Modification to the seismic case

On first consideration, the NW algorithm seems ill-suited to the seismic case, primarily because seismic amplitude data is continuous, not discrete. To utilize the machinery of the NW algorithm we consider the histogram of data amplitudes on the trace pair and form a set of bins. That is to say, all of the floating point amplitudes are partitioned into a small number of intervals (20 in the examples given below), and the similarity matrix is formed by the equation 10 operating on the binned amplitudes. An important aspect of the algorithm is that a global optimum alignment function is found independent of the similarity measure that is used. To test feasibility, we used a one point similarity that captures amplitude differences. However we could easily have worked with a twopoint measure to emphasize slope similarity, or three points to match curvature. Extending this idea, one could work with short window cross correlations to fill the similarity matrix similar Martinson and Hopper (1992). Clearly, any of these more ambitious similarity measures would increase the cost of the algorithm. In any case, the NW algorithm guarantees a global optimum alignment solution using any similarity matrix as input.

As a final comment, we note that any number of traces that require alignment can be processed as a cascaded series of pairwise problems. Thus there is no loss of generality in discussing just the pairwise problem.