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First consider a data volume *d*(*x*,*y*,*z*) where *x* and *y*
are the horizontal axes and *z* is the depth or time axis.
Building on Lomask (2003b), a vertical (time or depth) delay
function is estimated by minimizing the following
functional :
| |
(1) |

where *p*_{x} is the local step-out vector estimated in the *x* direction and
*p*_{y} is the local step-out vector estimated in the *y* direction. Both
vectors depend on , which makes the problem of finding the time/depth
delays non-linear.
In this paper, we propose solving for with a
quasi-Newton method called L-BFGS Guitton and Symes (2003). The
quasi-Newton method is an iterative process where the solution to the
problem is updated as follows:

| |
(2) |

where is the updated solution at iteration
*k*+1, the step length computed by a line search
that ensures a sufficient decrease of and
an approximation of the Hessian (or
second derivative.) One important property of L-BFGS is that it
requires the gradient of only to build the Hessian.
The gradient of can be found by introducing the Euler-Lagrange equation and is given by:
| |
(3) |

The 2-D case is a simple extension of this result where the terms in
*y* are dropped. In practice, the last four terms of the gradient in
equation (3) can be precomputed and evaluated at
when needed for the BFGS update. This saves a lot of
computational effort. Note that the relative vertical (time or depth) delays are
computed with respect to a reference trace chosen a priori in the data
volume. A weight that would throw-out fitting equations at fault
locations can also be incorporated easily in both the gradient and
objective function.
The most important components of this time/depth delay evaluation
technique are the dip fields *p*_{x} and *p*_{y}. In our implementation,
we use the method of Fomel (2002) to estimate both. This technique
estimates local dips from adjacent traces without slant-stacking.
It also gives one dip value only for each data point.
Next, 2-D and 3-D data examples illustrate the flattening technique.

** Next:** 2-D data examples
** Up:** Guitton et al.: Non-linear
** Previous:** Introduction
Stanford Exploration Project

5/3/2005