I test my methodology on two synthetic 2-D data sets. One shown in Figure 2(a) is a two-layer model with one reflector being horizontal and the other dipping at . The velocity increases with depth: v(z) = 2000 + 0.3z, which is shown in Figure 1. To make the synthetic data set more realistic, some random noise has also been added. Then I replace approximately of the traces in the offset dimension with zeros. The incomplete and sparse data set is shown in Figure 2(b). Then I perform DSR migration on both data sets to generate the SODCIGs; the corresponding migrated image cubes are shown in Figure 3. Comparing Figure 3(a) with Figure 3(b), we can see that even with the complete data set (Figure 2(a)), the SODCIGs suffer from the amplitude smearing effects caused by the offset truncation. The situation gets worse as the offset coverage is further reduced; there are severe amplitude smearing and aliasing artifacts in the SODCIGs as shown in Figure 3(b), and because of the interference of these artifacts in the offset domain, the resolution of the migrated image (i.e. offset=0) is also degraded. The effect is more obvious if we transform the SODCIGs into the ADCIGs, which are shown in Figure 4; there are some gaps in the middle of the ADCIGs (Figure 4(b)) obtained by migrating the incomplete data set, indicating that there are some illumination problems.
Figure 1 The velocity model for the two-layer model.
From this simple experiment, we intuitively understand that the amplitude smearing in the SODCIGs is another representation of poor illumination and that the more energy smearing we see in the SODCIGs, the more severe the illumination problem must be. Therefore, if we could make the energy more concentrated at zero-offset and penalize the energy at nonzero-offset, we would compensate for the illumination problem and fill the holes in the ADCIGs. To achieve this purpose, I first approximate the weighted Hessian matrix with equation (41), then solve the inversion problem based on the fitting goals (45) and (46). The reference image or is chosen to be the migrated image cube of the incomplete data, which is shown in Figure 2(b). The weight is created by demigrating and then migrating the demigrated image again. The mask weight is shown in Figure 5. As I apply the sparseness constraint along the offset dimension depth-by-depth and CMP-by-CMP, it would be inappropriate to use a global parameter to control the sparseness; therefore I apply locally, choosing for its value the mean value of the current offset vector. The final inversion result is shown in Figure 6(a); for comparison, Figure 6(b) shows the migration result. Figure 7 illustrates one single trace located at CMP= meters and offset= meters, Figure 7(a) is the result by migration, while Figure 7(b) is the result by inversion, where both (a) and (b) are normalized to compare their relative amplitude ratios. From the results we can clearly see that the DSO regularization term perfectly eliminates the energy at non-zero offset. The sparseness constraint also successfully penalizes weak amplitudes and consequently improves the resolution of the image. Figure 8 shows the comparison of ADCIGs between migration and inversion, where, as expected, the inversion result in Figure 8(a) fills the illumination gaps presented in Figure 8(b).
Figure 5 The computed mask weight from Figure 2(b). Black stands for ones, while grey stands for zeros.
The model with two reflectors in the previous example is simple. To test whether the inversion scheme works for complex models, I apply it to the Marmousi model, which is shown in Figure 9(a), again with about of the traces in the offset dimension replaced with zeros. The computed mask weight is shown in Figure 9(b). As before, I use the migrated image cube as the reference image cube for computing the weighting matrices and . The parameter is also chosen to be the mean value of the current offset vector. Because there are no good suggestions for the parameter ,it is chosen by trial and error to get a satisfactory result. Since I use only one reference velocity (the average between the maximum and the minimum velocities at each depth step) for the DSR-SSF algorithm, some steeply dipping faults are not well imaged, and because of the inaccuracy of the reference velocity, some locations are mispositioned, indicating there should be some residual moveout in both SODCIGs and ADCIGs.
The final inversion result is shown in Figure10 (b); for comparison, Figure10(a) is the migration result. By using the approximated inversion scheme, we suppress the weak and incoherent noise and obtain a much cleaner result, while also improving the resulotion to some extent. This is more obvious if we extract a single trace from the migration result and the inversion result to compare their relative amplitudes. Figure 11 shows the extracted trace located at CMP=4 km, offset= km, while Figure 12 shows the extracted trace located at CMP=7.5 km, offset= km. In both figures, (a) is obtained from the migration result, while (b) is obtained from the inversion result. From Figure 11 and Figure 12, we can see that small amplitudes and the sidelobes of the wavelets are penalized by the inversion scheme and the inversion result yields an image with higher resolution. But also notice that some weak reflections which are presented in the migration result are attenuated in the inversion result.
Figure 13 illustrates the SODCIGs for two different locations; (a) and (c) are the SODCIGs at CMP=4 km and CMP=7.5 km respectively obtained from the migration result, while (b) and (d) show the SODCIGs at the same CMP locations obtained from the inversion result. Because of the DSO regularization term in the inversion scheme, events that are far from zero-offset locations are penalized, making the energy more concentrated at zero-offset. The ADCIGs at the corresponding locations shown in Figure 14 explain this further, with the ADCIGs (Figure 14(b) and (d)) from the inversion result smoothed across angles and the illumination holes present in (a) and (c) filled in to some degree.
As mentioned above, because of the inaccuracy of the reference velocity, there are still some residual moveouts at some locations in both SODCIGs and ADCIGs, as seen in Figure 13(a) and Figure 14(a). One nice thing to see is by choosing a proper trade-off parameter , the proposed inversion scheme can successfully preserve the residual moveouts both in SODCIGs and ADCIGs, as shown in Figure 13(b) and Figure 14(b). The angle gathers even get cleaner, which makes it much easier to estimate the residual moveouts. Therefore, this approximated inversion scheme may have the potential to improve the accuracy of residual moveout estimation, and consequently improve velocity estimation results. However, this still needs further investigation.