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 used to describe the 3-dimensional convolution of the filter with the data does not need to be stored, and the inverse of 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