Instantaneous variation

The partial failure of the synthetic seismogram section to reproduce structural image of the seismic section is caused partly by the presence of unconformities, which limit the spatial extension of major reflectors, and also by the lateral changes in the amplitude and shape of the waveform in a given chronostratigraphic unit. Both the amplitude and temporal separation of the reflectivities associated with this unit can change laterally, but the spatial (depth) pattern is highly uniform, representing the periodic changes in the depositional environment. The impression of extensive lateral continuity in a seismic section comes from the specific patterns in the impedance transitions, not from the values of the transitions themselves.

Figure a shows two of the selected sonic logs. When looking for similarities between the logs, we can begin by focusing on their low-wavenumber behavior, as illustrated in Figure b. We than have two options: to match amplitudes or to match shapes, each of which leads to a different result. If we now analyze the behavior of the higher frequency components of the logs (Figure c), it becomes harder to find a match between the logs. A new look at a reveals that there are different patterns in the high frequency behavior, with strong oscillations in some parts, ``silent" zones (almost without oscillations), and some parts where the oscillations are predominantly to one side of the median.

To measure the degree of local variability of the logs, I used a filter that has the following form:

$$\begin{eqnarray} y_i & = & \sum_{j=i-L/2}^{i+L/2} {\vert x_j - \bar{x_j}\vert \o... ... \bar{x_j} & = & {1 \over L} \sum_{j=k-L/2}^{k+L/2} x_k. \nonumber\end{eqnarray}$$ (1)

The output is a measure of the relative variation of the input inside a window of length L. To keep a more linear relation between the amplitude of oscillation and the filter's output, equation (1) uses the absolute value instead of the usual definition of variation (difference between squares). Figure d corresponds to the output of the variability estimation filter applied to the same part of the logs as that represented in c. Clearly, the logs in d can be correlated better than the logs in c.

 

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Figure 5 (a) Sonic logs of wells 3476 and 3479 of Figure . (b) The same logs after smoothing by convolution with a triangle (400 points of length). (c) An interval of a after weak smoothing (40 points triangle). (d) Same interval of a convolved with a 40 points wide variability filter.

The same filter (using a larger window to avoid spatial aliasing) was applied to the sixteen wells, the results of which appear in Figure b. Figure a corresponds to the smoothing of the original logs with a triangle filter of length equivalent to the the length of the filter applied in b. In both cases, the background trend (very low-frequency components) was subtracted from the output to emphasize phase rather than amplitude. The local variability attribute shows a better lateral correlation over the whole interval than both the synthetic seismograms and the smoothed logs.

 

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Figure 6 (a) The logs shown in Figure a with subtraction of their strongly smoothed version. (b) Local variability estimation applied to the original logs.