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Data assimilation concepts and methods
March 1999
By F. Bouttier and P. Courtier
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Abstract
These training course lecture notes are an advanced and comprehensive presentation of most data assimilation methods that are considered useful in applied meteorology and oceanography today. Some are considered old-fashioned but they are still valuable for low cost applications. Others have never been implemented yet in realistic applications, but they are regarded as the future of data assimilation. A mathematical approach has been chosen, which allows a compact and rigorous presentation of the algorithms, though only some basic mathematical competence is required from the reader.
This document has been put together with the help of previous lecture notes, which are now superseded:
| • Variational analysis: use of observations, example of clear radiances, Jean Pailleux, 1989. |
| • Inversion methods for satellite sounding data, J. Eyre, 1991. (part 2 only) |
| • Methods of data assimilation: optimum interpolation, P. Undén, 1993. (except section 5) |
| • Data assimilation methods: introduction to statistical estimation, J. Eyre and P. Courtier, 1994. |
| • Variational methods, P. Courtier, 1995. (except sections 3.2-3.6, 4.5, 4.6) |
| • Kalman filtering, F, Bouttier, 1997. (except the predictability parts) |
Traditionally the lecture notes have been referring a lot to the assimilation and forecast system at ECMWF, rather than to more general algorithms. Sometimes ideas that had not even been tested found their way into the training course lecture notes. New notes had to be written every couple of years, with inconsistent notation.
In this new presentation it has been decided to stick to a description of the main assimilation methods used worldwide, without any reference to ECMWF specific features, and clear comparisons between the different algorithms. This should make it easier to adapt the methods to problems outside the global weather forecasting framework of ECMWF, e.g. ocean data assimilation, land surface analysis or inversion of remote-sensing data. It is hoped that the reader will manage to see the physical nature of the algorithms beyond the mathematical equations.
A first edition of these lecture notes was released in March 1998. In this second edition, some figures were added, and a few errors were corrected.
Thanks are due to J. Pailleux, J. Eyre, P. Undén and A. Hollingsworth for their contribution to the previous lecture notes, to A. Lorenc, R. Daley, M. Ghil and O. Talagrand for teaching the various forms of the statistical interpolation technique to the meteorological world, to D. Richardson for proof-reading the document, and to the attendees of training course who kindly provided constructive comments.
Table of contents
1 . Basic concepts of data assimilation
| 1.1 On the choice of model |
| 1.2 Cressman analysis and related methods |
| 1.3 The need for a statistical approach |
2 . The state vector, control space and observations
| 2.1 State vector |
| 2.2 Control variable |
| 2.3 Observations |
| 2.4 Departures |
3 . The modelling of errors
| 3.1 Using pdfs to represent uncertainty |
| 3.2 Error variables |
| 3.3 Using error covariances |
| 3.4 Estimating statistics in practice |
4 . Statistical interpolation with least-squares estimation
| 4.1 Notation and hypotheses |
| 4.2 Theorem: least-squares analysis equations |
| 4.3 Comments |
| 4.4 On the tangent linear hypothesis |
| 4.5 The point of view of conditional probabilities |
| 4.6 Numerical cost of least-squares analysis |
| 4.7 Conclusion |
5 . A simple scalar illustration of least-squares estimation
6 . Models of error covariances
| 6.1 Observation error variances |
| 6.2 Observation error correlations |
| 6.3 Background error variances |
| 6.4 Background error correlations |
| 6.5 Estimation of error covariances |
| 6.6 Modelling of background correlations |
7 . Optimal interpolation (OI) analysis
8 . Three-dimensional variational analysis (3D-Var)
9 . 1D-Var and other variational analysis systems
10 . Four-dimensional variational assimilation (4D-Var)
| 10.1 The four-dimensional analysis problem |
| 10.2 Theorem: minimization of the 4D-Var cost function |
| 10.3 Properties of 4D-Var |
| 10.4 Equivalence between 4D-Var and the Kalman Filter |
11 . Estimating the quality of the analysis
| 11.1 Theorem: use of Hessian information |
| 11.2 Remarks |
12 . Implementation techniques
| 12.1 Minimization algorithms and preconditioning |
| 12.2 Theorem: preconditioning of a variational analysis |
| 12.3 The incremental method |
| 12.4 The adjoint technique |
13 . Dual formulation of 3D/4D-Var (PSAS)
14 . The extended Kalman filter (EKF)
| 14.1 Notation and hypotheses |
| 14.2 Theorem: the KF algorithm |
| 14.3 Theorem: KF/4D-Var equivalence |
| 14.4 The Extended Kalman Filter (EKF) |
| 14.5 Comments on the KF algorithm |
15 . Conclusion
APPENDIX A A primer on linear matrix algebra
APPENDIX B Practical adjoint coding
APPENDIX C Exercises
APPENDIX D Main symbols
References
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