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Fractal Fract
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Fractional Equations and Calculation Methods in Exploration Seismology
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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: 31 December 2024 | Viewed by 6884

Special Issue Editors

Prof. Dr. Jidong Yang

E-Mail Website
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
Dr. Zeyu Zhao

E-Mail Website
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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

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

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Published Papers (5 papers)

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Research

14 pages, 8115 KiB  
Article
A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators
by Juan Chen, Fei Li, Ning Wang, Yinfeng Wang, Yang Mu and Ying Shi
Fractal Fract. 2024, 8(10), 574; https://doi.org/10.3390/fractalfract8100574 - 30 Sep 2024
Viewed by 528
Abstract
Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference [...] Read more.
Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods. Full article
(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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14 pages, 3192 KiB  
Article
A Stable Forward Modeling Approach in Heterogeneous Attenuating Media Using Reapplied Hilbert Transform
by Songmei Deng, Shaolin Shi and Hongwei Liu
Fractal Fract. 2024, 8(7), 434; https://doi.org/10.3390/fractalfract8070434 - 22 Jul 2024
Viewed by 859
Abstract
In the field of geological exploration and wave propagation theory, particularly in heterogeneous attenuating media, the stability of numerical simulations is a significant challenge for implementing effective attenuation compensation strategies. Consequently, the development and optimization of algorithms and techniques that can mitigate these [...] Read more.
In the field of geological exploration and wave propagation theory, particularly in heterogeneous attenuating media, the stability of numerical simulations is a significant challenge for implementing effective attenuation compensation strategies. Consequently, the development and optimization of algorithms and techniques that can mitigate these numerical instabilities are critical for ensuring the accuracy and practicality of attenuation compensation methods. This is essential to reveal subsurface structure information accurately and enhance the reliability of geological interpretation. We present a method for stable forward modeling in strongly attenuating media by reapplying the Hilbert transform to eliminate increasing negative frequency components. We derived and validated new constant-Q wave equation (CWE) formulations and a stable solving method. Our study reveals that the original CWE equations, when utilizing the analytic signal, regenerate and amplify negative frequencies, leading to instability. Implementing our method maintains high accuracy between analytical and numerical solutions. The application of our approach to the Chimney Model, compared with results from the acoustic wave equation, confirms the reliability and effectiveness of the proposed equations and method. Full article
(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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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 1327
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
(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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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 3 | Viewed by 1796
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
(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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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 2 | Viewed by 1394
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
(This article belongs to the Special Issue Fractional Equations and Calculation Methods in Exploration Seismology)
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