Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth

Abstract

The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy­Navier equation. Using a standard approach in seis mology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy­Navier equation into two non­coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations us ing the Mie representation. Those solutions are denoted by the Hansen vectors Ln,j , Mn,j , and Nn,j in geophysics. Next we apply the inverse Fourier trans form to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy­Navier equation. 2000 Mathematics Sub ject Classification: 35J05, 42C40.