Adaptive Discontinuous Galerkin Modeling of Intrinsic Attenuation Anisotropy for Fluid-Saturated Porous Media

Zhan, Qiwei; Zhuang, Mingwei; Liu, Qing

doi:10.1109/tgrs.2019.2948136

Cited by 8 publications

(4 citation statements)

References 34 publications

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“…We prove that the global error of FIR is reduced when α gets smaller, thus we can reduce the modes considerably in sum-ofexponential approximation when α is small, then the storage and the computational cost will decrease. This result successfully fits the discovery in engineering work [33][34][35].…”

Section: Discussionsupporting

confidence: 87%

“…In recent years, there has been an emerging interest in the field of fractional derivatives. Many phenomena in engineering have been described by models with fractional derivatives, such as the groundwater flow [2,3], the blood ethanol concentration system [24], the epidemic model [31], the magnetic hysteresis phenomena [6], and seismic wave propagation problems discussed in [34]. These complex media or processes in these fields need hereditary descriptions, while fractional derivative is an excellent mathematical tool for characterizing the memory effects [27,36].…”

Section: Introductionmentioning

confidence: 99%

“…Fortunately, there are plenty of works that propose fast evaluations of the Caputo derivative, including the second order implicit schemes [13], the fast evaluations with the sum-of-exponential approximation [16,32], as well as some efficient algorithms implemented in engineering works [33][34][35], [9,Chapter 2]. In this paper we focus on the fast schemes using the sum-of-exponential approximation, which is a widely-used method in speeding up the evaluation of the convolution integrals.…”

Section: Introductionmentioning

confidence: 99%

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Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

Zhang¹,

Zhan²,

Zhou³

2021

Preprint

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In this paper, we investigate fast algorithms to approximate the Caputo derivative C 0 D α t u(t) when α is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-ofexponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme we propose in this work, and the latter shows superiority when α is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as α gets small. Hence, only minimal cost for the fast evaluation is required in the small α regime, which matches prevailing protocols in engineering practice. We also present a stability and error analysis of FIDR for solving linear fractional diffusion equations. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when α is small.

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Section: Discussionsupporting

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

Zhang¹,

Zhan²,

Zhou³

2021

Preprint

Self Cite

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“…Nevertheless, the wave equations for these two models involve the convolution issue mentioned above. As an alternative, the generalized standard-linear-solid (SLS) model (also called the Zener model) and the generalized Maxwell model can be used to fit a given Q value in a frequency range of interest (Emmerich and Korn, 1987;Liu et al, 1976;Zhan et al, 2019aZhan et al, ,2019b. These two models are identical to each other due to the equivalence between the SLS and Maxwell elements (Tschoegl, 1989).…”

Section: Introductionmentioning

confidence: 99%

Viscoelastic wave propagation for nearly constant Q transverse isotropy

Hao

Greenhalgh

Huang

et al. 2022

Geophysical Prospecting

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Constant Q, also called frequency-independent Q, is a phenomenological assumption often used in practice for dissipative wave propagation in both forward and inverse seismic modelling. A constant Q dissipative model generally results in a temporal convolution-type wave equation, which is computationally costly to solve by a time-domain numerical modelling method. In this paper, we use the recently proposed weighting function method to derive the nearly constant Q wave equations in differential form for viscoelastic transversely isotropic media. These wave equations explicitly involve specified Q parameters, which is convenient for time-domain waveform modelling and inversion. As numerical examples, we analyse the nearly constant Q behaviour of the proposed model , nearly constant Q wave reflection and wave propagation in a heterogeneous case.

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Monitoring of water volume in a porous reservoir using seismic data: A 3D simulation study

Khalili,

Göransson,

Hesthaven

et al. 2024

Journal of Applied Geophysics

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Adaptive Discontinuous Galerkin Modeling of Intrinsic Attenuation Anisotropy for Fluid-Saturated Porous Media

Cited by 8 publications

References 34 publications

Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

Viscoelastic wave propagation for nearly constant Q transverse isotropy

Monitoring of water volume in a porous reservoir using seismic data: A 3D simulation study

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