# High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Method

#### 2.1. DFL Viscoelastic Wave Equation

#### 2.2. Boundary Treatment

#### 2.2.1. Free Surface Boundary

#### 2.2.2. Absorbing Boundary

#### 2.3. Numerical Implementation

- 1.
- Calculate the spatial derivatives

- 2.
- Calculate the fractional Laplacians term as follows:

- 3.
- Update the particle velocity and stress as follows:

- 4.
- Update the particle velocity and stress of the absorbing boundary:

## 3. Numerical Examples

#### 3.1. Homogeneous Half-Space Model

#### 3.1.1. Elastic Medium

#### 3.1.2. Viscoelastic Medium

#### 3.2. Two-Layer Model

#### 3.3. Marmousi Model

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Elastic Equation and Numerical Implementation

## Appendix B. The GSLS-Model-Based Wave Equation

## References

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**Figure 1.**(

**a**) Combined boundary condition structure schematic and (

**b**) layout of the wavefield variables and medium parameters on the staggered-grids mesh.

**Figure 2.**Comparison of seismograms computed using numerical (red dashed line) and analytical (black line) solutions at the offset of 600 m: (

**a**) ${v}_{x}$ component and (

**b**) ${v}_{z}$ component.

**Figure 3.**Elastic wavefield snapshots in a homogeneous half-space model at $t=$ 1 s. The left and right columns correspond to the ${v}_{x}$ and ${v}_{z}$ components, respectively. (

**a**,

**b**) Wavefields simulated using the SGFD method with $\Delta h=$ 2.5 m and $\Delta t=$ 0.5 ms; (

**c**,

**d**) Wavefields computed using the SGPS method with the same spatial and time intervals; (

**e**,

**f**) Wavefields generated using the SGFD method with $\Delta h=$ 1.25 m and $\Delta t=$ 0.25 ms.

**Figure 4.**Zoom-in section of Figure 3. The left and right columns correspond to the ${v}_{x}$ and ${v}_{z}$ components, respectively. (

**a**,

**b**) Wavefields simulated using the SGFD method with $\Delta h=$ 2.5 m and $\Delta t=$ 0.5 ms; (

**c**,

**d**) Wavefields computed using the SGPS method with the same spatial and time intervals; (

**e**,

**f**) Wavefields generated using the SGFD method with $\Delta h=$ 1.25 m and $\Delta t=$ 0.25 ms.

**Figure 5.**Seismograms recorded at (600 m, 0 m) computed using the SGFD and SGPS methods: (

**a**) ${v}_{x}$ component and (

**b**) ${v}_{z}$ component.

**Figure 6.**Comparisons between the elastic and viscoelastic wavefield snapshots in a homogeneous half-space model at $t=$ 0.5 s: (

**a**) elastic ${v}_{x}$ component, (

**b**) elastic ${v}_{z}$ component, (

**c**) viscoelastic ${v}_{x}$ component, and (

**d**) viscoelastic ${v}_{z}$ component.

**Figure 8.**Viscoelastic wavefield snapshots of the two-layer model at $t=$ 0.45 s: (

**a**) ${v}_{x}$ component and (

**b**) ${v}_{z}$ component.

**Figure 10.**(

**a**) Set of two-layer models and (

**b**) wavefield snapshots. I–III show the partial snapshots of Models 1–3 at $t=0.35$ s, respectively; IV shows the difference between Models 2 and 1; and V shows the difference between Models 3 and 1.

**Figure 11.**Comparison of seismograms computed by different Q values. (

**a**) Models 1 and 2; (

**b**) Models 1 and 3.

**Figure 13.**Marmousi model (

**a**) P-wave velocity, (

**b**) S-wave velocity, (

**c**) ${Q}_{P}$, and (

**d**) ${Q}_{S}$.

**Figure 14.**(

**a**) Wavefield snapshots at $t=$ 0.45 s and (

**b**) common-gathers. I–II are calculated using the SIM and proposed scheme, respectively. III shows the simulation results under absorption-only boundaries. IV represents the residuals between I and II, and V shows the residuals between II and III. VI–X show the common-gathers corresponding to wavefield snapshots.

**Figure 15.**The wavefield snapshots at $t=$ 0.45 s and the common-gather of particle velocity ${v}_{z}$ component. (

**a**,

**c**) display the simulation results obtained by solving the DFL equation using the SGPS method; (

**b**,

**d**) show the simulation results obtained by solving the GSLS equation using the SGFD method.

Method | Advantage | Disadvantage | Formulation |
---|---|---|---|

vacuum formalism | simple and easy to implement | low-accuracy and poor numerical stability | $\left\{\begin{array}{c}\rho (i,j-k)\to 0,\hfill \\ {v}_{P}(i,j-k)=0,\hfill \\ {v}_{S}(i,j-k)=0\hfill \end{array}\right.$ |

characteristic variable method | high accuracy | high computational load | $\left\{\begin{array}{c}{\dot{v}}_{x}^{new}={\dot{v}}_{x}^{old}+\frac{1}{\rho {c}_{S}}{\dot{\sigma}}_{xz}^{old}\hfill \\ {\dot{v}}_{z}^{new}={\dot{v}}_{z}^{old}+\frac{1}{\rho {c}_{P}}{\dot{\sigma}}_{zz}^{old}\hfill \\ {\dot{\sigma}}_{xx}^{new}={\dot{\sigma}}_{xx}^{old}-\frac{\tilde{\lambda}}{\tilde{\lambda}+2\tilde{\mu}}{\dot{\sigma}}_{zz}^{old}\hfill \\ {\dot{\sigma}}_{zz}^{new}=0,{\dot{\sigma}}_{xz}^{new}=0\hfill \end{array}\right.$ |

SIM | high computational efficiency and high-accuracy | only for plane waves and semi-infinite media | $\left\{\begin{array}{c}\frac{\partial {\sigma}_{xx}}{\partial t}=\frac{4\mu \left(\lambda +\mu \right)}{\lambda +2\mu}\frac{\partial {v}_{x}}{\partial t}\hfill \\ {\sigma}_{zz}\left(i,j\right)=0\hfill \\ {\sigma}_{zz}\left(i,j-k\right)=-{\sigma}_{zz}\left(i,j+k\right)\hfill \\ {\sigma}_{xz}\left(i,j-k\right)=-{\sigma}_{xz}\left(i,j+k-1\right)\hfill \end{array}\right.$ |

MS | high computational efficiency | complicated calculation | $\left\{\begin{array}{c}{\sigma}_{zz}=0,\phantom{\rule{0.277778em}{0ex}}{\rho}_{x}=0.5{\rho}_{0}\hfill \\ \lambda =0,\phantom{\rule{0.277778em}{0ex}}2\mu ={\mu}_{0}\hfill \end{array}\right.$ |

AEA | high computational efficiency | complicated calculation | $\left\{\begin{array}{c}{\sigma}_{zz}=0,\phantom{\rule{0.277778em}{0ex}}{\rho}_{x}=0.5{\rho}_{0}\hfill \\ \lambda =0,\phantom{\rule{0.277778em}{0ex}}2\mu =2{\mu}_{0}\hfill \end{array}\right.$ |

Depth (m) | $\mathit{\rho}$ (g/cm${}^{3}$) | ${\mathit{v}}_{\mathit{S}}$ (m/s) | ${\mathit{v}}_{\mathit{P}}$ (m/s) | ${\mathit{Q}}_{\mathit{S}}$ | ${\mathit{Q}}_{\mathit{P}}$ | |
---|---|---|---|---|---|---|

Layer 1 | 250 | 1.5 | 700 | 1500 | 30 | 50 |

Layer 2 | 500 | 2.0 | 1150 | 2000 | 90 | 150 |

Depth (m) | $\mathit{\rho}$ (g/cm${}^{3}$) | ${\mathit{v}}_{\mathit{S}}$ (m/s) | ${\mathit{v}}_{\mathit{P}}$ (m/s) | ${\mathit{Q}}_{\mathit{S}}$ | ${\mathit{Q}}_{\mathit{P}}$ | |
---|---|---|---|---|---|---|

Layer 1 | 15 | 1.5 | 600 | 2400 | 30 | 50 |

Layer 2 | 985 | 2.0 | 800 | 3000 | 90 | 150 |

**Table 4.**Parameters of the homogeneous viscoelastic half-space model. Note ${v}_{P}=2000$ m/s, ${v}_{S}=1000$ m/s, ${Q}_{P}=50$, ${Q}_{S}=30$ and $\rho =1.8\phantom{\rule{0.277778em}{0ex}}\mathrm{g}/\mathrm{c}{\mathrm{m}}^{3}$.

Points/Minimum Wavelength | Mesh | Spatial Step (m) | Time Step (ms) |
---|---|---|---|

10 | 200 × 160 | 5 | 1 |

12.5 | 250 × 200 | 4 | 0.8 |

25 | 500 × 400 | 2 | 0.4 |

50 | 1000 × 800 | 1 | 0.2 |

Equation | Spatial Step (m) | Computation Time (s) | ${\mathit{L}}_{2}$ misfit (%) |
---|---|---|---|

GSLS | 5 | 32 | 207 |

4 | 54 | 165 | |

2 | 208 | 27.49 | |

1 | 1075 | 1.52 | |

DFL | 5 | 62 | 42.67 |

4 | 180 | 29.02 | |

2 | 643 | 4.85 | |

1 | 3992 | 0.61 |

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**MDPI and ACS Style**

Wang, Y.; Lu, J.; Shi, Y.; Wang, N.; Han, L.
High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation. *Fractal Fract.* **2023**, *7*, 880.
https://doi.org/10.3390/fractalfract7120880

**AMA Style**

Wang Y, Lu J, Shi Y, Wang N, Han L.
High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation. *Fractal and Fractional*. 2023; 7(12):880.
https://doi.org/10.3390/fractalfract7120880

**Chicago/Turabian Style**

Wang, Yinfeng, Jilong Lu, Ying Shi, Ning Wang, and Liguo Han.
2023. "High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation" *Fractal and Fractional* 7, no. 12: 880.
https://doi.org/10.3390/fractalfract7120880