Inverse resonance problem for Love seismic surface waves

TL;DR

This paper addresses the inverse resonance problem for Love seismic surface waves in a half-solid, transforming it into a Schrödinger equation, and provides a method to recover shear modulus from resonances.

Contribution

It introduces a semi-classical approach to simplify the elastic wave problem and proves a one-to-one correspondence between potentials and Jost functions, enabling potential recovery.

Findings

Asymptotic analysis of resonances and wave numbers.

Proof of bijective mapping between potentials and Jost functions.

Algorithm for retrieving shear modulus from eigenvalues and resonances.

Abstract

In this paper we solve an inverse resonance problem for the half-solid with vanishing stresses on the surface: Lamb's problem. Using a semi-classical approach we are able to simplify this three-dimensional problem of the elastic wave equation for the half-solid as a Schr\"odinger equation with Robin boundary conditions on the half-line. We obtain asymptotic values on the number and the location of the resonances with respect to the wave number. Moreover, we prove that the mapping from real compactly supported potentials to the Jost functions in a suitable class of entire functions is one-to-one and onto and we produce an algorithm in order to retrieve the shear modulus from the eigenvalues and resonances.