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,

$${\bf r = A d_1 - d_2}$$ (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

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

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

$$a=\frac{1}{b}$$

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

$$\frac{\partial}{\partial a} \left\vert ab \, {\bf s} + a \, {\bf n_1} - {\bf s} - {\bf n_2} \right\vert^2 = 0$$ (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

$$a = \frac{b \, {\bf s}^2}{b^2 \, {\bf s}^2 + {\bf n_1}^2}$$ (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.