Discussion

1.

In the filtering method, PEFs are recomputed iteratively from the data residual. I think this solution is the method of choice since the PEF (squared) is the inverse noise covariance matrix ${\bf C_d^{-1}}$. For the subtraction method, however, the final result is driven by the orthogonality between the coherent noise operator and the signal operator Nemeth (1996). If the two operators can model similar parts of the data, the separation will not be efficient. Nemeth proposes introducing some regularization (Equation 15) to mitigate this difficulty. We could perhaps compute a prior coherent noise model from which we estimate the PEFs. This approach is related to Spitz's idea Spitz (1999), according to which a noise model is utilized to estimate the signal PEF. In any case, the strategy of computing the coherent noise operator (PEF) is of a vital importance for the quality of noise separation.

2.

The PEF estimation is one problem, but choosing the signal operator H is another. As said before, the two approaches perform noise attenuation (filtering method) or noise separation (subtraction method) along with a conventional signal processing step (velocity analysis here). The processing step should be chosen in agreement with the expected signal in the data. Basically, the processing operator H should mitigate the crosstalks between the signal and the coherent noise. Coherent noise comes in different flavors and H should reflect this heterogeneity. Harlan (1986) gives some guidelines alternating between migration, Slant-Stack, and offset-local Stack as a function of the coherent noise and of the signal. We should keep these guidelines in mind when dealing with different datasets, different problems.

3.

Because the data are not time-stationary, the coherent noise operator should be a function of time and space. This difficulty can be overcome using non-stationary filters. In particular, estimating space varying filters with coefficients smoothed along a radial direction proved efficient Crawley (1999). Nonetheless, Clapp and Brown (2000) experienced stability problems, making the computing of the inverse PEFs potentially unsafe.

4.

The two proposed methods have the advantage of performing noise attenuation (filtering method) or noise separation (subtraction method) along with a geophysical process (velocity inversion in this case). The two algorithms can be used at the same time. The fitting goals become

$$\begin{eqnarray} {\bf 0} &\approx& {\bf A_r(Hm_s+A_n^{-1}m_n - d)} \ {\bf 0} &... ...onumber \ {\bf 0} &\approx& \epsilon {\bf A_{m_n}m_n}. \nonumber\end{eqnarray}$$ (15)

${\bf A_r}$, ${\bf A_{m_s}}$, ${\bf A_{m_n}}$ and ${\bf A_n}$ are PEFs to be estimated. Note that if (1) ${\bf Hm_s=s}$ (the signal), (2) ${\bf A_n^{-1}m_n=n}$(the coherent noise), (3) ${\bf A_{m_n}=A_r=N}$ (the coherent noise PEF), (4) ${\bf A_{m_s}m_s=Ss}$ (S the signal PEF) and (5) ${\bf A_{m_n}m_n=Nn}$, then Equation 16 is exactly Abma's Equation 1995. Equation 16 gives a simple generalization of the methods proposed above and should be tested.

5.

A more general robust inversion scheme can be derived combining the two proposed methods along with a robust norm. In particular, it would be interesting to use the hybrid l²-l¹ Huber norm more routinely. Thus we would solve the noise problem in its totality, handling both outliers and coherent noise effects at the same time.

6.

The extension to 3-D data should be feasible. A problem arises in the choice of the operators and in the PEF estimation. For this method to be really efficient in 3-D, more 3-D operators should be used. The PEF's estimation in 3-D is theoretically a simple extension of the 1-D case. The shape of the PEF may be more difficult to anticipate, however. In addition, the irregular geometry, intrinsic to 3-D acquisitions, can make the estimation of the PEF very difficult.