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Claerbout approximates the derivatives of equation () with
2x2 finite difference stencils. Assuming that the grid spacing in
both the t and x directions are unity:
| |
(165) |
By convolving together these first-order stencils, we can construct
appropriate finite-difference stencils to approximate the second-order
differential operators of equation ():
| |
(166) |
| (167) |
| (168) |
The stencils of equations ()-() are
convolved with the data, . For simplicity, we can define the following
notation:
| |
(169) |
and rewrite equation () in matrix form:
| |
(170) |
The vector has the same dimension as the data, .If the data consists only of plane waves with slopes p1 and p2, then
equation () predicts values of from nearby values
of . If the data's slopes change in time and space, however, equation
() is valid only across local ``patches'' of the data.
We can rewrite equation () to reflect this fact:
| |
(171) |
Equation () 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 () 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