5D seismic data regularization by a damped least-norm Fourier inversion

Regularizing inadequate and irregularly sampled seismic data is one of important problems in seismic data processing. An improvement to existing methods to solve this problem is proposed by applying a 5D regularization/interpolation scheme with a damped least-norm Fourier inversion. Under the assumption of planar seismic events within small data windows, the spatial spectrum of regularized data for a fixed frequency should be sparse and have minimum damped norm. The inversion scheme consists in finding a set of regularly spaced spatial Fourier coefficients by minimizing its damped norm for each frequency, subject to the condition that the resulting spatial Fourier coefficients also faithfully reconstruct the original data. The damping factors are automatically derived from the amplitude spectra of the regularized low-frequency data. With the guidance of the damping factors and automatic adjustment of wavenumber ranges according to the Nyquist sampling theory, the proposed inversion algorithm naturally yields a one-step solution for both stabilization and antialiasing of the interpolation problem. A distinctive feature of the method is that it uses high-dimensional nonuniform fast Fourier transforms to evaluate expensive discrete Fourier transforms involved in conjugate gradient iterations. This improves the computational efficiency. The results of applying this algorithm to synthetic and field data demonstrate that it performs well when applied to highly irregular data and outperforms lower dimensional interpolation schemes.