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) |

The classical discrete solution Robinson and Treitel (1980) to
equation (1),
which minimizes in the *L _{2}* sense, can be written as

(2) |

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 *L _{2}*
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.

4/27/2000