Given quality local dip estimates of the facies template and seismic image, our approach may seem like overkill. The simplest alternative would be a direct subtraction approach, i.e., extract small windows the same size as the facies template from the seismic image's dip estimate, then take the RMS difference of it and the facies template's dip estimate and sum to get the attribute value. Such an approach would surely suffer from the extreme sensitivity of Claerbout's 1992 dip estimation technique to discontinuities in the data. The estimated local dip is anomalously high at discontinuities, and would thus skew a sum of squared differences. Bednar (1997) defined the dip estimate as a coherency attribute based on this property.
A second alternative method might use the fact that any decorrelating filter whitens the spectrum of the data. If a decorrelating filter is obtained for the facies template, then convolved with the seismic image, the local spectrum of the output should be whitest where the image most resembles the facies template. Unfortunately, this would require some measure of whiteness, i.e., how closely does the local autocorrelation of the residual resemble a spike, so in some sense we'd be back to square one. In any case, the authors have found through experience that direct interpretation of the residual to this end is often nonintuitive.
In practice, the method will interpret some regions which are obviously not the same geologic feature as the facies template as such. This brings up an important caveat: the program interprets data in terms of local dip spectrum only; not contextually. Such contextual interpretation is best done by a human, and certainly this will remain true for some time. Our method simply makes human interpretation of 3-D images feasible by directing the interpreter to the regions of the data which have a local dip spectrum which might match the facies template. Look for regions in Figure 9 which have a similar dip spectrum as the template of Figure 8 - you should be able to find many.
The algorithm we presented is still in prototype stage, but nontheless, we are encouraged by the performance characteristics. The algorithm required approximately five minutes on a single processor of our SGI Origin 200 to compute the real data example (Figure 9), including the dip estimation. We subsampled the 200x100-point output space by a factor of five, both spatially and temporally, for a total of 1600. These figures are not stunningly good, considering that the number of output points in many 3-D seismic images may number a million or more, but two facts leave room for improvement. First, since the output depends only on the input, the algorithm is highly parallelizable. Second, since the actual signal/noise separation panels are not output, we have found that the number of iterations per output point may be cut radically, say to five or less.
Our approach may also be useful in AVO analysis. AVO anomalies are relatively easy to spot, meaning a simple facies template, but the sheer volume of the data makes hand interpretation tedious.
Currently our approach operates on 2-D data only, and we believe that the extension to 3-D is possible, but there are theoretical issues to take stock of. Fomel (1999) discusses these issues in detail; I paraphrase his work here. In 2-D, the dip p is a scalar value; in 3-D it is a vector: . Plane waves in 3-D can be extinguished Schwab (1998) by a cascade of convolutional operators similar to the operator of equation (8). The matrix equivalent of such an operator is nonsquare and thus noninvertible. Unfortunately, for this application, we require the inverse, i.e., equation (7) Dubbing this composite operator , Fomel forms and performs spectral factorization to produce a single minimum phase operator.