Closed-Form Approximation of the Kirchhoff Integral for a General Class of Rough Surfaces

Most of the available analytical methods for the evaluation of the normalized radar cross section (NRCS) of randomly rough surfaces require the evaluation of the so-called Kirchhoff integral (KI). Numerical evaluation of this integral is computationally demanding and sometimes problematic. Therefore, analytical closed-form solutions have been proposed in the past. However, up to now they have been obtained only for a few special cases. In this article, based on a proper accurate approximation of the integrand, an analytical formulation of the KI is provided for a general class of rough surfaces that include those of currently available analytical solutions as particular cases. In the general case, the analytical formulation is expressed in terms of a couple of very rapidly convergent asymptotic series, that can be evaluated with a high accuracy by using very few terms in almost all cases. In the particular cases of Gaussian and exponential surface autocorrelation functions (ACFs), as well as for scattering in the specular direction, very compact closed-form expressions are obtained. The accuracy of the proposed solution is assessed via comparison with numerical “exact” evaluations of the KI. Comparisons with measurements are also presented.