Examples

We demonstrate the technique outlined in the preceding sections using a synthetic example. The model (Figure 4) consists of a body of high velocity incorporated in a background with strong but smooth lateral velocity variation.

 

model
Figure 4 Synthetic model. Reflectivity model (top left) and a few angle-gathers corresponding to the vertical grid in the upper plot (bottom left). Background slowness model (top right) and slowness perturbation (bottom right).

Our examples show the results of inversion for a regularized problem symbolically sumarized by the fitting goals:

$$\begin{eqnarray} \Delta R&\approx& {\bf L}\Delta s \nonumber \\ 0 &\approx& \AA \Delta s,\end{eqnarray}$$ (45)

where $\Delta R$ is the image perturbation, $\Delta s$ is the corresponding slowness perturbation, ${\bf L}$ is one of the linearized WEMVA operators and $\AA$ is a roughening operator, an isotropic Laplacian for our examples. After preconditioning Claerbout (1999), our fitting goals become

$$\begin{eqnarray} \Delta R&\approx& {\bf L}\AA^{-1} \Delta p, \nonumber \\ 0 &\approx& \Delta p\end{eqnarray}$$ (46)

where $\Delta p$ represents the preconditioned $\Delta s$.

We also note that since the operator ${\bf L}$ is large, similar in size to a migration operator, we cannot implement it in-core, and therefore we have to use out-of-core optimization Sava (2001).

For our experiments, we generate two kinds of image perturbations.

• The first kind is created from a given slowness perturbation $\Delta s$ using the linear operator in Equation (15). We refer to this type of image perturbation as linear , since it corresponds to the linearized Born operator. This type of image perturbation cannot be obtained in real applications, but serves as a reference when we investigate the Born approximation.
• The second kind is created by taking the difference of two images created using two slowness models ($\Delta R= R - R_o$). We refer to this type of image perturbation as non-linear , since it corresponds to the non-linear relation in Equation (7).

We analyze several examples where we change the magnitude of the slowness anomaly, but not its shape. We choose to test various magnitudes for the anomaly from $1\%$ to $50\%$ of the background slowness.

Figures 5, 7, 9, 11 show the image perturbations created by the slowness anomalies for the various levels of perturbation. In each figure, the left panels present the linear case, and the right panels the non-linear case. The top panels depict the stacked sections, and the bottom panels a few representative image gathers in the angle-domain Sava and Fomel (2000) corresponding to the locations of the vertical lines in the upper panels. For small values of the slowness perturbations, the two images should be similar, but for larger values we should see the image perturbation reaching and eventually breaking the Born approximation.

Figures 6, 8, 10, 12 present the results of inversion of the non-linear $\Delta R$ using the three WEMVA operators presented in the preceding section: the explicit (Born) operator (top), the bilinear operator (middle), and the implicit operator (bottom).

For the case of the small slowness perturbation ($1\%$), the linear and non-linear image perturbations are very similar, as seen in Figure (5). The corresponding slowness anomaly obtained by inversion is well focused, confirming that, for this case, even the Born approximation is satisfactory, as suggested by the theory.

The larger anomaly of $5\%$ of the background slowness shows the serious signs of breakdown for the Born approximation. For the case of the even larger slowness perturbation ($20\%$), the linear and non-linear image perturbations are not that similar anymore, indicating that we have already violated the limits of the Born approximation (Figure 9). Consequently, the inversion from the non-linear image perturbation using the Born operator blows-up. However, the WEMVA operators employing the bilinear and implicit approximations are still well-behaved, although the shape of the anomaly is slightly modified.

The case of the largest slowness anomaly ($40\%$), bring us closer to the limits of both the bilinear and implicit approximations. Although neither has blown-up yet, the shape of the anomalies is somewhat altered.