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Interferometric imaging condition

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 $W{ { \mathbf{y} } , { t } }$ is

\begin{displaymath}
W_{} \left ( { \mathbf{y} } , { t } \right) = {\int\limits_{...
...c{ {{ \mathbf{y} }_h} }{2}, { t } +\frac{ {{ t }_h} }{2}} \;,
\end{displaymath} (4)

where $Y$ and $T$ denote averaging windows in space and time, respectively. In general, $Y$ is three dimensional and $T$ is one dimensional. Then, the image $R_{IIC} \left ( { \mathbf{y} } \right)$ is obtained by extracting the time $ { t } =0$ from the pseudo WDF, $W_{} \left ( { \mathbf{y} } , { t } \right)$, of the wavefield $W{ { \mathbf{y} } , { t } }$:
\begin{displaymath}
R_{IIC} \left ( { \mathbf{y} } \right) = W_{} \left ( { \mathbf{y} } , { t } =0 \right) \;.
\end{displaymath} (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 $Y$ and $T$ 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

\begin{displaymath}
W_{} \left ( { \mathbf{y} } , { t } \right) =
\int d {{ t }...
...c{ {{ \mathbf{y} }_h} }{2}, { t } +\frac{ {{ t }_h} }{2}} \;,
\end{displaymath} (6)

where $W_T$ and $W_Y$ 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 $11$ grid points in space and $5$ grid points in time.


next up previous [pdf]

Next: Example Up: Micro-earthquake monitoring with sparsely-sampled Previous: Conventional imaging condition

2013-08-29