• \
    Start Submission Become a Reviewer

    Reading: Different approaches to model error formulation in 4D-Var: a study with high-resolution adve...

    Download

     A- A+
    Alt. Display

    Original Research Papers

    Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes

    Authors:

    S. Akella,

    Department of Earth & Planetary Sciences, The Johns Hopkins University, Baltimore, MD 21218; Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD 21218, US
    X close

    I. M. Navon

    Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, US
    X close

    Abstract

    All numerical models are imperfect. Weak constraint variational data assimilation (VDA), which provides a treatment of the modelling errors, is studied; building on the approach of Vidard et al. (Tellus, 56A, pp. 177–188, 2004). The evolution of model error (ME) is modelled using ordinary differential equations, which involve a scalar parameter. These approaches were tested using different high-resolution advection schemes. The first set of experiments were constructed to see if it is possible to account for (numerical) discretization error within such a framework. In other set of experiments, a systematic source of modelling error was introduced by deliberately specifying an incorrect value for the Coriolis parameter in the model. Results with observational state at half of the model state resolution, are also presented.We also discuss a method of estimating the scalar parameter in the ME through VDA. In all cases, the inclusion of ME provides reduction in forecasting errors. Also, our experiments indicate that different settings of the model (e.g. using different high-resolution advection schemes) would need different ME formulation. Results presented in this paper could be used to formulate sophisticated ME forms to account for systematic errors in higher dimensional models with complex advection schemes.

    How to Cite: Akella, S. and Navon, I.M., 2009. Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes. Tellus A: Dynamic Meteorology and Oceanography, 61(1), pp.112–128. DOI: http://doi.org/10.1111/j.1600-0870.2007.00362.x
    1
    Views
      Published on 01 Jan 2009
    Peer Reviewed
     CC BY 4.0
     Accepted on 14 Jul 2008            Submitted on 3 Sep 2007

    References

    1. Akella , S. and Navon , I. M. 2006 . A comparative study of the perfor-mance of high resolution advection schemes in the context of data assimilation . Int. J. Num. Meth. Fluids . 51 , 719 – 748 .  

    2. Buizza , R. , Milleer , M. and Palmer , T. N. 1999 . Stochastic representation of model uncertainties in the ECMWF ensembleprediction system. Q. J. R. MeteoroL Soc . 560 , 2887 - 2908 .  

    3. Byun , D. and Schere , K. L. 2006 . Review of the governing equations, computational algorithms, and other components ofthe models-3 Community Multiscale Air Quality (CMAQ) modeling system . AppL Mech. Rev . 59 , 51 – 77 .  

    4. Daley , R. 1992 . The effect of serially correlated observation and model error on atmospheric data assimilation . Mon. Wea. Rev . 120 , 164 – 177 .  

    5. Dee , D. and Da Silva , A. M. 1998 . Data assimilation in the presence of forecast bias . Q. J. R. Meteorol. Soc . 124 , 269 – 295 .  

    6. Derber , J. C. 1989 . A variational continuous assimilation technique. Mon. Wea. Rev . 117 , 2437 - 2446 .  

    7. Derber , J. C. and Bouttier , F. 1999 . A reformulation of the background error covariance in the ECMWF global data assimilationsystem. Tellus 51A , 195 - 221 .  

    8. Griffith , A. K. and Nichols , N. K. 1996 . Accounting for model error in data assimilation using adjoint methods. In: Computational Differenti-ation, Techniques, Applications, and Tools , (eds. M. Berz , C. Bischof , A. Griewank and G. Corliss ), SIAM, Philadelphia , 195 - 205 .  

    9. Griffith , A. K. and Nichols , N. K. 2000 . Adjoint methods in data assim-ilation for estimating model error . Flow Turb. Comb . 65 , 469 – 488 .  

    10. Hourdin , F. , Talagrand , J. and Idellcadi , A . 2006 . Eulerian backtracking of atmospheric tracers. 11, Numerical aspects. Q. J. R. Meteorol. Soc . 132 , 585 - 603 .  

    11. Kalnay , E. 2003 . Atmospheric Modeling, Data Assimilation and Pre-dictability , Cambridge University Press , Cambridge , UK , 364 pp .  

    12. LeDimet , E-X. and Shutyaev , V. 2005 . On deterministic error analysis in variational data assimilation . Nonlin. Proc. Geophy . 12 , 481 – 490 .  

    13. LeDimet , F.-X. and Talagrand , J. 1986 . Variational algorithms for anal-ysis and assimilation of meteorological observations-theoretical as-pects. Tellus 38A , 97 - 110 .  

    14. Lin , S. J. and Rood , R. B. 1997 . An explicit flux-form semi-lagrangian shallow-water model on the sphere. Q. J. R. Meteorol. Soc . 123 , 2477 - 2498 .  

    15. Lin , S.-J. , Chao , W. C. , Sud , Y. C. and Walker G. K. 1994 . A class of the van Leer transport schemes and its applications to the moisture transport in a general circulation model . Mon. Wea. Rev . 122 , 1575 – 1593 .  

    16. Liu , D. C. and Nocedal , J. 1989 . On the limited memory BFGS method for large-scale minimization . Math. Prog . 45 , 503 – 528 .  

    17. Lorene , A. 1986 . Analysis methods for numerical weather prediction . Q. J. R. Meteorol. Soc . 112 , 1177 – 1194 .  

    18. Martin , J. , Bell , M. and Nichols , N. 2002 . Estimation of systematic error in an equatorial ocean model using data assimilation. Int. J. Num. Meth. Fluids . 40 , 435 - 444 .  

    19. Nash , S. G. and Nocedal , J. 1991 . A numerical study of the limited memory BFGS method and the truncated-Newton method for large-scale optimization . SIAM J. Opt . 1 , 358 – 372 .  

    20. Navon , I. M. 1998 . Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography . Dyn. Atmos. Oceans 27 , 55 – 79 .  

    21. Navon , I. M. , Zou , X. , Derber , J. and Sela , J. 1992 . Variational data assimilation with an adiabatic version of the NMC spectral model . Mon. Wea. Rev . 120 , 1433 – 1446 .  

    22. Navon , I. M. , Daescu , D. N. and Liu , Z. 2005 . The impact of back-ground error on incomplete observations for 4d-var data assimilation with the FSU GSM. In: Computational Science-ICCS 2005, Part 2, LectureNotes in Computer Science 3515, (ed. V. S. Sunderam ), Springer-Verlag, Berlin, 837 - 844 .  

    23. Nichols , N. K. 2003 . Treating model error in 3-D and 4-D data assimila-tion (pp. 127-135). In: Data Assimilation for the Earth System (NATO science series IV: Earth and Environment Science, eds. R. Swinbanlc , V. Shutyaev and W. A. Lahoz ), Kluwer Academic Publishers, The Netherlands , 377 pp .  

    24. Suarez , M. J. and Talcacs , L. L. 1996 . Documentation of the ARIES/GEOS dynamical core . NASA Technical Memorandum 104606, NASA .  

    25. Trémolet , Y. 2006 . Accounting for an imperfect model in 4D-Var . Q. J. R. Meteorol. Soc . 132 , 2483 – 2504 .  

    26. Trémolet , Y. 2007 . Model error estimation in 4D-Var . Q. J. R. Meteorol. Soc . 133 , 1267 – 1280 .  

    27. Tsyrulnikov , M. D. 2005 . Stochastic modelling of model errors: a sim-ulation study . Q. J. R. Meteorol. Soc . 131 , 3345 – 3371 .  

    28. Vidard , P. A. , Blayo , E. , Le Dimet , E-X. and Piacentini , A. 2000 . 4D variational data analysis with imperfect model . Flow Turb. Comb . 65 , 489 - 504 .  

    29. Vidard , P. A. , Piacentini , A. and Le Dimet , F.-X. 2004 . Variational data analysis with control of the forecast bias . Tellus 56A , 177 – 188 .  

    30. Weaver , A. and Courtier , P. 2001 . Correlation modeling on the sphere using a generalized diffusion equation . Q. J. R. Meteorol. Soc . 127 , 1815 – 1846 .  

    31. Wergen , W. 1992 . The effect of model errors in variational assimilation . Tellus 44A , 297 – 313 .  

    32. Williamson , D. L. , Drake , J. B., Hack , J. J. Jakob , R. and Swarztrauber , P. N. 1992 . A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comp. Phys . 102 , 211 - 224 .  

    33. Zhu , Y. and Navon , I. M. 1999 . Impact of parameter estimation on the performance of the FSU Global Spectral Model using its full-physics adjoint . Mon. Wea. Rev . 127 , 1497 – 1517 .  

    34. Zupanski , D. 1999 . A general weak constraint applicable to operational 4DVAR data assimilation systems . Mon. Wea. Rev . 125 , 2274 – 2292 .  

    35. Zupanski , M. 1993 . Regional four-dimensional variational data assimi-lation in a quasi-operational forecasting environment . Mon. Wea. Rev . 121 , 2396 – 2408 .  

    36. Zupanski , D. and Zupanski , M. 2006 . Model error estimation employing an ensemble data assimilation approach . Mon. Wea. Rev . 134 , 1337 – 1354 .  

    37. Zupanski , M. , Zupanski , D. , Vulcicevic , T. , Eis , K. and Haar , T. 2005 . CIRA/CSU Four-Dimensional variational data assimilation system . Mon. Wea. Rev . 123 , 829 – 843 .  

    Jump to Discussions Related content
    comments powered by Disqus