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5D Regularization & Interpolation
Anti-leakage Fourier Transform
At first it estimates the spatial frequency content on an irregularly sampled input grid. Then, it reconstructs the seismic data on any desired grid. Three main difficulties arise in this process: the “spectral leakage” problem, the accurate estimation of Fourier components, and the effective antialiasing scheme used inside the algorithm. The antileakage Fourier transform algorithm can overcome the spectral leakage problem and handles aliased data. A weighting scheme is used to accurately estimate the Fourier components and a wavenumber domain oversampling inversion scheme is used to reduce Gibbs phenomenon.
Matching Pursuit Fourier Interpolation (MPFI) is a frequency-domain method. The data is transformed from space-time domain to the space-frequency domain by fast Fourier Transform (FFT). The following steps are performed for each frequency slice:
1. Compute the Fourier domain using discrete Fourier Transform (DFT)
2. Find the Fourier component with maximum energy
3. Add this Fourier component to the estimated spectrum
4. Transform this Fourier component back to the input locations
5. Subtract the result from step 4 from the input data for this iteration
6. The result of step 5 is the input for the next iteration
In each iteration, the estimated spectrum is built up. After a sufficient number of iterations, a direct inverse DFT can be used to obtain the interpolated data at the desired output locations. This is followed by an inverse FFT from frequency to time
- Xu, S., Zhang, Y., and Lambaré, G., 2010, Antileakage Fourier transform for seismic data regularization in higher dimensions: Geophysics, 74, 113-120.
- Zwartjes, P. M., and M. D. Sacchi, 2007, Fourier reconstruction of nonuniformly sampled, aliased seismic data: Geophysics, 72, 21-22.
- Liu, B., and M. D. Sacchi, 2004, Minimum weighted norm interpolation of seismic records: Geophysics, 69, 1560-1568.
- Mallat, S. G., and Z. Zhang, 1993, Matching pursuits with time-frequency dictionaries: IEEE Transactions on Signal Processing, 41, no.12, 3397-3415.
- Duijadam, A. J. W., and M. A. Schonewille, 1999, Nonuniform fast Fourier transform: Geophysics, 64, 539-551.