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.
@article{urn:eudml:doc:44513,
title = {Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {16},
year = {2003},
pages = {519-554},
zbl = {1066.86006},
mrnumber = {MR2032931},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44513}
}
Michel, Volker. Theoretical aspects of a multiscale analysis of the eigenoscillations of the Earth.. Revista Matemática de la Universidad Complutense de Madrid, Tome 16 (2003) pp. 519-554. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44513/