Time domain Green functions technique for a point source over a dissipative stratified half-space with a phase velocity mismatch at the surface

Abstract

A point source over a dissipative stratified half-space with a discontinuity for both the phase velocity and the dissipation coefficient at the surface is considered. The axial symmetry of the problem is exploited in a spatial Hankel transform. The Green functions technique based on wave splitting is used to solve both the direct and inverse problems in the time domain. A Green tensor is introduced which directly maps the physical incident field outside the inhomogeneous half-space to the internal split fields. The PDEs for the Green functions are derived. In the direct problem, the internal fields are calculated as well as the reflected field. The phase velocity is reconstructed in the inverse problem. Simultaneous reconstruction of the phase velocity and dissipation coefficient using two different values of the Hankel transform parameter is analyzed. Two special cases, namely a point source over a homogeneous non-dissipative half-space and over a homogeneous dissipative half-space, are discussed and it is shown that the solution for the reflected fields is given by Volterra equations of the second kind.