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

S. Akella; I. M. Navon

doi:10.1111/j.1600-0870.2007.00362.x

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

Tellus

Volume 61 (2009): Issue 1

By: S. Akella and I. M. Navon

Open Access

|Jan 2009

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.

References

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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Daley , R. 1992 . The effect of serially correlated observation and model error on atmospheric data assimilation . Mon. Wea. Rev . 120 , 164 – 177 .
Search in Google Scholar Back to article
Dee , D. and Da Silva , A. M. 1998 . Data assimilation in the presence of forecast bias . Q. J. R. Meteorol. Soc . 124 , 269 – 295 .
Search in Google Scholar Back to article
Derber , J. C. 1989 . A variational continuous assimilation technique. Mon. Wea. Rev . 117 , 2437 - 2446 .
Search in Google Scholar Back to article
Derber , J. C. and Bouttier , F. 1999 . A reformulation of the background error covariance in the ECMWF global data assimilationsystem. Tellus 51A , 195 - 221 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Griffith , A. K. and Nichols , N. K. 2000 . Adjoint methods in data assim-ilation for estimating model error . Flow Turb. Comb . 65 , 469 – 488 .
Search in Google Scholar Back to article
Hourdin , F. , Talagrand , J. and Idellcadi , A . 2006 . Eulerian backtracking of atmospheric tracers. 11, Numerical aspects. Q. J. R. Meteorol. Soc . 132 , 585 - 603 .
Search in Google Scholar Back to article
Kalnay , E. 2003 . Atmospheric Modeling, Data Assimilation and Pre-dictability , Cambridge University Press , Cambridge , UK , 364 pp .
Search in Google Scholar Back to article
LeDimet , E-X. and Shutyaev , V. 2005 . On deterministic error analysis in variational data assimilation . Nonlin. Proc. Geophy . 12 , 481 – 490 .
Search in Google Scholar Back to article
LeDimet , F.-X. and Talagrand , J. 1986 . Variational algorithms for anal-ysis and assimilation of meteorological observations-theoretical as-pects. Tellus 38A , 97 - 110 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Liu , D. C. and Nocedal , J. 1989 . On the limited memory BFGS method for large-scale minimization . Math. Prog . 45 , 503 – 528 .
Search in Google Scholar Back to article
Lorene , A. 1986 . Analysis methods for numerical weather prediction . Q. J. R. Meteorol. Soc . 112 , 1177 – 1194 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Navon , I. M. 1998 . Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography . Dyn. Atmos. Oceans 27 , 55 – 79 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Suarez , M. J. and Talcacs , L. L. 1996 . Documentation of the ARIES/GEOS dynamical core . NASA Technical Memorandum 104606, NASA .
Search in Google Scholar Back to article
Trémolet , Y. 2006 . Accounting for an imperfect model in 4D-Var . Q. J. R. Meteorol. Soc . 132 , 2483 – 2504 .
Search in Google Scholar Back to article
Trémolet , Y. 2007 . Model error estimation in 4D-Var . Q. J. R. Meteorol. Soc . 133 , 1267 – 1280 .
Search in Google Scholar Back to article
Tsyrulnikov , M. D. 2005 . Stochastic modelling of model errors: a sim-ulation study . Q. J. R. Meteorol. Soc . 131 , 3345 – 3371 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Vidard , P. A. , Piacentini , A. and Le Dimet , F.-X. 2004 . Variational data analysis with control of the forecast bias . Tellus 56A , 177 – 188 .
Search in Google Scholar Back to article
Weaver , A. and Courtier , P. 2001 . Correlation modeling on the sphere using a generalized diffusion equation . Q. J. R. Meteorol. Soc . 127 , 1815 – 1846 .
Search in Google Scholar Back to article
Wergen , W. 1992 . The effect of model errors in variational assimilation . Tellus 44A , 297 – 313 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Zupanski , D. 1999 . A general weak constraint applicable to operational 4DVAR data assimilation systems . Mon. Wea. Rev . 125 , 2274 – 2292 .
Search in Google Scholar Back to article
Zupanski , M. 1993 . Regional four-dimensional variational data assimi-lation in a quasi-operational forecasting environment . Mon. Wea. Rev . 121 , 2396 – 2408 .
Search in Google Scholar Back to article
Zupanski , D. and Zupanski , M. 2006 . Model error estimation employing an ensemble data assimilation approach . Mon. Wea. Rev . 134 , 1337 – 1354 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article