Abstract: Geomagnetic data assimilation aims at constraining the state of the geodynamo working at the Earth’s deep interior by sparse magnetic observations at and above the Earth’s surface. Due to difficulty separating the different magnetic field sources in the observations, spectral models of the geomagnetic field are generally used as inputs for data assimilation. However, the assimilation of raw pointwise observations can be relevant within certain configurations, specifically with paleomagnetic and historical geomagnetic data. Covariance localisation, which is a key ingredient to the assimilation performance in an ensemble framework, is relatively unexplored, and differs with respect to spectral and pointwise observations. This chapter introduces the main characteristics of geomagnetic data and magnetic field models, and explores the role of model and observation covariances and localisation in typical assimilation set-ups, focusing on the use of 3D dynamo simulations as the background model.

Coupled data assimilation is presented in detail. Starting from a coupled modeling system, a classification of coupled data assimilation based on coupling strength is defined. This includes uncoupled, weakly coupled, and strongly coupled data assimilation, and the coupling strength is quantified using mutual information. The most interesting aspects of coupled data assimilation can be related to a strongly coupled system in which the information exchange is maximized. The challenges of strongly coupled data assimilation include the account of the complex control variable and error covariance. The mentioned challenges can considerably increase in realistic high-dimensional applications. Additional issues that can hamper strongly coupled data assimilation include non-Gaussian errors and potentially different spatiotemporal scales of coupled system components. To improve understanding of strongly coupled data assimilation, a simple two-component system is introduced and analyzed. The theoretical assessment is followed by real-world examples of strongly coupled forecast error covariance. Finally, the coupled covariance localization is analyzed and a practical method to address it is described.

The previous chapter discussed data assimilation for the case in which the variables have known Gaussian distributions. However, in atmospheric and oceanic data assimilation, the distributions are neither Gaussian nor known, and the large number of state variables creates numerical challenges. This chapter discusses a class of algorithms, called Ensemble Square Root Filters, for performing data assimilation with high-dimensional, nonlinear systems. The basic idea is to use a collection of forecasts (called an ensemble) to estimate the statistics of the background distribution. In addition, observational information is incorporated by adjusting individual ensemble members (i.e., forecasts) rather than computing an entire distribution. This chapter discusses three standard filters: the Ensemble Transform Kalman Filter (ETKF), the Ensemble Square Root Filter (EnSRF), and the Ensemble Adjustment Kalman Filter (EAKF). However, ensemble filters often experience filter divergence, in which the analysis no longer tracks the truth. This chapter discusses standard approaches to mitigating filter divergence, namely covariance inflation and covariance localization.