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Stochastic Seismology
T.-K. Hong and B.L.N. Kennett
Stochastic models provide a convenient means of representing
fine scale structure and the general character of the high frequency
coda accompanying seismic phases. In seismic applications, the random
realisations of the stochastic model needed to apply ensemble techniques
are commonly assumed to be achieved by different spatial sampling.
Direct testing of ensemble averages, with multiple
numerical simulations for different realisations of the stochastic
model, indicate that spatial averaging may be sufficient to
gain an indication of the expected behaviour. However, there is not
a direct correspondence between the spatial and ensemble results.
Simulations for a single realisation of a stochastic model need not
conform to the expectation for the ensemble average, which can help
to explain the wide range in estimates of such quantities as the
cut-off angle in treatments of body-wave scattering.
A simple configuration for acoustic waves has been used to examine
the relation between spatial and ensemble averaging for receivers
within a zone of heterogeneity. Multiple realisations
of heterogeneous models described by a specific correlation function
for the seismic wavespeed, are generated using different random seeds
for the phase distribution. The properties of the wavefield are
then
compared with those obtained by spatial sampling within a
single model.
The model consists of a 40 km by 40 km domain, with a plane acoustic
wave generated across a line of sources. A set of 128 receivers are placed at
29.06 km away from the source line. The wavefield is calculated using
the wavelet based method of Hong & Kennett, which is
able to achieve high accuracy even in the presence of very strong
heterogeneity. This configuration allows the examination of the
change in wavefield characteristics with correlation distance and
perturbation level. A single heterogeneous model with a von Karman
spectrum specified by a Hurst number of 0.25, is used with
varying levels of perturbation (1%, 3.3% and 10%).
The common interpretation is that spatial averaging is equivalent
to an ensemble average because of the change of the characteristics
of the medium with position. As can be seem from figures 1 and 2
the characteristics of averages over 20 spatial samples are not dissimilar
to that from averages of 20 different realisations of the random medium
medium for a single point of recording. However the correspondence is not
exaxct and will depend on the nature of the assumed correlation
properties.
Figure 1:
Character of acoustic coda for an average of 20 spatial samples
from the same random medium.
Figure 2:
Character of acoustic coda for an ensemble average of 20 different
relaisations of the random medium recorded at the same point.
Multiple simulations of a stochastic medium are needed to gain the most
meaningful results from numerical simulations. It would appear that
about 20 such different media provides the potential for a stable
ensemble average without excessive computational effort.
The success of simple multiple scattering models of coda can be traced
to the fact that this part of the seismogram contains contributions from
a wide variety of different paths sampling the medium. The trend of the
amplitude described by an envelope function can be well explained,
but the detail of the seismogram will depend on the particular path
and hence the actual deterministic structure.
Most theoretical results and numerical simulations are carried
out for regions with consistent stochastic properties, but the
heterogeneity regimes in the crust and mantle are expected to be
quite different. For body waves a suitable representation is via
reflection and transmission operators for different portions of
the medium. At high frequencies a path oriented approximation
can be applied with convolutional properties in time. For guided
waves a modal description is more suitable and the scattering
attenuation can be represented through the sum of contributions
from the different zone of heterogeneity modulated by the
behaviour of the appropriate eigenfunctions.
Back to the beginning
Questions about this topic to Brian Kennett:
Brian.Kennett@anu.edu.au
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