Titlebook: Data Assimilation Fundamentals; A Unified Formulatio Geir Evensen,Femke C. Vossepoel,Peter Jan van Leeu Textbook‘‘‘‘‘‘‘‘ 2022 The Editor(s)

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Strong-Constraint 4DVarThis chapter introduces the . (SC-4DVar) method. By strong constraint, we refer to the dynamical model having no model errors. Hence, the model solution over the assimilation window is entirely determined by the model as soon as we give the initial conditions.

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Randomized-Maximum-Likelihood SamplingIn the following, we derive some methods for sampling the posterior conditional pdf in Eq.?(.). We aim to estimate the full pdf, not only finding its maximum. We will, in this chapter, use an approach named randomized maximum likelihood (RML) sampling.

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Fully Nonlinear Data AssimilationThis chapter provides an introduction to methods that, in theory, samples precisely the posterior pdf. Commonly-used ensemble data-assimilation methods, like the EnKF and EnRML, only sample the posterior pdf correctly in the Gauss-linear case and typically fail in cases with strong nonlinearity.

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DENT

EnKF for an Advection EquationThis chapter discusses a straightforward application of the EnKF with a linear advection equation. The example illustrates the smooth spatial update that the EnKF provides and how information propagates with the flow. Furthermore, we will see how the EnKF provides consistent error statistics.

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EnKF with the Lorenz EquationsThe chaotic Lorenz’63 model is a much-used testbed used to examine the capabilities of data-assimilation methods to handle nonlinear, unstable, and chaotic dynamics. This chapter will repeat some experiments that demonstrate the strengths of ensemble methods for highly nonlinear dynamics.

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Representer Method with an Ekman-Flow ModelEknes and Evensen (1997) solved the weak-constraint variational problem for a linear Ekman-flow model using the representer method. They computed the weak constraint solution for a long time series of velocity measurements. Additionally, they considered a parameter-estimation problem which rendered the problem nonlinear.

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