Matched-filtering Claerbout (1991) can simultaneously estimate a correction for static, phase and spectral differences between surveys. A cross-equalization operator, , can be designed to minimize the norm of the residual,

(1) |

Rickett (1997) solved for as a time domain
convolution operator by minimizing the residual in a least squares
(*L _{2}*) sense. The degree of spectral matching is then controlled by
the length of the time domain operator. By working with a short
operator of a similar length to the two wavelets being matched, the
operator can provide the ``right amount'' of spectral
shaping: a close enough spectral and phase match to compensate
for differences in wavelets and statics between the two surveys, while
avoiding over-match that can zero out differences in the data sets
caused by petrophysical changes during reservoir production.

As well as matching wavelets and static shifts, a matched-filter also
has an associated amplitude correction. However this amplitude
correction is biased by the presence of noise in . For
example, if is decomposed such that ,
where is a wavelet correction that matches the
spectrum in , and *a* is a scale factor, then

where is the common signal due to the geology that we are
trying to remove from the difference image, and
are the uncorrelated noise components, and *b* is a scalar
that quantifies the different signal amplitude present in the two data
windows. Ideally the operator scalar should be

(4) |

(5) |

The low value of *a* manifests itself in the low amplitude of . This was first noted while matched-filtering field data, and was
corrected empirically by a trace renormalization to equalize the
energy in traces between surveys.

10/9/1997