Bayesian inversion for the filtered flow at the Earth's core‐mantle boundary

Abstract

The inverse problem of determining the flow at the Earth's core-mantle boundary according to an outer core magnetic field and secular variation model has been investigated through a Bayesian formalism. To circumvent the issue arising from the truncated nature of the available fields, we combined two modeling methods. In the first step, we applied a filter on the magnetic field to isolate its large scales by reducing the energy contained in its small scales, we then derived the dynamical equation, referred as filtered frozen flux equation, describing the spatiotemporal evolution of the filtered part of the field. In the second step, we proposed a statistical parametrization of the filtered magnetic field in order to account for both its remaining unresolved scales and its large-scale uncertainties. These two modeling techniques were then included in the Bayesian formulation of the inverse problem. To explore the complex posterior distribution of the velocity field resulting from this development, we numerically implemented an algorithm based on Markov chain Monte Carlo methods. After evaluating our approach on synthetic data and comparing it to previously introduced methods, we applied it to a magnetic field model derived from satellite data for the single epoch 2005.0. We could confirm the existence of specific features already observed in previous studies. In particular, we retrieved the planetary scale eccentric gyre characteristic of flow evaluated under the compressible quasi-geostrophy assumption although this hypothesis was not considered in our study. In addition, through the sampling of the velocity field posterior distribution, we could evaluate the reliability, at any spatial location and at any scale, of the flow we calculated. The flow uncertainties we determined are nevertheless conditioned by the choice of the prior constraints we applied to the velocity field.

Decadal variability in core surface flows deduced from geomagnetic observatory monthly means

“…For example, our flows have a northward component beneath North America, westward flow beneath the northern-post Pacific Ocean and a southward component beneath eastern Asia which, if focussed, would form part of the eccentric planetary gyre characteristic of quasi-geostrophic flows discussed by Gillet et al (2009), amongst others. This feature of our flow resembles more closely that of Baerenzung et al (2014) which, although derived from satellite data, has a less spatially concentrated gyre. Baerenzung et al (2016) note that it is not necessary to make the quasi-geostrophic assumption to obtain flows with an eccentric planetary gyre.…”

“…Similarly, the small clockwise eddy beneath the south-western Pacific Ocean is not generally a feature of models derived from satellite data -in fact, the tangentially geostrophic flow of Holme and Olsen (2006) has a weak clockwise eddy, but their toroidal flow has an anti-clockwise eddy in the same location (and, as noted above, features of this size are not well resolved by observatory data). Besides generally decreasing power spectra for both toroidal and poloidal flow components, and an approximately order-of-magnitude difference between the toroidal and poloidal power, the dominant feature of the spectra is the loss of power at toroidal degree 3 (e.g Holme and Olsen 2006;Lesur et al 2010;Lesur et al 2015;Baerenzung et al 2014Baerenzung et al , 2016, regardless of whether or not the flow is assumed tangentially geostrophic. All these features are present in our spectra, for both snapshot and moderately TO-like flows (Fig.…”

“…Whaler (1986) simply compared the result with two different main field models. Rygaard-Hjalsted et al (2000) were the first to use a Monte Carlo Markov Chain approach, but computational resources at that time limited the applicability of the method; more recently, Baerenzung et al (2014) and Baerenzung et al (2016) used it to obtain a more reliable estimate of the posterior probability distribution. Lesur et al (2010) adopted an iterative approach, first estimating a field model from the data, then using it to determine a starting model for the flow in the traditional fashion, and finally co-estimating the field and flow iteratively from the starting field and flow models.…”

A statistical spatial power spectrum of the Earth's lithospheric magnetic field

“…In particular, the spectral form discussed in this paper is more energetic at the low SH degrees than the ones previously published (Jackson 1990(Jackson , 1994Voorhies 1998;Voorhies et al 2002). This suggests that the main field modelling error caused by the crustal field could have been so far underestimated (see Baerenzung et al 2014, for a recent consideration of the crustal field error). This aspect is important in the framework of geomagnetic data assimilation whose aim is to estimate the state vector of the Earth's outer core flow but that requires suitable error covariance matrices on the magnetic observations or on the core field models at low SH degrees .…”

Stochastic forecasting of the geomagnetic field from the COV-OBS.x1 geomagnetic field model, and candidate models for IGRF-12

“…The main source of errors, when imaging the flow from models of the secular variation ∂ t B r using Equation (3), comes from nonlinear interactions involving the unresolved parts of the flow and of the magnetic field (Baerenzung et al 2014;Eymin and Hulot 2005;Pais and Jault 2008). The projection of Equation (3) onto large length-scales (denoted by overlines) gives…”