Читать книгу Mantle Convection and Surface Expressions - Группа авторов - Страница 64

The results of a series of experiments or computations are typically inverted into sets of parameters that describe the variation of elastic properties as a function of finite strain and temperature. Different definitions of finite strain result in different functional forms of expressions for elastic properties. For equations of state, different definitions and assumptions give rise to a remarkable diversity in EOS formulations (Angel, 2000; Angel et al., 2014; Holzapfel, 2009; Stacey & Davis, 2004). The variation of components of the elastic stiffness tensor has been formulated using both the Lagrangian and Eulerian definitions of finite strain (Birch, 1947; Davies, 1974; Thomsen, 1972a). For interpolations between individual observations, i.e., experiments or computations, differences between formulations should be small since, in a semi‐empirical approach, finite‐strain parameters are chosen to best reproduce the observations. Extrapolations of elastic properties beyond observational constraints, however, can be extremely sensitive to the chosen formalism. Both different definitions of finite strain and expansions to different orders of finite strain can result in substantial deviations between different finite‐strain models when extrapolated beyond observational constraints. Owing to the prevalence of elasticity formalisms based on Eulerian finite strain (Davies & Dziewonski, 1975; Ita & Stixrude, 1992; Jackson, 1998; Sammis et al., 1970; Stixrude & Lithgow‐Bertelloni, 2005), this source of uncertainty is not commonly taken into account when computing seismic properties and tends to become less important as experiments and computations are being pushed towards the verges of the relevant pressure–temperature space. Additional uncertainties arise from extrapolating the thermal or vibrational properties of minerals beyond the limitations of underlying assumptions, such as the quasi-harmonic approximation. For example, anharmonic contributions to elastic properties at high temperatures are mostly ignored as they remain difficult to assess in experiments and computations. Cobden et al. (2008) have explored how different combinations of finite‐strain equations and thermal corrections, including anelastic contributions, affect the outcomes of mineral‐physical models.

Whether derived from experiments or computations, elastic properties are affected by uncertainties that need to be propagated into the uncertainties on finite‐strain parameters. Inverting elasticity data on a limited number of pressure–temperature combinations to find the optimal set of finite‐strain parameters will inevitably result in correlations between finite‐strain parameters. The uncertainties on derived finite‐strain parameters are only meaningful when the uncertainties on the primary data have been assessed correctly, a requirement that can be difficult to meet in particular for first‐principle computations. Uncertainties on modeled seismic properties of rocks arise from uncertainties on individual finite‐strain parameters and from the anisotropy of the rock‐forming minerals as captured by the bounds on the elastic moduli. When constructing mineral‐physical models, uncertainties have been addressed by varying the parameters that describe the elastic properties of minerals in a randomized way within their individual uncertainties (Cammarano et al., 2003; Cammarano et al., 2005a; Cobden et al., 2008). While capturing the combined variance of the models, randomized sampling of parameters cannot disclose how individual parameters or properties affect the model. Identifying key properties might help to define future experimental and computational strategies to better constrain the related parameters in mineral‐physical models.

To illustrate how modeled seismic properties of mantle rocks are affected by different sources of uncertainties, I first concentrate on the properties of monomineralic and isotropic aggregates of major minerals in Earth’s upper mantle, transition zone, and lower mantle, i.e., olivine, wadsleyite and ringwoodite, and bridgmanite. For these minerals, complete elastic stiffness tensors have been determined together with unit cell volumes for relevant compositions and at relevant pressures. Such data sets can be directly inverted for the parameters of the cold parts of finite‐strain expressions for the components of the elastic stiffness tensor. For olivine compositions with Mg/(Fe+Mg) = 0.9, i.e., San Carlos olivine, high‐pressure elastic stiffness tensors at room temperature (Zha et al., 1998) can be combined with recent experiments at simultaneously high pressures and high temperatures (Mao et al., 2015; Zhang & Bass, 2016) to self‐consistently constrain most anisotropic finite‐strain parameters. For wadsleyite and ringwoodite, recent experimental results on single crystals (Buchen et al., 2018b; Schulze et al., 2018) are combined with tabulated parameters for the isotropic thermal contributions (Stixrude & Lithgow‐Bertelloni, 2011). Similarly, high‐pressure elastic stiffness tensors of bridgmanite at room temperature (Kurnosov et al., 2017) are complemented with results of DFT computations (Zhang et al., 2013) and high‐pressure high‐temperature experiments on polycrystals (Murakami et al., 2012) that constrain the isotropic thermal contributions.

For each mineral, P‐ and S‐wave velocities are computed using the Voigt‐Reuss‐Hill averages for bulk and shear moduli of an isotropic aggregate. The explored pressures and temperatures are spanned by two adiabatic compression paths that start 500 K above and below a typical adiabatic compression path for each mineral. To examine the impact of uncertainties on a given finite‐strain parameter, P‐ and S‐wave velocities are then recalculated by first adding (+) and then subtracting (–) the respective uncertainty to a given parameter leaving all other parameters unchanged. The resulting difference in velocities Δv = v⁺ − v⁻ is then compared to the original velocity v^VRH as:

The effect of anisotropy is illustrated in the same way by using the Voigt (V) and Reuss (R) bounds on the moduli and setting Δv = v^V − v^R. The velocity variations d lnv are mapped over relevant pressures and temperatures for olivine, wadsleyite, ringwoodite, and bridgmanite. Note that, for each of these minerals, the results of experiments and computations need to be substantially inter‐ and extrapolated across the respective pressure–temperature space.

Figures 3.1 and 3.2 show relative variations d lnv of P‐ and S‐wave velocities, respectively, that arise from differences between the Voigt and Reuss bounds as well as from uncertainties on the parameters , , γ₀, q₀, and η₀. Varying the Debye temperature θ₀ within reported uncertainties does not change wave velocities by more than 0.5% for the minerals and conditions considered, and no corresponding plots have been included in Figures 3.1 and 3.2. Parameter values and uncertainties were taken from the respective references or, in the case of San Carlos olivine, derived by fitting finite‐strain equations to the combined data set of elastic stiffness tensors and unit cell volumes at high pressures and high temperatures (Mao et al., 2015; Zha et al., 1998; Zhang & Bass, 2016). The uncertainties can therefore be considered to reflect the precision of current experimental and computational methods. The resulting variations in P‐ and S‐wave velocities, however, display the impact of individual parameters rather than absolute uncertainties on wave velocities. This way of propagating uncertainties on individual parameters into uncertainties on wave velocities illustrates how the influence of a given parameter changes with pressure and temperature and allows to identify those parameters that exert the strongest impact on P‐ and S‐wave velocities, respectively.

Aggregate P‐ and S‐wave velocities of olivine, wadsleyite, and bridgmanite all show variations of more than 1% due to elastic anisotropy as reflected in the differences between the Voigt and Reuss bounds. The elastic anisotropy of ringwoodite single crystals is known to be fairly small (Mao et al., 2012; Sinogeikin et al., 1998; Weidner et al., 1984). As a result, aggregate P‐ and S‐wave velocities differ by less than 0.5% for ringwoodite. For minerals with significant elastic anisotropy, including the major mantle minerals olivine, wadsleyite, and bridgmanite, uncertainties on wave velocities that arise from averaging over grains with different orientations may contribute substantially to absolute uncertainties as the actual elastic response of an isotropic polycrystalline aggregate may fall somewhere in between the bounds on elastic moduli. Note that alternative bounding schemes might provide tighter bounds on elastic moduli than the Voigt and Reuss bounds (Watt et al., 1976).

In comparison to the impact of elastic anisotropy on aggregate elastic moduli and wave velocities, the pressure derivatives and do not appear to strongly affect wave velocities when varied within reported uncertainties. This observation reflects the common approach of experimental and computational methods to address the response of elastic properties to compression and hence to best constrain pressure derivatives. To a certain extent, the comparatively small leverage of pressure derivatives on elastic wave velocities justifies inter‐ and extrapolations of the finite‐strain formalism to pressures and temperatures not covered by experiments or computations. Along adiabatic compression paths of typical mantle rocks, for example, changes in elastic properties that result from compression or volume reduction surpass changes that result from the corresponding adiabatic increase in temperature.

With the exception of the Debye temperature, the parameters describing the quasi‐harmonic or thermal contribution to elastic properties all show significant impact on computed wave velocities. Both P‐ and S‐wave velocities of the minerals included in Figures 3.1 and 3.2 appear to be sensitive to variations in the Grüneisen parameter γ₀, in particular at low pressures and high temperatures. In general, the isotropic volume strain derivative q₀ = η_V0/γ₀ of the Grüneisen parameter has more impact on P‐wave velocities while the isotropic shear strain derivative η_S0 mostly affects S‐wave velocities. The S‐wave velocities of transition zone minerals seem to be particularly sensitive to η_S0. This sensitivity arises from comparatively large uncertainties on η_S0 for these minerals and from relatively high temperatures in the transition zone at comparatively small finite strains. The exothermic phase transitions from olivine to wadsleyite and from wadsleyite to ringwoodite are expected to raise the temperatures in the transition zone in addition to adiabatic compression (Katsura et al., 2010). For olivine and bridgmanite, adiabatic compression over extended pressure ranges somewhat mitigates the influence of thermoelastic parameters as compressional contributions to elastic moduli increase at the expense of thermal contributions.

A complete analysis of uncertainties on computed P‐ and S‐wave velocities would include correlations between finite‐strain parameters that are, however, not regularly reported. The derivation of an internally consistent matrix of covariances between finite‐strain parameters requires coherent data sets of elastic properties at high pressures and high temperatures, i.e., data sets than can be simultaneously inverted for all relevant finite‐strain parameters. Due to experimental challenges in performing measurements of sound wave velocities at combined high pressures and high temperatures, however, finite‐strain parameters for most mantle minerals have been derived from separate but complementary data sets. Elastic moduli and their pressure derivatives are often obtained from sound wave velocity measurements at high pressures but ambient temperatures. Thermoelastic parameters are then independently derived from a thermal EOS, from the results of separate measurements at high temperatures, or from a computational study. This approach does not always generate data sets that can be combined and jointly inverted for complete and consistent sets of finite‐strain parameters and their covariances. As a consequence, the results of different studies are combined in terms of finite‐strain parameters that have been derived from separate data sets. As an example, the inversion of the combined data set of elastic stiffness tensors and unit cell volumes of San Carlos olivine that have been determined in separate studies at high pressures (Zha et al., 1998) and at combined high pressures and high temperatures (Mao et al., 2015; Zhang and Bass, 2016) required fixing the components of the Grüneisen tensor γ_ii0 and the Debye temperature θ₀ in order to stabilize the inversion results. Correlations between finite‐strain parameters can be reduced by optimizing the sampling of the relevant volume–temperature space. Future studies that provide consistent elasticity data sets at combined high pressures and high temperatures will allow for analyzing parameter correlations and help to reduce uncertainties in computed P‐ and S‐wave velocities by integrating covariances into the propagation of uncertainties.

Figure 3.1 Variations in P‐wave velocities for isotropic polycrystalline aggregates of olivine (1st column), wadsleyite (2nd column), ringwoodite (3rd column), and bridgmanite (4th column) that result from propagating uncertainties on individual finite‐strain parameters. Respective parameters and uncertainties are given in each panel. Bold black curves show adiabatic compression paths separated by temperature intervals of 500 K at the lowest pressure for each mineral. See text for references.

Figure 3.2 Variations in S‐wave velocities for isotropic polycrystalline aggregates of olivine (1st column), wadsleyite (2nd column), ringwoodite (3rd column), and bridgmanite (4th column) that result from propagating uncertainties on individual finite‐strain parameters. Respective parameters and uncertainties are given in each panel. Bold black curves show adiabatic compression paths separated by temperature intervals of 500 K at the lowest pressure for each mineral. See text for references.