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## Cross-correlation and shaping filters

A simple numerical way to find a static shift between two traces is to find the maximum of their cross-correlation function. The relative shift is the corresponding cross-correlation lag. Shaping filters are closely related to the simple cross-correlation function, and can also be used to measure relative shifts.

The shaping filter designed to match a first dataset, , with a second dataset, , can be defined as the filter, that minimizes the norm of the objective function,
 (1)
where denotes convolution. Equation (1) is very general: it implies nothing about either the choice of norm, or the dimensionality of , or the filter .

The classical discrete solution Robinson and Treitel (1980) to equation (1), which minimizes in the L2 sense, can be written as
 (2)
In this paper, I will use the convention that a bold upper case letter represents the operator that describes convolution with the filter represented by the corresponding lower case letter. For example, represents the matrix which describes convolution with the dataset, , and describes the matrix which represents convolution with .Multi-dimensionality in equation (2) is built into the definition of the convolution matrices.

Equation (2) implies that the optimal shaping filter, , is given by the cross-correlation of with , filtered by the inverse of the auto-correlation of .Equation (2) provides an alternative method of computing a cross-correlation function: firstly calculate an L2 shaping filter to link one dataset with the other; secondly, recolor the filter with the auto-correlation of the first dataset.

It is not immediately clear why we would ever want to do this in practice, since the first step of computing a shaping filter is to compute a cross-correlation. However, shaping filter estimation can leverage the well-developed machinery of geophysical inversion Claerbout (1999) in a number of ways; for example, we may include non-stationarity, a different choice of norm, or different types of regularization in an alternative definition of a shaping filter.

The new algorithm for finding a warp function has three steps. First, estimate a non-stationary shaping filter. Second, recolor the shaping filter by convolving it with the autocorrelation of . Finally pick the maxima of the recolored shaping filters.

Next: Adaptive shaping filters Up: Theory Previous: Theory
Stanford Exploration Project
4/27/2000