Abstract: Hydrological sciences cover a wide variety of water-driven processes at the Earth’s surface, above, and below it. Data assimilation techniques in hydrology have developed over the years along many quite independent paths, following not only different data availabilities but also a plethora of problem-specific model structures. Most hydrologic problems that are addressed through data assimilation, however, share some distinct peculiarities: scarce or indirect observation of most important state variables (soil moisture, river discharge, groundwater level, to name a few), incomplete or conceptual modelling, extreme spatial heterogeneity, and uncertainty of controlling physical parameters. On the other side, adoption of simplified and scale-specific models allows for substantial problem reduction that partially compensates these difficulties, opening the path to the assimilation of very indirect observations (e.g. from satellite remote sensing) and efficient model inversion for parameter estimation. This chapter illustrates the peculiarities of data assimilation for state estimation and model inversion in hydrology, with reference to a number of representative applications. Sequential ensemble filters and variational methods are recognised to be the most common choices in hydrologic data assimilation, and the motivations for these choices are also discussed, with several examples.

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.

Abstract: There is a fundamental need to understand and improve the errors and uncertainties associated with estimates of seasonal snow analysis and prediction. Over the past few decades, snow cover remote sensing techniques have increased in accuracy, but the retrieval of spatially and temporally continuous estimates of snow depth or snow water equivalent remains challenging tasks. Model-based snow estimates often bear significant uncertainties due to model structure and error-prone forcing data and parameter estimates. A potential method to overcome model and observational shortcomings is data assimilation. Data assimilation leverages the information content in both observations and models while minimising inherent limitations that result from uncertainty. This chapter reviews current snow models, snow remote sensing methods, and data assimilation techniques that can reduce uncertainties in the characterisation of seasonal snow.

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.