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 VS 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 VS variations of d ln ρ/d ln VS = 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 VS variations in the lowermost mantle. The specific model used here ignores sensitivity to the density‐sensitive normal modes (data weight
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 viscosity profile) and the noise hyperparameter (Malinverno, 2002; Malinverno and Briggs, 2004). The rjMCMC method inherently includes an Occam factor, which penalizes overparameterization. Adding model parameters must be justified by a significant reduction in misfit. The result is a parsimonious parameterization of viscosity that balances data fit against model complexity. In general, incorporating additional data constraints or a priori information about mantle properties could lead to more complex solutions.
1.3 RESULTS
The power spectra of four recent global tomographic models are shown in Figure 1.5. While S362ANI+M contains little power above spherical harmonic degree 20 due to its long‐wavelength lateral parameterization with 362 evenly spaced spline knots, the other models contain significant power at shorter wavelengths that are beyond the scope of this study. In general, the models are dominated by longer wavelengths at all depths. The spectral slope for each model (up to degree 20) is shown in the rightmost panel of Figure 1.5. Increasingly negative spectral slopes indicate that heterogeneity is dominated by long‐wavelength features. All of the models generally show a more negative spectral slope in the transition zone than in the upper mantle or the shallow lower mantle, indicating the presence of more long‐wavelength VS heterogeneity within the transition zone and just below the 650 km discontinuity, which we attribute to the lateral deflection of slabs.
Figure 1.5 Power spectra of four recent global VS tomographic models. Because our focus is on long‐wavelength structure, and to ensure a more equitable comparison, we show only spherical harmonic degrees 1–20, though SEMUCB‐WM1, SEISGLOB2, and GLAD‐M15 contain significant power at shorter wavelengths. At the right, we show spectral slopes (defined in the text) for the four models. Here, increasingly negative values indicate a concentration of power in longer wavelengths/lower spherical harmonic degrees.
Figure 1.3 shows radial correlation plots for the four tomographic models. Here, the spherical harmonic expansions are truncated at degrees 2 (lower left triangle) and 4 (upper right triangle). At degrees 1–2 and 1–4, both SEMUCB‐WM1 and SEISGLOB2 show a clear change in the correlation structure near 1,000 km depth. On the other hand, S362ANI+M and GLAD‐M15 both show a change in correlation structure at 650 km.
We compared the character of heterogeneity in the geodynamic models with mantle tomography by calculating the power spectrum and spectral slope of each of the five geodynamic models. Because the depth‐variation of power in the geodynamic models does not have as straightforward an interpretation as the VS power spectra shown for tomographic models, we focus on the spectral slope of the geodynamic models, shown in Figure 1.4. We computed correlation coefficients between each of the geodynamic models and SEMUCB‐WM1, shown in Figure 1.4C.
For both of the tomographic models used to infer mantle viscosity, we carried out viscosity inversions (Figure 1.6) constrained by spherical harmonic degree 2 only, degrees 2 and 4 only, and degrees 2–7. The viscosity profiles are quite similar, regardless of which spherical harmonic degrees are used to constrain the inversion. However, we observe an increase in the overall complexity of the viscosity profile as more spherical harmonic degrees are included, as well as a tendency towards developing a low‐viscosity region below the 650 km discontinuity (red curves in Figure 1.6), considered in the context of parsimonious inversions, this tendency toward increased complexity can be attributed to the progressively greater information content of the data. The posterior ensembles from the viscosity inversions contain significant variability among accepted solutions, and the solid lines in