Global field reconstructions of the past are a key tool for understanding the dynamics of the Earth's magnetic field and the underlying processes in the Earth's core (e.g., Constable & Korte, 2015). This includes studying the evolution of field features, such as dipole decay, the South Atlantic Anomaly (SAA) and flux patches (Hartmann & Pacca, 2009;Jackson & Finlay, 2015). In the past, several techniques for constructing global field models have been developed and employed. Truncated spherical harmonics (SH) in the spatial domain combined with spline interpolation in time are widely used (Jackson et al., 2000;Korte et al., 2009). In the eighties, Constable and Parker (1988) first proposed using Gaussian processes to model the field dynamics, but until recently, the technique had not been applied to global field modeling. Only in the last years, statistical methods implementing this approach have been suggested (Hellio & Gillet, 2018;Nilsson & Suttie, 2021).Even though Bloxham and Jackson (1992) already discussed the accurate assessment of uncertainties, most following studies did not pursue the suggested analytical approaches, and either did not report uncertainties at

Summary For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective a priori assumptions. Rather than parameterizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. A priori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearisation that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 years is in general agreement with previous models while improved model uncertainty estimates are provided.

Existing models of the Earth's magnetic field (EMF) for the past millennia show a variety of time-dependent features: The evolution of the South Atlantic Anomaly, the observed dipole decay in recent centuries and the movement of flux patches all take place on timescales of several hundred years (see e.g., Hartmann & Pacca, 2009;Jackson & Finlay, 2015). To accurately describe and study these features, time-resolved models are necessary. Usually, these models are inferred from two classes of data: Data from materials with thermoremanent magnetization, such as volcanic rocks, bricks or burnt clay fragments from archeologic sites, and data from marine or lacustrine sediments with embedded magnetic particles. In this paper we focus on the former class and loosely refer to it as archeomagnetic data. Existing models differ in the approach to global modeling, but are usually constructed using inversion for spherical harmonics (SH) coefficients, truncated at a certain degree.

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In a previous study, a new snapshot modeling concept for the archeomagnetic field was introduced (Mauerberger et al., 2020). By assuming a Gaussian process for the geomagnetic potential, a correlation based algorithm was presented, which incorporates a closed form spatial correlation function. This work extends the suggested modeling strategy to the temporal domain. A space-time correlation kernel is constructed from the tensor product of the closed form spatial correlation kernel with a squared exponential kernel in time. Dating uncertainties are incorporated into the modeling concept using a noisy input Gaussian process. All but one modeling hyperparameters are marginalized, to reduce their influence on the outcome and to translate their variability to the posterior variance. The resulting distribution incorporates uncertainties related to dating, measurement and modeling process. Results from application to archeomagnetic data show less variation in the dipole than comparable models, but are in general agreement with previous findings.