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A brief theory of signal/noise separation

I present one possible formulation of the signal/noise separation problem as exemplified by Abma and Claerbout (1995); Brown and Clapp (2000); Guitton et al. (2001); Soubaras (1994). To simplify, we have the two following fitting goals:  
 \begin{displaymath}
\begin{array}
{ccccl}
 \bold 0& \approx & \bold{r_d}& =& \bo...
 ...prox & \epsilon \bold{r_s}& =& \epsilon \bold{Ss},
 \end{array}\end{displaymath} (8)
where the data $\bold d$ is equal to the sum of the noise $\bold n$and signal $\bold s$, $\bold N$ is the PEF for the noise ($\bold{Nn}\approx \bold{0}$), $\bold S$ is the PEF for the signal ($\bold{Ss}\approx \bold{0}$), and $\epsilon$ a constant. Solving for $\bold s$ we have
\begin{displaymath}
{\bf \hat{s}} = (\bold{N'N}+\epsilon^2\bold{S'S})^{-1}\bold{N'Nd}.\end{displaymath} (9)
The signal-noise attenuation is then separated into two distinct steps: first we estimate the noise and signal filters $\bold N$ and $\bold S$, then we estimate the signal $\bold s$ based on the fitting goals in equation (8).
next up previous print clean
Next: A 2-D example Up: Signal-noise separation with weighted Previous: Signal-noise separation with weighted
Stanford Exploration Project
7/8/2003