Correlation-based criterion

The usual approach to implement the imaging condition involves the zero-lag correlation between the reverse-propagated recorded wavefield and a function of the reverse-propagated modeled wavefield. The time correlation may also involve a weighting function (or a covariant operator) and a normalization factor. The general form is  

$$R(x,z;x_s) \; = \; {\int \; W(x,z,t;x_s) \; \phi^r(x,z,t;x_s... ... F[\phi^s(x,z,t;x_s)] \; dt \over \int \; W(x,z,t;x_s) \; dt}.$$ (9)

This criterion is obviously included in the category represented by (7), and implicitly requires the use of a smooth background velocity in the wavefield-extrapolation step. Ray-wavefield hybrid methods can be viewed as a particular case of (9), with $W=\delta(t-t_s)$,where t_s=t_s(x,z;x_s) is the ray-theoretical traveltime.

Jacobs (1982) compared the implementation of three different forms of equation (9), in a pre-stack profile migration in the $\omega-x$ domain: His conclusion was that R₃, though theoretically more correct for estimating the reflection coefficient, was too noise sensitive to be used in that migration scheme. There are two contradictory aspects between the imaging condition and the imaging criterion: while the former usually involves division by the modeled (incident) wavefield, the latter usually requires multiplication by the same wavefield to assure that only those estimations where the incident wavefield has significant energy will contribute effectively to the final expectation.

The implementation of the P-P reflectivity imaging-condition stated in equation 2 with the correlation criterion (9) uses the following definitions:

$$W(x,z,t;x_s) \;=\; [\phi^s]^2(x,z,t;x_s),$$

$$E(x,z;x_s) \;=\; \int \; W(x,z,t;x_s) \; dt$$

and

$$F[\phi^s(x,z,t;x_s)] \;=\; {1 \over \phi^s(x,z,t;x_s)},$$

which results in

$$\begin{eqnarray} R(x,z;x_s) & = & {\int \phi^r(x,z,t;x_s) \phi^s(x,z,t;x_s) \; d... ...,t;x_s) \, dt \over E_{cut}(x,z;x_s)}, \;\;\;\; \mbox{elsewhere,}\end{eqnarray}$$ (10)

and the local Snell parameter image as  

$$\check{p}(x,z;x_s) \; = \; {\int \; \mid \phi^r(x,z,t;x_s) \... ...int \; \mid \phi^r(x,z,t;x_s) \; \phi^s(x,z,t;x_s) \mid \, dt},$$ (11)

where $\tilde{p}(x,z,t;x_s)$ is computed from the two wavefields $\phi^r$and $\phi^s$, as described in appendix A.

In the above equations, E_cut(x,z;x_s) is a smooth function that should approximate a fraction of the inverse of the spherical divergence. Equation (10) gives the correct estimation of the reflection coefficient in the regions well illuminated by the source ($E(x,z;x_s) \geq E_{cut}(x,z;x_s)$) and a damped estimation in the dim regions.

 

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Figure 1 Elastic model used to generate the synthetic profiles that are used to illustrate the different imaging criteria. Only C₁₁ is represented in the figure. The model is isotropic, that is, C₁₁=C₃₃ for all layers, and the first layer is water (C₅₅=0).

Figures  and  show two images obtained with equations (10 and 11). The difference between these figures is that a smooth background model was used in the extrapolation step for Figure  and the original discontinuous model was used in the extrapolation step for Figure . The source is located at position 800 meters and the receivers extends from 980 meters to 2480 meters.

 

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Figure 2 (a) Image of the P-P reflection coefficient for a single synthetic shot profile, using the correlation criterion and a smooth background model. (b) Imaging of the local Snell parameter associated with the reflection-coefficient-imaging in a.

 

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Figure 3 (a) Image of the P-P reflection coefficient for a single synthetic shot profile, using the correlation criterion and without smoothing the background model. (b) Imaging of the local Snell parameter associated with the reflection-coefficient-imaging in (a).

Using a smooth background velocity avoids the contamination of the image by false reflectors originated by reflections in the extrapolation stage. In addition, the two wavefields involved in the migration correspond to unmixed modes (incident and reflected). One disadvantage is that transmission/conversion losses in the interfaces above the target are not compensated for in the extrapolation step. Another disadvantage is the loss in resolution caused by the higher dispersion in the wavefield extrapolation when a smooth background model is used.

When a non-smoothed background model is used in the extrapolation step the upward propagating modeled field and the downward propagating recorded field can be left out of the imaging computation, but the secondary reflections generated by them will generate false events in the final imaging. Also, at each interface (where the imaging computation is relevant) the two wavefields involved in the migration correspond to a superposition of several modes (incident, reflected, transmitted and converted) resulting in a contradictory application of the imaging principle. Nevertheless, there are two positive aspects in the use of a discontinuous model. First, the transmission/conversion losses are correctly compensated (at least for the incident wavefield $\phi^s$.), and second, the small dispersion results in a superior resolution.

Figure  shows the images obtained for the other modes using the correlation criterion implementation of equations (4), (5), and (6). In this figure, both -a and -b refer to the PS reflectivity but while -a was obtained using the true model, -b was generated using a smoothed model for P velocities and discontinuous (true) for S velocities. The other two images, which correspond to SP (-c) and SS (-d) modes, were also obtained using the true velocity model. All the mode-converted images obtained with a smoothed velocity model were completely unreliable because the mode-conversion dynamics of smooth models is notably different from the mode-conversion dynamics of discontinuous media. Even when the discontinuous model is used we can see false events which are not related to the generation of multiples as in the PP case.

Although the ocean floor interface should be absent from all the converted-mode images we can see from the figure that it is actually present in all of them. The reason is that the reverse-propagating P wavefield is partially converted to S in the liquid-solid boundary and the correlation of this converted wavefield with the modeled wavefield (which is also also partially converted to S) produces the false images. These false images of the ocean floor are actually deeper than the PP images because they are formed below the boundary. There are however some positive aspects in these converted-mode images. As expected the regions where the propagation-direction of the wavefield is nearly normal to the interface are not imaged, while some strong reflections can be observed at regions where the incidence angle is large.

 

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Figure 4 (a) Images of the P-S, S-P, and S-S reflection coefficients for the same synthetic gather of figure . (a) P-S reflectivity using the unsmoothed velocity model. (b) P-S reflectivity using a smoothed P velocity and the unsmoothed S velocity. (c) S-P reflectivity using the unsmoothed velocity model. (d) S-S reflectivity using the unsmoothed velocity model.