Wave-equation tomography by beam focusing

Uniform velocity error

I compare the results obtained for the uniform velocity error case with the corresponding gradient obtained from conventional full-waveform inversion. To compute the search directions $\Delta{s}$ for full-waveform inversion I applied the following expression:

$$\Delta{s}= -\sum_{x_{s}} \sum_{x_{g}} {\frac {\partial{\widetilde... ...t},x_{g},x_{s};{s}_{0}\right) - {{P}_{D} }\left({t},x_{g},x_{s}\right) \right].$$ (A-26)

Figures 1-2 show the search direction obtained by applying expression 26. Panels a) show the velocity perturbations for each $z$ and $x$ of the model; panels b) show the velocity perturbation averaged over the horizontal direction. As expected, the search direction computed using expression 26 is pointing in the correct direction when the velocity error is small ($0.1 \%$ ), Figure 1. However, when the velocity error is large the search direction has the wrong polarity ($8.0 \%$ ), Figure 2.

Figure 2 illustrates the limitations of full-waveform inversion when applied to estimating the background velocity. I therefore applied the proposed method to the case with a large velocity error, to demonstrate that it overcomes these limitations of full-waveform inversion.

Figure 3 shows the derivatives of the objective function with respect to the moveout parameters computed using equations 16-17 and 18-19. The plot in panel a) displays the derivatives with respect to the local curvature, ${\mu _C}$ ; the plot in panel b) displays the derivatives with respect to the time shifts of the beam centers, ${\mu _{\theta }}$ . In both cases the derivatives are plotted as a function of the beam center coordinate, ${\overline {x}}$ , for the source location at $x_{s}=2.56$ km; that is, in the middle of the model. As visible from the figure, I clipped to zero the derivatives for both parameters outside of the $-1.7\; {\rm km} \leq{\overline{x}}\leq 1.7$ km range to avoid edge effects.

Figure 3a clearly shows that the local curvature is an appropriate parameters to measure the effects of large-scale velocity errors. In contrast, the alternating signs of the time shift derivatives shown in Figure 3b demonstrates that such perturbations are not well captured by the global moveout parameters.

These observations are confirmed by Figures 4 and 5. Figure 4 shows the search direction computed using equation 24. In contrast with the search direction computed with conventional waveform inversion (Figure 2), the search direction shown in Figure 4 has the correct polarity. It provides a good search direction, similar to the one provided by full-waveform inversion with the small velocity error that is shown in Figure 1.

The search direction obtained by computing the gradient of the global objective function using equation 25 is shown in Figure 5. It shows strong edge effects and its horizontal average (Figure 5b)) has the wrong polarity.

Beam-DC-DT-large-neg Figure 3. Derivatives of the objective functions with respect to the moveout parameters plotted as a function of beam center coordinate, ${\overline {x}}$ , for the source location at $x_{s}=2.56$ km: a) derivatives of ${J_{\rm Local}}$ with respect to to the local beam curvatures, ${\mu _C}$ , b) derivatives of ${J_{\rm Global}}$ with respect to the time shifts of the beam centers, ${\mu _{\theta }}$ .

Avg-Dir-Beam-large-neg Figure 4. Search direction computed using the gradient of the local objective function ${J_{\rm Local}}$ with a positive large ($8.0 \%$ ) uniform slowness error: a) slowness perturbations and b) slowness perturbations averaged over the horizontal direction.

Avg-Dir-Glob-Beam-large-neg Figure 5. Search direction computed using the gradient of the global objective function ${J_{\rm Global}}$ with a positive large ($8.0 \%$ ) uniform slowness error: a) slowness perturbations and b) slowness perturbations averaged over the horizontal direction.

VH-Avg-DSlow-panom Figure 6. The slowness error that was assumed for the background slowness model ${s}_{0}$ : a) slowness perturbations, b) slowness perturbations averaged over the horizontal direction, and c) slowness perturbations averaged over the vertical direction.

Wave-equation tomography by beam focusing