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eismic images exhibit considerable local continuity.With the exception of unusual amplitude anomalies and faults, most seismic events can be characterized asexhibiting smoothly varying local dip. Bahorich andFarmer and Gersztenkorn and Marfurt (see “Suggestionsfor further reading”) show that deviations from local pla-nar behavior can be crucial in unraveling complex faultpatterns, improving interpretations, and delineatingreservoir extent.

As solutions to least squares optimization problems,local dip estimates are functions of the local spatial andtemporal derivatives of the 3-D seismic wavefield. Becausedifferentiation is a natural edge detector, unusual eventsin 3-D volumes frequently result in anomalous estimatesof local slowness. This article shows that the magnitudeof local dip estimates is an excellent detector of interest-ing events in 3-D seismic volumes.

Detection of plane waves. In his 1992 book Earth Sound-ing Analysis, Jon Claerbout shows how to use least squaresto calculate a vector (whose components are time averagesof spatial derivatives) that defines a local plane at eachpoint in the 3-D volume. Its magnitude is a scalar whichresponds in a somewhat dramatic fashion to relativelyminor deviations from local plane-wave assumptions. Thereasons for this response may not be completely clear. Fortime-migrated 3-D volumes, the x and y components of thevector are estimates of the post-migration dip. For a givenvelocity, they provide the basis for computing the zero-off-set point from which the local primary-reflection horizonmigrated. In this case, the dip magnitude is in some senseproportional to the distance to this zero-offset location. Forlarge dipping events, such as faults, this distance can bequite large.

Abrupt changes in amplitude or wavelet phase canalso result in major changes in dip magnitude. The vectorcomponents are estimates of differential change and sowill be significant whenever discontinuities are present inthe data volume.

Application. Figure 1 is a 3-D view of a small data volumewith good local plane wave behavior. While faults withlarge throw are clearly visible, those with smaller dis-placement are not. Claerbout’s least square method, witha three-point spatial and a 44-ms temporal smoother, wasused to derive the dip-magnitude cube shown in Figure2. Figure 2 is indicative of the fact that typical cross-sec-tional views of dip-magnitude cubes are not pleasing andreveal little about potential interesting events in the data.Figure 3 is a time slice through the dip-magnitude data setat 2.76 s. It should be compared with the equivalent inputtime slice in Figure 4. Interesting anomalies are now easily recognized. A careful review of the dip-magnitudevolume shows that many low-resolution events are high-lighted and much more easily recognized. The availabili-ty of dip-magnitude data has certainly increased the overallinformation content.

Enhancements. As suggested by S. Fomel, a possibleenhancement is the extension of Claerbout’s least squarescoherency to 3-D. In 2-D, this measure quantifies the quality

Least squares dip and coherency attributesJ. BEE BEDNAR, Houston, Texas

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Figure 1. Original 3-D data volume.

Figure 2. Dip-magnitude 3-D data volume.

Figure 3. Dip-magnitude slice at 2.76 s.

778 THE LEADING EDGE JUNE 1998

of the planar fit and is more statistically related to coheren-cy than the dip magnitude. A direct comparison of the twomethods clearly suggests that coherency will at least bebetter normalized. Figure 5 is a time slice at 2.76 s after cal-culation of the magnitude of the coherency at each pointin the volume displayed in Figure 1. In the view of theauthor, this display is a much improved version of Figure3. Moreover, the volume display in Figure 6 indicates someimprovement in the quality of vertical sections as well.Only a careful comparison of each volume in a valid inter-pretation setting will reveal which result is more useful.

In addition to the two approaches described here,other ideas may prove interesting. Estimates of local dipcan be used to orient calculation of other desirable statis-tical quantities. Given sufficient daring, one can considerthe computation of local curvature throughout the entire

volume. Displaying such values on a horizon slice throughthe volume should provide estimates of the local rough-ness and perhaps form a basis for fractal-based predictionof faulting below seismic resolution.

Conclusions. The identification of faults in 3-D data vol-umes can be enhanced through the use of simple least-squares-based edge-detection methods. The use of dip-magnitude and coherency based on local dip estimates isshown to be an effective approach to detection and iden-tification of both high- and low-relief fault patterns. Three-dimensional dip estimates can be an effective tool for theinterpretation of 3-D volumes.

Suggestions for further reading. “The coherency cube” by M. Bahorich and S. Farmer (TLE, October 1995). “Eigen-structure based coherence computations” by A. Ger-sztenkorn and K. Marfurt (SEG Expanded Abstracts, 1996).

Acknowledgments: Thanks go to Jon Claerbout for providing the oppor-tunity to participate in the Stanford Exploration Project during thesummer of 1997, to Sergey Fomel for the coherency suggestion, and toChristine Ecker for testing my changes after I left SEP.

Corresponding author: Bee Bednar, email bednarb@3dbee.com

Figure 4. Original slice at 2.76 s. Figure 5. Coherency slice at 2.76 s.

Figure 6. Coherency 3-D data volume.

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