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Variational Assimilation. Adjoint Equations

  • Conference paper
Data Assimilation for the Earth System

Part of the book series: NATO Science Series ((NAIV,volume 26))

Abstract

The basic theory of statistical linear estimation has been described in the chapter Bayesian Estimation. Optimal Interpolation. Statistical Linear Estimation (which will hereafter be referred to as Part I). A number of formulae, which are mutually equivalent, have been established for the BLUE x a of the state vector, given a vector of data z. Some of these formulae express x a as the minimiser of a scalar objective function defined on state space.

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References

  • Bergot, T. and Doerenbecher, A. (2002) A study on the optimization of the deployment of targeted observations using adjoint-based methods, Q. J. R. Meteorol Soc., 128, 1689–1712.

    Article  Google Scholar 

  • Bonnans, J.-F., Gilbert, J.-C., Lemaréchal, C. and Sagatizabal, C. (2002) Numerical Optimization-Theoretical and Practical Aspects, Springer Verlag, Berlin.

    Google Scholar 

  • Courtier, P. and Talagrand, O. (1987) Variational assimilation of meteorological observations with the adjoint vorticity equation. II: Numerical results, Q. J. R. Meteorol. Soc. 113, 1329–1347.

    Article  Google Scholar 

  • Egbert, G. D., Bennett, A. F. and Foreman, M. G. C. (1994) Topex/Poseidon tides estimated using a global inverse model, J. Geophys. Res. 99, 24,821–24, 852.

    Google Scholar 

  • Elbern, H., Schmidt, H. and Ebel, A. (1997) Variational Data Assimilation for Tropospheric Chemistry Modeling, J. Geophys. Res. 102 (D13), 15,967–15, 985.

    Article  Google Scholar 

  • Errera, Q. and Fonteyn, D. (2001) Four-dimensional variational chemical assimilation of CRISTA stratospheric measurements, J. Geophys. Res. 106 (D11), 12,253–12, 265.

    Article  Google Scholar 

  • Errico, R. M and Vukisevic, T. (1992) Sensitivity Analysis Using an Adjoint of the PSU-NCAR Mesoscale Model, Mon. Wea. Rev. 120, 1644–1660.

    Article  Google Scholar 

  • Farrell, B. F. (1989) Optimal Excitation of Baroclinic Waves, J. Atmos. Sci. 46, 1193–1206.

    Article  Google Scholar 

  • Fisher, M. and Lary, D. J. (1995) Lagrangian four-dimensional variational data assimilation of chemical species, 0. J. R. Meteorol. Soc. 121, 1681–1704.

    Article  Google Scholar 

  • Giering, R. and Kaminski, T. (1998) Recipes for Adjoint Code Construction, Trans. Math. Software 24, 437–474.

    Article  Google Scholar 

  • Greiner, É. and Périgaud, C. (1996) Assimilation of Geosat Altimetric Data in a Nonlinear S hallow-Water Model of the Indian Ocean by Adjoint Approach. Part II: Some Validation and Interpretation of the Assimilated Results, J. Phys. Oceanogr. 26, 1735–1746.

    Article  Google Scholar 

  • Gustafsson, N., Kållen, E. and Thorsteinsson, S. (1998) Sensitivity of forecast errors to initial and lateral boundary conditions, Tellus 50A, 167–185.

    Google Scholar 

  • Hall, M. C. G., Cacuci, D. G. and Schlesinger, M. E. (1982) Sensitivity analysis of a radiative-convective model by the adjoint method, J. Atmos. Sci. 39, 2038–2050.

    Article  Google Scholar 

  • Hoffman, R. N. (1986) A four-dimensional analysis exactly satisfying equations of motion, Mon. Wea. Rev. 114, 388–397.

    Article  Google Scholar 

  • Klinker, E., Rabier, F., Kelly, G. and Mahfouf, J.-F. (2000) The ECMWF operational implementation of four-dimensional variational assimilation. III: Experimental results and diagnostics with operational configuration, Q. J. R. Meteorol. Soc. 126, 1191–1215.

    Article  Google Scholar 

  • Lacarra, J.-F. and Talagrand, O. (1988) Short-range evolution of small perturbations in a barotropic model, Tellus 40A, 81–95.

    Article  Google Scholar 

  • Langland, R. H., Gelaro, R., Rohaly, G. D. and Shapiro, M. A. (1999) Targeted observations in FASTEX: Adjoint-based targeting procedures and data impact experiments in IOP17 and IOP18, Q. J. R. Meteorol. Soc. 125, 3241–3270.

    Article  Google Scholar 

  • Le Dimet, F.-X. and Talagrand, O. (1986) Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus 38A, 97–110.

    Article  Google Scholar 

  • Lewis, J. M. and Derber, J. C. (1985) The use of adjoint equations to solve a variational adjustment problem with advective constraints, Tellus 37A, 309–322.

    Article  Google Scholar 

  • Lions, J.-L. (1971) Optimal control of systems governed by partial differential equations (translated from the French), Springer-Verlag, Berlin.

    Book  Google Scholar 

  • Louvel, S. (2001) Implementation of a dual variational algorithm for assimilation of synthetic altimeter data in the oceanic primitive equation model MICOM, J. Geophys. Res. 106 (C5), 9199–9212.

    Article  Google Scholar 

  • Marchuk, G. I. (1974) Numerical solution of the problems of dynamics of the atmosphere and the ocean (in Russian), Gidrometeoizdat, Leningrad.

    Google Scholar 

  • Marotzke, J., Giering, R., Zhang, K. Q., Stammer, D., Hill, C. and Lee, T. (1999) Construction of the adjoint MIT ocean general circulation model and application to Atlantic heat transport sensitivity, J. Geophys. Res. 104 (C12), 29,529–29, 547.

    Article  Google Scholar 

  • Morrow, R. and De Mey, P. (1995) Adjoint assimilation of altimetric, surface drifter and hydrographie data in a QG model of the Azores current, J. Geophys. Res. 100 (C12), 25,007–25, 025.

    Article  Google Scholar 

  • Mu M. (2000) Nonlinear singular vectors and nonlinear singular values, Science in China (ser. D) 43, 375–385.

    Article  Google Scholar 

  • Penenko, V. V. and Obraztsov, N. N. (1976) A variational initialization method for the fields of the meteorological elements (English translation), Soviet Meteorol. Hydrol. no 11, 1–11.

    Google Scholar 

  • Rabier, F., Courtier, P. and Talagrand, O. (1992) An Application of Adjoint Models to Sensitivity Analysis, Beitr. Phys. Atmosph. 65, 177–192.

    Google Scholar 

  • Rostaing, N., Dalmas, S. and Galligo, A. (1993) Automatic Differentiation in Odyssée, Tellus 45A, 558–568.

    Google Scholar 

  • Talagrand, O. (1991) The Use of Adjoint Equations in Numerical Modeling of the Atmospheric Circulation, in A. Griewank and G. F. Corliss (eds), Automatic Differentiation of Algorithms: Theory, Implementation, and Application, Society for Industrial and Applied Mathematics, Philadelphia, United States, pp. 169–180.

    Google Scholar 

  • Thacker, W. C. and Long, R. B. (1988) Fitting dynamics to data, J. Geophys. Res. 93, 1227–1240.

    Article  Google Scholar 

  • Thépaut, J.-N. and Courtier, P. (1991) Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model, Q. J. R. Meteorol. Soc. 117, 1225–1254.

    Article  Google Scholar 

  • Urban, B. (1993) A method to determine the theoretical maximum error growth in atmospheric models, Tellus 45A, 270–280.

    Google Scholar 

  • Waelbroeck, C. and Louis, J.-F. (1995) Sensitivity analysis of a model of CO2 exchange in tundra ecosystems by the adjoint method, J. Geophys. Res. 100 (D2), 2801–2816.

    Article  Google Scholar 

  • Weaver, A. T. and Anderson, D. L. T. (1997) Variational Assimilation of Altimeter Data in a Multilayer Model of the Tropical Pacific Ocean, J. Phys. Oceanogr. 27, 664–682.

    Article  Google Scholar 

  • Zhang, Y., Bischof, C. H., Easter, R. C. and Wu, P.-T. (1998) Sensitivity analysis of a mixed-phase chemical mechanism using automatic differentiation, J. Geophys. Res. 103 (D15), 18,953–18, 979.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire de Météorologie Dynamique, CNRS École Normale Supérieure, Paris, France

    O. Talagrand

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  1. O. Talagrand
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Editors and Affiliations

  1. Met Office, Bracknell, UK

    Richard Swinbank

  2. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    Victor Shutyaev

  3. Data Assimilation Research Centre, University of Reading, Reading, UK

    William Albert Lahoz

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© 2003 Springer Science+Business Media Dordrecht

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Talagrand, O. (2003). Variational Assimilation. Adjoint Equations. In: Swinbank, R., Shutyaev, V., Lahoz, W.A. (eds) Data Assimilation for the Earth System. NATO Science Series, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0029-1_4

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  • DOI: https://doi.org/10.1007/978-94-010-0029-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-1593-9

  • Online ISBN: 978-94-010-0029-1

  • eBook Packages: Springer Book Archive

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