What is the correct cost functional for variational data assimilation?

  • Jochen Bröcker


Variational approaches to data assimilation, and weakly constrained four dimensional variation (WC-4DVar) in particular, are important in the geosciences but also in other communities (often under different names). The cost functions and the resulting optimal trajectories may have a probabilistic interpretation, for instance by linking data assimilation with maximum aposteriori (MAP) estimation. This is possible in particular if the unknown trajectory is modelled as the solution of a stochastic differential equation (SDE), as is increasingly the case in weather forecasting and climate modelling. In this situation, the MAP estimator (or “most probable path” of the SDE) is obtained by minimising the Onsager–Machlup functional. Although this fact is well known, there seems to be some confusion in the literature, with the energy (or “least squares”) functional sometimes been claimed to yield the most probable path. The first aim of this paper is to address this confusion and show that the energy functional does not, in general, provide the most probable path. The second aim is to discuss the implications in practice. Although the mentioned results pertain to stochastic models in continuous time, they do have consequences in practice where SDE’s are approximated by discrete time schemes. It turns out that using an approximation to the SDE and calculating its most probable path does not necessarily yield a good approximation to the most probable path of the SDE proper. This suggest that even in discrete time, a version of the Onsager–Machlup functional should be used, rather than the energy functional, at least if the solution is to be interpreted as a MAP estimator.


Variational data assimilation Onsager–Machlup functional Stochastic differential equations 


  1. Apte A, Hairer M, Stuart AM, Voss J (2007) Sampling the posterior: an approach to non-gaussian data assimilation. Phys D Nonlinear Phenom 230(1):50–64. (Data Assimilation. ISSN 0167-2789)
  2. Breiman Leo (1973) Probability. Addison-Wesley, Reading, MassGoogle Scholar
  3. Cotter SL, Dashti M, Robinson JC, Stuart AM (2009) Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Probl 25(11):115008, 43. ( ISSN 0266-5611)
  4. Derber JC (1989) A variational continuous assimilation technique. Monthly Weather Rev 117(11):2437–2446CrossRefGoogle Scholar
  5. Dutra DA, Teixeira BOS, Aguirre LA (2014) Maximum a posteriori state path estimation: Discretization limits and their interpretation. Automatica 50(5):1360–1368. ISSN 0005-1098.
  6. Evensen Geir (2007) Data assimilation. The ensemble Kalman filter. Springer, New YorkGoogle Scholar
  7. Franzke Christian LE, O’Kane Terence J, Judith Berner, Williams Paul D, Valerio Lucarini (2015) Stochastic climate theory and modeling. Wiley Interdiscip Rev Clim Change 6(1):63–78. (ISSN 1757-7799)CrossRefGoogle Scholar
  8. Gallot S, Hulin D, Lafontaine J (2004) Riemannian geometry. Universitext. Springer, Berlin, third edition. (ISBN 3-540-20493-8)
  9. Ide K, Courtier P, Ghil M, Lorenc AC (1997) Unified notation for data assimilation: operational, sequential and variational. J Meteorol Soc Jpn 75(1B):181–189CrossRefGoogle Scholar
  10. Ikeda N, Watanabe S (1989) Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second editionGoogle Scholar
  11. Imkeller P, von Storch J-S (eds) (2001) Stochastic climate models, volume 49 of Progress in Probability. Birkhäuser Verlag, Basel. (ISBN 3-7643-6520-X)
  12. Jazwinski AH (1970) Stochastic processes and filtering theory volume 64 of mathematics in science and engineering. Academic Press, New York (ISBN 9780123815507)Google Scholar
  13. Kalnay Eugenia (2001) Atmospheric modeling, data assimilation and predictability, 1st edn. Cambridge University Press, CambridgeGoogle Scholar
  14. Kloeden PE, Platen E (1992) Numerical solution of Stochastic differential equations. Springer, BerlinCrossRefGoogle Scholar
  15. Milstein GN (1995) Numerical integration of stochastic differential equations, volume 313 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht. (Translated and revised from the 1988 Russian original. ISBN 0-7923-3213-X)
  16. Mortensen RE (1968) Maximum-likelihood recursive nonlinear filtering. J Optim Theory Appl 2:386–394CrossRefGoogle Scholar
  17. Mörters P, Peres Y (2010) Brownian motion, volume 30 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. (With an appendix by Oded Schramm and Wendelin Werner. ISBN 978-0-521-76018-8)
  18. Øksendal B (1998) Stochastic differential equations. Universitext. Springer, Berlin, fifth edition. (An introduction with applications.ISBN 3-540-63720-6)
  19. Stuart AM (2010) Inverse problems: a bayesian perspective. Acta Numer 19:451–559. CrossRefGoogle Scholar
  20. Sugiura N (2017) The Onsager–Machlup functional for data assimilation. Nonlinear Process Geophys 24(4):701–712. CrossRefGoogle Scholar
  21. Vanden-Eijnden E, Weare J (2013) Data assimilation in the low noise regime with application to the kuroshio. Monthly Weather Rev 141(6):1822–1841, 6 2013. (ISSN 0027-0644)
  22. Zeitouni O, Dembo A (1987) A maximum a posteriori estimator for trajectories of diffusion processes. Stochastics 20(3):221CrossRefGoogle Scholar
  23. Zeitouni O, Dembo A (1988) An existence theorem and some properties of maximum a posteriori estimators of trajectories of diffusions. Stochastics 23(2):197. (ISSN 0090-9491)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesUniversity of ReadingReadingUK

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