Modern Techniques in Seismic Tomography

Abstract

This chapter is focussed on two local inverse kinematic problems of seismology, concerning reflected rays and refracted rays. Both model problems are reduced to a sequence of 2D problems, where theoretical and numerical results are offered. In the case of reflected rays, it is shown how to select a stable problem of recovering a velocity distribution in a layer, by using travel time measurements along rays with one reflection on the boundary. This way, a simple inversion algorithm is obtained for the linearized near a constant velocity case. In the case of refracted rays, a Newton-type algorithm for finding the 3D velocity distribution from 3D travel time measurements is constructed for the local inverse kinematic problem. To this end, a sound velocity that increases linearly with depth is chosen as a first approximation. With this particular choice for the linearization, the underlying problem reduces to a sequence of 2D Radon transforms in discs.