Fractional Equations and Calculation Methods in Exploration Seismology

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 27 June 2024 | Viewed by 3659

Special Issue Editors


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Guest Editor
School of Geosciences, China University of Petroleum (East China), Qingdao 266580, China
Interests: exploration seismology; earthquake seismology; computational seismology
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
School of Earth and Space Sciences, Peking University, Beijing 100871, China
Interests: exploration geophysics; seismic imaging; seismology; inverse problems; high-performance computing

Special Issue Information

Dear Colleagues,

Exploration seismology is an interdisciplinary subject involving mathematics, physics, and computer science that aims to utilize the properties of seismic waves to detect the hydrocarbon and mineral resources of the Earth. Fractional equations have been extensively employed in exploration seismology, such as seismic wave simulation, imaging and inversion in viscoacoustic/viscoelastic media, quasi-P and S wave separation in anisotropic media and related applications, and one-way wave approximation to the full two-way wave  equation. Accurately and efficiently calculating fractional wave equations can provide a power engine for seismic imaging and model parameter building in complex media, which are vital in exploration seismology.

The aim of this Special Issue is to present the state-of-the-art fractional equations and calculation methods in exploration seismology. The scope of this Special Issue includes, but is not limited to, the following:

  • Seismic wave simulation using fractional partial differential equations
  • Advanced calculation methods for fractional equations
  • Accurate one-way approximation for acoustic/elastic wave equations
  • Seismic imaging in viscous and anisotropic media involving fractional equations
  • Seismic inversion with fractional wave equations

Prof. Dr. Jidong Yang
Dr. Zeyu Zhao
Guest Editors

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Keywords

  • fractional wave equation
  • fractional Laplacian
  • calculation of fractional equation
  • seismic modeling, imaging and inversion

Published Papers (3 papers)

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Research

17 pages, 18858 KiB  
Article
Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil
by Shanyuan Qin, Jidong Yang, Ning Qin, Jianping Huang and Kun Tian
Fractal Fract. 2024, 8(3), 174; https://doi.org/10.3390/fractalfract8030174 - 18 Mar 2024
Viewed by 862
Abstract
In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse [...] Read more.
In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method. Full article
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19 pages, 6816 KiB  
Article
High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation
by Yinfeng Wang, Jilong Lu, Ying Shi, Ning Wang and Liguo Han
Fractal Fract. 2023, 7(12), 880; https://doi.org/10.3390/fractalfract7120880 - 12 Dec 2023
Cited by 1 | Viewed by 1198
Abstract
The propagation of Rayleigh waves is usually accompanied by dispersion, which becomes more complex with inherent attenuation. The accurate simulation of Rayleigh waves in attenuation media is crucial for understanding wave mechanisms, layer thickness identification, and parameter inversion. Although the vacuum formalism or [...] Read more.
The propagation of Rayleigh waves is usually accompanied by dispersion, which becomes more complex with inherent attenuation. The accurate simulation of Rayleigh waves in attenuation media is crucial for understanding wave mechanisms, layer thickness identification, and parameter inversion. Although the vacuum formalism or stress image method (SIM) combined with the generalized standard linear solid (GSLS) is widely used to implement the numerical simulation of Rayleigh waves in attenuation media, this type of method still has its limitations. First, the GSLS model cannot split the velocity dispersion and amplitude attenuation term, thus limiting its application in the Q-compensated reverse time migration/full waveform inversion. In addition, GSLS-model-based wave equation is usually numerically solved using staggered-grid finite-difference (SGFD) method, which may result in the numerical dispersion due to the harsh stability condition and poses complexity and computational burden. To overcome these issues, we propose a high-accuracy Rayleigh-waves simulation scheme that involves the integration of the fractional viscoelastic wave equation and vacuum formalism. The proposed scheme not only decouples the amplitude attenuation and velocity dispersion but also significantly suppresses the numerical dispersion of Rayleigh waves under the same grid sizes. We first use a homogeneous elastic model to demonstrate the accuracy in comparison with the analytical solutions, and the correctness for a viscoelastic half-space model is verified by comparing the phase velocities with the dispersive images generated by the phase shift transformation. We then simulate several two-dimensional synthetic models to analyze the effectiveness and applicability of the proposed method. The results show that the proposed method uses twice as many spatial step sizes and takes 0.6 times that of the GSLS method (solved by the SGFD method) when achieved at 95% accuracy. Full article
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15 pages, 7554 KiB  
Article
A Multi-Task Learning Framework of Stable Q-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation
by Zongan Xue, Yanyan Ma, Shengjian Wang, Huayu Hu and Qingqing Li
Fractal Fract. 2023, 7(12), 874; https://doi.org/10.3390/fractalfract7120874 - 10 Dec 2023
Cited by 1 | Viewed by 1032
Abstract
Q-compensated reverse time migration (Q-RTM) is a crucial technique in seismic imaging. However, stability is a prominent concern due to the exponential increase in high-frequency ambient noise during seismic wavefield propagation. The two primary strategies for mitigating instability in Q [...] Read more.
Q-compensated reverse time migration (Q-RTM) is a crucial technique in seismic imaging. However, stability is a prominent concern due to the exponential increase in high-frequency ambient noise during seismic wavefield propagation. The two primary strategies for mitigating instability in Q-RTM are regularization and low-pass filtering. Q-RTM instability can be addressed through regularization. However, determining the appropriate regularization parameters is often an experimental process, leading to challenges in accurately recovering the wavefield. Another approach to control instability is low-pass filtering. Nevertheless, selecting the cutoff frequency for different Q values is a complex task. In situations with low signal-to-noise ratios (SNRs) in seismic data, using low-pass filtering can make Q-RTM highly unstable. The need for a small cutoff frequency for stability can result in a significant loss of high-frequency signals. In this study, we propose a multi-task learning (MTL) framework that leverages data-driven concepts to address the issue of amplitude attenuation in seismic records, particularly when dealing with instability during the Q-RTM (reverse time migration with Q-attenuation) process. Our innovative framework is executed using a convolutional neural network. This network has the capability to both predict and compensate for the missing high-frequency components caused by Q-effects while simultaneously reconstructing the low-frequency information present in seismograms. This approach helps mitigate overwhelming instability phenomena and enhances the overall generalization capacity of the model. Numerical examples demonstrate that our Q-RTM results closely align with the reference images, indicating the effectiveness of our proposed MTL frequency-extension method. This method effectively compensates for the attenuation of high-frequency signals and mitigates the instability issues associated with the traditional Q-RTM process. Full article
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