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[1] Geodynamic models commonly assume equations of state as a function of pressure and temperature. This form is legitimate for homogenous materials, but it is impossible to formulate a general equation of state for a polyphase aggregate, e.g., a rock, as a function of pressure and temperature because these variables cannot distinguish all possible states of the aggregate. In consequence, the governing equations of a geodynamic model based on a pressure-temperature equation of state are singular at the conditions of low-order phase transformations. An equation of state as a function of specific entropy, specific volume, and chemical composition eliminates this difficulty and, additionally, leads to a robust formulation of the energy and mass conservation equations. In this formulation, energy and mass conservation furnish evolution equations for entropy and volume and the equation of state serves as an update rule for temperature and pressure. Although this formulation is straightforward, the computation of phase equilibria as a function of entropy and volume is challenging because the equations of state for individual phases are usually expressed as a function of temperature and pressure. This challenge can be met by an algorithm in which continuous equations of state are approximated by a series of discrete states: a representation that reduces the phase equilibrium problem to a linear optimization problem that is independent of the functional form used for the equations of state of individual phases. Because the efficiency of the optimization decays as an exponential function of the dimension of the function to be optimized, direct solution of the linearized optimization problem is impractical. Successive linear programming alleviates this difficulty. A pragmatic alternative to optimization as an explicit function of entropy and volume is to calculate phase relations over the range of pressure-temperature conditions of interest. Numerical interpolation can then be used to generate tables for any thermodynamic property as a function of any choice of independent variables. Regardless of the independent variables of the governing equations, a consistent definition of pressure, and the coupling of equilibrium kinetics to deformation, is only possible if the continuity equation accounts for dilational strain.

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Aims. We present an inversion method based on Bayesian analysis to constrain the interior structure of terrestrial exoplanets, in the form of chemical composition of the mantle and core size. Specifically, we identify what parts of the interior structure of terrestrial exoplanets can be determined from observations of mass, radius, and stellar elemental abundances. Methods. We perform a full probabilistic inverse analysis to formally account for observational and model uncertainties and obtain confidence regions of interior structure models. This enables us to characterize how model variability depends on data and associated uncertainties. Results. We test our method on terrestrial solar system planets and find that our model predictions are consistent with independent estimates. Furthermore, we apply our method to synthetic exoplanets up to 10 Earth masses and up to 1.7 Earth radii, and to exoplanet Kepler-36b. Importantly, the inversion strategy proposed here provides a framework for understanding the level of precision required to characterize the interior of exoplanets. Conclusions. Our main conclusions are (1) observations of mass and radius are sufficient to constrain core size; (2) stellar elemental abundances (Fe, Si, Mg) are principal constraints to reduce degeneracy in interior structure models and to constrain mantle composition; (3) the inherent degeneracy in determining interior structure from mass and radius observations does not only depend on measurement accuracies, but also on the actual size and density of the exoplanet. We argue that precise observations of stellar elemental abundances are central in order to place constraints on planetary bulk composition and to reduce model degeneracy. We provide a general methodology of analyzing interior structures of exoplanets that may help to understand how interior models are distributed among star systems. The methodology we propose is sufficiently general to allow its future extension to more complex internal structures including hydrogen-and water-rich exoplanets.

[1] A combined geophysical-petrological methodology to study the thermal, compositional, density, and seismological structure of lithospheric/sublithospheric domains is presented. A new finite-element code (LitMod) is used to produce 2-D forward models from the surface to the 410-km discontinuity. The code combines data from petrology, mineral physics, and geophysical observables within a self-consistent framework. The final result is a lithospheric/sublithospheric model that simultaneously fits all geophysical observables and consequently reduces the uncertainties associated with the modeling of these observables alone or in pairs, as is commonly done. The method is illustrated by applying it to both oceanic and continental domains. We show that anelastic attenuation and uncertainties in seismic data make it unfeasible to identify compositional variations in the lithospheric mantle from seismic studies only. In the case of oceanic lithosphere, plates with thermal thicknesses of 105 ± 5 km satisfy geophysical and petrological constraints. We find that Vp are more sensitive to phase transitions than Vs, particularly in the case of the spinel-garnet transition. A low-velocity zone with absolute velocities and gradients comparable to those observed below ocean basins is an invariable output of our oceanic models, even when no melt effects are included. In the case of the Archean subcontinental lithospheric mantle, we show that ''typical'' depleted compositions (and their spatial distribution) previously thought to be representative of these mantle sections are compatible neither with geophysical nor with petrological data. A cratonic keel model consisting of (1) strongly depleted material (i.e., dunitic/harzburgitic) in the first 100-160 km depth and (2) less depleted (approximately isopycnic) lower section extending down to 220-300 km depth is necessary to satisfy elevation, geoid, SHF, seismic velocities, and petrological constraints. This highly depleted (viscous) upper layer, and its chemical isolation, may play a key role in the longevity and stability of cratons.

Volatiles, most notably CO2, are recycled back into the Earth's interior at subduction zones. The amount of CO2 emitted from arc volcanism appears to be less than that subducted, which implies that a significant amount of CO2 either is released before reaching the depth at which arc magmas are generated or is subducted to deeper depths. Few high-pressure experimental studies have addressed this problem and therefore metamorphic decarbonation in subduction zones remains largely unquantified, despite its importance to arc magmatism, palaeoatmospheric CO2 concentrations and the global carbon cycle. Here we present computed phase equilibria to quantify the evolution of CO2 and H2O through the subduction-zone metamorphism of carbonate-bearing marine sediments (which are considered to be a major source for CO2 released by arc volcanoes). Our analysis indicates that siliceous limestones undergo negligible devolatilization under subduction-zone conditions. Along high-temperature geotherms clay-rich marls completely devolatilize before reaching the depths at which arc magmatism is generated, but along low-temperature geotherms, they undergo virtually no devolatilization. And from 80 to 180 km depth, little devolatilization occurs for all carbonate-bearing marine sediments. Infiltration of H2O-rich fluids therefore seems essential to promote subarc decarbonation of most marine sediments. In the absence of such infiltration, volatiles retained within marine sediments may explain the apparent discrepancy between subducted and volcanic volatile fluxes and represent a mechanism for return of carbon to the Earth's mantle.

[1] Traditional inversion techniques applied to the problem of characterizing the thermal and compositional structure of the upper mantle are not well suited to deal with the nonlinearity of the problem, the trade-off between temperature and compositional effects on wave velocities, the nonuniqueness of the compositional space, and the dissimilar sensitivities of physical parameters to temperature and composition. Probabilistic inversions, on the other hand, offer a powerful formalism to cope with all these difficulties, while allowing for an adequate treatment of the intrinsic uncertainties associated with both data and physical theories. This paper presents a detailed analysis of the two most important elements controlling the outputs of probabilistic (Bayesian) inversions for temperature and composition of the Earth's mantle, namely the a priori information on model parameters, (m), and the likelihood function, L(m). The former is mainly controlled by our current understanding of lithosphere and mantle composition, while the latter conveys information on the observed data, their uncertainties, and the physical theories used to relate model parameters to observed data.[2] The benefits of combining specific geophysical datasets (Rayleigh and Love dispersion curves, body wave tomography, magnetotelluric, geothermal, petrological, gravity, elevation, and geoid), and their effects on L(m), are demonstrated by analyzing their individual and combined sensitivities to composition and temperature as well as their observational uncertainties. The dependence of bulk density, electrical conductivity, and seismic velocities to major-element composition is systematically explored using Monte Carlo simulations. We show that the dominant source of uncertainty in the identification of compositional anomalies within the lithosphere is the intrinsic nonuniqueness in compositional space. A general strategy for defining (m) is proposed based on statistical analyses of a large database of natural mantle samples collected from different tectonic settings (xenoliths, abyssal peridotites, ophiolite samples, etc.). This strategy relaxes more typical and restrictive assumptions such as the use of local/limited xenolith data or compositional regionalizations based on age-composition relations. We demonstrate that the combination of our (m) with a L(m) that exploits the differential sensitivities of specific geophysical observables provides a general and robust inference platform to address the thermochemical structure of the lithosphere and sublithospheric upper mantle. An accompanying paper deals with the integration of these two functions into a general 3-D multiobservable Bayesian inversion method and its computational implementation.