Seismic wave extrapolation using lowrank symbol approximation

INTRODUCTION

Wave extrapolation in time plays an important role in seismic imaging (reverse-time migration), modeling, and full waveform inversion. Conventionally, extrapolation in time is performed by finite-difference methods (Etgen, 1986). Spectral methods (Reshef et al., 1988; Tal-Ezer et al., 1987) have started to gain attention recently and to become feasible in large-scale 3-D applications thanks to the increase in computing power. The attraction of spectral methods is in their superb accuracy and, in particular, in their ability to suppress dispersion artifacts (Chu and Stoffa, 2008; Etgen and Brandsberg-Dahl, 2009).

Theoretically, the problem of wave extrapolation in time can be reduced to analyzing numerical approximations to the mixed-domain space-wavenumber operator (Wards et al., 2008). In this paper, we propose a systematic approach to designing wave extrapolation operators by approximating the space-wavenumber matrix symbol with a lowrank decomposition. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. The optimized separable approximation or OSA (Song, 2001) was previously employed for wave extrapolation (Du et al., 2010; Zhang and Zhang, 2009) and can be considered as another form of lowrank decomposition. However, the decomposition algorithm in OSA is significantly more expensive, especially for anisotropic wave propagation, because it involves eigenfunctions rather than rows and columns of the original extrapolation matrix. Our algorithm can also be regarded as an extension of the wavefield interpolation algorithm of Etgen and Brandsberg-Dahl (2009), with optimally selected reference velocities and weights. Another related method is the Fourier finite-difference (FFD) method proposed recently by Song and Fomel (2011). FFD may have an advantage in efficiency, because it uses only one pair of multidimensional forward and inverse FFTs (fast Fourier transforms) per time step. However, it does not offer flexible controls on the approximation accuracy.

Our approach to wave extrapolation is general and can apply to different types of waves, including both acoustic and elastic seismic waves, as well as velocity continuation (Fomel, 2003b), offset continuation (Fomel, 2003a), prestack exploding reflector extrapolation (Alkhalifah and Fomel, 2010), etc.

The paper is organized as follows. We first present the theory behind the proposed algorithm, then describe the algorithm and test its accuracy on a number of synthetic benchmark examples of increasing complexity.

Seismic wave extrapolation using lowrank symbol approximation