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

Numerical Examples

Figure 4a-4c shows wavefield snapshots (depth, inline, and crossline) in a vertical orthorhombic medium with constant parameters: $v_v=2 km/s$, $v_1=2.1 km/s$, $v_2=2.05 km/s$, $\eta_1=0.3$, $\eta_2=0.1$, and $\gamma=1$. The time-step size is 1 ms and the space grid sizes in three directions are all 25 m. As the model is homogeneous, the rank is 1 for the lowrank decomposition. The depth slice is anelliptical, whereas the inline and crossline display different diamond shapes, indicating different VTI properties. In Figures 4a-4c, red dashed lines are calculated using ray tracing. Note that the red dashed lines match the wavefront from the lowrank method very well.

wavexy,waveyz,wavexz
Figure 4. Three slices of the wavefield snapshot based on the dispersion relation 12 at 1 second in a vertical orthorhombic medium: (a) Depth Slice; (b) Inline Slice; (c) Crossline Slice. Also plotted are red curves representing the wavefront at that time calculated using raytracing.

To show that the lowrank approximation method can handle rough velocity models, we use a two-layer velocity model with high velocity contrast. The first layer has lower velocity parameters: $v_v=1.5 km/s$, $v_1=1.6 km/s$, $v_2=1.7 km/s$, while the values in the other layer are much higher: $v_v=3.5 km/s$, $v_1=4.1 km/s$, $v_2=4.2 km/s$. And we use the same anisotropic parameters for both layers: $\eta_1=0.3$, $\eta_2=0.1$, and $\gamma=1$. For this test, we use a time step size of 1 ms and a space grid size of 25 m. The rank is 2 calculated by the lowrank decomposition within an error level of $10^{-5}$. Figure 5a displays the depth slice above the reflector at 0.6 second. Note the snapshot shows the reflection from the velocity contrast. Figure 5b and 5c show the inline and crossline slices, which indicate strong anisotropy in the medium.

wavexyt,waveyzt,wavexzt
Figure 5. Three slices of the wavefield snapshot by the dispersion relation 12 at 0.6 second in a 2-layer vertical orthorhombic model (high velocity contrast): (a) Depth Slice; (b) Inline Slice; (c) Crossline Slice.

snapxy4,snapyz4,snapxz4
Figure 6. Wavefield snapshots based on the dispersion relation 12 in an rotated and tilted orthorhombic medium ( $\theta =\phi =45\,^{\circ }$) with variable velocity shown in Figure 1a-1c: (a) Depth Slice; (b) Inline Slice; (c) Crossline Slice

Our next example is wavefield snapshots in an orthorhombic model with smoothly varying velocity, shown in Figure 1a-1c: $v_1$: 1500-3088 m/s, $v_2$: 1500-3686 m/s, $v_v$: 1500-3474 m/s, $\eta_1=0.3$, $\eta_2=0.1$, and $\gamma=1.03$. The time-step size is 4 ms. We also rotate the model ( $\theta =\phi =45\,^{\circ }$). Figure 6a-6c shows corresponding wavefield snapshots by the dispersion relation 12 in depth, inline, and crossline slices through the central source location. The inline section (Figure 6b) displays the strongest anisotropic property, because $\eta_1$ is as large as $0.3$. Note that the snapshots are free of dispersion and that there is no coupling of qSV and qP waves in the middle. Lowrank parameters were $M=7$ and $N=7$. Therefore, the cost is 7 FFTs at each time step.

Table 1 displays rank $N$ required for maintaining an error level of $10^{-5}$ with different time step size $\Delta t$. From table 1, one could find for this smooth model, $\Delta t=4$ ms and $N=7$ is the optimal choice for cost consideration. For models with very wide range of parameters and rather complicated structures, the resulting rank may be high, because more space locations and wavenumbers are required to properly represent the original mixed-domain matrix. In order to reduce the computational cost, one may consider Lowrank Finite differences proposed by Song et al. (2013), which is a space-domain finite-difference scheme in which the coefficients of the Laplacian finite-difference stencil is derived from the lowrank approximation.

$\Delta t$ (ms)	0.5	1	2	3	4	5
$Rank N$	5	5	7	7	7	12

Table 1. Rank $N$ calculated from the lowrank approximation of the propagation matrix for a 2D smooth orthorhombic model with different time step size $\Delta t$ for a given error level $10^{-5}$.

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