Recursive integral time extrapolation of elastic waves using low-rank symbol approximation

Discussion

In this paper, we have considered second-order displacement equations for elastic wave extrapolation. However, the framework presented here is not limited to such a formulation. For example, second-order stress equations can be used, which has the advantage of not calculating spatial derivative of stiffnesses. Other first-order formulations should also be investigated.

Our formulation can be viewed as an extension of the one-step extrapolation method by Zhang and Zhang (2009) to elastic anisotropic media. It has been shown by Bleistein et al. (2008) that the one-step solution is asymptotically true amplitude, i.e., it provides the same traveltime and leading order amplitude as conventional acoustic wave propagation. It is reasonable to expect that the new formulation for elastic waves has similar behavior. However, more rigorous theoretical proof is required to arrive at a definitive conclusion.

In the last numerical example, we have demonstrated that our method is advantageous in providing directional information about the wavefield. This allows for efficient computation of wavefield up-down separation, RTM angle gathers and absorbing boundary conditions (Sun et al., 2016b; Hu et al., 2016; Shen and Albertin, 2015) in the context of elastic imaging.

Recursive integral time extrapolation of elastic waves using low-rank symbol approximation