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Next: Field data example Up: Sava and Biondi: WEMVA Previous: Differential image perturbation

Synthetic example

We demonstrate the method on a synthetic example consisting of several dipping reflectors embedded in laterally varying slowness. Figure [*] shows from top to bottom the correct slowness model, the stacked reflectivity model and a few selected angle-gathers. We use this model to create a synthetic dataset.

 
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Figure 1
Correct model: slowness (top), stacked image (middle) and selected angle-gathers (bottom).
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Figure [*] shows from top to bottom the background slowness model, the stacked reflectivity and a few selected angle-gathers. Since we do not use the correct slowness, the angle gathers are not flat and the image is distorted.

Figure [*] shows from top to bottom the slowness perturbation between the true and the background slowness models, and the image perturbation created using the forward linear WEMVA operator: the stacked image in the middle panel and a few selected angle-gathers (bottom). This image is, by definition, consistent with the Born approximation. In practice, however, we need to go backward and compute a slowness perturbation from an image perturbation.

The problem with the Born approximation is illustrated in Figure [*]. The image perturbation obtained as a difference between the background image and an improved version of it is presented in the bottom two panels. The phase difference between corresponding events is larger than a fraction of the wavelet, and clearly violates the Born approximation. The inverted slowness anomaly (top panel) shows the characteristic sign changes usually seen in wave-equation tomography when the limits of the Born approximation are violated.

 
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Figure 2
Background model for WEMVA: slowness (top), stacked image (middle) and selected angle-gathers (bottom).
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Figure 3
Image perturbation by the forward WEMVA operator: slowness perturbation (top), stacked image (middle) and selected angle-gathers (bottom).
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Figure [*] illustrates our method for computing the image perturbation from the background data. We run residual migration as indicated in the preceding section and then pick at every location in the image the best velocity ratio $\rho$which corresponds to flat gathers. We also compute and image derivative according to Equation (12) without the $\Delta \rho$ scaling. Figure [*] shows the stacked image derivative and a few selected angle-gathers. The shape of this image is similar to that of the background image, with some phase and amplitude differences introduced by the derivative process.

 
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Figure 4
Difference image perturbation: inverted slowness perturbation (top), stacked image (middle) and selected angle-gathers (bottom).
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Figure 5
Differential image perturbation: picked $\Delta \rho$ map (top), stacked image differential (middle) and selected angle-gathers (bottom).
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We use Equation (12) and the $\Delta \rho$ weight (Figure [*]) to create the differential image perturbation (Figure [*]). This image perturbation is comparable in shape and magnitude to the ideal perturbation (Figure [*]). This indicates that we have succeeded in computing from the background image an image perturbation which is consistent with the Born approximation, and, therefore, we can use the linearized WEMVA operator without the danger of divergence due to images going out of phase.

 
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Figure 6
Differential image perturbation: inverted slowness perturbation (top), stacked image (middle) and selected angle-gathers (bottom).
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Figure 7
Image perturbation comparison: image perturbation computed by the forward WEMVA operator (top), image perturbation computed as a difference between the background image and a residually migrated one, which violates the Born approximation (middle), image perturbation computed by our differential method (bottom).
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For comparison, Figure [*] shows, from top to bottom, the image perturbations computed with the forward WEMVA operator, the one computed as a simple difference between the background image and an improved version of it, and the one computed by our differential procedure. We observe in the middle panel events which are largely out of phase, indicating that we cannot use the Born linearization. In contrast, the differential image perturbation is fully consistent with the one computed by the forward operator.

We use the image perturbation depicted in the bottom two panels of Figure [*] to invert for the corresponding slowness perturbation. We use 10 linear iterations for this example, with only one non-linear iteration.

Finally, Figure [*] shows from top to bottom the updated slowness model, and the updated staked image in the middle panel and a few selected angle-gathers in the bottom panel, which are much flatter than the ones of the background image (Figure [*]).

The key to the success of WEMVA inversion is in the appropriate definition of the image perturbation. The linear WEMVA operator does not account for reflector movement away from the reference image. Therefore, if the updated image incorporates such movement, we risk breaking the Born limitations quite easily, as illustrated in Figure [*]. The differential image perturbation operator, however, does not pose such a risk, since we construct the image perturbation on top of the reference image, in compliance with the Born approximation.

 
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Figure 8
Estimated model after WEMVA: slowness (top), stacked image (middle) and selected angle-gathers (bottom).
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next up previous print clean
Next: Field data example Up: Sava and Biondi: WEMVA Previous: Differential image perturbation
Stanford Exploration Project
7/8/2003