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, ${\bf d}_1$, with a second dataset, ${\bf d}_2$, can be defined as the filter, ${\bf a}$ that minimizes the norm of the objective function,  

$$O({\bf a}) = \left\vert\left\vert \; {\bf a} \ast {\bf d}_1 - {\bf d}_2 \; \right\vert\right\vert,$$ (1)

where $\ast$ denotes convolution. Equation (1) is very general: it implies nothing about either the choice of norm, or the dimensionality of ${\bf d}_1$,${\bf d}_2$ or the filter ${\bf a}$.

The classical discrete solution Robinson and Treitel (1980) to equation (1), which minimizes $O({\bf a})$ in the L₂ sense, can be written as  

$${\bf a} = \left( {\bf D}_1^{\rm T} {\bf D}_1 \right)^{-1} \; {\bf D}_1^{\rm T} \, {\bf d}_2.$$ (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, ${\bf D}_1$ represents the matrix which describes convolution with the dataset, ${\bf d}_1$, and ${\bf D}_2$ describes the matrix which represents convolution with ${\bf d}_2$.Multi-dimensionality in equation (2) is built into the definition of the convolution matrices.

Equation (2) implies that the optimal shaping filter, ${\bf a}$, is given by the cross-correlation of ${\bf d}_1$ with ${\bf d}_2$, filtered by the inverse of the auto-correlation of ${\bf d}_1$.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 ${\bf d}_1$. Finally pick the maxima of the recolored shaping filters.