On the basis of the stochastic functional approach, an analytical derivation for the incoherent scattering angular distribution contributed from the double Kirchhoff solution is presented for very rough surface, that is, with both the surface roughness and the correlation length comparable to the wavelength. By the second-order Kirchhoff approximation and using the propagation shadowing function, the incoherent scattering distribution can be numerically calculated by only the two-dimensional integrals. The numerical calculations are made for the normalized roughness σ/λ = 1.0, 1.5, and 3.5 and the normalized correlation length ℓ/λ = 1.8 and 5.0. The results show that the backscattering enhancement comes from the cross terms in the second-order Kirchhoff approximation. However, in order to obtain the calculations agreement well with the Monte Carlo simulation, the angular shadowing effect should also be included, in addition to the propagation shadowing effect.
 In recent years, the scattering of waves from very rough random surfaces has been studied analytically, numerically, and experimentally by many authors [Chen and Ishimaru, 1990; Jin and Lax, 1990; Maystre and Dainty, 1991; Ishimaru et al., 1991, 2002; Tsang et al., 1994; Ogura and Wang, 1997; Torrungrueng et al., 2002], with the emphasis on the effect of backscattering enhancement. Very rough surfaces are those with steep slope and RMS roughness characteristics comparable to the wavelength of the incident wave, and the conventional theories based on the Kirchhoff approximation and the perturbation method are not valid for them. It is quite difficult to treat the problem of scattering from very rough surfaces analytically because of the strong multiple scattering effects, and few analytical theories have been proposed for very rough surfaces. The modified second-order Kirchhoff approximation proposed by [Ishimaru and Chen, 1990] can give the results that agree well with those obtained by the Monte Carlo numerical simulations and in experiments, using the angular and the propagation shadowing functions introduced in their solutions for accounting for the effects of the higher-order scattering. The small-slope approximation first proposed by [Voronovich, 1994] is valid for arbitrary roughness but requiring relatively small slopes.
 By means of the stochastic functional approach [Ogura and Wang, 1994, 1996; Wang et al., 1995], we have recently proposed a theory of scattering from very rough random surfaces [Ogura and Wang, 1997], but merely the first-order Kirchhoff approximation has been considered. For a plane wave incidence, the random wave field is written in the stochastic Floquet form in terms of a random spectral measure and then represented as an integral over a random distributed source through a Green's function or a propagator whose form depends on the scattering structure studied. The consideration on the level shift invariance leads to the extended Voronovich form for the stochastic boundary value solutions, and the use of the boundary condition leads to a stochastic integral equation satisfying by the unknown random distributed source (the boundary value).
 The stochastic integral equation can be a Fredholm integral equation of the first or second kind, depending on the boundary value being considered is the field itself or its derivative. For the Fredholm integral equation of the first kind, the unknown surface scattered field in the extended Voronovich form may be further represented as a series expansion based on small values of surface slopes and be determined by a similar procedure in the small-slope approximation method [Voronovich, 1994]. For the Fredholm integral equation of the second kind, the inhomogeneous term automatically gives the solution of the Kirchhoff approximation for infinite surface. The iterative substitution of Kirchhoff solution into the integral equation would then give the solution of the “double” (the second-order) Kirchhoff approximation, but the direct solution of the double Kirchhoff approximation leads to a five-dimensional integral for the incoherent scattering distribution, which is difficult to be performed numerically. In the paper, we show that the incoherent scattering angular distribution can be numerically calculated by only the two-dimensional integrals, if we make use of an approximation in which the correlation between the forward and backward scattering process is neglected and use of the propagation shadowing function [Ishimaru and Chen, 1990] of a very rough surface to obtain the incoherent scattering distribution from the second-order Kirchhoff terms.
2. Stochastic Boundary Value Integral Equations
2.1. Gaussian Random Rough Surface
 We consider here the simplest problem of rough surface scattering theory: the scattering of a scalar wave from a one-dimensional (1D) random rough surface with the Dirichlet boundary condition, as shown in Figure 1. The rough surface is assumed to be a stationary Gaussian random process and described by z = f(x; ω), where f(x; ω) is a random function with its mean 〈f(x; ω)〉 = 0. In this notation, ω denotes a sample point in the sample space Ω, which is the ensemble of the realizations of f, and the angle brackets 〈 〉 indicate the probabilistic average over Ω. The homogeneous Gaussian random surface can be expressed by a spectral representation [Ogura and Wang, 1997]:
The shift operator Ta denotes a measure-preserving transformation on Ω representing the invariance of the probability measure with respect to the translation on the one-dimensional space R1 and is assumed to be ergodic. dB(λ) denotes the 1D complex Gaussian random measure with the property: 〈dB* (λ)dB(λ′)〉 = δ (λ − λ′)dλdλ′, where the probability parameter ω is suppressed for brevity. The correlation function can be given by
Where f(ω) ≡ f(0; ω), and ∣F(λ)∣2 with the property F* (λ) = F(−λ) denotes the spectral density. For a Gaussian random rough surface, the power spectrum is given as [Wang et al., 1995]
so that the correlation function is expressed as
where ℓ is the correlation length of the rough surface, and σ2 ≡ R(0) the variance, namely, the parameter describing the surface roughness.
2.2. Stochastic Boundary Value Integral Equations
 The scattered wave field can be expressed in terms of the unknown random measure as [Ogura and Wang, 1997]
where μ0 ≡ . The random measure h(Tx ω; v) is the Fourier transform of the undetermined random distributed source in the z direction (perpendicular to the surface), and is related to another random measure j (Tx ω) by the extended Voronovich form
so that the random measure j(ω; v) is actually stochastic functional depending only on the derivative f′(Tx ω) or the increments f(Tx ω) − f(ω). The Green function g(x − x0, v) that plays a role of propagator along the surface is given as
where Λ(v) ≡ and θ(x) denotes the sign function which actually describes the positive and negative traveling waves in the scattering process.
 Substituting the scattered wave field of equation (5) or (6) into the Dirichlet boundary condition on the random rough surface, we then get the following integral equations for the boundary values of the unknown source functions j(Tx ω) or h(Tx ω, v):
where the first terms with the subscript K stand for the solutions of the Kirchhoff approximation [Ogura and Wang, 1997] and are given by
To give the physical interpretation of this integral equation, we may call h(Tx ω; μ∣μ0, λ0) the kernel of the scattering process (−μ0, λ0) → (μ, Λ (μ)), and hK as the first-order Kirchhoff (scattering) kernel. Thus the second term (the “double” scattering kernel) in (13) can be interpreted in such a way that the scattering of the incident wave at x0 described by the kernel h(Tx ω; v∣μ0, λ0) propagates to the observation point x (from the sides of the point x, that is, x0 < x and x0 > x, respectively) by the Green function g(x − x0; v) with the wave vector (ν, θ(x − x0)Λ (μ)), and then scattered by the Kirchhoff kernel hK (Tx ω; μ∣ −v, θ(x − x0) Λ(v)) into the direction (μ, Λ (μ)).
3.1. “Double” Kirchhoff Approximation
 If h in the integral kernel of equation (13) is approximated by hK we then get the “double” Kirchhoff approximation for h, which we write
where we have put the “double” Kirchhoff integral,
where R1+, R1− stands for the region x0 < x, x0 > x, respectively.
 The second-order Kirchhoff term (15) implies in the similar way that the incident wave with direction (−μ0, λ0) is scattered by the first Kirchhoff kernel at x0, that is, by hK (Tx0ω; v∣μ0, λ0), that it propagates by the propagator or the partial Green function g(x − x0, v) from x0 (<x, >x) to the observation point x, and that it is again scattered into the direction (μ, Λ (μ)) by the second Kirchoff factor, hK (Tx ω; μ∣ −v, ± Λ (v)), which depends on the incident direction (v, Λ (v)) in R1+ (x0 < x) or (v, − Λ(v)) in R1− (x0 > x), as shown in Figure 2. Consequently the “double” Kirchhoff integral consists of all such Kirchhoff scattering processes at location x0 with propagating directions (v, ±Λ (v)) of the intermediate processes.
3.2. Spectral Representation for h2K
 For the calculation of the incoherent scattering distribution (see equation (28) below) contributed from the second-order Kirchhoff term, we need to calculate the average:
To make the following calculations for the incoherent scattering distribution easier, we introduce a spectral representation for a homogeneous random field defined by
The spectral density SI equals the Fourier transformation of the correlation function RI (x; ν, ν′) of the random field I(Tx ω, ν), and is expressed as
where R(x) is the correlation function of the rough surface given by equation (2) or (4).
 Making use of equation (20), we can obtain the spectral representation for the first-order Kirchhoff approximation term as
and for the second-order Kirchhoff approximation term as
4. Incoherent Scattering Angular Distribution
 On the basis of the scattered wave field in the far region (z > f(Tx ω), ∀ (x, ω)), the angular distribution of the power scattered incoherently into the unit solid angle can be expressed in terms of h as
where λs and μs = are the parameters related to the scattering direction. When we use the “double” Kirchhoff approximation (14) for h to calculate the incoherent scattering angular distribution, we need to calculate the average of the first-order Kirchhoff approximation term hK and the second-order Kirchhoff approximation term h2K as well as the correlation between these two terms. The incoherent scattering angular distribution PicKA1 (λs∣λ0) contributed by the first-order Kirchhoff term can be found in work by Ogura and Wang .
 By making use of equation (27), the average of the second-order Kirchhoff approximation terms h2K+ and h2K− are given as
for the ladder terms, and
for the cross terms. It has to be pointed out that we have made use of the approximation
with D = 11.13 ℓ representing the shadowing distance (this is chosen to conserve the energy, see details in work by Ishimaru and Chen ), we then get the final expressions for the incoherent scattering angular distribution contributed from the second-order Kirchhoff approximation terms as
for the ladder terms, and
for the cross terms, where η(ν, ν′) = (Λ(v) − Λ(v′))/2 and γ (ν, ν′) = (Λ(v) + Λ(v′))/2. Similarly, the incoherent scattering angular distribution contributed by the correlation term between the first-order and the second-order Kirchhoff approximation terms is given as
Thus the total incoherent scattering angular distribution is given by
Numerical calculations have been performed for the incoherent scattering angular distribution using equations (34)–(36) and equation (42) of Ogura and Wang , for the normalized roughness σ/λ = 1.0, 1.5, and 3.5 and the normalized correlation length ℓ/λ = 1.8 and 5.0. These parameters indicate that the surfaces are very rough. Figures 4 and 5 show the incoherent scattering intensities versus the observed angle θs for the oblique incident angle θi = 20° and θi = 40°, respectively. It is found that the backscattering enhancement occurring at λs + λ0 = 0 comes from the cross terms in the second-order Kirchhoff approximation and the enhanced peak is produced by the factor e−D(λ + λ + Λ(ν)−Λ(ν′))/8 in (35). The correlation term KA12 between the first-order and the second-order Kirchhoff scattering can be almost neglected for all of the cases. However, our results do not agree well with those of the Monte Carlo simulations and the numerical calculations of Ishimaru and Chen . The reason for this is because we have included merely the propagation shadowing effect by (33) but have not considered yet the angular shadowing effect. As stated by Ishimaru and Chen , these shadowing effects represent the effects of the higher_order scattering processes. Figure 6 compared the incoherent scattering of the first-order KA1 with and without the angular shadowing. It is demonstrated that with the angular shadowing our results are almost same with those of Ishimaru and Chen .
 We present here an analytical derivation for the incoherent scattering distribution from a very rough random surface, by means of the double Kirchhoff solution of equation (14), based on the stochastic functional approach. With the help of the approximation that the terms shown by the physical processes in Figure 3 are dominant and the use of the propagation shadowing function of equation (33), only the two-dimensional integrals have to be calculated numerically for the incoherent scattering distribution. The numerical calculations are made for the normalized roughness σ/λ = 1.0, 1.5, and 3.5 and the normalized correlation length ℓ/λ = 1.8 and 5.0. The results show that the backscattering enhancement comes from the cross terms in the second-order Kirchhoff approximation. However, in order to obtain the calculations agreement well with the Monte Carlo simulation, the angular shadowing effect should also be included, in addition to the propagation shadowing effect.
 This work was supported in part by the China State Key Basis research project 2001CB309401-5 and China Natural Science Foundation 60171009.