Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

Abstract

The Fourier method for one-way wave propagation is efficient but potentially inaccurate in complex media. The implicit finite-difference (FD) method can handle arbitrarily complex media, but it can be inefficient in 3D and has limited dip bandwidth. We have adopted a new Fourier method based on the Chebyshev expansion of the second kind. Theoretical analyses and numerical experiments indicate that our method is comprehensively superior to a similar method based on the Chebyshev expansion of the first kind in terms of balanced amplitude and error tolerance. Within the dip bandwidth from 0° to 65°, the fourth-order form of our method has an error tolerance of 2%, which is approximately one-third that of the Chebyshev expansion of the first kind. Our method also is superior to the implicit FD method in several important aspects: effective bandwidth, computational efficiency, numerical dispersion, and two-way splitting error. It can be easily extended from 2D to 3D compared with the FD method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. Our method finds better imaging results of the SEG/EAGE model by providing a well-focused salt dome, flank, and bottom as well as the detailed structures beneath the salt body, compared with the implicit FD method and the Chebyshev expansion of the first kind; meanwhile, our method has fewer imaging artifacts because it can better position the reflectors.