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# Simple WEMVA example

The first example concerns linearized image perturbations computed for prestack images. I use a simple model with flat reflectors and constant velocity. The image perturbation methodology is identical to the one outlined in wemva. The main point of this example is to illustrate the WEMVA methodology in a situation when the requirements of the first-order Born approximation are clearly violated. In this case, the slowness perturbation is of the background slowness.

 WEPf.imag Figure 1 Comparison of common image gathers for image perturbations obtained as a difference between two migrated images (c), as the result of the forward WEMVA operator applied to the known slowness perturbation (d), and as the result of the differential image perturbation operator applied to the background image (e). Panel (a) depicts the background image corresponding to the background slowness, and panel (b) depicts an improved image obtained from the background image using residual migration.

WEPf.imag shows representative common image gathers in the angle-domain (adcig) for the background image (a), the correct image (b), the image perturbation obtained as a difference of the two images (c), the image perturbation obtained using the forward WEMVA operator (d), and the linearized image perturbation (e). Panels (d) and (e) are identical within numeric precision, indicating that the WEMVA methodology can successfully be employed to create correct image perturbations well beyond the limits of the first-order Born approximation.

Next, I apply the wave-equation migration velocity analysis algorithm (wemva) to the example in WEPf.imag. First, I compute the background wavefield represented by the background image (WEPf.imaga and zflat.backa). Next, I compute the linearized image perturbation, shown in zflat.perta (stack) and in WEPf.image (angle gather from the middle of the image).

From this image perturbation, I invert for the slowness perturbation (zflat.pertb). I stop the inversion after 19 linear iterations when the data residual has stopped decreasing (zflat.pertc). The slowness updates occur in the upper half of the model. Since no reflectors exist in the bottom part of the model, no slowness update is computed for this region.

Finally, I remigrate the data using the updated slowness and obtain the image in zflat.backb. For comparison, zflat.backc depicts the image obtained after migration with the correct slowness. The two images are identical in the upper half where we have updated the slowness model. Further updates to the model would require more non-linear iterations.

zflat.back
Figure 2
WEMVA applied to a simple model with flat reflectors. The background image (a), the image updated after one non-linear iteration (b), and the image computed with the correct slowness (c).

zflat.pert
Figure 3
WEMVA applied to a simple model with flat reflectors. The zero-offset of the image perturbation (a), the slowness update after the first non-linear iteration (b), and the convergence curve of the first linear iterations (c).

The main point of this simple example is to illustrate that linearized image perturbations are capable of handling large velocity perturbations. In this example, the velocity change is so large that corresponding reflectors are not even close to one another, let alone within a fraction of the wavelet. Straight image differences are not able to handle correctly even this simple example. However, linearized image perturbations handle such large perturbations correctly and the velocity updates correct the starting image. Next sections illustrate the WEMVA methodology with more complex examples.

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Stanford Exploration Project
11/4/2004