Geomagnetic data assimilation utilizing the ensemble Kalman smoother

In data assimilation, smoothers improve estimates of the system state by incorporating future observations. However, in geomagnetic data assimilation, the application of smoothers requires solving complex adjoint operators associated with the full nonlinear MHD equations, and the computation of gradients of the objective function is computationally expensive. Here, we employ the ensemble Kalman smoother (EnKS), which exploits ensemble-based statistical correlations across different times and thereby avoids the explicit construction of adjoint operators. We evaluate the performance of EnKS using synthetic observation experiments in moderately nonlinear models and compare it with Ensemble Kalman Filter (EnKF). The results show that both methods recover similar velocity field structures. EnKS exhibits velocity intensities closer to the reference model and performs better in the recovery of the surface flows. However, EnKS is more sensitive to sampling errors, which lead to filter divergence in the magnetic field. We further examine the impact of model error on EnKS, where the model error only arises from variations in viscous effects. The results show that model error causes the loss recovery of some dominant velocity field modes in the recovered solution and ultimately leads to filter divergence. Overall, our results indicate that EnKS can further improve recovery quality in regimes where EnKF already achieves reasonable performance, but may perform worse in regions strongly affected by sampling errors.