Tests

I test my irregular-geometry dip estimation method on a decimated subset of the ``quarter dome'' model Claerbout (1999), shown in Figure . Only 100 out of 1024 input traces are assumed known, for a decimation rate of over 90 percent. The dip estimated by my method will then be used to interpolate the missing trace locations by solving the system (7)-(9).

 

qdome-known
Figure 2 Left: 32x32-trace subset of the quarter dome model. The subset was selected to be less sensitive to spatial aliasing than the steeper-dipping portions of the model. Right: Missing data test, with 100 known traces and 924 missing traces.

The left panels of Figure show the quarter dome model's ``known'' dip, which was computed using a variant of Fomel's 2002 dip estimation method. The right panels show the dip estimated from the irregularly-sampled traces shown in Figure , using the method described herein. To smooth the dip estimates between master trace locations, I use an expanding-window smoothing program.

While the dip estimates are decidedly imperfect, they nontheless do contain the general trends seen in the known dip fields. Particularly note the unconformity deep in the section. We see that in the more steeply-dipping parts of the section, my method tends to underestimate the reflector dip. Since the decimation is severe, spatial aliasing may arise, even if the dips are not severe, because my method measures dip directly between two arbitrary traces. Claerbout's puck method is known to be sensitive to spatial aliasing. Fomel (2002) discusses ways to de-sensitize dip estimation with respect to spatial aliasing.

 

qdome-dipcomp
Figure 3 Top row, L-to-R: Known quarter dome x-direction dip; Estimated x-direction dip. Bottom row, L-to-R: Known quarter dome y-direction dip; Estimated x-direction dip.

Figure compares the result of using: a) the known dip, b) the dip estimated from the irregular data, and c) zero dip in solving equations (7)-(9) for an infilled model. We see that in spite of the imperfections of the dip estimated from the irregular data, that it definitely leads to a better inverse interpolation result that the zero dip result, which just smoothes laterally. All results were computed using 20 iterations of a linear conjugate gradient solver.

 

qdome-cubecomp
Figure 4 Left to right: Known result; Decimated data; Data filled with known-dip steering filters; Data filled with estimated-dip steering filters; Data filled with spatial gradient only (zero dip).