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.
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References
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
Anderson BDO and Moore JB (1979). Optimal filtering. Prentice-Hall, Englewood Cliffs, NJ
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
Bengtsson T, Snyder C, Nychka D (2003) Toward a nonlinear ensemble filter for high dimensional systems. J Geophys Res-Atmos 108(D24):STS 2-1-10
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
Burgers G, Evensen G and Leeuwen PJ (1998). Analysis scheme in the ensemble Kalman filter. Mon Weather Rev 126: 1719–1724
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
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
Doucet A, de Freitas N, Gordon N (2001) (eds), Sequential Monte Carlo in practice. Springer
Evensen G (2004). Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn 54: 539–560
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
Evensen G (2003). The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53: 343–367
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
Harvey AC (1989) Forecasting, structural time series models and the Kalman filter. Cambridge
Houtekamer P and Mitchell HL (1998). Data assimilation using an ensemble Kalman filter technique. Monthly Weather Review 126(3): 796–811
Jazwinski AH (1970). Stochastic processes and filtering theory. Academic Press, New York
Johansen TA (1997). On Tikhonov regularization, bias and variance in nonlinear system identification. Automatica 33(3): 441–446
Johns CJ and Shumway RH (2005). A state-space model for censored data. Environmetrics 16: 167–180
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 158–188
Kalman RE (1960). A new approach to linear filtering and prediction problems. Trans ASME - J Basic Eng Ser D 82: 35–45
Kalman RE and Bucy RS (1961). New results in filtering and prediction theory. Trans ASME - J Basic Eng 83: 95–108
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
Kitagawa G (1987). Non-Gaussian state-space modeling of nonstationary time series, (with discussion). J Am Stat Assoc 400: 1032–1044
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. 725–731
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.pdf
Meinhold R and Singpurwalla N (1983). Understanding the Kalman filter. American Statistician 37: 123–127
Meinhold R and Singpurwalla N (1989). Robustification of Kalman filter models. J Am Stat Soc 84: 479–486
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
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
Tippett MK, Anderson JL, Bishop CH, Hamill TM and Whitaker JS (2003). Ensemble square root filters. Monthly Weather Review 131: 1485–1490
Leeuwen P (2003). A variance-minimizing filter for large-scale applications. Mont Weather Rev 131(9): 2071–2084
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Johns, C.J., Mandel, J. A two-stage ensemble Kalman filter for smooth data assimilation. Environ Ecol Stat 15, 101–110 (2008). https://doi.org/10.1007/s10651-007-0033-0
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DOI: https://doi.org/10.1007/s10651-007-0033-0