Examples

In this section, we demonstrate the method on a synthetic example. The reflectivity model (Figure 1) consists of two flat reflectors surrounding a set of reflectors dipping at 45 degrees.

 

rr Figure 1 Reflectivity model.

The background velocity (Figure 2) is characterized by strong lateral variation to provide a somewhat complex background wavefield.

 

s1 Figure 2 Background slowness model.

The image perturbation (Figure 3) consists of a rectangular block in the upper part of the section, which creates perturbations in the image both on the flat and on the dipping reflectors. We change the magnitude of this anomaly from small values that satisfy the Born approximation, to high values that clearly violate the linear assumptions.

 

ds Figure 3 Slowness perturbation.

We use the true slowness to model the data, and then we use the background slowness to obtain the reference image and the reference wavefield. From the slowness perturbation, we use the forward WEMVA operator to create the ideal image perturbation (Figure 4). Backprojection by the adjoint WEMVA operator gives the slowness update in Figure 5, and inversion gives the slowness in Figure 6.

 

di.wei.01 Figure 4 Ideal image perturbation obtained by the forward WEMVA operator.

 

bds.wei.01 Figure 5 Slowness backprojection obtained from the ideal image perturbation using the adjoint WEMVA operator.

 

ids.wei.01 Figure 6 Slowness inversion obtained from the ideal image perturbation using the WEMVA operator.

Figures 4 and 5 represent the benchmark against which we want to compare our new image perturbation method.

Figure 7 represents the image perturbation created using Equation (12) where we consider that $\Delta \gamma=1$. Of course, this is not our image perturbation, but it only gives us a rough idea of how we should pick the actual perturbation from after weighting with $\Delta \gamma$.As expected, the magnitude of this component of the image increases with depth.

 

di.ana.full Figure 7 Image perturbation produced by Equation (12). The magnitude of the perturbation increases with depth.

Next we multiply the image in Figure 7 with the $\Delta \gamma$ weight (Figure 8) and obtain an image perturbation (Figure 9) which is comparable in shape and magnitude with the ideal perturbation (Figure 4). Backprojection by the adjoint WEMVA operator gives the slowness update in Figure 10, and inversion gives the slowness in Figure 11.

 

mask Figure 8 $\Delta \gamma$ in Equation (12).

 

di.ana.01 Figure 9 Analytical image perturbation obtained by multiplication of the image in Figure 7 with the weight in Figure 8.

 

bds.ana.01 Figure 10 Slowness backprojection obtained from the analytical image perturbation using the adjoint WEMVA operator.

 

ids.ana.01 Figure 11 Slowness inversion obtained from the analytical image perturbation using the WEMVA operator.

Finally, Figures 12, 13, and 14 are comparisons of the various methods used to compute the image perturbation, and the corresponding slowness model. All figures are presented in landscape mode and have three columns and four rows.

• The first column contains the real part of the image perturbation ($\Delta \mathcal R$), the second column contains the slowness model ($\Delta s$) obtained using backprojection of the image perturbation, and the third column contains the inverted slowness model obtained from the same image perturbation.
• The first row represents $\Delta \mathcal R$ and $\Delta s$ computed using the WEMVA forward and adjoint operators, the second row represents $\Delta \mathcal R$ and $\Delta s$in the case in which we narrow down the angular aperture in the image perturbation, the third row represents $\Delta \mathcal R$ and $\Delta s$ of our new approach, and the fourth row represents $\Delta \mathcal R$ and $\Delta s$ obtained using the PSRM method.

Each of the Figures 12, 13, and 14 corresponds to a different magnitude of the slowness anomaly, everything else remaining unchanged. The main comparison is between rows 2 and 3 of the composite image. We can observe that as we increase the magnitude of the anomaly, the shapes of $\Delta \mathcal R$ and $\Delta s$ remain roughly unchanged, although the magnitudes of both the image and slowness perturbations increase. Since, by construction, we obey the Born approximation, we do not observe any out-of-phase effects.

In contrast, when we compare rows 3 and 4, that is our new method and the PSRM approach, we observe that, for the small anomalies, the image and slowness perturbations are very similar, but at high values of the anomaly, the PSRM $\Delta \mathcal R$ gets out of phase with respect to the background wavefield, and therefore the backprojected or inverted $\Delta s$ either cancels or changes sign altogether. This is exactly the kind of effect we are trying to avoid using our new approach.

 

block.01
Figure 12 View as 4 rows and 3 columns. The left column represents the image perturbations, the middle represents backprojection using the WEMVA operator, and the right column represents unconstrained inversion in 3 iterations using the WEMVA operator. The top row is using the ideal image perturbation, the second row is using the narrow aperture image perturbation, the third row is using our new approach, and the fourth row the PSRM approach.

 

block.06
Figure 13 View as 4 rows and 3 columns. The left column represents the image perturbations, the middle represents backprojection using the WEMVA operator, and the right column represents unconstrained inversion in 3 iterations using the WEMVA operator. The top row is using the ideal image perturbation, the second row is using the narrow aperture image perturbation, the third row is using our new approach, and the fourth row the PSRM approach.

 

block.11
Figure 14 View as 4 rows and 3 columns. The left column represents the image perturbations, the middle represents backprojection using the WEMVA operator, and the right column represents unconstrained inversion in 3 iterations using the WEMVA operator. The top row is using the ideal image perturbation, the second row is using the narrow aperture image perturbation, the third row is using our new approach, and the fourth row the PSRM approach.