Spline-based geomagnetic field  modeling revisited

Many existing global geomagnetic field models for historical and longer timescales are based on spherical harmonics for the spatial part and low order basis splines for the temporal dynamics. The related modeling procedure is well established and implements a regularized non-linear least squares inversion. Early papers on the methodology already mention uncertainty estimation in this context. Even though they use a different language, this connects the regularized least squares approach to Bayesian statistics. However, most likely due to computational costs, the ideas from the early papers were lost in subsequent studies and most presented models rely on bootstrapping methods for uncertainty estimation or do not include uncertainties at all. In a statistical setting, this translates to presenting a point estimate (mean or mode) of the posterior distribution. We revised the established spline based modeling formalism in order to excavate the procedure for uncertainty estimation and complement the point estimate by covariance matrices. This way, uncertainty estimates can be provided for many existing models retrospectively. Using ensemble techniques, the estimated uncertainty can also be propagated to derived quantities like the PSV-index or geomagnetic shielding. Finally, the statistical view provides a new way of estimating the regularization parameters using marginal likelihood optimization. The presented method is applicable to satellite based models in principle, although it is less relevant there due to the large amount of data and uncertainties from other sources (e.g. external fields).