Viscoelastic Wave Equation Research Articles

Nearly perfectly matched layer absorber for viscoelastic wave equations

We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.

Q-Compensated Pre-stack Reverse Time Migration in Viscoelastic Media with an Irregular Free Surface

Pre-stack reverse time migration (RTM) is a powerful tool that provides better imaging of complex geological structures than traditional migration methods. An irregular free surface has significant impacts on the characteristics of seismic wave propagation and seismic imaging in mountain or basin areas. In addition, viscoelastic media produce a better representation of the earth than acoustic or elastic media. A main feature of RTM in viscoelastic media is that it can consider compensation for P- and S-wave attenuation effects. In this study, we develop an approach for Q-compensated pre-stack RTM in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming method, the viscoelastic model can be discretized to better match the irregular surface and interfaces. The 2D second-order viscoelastic equations and free surface boundary conditions are converted from the Cartesian coordinate system to the curvilinear coordinate system. The finite-difference operators are applied to approximate the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, the convolutional perfectly matched layer (CPML) absorbing boundary condition is selected to effectively suppress artificial reflections from the model boundaries. In the numerical examples, the snapshots, shot gathers and Q-compensated pre-stack RTM results show that our algorithm can obtain an accurate structure image in viscoelastic media in the presence of surface topography.

General decay of energy for a viscoelastic wave equation with a distributed delay term in the nonlinear internal dambing

In this article, we consider a viscoelastic wave equation of Kirchhoff type in a bounded domain with a distributed delay term present in the nonlinear internal dambing. By introducing a functional energy and suitable Lyapunov functional, under suitable assumptions, we establish a general decay result from which the exponential and polynomial decay are only special cases.

Well-posedness and stability for a viscoelastic wave equation with density and time-varying delay in $\mathbb {R}^n$

This paper concerns the well-posedness and energy decay of a linear wave equation with density, infinite memory and time-varying delay in the whole space $\mathbb {R}^n$ $(n\geq 3)$. We consider the weighted spaces $\mathcal {D}^{1,2}(\mathbb {R}^n)$ and $L^2_{\rho }(\mathbb {R}^n)$ introduced by Karachalios and Stavrakakis (1999) to overcome the difficulty that some operators on $\mathbb {R}^n$ are not compact. We prove the global well-posedness of the Cauchy problem by using Faedo-Galerkin approximation and establish the exponential decay of energy when the amplitude of the time delay term is small by using suitable Lyapunov functional.

Decay rates for a viscoelastic wave equation with Balakrishnan-Taylor and frictional dampings

In this paper we are concerned with a viscoelastic wave equation with Balakrishnan-Taylor damping and frictional damping. By using the multiplier method and some properties of convex functions, we establish general energy decay rates of the equation without imposing any growth assumption near the origin on the frictional term and strongly weakening the usual assumptions on the relaxation term. Our stability result generalizes the earlier related results.

Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

The finite speed of propagation for solutions to stochastic viscoelastic wave equation

In this paper, a class of second order stochastic evolution equations with memory \t\t\tutt(t,x)−Δu(t,x)+∫0tg(t−s)Δu(s,x)ds+f(u)=σ(u)∂W(x,t)∂t,x∈D⊂Rn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}(t,x)-\\Delta u(t,x)+ \\int _{0}^{t} g(t-s)\\Delta u(s,x)\\,ds+f(u)= \\sigma (u)\\frac{\\partial W(x,t)}{\\partial t}, \\quad x\\in D\\subset \\mathbb{R}^{n}, $$\\end{document} is considered, where f is a continuous function with polynomial growth of order less than or equal to n/(n-2) and σ is Lipschitz with sigma (0)=0. By Tartar’s energy method, we prove that for any solution to the equation the propagate speed is finite.

Attenuation compensation for time-reversal imaging in VTI media

Time reversal is a key component in reverse-time migration (RTM) and source localization using wavefield extrapolation. The successful implementation of time reversal depends on the time symmetry (reversibility) of the wave equation in acoustic and elastic media. This symmetry in time, however, is no longer valid in attenuative media, and attenuation is often anisotropic. Here, we employ a viscoelastic anisotropic wave equation that decouples the influence of energy dissipation and velocity dispersion. That equation helps compensate for anisotropic attenuation and restore the time symmetry by changing the signs of the dissipation-dominated terms in time-reversed propagation, while keeping the dispersion-related terms unchanged. We test the [Formula: see text]-compensated time-reversal imaging algorithm on synthetic microseismic data from a 2D transversely isotropic medium with a vertical symmetry axis (VTI). After back-propagating multicomponent data acquired in a vertical borehole, we image microseismic sources using wavefield focusing. The source excitation times are estimated by picking the maximum amplitude of the squared shear strain component [Formula: see text] at the source locations. Accounting for attenuation anisotropy produces superior source images and more accurate excitation times compared to those obtained without attenuation compensation or with purely isotropic attenuation coefficients. The algorithm is also applied to a modified BP TI model to investigate the influence of such factors as survey geometry, errors in velocity and attenuation, noise, and limited aperture.

Q-compensated viscoelastic reverse time migration using mode-dependent adaptive stabilization scheme

The [Formula: see text]-compensated viscoelastic reverse time migration ([Formula: see text]-ERTM) method counteracts the subsurface quality-factor ([Formula: see text]) filtering effect for attenuated multicomponent seismic data to produce high-quality migrated images. Compared with [Formula: see text]-compensated viscoacoustic reverse time migration ([Formula: see text]-ARTM), [Formula: see text]-ERTM provides more informative geologic and structural characterization of the subsurface, but it poses greater challenges on viscoelastic wavefield decomposition and stabilization. On the basis of our previously proposed stabilization operator for [Formula: see text]-ARTM, we have developed a mode-dependent adaptive stabilization scheme for [Formula: see text]-ERTM, which has the ability to handle the numerical instability issue arising from viscoelastic compensation. The stabilization scheme exhibits superior properties of time variance and [Formula: see text] dependence over the commonly used low-pass filtering method. In the context of the viscoelastic wave equation with decoupled fractional Laplacians, we have thoroughly investigated the staggered-grid pseudospectral approach for viscoelastic simulation, vector-based wavefield decomposition for imaging, and mode-dependent adaptive stabilization for compensation. These indispensable modules eventually form the whole framework for stable and accurate [Formula: see text]-ERTM. The [Formula: see text]-ERTM results including PP- and PS-images from synthetic and field data sets are provided to verify the feasibility and superiority of our approach in terms of fidelity and stability.

Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations

In this work, we consider an initial-boundary value problem related to the nonlinear coupled viscoelastic equations \[ \left\{ \begin{array}{c} \left\vert u_{t}\right\vert ^{j}u_{tt}-\Delta u_{tt}-div\left( \left\vert \nabla u\right\vert ^{\alpha -2}\nabla u\right) -\Delta u+\int\limits_{0}^{t}g\left( t-s\right) \Delta uds+\left\vert u_{t}\right\vert ^{m-1}u_{t}=f_{1}\left( u,v\right) ,\text{ } \\ \left\vert v_{t}\right\vert ^{j}v_{tt}-\Delta v_{tt}-div\left( \left\vert \nabla v\right\vert ^{\beta -2}\nabla v\right) -\Delta v+\int\limits_{0}^{t}h\left( t-s\right) \Delta vds+\left\vert v_{t}\right\vert ^{r-1}v_{t}=f_{2}\left( u,v\right) .\text{ } \end{array} \right. \] We will show the exponential growth of solutions with positive initial energy.

A reduced‐order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two‐dimensional viscoelastic wave equations

AbstractIn order to reduce the order of the coefficient vectors of the solutions for the classical Crank–Nicolson collocation spectral (CNCS) method of two‐dimensional (2D) viscoelastic wave equations via proper orthogonal decomposition, we first establish a reduced‐order extrapolated CNCS (ROECNCS) method of the 2D viscoelastic wave equations so that the ROECNCS method has the same basis functions as the classical CNCS method and maintains all advantages of the classical CNCS method. Then, by means of matrix analysis, we discuss the existence, stability, and convergence for the ROECNCS solutions so that the theory analysis becomes very concise. Finally, we utilize some numerical experiments to validate that the consequences of numerical computations are accordant with the theoretical analysis such that the effectiveness and viability of the ROECNCS method are further verified. Therefore, both theory and method of this paper are new and totally different from the existing reduced‐order methods.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

General decay rate for weak viscoelastic wave equation with Balakrishnan–Taylor damping and time-varying delay

A weak viscoelastic wave equation with Balakrishnan–Taylor damping and internal time-varying delay in a bounded domain is considered. The Balakrishnan–Taylor damping appears in the flutter panel equation and the spillover problem which was initially proposed by Balakrishnan and Taylor in 1989. The presence of delay may be a source of instability. Under appropriate assumptions on the source term, with the relaxation function and certain initial data and by adopting the perturbed energy method, we establish general decay estimate for the energy, which depends on the behavior of the relation function, and which is not necessarily decaying in a polynomial or exponential fashion..

Upper and lower bounds for the blow-up time for a viscoelastic wave equation with dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation with dynamical boundary conditions. Under certain assumptions, we give the upper and lower bounds for the blow-up time according to the exponent numbers m and p of the nonlinear boundary damping term and the source term. For the case 2 ≤ m < p, we extend the earlier exponentially growth result in Gerbi and Said-Houari (Adv. Non-linear Analysis, 2013) to a blow-up in finite time result with positive initial energy and get the upper bound for the blow-up time. For the case m = 2, by using the concavity method, we prove a finite time blow-up result and get the upper bound for the blow-up time, which is a supplement to Gerbi and Said-Houari (Adv. Nonlinear Analysis, 2013). Moreover, for the case m ≥ 2, under certain conditions on the data, we give a lower bound for the blow-up time when blow-up occurs.

3D viscoelastic full waveform inversion of seismic waves for geotechnical site investigation

Full waveform inversion (FWI) methods are becoming efficient geophysical tools for geotechnical site investigation. They rely on a mathematical description of wave propagation phenomena to generate synthetic wave-fields, which are then matched with measured wave-fields for extraction of material properties. Due to the inherent computational challenges in implementing a viscoelastic treatment of the wave propagation phenomena, most of the existing 3D FWI algorithms are based on an elastic treatment, and hence ignore the anelastic behavior of materials. The focus of this study is to account for the anelastic effect of materials in the inversion analysis to improve inverted results. Therefore, a 3D viscoelastic FWI method is developed and tested on synthetic and field datasets. The method is based on an existing solution of 3D viscoelastic wave equations (finite-difference solver SOFI3D) and the adjoint-state inversion approach. The results show that the viscoelastic waveform analysis was able to characterize low- and high-velocity synthetic layers, and real variable soil layers. The S-wave velocity (Vs) profiles from field experiment generally agree with invasive standard penetration test (SPT) N-values, including identification of a low-velocity zone. Vs profiles obtained from a 3D elastic FWI are also included for comparison, and results from the viscoelastic inversion are more consistent with the SPT N-values in both trend and magnitudes.

Blow‐up of solutions for a viscoelastic wave equation with variable exponents

In this paper, we consider a viscoelastic wave equation with variable exponents: urn:x-wiley:mma:media:mma5501:mma5501-math-0001 where the exponents of nonlinearity p(·) and m(·) are given functions and a,b &gt; 0 are constants. For nonincreasing positive function g, we prove the blow‐up result for the solutions with positive initial energy as well as nonpositive initial energy. We extend the previous blow‐up results to a viscoelastic wave equation with variable exponents.

No existencia global para un sistema de ecuaciones de onda viscoelástica no lineal con operador p-Laplaciano

Consideramos un problema mixto para un sistema de ecuaciones de onda viscoelástica no lineal con operador p-Laplaciano. Bajo ciertas condiciones sobre las funciones de relajación y los términos de fuente, utilizando el método indirecto, probamos un resultado de no existencia global para ciertas soluciones con energía inicial no positiva así como con energía inicial positivos.

Uniform decay for a viscoelastic wave equation with density and time-varying delay in Rn

A linear viscoelastic wave equation with density and a time-varying delay term in the internal feedback is considered. Under suitable assumptions on the relaxation function, we establish a decay result of solution for by using energy perturbation method in the space Rn (n &gt; 2). We extend a recent result in Feng [10].

Boundary Observability for the Viscoelastic Wave Equation

In this paper, we give a proof of the observability inequality from a portion of the boundary for the viscoelastic wave equation (also called the wave equation with memory kernel) in an arbitrary space dimension. Previous approaches have been limited to one dimension with a recent extension to dimension three by Pandolfi. We complete these results arguing by perturbation from the standard wave equation to show that the corresponding harmonic system is a Riesz sequence. Also, our result preserves the control/observation time from the wave equation. We include a new proof of the boundary observability inequality for the wave equation in line with this Riesz sequence perspective, but again, in arbitrary space dimensions, since it is the crucial result from which ours is derived.

Blow-Up of Solution for A Viscoelastic Wave Equation with Delay

In this paper, we consider the following viscoelastic wave equation with delay $${\left| {{u_t}} \right|^\rho }{u_{tt}} - \Delta u - \Delta {u_{tt}} + \int_0^t {g\left( {t - s} \right)} \Delta u\left( s \right)ds + {\mu _1}{\mu _t}\left( {x,t} \right) + {\mu _2}{\mu _t}\left( {x,t - \tau } \right) = b{\left| u \right|^{p - 2}}u$$ in a bounded domain. Under appropriate conditions on μ1, μ2, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.