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Lowrank approximation

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:

$\displaystyle W(\mathbf{x},\mathbf{k}) \approx \sum\limits_{m=1}^M \sum\limits_{n=1}^N W(\mathbf{x},\mathbf{k}_m) a_{mn} W(\mathbf{x}_n,\mathbf{k}).$ (27)

The difference is that now the lowrank decomposition is implemented for complex matrices or linear operators instead of real ones. The computation of $ p(\mathbf{x},t+\Delta t)$ then becomes:
$\displaystyle p(\mathbf{x},t+\Delta t) \approx \sum\limits_{m=1}^M W(\mathbf{x}...
...{k}} W(\mathbf{x}_n,\mathbf{k}) P(\mathbf{k},t) d\mathbf{k} \right) \right)\; .$     (28)

The computational cost of representation 28 is effectively equivalent to applying $ N$ inverse fast Fourier transforms per time step. In practice, $ N$ is a small number, typically less than 5 for isotropic media. $ N$ 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 $ O(N_x^2)$ to $ O(N N_x \log{N_x})$ , where $ N_x$ is the total size of the spatial grid.


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Next: RTM imaging conditions Up: Theory Previous: Direction-dependent absorbing boundary conditions

2016-11-16