Applications of the Kirchhoff-Helmholtz integral to problems in seismology

Abstract

Summary. A numerical method for evaluating the Kirchhoff-Helmholtz integral is described. The Kirchhoff response is calculated by discretizing the surface, specifying simple point sources on each element of the surface, and summing the contribution from the elements. The results of the method are compared to those of an asymptotic, first motion approximation of the analytical solution of SH-waves impinging on a rigid sphere. The agreement between the results of the two methods is excellent for source and receiver distances which are large compared to the radius of the sphere. The method is applied to the calculations of reflections from mountain topography and a planar surface with an aperture. The phase shifts of pulses are consistent with optics; the amplitudes are not. The method does predict frequency dependence of the scattered amplitudes. Calculations are presented to model spall which produce travel-time and amplitude anomalies consistent with observations from nuclear blasts.