Plane-wave misfit by correlation

The direct comparison between a local image patch and its best-fitting plane wave offers another alternative discontinuity attribute. How do I estimate the best-fitting plane wave of an image region $f({\bf x})$? In the previous section, I showed how to estimate the best-fitting plane-wave normal ${\bf p}$ for a given image region. Given its normal estimate ${\bf p}$, stacking the image f along the hyper-planes orthogonal to ${\bf p}$ yields an estimate of the one-dimensional waveform . The adjoint of the stacking procedure yields an estimate of the plane-wave volume .The stacking operator and its adjoint use weights to compensate for the variable number of contributions for each waveform element.

The plane wave is not best-fitting the original in a least-square sense of the expression (9). Instead, the normal ${\bf p}$ is a least-squares approximation of the true normal. The waveform is an estimate that depends on the normal ${\bf p}$ (rather than being estimated simultaneously). If the original image is a plane wave, my process, of course, yields the correct estimate.

The reformulation of the problem greatly simplifies the plane-wave estimation. I reformulate a highly nonlinear, non-convex cost function into a linear estimation of the plane-wave normal ${\bf p}$ and the linear estimation of the waveform f.

Many geophysical problems are conveniently expressed as optimization problems. Unfortunately, the computational costs of these optimization problems (often expressed by the number of iterations necessary to converge) deliver these optimal formulations often unrealistic for routine computations. I believe, a central challenge for geophysicists today is the mathematical reformulation of geophysical optimization problems into computationally manageable procedures. The crossproduct formulation of the plane-wave estimation problem is such a reformulation.

Figure 31 displays the composite image of the local best-fitting plane waves for the synthetic test case. I subdivide the original synthetic data of Figure 4 into local patches without overlap, estimate in each patch the best-fitting local plane wave, and merge the local plane waves to a composite image. The plane-wave components in the dipping and horizontal half-space are correctly estimated. Along the boundary of the two half-spaces, however, the synthetic data deviates from the single plane-wave assumption. The least-squares estimate of the dip averages the correct values of either side. The combination of stacking and its adjoint nevertheless fills each patch with a plane wave volume, which, of course, does not resemble the original patch contents of two plane waves.

 

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Figure 31 Locally best-fitting plane-wave volumes in the synthetic image 4. Away from the fault surface, each patch's plane wave is correctly estimated. For patches along the fault surface, the single plane-wave assumption is violated and the dip and local wavefield estimates are incorrect.

To compare the best-fitting plane wave and the original local image region I correlate the patches' corresponding individual traces (the patches' individual time intervals). The discontinuity attribute divides the image into patches, estimates the best-fitting plane wave of each patch, computes a correlation coefficient for each trace of a patch, sets all samples of a trace to the correlation coefficient, merges such patches - now indicating correlation - into a single image volume.

The correlation of two traces is

where i loops over the samples of the one-dimensional trace f and . The coefficient is bounded by [-1,1] and yields 1 only if the two traces are identical.

In general, correlation is a robust estimate of similarity as it averages over an ensemble of samples. The discontinuity attribute could loop over all kind of subsets of an individual patch to compute correlation. The entire patch could be thought as a supertrace, which would be consistent with the plane-wave estimation of a single patch. I choose a single coefficient per trace since it averages over time samples and, thereby, looses vertical resolution, but preserves lateral resolution. Such a tradeoff seems to be well-suited for seismic data, which is usually oversampled along the time axis, but sparsely sampled in space, and which often depicts horizontal strata. Nevertheless, the overall lateral resolution of the attribute is limited by the attributes' ability to estimate the normal and waveform at discontinuities (as described above).

As with the display of the horizontal correlation attribute, I simplify the attribute maps by linearly transforming the correlation values to non-negative numbers. The optimal correlation of 1 is displayed as amplitude, the better correlated the image, the smaller the attribute value. In comparison, the correlation (6) in this section estimates the similarity between two one-dimensional arrays. The correlation (7) used in the horizontal correlation coefficient, on the other hand, estimates the similarity among a set of one-dimensional arrays.