DISCRETE INVERSE AND STATE ESTIMATION PROBLEMS

Transcription

1 DISCRETE INVERSE AND STATE ESTIMATION PROBLEMS With Geophysical The problems of making inferences about the natural world from noisy observations and imperfect theories occur in almost all scientific disciplines. This book addresses these problems using examples taken from geophysical fluid dynamics. It focuses on discrete formulations, both static and time-varying, known variously as inverse, state estimation or data assimilation problems. Starting with fundamental algebraic and statistical ideas, the book guides the reader through a range of inference tools including the singular value decomposition, Gauss Markov and minimum variance estimates, Kalman filters and related smoothers, and adjoint (Lagrange multiplier) methods. The final chapters discuss a variety of practical applications to geophysical flow problems. Discrete Inverse and State Estimation Problems: With Geophysical is an ideal introduction to the topic for graduate students and researchers in oceanography, meteorology, climate dynamics, geophysical fluid dynamics, and any field in which models are used to interpret observations. It is accessible to a wide scientific audience, as the only prerequisite is an understanding of linear algebra. is Cecil and Ida Green Professor of Physical Oceanography at the Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology. After gaining his Ph.D. in geophysics in 1966 at MIT, he has risen through the department, becoming its head for the period between He subsequently served as Secretary of the Navy Research Professor and has held senior visiting positions at many prestigious universities and institutes across the world. His previous books include Ocean Acoustic Tomography (Cambridge University Press, 1995) with W. Munk and P. Worcester, and The Ocean Circulation Inverse Problem (Cambridge University Press, 1996).

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3 DISCRETE INVERSE AND STATE ESTIMATION PROBLEMS With Geophysical CARL WUNSCH Department of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology

4 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge cb2 8ru, UK Published in the United States of America by Cambridge University Press, New York Information on this title: / C. Wunsch 2006 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2006 First paperback edition 2011 A catalogue record for this publication is available from the British Library isbn Hardback isbn Paperback Additional resources for this publication at / Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 To Walter Munk for decades of friendship and exciting collaboration.

6 Colour plate section appears between pages 182 and 183, and is also available for download in colour from www. cambridge.org/

7 Contents Preface page ix Acknowledgements xi Part I Fundamental machinery 1 1 Introduction Differential equations Partial differential equations More examples Importance of the forward model 17 2 Basic machinery Background Matrix and vector algebra Simple statistics: regression Least-squares The singular vector expansion Combined least-squares and adjoints Minimum variance estimation and simultaneous equations Improving recursively Summary 143 Appendix 1. Maximum likelihood 145 Appendix 2. Differential operators and Green functions 146 Appendix 3. Recursive least-squares and Gauss Markov solutions Extensions of methods The general eigenvector/eigenvalue problem Sampling Inequality constraints: non-negative least-squares Linear programming Empirical orthogonal functions Kriging and other variants of Gauss Markov estimation 170 vii

8 viii Contents 3.7 Non-linear problems The time-dependent inverse problem: state estimation Background Basic ideas and notation Estimation Control and estimation problems Duality and simplification: the steady-state filter and adjoint Controllability and observability Non-linear models Forward models A summary 250 Appendix. Automatic differentiation and adjoints Time-dependent methods Monte Carlo/ensemble methods Numerical engineering: the search for practicality Uncertainty in Lagrange multiplier method Non-normal systems Adaptive problems 273 Appendix. Doubling 274 Part II Applications Applications to steady problems Steady-state tracer distributions The steady ocean circulation inverse problem Property fluxes Application to real oceanographic problems Linear programming solutions The β-spiral and variant methods Alleged failure of inverse methods Applications of empirical orthogonal functions (EOFs) (singular vectors) Non-linear problems Applications to time-dependent fluid problems Time-dependent tracers Global ocean states by Lagrange multiplier methods Global ocean states by sequential methods Miscellaneous approximations and applications Meteorological applications 356 References 357 Index 367 Colour plates between pp. 182 and 183.

9 Preface This book is to a large extent the second edition of The Ocean Circulation Inverse Problem, but it differs from the original version in a number of ways. While teaching the basic material at MIT and elsewhere over the past ten years, it became clear that it was of interest to many students outside of physical oceanography the audience for whom the book had been written. The oceanographic material, instead of being a motivating factor, was in practice an obstacle to understanding for students with no oceanic background. In the revision, therefore, I have tried to make the examples more generic and understandable, I hope, to anyone with even rudimentary experience with simple fluid flows. Also many of the oceanographic applications of the methods, which were still novel and controversial at the time of writing, have become familiar and almost commonplace. The oceanography, now confined to the two last chapters, is thus focussed less on explaining why and how the calculations were done, and more on summarizing what has been accomplished. Furthermore, the time-dependent problem (here called state estimation to distinguish it from meteorological practice) has evolved rapidly in the oceanographic community from a hypothetical methodology to one that is clearly practical and in ever-growing use. The focus is, however, on the basic concepts and not on the practical numerical engineering required to use the ideas on the very large problems encountered with real fluids. Anyone attempting to model the global ocean or atmosphere or equivalent large scale system must confront issues of data storage, code parallelization, truncation errors, grid refinement, and the like. Almost none of these important problems are taken up here. Before constructive approaches to the practical problems can be found, one must understand the fundamental ideas. An analogy is the need to understand the implications of Maxwell s equations for electromagnetic phenomena before one undertakes to build a high fidelity receiver. The effective engineering of an electronic instrument can only be helped by good understanding ix

10 x Preface of how one works in principle, albeit the details of making one work in practice can be quite different. In the interests of keeping the book as short as possible, I have, however, omitted some of the more interesting theoretical material of the original version, but which readers can find in the wider literature on control theory. It is assumed that the reader has a familiarity at the introductory level with matrices and vectors, although everything is ultimately defined in Chapter 2. Finally, I have tried to correct the dismaying number of typographical and other errors in the previous book, but have surely introduced others. Reports of errors of any type will be gratefully received. I thank the students and colleagues who over the years have suggested corrections, modifications, and clarifications. My time and energies have been supported financially by the National Aeronautics and Space Administration, and the National Science Foundation through grants and contracts, as well as by the Massachusetts Institute of Technology through the Cecil and Ida Green Professorship.

11 Acknowledgements The following figures are reproduced by permission of the American Geophysical Union: 4.8, , , 6.31, and Figures 2.16, 6.21, 6.27, 6.32, 7.1 and 7.11 are reproduced by permission of the American Meteorological Society. Figure 6.22 is reproduced by permission of Kluwer. Errata A correction list compiled by the author can be found here: xi