RELATION TO CLAERBOUT'S AND SPITZ`S SCHEMES

Recently Claerbout (1991) and Spitz (1991) showed comparable results in trace interpolation in the t-x and the f-x domains, respectively. Claerbout gives several schemes in his book, however, in order to compare methods, I consider only the two-step linearized scheme. Both Claerbout and Spitz used prediction-error filters to interpolate traces. Since a prediction-error filter itself has a spectrum that is inverse to the input, the input spectra of their prediction-error filters will help us to understand how both methods estimate the spectrum we want for the interpolated data.

Claerbout found a two-dimensional prediction-error filter with subsampling in the time direction. According to the stretching theorem of the Fourier transform, subsampling in the time is stretching in the spectrum in the frequency. This means that his decimated input spectrum for a prediction-error filter is a spectrum stretched by a factor of two. If the original spectrum has no energy beyond the half the temporal Nyquist frequency, the spectrum stretched by a factor of two is equivalent to taking the original spectrum only from zero to half temporal Nyquist frequency and considering it as a spectrum for the full frequency range.

Spitz computed one-dimensional prediction-error filters along the x direction for each frequency from zero to half the Nyquist frequency and applied each filter to a frequency whose magnitude is twice the original frequency. This approach also implies that he considered the spectrum from zero to half the temporal Nyquist frequency as a whole spectrum after first-order interpolation.