The three-dimensional extension of f-x prediction

The extension of f-x prediction into 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 3-dimensional f-x prediction shown here, I computed a complex-valued 2-dimensional filter at each frequency with a conjugate-gradient routine. While other techniques for computing this filter exist, they should produce similar results. The advantage of this approach is that the huge matrix $\st X$ used to describe the 3-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 2-dimensional filter used to predict numbers in a 2-dimensional frequency slice has the form  

$$\begin{array} {ccccc} c_{-2,0} & c_{-2,1} & c_{-2,2} & c_{-... ...1,4} \\ 0 & c_{2,1} & c_{2,2} & c_{2,3} & c_{2,4} \end{array},$$ (71)

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