First we calculate the dip. Dip can be easily calculated using a plane-wave destructor as described in Claerbout (1992).
For the dip in the x direction of a seismic cube with a wave field represented by u(x,y,t), at each sample we calculate:
Our main objective is to find an absolute time (t) at each sample in the seismic data cube. Because the dip can be thought of as the gradient (), the dip in the x direction (px) is the x component of the gradient. Similarly, the dip in the y direction (py) is the y component of the gradient. Using our integration method described below, we first apply the divergence () to the gradient. Then we convert to Fourier space where we integrate twice by dividing by the Laplacian. Then we convert back to the time domain. The resulting t can be thought of as the absolute time for each point in the data.
Beginning with our input dip data:
The analytical solution is found with:
The denominator is the Z-transform of the 3D Laplacian. The zero frequency term of the Z-transform of the denominator is neglected. This means that the resulting surface in space will have an unknown constant shift applied to it. However, by adding the t dimension and assuming the gradient in the t direction to be all ones, we are insuring that the integrated time varies smoothly in the t direction.
Integrating in three dimensions enforces vertical smoothness. The dip in the t direction is all ones. This can be thought of intuitively as imagining that the dip in the x direction is the derivative of x with respect to t. So dip in the t direction is the derivative of t with respect to t, therefore it is always one. By integrating in 3D, we prevent our method from swapping sample positions in time.