Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation |

For orthorhombic media, the mixed-domain phase operator, , is given by equation 12.
Considering inhomogeneous media, we choose lowrank approximation (Fomel et al., 2010,2012) to implement the mixed-domain operator.

Fomel et al. (2010,2012) showed that mixed-domain matrix
, which appears in wavefield extrapolation,
can be decomposed using a separable representation:

In the case of smooth models, the mixed-domain operator can be decomposed by a low-rank approximation. In models with serious roughness and randomness, the time step may be restricted to small values or otherwise; the rank will end up high. As a result, the computational cost maybe high.

To perform a linear-time lowrank decompositon as proposed by Fomel et al. (2012),
we first need to restrict the mixed-domain to randomly selected rows.
In practice, can be scaled as and is the numerical rank of .
Then, we perform pivoted QR algorithm (Golub and Van Loan, 1996) to find the corresponding columns for
.
To find the rows for
, we apply the pivoted QR algorithm to .

Representation 15 speeds up the computation of
because

Evaluation of the last formula requires inverse FFTs. Correspondingly, with lowrank approximation, the cost can be reduced to , where is the model size and is a small number, related to the rank of the above decomposition and it is automatically calculated for some given error level ( ) with a pre-determined .

Figure 1a-1c shows an orthorhombic model with smoothly varying velocity - : 1500-3088 m/s, : 1500-3686 m/s, : 1500-3474 m/s, , , and . The time step . Figure 2 display error of lowrank decomposition for at the location (-1.925 km, -1.925 km, -1.925 km) with relatively high velocity values, km/s, km/s, km/s. One can find the error level is around . Figure 3 display error of lowrank decomposition for at the location (0.575 km, 0.575 km, 0.575 km) with relatively low velocity values, km/s, km/s, km/s. One can find the error is also well controlled.

velxfig,velyfig,velzfig
An orthorhombic model with smoothly varying velocity: (a) : 1500-3088 m/s; (b) : 1500-3686 m/s; (c) : 1500-3474 m/s.
Figure 1. |
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errfig1
Error plot for the lowrank approximation for
at the location (-1.925 km, -1.925 km, -1.925 km) with relatively high velocity values, km/s, km/s, km/s.
Figure 2. |
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errfig3
Error plot for the lowrank approximation for
at the location (0.575 km, 0.575 km, 0.575 km) with relatively low velocity values, km/s, km/s, km/s.
Figure 3. |
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We propose using the above lowrank approximation algorithm to handle mixed-domain operator
in equation 12 for wave extrapolation in orthorhombic media.

Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation |

2013-07-26