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. Following a similar model to that used to describe the failings of matched-filtering we can consider the 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:

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

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

$$\nu = \frac{\left\vert{\bf s + n_1}\right\vert}{\left\vert{\bf s + n_2}\right\vert}$$ (8)

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

$$\nu \approx \frac{\sqrt{{\bf s}^2 + {\bf n_1}^2}}{\sqrt{{\bf... ...} = \sqrt{\frac{1+\frac{1}{(s:n_1)^2}}{1+\frac{1}{(s:n_2)^2}}}$$ (9)

where s:n₁ and s:n₂ are the signal-to-noise levels in the two datasets.

The value of $\nu$ applied to ${\bf d_1}$ has a large effect on the amplitude of coherent events in the difference section, and consequently may significantly affect the interpretation of the 4D data. Unfortunately, the value of $\nu$ cannot be obtained directly from the data without an a priori assumption about the nature of the noise. Also the optimum value of $\nu$ may be time and space varying, which further complicates its determination.

Fortunately, for seismic data with high signal-to-noise ($s:n_1, s:n_2 \gg 1$), the equal energy ($\nu=1$) balancing will be valid since $\Delta \nu \equiv \nu - 1$ will be very small.

$$\Delta \nu \approx \frac{1}{2}\left(\frac{1}{(s:n_1)^2} - \frac{1}{(s:n_2)^2} \right) \ll 1$$ (10)

In the past we have been estimating a value for $\nu$ by eye from difference sections by sweeping through a range of scalar values and finding one that minimizes the non-reservoir coherent energy in the difference sections. This is unsatisfactory in that it can be overly subjective. However an alternative approach is to make a priori estimations of the signal-to-noise ratio's, and use Equation 9 to calculate $\nu$.

For the synthetic example described below, three independent ``eyeball'' estimates of $\nu$ were 0.77, 0.81 and 0.80. Knowing the noise fields exactly, the value obtained with the formula above was 0.808.

In production environments we expect to use an F-X decon method to estimate noise levels present in the two surveys. Alternatively, the amplitude of a marker horizon above the reservoir could be used to scale the datasets to the correct level.