Normal modes of layered elastic media and application to diffuse fields

TL;DR

This paper derives the spectral decomposition of elastic wave operators in layered media, enabling analysis of diffuse fields and energy ratios, with applications to seismic wave behavior near free surfaces and soft layers.

Contribution

It introduces a generalized normal mode expansion for layered elastic media using functional analysis, providing a new framework for analyzing diffuse seismic fields.

Findings

Validated the normal mode expansion against classical methods.

Calculated vertical to horizontal energy ratios in layered media.

Demonstrated the impact of local velocity structure on energy partitioning.

Abstract

The spectral decomposition of the elastic wave operator in a layered isotropic half-space is derived by means of standard functional analytic methods. Particular attention is paid to the coupled $P$-$SV$ waves. The problem is formulated directly in terms of displacements which leads to a $2 \times 2$ Sturm-Liouville system. The resolvent kernel (Green function) is expressed in terms of simple plane-wave solutions. Application of Stone's formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity. The generalized normal mode expansion is employed to define a diffuse field as a white noise process in modal space. By means of a Wigner transform, we calculate vertical to horizontal kinetic energy ratios in layered media, as a function of depth and frequency. Several illustrative examples are considered including energy ratios near a free surface, in the presence of a soft layer. Numerical comparisons between the generalized eigenfunction summation and a classical locked-mode approximation demonstrate the validity of the approach. The impact of the local velocity structure on the energy partitioning of a diffuse field is illustrated.