# Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Pure Quasi-P-Wave Equation in TTI Media

#### 2.2. Approximating the Pseudo-Differential Operator Using a Generalized Fractional Convolution Stencil

#### 2.3. Numerical Implementation in Modeling and RTM

- In regions with non-elliptical anisotropy, load the convolution stencils from the fractional convolution stencil library based on the anisotropic parameters.
- Convolve the convolution stencil with the stress field as $\Lambda (x,z,\epsilon ,\delta ,\theta )\ast p$ to correct the non-elliptically anisotropic effects.
- Use the central finite-difference scheme to calculate $\frac{{\partial}^{2}}{\partial {x}^{2}}$, $\frac{{\partial}^{2}}{\partial {z}^{2}}$ and $\frac{{\partial}^{2}}{\partial x\partial z}$ of the corrected stress field $p+\Lambda (x,z,\epsilon ,\delta ,\theta )\ast p$.
- Update the wavefield using the second-order equation finite in time ${p}^{n+1}=2{p}^{n}-{p}^{n-1}+\frac{1}{2}\Delta {t}^{2}{v}^{2}\left[(1+2\epsilon cos{\theta}^{2})\frac{{\partial}^{2}}{\partial {x}^{2}}+2\epsilon sin{\theta}^{2}\frac{{\partial}^{2}}{\partial {z}^{2}}-4\epsilon sin\theta cos\theta \frac{{\partial}^{2}}{\partial x\partial z}\right]\left(p+\Lambda (x,z,\epsilon ,\delta ,\theta )\ast p\right)$.

## 3. Numerical Examples

#### 3.1. Homogeneous Model

#### 3.2. Overthrust TTI Model

#### 3.3. 2007 BP TTI Model

#### 3.4. Marmousi TTI Model

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the true and approximated pseudo-differential operators with the size of 11 × 11 for a case with $\epsilon $ = 0.2, $\delta $ = 0.1 and $\theta =\frac{\pi}{6}$. (

**a**) The generalized fractional convolution stencil in the space domain, (

**b**) the true fractional pseudo-differential operator in the wavenumber domain, (

**c**) the approximated result in the wavenumber domain and (

**d**) the difference between (

**c**) and (

**d**).

**Figure 2.**Comparisons of the true and approximated phase velocity for a case with v = 3000 m/s, $\epsilon =0.2$, $\delta =0.1$ and $\theta ={30}^{\circ}$. (

**a**) The phase velocity calculated using the exact operator, (

**b**) the phase velocity calculated using the approximated operator and (

**c**) the difference between (

**a**) and (

**b**).

**Figure 3.**Wavefields of isotropic, VTI and TTI modeling at 0.6 s for a case with v = 3000 m/s, $\epsilon =0.2$, $\delta =0.1$ and $\theta ={30}^{\circ}$. (

**a**) Isotropic wavefield, (

**b**) VTI wavefield simulated using the coupled quasi-P wave equations, (

**c**) VTI wavefield simulated using the proposed approach and (

**d**) TTI wavefield simulated using the proposed approach.

**Figure 4.**P-wave velocity (

**a**), $\epsilon $ (

**b**), $\delta $ (

**c**) and $\theta $ (

**d**) for the overthrust TTI model.

**Figure 6.**Migration images of the overthrust TTI model using various RTM approaches. (

**a**) Image produced using the isotropic RTM approach, (

**b**) image produced using the proposed VTI RTM approach and (

**c**) image produced using the proposed TTI RTM approach.

**Figure 7.**P-wave velocity (

**a**), $\epsilon $ (

**b**), $\delta $ (

**c**) and $\theta $ (

**d**) for the 2007 BP TTI model.

**Figure 9.**Migration images of the 2007 BP TTI model using various RTM approaches. (

**a**) Image produced using the isotropic RTM approach, (

**b**) image produced using the proposed VTI RTM approach and (

**c**) image produced using the proposed TTI RTM approach.

**Figure 10.**P-wave velocity (

**a**), $\epsilon $ (

**b**), $\delta $ (

**c**) and $\theta $ (

**d**) for the Marmousi TTI model.

**Figure 12.**Migration images of the Marmousi TTI model using various RTM approaches. (

**a**) Image obtained using the isotropic RTM approach, (

**b**) image obtained using the proposed VTI RTM approach and (

**c**) image obtained using the proposed TTI RTM approach.

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**MDPI and ACS Style**

Qin, S.; Yang, J.; Qin, N.; Huang, J.; Tian, K.
Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil. *Fractal Fract.* **2024**, *8*, 174.
https://doi.org/10.3390/fractalfract8030174

**AMA Style**

Qin S, Yang J, Qin N, Huang J, Tian K.
Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil. *Fractal and Fractional*. 2024; 8(3):174.
https://doi.org/10.3390/fractalfract8030174

**Chicago/Turabian Style**

Qin, Shanyuan, Jidong Yang, Ning Qin, Jianping Huang, and Kun Tian.
2024. "Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil" *Fractal and Fractional* 8, no. 3: 174.
https://doi.org/10.3390/fractalfract8030174