Land data example

Figure displays a CMP gather from a land acquisition. This gather is contaminated by a strong ground-roll with weak signal. The goal is to test the two proposed techniques with the same operators as in the preceding example, i.e., hyperbolic radon transform to obtain velocity information and a $5 \times 3$ PEF for the filtering and modeling of the noise.

 

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Figure 7 A CMP gather from a land survey. The ground-roll is very strong throughout the section.

By doing inversion without taking care of the coherent noise, the velocity panel in Figure a is obtained. The model space is contaminated with strong noise coming from the residual in Figure c. When noise is present in the data, least-squares with a simple damping can help to mitigate the noise effects. This was not done in this case, however. The reconstructed data in Figure b is relatively noise free, which indicates that most of the energy in the model space (Figure a) is in the null space of the operator.

 

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Figure 8 Result of least-squares inversion of the data in Figure . (a) Estimated model space. The noise level is too high to see any useful information. (b) The remodeled data. The low noise level in this panel indicates that most of the energy in (a) resides in the null space of the operator. (c) Data residual. The residual is not IID.

For the filtering and modeling techniques to work, a noise model is needed. As suggested in a preceding section, the noise model is estimated from the residual of least-squares inversion assuming that ${\bf A=I}$ (i.e., Figure c). From this noise model, a $5 \times 3$ PEF is estimated. Figure a displays the result of the inversion after 40 iterations of CG. The few hyperbolas present in the data are clearly mapped in the model space without any artifact. The remodeled data in Figure b show almost no ground-roll remaining. The unweighted residual in Figure c (${\bf Hm-d}$) contains only coherent noise and the residual in Figure d is almost IID: a little bit of ground-roll is still present.

 

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Figure 9 Inversion result with the filtering technique. (a) Estimated model. (b) Remodeled data. (c) Unweighted residual, showing the ground-roll only. (d) Residual. The residual is not exactly IID. Some coherent energy remains.

The modeling technique leads to similar results to the filtering technique, as shown in Figure . The main difference comes from the residual in Figure d. With the modeling method, the residual is smaller than with the filtering technique and is IID. As a final comparison between the two techniques, Figure shows how the objective function (normalized) decreases in both cases. The modeling technique has the best convergence properties.

 

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Figure 10 Inversion result with the modeling technique. (a) Estimated model ${\bf m_s}$. (b) Remodeled data ${\bf Hm_s}$.(c) Remodeled noise ${\bf A^{-1}m_n}$.(d) Residual. The residual is IID.

 

amococonv Figure 11 A comparison of convergence between the filtering and modeling approach for the land data example.