A series approximation to the Kirchhoff integral for Gaussian and exponential roughness covariance functions

The Kirchhoff integral is a fundamental integral in scattering theory, appearing in both the Kirchhoff approximation and the small slope approximation. In this work, a functional Taylor series approximation to the Kirchhoff integral is presented, under the condition that the roughness covariance function follows either an exponential or Gaussian form—in both the one-dimensional and two-dimensional cases. Previous approximations to the Kirchhoff integral [Gragg, Wurmser, and Gauss (2001) J. Acoust. Soc. Am. 110(6), 2878-2901; Drumheller and Gragg (2001) J. Acoust. Soc. Am. 110(5), 2270-2275] assumed that the outer scale of the roughness was very large compared to the wavelength, whereas the proposed method can treat arbitrary outer scales. Assuming an infinite outer scale implies that the root mean square (rms) roughness is infinite. The proposed method can efficiently treat surfaces with finite outer scale and therefore finite rms height. This series is shown to converge independently of roughness or acoustic parameters and converges to within roundoff error with a reasonable number of terms for a wide variety of dimensionless roughness parameters. The series converges quickly when the dimensionless rms height is small and slowly when it is large.