Figure 2(a) shows the modeled data from the velocity-stack panel obtained using the conventional CG algorithm for LS solution. The inversion was obtained with 30 iterations and the same number of iteration was used for all the other examples including the real data cases. We can clearly see the limit of L^{2}-norm minimization. The noise with Gaussian statistics is removed quite well, but some spurious events are generated around the bursty noise spikes and noisy trace. Figure 2(b) shows the modeled data from the velocity-stack panel obtained using the IRLS algorithm in an L^{1}-norm sense. In Figure 2(b) we can see the robustness of L^{1}-norm minimization. The nursty noise is reduced significantly, but the removal of the background noise seems to be worse than the result of L^{2}-norm minization.
Figure 2(c) shows the modeled data from the velocity-stack panel obtained using the CGG method, with the iteratively reweighted residual in an L^{1}-norm sense. The result is comparable to the IRLS inversion ( Figure 2(b)). This tells us that guiding the gradient vector toward the L^{1}-norm gradient gives a solution similar to the L^{1}-norm solution with the IRLS method. Figure 2(d) shows the modeled data from the velocity-stack panel obtained using the CGG method with iteratively reweighted gradient as follows:
that is the diagonal element of the i-th iteration weighting matrix is the absolute value of the model vector from the previous iteration, raised to the power 1.5. The result shows that the Gaussian background noise and the very noisy trace are better removed than with any of the L^{1}-norm approaches, either in CGG or IRLS. However, some spurious events around the bursty spikes still exist. This is because of the high amplitude of the noise. Since the weight is dependent on the amplitude of the model vector, and because the high amplitude in the CMP gather is also mapped to a high amplitude in the velocity-stack panel, the bursty noise with high amplitude would have higher weighting than the noise with low amplitude. However, this kind of artifact would easily be removed if the bursty noise had an amplitude similar to the rest of signal, which is the case when AGC (automatic gain control) is applied to the data.Figure 3 shows the modeled data obtained using the CGG algorithm with both residual and gradient weightings. In this case, the result looks like modeled data without any noise, because the bursty noise is reduced with residual weighting (L^{1}-norm criteria), and background noise is removed with gradient weighting.
fig3
Figure 3 Remodeled data from the CGG method, with iteratively reweighted residual and gradient together. |
The velocity stacks obtained from the various inversions are shown in Figure 4. From left to right, the velocity-stack panels correspond to the results from LS inversion, IRLS inversion, CGG with residual weighting, CGG with gradient weighting, and CGG with both residual and gradient weighting. From these velocity-stack panels, we can deduce why different inversion methods were successful with the different noise styles. If we want an application that distinguishes signals in the model space, we can see that gradient weighting is the preferred method, because it gives a more parsimonious representation than the others.