Flow, Turbulence and Combustion

, Volume 65, Issue 3–4, pp 469–488 | Cite as

Adjoint Methods in Data Assimilation for Estimating Model Error

  • A.K. Griffith
  • N.K. Nichols
Article

Abstract

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure.

data assimilation adjoint methods model error bias estimation nonlinear shallow water equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bennett, A.F., Inverse Methods in Physical Oceanography. Cambridge University Press, Cambridge (1992).Google Scholar
  2. 2.
    Courtier, P., Dual formulation of four-dimensional variational assimilation. Quarterly Journal of the Royal Meteorological Society 123 (1997) 2449–2461.CrossRefGoogle Scholar
  3. 3.
    Dalcher, A. and Kalnay, E., Error growth and predictability in operational ECMWF forecasts. Tellus 39A (1987) 474–491.Google Scholar
  4. 4.
    Daley, R., Atmospheric Data Analysis. Cambridge University Press, Cambridge (1991).Google Scholar
  5. 5.
    Derber, J.C., A variational continuous assimilation technique. Monthly Weather Review 117 (1989) 2437–2446.CrossRefGoogle Scholar
  6. 6.
    Gilbert, J.Ch. and Lemarechal, C., Some numerical experiments with variable-storage quasi-Newton algorithms. Mathematical Programming 45 (1989) 407–435.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Griffith, A.K., Data assimilation for numerical weather prediction using control theory. Ph.D. Thesis, The University of Reading, Department of Mathematics (1997).Google Scholar
  8. 8.
    Griffith, A.K. and Nichols, N.K., Accounting for model error in data assimilation using adjoint methods. In: Berz, M., Bischof, C., Corliss, G. and Greiwank, A. (eds), Computational Differentiation: Techniques, Applications and Tools. SIAM, Philadelphia (1996) pp. 195–204.Google Scholar
  9. 9.
    Griffith, A.K. and Nichols, N.K., Adjoint methods for treating error in data assimilation. In: Baines, M.J. (ed.), Numerical Methods for Fluid Dynamics VI. ICFD, Oxford (1998) pp. 335–344.Google Scholar
  10. 10.
    Griffith, A.K., Martin, M.J. and Nichols, N.K., Techniques for treating systematic model error in 3D and 4D data assimilation. In: Proceedings of the Third WMO International Symposium on Assimilation of Observations in Meteorology and Oceanography, World Meteorological Organization, WWRP Report Series No. 2, WMO/TD No. 986 (2000) pp. 9–12.Google Scholar
  11. 11.
    Kalman, R.E., A new approach to linear filtering and prediction problems. Transactions of the ASME, Series D 83 (1961) 35–44.MathSciNetGoogle Scholar
  12. 12.
    Lorenc, A.C., Analysis methods for numerical weather prediction. Quarterly Journal of the Royal Meteorological Society 112 (1986) 1177–1194.CrossRefADSGoogle Scholar
  13. 13.
    Lorenc, A.C., Optimal nonlinear objective analysis. Quarterly Journal of the Royal Meteorological Society 114 (1988) 205–240.CrossRefADSGoogle Scholar
  14. 14.
    Menard, R. and Daley, R., The application of Kalman smoother theory to the estimation of 4DVAR error statistics. Tellus 48A (1996) 221–237.Google Scholar
  15. 15.
    Parrett, C.A. and Cullen, M.J.P., Simulation of hydraulic jumps in the presence of rotation and mountains. Quarterly Journal of the Royal Meteorological Society 110 (1984) 147–165.CrossRefADSGoogle Scholar
  16. 16.
    Rabier, F., Järvinen, H., Klinker, E., Mahfouf, J.-F. and Simmons, A., The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics. Quarterly Journal of the Royal Meteorological Society 126 (2000) 1143–1170.CrossRefGoogle Scholar
  17. 17.
    Talagrand, O. and Courtier, P., Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Quarterly Journal of the Royal Meteorological Society 113 (1997) 1311–1328.CrossRefADSGoogle Scholar
  18. 18.
    Thacker, W.C., Relationships between statistical and deterministic methods of data assimilation. In: Sasaki, Y.K. (ed.), Variational Methods in the Geosciences. Elsevier, New York (1996).Google Scholar
  19. 19.
    Tribbia, J.J. and Baumhefner, D.P., The reliability of improvements in deterministic short-range forecasts in the presence of initial state and modeling deficiencies. Monthly Weather Review 108 (1988) 2276–2288.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • A.K. Griffith
    • 1
  • N.K. Nichols
    • 1
  1. 1.Department of MathematicsThe University of ReadingReadingU.K.

Personalised recommendations