Abstract: Geomagnetic data assimilation is a recently established research discipline in geomagnetism. It aims to optimally combine geomagnetic observations and numerical geodynamo models to better estimate the dynamic state of the Earth’s outer core, and to predict geomagnetic secular variation. Over the past decade, rapid advances have been made in geomagnetic data assimilation on various fronts by several research groups around the globe, such as using geomagnetic data assimilation to understand and interpret the observed geomagnetic secular variation, estimating part of the core state that is not observable on the Earth’s surface, and making geomagnetic forecasts on multi-year time scales. In parallel, efforts have also been made on proxy systems for understanding fundamental statistical properties of geomagnetic data assimilation, and for developing algorithms tailored specifically for geomagnetic data assimilation. In this chapter, we provide a comprehensive overview of these advances, as well as some of the immediate challenges of geomagnetic data assimilation, and possible solutions and pathways to move forward.

The role of forecast error covariance in practical ensemble and variational data assimilation is described following algebraic and dynamical views. This is used to introduce a motivation for ensemble data assimilation. It is shown how a dynamically induced and anisotropic ensemble error covariance can benefit data assimilation, compared to climatological (static) and isotropic error covariance used in variational methods. In addition to the standard ensemble Kalman filter (EnKF), more practical square root EnKF equations are also presented. Direct transform ensemble methods are also introduced and their connection with both ensemble and variational methods described. Error covariance localization in terms of the Schur product, a standard component of any realistic ensemble-based data assimilation, is also introduced and discussed. Following that, hybrid data assimilation and in particular the ensemble-variational (EnVar) methods are introduced and presented in relation to pure ensemble and variational methods. As a particular example of hybrid methods the maximum likelihood ensemble filter (MLEF) is introduced.

This chapter focuses on assimilation of observations from satellites, which is a dominant source of observation information in weather and climate. This includes satellite radiances, both clear sky and all-sky. The most important challenges of all-sky radiances come from their connection to cloud microphysics, which potentially implies nonlinear, non-Gaussian, and nondifferentiable processes that are difficult for data assimilation. The complexity of error covariance with microphysical variables is illustrated in a few real-world examples. An additional difficulty with assimilating all-sky radiances comes from correlated observation errors that require special attention in data assimilation. Practical ways to deal with correlated observation errors are described. Nonlinearity and nondifferentiability of observation operators for all-sky radiances is also briefly explained. Since satellite radiance observations and observation operators generally contain bias, a common formulation of radiance bias correction methods is also presented. The observations from satellites also include radio occultation and lightning observations, as well as satellite products.