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Introduction

The task of wave extrapolation is propagation of waves in depth or time, which finds applications in seismic modeling and migration. Conventionally, it is implemented by finite differences (FD). FD methods have a low computational cost but suffer from dispersion artifacts and instabilities (Kosloff and Baysal, 1982). Instead, numerical differentiation of space coordinates in wave equations can be implemented by employing the Fourier transform, which are known as pseudo-spectral methods (Fornberg, 1998). Pseudo-spectral methods have higher accuracy and can suppress numerical dispersion (Virieux et al., 2011; Reshef et al., 1988). On the other hand, they are limited to small time steps and may be subject to dispersion because of finite-difference approximations used for the time derivatives. Recently, alternative strategies have been proposed to propagate waves by mixed-domain space-wavenumber operators (Fomel et al., 2013; Liu et al., 2009; Pestana and Stoffa, 2010; Du et al., 2010; Soubaras and Zhang, 2008; Wu and Alkhalifah, 2014; Zhang and Zhang, 2009; Chu and Stoffa, 2011; Etgen and Brandsberg-Dahl, 2009; Wards et al., 2008). Fowler et al. (2010b) and Du et al. (2014) referred to these methods as recursive integral time extrapolation (RITE) methods.

RITE methods are designed to make time extrapolation both stable and dispersion free in heterogeneous media for large time steps, even beyond the Nyquist limit (Du et al., 2014), and therefore are particularly suitable for reverse-time migration (RTM), a depth-migration method in which waves are extrapolated in time (Farmer et al., 2006; Fletcher et al., 2009; McMechan, 1983; Baysal et al., 1983; Fowler et al., 2010a; Whitmore, 1983). RTM performs imaging by solving the two-way wave equation in the most straightforward manner compared with other methods, and therefore is capable of handling complicated wave forms, such as prismatic waves, and generating images free of artifacts that are caused by approximations of the physics of wave propagation in other methods (Leveille et al., 2011; Etgen et al., 2009). The efficiency of RTM thus relies on the wave propagation engine.

Among the different RITE approaches, the lowrank wave extrapolation (Fomel et al., 2013,2010) distinguishes itself by its high efficiency and flexible control over approximation accuracy. Lowrank approximation has been implemented under different frameworks, including lowrank finite differences and lowrank Fourier finite differences (Song et al., 2013; Fang et al., 2014). A lowrank algorithm decomposes the original mixed-domain wave propagation matrix into a small set of representative spatial locations and a small set of representative wavenumbers. Similar to other spectral methods, the cost of computation per time step with the lowrank method is $ O(N N_x \log{N_x})$ , where $ N_x$ is the total size of the computational grid and $ N$ is a small number (the rank of the approximation) controlling the trade-off between accuracy and efficiency.

Fomel et al. (2013) implemented lowrank wave extrapolation using a two-step time marching scheme, which involves a real-valued wave propagation operator. The application of a two-step scheme is constrained by its requirement of a real-valued phase function. In this paper, we propose adopting a one-step scheme, which propagates a complex-valued wavefield using a complex phase function. Following Zhang and Zhang (2009), we show that separation of forward- and backward-propagating waves can be achieved by constructing a complex wavefield, with its imaginary part being the Hilbert transform of the real part. The complex wavefield corresponds to the analytical signal (Taner et al., 1979). In practice, the proposed one-step scheme demonstrates significantly improved stability. Its ability to extrapolate waves using large time steps should help reduce the cost of computationally intensive tasks, such as RTM and time-domain full waveform inversion (FWI). A complex-valued phase function is capable of incorporating modified forms of dispersion relations. In particular, we show that, by including a velocity gradient term into the approximation of phase function, the accuracy of wave extrapolation can be improved in media with large velocity variations. Additionally, we propose a propagation-direction-dependent absorbing boundary condition that can be incorporated into the complex-valued phase function. This condition behaves like an anisotropic attenuation term, which attenuates waves preferencially in the direction perpendicular to the absorbing boundary, thus reducing artificial reflections at wide incident angles. Such modifications of the phase function take advantage of the complex-valued symbol in the one-step scheme. The proposed method is easily extended to anisotropic wave propagation, such as tilted transversely isotropic (TTI) or orthorhombic media, without producing unwanted residual shear-wave energy (Du et al., 2014). We use numerical examples with synthetic models to test the accuracy, efficiency and stability of the proposed lowrank one-step wave extrapolation method. Finally, we apply lowrank one-step RTM to the BP 2007 TTI synthetic data set to test its ability of producing high-quality seismic images.


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Next: Theory Up: Sun et al.: Lowrank Previous: Sun et al.: Lowrank

2016-11-16