Match-filter amplitudes

Match-filtering can simultaneously estimate a correction for static, phase and spectral differences between surveys. Typically an operator, ${\bf A}$, is 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 regions to be matched.

Rickett (1997) solved for as a time domain convolution operator by minimizing the residual in a least squares (L2) sense. The degree of spectral shaping 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 will provide the right amount of spectral shaping, and will preserve the details of the spectra that are associated with the temporally long reflectivity function.

As well as matching wavelets and static shifts, a match-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 broken into a wavelet correction, ${\bf A_w}$that preserves the energy in ${\bf d_1}$, and a scale factor, a, then

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

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 ${\bf n_1}$ are the uncorrelated components, and b is a scalar that captures the different level of signal present in the two regions. Ideally the operator scalar,

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

then the residual will simply be a purely a function of the uncorrelated 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, then

$$a = \frac{b {\bf s}^2}{b^2 {\bf s}^2 + {\bf n_1}^2}$$ (5)

which will only be unbiased if ${\bf n_1 = 0}$

The low value of a manifests itself in the low amplitude of ${\bf A d_1}$. This was first noted while match-filtering field data, and was corrected empirically by a trace renormalization to equalize the energy in traces between surveys.