Trace interpolation using spectral estimation

Jun Ji

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ABSTRACT

A scheme for missing-trace interpolation of linear events is proposed. For a two-dimensional dataset which contains linear events, a post-interpolation spectrum can be estimated from a portion of the original aliased spectrum. The restoration of missing trace data is accomplished by minimizing the energy after applying a filter which has an amplitude spectrum that is inverse to the estimated spectrum.

Jon Claerbout (1991, p.175) has given us a direction for interpolation in his new book as follows: ``A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy.'' He has also suggested choosing the filter to have an amplitude spectrum that is inverse to the spectrum we want for the interpolated data. Following his direction, we should predict the spectrum for the interpolated data. For a one-dimensional dataset that is regularly sampled and aliased, we cannot use the spectrum of the known data as a spectrum for the interpolated data because we cannot distinguish the unaliased spectrum from the aliased spectrum. Instead, we use the original spectrum, spectrally weighted. This approach gives low weighting to high frequency, where most of aliasing occurs. Such spectral weighting, however, is extremely subjective. On the other hand, for a two-dimensional dataset, we can estimate a spectrum which is supposed to be the result of interpolation when all events are linear.