Amplitude balancing

Whether an AGC window or a more careful geometric spreading correction has been applied, two generations of seismic survey will, in general, have different time-varying gain functions applied to them. If not compensated for correctly, this may lead to a systematic leakage of non-reservoir events into the difference section. Although an amplitude correction may need to be time and space-varying, it should be constrained to vary very slowly, so it is not influenced by changes in the reservoir zone.

The simplest approach to amplitude balancing is to scale the data based on the r.m.s. energy in the two surveys. However, this assumes that the energy present in the noise fields are the same in both datasets, or of much smaller magnitude than the signal energy. As an illustration we can consider two normalized datasets, ${\bf d_1}$ and ${\bf d_2}$, to consist of some shared signal, ${\bf s}$, and uncorrelated ``noise'' components, ${\bf n_1}$ and ${\bf n_2}$, which include the reservoir difference anomaly we seek:

$$\begin{eqnarray} {\bf d_1} & = & \frac{1}{\left\vert{\bf s + n_1}\right\vert} \l... ...{1}{\left\vert{\bf s + n_2}\right\vert} \left({\bf s + n_2}\right)\end{eqnarray}$$ (2) (3)

In order to rescale the signals to the same level, we need to apply a scale factor, $\nu$ to ${\bf d_1}$, where

$$\nu = \frac{\left\vert{\bf s + n_1}\right\vert}{\left\vert{\bf s + n_2}\right\vert}$$ (4)

or again assuming the noise fields are weakly correlated with the geological signal  

$$\nu \approx \frac{\sqrt{{\bf s}^2 + {\bf n_1}^2}}{\sqrt{{\bf... ... n_2}^2}} = \sqrt{\frac{1+\frac{1}{s_1^2}}{1+\frac{1}{s_2^2}}}$$ (5)

where s₁ and s₂ are the signal-to-noise levels in the two datasets. For high ($s_1 \gg 1$ and $s_2 \gg 1$), or similar ($s_1 \approx s_2$), signal-to-noise levels $\nu$ reduces to unity, and the equal energy condition is valid.

For the field examples in this paper, the equal energy condition was used to balance the filter amplitudes. This is a reasonable assumption for many examples, and does not require independent estimates of the signal-to-noise ratio.