Lowrank approximation for first-order extrapolation operators

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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 $\displaystyle \frac{\partial p(\mathbf{x},t)}{\partial^+x}$ as an example,

$\displaystyle W_x(\mathbf{x}, \mathbf{k}) = k_x$ sinc $\displaystyle (v(\mathbf{x})\left\vert\mathbf{k}\right\vert\Delta t/2),$

(11)

can be efficiently decomposed into a separated representation as follows:

$\displaystyle W_x(\mathbf{x}, \mathbf{k}) \approx \sum\limits_{m=1}^M\sum\limits_{n=1}^N W_x(\mathbf{x}, \mathbf{k}_m)a_{mn}W_x(\mathbf{x}_n, \mathbf{k}),$

(12)

where $W_x(\mathbf{x}, \mathbf{k}_m)$ is a submatrix of $W_x(\mathbf{x}, \mathbf{k})$ which consists of selected columns associated with $\mathbf{k}_m$ ; $W_x(\mathbf{x}_n, \mathbf{k})$ is another submatrix that contains selected rows associated with $\mathbf{x}_n$ ; and $a_{nm}$ 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 $\displaystyle \frac{\partial p(\mathbf{x},t)}{\partial^+x}$ using a small number of fast Fourier transforms (FFTs), because

$\displaystyle \displaystyle \frac{\partial p(\mathbf{x},t)}{\partial^+x} \appro... ...x/2}W_x(\mathbf{x}_n, \mathbf{k})\mathbf{F}\left[p(\mathbf{x},t)\right]\right].$

(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 $\Delta t$ . Thus a complex model or increasing the time interval size $\Delta t$ 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.

Next: Lowrank FD for first-order Up: Theory Previous: Second-order and first-order mixed-domain

2014-06-02