Interferometric imaging condition for wave-equation migration

Multi-offset interferometric imaging condition

The imaging procedure in equations 5-8 can be generalized for imaging prestack (multi-offset) data (Figure 1(b)). The conventional imaging procedure for this type of data consists of two steps (Claerbout, 1985): wavefield simulation from the source location to the image coordinates ${ \mathbf{y} }$ and wavefield reconstruction at image coordinates ${ \mathbf{y} }$ from data recorded at receiver coordinates ${ \mathbf{x} }$, followed by an imaging condition evaluating the match between the simulated and reconstructed wavefields.

Let $U_{S} \left ( { \mathbf{y} } , { t } \right)$ be the source wavefield constructed from the location of the seismic source and $U_{R} \left ( { \mathbf{y} } , { t } \right)$ the receiver wavefield reconstructed from the receiver locations. A conventional imaging procedure produces a seismic image as the zero-lag of the time cross-correlation between the source and receiver wavefields. Mathematically, we can represent this operation as

$$R_{} \left ( { \mathbf{y} } \right) = {\int\limits_ { t } \!... ... t } \right) U_{R} \left ( { \mathbf{y} } , { t } \right) \;,$$ (9)

where $R_{} \left ( { \mathbf{y} } \right)$ represents the seismic image for a particular seismic experiment at coordinates ${ \mathbf{y} }$. When multiple seismic experiments are processed, a complete image is obtained by summation of the images constructed for individual experiments. The actual reconstruction methods used to produce the wavefields $U_{S} \left ( { \mathbf{y} } , { t } \right)$ and $U_{R} \left ( { \mathbf{y} } , { t } \right)$ are irrelevant for the present discussion. As in the zero-offset/exploding reflector case, we use time-domain finite-difference solutions to the acoustic wave-equation, but any other reconstruction technique can be applied without changing the imaging approach.

When imaging in random media, the data recorded at the surface incorporates phase delays caused by the velocity variations encountered while waves propagate in the subsurface. In a typical seismic experiment, random phase delays accumulate both on the way from the source to the reflectors, as well as on the way from the reflectors to the receivers. Therefore, the receiver wavefield reconstructed using the background velocity model is characterized by random fluctuations, similar to the ones seen for wavefields reconstructed in the zero-offset situation. In contrast, the source wavefield is simulated in the background medium from a known source position and, therefore, it is not affected by random fluctuations. However, the zero-lag of the cross-correlations between the source wavefields (without random fluctuations) and the receiver wavefield (with random fluctuations), still generates image artifacts similar to the ones encountered in the zero-offset case.

Statistically stable imaging using pseudo WDFs can be obtained in this case, too. What we need to do is attenuate the phase errors in the reconstructed receiver wavefield and then apply a conventional imaging condition. Therefore, a multi-offset interferometric imaging condition can be formulated as

$$R_{} \left ( { \mathbf{y} } \right) = {\int\limits_ { t } \!... ... t } \right) W_{R} \left ( { \mathbf{y} } , { t } \right) \;,$$ (10)

where $W_{R} \left ( { \mathbf{y} } , { t } \right)$ represents the pseudo WDF of the receiver wavefield $U_{R} \left ( { \mathbf{y} } , { t } \right)$ which can be constructed, in principle, either with parametrization relative to data coordinates, according to equations 4-6, or relative to image coordinates, according to equation 5. Of course, our choice is to use image-space parametrization for computational efficiency reasons.

Interferometric imaging condition for wave-equation migration