    Lowrank seismic wave extrapolation on a staggered grid  Next: Lowrank FD for first-order Up: Theory Previous: Second-order and first-order mixed-domain

## Lowrank approximation for first-order extrapolation operators

In this section, we apply lowrank decomposition to approximate the extrapolation operator in equation 9. As indicated by Fomel et al. (2013b,2010), the mixed-domain matrix in equation 9, taking as an example, sinc (11)

can be efficiently decomposed into a separated representation as follows: (12)

where is a submatrix of which consists of selected columns associated with ; is another submatrix that contains selected rows associated with ; and stands for middle matrix coefficients. The numerical construction of the separated representation in equation 12 follows the method of Engquist and Ying (2009).

Using representation 12, we can calculate using a small number of fast Fourier transforms (FFTs), because (13)

The same lowrank decomposition approach can be applied to the remaining three partial derivative operators in equation 9. Evaluation of equation 13 only needs inverse FFTs, whose computational cost is . However a straightforward application of equation 9 needs computational cost of , where is the total size of the space grid. is related to the rank of the decomposed mixed-domain matrix 12, which is usually significantly smaller than . Note that the number of FFTs also depends on the given error level of lowrank decomposition with a predetermined . Thus a complex model or increasing the time interval size may increase the rank of the approximation matrix and correspondingly . In the numerical examples in this paper, the values of rank are usually between and . Lowrank decomposition saves cost in calculating equations 9 and 10. We propose to apply it for seismic wave extrapolation in variable velocity and density media on a staggered grid. We call this method staggered grid lowrank(SGL) method.    Lowrank seismic wave extrapolation on a staggered grid  Next: Lowrank FD for first-order Up: Theory Previous: Second-order and first-order mixed-domain

2014-06-02