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.
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 which corresponds to flat gathers. We also compute and image derivative according to Equation (12) without the 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.
We use Equation (12) and the 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.
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.