3D wavefield extrapolation with optimum split-step Fourier method

Abstract

A one-way propagator is proposed for more accurately modeling wide-angle wavefields in the presence of severe lateral variations of the velocity. The method adds a higher-order correction to improve the split-step Fourier method by directly designing a cascaded operator that matches the exact phase-shift operator of a varying velocity. Using an optimization scheme, the coefficients in the cascaded operator are determined according to the local velocity distribution and the prescribed angular range of wavefield propagation. The proposed algorithm is implemented alternately in spatial and wavenumber domains using fast Fourier transforms, as in the split-step Fourier and generalized-screen methods. This algorithm can achieve higher accuracy than the generalized-screen method for wide-angle wavefields, although the same numerical scheme is used with comparable computational cost. No extra error arises for the proposed algorithm when used for 3D wave propagation, in contrast to methods that introduce an implicit finite–difference higher-order correction to the split-step Fourier method, such as the Fourier finite difference (FFD) and wide-angle screen methods. A detailed comparison of the proposed one-way propagator with the split-step Fourier, generalized-screen, and FFD methods is presented. The 2D Marmousi and 3D SEG/EAEG overthrust data sets are used to test the prestack depth-migration schemes developed based on the proposed one-way propagators.

Optimum split‐step Fourier one‐way operators for seismic modeling and imaging in 3D VTI media

“…Usually, the minimum V P (x, y, z), e(x, y, z) and d(x, y, z) are chosen as V P 0 (z), e 0 (z) and d 0 (z). In contrast to the isotropic optimum split-step Fourier one-way operators [Liu and Zhang, 2006], the additional terms, i.e., the third terms in the brace of equation (5), are designed to account for the lateral variations of the Thomsen parameters. The expressions of those terms are derived from the rule of the same dimension.…”

“…Following the optimum splitstep Fourier method [Liu and Zhang, 2006], we define the split cascaded operators for propagation of downgoing qPwaves and downward continuation of upgoing qP-waves in VTI media as…”

“…[11] Following the isotropic recursive depth extrapolations based on the dual-domain one-way operators [e.g., Liu and Zhang, 2006], we have…”

“…Then we find the unknown coefficients such that the split cascaded operator matches the spatially varying, exact anisotropic phase-shift operator over a prescribed angular range of wave propagation as accurately as possible. Following Liu and Zhang [2006], we refer to the proposed method as optimum split-step Fourier one-way operators. The splitting in the one-way operators means that the resulting algorithm can be implemented alternately in the space and wavenumber domains using the fast Fourier transforms.…”

“…The first formal dual-domain one-way operator used in seismic imaging is the split-step Fourier [Stoffa et al, 1990]. Since then the Fourier finite-difference [Ristow and Rühl, 1994], generalized-screen [Le Rousseau and de Hoop, 2001b], and optimum split-step Fourier [Liu and Zhang, 2006] have been developed by introducing higherorder corrections to the split-step Fourier operator. The anisotropic schemes proposed by Le Rousseau and de Hoop [2001a] and Han and Wu [2005] are of the dual-domain one-way operators.…”