An algorithm for interpolation in the pyramid domain‡

With the pyramid transform, 2D dip spectra can be characterized by 1D prediction‐error filters (pefs) and 3D dip spectra by 2D pefs. These filters, contrary to pefs estimated in the frequency‐space domain (ω, x) , are frequency independent. Therefore, one pef can be used to interpolate all frequencies. Similarly, one pef can be computed from all frequencies, thus yielding robust estimation of the filter in the presence of noise. This transform takes data from the frequency‐space domain (ω, x) to data in a frequency‐velocity domain (ω, u=ω·x) using a simple mapping procedure that leaves locations in the pyramid domain empty. Missing data in (ω, x) ‐space create even more empty bins in (ω, u) ‐space. We propose a multi‐stage least‐squares approach where both unknown pefs and missing data are estimated. This approach is tested on synthetic and field data examples where aliasing and irregular spacing are present.