Field data results

The proposed algorithm is now tested on a field data example. For this purpose, a shot gather from a land-data survey in the Middle East is selected. The trajectories of the events in Figure look ``hyperbolic'' enough to be inverted with our method. Note that in theory, the data should be sorted into CMP gathers before doing the inversion. This dataset is particularly interesting because it has amplitude anomalies at short offset and a low velocity coherent noise that is probably due to guided energy in the near-surface. I could attenuate the amplitude anomalies by applying an Automatic Gain Control (AGC) on the data before inversion. A better approach is by weighting the residual with a function mimicking AGC.

 

model2 Figure 7 The field data used for the inversion. Notice the amplitude anomalies at near offset and the time shift near offset 2 km.

This gather is first inverted with the $\ell^2$ norm without regularization (Figure ). The left panel displays the velocity domain obtained after the least-squares inversion. The main velocity event is masked with horizontal stripes coming from the short-offset amplitude anomalies. The reconstructed data (middle panel) show spurious noise at large offset and other inversion artifacts. I now show in Figure the result of the damped least-squares. The inverted slowness field is much cleaner, but the horizontal stripes remain. We also have the same velocity from the top to the bottom of Figure a. This shows that the data are contaminated with multiples generated in the near surface. Figure displays the inversion result with the Huber norm and demonstrates the robustness of this method. A very well focused velocity corridor is obtained as opposed to the $\ell^2$ result in Figure a. In addition, the horizontal stripes have disappeared.

The synthetic and field data examples demonstrate that the Huber norm can be an efficient alternative to the $\ell^2$ norm when outliers (or non-gaussian noise) are present in the data.

 

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Figure 8 The result of the inversion with the $\ell^2$ norm for the field data. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure ) and remodeled data. The horizontal stripes in the velocity panel are created by the amplitude anomalies at short offsets.

 

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Figure 9 The result of the inversion with the $\ell^2$ norm and regularization for the field data. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure ) and remodeled data. The model is much cleaner that in Figure a, but the horizontal events remain.

 

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Figure 10 The result of the robust inversion with the Huber norm for the field data. (a) Inverted slowness field. (b) Remodeled data. (c) Difference between the input (Figure ) and remodeled data.