MULTIPLE SUPPRESSION USING SIGNAL-NOISE SEPARATION

Signal to noise separation has a long history at SEP Claerbout (1991); Harlan (1986); Kostov (1990). The method we use is similar to Abma's 1995 formation. Abma (1995) proposed solving the set of equations:

$$\begin{eqnarray} \bf N\bf n&\approx\ \bf 0\nonumber \\ \epsilon \bf S\bf s&\app... ...f 0 \\ \mbox{subject to} &\leftrightarrow& \bf d = s+n \nonumber \end{eqnarray}$$ (1)

where the operators $\bf N$ and $\bf S$ represent t-x domain convolution with (PEF's) which decorrelate the unknown noise $\bf n$ and signal $\bf s$, respectively, and the factor $\epsilon$ balances the energies of the residuals.

For his problem he assumed that the noise was uncorrelated, therefore $\bf N$becomes the identity and $\bf S$ is the PEF that best predicted the data in a given window [patching approach Claerbout (1992); Schwab and Claerbout (1995)].

In the multiple problem the noise is not uncorrelated so we must find another way to find $\bf N$. Spitz (1999) proposed defining $\bf S$ as $\bf S= \bf D\bf N^{-1}$ where $\bf D$ is a filter that characterizes the data rather than the signal. Using this new definition we get a new set of fitting goals:

$$\begin{eqnarray} \bf N\bf s&\approx& \bf N\bf d\nonumber \\ \epsilon \bf D\bf N^{-1} &\approx& \bf 0.\end{eqnarray}$$ (2)

Following Fomel et al. (1997) we can set up the conversion by reformulating it as a preconditioned problem by a simple change of variables ($\bf p=\bf D\bf N^{-1}$)

$$\begin{eqnarray} \bf N\bf N\bf D^{-1} \bf p&\approx& \bf \bf N\bf d\nonumber \\ \epsilon \bf p&\approx& \bf 0 ,\end{eqnarray}$$ (3)

where $\bf p$ is just a dummy preconditioning variable.

Instead of using patching we followed the methodology of Crawley et al. (1998) and constructed and estimated a space varying filter.

$$\begin{eqnarray} \bf 0&\approx& \bf D\bf A^{-1}\bf p\nonumber \\ \bf 0&\approx& \bf p\end{eqnarray}$$ (4)

where $\bf A$ is a radial smoother Clapp et al. (1999). For $\bf N$ we follow a similar procedure assuming an a priori model for the noise.