the horizontally polarized cross sections are much more sensitive to tilt modulation than the vertically polarized cross sections [Collin, 1992; Bahar and Crittenden, 2008].

Appendix A

[60] The terms that were neglected by replacing (49) by (51) are expressed in their Taylor series expansions for rdlcslcl as follows:

equation image

In (A1),

equation image

For 0 < rd < lcslcl, (1 − Rl) → 0 and Rs < 1; therefore, the contribution of (A1) to (49) is negligible.

Acknowledgments

[61] The author wishes to thank R. E. Collin for his comments and suggestions regarding the variational approach. M. Craig, R. Odom, and L. Smith prepared this manuscript.