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Theory and practice of interpolation in the pyramid domain

Antoine Guitton and Jon Claerbout


With the pyramid transform, 2-D dip spectra can be characterized by 1-D prediction-error filters (pefs) and 3-D dip spectra by 2-D pefs. This transform takes data from $(\omega ,x)$-space to data in $(\omega,u=\omega\cdot x)$-space using a simple mapping procedure that leaves empty locations in the pyramid domain. Missing data in $(\omega ,x)$-space create even more empty bins in $(\omega ,u)$-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.