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    Data Assimilation (2007)
    00 bookname
    00 front-matter
    01 Introduction
    02 Statistical definitions
    03 Analysis scheme
    04 Sequential data assimilation
    05 Variational inverse problems
    06 Nonlinear variational inverse problems
    07 Probabilistic formulation
    08 Generalized Inverse
    09 Ensemble methods
    10 Statistical optimization
    11 Sampling strategies for the EnKF
    12 Model errors
    13 Square Root Analysis schemes
    14 Rank issues
    15 An ocean prediction system
    16 Estimation in an oil reservoir simulator
    17 back-matter
    Geir Evensen Data Assimilation
    Geir Evensen Data Assimilation The Ensemble Kalman Filter With 63 Figures
    PROF. GEIR EVENSEN Hydro Research Centre, Bergen PO Box 7190 N 5020 Bergen Norway and Mohn-Sverdrup Center for Global Ocean Studies and Operational Oceanography at Nansen Environmental and Remote Sensing Center Thormølensgt 47 N 5600 Bergen Norway e-mail: Geir.Evensen@hydro.com Library of Congress Control Number: 2006932964 3-540-38300-0-X Springer Berlin Heidelberg New York ISBN-10 978-3-540-38300-0 Springer Berlin Heidelberg New York ISBN-13 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner Typesetting: camera-ready by the author Production: Christine Adolph Printing: Krips bv, Meppel Binding: Stürtz AG, Würzburg Printed on acid-free paper 30/2133/ca 5 4 3 2 1 0
    To Tina and Endre
    Preface The aim of this book is to introduce the formulation and solution of the data assimilation problem. The focus is mainly on methods where the model is allowed to contain errors and where the error statistics evolve through time. So-called strong constraint methods and simple methods where the error statistics are constant in time are only briefly explained, and then as special cases of more general weak constraint formulations. There is a special focus on the Ensemble Kalman Filter and similar meth- ods. These are methods which have become very popular, both due to their simple implementation and interpretation and their properties with nonlinear models. The book has been written during several years of work on the development of data assimilation methods and the teaching of data assimilation methods to graduate students. It would not have been completed without the continuous interaction with students and colleagues, and I particularly want to acknowl- edge the support from Laurent Bertino, Kari Brusdal, Fran¸cois Counillon, Mette Eknes, Vibeke Haugen, Knut Arild Lisæter, Lars Jørgen Natvik, and Jan Arild Skjervheim, with whom I have worked closely for several years. Laurent Bertino and Fran¸cois Counillon also provided much of the material for the chapter on the TOPAZ ocean data assimilation system. Contributions from Laurent Bertino, Theresa Lloyd, Gordon Wilmot, Martin Miles, Jennifer Trittschuh-Vall`es, Brice Vall`es and Hans Wackernagel, on proof-reading parts of the final version of the book are also much appreciated. It is hoped that the book will provide a comprehensive presentation of the data assimilation problem and that it will serve as a reference and textbook for students and researchers working with development and application of data assimilation methods. Bergen, June 2006 Geir Evensen
    Contents List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 Statistical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Statistical moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Working with samples from a distribution . . . . . . . . . . . . . . . . . . 9 2.3.1 Sample mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Sample variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Sample covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Statistics of random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Sample mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.2 Sample variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 Sample covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Analysis scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 Scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 State-space formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Bayesian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Extension to spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Basic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.2 Euler–Lagrange equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Representer solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.4 Representer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
    x Contents 4 3.2.5 Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.6 Uniqueness of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.7 Minimization of the penalty function . . . . . . . . . . . . . . . . . 23 3.2.8 Prior and posterior value of the penalty function . . . . . . 24 3.3 Discrete form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Sequential data assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.1 Kalman filter for a scalar case . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.2 Kalman filter for a vector state . . . . . . . . . . . . . . . . . . . . . . 29 4.1.3 Kalman filter with a linear advection equation . . . . . . . . 29 4.2 Nonlinear dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.1 Extended Kalman filter for the scalar case . . . . . . . . . . . . 32 4.2.2 Extended Kalman filter in matrix form . . . . . . . . . . . . . . . 33 4.2.3 Example using the extended Kalman filter . . . . . . . . . . . . 35 4.2.4 Extended Kalman filter for the mean . . . . . . . . . . . . . . . . 36 4.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Ensemble Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.1 Representation of error statistics . . . . . . . . . . . . . . . . . . . . 38 4.3.2 Prediction of error statistics . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3.3 Analysis scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.5 Example with a QG model . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Variational inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Simple illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Linear inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2.1 Model and observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Measurement functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.3 Comment on the measurement equation . . . . . . . . . . . . . . 51 5.2.4 Statistical hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.5 Weak constraint variational formulation . . . . . . . . . . . . . . 52 5.2.6 Extremum of the penalty function . . . . . . . . . . . . . . . . . . . 53 5.2.7 Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.8 Strong constraint approximation . . . . . . . . . . . . . . . . . . . . 55 5.2.9 Solution by representer expansions. . . . . . . . . . . . . . . . . . . 56 5.3 Representer method with an Ekman model . . . . . . . . . . . . . . . . . 57 5.3.1 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3.2 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3.3 Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3.4 Representer solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.5 Example experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3.6 Assimilation of real measurements . . . . . . . . . . . . . . . . . . . 64 5.4 Comments on the representer method . . . . . . . . . . . . . . . . . . . . . . 67
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