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Register a new userNearly constant Q dissipative models and wave equations for general viscoelastic anisotropy
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The quality factor (Q) links seismic wave energy dissipation to physical properties of the Earths interior, such as temperature, stress and composition. Frequency independence of Q, also called constant Q for brevity, is a common assumption in practice for seismic Q
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Time-domain seismic forward and inverse modeling for a dissipative medium is a vital research topic to investigate the attenuation structure of the Earth. Constant Q, also called frequency independence of the quality factor, is a common assumption for seismic Q inversion. We propose the first- and second-order nearly constant Q dissipative models of the generalized standard linear solid type, using a novel Q-independent weighting function approach. The two new models, which originate from the Kolsky model (a nearly constant Q model) and the Kjartansson model (an exactly constant Q model), result in the corresponding wave equations in differential form. Even for extremely strong attenuation (e.g., Q = 5), the quality factor and phase velocity for the two new models are close to those for the Kolsky and Kjartansson models, in a frequency range of interest. The wave equations for the two new models involve explicitly a specified Q parameter and have compact and simple forms. We provide a novel perspective on how to build a nearly constant Q dissipative model which is beneficial for time-domain large scale wavefield forward and inverse modeling. This perspective could also help obtain other dissipative models with similar advantages. We also discuss the extension beyond viscoacousticity and other related issues, for example, extending the two new models to viscoelastic anisotropy.
Hysteretic damping is often modeled by means of linear viscoelastic approaches such as nearly constant Attenuation (NCQ) models. These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving non-linearity. This formulation is implemented into a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).
We pursue here the development of models for complex (viscoelastic) fluids in shallow free-surface gravity flows which was initiated by [Bouchut-Boyaval, M3AS (23) 2013] for 1D (translation invariant) cases. The models we propose are hyperbolic quasilinear systems that generalize Saint-Venant shallow-water equations to incompressible Maxwell fluids. The models are compatible with a formulation of the thermo-dynamics second principle. In comparison with Saint-Venant standard shallow-water model, the momentum balance includes extra-stresses associated with an elastic potential energy in addition to a hydrostatic pressure. The extra-stresses are determined by an additional tensor variable solution to a differential equation with various possible time rates. For the numerical evaluation of solutions to Cauchy problems, we also propose explicit schemes discretizing our generalized Saint-Venant systems with Finite-Volume approximations that are entropy-consistent (under a CFL constraint) in addition to satisfy exact (discrete) mass and momentum conservation laws. In comparison with most standard viscoelastic numerical models, our discrete models can be used for any retardation-time values (i.e. in the vanishing solvent-viscosity limit). We finally illustrate our hyperbolic viscoelastic flow models numerically using computer simulations in benchmark test cases. On extending to Maxwell fluids some free-shear flow testcases that are standard benchmarks for Newtonian fluids, we first show that our (numerical) models reproduce well the viscoelastic physics, phenomenologically at least, with zero retardation-time. Moreover, with a view to quantitative evaluations, numerical results in the lid-driven cavity testcase show that, in fact, our models can be compared with standard viscoelastic flow models in sheared-flow benchmarks on adequately choosing the physical parameters of our models. Analyzing our models asymptotics should therefore shed new light on the famous High-Weissenberg Number Problem (HWNP), which is a limit for all the existing viscoelastic numerical models.
In this paper, we consider a long-wave equivalent medium to a finely parallel-layered inhomogeneous medium, obtained using the Backus average. Following the work of Postma and Backus, we show explicitly the derivations of the conditions to obtain the equivalent isotropic medium. We demonstrate that there cannot exist a transversely isotropic (TI) equivalent medium with the coefficients $c^{overline{rm TI}}_{1212} eq c^{overline{rm TI}}_{2323}$, $c^{overline{rm TI}}_{1111} = c^{overline{rm TI}}_{3333}$ and $c^{overline{rm TI}}_{1122} = c^{overline{rm TI}}_{1133}$. Moreover, we consider a new parameter, $varphi$, indicating the anisotropy of the equivalent medium, and we show its range and properties. Subsequently, we compare $varphi$ to the Thomsen parameters, emphasizing its usefulness as a supportive parameter showing the anisotropy of the equivalent medium or as an alternative parameter to $delta$. We argue with certain Berryman et al. considerations regarding the properties of the anisotropy parameters $epsilon$ and $delta$. Additionally, we show an alternative way---to the one mentioned by Berryman et al.---of indicating changing fluid content in layered Earth.
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form, thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparlings tetrad-dependent differential forms, and our wave equation governs the propagation of Sparlings 2-form, which in the ``time-gauge is built linearly from the ``extrinsic curvature 1-form. The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum.
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