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Optimal signal extraction

The frequency domain representation of the Wiener optimal reconstruction filter for uncorrelated signal and noise is Castleman (1996); Leon-Garcia (1994)  
\bold H = \frac{\bf P_s}{\bf P_s + P_n},
 \end{displaymath} (8)
where $\bf P_s$ and $\bf P_n$ are the power spectra of the unknown signal and noise, respectively. Multiplication of $\bf H$ with the data spectrum extracts the signal spectrum which is optimal in the least squares sense.

Abma 1995 solved a constrained least squares problem to separate signal from spatially uncorrelated noise.
\bf Nn &\approx& 0 \nonumber \\  \bf \epsilon Ss &\approx& 0
\\  \mbox{subject to} &\leftrightarrow& \bf d = s+n \nonumber
which reduces to equation (2). Explicitly minimizing the quadratic objective function suggested by equations (9) or (2) leads to the following expression for the predicted signal:
\bf s = \left[\frac{ N'N}{ N'N + \epsilon^2 S'S}\right] \ d
 \end{displaymath} (10)
Recalling that the frequency response of the PEF is a smoothed approximation to the inverse spectrum of the data from which it was estimated
\bf P_s &\approx& \bf {\cal F}\left\{\frac{1}{S'S}\right\}
\\  ...
 ..._n &\approx& \bf {\cal F}\left\{\frac{1}{N'N}\right\} \nonumber 
 \end{eqnarray} (11)
it is easy to show that the Wiener reconstruction result is equivalent to Abma's. Claerbout 1998a uses this approach, and we extended a variation of it to obtain the results obtained in this paper. When spatially coherent events cross, as they do with ground roll and primaries, they are not uncorrelated. We believe that in order to maintain a high degree of rigor in our future formulation of this problem, the correlation between signal and noise should be accounted for. A more general form of the Wiener optimal reconstructor Castleman (1996); Leon-Garcia (1994) is  
\bold H = \frac{\bf P_{ds}}{\bf P_d},
 \end{displaymath} (12)
where $\bf P_{ds}$ is the Fourier transform of the cross-correlation of the data and the unknown signal. Such a formulation is considerably less intuitive than equation (8), and we currently have no concrete ideas as to a starting point.

Nemeth 1996 presents a more rigorous formulation for the separation of coherent noise and signal. As it is collected, the data is composed of overlapping signal and noise events, so the goal is to map the data to a domain where the signal and noise are uncorrelated, and thus separable without crosstalk. Nemeth's model is a composite vector, $[{\bf m_s \;\; m_n}]^T$, consisting of the independent signal and noise model in the transformed (migrated) domain. His composite modeling operator, $[{\bf L_s \;\; L_n}]$, is adjoint to migration, so his method accounts for the arbitrary moveout of real data - not the idealized hyperbolic moveout assumed for Radon-family transforms. The least squares inverse for Nemeth's model is
 \bold m_s \\  \bold m_n
 \bold L^T_s \\  \bold L^T_n
 \end{array}\right] \bf d
 \end{displaymath} (13)
In the context of prediction-error filtering, the ``model'' would be the residual of some PEF convolved with the data, such that the signal and noise are separated in the model space. The cross terms in Nemeth's ``inverse model covariance matrix,'' ${\bold L^T \bold L}^{-1}$, account for correlation between signal and noise. In practice, Nemeth's method is weakened by the need for an explicit ground roll model. However, if Nemeth's migration operators are cleverly replaced with prediction error filters, a similarly powerful formulation could probably be derived, one which is free from the need to model ground roll explicitly.

In any case, Spitz' 1999 choice of signal predictor, ${\bf S = DN}^{-1}$ gives good results, and probably handles the correlation between signal and noise correctly. Though we don't show the result in this paper, we have found that when the actual noise model is used (synthetic data), the estimated signal is nearly perfect.

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