Environmental and Ecological Statistics

, Volume 15, Issue 1, pp 101–110

A two-stage ensemble Kalman filter for smooth data assimilation

Article

Abstract

The ensemble Kalman Filter (EnKF) applied to a simple fire propagation model by a nonlinear convection-diffusion-reaction partial differential equation breaks down because the EnKF creates nonphysical ensemble members with large gradients. A modification of the EnKF is proposed by adding a regularization term that penalizes large gradients. The method is implemented by applying the EnKF formulas twice, with the regularization term as another observation. The regularization step is also interpreted as a shrinkage of the prior distribution. Numerical results are given to illustrate success of the new method.

Keywords

Data assimilation Ensemble Kalman filter State-space model Penalty Tikhonov regularization Wildfire Convection-reaction-diffusion Shrinkage Bayesian 

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References

  1. Anderson JL and Anderson SL (1999). A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon Weather Rev 127: 2741–2758 CrossRefGoogle Scholar
  2. Anderson BDO and Moore JB (1979). Optimal filtering. Prentice-Hall, Englewood Cliffs, NJ Google Scholar
  3. Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999). LAPACK users’ guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA Google Scholar
  4. Bengtsson T, Snyder C, Nychka D (2003) Toward a nonlinear ensemble filter for high dimensional systems. J Geophys Res-Atmos 108(D24):STS 2-1-10Google Scholar
  5. Blackford LS, Choi J, Cleary A, D’Azevedo E, Demmel J, Dhillon I, Dongarra J, Hammarling S, Henry G, Petitet A, Stanley K, Walker D and Whaley RC (1997). ScaLAPACK users’ guide. Society for Industrial and Applied Mathematics, Philadelphia, PA Google Scholar
  6. Burgers G, Evensen G and Leeuwen PJ (1998). Analysis scheme in the ensemble Kalman filter. Mon Weather Rev 126: 1719–1724 CrossRefGoogle Scholar
  7. Carlin JB, Polson N and Stoffer DS (1992). A Monte Carlo approach to nonnormal and nonlinear state-space modeling. J Am Stat Assoc 418: 494–500 Google Scholar
  8. Dongarra JJ, Croz JD, Hammarling S and Duff I (1990). A set of level 3 basic linear algebra subprograms. ACM Trans Math Softw 16: 1–17 CrossRefGoogle Scholar
  9. Doucet A, de Freitas N, Gordon N (2001) (eds), Sequential Monte Carlo in practice. SpringerGoogle Scholar
  10. Evensen G (2004). Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn 54: 539–560 CrossRefGoogle Scholar
  11. Evensen G (1994). Sequential data assimilation with nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 99(C5)(10): 143–162 Google Scholar
  12. Evensen G (2003). The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53: 343–367 CrossRefGoogle Scholar
  13. Hansen PC (1998). Rank-deficient and discrete ill-posed problems. SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia PA Google Scholar
  14. Harvey AC (1989) Forecasting, structural time series models and the Kalman filter. CambridgeGoogle Scholar
  15. Houtekamer P and Mitchell HL (1998). Data assimilation using an ensemble Kalman filter technique. Monthly Weather Review 126(3): 796–811 CrossRefGoogle Scholar
  16. Jazwinski AH (1970). Stochastic processes and filtering theory. Academic Press, New York Google Scholar
  17. Johansen TA (1997). On Tikhonov regularization, bias and variance in nonlinear system identification. Automatica 33(3): 441–446 CrossRefGoogle Scholar
  18. Johns CJ and Shumway RH (2005). A state-space model for censored data. Environmetrics 16: 167–180 CrossRefGoogle Scholar
  19. Jones RH (1984) Fitting multivariate models to unequally spaced data. In: Parzen E (ed) Time series analysis of irregularly spaced data, no. 25 in Lecture Notes in Statistics. Springer-Verlag, pp 158–188Google Scholar
  20. Kalman RE (1960). A new approach to linear filtering and prediction problems. Trans ASME - J Basic Eng Ser D 82: 35–45 Google Scholar
  21. Kalman RE and Bucy RS (1961). New results in filtering and prediction theory. Trans ASME - J Basic Eng 83: 95–108 Google Scholar
  22. Kim KY, Kang SI, Kim MC, Kim S, Lee YJ and Vauhkonen M (2002). Dynamic image reconstruction in electrical impedance tomography with known internal structures. IEEE Trans Magn 38: 1301–1304 CrossRefGoogle Scholar
  23. Kitagawa G (1987). Non-Gaussian state-space modeling of nonstationary time series, (with discussion). J Am Stat Assoc 400: 1032–1044 CrossRefGoogle Scholar
  24. Mandel J, Chen M, Franca LP, Johns C, Puhalskii A, Coen JL, Douglas CC, Kremens R, Vodacek A, Zhao W (2004a) A note on dynamic data driven wildfire modeling. In: Computational Science - ICCS 2004 Bubak M, van Albada GD, Sloot PMA, Dongarra JJ (eds), vol 3038 of Lecture Notes in Computer Science. Springer pp. 725–731Google Scholar
  25. Mandel J, Chen M, Franca LP, Johns C, Puhalskii A, Coen JL, Douglas CC, Kremens R, Vodacek A, Zhao W (2004b) Dynamic data driven wildfire modeling. UCD CCM Report 208, March 2004, http://www-math.cudenver.edu/ccm/reports/rep208.pdfGoogle Scholar
  26. Meinhold R and Singpurwalla N (1983). Understanding the Kalman filter. American Statistician 37: 123–127 CrossRefGoogle Scholar
  27. Meinhold R and Singpurwalla N (1989). Robustification of Kalman filter models. J Am Stat Soc 84: 479–486 Google Scholar
  28. Paige CC and Saunders MA (1977). Least squares estimation of discrete linear dynamic systems using orthogonal transformations. SIAM J Numer Anal 14(2): 180–193 CrossRefGoogle Scholar
  29. Shumway RH and Stoffer DS (1982). An approach to time series smoothing and forecasting using the EM algorithm. J Time Ser Anal 3: 253–264 Google Scholar
  30. Tippett MK, Anderson JL, Bishop CH, Hamill TM and Whitaker JS (2003). Ensemble square root filters. Monthly Weather Review 131: 1485–1490 CrossRefGoogle Scholar
  31. Leeuwen P (2003). A variance-minimizing filter for large-scale applications. Mont Weather Rev 131(9): 2071–2084 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Milliman, Inc.DenverUSA
  2. 2.Department of Mathematical SciencesUniversity of Colorado at Denver and Health Sciences CenterDenverUSA

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