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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:
 

 (1) 
 
where the operators and represent tx domain convolution with
(PEF's) which decorrelate the unknown noise and signal , respectively, and the factor balances the energies of the residuals.
For his problem he assumed that the noise was uncorrelated, therefore becomes the identity and 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 . Spitz (1999) proposed defining
as where is a filter that
characterizes the data rather than the signal. Using this new definition
we get a new set of fitting goals:
 

 (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 ()
 

 (3) 
where 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.
 

 (4) 
where is a radial smoother Clapp et al. (1999).
For we follow a similar procedure assuming an a priori model for the
noise.
Next: Synthetic example
Up: Clapp & Brown: Multiple
Previous: INTRODUCTION
Stanford Exploration Project
10/25/1999