Fourier reconstruction of signal

Akshay Gulati


This thesis addresses the problem of interpolating missing samples in seismic data. Interpolation is an important step in the seismic data processing flow, since most of the pre-processing algorithms are designed to work with regularly sampled data. In this thesis, I will focus on the Fourier reconstruction techniques which do not require a geological model as input, but use a priori information of the signal to reconstruct the wavefield. This reconstruction requires that the signal be band limited. Most presently available algorithms are slow, because of inability of fast Fourier transform (FFT) to work with irregular samples. In case of the irregularity, the Discrete Fourier transform (DFT) kernel is used for domain mapping. The computational complexity of the DFT is O(N2) as compared to O(NlogN) for the FFT. This extra computing time can cause processing delays so seismic data processors to use simple regularization techniques which compromise accuracy for speed. To address this problem, Non uniform Fast Fourier Transform (NFFT) is implemented, which reduces the complexity of the DFT to O(NlogN) comparable to the FFT. It has been used for seismic data regularization before using a Gaussian convolution kernel. Our newly proposed approach uses a Kaiser Bessel filter for convolution, which gives a better result. We applied this technique to solve the problem of clipped amplitudes in Ground Penetrating Radar data. The NFFT is hybridized with the POCS (Projection on Convex Sets) method to restore clipped peaks from an acquired GPR data set.

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