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Matched-filter amplitudes

Matched-filtering Claerbout (1991) can simultaneously estimate a correction for static, phase and spectral differences between surveys. A cross-equalization operator, ${\bf A}$, can be designed to minimize the norm of the residual,
{\bf r = A d_1 - d_2}\end{displaymath} (1)
where ${\bf d_1}$ and ${\bf d_2}$ are the operator-design windows of the two data sets to be matched.

Rickett (1997) solved for ${\bf A}$ as a time domain convolution operator by minimizing the residual in a least squares (L2) 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 ${\bf d_1}$. For example, if ${\bf A}$ is decomposed such that ${\bf A} = a{\bf A_w}$, where ${\bf A_w}$ is a wavelet correction that matches the spectrum in ${\bf d_1}$, and a is a scale factor, then

a\, {\bf A_w d_1} & = & a b\, {\bf s} + a\,{\bf n_1} \\ {\bf d_2} & = & {\bf s + n_2}\end{eqnarray}

where ${\bf s}$ is the common signal due to the geology that we are trying to remove from the difference image, ${\bf n_1}$ and ${\bf n_2}$ 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


and then the residual will be simply the difference of the (scaled) uncorrelated noise fields. However if the residual, ${\bf
r}$ is minimized in an L2 sense then
\frac{\partial}{\partial a} \left\vert ab \, {\bf s} + a \, {\bf n_1} - {\bf
s} - {\bf n_2} \right\vert^2 = 0\end{displaymath} (4)
and if the noise fields are largely uncorrelated with the geology and each other, ${\bf n_1 \cdot n_2 = n \cdot s = 0}$, then
a = \frac{b \, {\bf s}^2}{b^2 \, {\bf s}^2 + {\bf n_1}^2}\end{displaymath} (5)
which will only be unbiased if ${\bf n_1 = 0}$.Note that as the energy of the noise field ${\bf n_1}$ increases, the L2 estimate of a will be biased increasingly lower than the optimal value 1/b.

The low value of a manifests itself in the low amplitude of ${\bf A
d_1}$. 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.

previous up next print clean
Next: Amplitude balancing Up: Rickett, et al.: Amplitude Previous: Introduction
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