Seismic wave extrapolation using lowrank symbol approximation |

We start with a simple 1-D example. The 1-D velocity model contains a linear increase in velocity, from 1 km/s to 2.275 km/s. The extrapolation matrix, , or pseudo-Laplacian in the terminology of Etgen and Brandsberg-Dahl (2009), for the time step s is plotted in Figure 1a. Its lowrank approximation is shown in Figure 1b and corresponds to . The locations selected by the algorithm correspond to velocities of 1.59 and 2.275 km/s. The wavenumbers selected by the algorithm correspond to the Nyquist frequency and 0.7 of the Nyquist frequency. The approximation error is shown in Figure 1c. The relative error does not exceed 0.34%. Such a small approximation error results in accurate wave extrapolation, which is illustrated in Figure 2. The extrapolated wavefield shows a negligible error in wave amplitudes, as demonstrated in Figure 2c.

prop,prod,proderr
Wave
extrapolation matrix for 1-D wave propagation with linearly
increasing velocity (a), its lowrank approximation (b), and
Approximation error (c).
Figure 1. |
---|

wave2,awave2,waverr
(a) 1-D wave
extrapolation using the exact extrapolation symbol. (b) 1-D wave
extrapolation using lowrank approximation. (c) Difference between
(a) and (b), with the scale amplified 10 times compared to (a) and (b).
Figure 2. |
---|

wavefd,wave
Wavefield snapshot in a smooth
velocity model computed using (a) fourth-order finite-difference method
and (b) lowrank approximation. The velocity model is
.
The wave source is a point-source Ricker wavelet, located in the
middle of the model. The finite-difference result exhibits
dispersion artifacts while the result of the lowrank approximation,
similarly to that of the FFD method, is dispersion-free.
Figure 3. |
---|

slicefd,slice
Horizontal slices through wavefield snapshots in Figure 3
Figure 4. |
---|

Our next example (Figures 3 and 4) corresponds to wave extrapolation in a 2-D smoothly variable isotropic velocity field. As shown by Song and Fomel (2011), the classic finite-difference method (second-order in time, fourth-order in space) tends to exhibit dispersion artifacts with the chosen model size and extrapolation step, while spectral methods exhibit high accuracy. As yet another spectral method, the lowrank approximation is highly accurate. The wavefield snapshot, shown in Figures 3b and 4b, is free from dispersion artifacts and demonstrates high accuracy. The approximation rank decomposition in this case is , with the expected error of less than . In our implementation, the CPU time for finding the lowrank approximation was 2.45 s, the single-processor CPU time for extrapolation for 2500 time steps was 101.88 s or 2.2 times slower than the corresponding time for the finite-difference extrapolation (46.11 s).

fwavefd,fwave
Wavefield snapshot in a simple two-layer
velocity model using (a) fourth-order finite-difference method and
(b) lowrank approximation. The upper-layer velocity is 1500 m/s, and
the bottom-layer velocity is 4500 m/s. The finite-difference result
exhibits clearly visible dispersion artifacts while the result of
the lowrank approximation is dispersion-free.
Figure 5. |
---|

To show that the same effect takes place in case of rough velocity model, we use first a simple two-layer velocity model, similar to the one used by Fowler et al. (2010). The difference between a dispersion-infested result of the classic finite-difference method (second-order in time, fourth-order in space) and a dispersion-free result of the lowrank approximation is clearly visible in Figure 5. The time step was 2 ms, which corresponded to the approximation rank of 3. In our implementation, the CPU time for finding the lowrank approximation was 2.69 s, the single-processor CPU time for extrapolation for 601 time steps was 19.76 s or 2.48 times slower than the corresponding time for the finite-difference extrapolation (7.97 s). At larger time steps, the finite-difference method in this model becomes unstable, while the lowrank method remains stable but requires a higher rank.

sub
Portion of BP-2004 synthetic isotropic velocity
model.
Figure 6. |
---|

snap
Wavefield snapshot for the velocity
model shown in Figure 6.
Figure 7. |
---|

Next, we move to isotropic wave extrapolation in a complex 2-D velocity field. Figure 6 shows a portion of the BP velocity model (Billette and Brandsberg-Dahl, 2005), containing a salt body. The wavefield snapshot (shown in Figure 7) confirms the ability of our method to handle complex models and sharp velocity variations. The lowrank decomposition in this case corresponds to , with the expected error of less than . Increasing the time step size does not break the algorithm but increases the rank of the approximation and correspondingly the number of the required Fourier transforms. For example, increasing from 1 ms to 5 ms leads to .

salt
SEG/EAGE 3-D salt model.
Figure 8. |
---|

wave3
Snapshot of a point-source wavefield propagating in the SEG/EAGE 3-D salt model.
Figure 9. |
---|

Our next example is isotropic wave extrapolation in a 3-D complex velocity field: the SEG/EAGE salt model (Aminzadeh et al., 1997) shown in Figure 8. A dispersion-free wavefield snapshot is shown in Figure 9. The lowrank decomposition used , with the expected error of .

vpend2,vxend2,etaend2,thetaend2
Portion of BP-2007 anisotropic benchmark model. (a) Velocity along the axis of symmetry. (b) Velocity perpendicular to the axis of symmetry. (c) Anellipticity parameter
. (d) Tilt of the symmetry axis.
Figure 10. |
---|

snap4299
Wavefield snapshot for the velocity
model shown in Figure 10.
Figure 11. |
---|

Finally, we illustrate wave propagation in a complex anisotropic
model. The model is a 2007 anisotropic benchmark dataset from
BP^{}. It exhibits a strong TTI (tilted transverse isotropy)
with a variable tilt of the symmetry axis
(Figure 10). A wavefield
snapshot is shown in Figure . Because of the
complexity of the wave propagation patterns, the lowrank decomposition
took
in this case and required 10 FFTs per time step. In a
TTI medium, the phase velocity
from
equation (10) can be expressed with the help of the
acoustic approximation
(Fomel, 2004; Alkhalifah, 19982000)

where is the P-wave phase velocity in the symmetry plane, is the P-wave phase velocity in the direction normal to the symmetry plane, is the anellipticity parameter (Alkhalifah and Tsvankin, 1995), and and stand for the wavenumbers evaluated in a rotated coordinate system aligned with the symmetry axis:

where is the tilt angle measured with respect to horizontal.

Seismic wave extrapolation using lowrank symbol approximation |

2013-08-31