is a small plane of numbers that is convolved over a big data plane of numbers. One-dimensional convolution can use the mathematics of polynomial multiplication, such as Y(Z)=X(Z)F(Z), whereas two-dimensional convolution can use something like Y(Z1,Z2)=X(Z1,Z2)F(Z1,Z2). Polynomial mathematics is appealing, but unfortunately it implies transient edge conditions, whereas here we need different edge conditions, such as those of the dip-rejection filters discussed in Chapter , which were based on simple partial differential equations. Here we will examine spatial prediction-error filters (2-D PE filters) and see that they too can behave like dip filters.
The typesetting software I am using has no special provisions for two-dimensional filters, so I will set them in a little table. Letting ``'' denote a zero, we denote a two-dimensional filter
that can be a dip-rejection filter as
Fitting the filter to two neighboring traces that are identical but for a time shift, we see that the filter (a,b,c,d,e) should turn out to be something like (-1,0,0,0,0) or (0,0,-.5,-.5, 0), depending on the dip (stepout) of the data. But if the two channels are not fully coherent, we expect to see something like (-.9,0,0,0,0) or (0,0,-.4,-.4,0). For now we will presume that the channels are fully coherent.