The three-dimensional extension of f-x prediction

The extension of f-x prediction to three-dimensions is more difficult than that of t-x prediction. For each frequency, instead of a prediction along a vector, the prediction of a set of complex numbers within a plane is required. For the examples of three-dimensional f-x prediction shown here, we used a complex-valued two-dimensional filter calculated with a conjugate-gradient routine for each frequency. While other techniques for computing this filter exist, they should produce similar results. The advantage of our approach is that the huge matrix $\st X$ used to describe the three-dimensional convolution of the filter with the data does not need to be stored, and the inverse of $\st X^{\dagger} \st X$does not need to be computed, which simplifies the problem significantlyClaerbout (1992a).

The shape of the two-dimensional filter used to predict numbers in a two-dimensional frequency slice has the form  

$$\begin{array} {ccccc} c_{-2,0} & c_{-2,1} & c_{-2,2} & c_{-... ...1,4} \\ 0 & c_{2,1} & c_{2,2} & c_{2,3} & c_{2,4} \end{array},$$ (9)

where all the coefficients are complex-valued. The justification for this shape is more fully discussed on page 198 ofClaerbout (1992a). This form may be compared to a horizontal slice through the center of the filter shown in Figure 3, and if the collection of the filters for all frequencies is Fourier transformed in time, a filter similar to the one shown in Figure 3, but extended in time, is formed.