But this is merely the best alignment via static shift. In a complex data set, such as a CMP gather, the human eye can associate events that a global correlation alignment will not honor. What the eye is able to do is a time-variant, nonlinear association of events. To carry out this alignment process requires the determination and application of a nonlinear mapping between the trace samples - a combination of compression, stretching, and translation, all varying with time.
If it were possible to do this kind of alignment, what would be the use of it? In a sense, it is already done everyday in seismic processing. A collection of traces are analyzed for a set of coefficients which drive a nonlinear stretch to make all events flat at all times. This process is, of course, normal moveout. But NMO is a model-driven process, with the model being the NMO equation. A general nonlinear trace alignment algorithm would make it possible to flatten all events in a CMP gather with no knowledge of the NMO equation. We are not advocating such a procedure, but making a point. There is value in using the NMO equation to flatten events, including the fact that it leaves multiples non-flat and therefore removable (at least partially). However, a the general alignment algorithm may be useful as a final flattening procedure for any type of gather (CMP, CIG, CAG, etc.). It could also have application in alignment of synthetic seismograms with field data, associating events on P-P and P-Sv data cubes, etc. In short it would be a useful, general utility.
While the literature on trace interpolation and estimation of missing data is vast, there is very little published work on nonlinear trace alignment. To our knowledge the only published work directly on point with the current study is Martinson et al. (1982) and a derivative paper Martinson and Hopper (1992). In the second work, an iterative, linear inverse approach is used to determine a set of coefficients describing a mapping function which relates features on one trace with those on another. The process is driven by maximizing the correlation or coherence between the modified traces, and used as a trace interpolation technique. This method is similar in spirit to our approach, but owing to the use of linear inverse theory it tends to be expensive, sensitive to the starting model, and does not guarantee a global solution.
Our solution to the pairwise trace alignment problem borrows a concept and algorithm from computational biology and modifies it to the seismic case. The concept is pairwise alignment of amino acid sequences, and the algorithm is due to Needleman and Wunsch (1970).