Methodology

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.