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If we have an estimate *S*_{0} of the interfering signal the
following subtraction can be applied to the traces (Jenkins & Watts,1968):

| |
(2) |

where *X*_{ir} is the residual, is the smoothed
cross-spectrum between the
*i*-th trace and the estimate, and is
the smoothed power spectrum of the estimate.
The problem with (2) is that often it is difficult
to obtain a good estimate *S*_{0}. Now assume *S*_{i}>>*U*_{i} and
substitute into (2) instead of *S*_{0} *X*_{k}-just one trace.
Remembering that *X*_{k}=*U*_{k}+*S*_{k} we get:

| |
(3) |

and, rewriting the result:
| |
(4) |

Here *U*_{k}/*S*_{k} is a complex constant. The contribution of
the interfering source to each trace
has been reduced by the absolute value of this
constant (which is <<1 under our assumption), but the phase shift
between channels is preserved. To summarize, the magnitude of the
interfering source was reduced significantly without adding major
artifacts, and location of the weaker source is now more feasible.

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** Up:** DESCRIPTION OF THE SUBTRACTION
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Stanford Exploration Project

12/18/1997