Seismic wave extrapolation using lowrank symbol approximation |

Let be the seismic wavefield at location and time . The wavefield at the next time step can be approximated by the following mixed-domain operator (Wards et al., 2008)

where is the spatial Fourier transform of

where is the spatial wavenumber. To define the phase function , which appears in equation (1), one can substitute approximation (1) into the wave equation and extract the geometrical (high-frequency) asymptotic of it. In case of seismic wave propagation, this leads to the eikonal-like equation

where is the phase velocity, and the choice of the sign corresponds, in the case of a point source, to expanding or contracting waves. In the isotropic case, does not depend on . The initial condition for equation (3) is

which turns equation (1) into the simple inverse Fourier transform operation.

Assuming small steps in equation (1), one can build successive approximations for the phase function by expanding it into a Taylor series. In particular, let us represent the phase function as

Correspondingly,

Substituting expansions (5) and (6) into equation (3) and separating terms with different powers of , we find that

When either the velocity gradient or the time step are small, the Taylor expansion (5) can be reduced to only two terms, which in turn reduces equation (1) to the familiar expression (Etgen and Brandsberg-Dahl, 2009)

or

In rough velocity models, where the gradient does not exist, one can attempt to solve the eikonal equation 3 numerically or to apply approximations other than the Taylor expansion (5). In the examples of this paper, we used only the term.

Note that the approximations that we use, starting from equation (1), are focused primarily on the phase of wave propagation. As such, they are appropriate for seismic migration but not necessarily for accurate seismic modeling, which may require taking account of amplitude effects caused by variable density and other elastic phenomena.

The computational cost for a straightforward application of equation (1) is , where is the total size of the three-dimensional grid. Even for modest-size problems, this cost is prohibitively expensive. In the next section, we describe an algorithm that reduces the cost to , where is a small number.

Seismic wave extrapolation using lowrank symbol approximation |

2013-08-31