METHOD 2: A subtraction method

Instead of removing the noise by filtering, we can remove it by subtraction. If an operator is unable to model all the information embedded in the data, then the residual is not IID. The second formulation I propose is based on the idea that if we can model the coherent noise with another operator, then the residual components become IID. Let us consider that we have

$${\bf d} ={\bf s}+{\bf n}$$

and that there exists an operator L such that

$${\bf L}=[{\bf H}\;\;\; {\bf L_n}].$$

We assume that H is the modeling operator for the signal s and that ${\bf L_n}$ is the modeling operator for the coherent noise n. Following this decomposition, we can write

$${\bf m}=\left [ \begin{array} {c} {\bf m_s}\ {\bf m_n} \end{array} \right ]$$

where ${\bf m_n}$ is the noise-model and ${\bf m_s}$ is the signal-model. Starting from

$${\bf 0} \approx {\bf Lm - d},$$ (12)

the fitting goal then becomes  

$${\bf 0} \approx {\bf Hm_s+L_nm_n - d} .$$ (13)

Because we have to find ${\bf m_s}$ and ${\bf m_n}$, this system is clearly under-determined and some regularization is needed. Thus, we end up with the following fitting goals

$$\begin{eqnarray} {\bf 0} &\approx& {\bf Hm_s+L_nm_n - d} \nonumber \ {\bf 0} &... ..._s} \nonumber \ {\bf 0} &\approx& \epsilon {\bf Im_n}. \nonumber\end{eqnarray}$$

Because there should be a different operator ${\bf L_n}$ for each different coherent noise pattern, the cost of this method increases considerably. Fortunately, we can use multi-dimensional PEFs to estimate the coherent noise operator. This estimation is possible if we assume that the coherent noise is predictable, i.e., made up of the superposition of local plane wave segments Claerbout (1992). If we can estimate PEFs from the coherent noise, then the inverse PEF should be our coherent noise modeling operator ${\bf L_n = A_n^{-1}}$. Computing the inverse of multi-dimensional PEFs is now possible via the helix. In addition, with the helical boundary conditions, computing the inverse of multi-dimensional PEFs is as easy as computing the inverse of 1-D filters. We have then

$$\begin{eqnarray} {\bf 0} &\approx& {\bf Hm_s+A_n^{-1}m_n - d} \nonumber \ {\bf... ...ilon{\bf Im_s} \ {\bf 0} &\approx& \epsilon{\bf Im_n}, \nonumber\end{eqnarray}$$ (14)

where ${\bf A_n}$ is the noise PEF. This approach is similar to Tamas Nemeth's approach 1996. The difference emerges in the choice of the operators ${\bf L_n}$ and ${\bf H}$. Whereas Nemeth (1996) imposes one operator ${\bf L_n}$ to model the noise, we estimate a PEF ${\bf A_n}$ and use it in the fitting goals (Equation 15). Because PEFs (with appropriate dimensions) whiten the spectrum of many different plane-waves, this strategy is more flexible (no assumptions regarding the moveout of the noise). This method should give IID residual variables as long as we are able to estimate PEFs for the coherent noise. This is the main difficulty and challenge of this method. The minimization of the objective function in a least-squares sense for the fitting goals in Equation 15 can be done again with a fast conjugate gradients method.

I did not develop any specific algorithm to solve this inverse problem. I assume that we have a strategy that allows us to estimate the operator ${\bf A_n}$. We can then minimize the objective function for the fitting goals given in Equation 15 in a least-squares sense, for example.