PS regularization

We previously discussed that by formulating the irregular geometries problem in the least-squares sense it is possible to solve for gaps in the data using a regularization operator. The significant element of the previous section is the use of the AMO operator as the regularization term in the solution of the least-squares problem.

Recently, Rosales and Biondi developed and implemented an AMO operator for converted waves (PSAMO). This operator acts in the Fourier domain and also handles the amplitudes properly. Due to this new PSAMO operator, it is now possible to solve for the irregular geometries problem on converted wave data by following the same procedure as in the previous section.

Partial stacking requires the data to be coherent among the traces. NMO obtains this coherency well for PP data. However, for converted waves we know that the moveout is not a perfect hyperbola, even in constant velocity media.

On conventional PP processing, the AMO operator is velocity independent. However, for converted waves the PSAMO operator depends on the ratio between the P and the S velocities ($\gamma$). Therefore, we need a priori velocity estimation. This fact suggests that for different $\gamma$ values we will have different regularization results.

Traditional PS processing intends to first sort the data in the common conversion point (CCP) domain. This process has always been dependent on the $\gamma$ value; therefore, the PS processing community performs iterative processing (CCP binning, velocity analysis) until obtaining a satisfactory result.

The PSAMO operator that we use has the advantage of not demanding the data in the CCP domain. This operator is a cascade operation of converted wave dip moveout () (PSDMO) and inverse PSDMO. The input for the PSDMO operator is in the CMP domain after NMO, since this operator performs the lateral shift correction.

After performing NMO on the PS data and the PP data, the $\gamma$ value is ():  

$$\gamma = \frac{v_p^2}{v_{eff}^2},$$ (103)

where v_eff is the NMO velocity of the PS section.

In order to proceed with the PS data regularization, a process that depends on the $\gamma$ value, we need to have the PP section regularized as well as the RMS velocity model. We proceed with the following algorithm:

1.

Sort the data in the CMP domain.

2.

Estimate velocity model on the PS section.

3.

Estimate the $\gamma$ section with equation ().

4.

If it is not the first iteration, compare the previous and the actual $\gamma$ sections and:

(a)

if they are the same, finish the process.

(b)

if they are not, continue.

5.

Apply NMO on the PS section.

6.

Apply PSAMO regularization.

7.

Apply inverse NMO.

8.

Go back to step 2.

This is our main methodology to correctly regularize PS data.