To implement wave propagation numerically, we employ the lowrank approximation method of Fomel et al. (2013) to decompose the wave extrapolation matrix 26 into the following separated representation:
(27)
The difference is that now the lowrank decomposition is implemented for complex matrices or linear operators instead of real ones. The computation of
then becomes:
(28)
The computational cost of representation 28 is effectively equivalent to applying
inverse fast Fourier transforms per time step. In practice,
is a small number, typically less than 5 for isotropic media.
may grow with increasing model complexity, such as the introduction of anisotropy. Compared with a naive straightforward implementation of equation 17, the number of floating point operations per time step is reduced from
to
, where
is the total size of the spatial grid.
Lowrank one-step wave extrapolation for reverse-time migration