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 method can handle arbitrarily complex media, but can be inefficient in 3D and has limited dip bandwidth. We proposed a new Fourier method based on Chebyshev expansion of the second kind. Both theoretical analyses and numerical experiments show that the proposed method is comprehensively superior to a similar method based on 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 about one-third that of Chebyshev expansion of the first kind. Our method is also superior to the implicit finite-difference 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 finite-difference method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. The proposed method shows 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 finite-difference method and Chebyshev expansion of the first kind; meanwhile, our method has less imaging artifacts since it can better position the reflectors.

Full-Matrix Imaging in Fourier Domain towards Ultrasonic Inspection with Wide-Angle Oblique Incidence for Welded Structures

“…Specifically, we utilize Chebyshev polynomials of the second kind. As demonstrated by Song et al [38], a significant enhancement in overall accuracy has been reported with the corresponding Fourier method. The Chebyshev polynomials of the second kind are shown as follows…”

“…The introduced error in approximation typically hinges on the retained terms or the order of expansion. Diverging from the conventional expansion over slowness perturbation, we opt to directly expand the wavenumber k z [38]. Assuming s is equal to ck x /ω, the wavenumber k z is denoted as…”

“…However, the accuracy and stability of Zhang et al's method are still constrained, particularly in the context of large-angle oblique incidence in UNDT. In 2022, Song et al introduced the direct Chebyshev expansion of the second kind for wide-angle approximation without global optimization [38]. The second-order Chebyshev expansion achieves accuracy up to 60 • , and the fourth-order is accurate up to 80 • .…”