Synthetic data results

Figure displays the synthetic model. In Figure a I show the ideal velocity space with five events at different slownesses. In Figure b the five corresponding hyperbolas in the CMP domain are shown. Finally in Figure c, four spikes are added to the CMP gather in Figure b to make gaussian statistics unsuitable. The energy of the four spikes is five times the energy of the five hyperbolas. Our goal is to find the velocity field ${\bf m}$ that will best fit a CMP gather ${\bf d}$ via the HRT.

 

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Figure 2 Synthetic data used for the inversion. (a) The true velocity model represented in slowness. (b) The noise-free input data used for the inversion. (c) The same data but with noise added (four spikes).

Figure shows the result of the inversion when the $\ell^2$ 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.

 

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Figure 3 Result of the inversion with the $\ell^2$ norm for the noise-free data. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure b) and remodeled data.

I now use the same inversion scheme with the $\ell^2$ 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 $\ell^2$ 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.

 

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Figure 4 Result of the inversion with the $\ell^2$ norm for the data with noise. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure c) and remodeled data. The four spikes create artifacts in both inverted model and reconstructed data space.

 

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Figure 5 Result of the inversion with the $\ell^2$ norm with damping for the data with noise. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure c) and remodeled data. The three panels are cleaner than in Figure but some artifacts remain, however.

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 $\ell^1$ 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.

 

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Figure 6 Result of the robust inversion with the Huber norm for the data with noise. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure c) and remodeled data. The Huber norm behaves like a pure $\ell^1$ norm since all artifacts have disappeared.