Extension to seismic trace alignment

There are two obvious problems with a straightforward implementation of the NW algorithm. Both are related to the fact that our data is continuous, rather then discrete. We do not have the limited alphabet of the DNA example, so our similarity matrix can't simply be matching of values. Therefore, a more appropriate measure of similarity in the seismic case is a short-window correlation. We capture this idea in a similarity function as

 

$$\sigma (i,j) = \frac{\sum_{a=-n}^n x_{i+a} y_{j+a}} { \sum_{a=-n}^n x_{i+a} \sum_{a=-n}^n y_{j+a} }$$ (6)

or a semblance measure

$$\sigma (i,j) = \frac{\sum_{a=-n}^n \frac{( x_{i+a} + y_{j+a})^2} {( x_{i+a}^2 + y_{j+a}^2)} } {2*n}$$ (7)

where x and y are again the two traces, and n is the correlation length.

The second problem is how to use our alignment data. We are not dealing with discrete points but smooth functions. We can estimate a prediction error filter Claerbout (1998) or a time variant, non-stationary prediction error filter upon the original data to describe the wavelet. We have adopted this approach, allowing us to estimate our aligned model $\bf m$ from our unaligned data $\bf d$ by minimizing

$${\rm min}_{\bf m} = \vert\vert {\bf m} - {\bf d} \vert\vert^2 + \epsilon^2~ \vert\vert {\bf A} {\bf m} \vert\vert^2$$ (8)

where $\bf A$ is filtering with the estimated prediction error filter and $\epsilon$ is a scalar controlling how much weight to give the aligning. This controls the relative importance of the alignment versus fitting the wavelet.