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In the multiple attenuation case, most of the problems
encountered with the adaptive subtraction
technique stem from the correlation that might exist
between the noise and signal. In addition, the minimum
energy assumption forces the residual (the signal) to be white, which
is not valid all the time. Fortunately we can derive a
fitting goal that can cope with interfering noise and signal and
non-white spectrum.
In Guitton (2002), I presented a
method that approximates covariance operators with pef. The goal was
basically to obtain independent and identically distributed (iid)
residual components. I propose using the same approach for the
filter estimation in adaptive filtering.

Following this idea, I have the new fitting goal

| |
(4) |

where is a pef that whitens the signal spectrum only.
The corresponding least-squares estimate for becomes
| |
(5) |

If the signal has a white spectrum, then this new estimate
of the filter is identical to the estimate in equation
(3). If the signal is not white, then this new estimate
is going to be more accurate than the estimate in equation
(3). More specifically, the noise and signal do not
have to be orthogonal any more.
I call this scheme hybrid because it puts back together two
worlds: the world of adaptive subtraction and the world of pef.
Nonetheless, this method is not a pattern-based
technique because the multiples and primaries are not separated
according to their spatial predictability. I am only proposing to
unbias the filter estimation.

Once the filter has been estimated [equation (5)] I
compute the noise and signal as follows:

| |
(6) |

One unsolved problem is the pef estimation. I give few guidelines in
the next section.

** Next:** How to estimate the
** Up:** Improving adaptive subtraction
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Stanford Exploration Project

6/7/2002