This paper compares several commonly used state-of-the-art ensemble-based data assimilation methods in a coherent mathematical notation. The study encompasses different methods that are applicable to high-dimensional geophysical systems, like ocean and atmosphere and provide an uncertainty estimate. Most variants of Ensemble Kalman Filters, Particle Filters and second-order exact methods are discussed, including Gaussian Mixture Filters, while methods that require an adjoint model or a tangent

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... inear formulation of the model are excluded. The detailed description of all the methods in a mathematically coherent way provides both novices and experienced researchers with a unique overview and new insight in the workings and relative advantages of each method, theoretically and algorithmically, even leading to new filters. Furthermore, the practical implementation details of all ensemble and particle filter methods are discussed to show similarities and differences in the filters aiding the users in what to use when. Finally, pseudo-codes are provided for all of the methods presented in this paper. model in time. However, due to technological and scientific advances, three significant developments have occurred in the last decade that force us to look beyond standard Ensemble Kalman Filtering, which is based on linear and/or Gaussian assumptions. Firstly, continuous increase in computational capability has recently allowed to run operational models at high resolutions so that the dynamical models have become increasingly non-linear due to the direct resolution of small-scale non-linear processes in these models, e.g. small-scale turbulence. Secondly, in several geoscientific applications, such as atmosphere, ocean, land surface, hydrology and sea -i.e. it is of interest to estimate variables or parameters that are bounded requiring DA methods that can deal with non-Gaussian distributions. Thirdly, the observational network around the world has increased manyfold for weather, ocean and land surface areas providing more information about the real system with greater accuracy and higher spatial and temporal resolution. Often the so-called observation operators that connect model states to observations of these new observations are non-linear, again asking for non-Gaussian DA methods. Thus, the research in non-linear DA Tellus A: 2018.