Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation

Blanc, Emilie; Komatitsch, Dimitri; Chaljub, Emmanuel; Lombard, Bruno; Xie, Zhinan

doi:10.1093/gji/ggw024

Cited by 72 publications

(48 citation statements)

References 48 publications

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“…These observations are even more pronounced in the case α = 0.7, where the signal at offset 16 m has almost disappeared at the scale of the figure. The behaviors illustrated on these seismograms are similar to those observed in viscoelasticity, where typical models of attenuation involve quality factors Q(ω) ∼ Q 0 ω −α [7]. 6.3.…”

Section: General Casesupporting

confidence: 63%

“…More generally, the diffusive approach can be applied to a wide range of pseudodifferential time operators with a hereditary behavior. Examples may be found in mechanics for the modeling of viscoelasticity [14] and poroelasticity [5,6,7]. Other models can be investigated in electromagnetism, to describe dispersive media; see [43] and references therein for a review.…”

Section: Discussionmentioning

confidence: 99%

“…In this paper, we use optimization with constraints of positivity, which provides a great improvement of accuracy compared with the aforementionned quadrature methods. This type of optimization has already been used with success in the context of poroelasticity [5], viscoelasticity [7], and recently for Chester's equation describing nonlinear acoustic waves in a guide [2].…”

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confidence: 99%

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Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

Lombard¹,

Matignon²

2016

SIAM J. Appl. Math.

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A fractional time derivative is introduced into Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.

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Section: General Casesupporting

confidence: 63%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

Lombard¹,

Matignon²

2016

SIAM J. Appl. Math.

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“…In the presence of (strong) attenuation, the wavefield simulations (including forward and backward simulations) could be inaccurate or unstable if not well treated. For example, the instability problem could arise in forward modeling when some of the anelastic coefficients are negative in the viscoelastic system (Blanc et al, ; Komatitsch et al, ). Though several techniques can deal with this issue, such as by modifying the wave equation to introduce stabilizing terms (Ammari et al, ; Komatitsch et al, ) or by using optimal checkpointing solution (Griewank & Walther, ), we propose that a strong stability preserving time discretization scheme may be efficient (and easy to implementation) to solve the instability problems for both forward and backward anelastic simulations.…”

Section: Introductionmentioning

confidence: 99%

Modeling Three‐Dimensional Wave Propagation in Anelastic Models With Surface Topography by the Optimal Strong Stability Preserving Runge‐Kutta Method

Wang

Borisov

et al. 2019

JGR Solid Earth

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Accurate and efficient forward modeling methods are important for simulation of seismic wave propagation in 3D realistic Earth models and crucial for high-resolution full waveform inversion. In the presence of attenuation, wavefield simulation could be inaccurate or unstable over time if not well treated, indicating the importance of the implementation of a strong stability preserving time discretization scheme. In this study, to solve the anelastic wave equation, we choose the optimal strong stability preserving Runge-Kutta (SSPRK) method for the temporal discretization and apply the fourth-order MacCormack scheme for the spatial discretization. We approximate the rheological behaviors of the Earth by using the generalized Maxwell body model and use an optimization procedure to calculate the anelastic coefficients determined by the Q(ω) law. This optimization constrains positivity of the anelastic coefficients and ensures the decay of total energy with time, resulting in a stable viscoelastic system even in the presence of strong attenuation. Moreover, we perform theoretical and numerical analyses of the SSPRK method, including the stability criteria and the numerical dispersion. Compared with the traditional fourth-order Runge-Kutta method, the SSPRK has a larger stability condition number and can better suppress numerical dispersion. We use the complex-frequency-shifted perfectly matched layer for the absorbing boundary conditions based on the auxiliary difference equation and employ the traction image method for the free-surface boundary condition on curvilinear grids representing the surface topography. Finally, we perform several numerical experiments to demonstrate the accuracy of our anelastic modeling in the presence of surface topography. Key Points:• We develop a new optimal strong stability preserving Runge-Kutta (SSPRK) method for solving the anelastic wave equation • We perform theoretical and numerical analyses of the SSPRK method • We demonstrate the accuracy and efficiency of the SSPRK method in anelastic wavefield modeling in the presence of surface topographySupporting Information:• Supporting Information S1 ). Modeling three-dimensional wave propagation in anelastic models with surface topography by the optimal strong stability preserving Runge-Kutta method. Withers et al., 2015) for a weak frequency dependence of Q is sometimes of interest. We follow a conventional way to calculate the anelastic coefficients determined by the Q(ω) law (Emmerich & Korn, 1987;Käser et al., 2007;Kristek & Moczo, 2003), but we constrain the coefficients to be positive (Blanc et al., 2016;Yang et al., 2016). This positivity ensures the decay of total energy over time as proved in the supporting information (SI) section A. Assuming the attenuation factors at a reference frequency f r for P and S waves is Q P and Q S , the frequency range of interest in the classical approach to calculate the anelastic coefficients is defined as [f min , f max ] = [f r /f amp , f r × f amp ],

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“…Table 2: Table lists the relaxation times (two mechanisms) used with the shell layer. Relaxation times are computed using the nonlinear optimization method discussed in detail in [34]. τ The derived wave speeds for each subdomain are given in Table 3.…”

Section: Model Setupmentioning

confidence: 99%

Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography

Lähivaara

Kärkkäinen

Huttunen

et al. 2018

The Journal of the Acoustical Society of America

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The feasibility of data based machine learning applied to ultrasound tomography is studied to estimate water-saturated porous material parameters. In this work, the data to train the neural networks is simulated by solving wave propagation in coupled poroviscoelastic-viscoelastic-acoustic media. As the forward model, a high-order discontinuous Galerkin method is considered, while deep convolutional neural networks are used to solve the parameter estimation problem. In the numerical experiment, the material porosity and tortuosity is estimated, while the remaining parameters which are of less interest are successfully marginalized in the neural networks-based inversion. Computational examples confirm the feasibility and accuracy of this approach.

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Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation

Cited by 72 publications

References 48 publications

Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

Modeling Three‐Dimensional Wave Propagation in Anelastic Models With Surface Topography by the Optimal Strong Stability Preserving Runge‐Kutta Method

Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography

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