INTERPOLATION

The interpolation considered in this paper is first-order interpolation, achieved by interlacing the space axis. Higher-order interpolation can be obtained by applying the first-order interpolation scheme several times. Now we know the spectrum which is supposed to result after interpolation, our next step is to find a filter which has an amplitude spectrum that is inverse to the spectrum we estimate. The filter can be found in both the t-x domain and the f-x domain. In this paper, I choose the f-x domain for its simplicity.

The procedure for an interpolation can be summarized as follows:

1.

I pad zeros in the time domain with the same number of original time samples, and Fourier transform over time to $(\omega, x)$.

2.

For each frequency from zero to half Nyquist, I find a minimum phase wavelet whose inverse spectrum is the spectrum of the signal along the x axis. Here, I used Kolmogoroff spectral factorization for finding minimum phase filters.

3.

For each frequency, I find missing data by minimizing the filtered output in the least-squares sense using a conjugate-gradient algorithm. The filter applied is the wavelet which comes from the half frequency.

Another approach for finding filters is to obtain a prediction-error filter for each frequency, because the prediction-error filter has a spectrum that is inverse to the input (Claerbout, 1991). If you want to limit the length of filter, the prediction-error filter is more attractive than spectral factorization.