Full paper in PDF:
$% V. Michel, Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth , Rev. Mat. Complut. 16 (2003), 519–554.%$

Theoretical Aspects of a Multiscale Analysis of the Eigenoscillations of the Earth

Volker MICHEL
Geomathematics Group
Department of Mathematics
University of Kaiserslautern
P.O. Box 3049
D-67653 Kaiserslautern — Germany
michel@mathematik.uni-kl.de

Received: July 16, 2002
 Accepted: October 21, 2002
ABSTRACT

The elastic behavior of the Earth, including its eigenoscillations, is usually described by the Cauchy-Navier equation. Using a standard approach in seismology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy-Navier equation into two non-coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations using 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 transform 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 Cauchy-Navier equation.

Key words: Cauchy-Navier equation, wavelets, multiresolution, Helmholtz equation, Hansen vectors, geomathematics.
2000 Mathematics Subject Classification: 35J05, 42C40.