Interpolation with spatial predictors

A two-dimensional filter

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 ``$\cdot$'' denote a zero, we denote a two-dimensional filter

that can be a dip-rejection filter as  

$$\begin{array} {ccccc} a &b &c &d &e \ \cdot &\cdot &1 &\cdot &\cdot \end{array}$$ (6)

where the coefficients (a,b,c,d,e) are to be estimated by least squares in order to minimize the power out of the filter. (In the table, the time axis runs horizontally, as on data displays.)

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.