A dip filter is a two-dimensional filter that is intended to weight information depending on its dip and not on its frequency content. Dip filters may be designed by Fourier analysis which tends to make filters of great extent, or by partial differential operators which makes filters that are short, but imperfect near Nyquist frequencies. For example the 1-D filter (1,-1) is like a time derivative only far from Nyquist. Likewise the 2-D filter made by expressing the partial differential operator as a differencing star on a mesh has its expected dip behavior only far from Nyquist. The derivative filter creates an output with a vanishing zero-frequency component. The dip filter creates an output which vanishes at dip p in dip space. (For more details about dip space, see IEI.)
When the mesh is refined, say by interlacing a new row and a new column between each existing row and column, two interesting questions arise. First we'll consider the questions independently, and then we'll consider them together. The first question is how to reexpress the filter on the refined mesh. Since refining both the t mesh and the x mesh simultaneously preserves the apparent dip, the values on the differencing star of a dip filter should remain unchanged--you just push them closer together. The second question is how to find new data values that interlace the given data values. This missing-data problem is one of linear least squares that is straightforward in principle. As I described in an earlier chapter the method fills out the data space by minimizing the output energy from the given filter. Interpolating the time axis is a boring question, but interpolating the space axis in seismology is a question of great interest because we often encounter spatially aliased data which frustrates many of our data analysis procedures.