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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 socalled Wiener estimator
is a filter, which when applied to the data, yields an optimal (leastsquares 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 tx domain convolution with
nonstationary 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 onedimensional PEF's in the fx domain, but our use
of the helix transform Claerbout (1998) permits stable inverse filtering
with multidimensional tx domain filters.
Substituting and applying the constraint to equation (2) gives
 

 (5) 
Iterative solutions to leastsquares 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