# Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.

• Volume: 16, Issue: 2, page 519-554
• ISSN: 1139-1138

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## Abstract

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The elastic behaviour 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.

## How to cite

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Michel, Volker. "Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.." Revista Matemática Complutense 16.2 (2003): 519-554. <http://eudml.org/doc/44513>.

@article{Michel2003,
abstract = {The elastic behaviour 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.},
author = {Michel, Volker},
journal = {Revista Matemática Complutense},
keywords = {Ecuaciones diferenciales elípticas; Ecuación de Helmholtz; Ondículas; Análisis de Fourier; Cauchy-Navier equation; Wavelets; Multiresolution; Helmholtz equation; Hansen vector},
language = {eng},
number = {2},
pages = {519-554},
title = {Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.},
url = {http://eudml.org/doc/44513},
volume = {16},
year = {2003},
}

TY - JOUR
AU - Michel, Volker
TI - Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.
JO - Revista Matemática Complutense
PY - 2003
VL - 16
IS - 2
SP - 519
EP - 554
AB - The elastic behaviour 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.
LA - eng
KW - Ecuaciones diferenciales elípticas; Ecuación de Helmholtz; Ondículas; Análisis de Fourier; Cauchy-Navier equation; Wavelets; Multiresolution; Helmholtz equation; Hansen vector
UR - http://eudml.org/doc/44513
ER -

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