Any interpolation method fills missing traces using a prior estimate of the missing data's spatial correlation; obtaining this correlation is often the main challenge. When the data geometry is regular, many existing methods can resample the data onto a finer grid. Sinc interpolation (e.g.: Bracewell (1986)) is optimal for the regular resampling of band-limited data. Crawley (2000) estimates a nonstationary prediction-error filter (PEF) on regularly sampled, spatially-aliased seismic data and uses inverse interpolation Claerbout (1999) to fill missing traces on a finer grid. Fomel (2001) solves the same problem, but substitutes ``plane-wave-destructor'' filters, derived from estimated reflector dip, for a PEF.
Unfortunately, irregular data geometry renders most conventional methods, including those mentioned previously, inapplicable. To reliably estimate autoregressive filters, like the PEF, all points in the filter stencil must, but generally do not, fall on known data locations. A multi-scale autoregression technique Curry and Brown (2001); Curry (2002) has yielded some success by estimating the PEF simultaneously from data subsampled to a series of different resolutions. Biondi and Vlad (2001) uses azimuth moveout to transform known data to arbitrary azimuth/offset bins to constrain missing data. Other techniques (e.g: Liu and Sacchi (2001) and Zwartjes and Hindriks (2001)) solve an inverse interpolation problem in the Discrete Fourier Transform domain by regularizing the unknown coefficients. Nonetheless, a industry-standard technique does not yet exist.
In this short note, I present a different way to obtain the correlation between irregularly-sampled traces in three dimensions. Given a pair of traces, as shown in Figure , reflector dip can be measured along an arbitrary azimuth, by (for instance) Claerbout's ``puck'' method Claerbout (1992). If the dip between a master trace and two or more neighbors is measured along two distinct azimuths, we can solve a simple least-squares problem for the dip in the x- and y-directions at the master trace's location. Both geology and survey geometry often dictate that the reflector dip should vary smoothly in space. Therefore it is both natural and intuitive to extend the estimated reflector dip to missing data locations.
I test my dip estimation scheme on a sparsely and irregularly sampled 3-D synthetic model. I use the estimated dip to compute ``steering filters'' Clapp et al. (1997) which regularize an inverse interpolation problem. This choice of regularization leads to a far better result than the spatial gradient operator, which corresponds to the assumption of zero dip between traces.