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| Micro-earthquake monitoring with sparsely-sampled data | |
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Migration with an interferometric imaging condition (IIC) uses the
same generic framework as the one used for the conventional imaging
condition, i.e. wavefield reconstruction followed by an imaging
condition. However, the difference is that the imaging condition is
not applied to the reconstructed wavefield directly, but it is applied
to the wavefield which has been transformed using pseudo Wigner
distribution functions (WDF) (Wigner, 1932). By definition, the
zero frequency pseudo WDF of the reconstructed wavefield
is
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(4) |
where and denote averaging windows in space and time,
respectively. In general, is three dimensional and is one
dimensional. Then, the image
is obtained by extracting
the time from the pseudo WDF,
, of the
wavefield
:
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(5) |
The interferometric imaging condition represented by equations 4 and
5 effectively reduces the artifacts caused by the random
fluctuations in the wavefield by filtering out its rapidly varying
components (Sava and Poliannikov, 2008). In this paper, I use this
imaging condition to attenuate noise caused by sparse data sampling or
noise caused by random velocity variations. As suggested earlier, the
interferometric imaging condition attenuates both types of noise at
once, since it does not explicitly distinguish between the various
causes of random fluctuations.
The parameters and defining the local window of the pseudo WDF
are selected according to two criteria (Cohen, 1995). First,
the windows have to be large enough to enclose a representative
portion of the wavefield which captures the random fluctuation of the
wavefield. Second, the window has to be small enough to limit the
possibility of cross-talk between various events present in the
wavefield. Furthermore, cross-talk can be attenuated by selecting
windows with different shapes, for example Gaussian or
exponentially-decaying. Therefore, we could in principle
define the transformation in equation 4 more generally as
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(6) |
where and are weighting functions which could represent
Gaussian, boxcar or any other local functions (Artman 2011, personal
communication). For simplicity, in all examples presented in this
paper, the space and time windows are rectangular with no tapering and
the size is selected assuming that micro-earthquakes occur
sufficiently sparse, i.e. the various sources are located at least
twice as far in space and time relative to the wavenumber and
frequency of the considered seismic event. Typical window sizes used
here are grid points in space and grid points in
time.
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| Micro-earthquake monitoring with sparsely-sampled data | |
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Next: Example
Up: Micro-earthquake monitoring with sparsely-sampled
Previous: Conventional imaging condition
2013-08-29