This chapter 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, based on Treitel's complex series prediction workTreitel (1974). This technique divides the two-dimensional filtering problem into many one-dimensional filtering problems in space, one for each frequency. These f-x prediction techniques were significant improvements over the other noise attenuation methods available at that time. The name Gulunay used for this process was FXDECON, which stood 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 chapter, 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 presented in this chapter 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.
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, its many free filter coefficients allow some random noise to be passed and spurious events to be generated. Since the length in time of the t-x prediction filter can be controlled, t-x prediction avoids the disadvantages of f-x prediction.
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 Abma (1993); Chase (1992); Gulunay et al. (1993). . I found that extending these prediction techniques from two dimensions into three dimensions produces better results than two passes of the 2-dimensional processes in the inline and crossline directions.
This chapter examines these two prediction techniques and compares the results in both two and three dimensions. I also show some of the advantages that 3-dimensional prediction has over the 2-dimensional applications of these techniques.