## Data Assimilation: The Ensemble Kalman Filter by Geir Evensen

By Geir Evensen

Data Assimilation comprehensively covers info assimilation and inverse equipment, together with either conventional country estimation and parameter estimation. this article and reference specializes in quite a few well known info assimilation tools, reminiscent of vulnerable and powerful constraint variational tools and ensemble filters and smoothers. it really is established how different equipment will be derived from a standard theoretical foundation, in addition to how they vary and/or are concerning one another, and which homes symbolize them, utilizing numerous examples.

It offers the mathematical framework and derivations in a fashion that's universal for any self-discipline the place dynamics is merged with measurements. the math point is inconspicuous, even though it calls for wisdom of uncomplicated spatial records, Bayesian records, and calculus of diversifications. Readers also will relish the creation to the mathematical equipment used and particular derivations, which can be effortless to stick to, are given in the course of the publication. The codes utilized in a number of of the knowledge assimilation experiments can be found on an internet page.

The specialize in ensemble equipment, corresponding to the ensemble Kalman clear out and smoother, additionally makes it a superior connection with the derivation, implementation and alertness of such options. a lot new fabric, specifically concerning the formula and answer of mixed parameter and country estimation difficulties and the final houses of the ensemble algorithms, is accessible the following for the 1st time.

The second variation contains a partial rewrite of Chapters thirteen an 14, and the Appendix. moreover, there's a thoroughly new bankruptcy on "Spurious correlations, localization and inflation", and an up to date and more desirable sampling dialogue in Chap 11.

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**Example text**

1998) it is essential that the observations are treated as random variables having a distribution with mean equal to the first guess observations and covariance equal to C . 35) where j counts from 1 to the number of ensemble members N . It is ensured that the simulated random measurement errors have mean equal to zero. 36) and, of course, in the limit of an infinite ensemble size this matrix will converge towards the prescribed error covariance matrix C used in the standard Kalman filter. The following discussion is valid both using an exactly prescribed C and an ensemble representation C e of C .

The EKF applies a closure scheme where third- and higher-order moments in the error covariance evolution equation are discarded. This results in an unbounded error variance growth or linear instability in the error covariance equation in some dynamical models. If an exact error covariance evolution equation could be used all linear instabilities will saturate due to nonlinear effects. This saturation is missing in the EKF, as was later confirmed by Miller et al. (1994), Gauthier et al. (1993) and Bouttier (1994).

7 6 Solution 5 4 3 2 1 0 0 200 400 600 800 1000 x-axis Fig. 1. Dev. Dev. Dev. 7 6 Solution 5 4 3 2 1 0 0 200 400 600 800 1000 x-axis Fig. 2. Kalman filter experiment when system noise is included: reference solution, measurements, estimate and standard deviation at three different times t = 5 (top), t = 150 (middle), and t = 300 (bottom) 32 4 Sequential data assimilation times during the experiment, and show how information from measurements is propagated with the advection speed and how the error variance is reduced every time measurements are assimilated.