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## Discretizing the problem

Claerbout approximates the derivatives of equation (1) with 2x2 finite difference stencils. Assuming that the grid spacing in both the t and x directions are unity:
 (4)
By convolving together these first-order stencils, we can construct appropriate finite-difference stencils to approximate the second-order differential operators of equation (3):
 (5) (6) (7)
The stencils of equations (5)-(7) are convolved with the data, . For simplicity, we can define the following notation:
 (8)
and rewrite equation (3) in matrix form:
 (9)
The vector has the same dimension as the data, .If the data consists only of plane waves with slopes p1 and p2, then equation (9) predicts values of from nearby values of . If the data's slopes change in time and space, however, equation (9) is valid only across local patches'' of the data. We can rewrite equation (9) to reflect this fact:
 (10)
Equation (10) denotes the convolution of the respective finite-difference stencils over a data patch of size n, where n may be as large as the entire data, or as small as

While it is tempting to make a change of variables (a=p1+p2, b=p1 p2) and treat equation (10) as a linear relationship, I have found that this approach produces trivial coupled estimates of the true slopes. This problem is inherently nonlinear.

Next: Dip Estimation Up: The Method Previous: The Method
Stanford Exploration Project
11/11/2002