An example of three-dimensional prediction's ability to predict nonlinear events is shown in Figure 4, where the input consists of several dipping layers cut by a fault in the crossline direction, so that events are nonlinear in the inline direction. The one-pass three-dimensional prediction did not smear the fault, because the calculated three-dimensional filter created a prediction in the crossline direction that preserved the discontinuity in the inline direction. With the two-pass prediction, the inline pass smeared the reflections across the fault. For the noiseless case of Figure 4, the f-x prediction is not shown because it gave the same results as the t-x prediction.
The results of applying f-x prediction and t-x prediction to a three-dimensional land survey provided by ARCO are shown in Figures 5 to 9 to demonstrate the differences between the two processes. This data set is interesting because it has a significant noise level with fairly flat, predictable events.
For both two-pass applications, the filter size was five elements in the spatial direction. For both three-dimensional one-pass applications, we employed a filter with five elements in both spatial directions. The t-x prediction used a five-element filter length in the time direction for both one- and two-pass applications. The window sizes were 60 traces in the inline direction, 60 traces in the crossline direction, and 200 samples, or 0.4 seconds in time.
Both the two-pass and the one-pass t-x prediction results in Figures 8 and 9 show less noise than the corresponding f-x results; otherwise the results are similar. While the one-pass t-x prediction and f-x prediction results are much the same, the t-x prediction output shows somewhat less noise.
The advantage of using three-dimensional lateral prediction is especially clear in the results shown in Figures 7 and 9. Both the one-pass results show significantly less smearing of the structure. On the top faces of the cubes in Figures 7 and 9, the one-pass results appear clean and reasonable, whereas the two-pass results show smearing along the inline and crossline directions. An example of the smearing of the detail can be seen in point A of the figures, where a small doughnut-shaped feature is badly smeared in both the two-pass results. The front face of the cubes in Figures 6 to 9 are significantly different for the one- and two-pass results; with the one-pass results showing much more detail. The features at point B in the figures once again demonstrate the loss of detail. Although the differences between the three-dimensional t-x and f-x results in Figures 7 and 9 are less than those between the two-dimensional t-x and f-x results in Figure 2, the results of the three-dimensional t-x prediction appear slightly cleaner than those of the three-dimensional f-x prediction.