  The Kirchhoff integral is a fundamental integral in scattering theory,
appearing in both the Kirchhoff approximation, as well as the small slope
approximation. In this work, a functional Taylor series approximation to
the-eps-converted-to.pdf 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 et al. J. Acoust. Soc. Am.
2001, Drumheller and Gragg J. Acoust. Soc. Am., 2001] 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.
