Figure a shows a synthetic section containing three linear events, each of which has a different wavelet. The steepest event has a high-pass wavelet and is almost completely aliased in space. The event with its amplitude decaying along its trajectory has an all-pass wavelet. Finally, the flattest event has a low-pass wavelet. Figure a shows the 2-D spectrum of this section. Clearly, the dip structure in this example varies with respect to frequencies. Figures b and c show the first-order interpolation with Spitz's algorithm and our algorithm, respectively. We can see that both algorithms correctly interpolate two gentle-dip events, even though one of the events has amplitude variation. Spitz's formulation does not explicitly consider the amplitude variations of events, but it is valid when the amplitude variations do not depend on frequency; frequency-dependent amplitude variations may cause some minor problems. Spitz's algorithm fails, however, to interpolate in the correct dip direction of the steepest event that is completely aliased in space because of the frequency-dependent dip structure of data. In contrast, our algorithm interpolates the event correctly. Figures b and c show the 2-D spectra of the interpolated sections, which confirm our observations in the time-space domain. Figure c also displays, at frequency 40 Hz and wavenumber 8 (1/km), a gap in one event, which is a noticeable pitfall of our algorithm. This occurs because our zero-searching routine assumes a single-zero presence at each location. Hence, the neural net may reject a genuine zero if it coincides with a fake zero. Because this problem occurs at only a few frequencies, we do not expect it to have strong impact on the later processing of the interpolated data.

This synthetic example represents an extreme case that may or may not exist in practice. However, it clearly reveals the important assumptions that Spitz's algorithm entails.

Figure 4

Figure 5

11/18/1997