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The Generalized Standard Linear Solid (GSLS) model successfully fits seismic-attenuation and phase-velocity or moduli dispersion spectra observed in many experiments. Here, we call this general property of the spectra 'GSLS-equivalence' and try determining its physical causes and consequences. Three criteria must be satisfied by the attenuation/dispersion spectra of the system (rock and experimental apparatus) in order to be explainable by a GSLS: (1) non-negative wave-velocity or empirical-modulus dispersion at all frequencies, (2) typically band-limited attenuation, that is negligible inverse quality factor Q −1outside of some frequency band and (3) slopes in Q −1(f) spectra not steeper than proportional to f ±1. Although the well-known (Kramers–Krönig) relation of velocity dispersion to the Q −1(f) is usually attributed to causality, in practical band-limited observations, this relation is actually more specific and corresponds to the GSLS-equivalence. The physical significance of this property can be understood by considering mechanical models of the system. Although the GSLS-equivalence is tacitly implied in most attenuation models, this property is not automatic and suggests that the system possesses a specific mechanical structure in which the friction (viscosity) is only associated with certain types of internal deformation analogous to pore-fluid flows. This observation is illustrated on three examples. First, bitumen-sand samples in P -wave attenuation experiments are GSLS-equivalent, which means that their mechanical properties in arbitrary deformations can be described by 6 × 6 matrices of elasticity and viscosity parameters. By contrast, for spectra containing frequency ranges of negative velocity dispersion, such as predicted by some models of fluid-saturated rock, the GSLS-equivalence is absent, and velocity dispersion cannot be inferred from band-limited Q −1(f) data. Steep empirical Q −1(f) dependencies often found in local-earthquake observations are also GSLS-nonequivalent, which suggests problems with such measurements. To avoid physically problematic empirical attenuation/dispersion models, first-principle physics-based models such as solid viscosity, thermo- and poroelasticity should be used. [ABSTRACT FROM AUTHOR]

In many geophysical applications, viscoelastic (VE) model is applied to heterogeneous media, such as in studies of seismic waves within the layered Earth, models of tidal deformations of planetary bodies or mechanical testing of rock samples in the laboratory. However, for heterogeneous media or finite bodies, VE model contains one important omission, which consists in disregarding the boundary conditions (BCs) for internal-deformation variables. Internal variables are present in almost all anelastic rheologies, and their examples include filtration flows in porous rock and memory variables commonly used in VE numerical modelling. Similar to poroelasticity, internal variables in arbitrary VE models are sensitive to BCs on material-property contrasts. With several selections of BCs typical in mechanics ('closed', 'open', 'mixed' and 'continuity'), the effective modulus (stress/strain ratio) and Q −1(strain–stress phase lag) vary from near-elastic to significantly anelastic. For closed and mixed BCs, the Q −1can be strongly spatially variable and negative within parts of the body and at certain frequencies. These observations are illustrated by rigorous solutions for an anelastic layer with Maxwell's or Standard Linear Solid rheologies enclosed within an elastic space. In the conventional VE model, internal BCs are not stated explicitly but nevertheless implied by assuming the absence of non-VE deformation modes, which means internal variables not interacting with material-property contrasts and boundaries of the body. However, non-VE modes should exist, and they strongly modify the behaviour of the composite medium. The uncertainties of BCs in the VE model are somewhat analogous to the well-known difficulty of differentiating between the intrinsic and scattering Q s for seismic waves. Thus, although predicting the basic attenuation phenomena, the VE model describes heterogeneous anelastic media incompletely and consequently inaccurately. To overcome such shortcomings of the VE model, first-principle physical approaches are required for describing rock anelasticity. [ABSTRACT FROM AUTHOR]

A geometrical attenuation model is proposed as an alternative to the conventional frequency-dependent attenuation law Q( f) = Q0( f/ f0) η. The new model provides a straightforward differentiation between the geometrical and effective attenuation ( Qe) which incorporates the intrinsic attenuation and small-scale scattering. Unlike the ( Q0, η) description, the inversion procedure uses only the spectral amplitude data and does not rely on elaborate theoretical models or restrictive assumptions. Data from over 40 reported studies were transformed to the new parametrization. The levels of geometrical attenuation strongly correlate with crustal tectonic types and decrease with tectonic age. The corrected values of Qe are frequency-independent and generally significantly higher than Q0 and show no significant correlation with tectonic age. Several case studies were revisited in detail, with significant changes in the interpretations. The absorption-band and the ‘10-Hz transition’ are not found in the corrected Qe data, and therefore, these phenomena are interpreted as related to geometrical attenuation. The absorption band could correspond to changes in the dominant mode content of the wavefield as the frequency changes from about 0.1 to 100 Hz. Alternatively, it could also be a pure artefact related to the power-law Q( f) paradigm above. The explicit separation of the geometrical and intrinsic attenuation achieves three goals: (1) it provides an unambiguous, assumption- and model-free description of attenuation, (2) it allows relating the observations to the basic physics and geology and (3) it simplifies the interpretation because of reduced emphasis on the apparent Q( f) dependence. The model also agrees remarkably well with the initial attempts for finite-difference short-period coda waveform modelling. Because of its consistency and direct link to the observations, the approach should also help in building robust and transportable coda magnitudes and in seismic regionalization. [ABSTRACT FROM AUTHOR]