Full wave scatter cross sections for two-dimensional random rough surfaces using joint conditional surface height characteristic functions

The bistatic radar scatter cross sections for two-dimensional random rough surfaces are obtained using the full wave approach. The formal solution is expressed as a 10-dimensional integral over the random surface heights h₁, h₂ and slopes h(x1), h _x2, h_z1, h_z2 and the surface variables x_s1, x_x2, z_s1, z_z2. On averaging over the surface heights, the joint conditional surface height joint characteristic functions χ(a,blh(x1), h_x2, h_z1 , h_z2) are introduced and the l0-dimensional integral reduces to an 8-dimensional integral. For homogeneous isotropic rough surfaces, χ is a function of r_d=√(x_d²+z_d² ) where x_d≡x_s1-x_s2 and z _d≡z_x1-z_x2 and the solution reduces to a 5-dimensional integral over h(x1), h_x2, h_z1 , h_z2 and r_d. If the radius of the curvature of the surface is large compared to the wavelength of the incident wave, the surface slopes at two neighboring points are approximately equal. Thus the 5-dimensional integral can be expressed as a 3-dimensional integral. These full wave results can be further simplified if the mean square surface slopes are small (≪0.15), in which case it can be assumed that the surface height and slopes are uncorrelated and the above 3-dimensional integral reduces to the product of a 2 and a 1-dimensional integral. In the low frequency limit when the surface height and slopes are of the same order of smallness, the full wave solution reduces to the small perturbation solution. For high frequencies, the 3-dimensional integral reduces to the 1-dimensional physical optics integral. In the high frequency limit, it reduces to the geometrical optics solution since in this case the Fourier transform of χ(a,b|h_x,h_z) is a Dirac delta function of the slopes at the specular points. In the illustrative examples, the full wave results are compared with the associated small perturbation and physical/geometrical optics results for a large range of frequencies