Examples

This section presents several examples of the dip estimation problem with bound constraints. All of these examples were computed with double precision arithmetic. For most of these examples the program stopped because the objective function was not decreasing enough, thus indicating that a possible minimum was found.

Figures to show in (a) the input data, in (b) the estimated dips with L-BFGS-B and no constraints, in (c) the estimated positive dips and in (d), the estimated negative dips. The same clip is applied for all the color plots within each Figure. Warm colors represent positive dips whereas cold colors represent negative dips. Figures and illustrate that the dip estimation program with bounds work as expected, but they do not represent real challenges where, for example, multiple events with different slopes overlap.

Figure shows how the bounds can improve the local dip values. In Figure a, when no bounds are used, a clear cut difference between positive and negative dips is visible. Applying bounds in Figures b and c, the dip estimation program is able to locate the dips of interest when aliasing is present.

On a CMP gather in Figure a, two lines cross at a location where the dips should be positive. If no bounds are applied, Figure b shows that negative dips are found instead. Applying bounds in Figure c, strong positive dips are now recovered. We are able to estimate positive dips beyond aliasing, thus improving on the existing program. Similar conclusions can be made on Figure .

It might happen that we cannot separate dips as easily as we could in Figures , and . For example, Figure displays some earthquake data from Professor Peter Shearer for which positive and negative dips do not clearly separate (see, for instance, where the two black lines cross). The problem here stems from the fact that the event with negative dips (in blue) is much stronger than the overlapping event with positive dips at this location.

Finally, Figure shows a dip decomposition of the data in Figure a. This illustrates the ability to select a small range of dips that goes beyond the simple positive/negative constraints of the preceding examples.