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In contrast to the Wienerlike method which filters the noise, the
following method aims to model the noise and then subtract it from
the input data. In this section, we show that the formalism used
by Nemeth (1996) can help to better separate correlated noise and signal.
But first, we detail the similarities and differences between
the Wienerlike and the subtraction method.
In equation (2), the noise and signal PEFs filter
the data components. Alternatively, building on Nemeth (1996), the noise
and signal nonstationary PEF can predict the data components via a
deconvolution as follows:
 
(4) 
We call the signal model component and the noise model
component (not to be confused with the noise model that we use to compute the
noise PEF). Clearly, models the noise vector n
and the signal vector s.
Because we use PEFs in equation (4), this approach is
patternbased in essence.
With and ,using linear algebra, we can prove that the leastsquares
solution of and is
 
(5) 
with
 

 (6) 
The operators and can be
seen as signal and noise filters respectively since
and
are the data resolution operators for
the signal and the noise, respectively. In the appendix B, we give a geometrical
interpretation for both and .
The degree of orthogonality between the noise operator and
the signal operator restricts the existence of
and
in equation (5). If the two
operators overlap completely, the Hessians and are not
invertible. If the two operators overlap only partially,
Nemeth (1996) proves that the separability of the signal
and noise can be improved if we introduce a regularization term.
If we use a model space regularization Fomel (1997),
we have
 
(7) 
with and the regularization operators
for the noise model and the signal model .A data space regularization can also improve the separation but
will not be considered here.
In equation (4),
the outcome of the inversion is and .
The estimated signal is then easily derived as follows:
 
(8) 
We call this new method the subtraction method. In the next two sections,
we compare the Wienerlike approach and the subtraction
method for groundroll and multiple attenuation.
Next: Approximating the signal PEF
Up: Theory review
Previous: Wienerlike method
Stanford Exploration Project
4/29/2001