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  1. Despite progress in tomographic imaging of Earth’s interior, a number of critical questions regarding the large-scale structure and dynamics of the mantle remain outstanding. One of those questions is the impact of phase-boundary undulations on global imaging of mantle heterogeneity and on geodynamic (i.e. convection-related) observables. To address this issue, we developed a joint seismic-geodynamic-mineral physical tomographic inversion procedure that incorporates lateral variations in the depths of the 410- and 660-km discontinuities. This inversion includes S-wave traveltimes, SS precursors that are sensitive to transition-zone topography, geodynamic observables/data (free-air gravity, dynamic surface topography, horizontal divergence of tectonic plates and excess core-mantle boundary ellipticity) and mineral physical constraints on thermal heterogeneity. Compared to joint tomography models that do not include data sensitivity to phase-boundary undulations in the transition zone, the inclusion of 410- and 660-km topography strongly influences the inference of volumetric anomalies in a depth interval that encompasses the transition zone and mid-mantle. It is notable that joint tomography inversions, which include constraints on transition-zone discontinuity topography by seismic and geodynamic data, yield more pronounced density anomalies associated with subduction zones and hotspots. We also find that the inclusion of 410- and 660-km topography may improve the fit to the geodynamic observables, depending on the weights applied to seismic and geodynamic data in the inversions. As a consequence, we find that the amplitude of non-thermal density anomalies required to explain the geodynamic data decreases in most of the mantle. These findings underline the sensitivity of the joint inversions to the inclusion of transition-zone complexity (e.g. phase-boundary topography) and the implications for the inferred non-thermal density anomalies in these depth regions. Finally, we underline that our inferences of 410- and 660-km topography avoid a commonly employed approximation that represents the contribution of volumetric heterogeneity to SS-wave precursor data. Our results suggest that this previously employed correction, based on a priori estimates of uppermantle heterogeneity, might be a significant source of error in estimating the 410- and 660-km topography. 
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