Horizontal correlation

In the special case, that all sedimentary layers are horizontal, a time slice of an undisturbed horizontal layer shows a constant amplitude, and, within the time slice, discontinuities create standard amplitude edges. The amplitude edges could be enhanced by standard edge enhancement techniques applied to individual time slices. Instead, I chose trace correlation to compute discontinuity attributes of horizontally layered seismic images.

Correlation measures horizontal alignment of one-dimensional arrays (traces). The standard normalized correlation between two one-dimensional arrays, f₁ and f₂, is   where i is the arrays' sample index. Neidell and Taner 1971 generalize the correlation coefficient to measure alignment among a set of one-dimensional arrays:   where N is the length of a single trace in samples and f_ik is the k-th sample of the i-th trace. The correlation coefficients are bounded by -1 and 1. The coefficient c is 1 only if all the traces are identical.

To simplify plots of correlation coefficients, I map the coefficient range from [-1,1] to [0,2] so that pixels of correlation 1 - perfect correlation - plot as and less well-correlated regions show positive amplitudes. Hence, I first subtract 1 from each pixel value and then multiply it by -1.

To compute the local correlation of a nonstationary image volume, I split the image into small patches, compute the generalized correlation within the patch, set all patch pixels to its correlation value c and merge the individual patches to a single output quilt. The correlation within a patch measures the similarity among its traces f_i, the one-dimensional vertical array of all pixels of identical horizontal location. Perfectly horizontal layers result in a correlation coefficient of 1 and are mapped to a pixel value of .

The patch size is determined by a trade-off between resolution and data nonstationarity on one side and the need to capture sufficient statistical information in a patch on the other. Since each patch is filled with a single correlation coefficient, the procedure can only resolve two discontinuities if they are separated by more than the size of a patch. Besides loss of resolution, a large patch may capture nonstationary data and yield incorrect statistical estimates. On the other hand, a small patch may not contain enough data to gather reliable statistics in the presence of noise.