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# Theory of time/depth delays estimation

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 px is the local step-out vector estimated in the x direction and py 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 px and py. 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