Most seismic reflections are recognized by their lateral continuity, and this continuity is used to distinguish events of interest from the background of noise. In this discussion, we assume the events of interest are linear, that is, events are lines in a two-dimensional image and planes in a three-dimensional image. For nonlinear events, the images are subdivided, or windowed, into smaller sections where the events of interest are approximately linear.
This paper discusses two approaches to predicting linear events: a frequency-space, or f-x, prediction technique, and a time-space, or t-x, prediction technique. The f-x prediction technique was introduced by Canales 1984 and further developed by Gulunay 1986. The standard industry name for this method is FXDECON, which stands for frequency-space domain predictive deconvolution. The reference to deconvolution is something of a misnomer, since deconvolution refers to the removal of predictable information, whereas in this paper, the data of interest are the predictable parts of the input. After a transformation of the data from the time-space domain into the frequency-space domain, the process of predicting a linear event can be divided into many smaller problems of predicting periodic events within a frequency. The t-x prediction process introduced in this paper is done with a single prediction filter calculated in the time-space domain using a conjugate-gradient method. The conjugate-gradient method and programs for filter calculations similar to the ones used here are discussed in Claerbout 1992a and Claerbout 1992b.
While the two methods generally produce similar results, t-x prediction has several advantages over the older f-x prediction. These advantages allow t-x prediction to pass less random noise than the f-x prediction method. Most of the extra random noise is passed because the f-x prediction technique, while dividing the prediction problem into separate problems for each frequency, produces a filter as long as the data series in time when the collection of filters is transformed into a single filter in the t-x domain. Because the f-x prediction filter is very long in the time direction, it allows some random noise to be predicted. Since the length in time of the t-x prediction filter can be controlled, t-x prediction avoids passing this excess noise.
In three-dimensions, better results are expected for both techniques since more data goes into every prediction and since some of the linearity assumptions can be relaxed in three dimensions (Chase 1992; Gulunay et al., 1993). We find that extending these prediction techniques from two dimensions into three dimensions produces better results than two passes of the two-dimensional processes in the inline and crossline directions.
This paper examines these two prediction techniques and compares the results in both two and three dimensions. We also show some of the advantages that three-dimensional prediction has over the two-dimensional applications of these techniques.