# A Multi-Task Learning Framework of Stable Q-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fractional Laplacian Viscoacoustic Wave Equation

**x**for $\gamma $. Then, $\eta =-{c}_{0}^{2\gamma}{\omega}_{0}^{-2\gamma}\mathrm{cos}\left(\pi \gamma \right)$, $\tau =-{c}_{0}^{2\gamma -1}{\omega}_{0}^{-2\gamma}\mathrm{sin}\left(\pi \gamma \right)$, ${c}^{2}=K/\rho ={c}_{0}^{2}{\mathrm{cos}}^{2}\left(\pi \gamma \right)$, ${c}_{0}$ and $\rho $ are the reference velocity and density, ${\omega}_{0}$ is the reference angular frequency, and $K$ represents the bulk modulus. In the time–wavenumber domain, Equation (1) can be written as follows.

#### 2.2. Network Architecture of Frequency Extension Based on MTL

#### 2.3. Data Preparation

## 3. Examples

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

_{2}geological sequestration in hydrocarbon accumulation area, grant number DD20230025”.

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The 14th (

**a**) phase-dispersion-only seismic data obtained using Equation (3) and their (

**b**) filtered low-frequency component and (

**d**) filtered high-frequency component. (

**d**) The viscoacoustic seismic data obtained using Equation (1) with the low-frequency component cut off. Note that all the figures show data plotted after the implementation of the AGC algorithm.

**Figure 8.**The combination of low-frequency (

**a**), medium-frequency (

**b**), and high-frequency (attenuated because of the Q effect) (

**c**) data patches.

**Figure 9.**Comparison of prediction results from our MTL neural network. The upper and lower parts stand for the high-frequency and low-frequency seismic data. From left to right are the simulated PDO seismic data, real low- or high-frequency seismic data, predicted low- or high-frequency seismic data, and residual errors between the real and predicted seismic data.

**Figure 10.**Comparison of (

**a**) the high-frequency, (

**b**) low-frequency, and (

**c**) full frequency band single traces. The red and blue lines represent the reconstructed and real parts.

**Figure 11.**(

**a**) Acoustic RTM image of acoustic data, reference imaging; (

**b**) acoustic RTM image of viscoacoustic data, without Q compensation; and (

**c**) Q-RTM image of viscoacoustic data, our proposed method.

**Figure 12.**Waveform plot of traces located at (

**a**) $x=1500$ m and (

**b**) $x=2000$ m. The red line is from the acoustic RTM image for acoustic data, the blue and green lines are from the images of our proposed compensated Q-RTM and non-compensated RTM, respectively.

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**MDPI and ACS Style**

Xue, Z.; Ma, Y.; Wang, S.; Hu, H.; Li, Q.
A Multi-Task Learning Framework of Stable *Q*-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation. *Fractal Fract.* **2023**, *7*, 874.
https://doi.org/10.3390/fractalfract7120874

**AMA Style**

Xue Z, Ma Y, Wang S, Hu H, Li Q.
A Multi-Task Learning Framework of Stable *Q*-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation. *Fractal and Fractional*. 2023; 7(12):874.
https://doi.org/10.3390/fractalfract7120874

**Chicago/Turabian Style**

Xue, Zongan, Yanyan Ma, Shengjian Wang, Huayu Hu, and Qingqing Li.
2023. "A Multi-Task Learning Framework of Stable *Q*-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation" *Fractal and Fractional* 7, no. 12: 874.
https://doi.org/10.3390/fractalfract7120874