Mantle Convection and Surface Expressions. Группа авторов

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Figure 1.4 (A) Viscosity profiles used in our geodynamic models. For comparison, we also show viscosity profiles obtained in a joint inversion constrained by glacial isostatic adjustment (GIA) and convection‐related observables (Mitrovica and Forte (2004) Figure 1.4B), a combination of geoid, GIA, geodynamic constraints (Case C from Steinberger and Holme (2008)) and a joint inversion of GIA data including the Fennoscandian relaxation spectrum (Fig. 12 from Lau et al. (2016)). (B) Spectral slope vs. depth computed from the dimensionless temperature field of the geodynamic models. (C) Correlation at spherical harmonic degrees 1–4 between each of the geodynamic models and SEMUCB‐WM1.

We inferred density anomalies from two different tomographic models. First, we used SEMUCB‐WM1 and scaled Voigt V_S anomalies to density using a depth‐dependent scaling factor for a pyrolitic mantle composition along a 1600 K mantle adiabat, calculated from thermodynamic principles using HeFESTo Stixrude and Lithgow‐Bertelloni, 2011). Second, we used a whole‐mantle model of density variations constrained by full‐spectrum tomography (Moulik and Ekström, 2016). This model, hereafter referred to as ME16‐160, imposes a best data‐fitting scaling factor between density and V_S variations of d ln ρ/d ln V_S = 0.3 throughout the mantle. We note that Moulik and Ekström (2016) present a suite of models with different choices for data weighting and preferred correlation between density variations and V_S variations in the lowermost mantle. The specific model used here ignores sensitivity to the density‐sensitive normal modes (data weight = 0) and imposes strong V_S − ρ correlation in the lowermost mantle (). While this model is not preferred by seismic data, these choices produce a density model that closely resembles a scaled V_S model in the lowermost mantle. We note that the assumption of purely thermal contributions to density is unlikely to be correct in the lowermost mantle, where temperature and compositional variations both contribute to density variations. However, this assumption should not affect our inferences of viscosity, for two reasons. First, previous viscosity inversions found that removing all buoyancy structure from the bottom 1,000 km of the mantle did not significantly influence the retrieved viscosity profile (Rudolph et al., 2015). This is confirmed by inversions for a four‐layer viscosity structure constrained by thermal vs. thermochemical mantle buoyancy Liu and Zhong (2016). Second, geodynamic models of thermochemical convection suggest that the thermal buoyancy above the LLSVPs counteracts the thermal and chemical buoyancy of the compositionally distinct LLSVPs, resulting in only a very small net contribution to the long‐wavelength geoid from the bottom 1,000 km of the mantle (Liu and Zhong, 2015).

In the inversions using SEMUCB‐WM1, we assume a diagonal covariance matrix to describe the data and forward modeling uncertainty on geoid coefficients, i.e., uncorrelated errors and uniform error variance at all spherical harmonic degrees (because the corresponding posterior covariance matrix is not available). For the inversions using ME16‐160, we first sample the a posteriori covariance matrix of the tomographic model, generating a collection of 10⁵ whole‐mantle models of density and wavespeed variations. For each of these models, we calculate a synthetic geoid assuming a reference viscosity profile (Model C from Steinberger and Holme (2008)). This procedure yields a sample of synthetic geoids from which we calculate a sample covariance matrix that is used to compute the Mahalanobis distance as a measure of viscosity model misfit.

      In all of the inversions shown in this chapter, we include a hierarchical hyperparameter that scales the covariance matrix. This parameter has the effect of smoothing the misfit function in model space, and the value of the hyperparameter is retrieved during the inversion, along with the other model parameters. The inversion methodology, described completely in Rudolph et al. (2015), uses a reversible‐jump Markov‐Chain Monte Carlo (rjMCMC) method (Green, 1995) to determine the model parameters (the depths and viscosity values of control points describing the piecewise‐linear СКАЧАТЬ