Workshop:
" Wave Phenomena - Mathematical modeling and numerical simulation
"
10.-11.02.2005 in Hamburg
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[
Program]
[
Participants]
Volker Michel
A Multiscale Method for the Modelling and Interpolation of
Earthquake Wave Propagation
Earthquake waves can be modelled by the Cauchy-Navier equation.
In the simplified case of a homogeneous, isotropic, and ball-shaped
Earth a fundamental system is known for the Fourier-transformed
equation. This system of Hansen vectors is used to derive the theoretical
eigenfrequencies of the Earth. In case of major earthquakes such as the
1960 Chile event such eigenoscillations, which can last several days after
the event, have already been found in seismograms.
In this talk the inverse Fourier-transformed function system is derived
yielding a time-dependent function system to describe not only the
eigenoscillations but also the propagation of regular earthquake waves.
Based on this result scaling functions and wavelets are constructed to
obtain a multiresolution of the solution space of the time-dependent
Cauchy-Navier equation. This method offers a new approach to analyse
the propagation of seismic waves at different (temporal and spatial)
resolution scales. Moreover, the obtained kernels have a localising
character. This is important since the distribution of seismographs over
the Earth is extremely inhomogeneous. Hence, the new method will
allow the interpolation of waves at comparatively high resolution in
regions with high seismograph density such as California without being
seriously influenced by areas with less data.
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