F-x prediction expressed as t-x prediction

As shown in Abma and Claerbout 1993 and in chapter  of this thesis, the similarities and differences between f-x prediction and t-x prediction are explained by comparing the form of the t-x prediction filter and the form of the effective t-x domain filter created by f-x prediction. This comparison, which is also shown in chapter , is quickly reviewed here.

For each frequency, f-x prediction generates a prediction filter over the samples in space. The individual filters calculated by the f-x prediction have the form  

$$\begin{array} {ccccc} 1 & a_{1} & a_{2} & a_{3} & a_{4} \end{array},$$ (72)

where a₁,a₂,a₃, and a₄ are complex numbers. Collecting these filters for all frequencies and presenting them in the frequency-space domain produces the following composite filter:

$$\begin{array} {ccccc} \vdots & \vdots & \vdots & \vdots & \... ...,4} \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{array}.$$ (73)

Transforming this filter from the frequency-space domain into the time-space domain gives the effective t-x domain filter,  

$$\begin{array} {ccccc} \vdots & \vdots & \vdots & \vdots & \... ...,4} \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{array},$$ (74)

where the number of rows is the number of time samples in the data being considered, i.e. the design window. The Fourier transform has converted the first column of 1s into a single 1 at the output position, creating a time-space filter.

A typical t-x prediction filter appears as:  

$$\begin{array} {ccccc} 0 & b_{-1,1} & b_{-1,2} & b_{-1,3} & ... ...0,4} \\ 0 & b_{1,1} & b_{1,2} & b_{1,3} & b_{1,4} \end{array}.$$ (75)

This filter has the same form as the f-x prediction filter shown in () but with the number of rows greatly decreased. The action of an f-x prediction filter can be seen to be the action of a t-x prediction filter with a very long time-length.