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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 *p*_{1} and *p*_{2}, 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*=*p*_{1}+*p*_{2}, *b*=*p*_{1} *p*_{2})
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.

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** Up:** The Method
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Stanford Exploration Project

11/11/2002