Interferometric imaging condition for wave-equation migration

Wigner distribution functions

One possible way to address the problem of random fluctuations in reconstructed wavefields is to use Wigner distribution functions (Wigner, 1932) to pre-process the wavefields prior to the application of the imaging condition. Appendix C provides a brief introduction for readers unfamiliar with Wigner distribution functions. More details about this topic are presented by Cohen (1995).

Wigner distribution functions (WDF) are bi-linear representations of multi-dimensional signals defined in phase space, i.e. they depend simultaneously on position-wavenumber ( ${ \mathbf{y} } - { \mathbf{k} }$) and time-frequency ( ${ t } - {\omega}$). Wigner (1932) developed these concepts in the context of quantum physics as probability functions for the simultaneous description of coordinates and momenta of a given wave function. WDFs were introduced to signal processing by Ville (1948) and have since found many applications in signal and image processing, speech recognition, optics, etc.

A variation of WDFs, called pseudo Wigner distribution functions are constructed using small windows localized in space and/or time (Appendix C). Pseudo WDFs are simple transformations with efficient application to multi-dimensional signals. In this paper, we apply the pseudo WDF transformation to multi-dimensional seismic wavefields obtained by reconstruction from recorded seismic data. We use pseudo WDFs for decomposition and filtering of extrapolated space-time signals as a function of their local wavenumber-frequency. In particular, pseudo WDFs can filter reconstructed wavefields to retain their coherent components by removing high-frequency noise associated with random fluctuations in the wavefields due to random fluctuations in the model.

The idea for our method is simple: instead of imaging the reconstructed wavefields directly, we first filter them using pseudo WDFs to attenuate the random phase noise, and then proceed to imaging using a conventional or an extended imaging conditions. Wavefield filtering occurs during the application of the zero-frequency end-member of the pseudo WDF transformation, which reduces the random character of the field. For the rest of the paper, we use the abbreviation WDF to denote this special case of pseudo Wigner distribution functions, and not its general form.

As we described earlier, we can distinguish two options. The first option is to use wavefield parametrization as a function of data coordinates ${ \mathbf{x} }$. In this case, we can write the pseudo WDF of the reconstructed wavefield $V\left ( { \mathbf{x} } , { \mathbf{y} } , { t } \right)$ as

$$V_{x} \left ( { \mathbf{x} } , { \mathbf{y} } , { t } \right... ...{ \textcolor{darkgreen}{ {{ t }_h}} }{2} \right) \right]\;,$$ (4)

where $\textcolor{blue} { {{ \mathbf{x} }_h}}$ and $\textcolor{darkgreen}{ {{ t }_h}}$ are variables spanning space and time intervals of total extent $X$ and $T$, respectively. For 3D surface acquisition geometry, the 2D variable $\textcolor{blue} { {{ \mathbf{x} }_h}}$ is defined on the acquisition surface. The second option is to use wavefield parametrization as a function of image coordinates ${ \mathbf{y} }$. In this case, we can write the pseudo WDF of the reconstructed wavefield $U\left ( { \mathbf{y} } , { t } \right)$ as

$$W_{y} \left ( { \mathbf{y} } , { t } \right) = {\int\limits_... ... } +\frac{ \textcolor{darkgreen}{ {{ t }_h}} }{2} \right) \;,$$ (5)

where $\textcolor{red} { {{ \mathbf{y} }_h}}$ and $\textcolor{darkgreen}{ {{ t }_h}}$ are variables spanning space and time intervals of total extent $Y$ and $T$, respectively. For 3D surface acquisition geometry, the 3D variable $\textcolor{red} { {{ \mathbf{y} }_h}}$ is defined around image positions.

For the examples used in this section, we employ $41$ grid points for the interval $X$ centered around a particular receiver position, $5 \times 5$ grid points for the interval $Y$ centered around a particular image point, and $21$ grid points for the interval $T$ centered around a particular time. These parameters are not necessarily optimal for the transformation, since they characterize the local WDF windows and depend on the specific implementation of the pseudo WDF transformation. The main criterion used for selecting the size of the space-time window for the pseudo WDF transformation is that of avoiding cross-talk between nearby events, e.g. reflections. Finding the optimal size of this window is an important consideration for our method, although its complete treatment falls outside the scope of the current paper and we leave it for future research. Preliminary results on optimal window selection are discussed by Borcea et al. (2006a).

uxx1,wxx1
Figure 4. Reconstructed seismic wavefield as a function of data coordinates (a) and its pseudo Wigner distribution function (b) computed as a function of data coordinates ${ \mathbf{x} }$ and time ${ t }$. The wavefield is reconstructed using the background model from data simulated in the random model.

uyy1,wyy1
Figure 5. Reconstructed seismic wavefield as a function of image coordinates (a) and its pseudo Wigner distribution function (b) computed as a function of image coordinates ${ \mathbf{y} }$ and time ${ t }$.The wavefield is reconstructed using the background model from data modeled in the random model.

Figure 4(b) depicts the results of applying the pseudo WDF transformation to the reconstructed wavefield in Figure 4(a). For the case of modeling in the random model and reconstruction in the background model, the pseudo WDF attenuates the random character of the wavefield significantly, Figure 4(b). The random character of the reconstructed wavefield is reduced and the main events cluster more closely around time ${ t } =0$. Similarly, Figure 5(b) depicts the results of applying the pseudo WDF transformation to the reconstructed wavefields in Figure 5(a). For the case of modeling in the random model and reconstruction in the background model, the pseudo WDF also attenuates the random character of the wavefield significantly, Figure 5(b). The random character of the reconstructed wavefields is also reduced and the main events focus at the correct image location at time ${ t } =0$.

Interferometric imaging condition for wave-equation migration