On the orthogonality of anti‐leakage Fourier transform based seismic trace interpolation

Abstract

Seismic data regularization, which aims to estimate the seismic traces on a spatially regular grid using the acquired irregular sampled data, is an interpolation/extrapolation problem. Sampling theory offers the basic conditions for all the seismic data regularization implementations. In sampling theory, Fourier transform plays a crucial role in the analysis of the reconstruction/interpolation basis (interpolant); it estimates the frequency components in frequency/wave‐number domain, and its inverse transform creates the seismic data on the desired regular grid. Difficulties arise from the non‐orthogonality of the global Fourier basis on an irregular grid, which results in the energy from one frequency component leaks onto others. This well‐known phenomenon is called “spectral leakage”. The updated Fourier transform: Anti‐leakage Fourier transform (ALFT) offers to overcome the above mentioned difficulties. It estimates the spatial frequency content on an irregularly sampled grid with significantly reduced frequency leakage. In this paper, we investigate the properties of ALFT and give an insight on how it works. The interpolants are numerically calculated and analyzed in detail. The orthogonality condition of the interpolants is discussed, which demonstrates that the ALFT data reconstruction meets the two most important interpolation conditions (e.g. orthogonal condition and unity condition). With the amplitude analysis on interpolants, the stability of ALFT algorithm is also addressed.