Next: Filter estimation Up: METHODOLOGY Previous: METHODOLOGY

## Signal to noise separation

Consider the recorded data to be the simple superposition of signal'' , i.e., reflection events, and noise'' , i.e., multiples: . For the special case of uncorrelated signal and noise, the so-called Wiener estimator is a filter, which when applied to the data, yields an optimal (least-squares sense) estimate of the embedded signal Castleman (1996). The frequency response of this filter is
 (1)
where and are the signal and noise power spectra, respectively. Abma (1995) and Claerbout (1999) solved a constrained least squares problem to separate signal from spatially uncorrelated noise:
 (2)
where the operators and represent t-x domain convolution with non-stationary PEF which whiten the unknown noise and signal , respectively, and is a Lagrange multiplier. Minimizing the quadratic objective function suggested by equation (2) with respect to leads to the following expression for the estimated signal:
 (3)
By construction, the frequency response of a PEF approximates the inverse power spectrum of the data from which it was estimated. Thus, we see that the approach of equation (2) is similar to the Wiener reconstruction process.

Spitz (1999) showed that for uncorrelated signal and noise, the signal can be expressed in terms of a PEF, , estimated from the data , and a PEF, , estimated from the noise model:
 (4)
Spitz' result applies to one-dimensional PEF's in the f-x domain, but our use of the helix transform Claerbout (1998) permits stable inverse filtering with multidimensional t-x domain filters.

Substituting and applying the constraint to equation (2) gives
 (5)
Iterative solutions to least-squares problems converge faster if the data and the model being estimated are both uncorrelated. To precondition this problem, we again appeal to the Helix transform to make the change of variables or and apply it to equation (5):
 (6)
After solving equation (6) for the preconditioned solution , we obtain the estimated signal by reversing the change of variables: .

Next: Filter estimation Up: METHODOLOGY Previous: METHODOLOGY
Stanford Exploration Project
4/27/2000