APPLICATIONS OF THE DISCONTINUITY METHOD IN GEOPHYSICS AND TOMOGRAPHY

Abstract

In many applications (geophysics, acoustics, or tomography) one encounters integral transformations. In geophysical applications, the transformation describes numerous procedures for time-shifted stacking of seismograms and also expresses the correlation between arrival times for transmitted waves and the refractive index in a nonlinear formulation of the direct kinematic problem. In analyzing the procedures for time-shifted stacking, we must determine how the operator P acts on regular waves present in the wave field u0. It is convenient to assign to each wave a certain discontinuity of the field u0 and find the corresponding discontinuity in the field u1. This procedure is justified by the fact that the propagation of discontinuities in solutions of hyperbolic equations is governed by the same formulas as the propagation of waves in a ray approximation.