The optimal Kalman gain was analyzed in a rigorous statistical framework. Emphasis was placed on a comprehensive understanding and interpretation of the current algorithm, especially when the measurement function is nonlinear. It is argued that when the measurement function is nonlinear, the current ensemble Kalman Filter algorithm seems to contain implicit assumptions: the forecast of the measurement function is unbiased or the nonlinear measurement function is linearized. While the forecast of the model state is assumed to be unbiased, the two assumptions are actually equivalent.
On the above basis, we present two modified Kalman gain algorithms. Compared to the current Kalman gain algorithm, the modified ones remove the above assumptions, thereby leading to smaller estimated errors. This outcome was confirmed experimentally, in which we used the simple Lorenz 3-component model as the test-bed. It was found that in such a simple nonlinear dynamical system, the modified Kalman gain can perform better than the current one. However, the application of the modified schemes to realistic models involving nonlinear measurement functions needs to be further investigated.
ensemble Kalman filter measurement function data assimilation
This is a preview of subscription content, log in to check access.
Ambadan, J. T., and Y. M. Tang, 2011: Sigma-point particle filter for parameter estimation in a multiplicative noise environment. Journal of Advances in Modeling Earth Systems, 3, M12005, doi:10.1029/2011MS000065.CrossRefGoogle Scholar
Anderson, J. L., 2001: An ensemble adjustment kalman filter for data assimilation. Mon. Wea. Rev., 129, 2884–2903.CrossRefGoogle Scholar
Anderson, J. L., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210–224.CrossRefGoogle Scholar
Courtier, P., and Coauthors, 1998: The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation. Quart. J. Roy. Meteor. Soc., 124, 1783–1807.Google Scholar
Deng, Z., Y. Tang, D. Chen, and G. Wang 2012: A hybrid ensemble generation method for EnKF for Argo data assimilation. Atmos-Ocean, 50, 129–145.CrossRefGoogle Scholar
Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99(C5), 10 143–10 162.CrossRefGoogle Scholar
Evensen, G., 1997: Advanced data assimilation for strongly nonlinear dynamics. Mon. Wea. Rev., 125, 1342–1354.CrossRefGoogle Scholar
Evensen, G., 2003: The Ensemble Kalman Filter: Theoretical formulation and practical implementation. Ocean Dynamics, 53, 343–367.CrossRefGoogle Scholar
Furrer, R., and T. Bengtsson, 2007: Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. Journal of Multivariate Analysis, 98, 227–255.CrossRefGoogle Scholar
Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble kalman filter for atmospheric data assimilation. Mon.Wea. Rev., 129, 123–137.CrossRefGoogle Scholar
Houtekamer, P. L., H. L. Mitchell, G. Pellerin, M. Buehner, M. Charron, L. Spacek, and B. Hansen, 2005: Atmospheric data assimilation with an Ensemble Kalman Filter: Results with real observations. Mon. Wea. Rev., 133, 604–620.CrossRefGoogle Scholar
Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform kalman filter. Physica D, 230, 112–126.CrossRefGoogle Scholar
Ito, K., and K. Q. Xiong, 2000: Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Contr., 45(5), 910–927.CrossRefGoogle Scholar
Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, 375pp.Google Scholar
Julier, S. J., J. K. Uhlmann, and H. F. Durrant-Whyte, 1995: A new approach for filtering nonlinear systems. Proc. of the 1995 American Control Conference, Seattle WA, USA, 1628–1632.Google Scholar
Kalman, R. E., 1960: A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82, 35–45.CrossRefGoogle Scholar
Klinker, E., F. Rabier, G. Kelly, and J.-F. Mahfouf, 2000: The ECMWF operational implementation of four-dimensional variational assimilation. III: Experimental results and diagnostics with operational configuration. Quart. J. Roy. Meteor. Soc., 126, 1191–1215.CrossRefGoogle Scholar
Le Dimet, F. X., and O. Talagrand, 1986: Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus, 38A, 97–110.CrossRefGoogle Scholar