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In Lomask and Claerbout (2002), we found that we could integrate local dip information () into total time shifts () quickly in the Fourier domain with:
| |
(1) |

where
.
We also found that if we initialized the dips in the *y* direction () to zero, then this equation would apply some kind of regularization in the *y* direction:

| |
(2) |

where
, , and.This would cause the integration to be smooth in the y direction. However, we were not able to control how smooth it would be.
Here we will add an adjustable regularization parameter () to equation (2). We begin with the fitting goal:

| |
(3) |

We can minimize the difference between the estimated slope and the theoretical slope with:
| |
(4) |

Next, we write the quadratic form to be minimized as:

| |
(5) |

Because the gradient is (), we can write:

| |
(6) |

This can be rewritten as:
| |
(7) |

The second term in equation (7) is the regularization term and only needs a scalar parameter to adjust its weight relative to the first term. Now we have:
| |
(8) |

Working backwards we see that it is now necessary to define a gradient operator that has an epsilon weight applied to one direction as:
| |
(9) |

It is also necessary to apply the scalar to the dip in the y direction as:
| |
(10) |

Lastly, the y components of the z-transform in the denominator of equation (2) also need to be scaled. The final analytical solution with an adjustable regularization parameter is:

| |
(11) |

where
and.

** Next:** Examples
** Up:** Lomask and Guitton: Adjustable
** Previous:** Introduction
Stanford Exploration Project

5/23/2004