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

Abstract

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 h/sub 1/, h/sub 2/ and slopes h(x1), h/sub x2/, h/sub z1/, h/sub z2/ and the surface variables x/sub s1/, x/sub x2/, z/sub s1/, z/sub z2/. On averaging over the surface heights, the joint conditional surface height joint characteristic functions /spl chi/(a,blh(x1), h/sub x2/, h/sub z1/, h/sub z2/) are introduced and the l0-dimensional integral reduces to an 8-dimensional integral. For homogeneous isotropic rough surfaces, /spl chi/ is a function of r/sub d/=/spl radic/(x/sub d//sup 2/+z/sub d//sup 2/) where x/sub d//spl equiv/x/sub s1/-x/sub s2/ and z/sub d//spl equiv/z/sub x1/-z/sub x2/ and the solution reduces to a 5-dimensional integral over h(x1), h/sub x2/, h/sub z1/, h/sub z2/ and r/sub 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 (/spl Lt/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 /spl chi/(a,. >