From filtering to covariance-pattern based approach

The filtering approach for coherent noise removal can lead to a covariance-pattern based approach. This technique is often called pattern-based approach for short. The basic idea is to separate the noise and signal present in the data according to their pattern, or multivariate spectra. To identify and approximate these patterns, multidimensional PEFs can be used.

I now show briefly how this technique works. Building on equation () we introduce a regularization operator to obtain  

$$\begin{array} {rclcl} {\bf 0} &\approx& {\bf r_d} &=& {\bf ... ... &\approx& \epsilon{\bf r_m} &=& \epsilon{\bf Rm}, \end{array}$$ (28)

and minimize

$$f({\bf m}) = \Vert{\bf r_d}\Vert^2+\epsilon^2\Vert{\bf r_m}\Vert^2.$$ (29)

In this case, $\epsilon$ is a trade-off parameter between data fitting and model smoothing. In the pattern-based approach, the unknown parameter is the signal ${\bf m=s}$ and the modeling operator ${\bf L}$ becomes the identity. This choice of ${\bf L}$ comes from the fact that both the signal and the noise reside simultaneously in the data space. In the filtering approach, ${\bf A}$ is a PEF that annihilates the noise components. Let us call it ${\bf A=N}$. For the regularization operator, we need to find an operator for the signal such that ${\bf Rs \approx 0}$. By definition, a PEF ${\bf S}$that annihilates the signal components is a good candidate for ${\bf R}$. From these choices, we end up with  

$$\begin{array} {rclcl} {\bf 0} &\approx& {\bf r_d} &=& {\bf ... ... &\approx& \epsilon{\bf r_s} &=& \epsilon{\bf Ss}. \end{array}$$ (30)

Solving equation () in a least-squares sense gives for the estimated signal Abma (1995):  

$${\bf \hat{s}}=({\bf N'N} + \epsilon^2{\bf S'S})^{-1}{\bf N'Nd}.$$ (31)

Note that ${\bf N'N}$ and ${\bf S'S}$ are the inverse multidimensional spectra for the noise and signal, respectively, which explains why this technique of signal/noise separation is called pattern-based. As stated by Claerbout and Fomel (2000), PEFs are important where they are small. Analyzing the noise filter in equation ()

$${\bf F}=({\bf N'N} + \epsilon^2{\bf S'S})^{-1}{\bf N'N},$$ (32)

we see that

$$\begin{array} {ccccl} {\bf F}&=&{\bf 0}&\mbox{ if }&{\bf N'... ...1}&\mbox{ if }&{\bf N'N}\gt\gt\epsilon^2{\bf S'S}. \end{array}$$ (33)

The first condition is met when the noise is strong (PEF ${\bf N}$small), thus attenuating the noise, and the second when the signal is strong (PEF ${\bf S}$ small), thus preserving it. The filter ${\bf F}$is also a projection filter Sacchi and Kuehl (2001); Soubaras (1994).

In the pattern-based approach, ${\bf N}$ and ${\bf S}$ need to be estimated before the actual separation. This is the main challenge of this technique, i.e, how to choose good noise and signal models. Chapter and illustrate the pattern-based technique for surface-related multiple attenuation. In this case, strategies exist to estimate the noise and signal PEFs. In the next section, I illustrate the filtering and modeling techniques on synthetic and field data examples.