Figure 2

Figure shows the result of the inversion when the norm is used for the noise-free data. On the left the velocity space is displayed with its five events. The focusing is not perfect and some artifacts appear Sacchi and Ulrych (1995). The middle panel shows the remodeled data after inversion and the right panel displays the residual (difference between the input and remodeled data). I conclude that the inversion reached a minimum since no coherent energy is left in the residual: the data fitting is very good. Because the input data have gaussian statistics, the performance of the least-squares inversion was expected.

Figure 3

I now use the same inversion scheme with the norm but with the ``contaminated'' CMP gather (Figure c). Notice that no regularization was applied with the least-squares method. The final result is shown in Figure . In this case, as expected, the inversion creates a number of artifacts both in the model and data space. In the model space (Figure a), the spikes are mapped into curves whereas the hyperbola are mapped into nearly focused points. In the data space, after remodeling (Figure b), spikes are not remodeled correctly and are smearing on the neighboring traces. In Figure I use least-squares with a simple damping in the regularization. The model and data space are cleaner but the difference between the input and the remodeled data or residual is still large (comparing Figure c to Figure c.) In addition, artifacts in the inverted slowness field and the reconstructed data can be seen.

Figure 4

Figure 5

Figure displays the result of the inversion with the Huber norm. The outcome of the inversion is insensitive to the spiky events, like a pure norm misfit function. The residual (Figure c) exhibits the four spikes very clearly. This result demonstrates that the proposed algorithm, although not specifically designed to minimize the Huber function, converges to a satisfying solution. The next section shows inversion results with noisy field data.

Figure 6

5/5/2005