Semiclassical inverse problems for elastic surface waves in isotropic media

Alexei Iantchenko, Malmö University, Sweden

Abstract

We carry out a semiclassical analysis of surface waves in Earth which is stratified near its boundary at some scale comparable to the wave length.

Propagation of such waves is governed by effective Hamiltonians which are nonhomogeneous principal symbols of some pseudodifferential operators. Each Hamiltonian is identified with an eigenvalue in the discreet spectrum of a locally 1D Schrödinger-like operator on the one hand, and generates a flow identified with surface wave bicharacteristics in the two-dimensional boundary on the other hand.

The eigenvalues exist under certain assumptions reflecting that wave speeds near the boundary are smaller than in the deep interior. This assumption is naturally satisfied by the structure of Earth’s crust and mantle.

Using these Hamiltonians, we obtain pseudodifferential surface wave equations. In case of isotropic medium the equations decouple into Rayleigh and Love waves. In both cases we perform a comprehensive analysis of the recovery of the S-wavespeed from the semiclassical spectrum.

The approach follows the ideas of Colin de Verdière on acoustic surface waves and is joint work with Maarten V. de Hoop, Jian Zhai, Rice University, and Gen Nakamura, Hokkaido University.