    Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation  Next: Numerical Examples Up: Song & Alkhalifah: Orthorhombic Previous: Tilted Orthorhombic Anisotropy

# 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: (15) is a submatrix of that consists of a few columns associated with , is another submatrix that contains some rows associated with , and stands for the coefficients. The construction of the separated form 15 follows the method of Engquist and Ying (2009). The main observation is that the columns of are able to span the column space of the original matrix and that the rows of can span the row space as well as possible.

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  (16)

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
Figure 1.
An orthorhombic model with smoothly varying velocity: (a) : 1500-3088 m/s; (b) : 1500-3686 m/s; (c) : 1500-3474 m/s.        errfig1
Figure 2.
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.    errfig3
Figure 3.
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.   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  Next: Numerical Examples Up: Song & Alkhalifah: Orthorhombic Previous: Tilted Orthorhombic Anisotropy

2013-07-26