Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media

2D two-layer VTI/TTI model

The first example is on a 2D two-layer model, in which the first layer is a VTI medium with $v_{p0}=2500 m/s$ , $v_{s0}=1200 m/s$ , $\epsilon=0.2$ , and $\delta=-0.2$ , and the second layer is a tilted TI (TTI) medium with $v_{p0}=3600 m/s$ , $v_{s0}=1800 m/s$ , $\epsilon=0.2$ , $\delta=0.1$ and $\theta=30^{\circ}$ . A point source is placed at the center of this model. Firstly, we compare the synthetic elastic wavefields by solving the elastic displacement wave equation using the 10th-order explicit finite-difference (FD) and low-rank pseudo-spectral schemes (with or without the $k$ -space adjustment), respectively. Figure 1 shows the wavefield snapshots at the time of 0.3 s using the spatial sampling $\Delta{x}=\Delta{z}=5 m$ and time-step $\Delta{t}=0.5 ms$ . Only the low-rank pseudo-spectral solutions with the $k$ -space adjustment are displayed because the three schemes produce very similar results. The vertical slices through the z-components of the elastic wavefields show little differences among them (Figure 2). For the low-rank pseudo-spectral scheme, the ranks are all $2$ for the decomposition of the mixed-domain matrices $w_{xx}$ , $w_{zz}$ and $w_{xz}$ in equation 21, and the $k$ -space adjustment doesn't change the ranks. It takes CPU time of 0.20, 0.23 and 0.23 seconds for them to finish the wavefield extrapolation of one time-step. Additional 4.3 and 8.2 seconds have been used to finish the low-rank decomposition of the involved mixed-domain matrices before wavefield extrapolation. We observe the FD scheme unstable if the time-step is increased to 1.0 ms and the low-rank pseudo-spectral scheme unstable if the time-step is increased to 2.0 ms (with unchanged spatial sampling). However, the low-rank pseudo-spectral solution using the $k$ -space adjustment produces acceptable results even the time-step is increased to 3.0 ms and the maximum time exceeds 3 s. Figure 3 and Figure 4 compare the wavefield snapshots and the vertical slices at the time of 0.6 s using the three schemes with the increased spatial sampling (namely $\Delta {x}=\Delta {z}=10$ m). The FD scheme tends to exhibit dispersion artifacts with the chosen model size and extrapolation step, while low-rank pseudo-spectral scheme exhibit acceptable accuracy. The $k$ -space adjustment permits larger time-steps without reducing accuracy or introducing instability. For this example, it has produced the best results with less numerical dispersion. Thanks to the larger spatial and temporal sampling, the same CPU time is used for each scheme as in Figure 1. In addition, only the ranks for the low-rank decomposition of the matrix $w_{12}$ reduce to $1$ when we change the tilt angle of the second layer to 0 .

ElasticxKSSInterf,ElasticzKSSInterf
Figure 1. Horizontal and vertical components of the elastic wavefields at the time of 0.3 s synthesized by solving the 2nd-order elastic wave equation with $\Delta {x}=\Delta {z}=5$ m and $\Delta {t}=0.5$ ms.

ElasticzFDwave,ElasticzPSLRwave,ElasticzKSwave
Figure 2. Vertical slices through the vertical components of the synthetic elastic wavefields at $x=0.75$ km: (a) 10th-order FD, (b) low-rank pseudo-spectral and (c) low-rank pseudo-spectral using the $k$ -space adjustment.

ElasticzFDInterfC,ElasticzPSLRInterfC,ElasticzKSSInterfC
Figure 3. Vertical components of the elastic wavefields at the time of 0.6 s synthesized using three schemes with the same spatial sampling $\Delta {x}=\Delta {z}=10$ m: (a) 10th-order FD and (b) low-rank pseudo-spectral with $\Delta {t}=1.5$ ms, and (c) low-rank pseudo-spectral solution using the $k$ -space adjustment with $\Delta {t}=3.0$ ms.

ElasticzFDwave,ElasticzPSLRwave,ElasticzKSSwave
Figure 4. Vertical slices through the vertical components at $x=1.5$ km in Figure 3: (a) 10th-order FD, (b) low-rank pseudo-spectral and (c) low-rank pseudo-spectral using the $k$ -space adjustment.

Secondly, we compare two approaches to get the decoupled elastic wavefields during time extrapolation. The first approach uses the low-rank pseudo-spectral algorithm to synthesize the elastic wavefields and then apply the low-rank vector decomposition algorithm (Cheng and Fomel, 2014) to get the vector qP- and qSV-wave fields (Figure 5). The second extrapolates the decoupled qP- and qSV-wave fields using the proposed low-rank mixed-domain integral operations (Figure 6). Extrapolation steps of $\Delta {x}=\Delta {z}=10$ m and $\Delta{t}=1.0$ ms are used in this example. The ranks are still $2$ for the involved low-rank decomposition of the propagation matrices defined in equation 20. The Two approaches produce comparable elastic wavefields, in which we can observe all transmitted and reflected waves including mode conversions. For one step of time extrapolation, it takes the CPU time of 0.6 ms for the first approach and 0.5 ms for the second. This means that merging time extrapolation and vector decomposition into a unified Fourier integral framework provides more efficient solution than operating them in sequence for anisotropic media thanks to the reduced number of forward and inverse FFTs.

ElasticxPSLR1Interf,ElasticzPSLR1Interf,ElasticPxPSLR1Interf,ElasticPzPSLR1Interf,ElasticSxPSLR1Interf,ElasticSzPSLR1Interf
Figure 5. Elastic wavefields at the time of 0.6 s synthesized by using low-rank pseudo-spectral solution of the displacement wave equation followed with low-rank vector decomposition: (a) x- and (b) z-components of the displacement wavefields; (c) x- and (d) z-components of the qP-wave fields; (e) x- and (f) z-components of the qSV-wave fields.

ElasticPxPSLRInterf,ElasticPzPSLRInterf,ElasticSxPSLRInterf,ElasticSzPSLRInterf,ElasticxPSLRInterf,ElasticzPSLRInterf
Figure 6. Elastic wavefields at the time of 0.6 s synthesized by using low-rank pseudo-spectral operators for extrapolating and decomposing the elastic waves simultaneously: (a) x- and (b) z-components of the qP-wave displacement wavefields; (c) x- and (d) z-components of the qSV-wave displacement wavefields; (e) x- and (f) z-components of the total elastic wavefields.

Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media