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## Causality in multiple dimensions

In order for the PEF to contain the inverse of the data spectrum, it has to be causal. The notion of causality is most obvious when there is a single axis labeled ``time,'' but it extends readily enough to two or more dimensions. Your eye scans the page from left to right and top to bottom; the words previous to the current one are to the left on the same line and everywhere on the lines above. This gives a sort of two-dimensional causal region. Including all the words on previous pages gives a sort of three-dimensional causal region. This is somewhat arbitrary. In some other language the fast and slow axes might be swapped or reversed. The important thing is that along a line parallel to any axis and going through the current point (the zero lag, the value 1z0 in equation pefdef), a causal region lies only to one side of the current location. Figure causal shows an illustration of a two-dimensional causal region. We can make a causal prediction of the data value in the square labeled ``1'' from any or all of the other shaded squares. The shaded squares are labeled to show the layout of of a PEF with four adjustable coefficients. The ``1'' is the value 1z0 in equation pefdef, and the a(n) are the adjustable coefficients.

causal
Figure 1
A two-dimensional causal region. We can predict the data value in the square labeled ``1'' from any or all of the other shaded squares.

Figure 2dpef shows a picture of a 2-D PEF, and Figure 3dpef shows a picture of a 3-D PEF. In both cases, the dark shaded block holds the 1, and the lighter blocks are the coefficients a(n) calculated from the data.

 2dpef Figure 2 Form of a 2-D prediction error filter. The shaded box holds the zero-lag coefficient, with a fixed value of 1.

 3dpef Figure 3 Form of a 3-D prediction error filter. The shaded box holds the zero-lag coefficient, with a fixed value of 1.

Next: Scale invariance and filling Up: Prediction error filters Previous: PEF spectrum equals the
Stanford Exploration Project
1/18/2001