集合卡尔曼滤波同化方法的研究
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摘要
在大气海洋的科学研究中，随着卫星雷达等各种非常规资料迅速增加和模式的模拟能力加强，作为“把各种时空上不规则的零散分布的观测融合到基于物理规律的模式当中”的同化方法，在大气和海洋的研究中越来越重要。近几年来，一种新的同化方法—集合卡尔曼滤波同化方法受到广泛的关注和研究。现有的研究表明，集合卡尔曼滤波同化方法是一种具有业务应用潜力的同化方法。本文简单回顾了这种资料同化方法的发展历史和研究现状，介绍了其在气象中应用的基本原理。
集合卡尔曼滤波的主要优点是利用随天气流型演变的背景场误差协方差来进行资料分析，这是在目前变分同化中难以实现的，也是变分同化中存在的主要问题之一。但是集合卡尔曼滤波也存在滤波发散，分析量不平衡等问题。本文利用浅水模式和实际大气预报模式(MM5)，通过一系列数值试验来对集合卡尔曼滤波的主要理论和方法进行了研究，包括集合卡尔曼滤波中随流型演变的背景场误差协方差，集合数对集合卡尔曼滤波的影响等，并和三维变分同化结果进行了初步比较和分析。得到以下结论：集合卡尔曼滤波同化的效果优于三维变分同化的效果，而且其误差是收敛的，造成差异的根本原因是三维变分无法实现更新误差协方差，而集合卡尔曼滤波的误差协方差是随流型演变的；随着集合数的增加，集合卡尔曼滤波同化的效果得到改善，主要原因是集合卡尔曼滤波的统计误差相关场存在虚假的相关，而集合数的增加会减少这种虚假的相关。
"The blending of the existing noisy observation irregularly distributed in space and time into numerical models based on the physical laws that govern atmospheric flows became known as model assimilation of the data or data assimilation". It has become more and more important in the meteorology and oceanography because of the increasing observation and computation ability. In the recent years, a new data assimilation method called ensemble Kalman filter (EnKF) has aroused people's attention. The available result indicates that EnKF owns the potential ability of becoming an operational data assimilation method. This article simply reviews the developmental history and study actuality of this advanced data assimilation method.
The main virtue of the EnKF is that its background error covariance is flow-dependent. But the flow-dependent background error covariance is not easy to obtain in the operational variation analysis at present. But the EnKF aIso has drawbacks, for example, filter divergence, unbalance between the analysis variables et al. A serial experiment using the shallow water equation and the real atmosphere model (MM5) were carried out in order to study the EnKF theory, including the flow-dependent background error covariance in EnKF, the ensemble number's effect on the EnKF system and the comparison between EnKF and 3D-VAR. From the experiments it is seen that: the EnKF is superior to the 3D-VAR and its error is convergent. The main reason is that the 3D-VAR can't update its background error variance but the EnKF's background error covariance is flow-dependent. As the number of ensemble increasing the EnKF's analysis quality is improved because the increasing ensemble numbers will reduce the spurious error correlations.
集合卡尔曼滤波的主要优点是利用随天气流型演变的背景场误差协方差来进行资料分析，这是在目前变分同化中难以实现的，也是变分同化中存在的主要问题之一。但是集合卡尔曼滤波也存在滤波发散，分析量不平衡等问题。本文利用浅水模式和实际大气预报模式(MM5)，通过一系列数值试验来对集合卡尔曼滤波的主要理论和方法进行了研究，包括集合卡尔曼滤波中随流型演变的背景场误差协方差，集合数对集合卡尔曼滤波的影响等，并和三维变分同化结果进行了初步比较和分析。得到以下结论：集合卡尔曼滤波同化的效果优于三维变分同化的效果，而且其误差是收敛的，造成差异的根本原因是三维变分无法实现更新误差协方差，而集合卡尔曼滤波的误差协方差是随流型演变的；随着集合数的增加，集合卡尔曼滤波同化的效果得到改善，主要原因是集合卡尔曼滤波的统计误差相关场存在虚假的相关，而集合数的增加会减少这种虚假的相关。
"The blending of the existing noisy observation irregularly distributed in space and time into numerical models based on the physical laws that govern atmospheric flows became known as model assimilation of the data or data assimilation". It has become more and more important in the meteorology and oceanography because of the increasing observation and computation ability. In the recent years, a new data assimilation method called ensemble Kalman filter (EnKF) has aroused people's attention. The available result indicates that EnKF owns the potential ability of becoming an operational data assimilation method. This article simply reviews the developmental history and study actuality of this advanced data assimilation method.
The main virtue of the EnKF is that its background error covariance is flow-dependent. But the flow-dependent background error covariance is not easy to obtain in the operational variation analysis at present. But the EnKF aIso has drawbacks, for example, filter divergence, unbalance between the analysis variables et al. A serial experiment using the shallow water equation and the real atmosphere model (MM5) were carried out in order to study the EnKF theory, including the flow-dependent background error covariance in EnKF, the ensemble number's effect on the EnKF system and the comparison between EnKF and 3D-VAR. From the experiments it is seen that: the EnKF is superior to the 3D-VAR and its error is convergent. The main reason is that the 3D-VAR can't update its background error variance but the EnKF's background error covariance is flow-dependent. As the number of ensemble increasing the EnKF's analysis quality is improved because the increasing ensemble numbers will reduce the spurious error correlations.
引文
[1]Bjerknes,V.1911:Dynamic meteorology and hydrography.Part Ⅱ.Kinematics (New York:Carnegie Institute.Gibson Bros.)
[2]Daley.1991:Atmospheric data analysis.Cambridge Univ.Press
[3]Jones,Robert W.1964:On Improving initial Data for Numerical Forecasts of Hurricane Trajectories by the Steering Method.Journal of Applied Meteorology:Vol.3,No.3,pp.277-284.
[4]Nagle,Roland E.,Clare,James R.,Holl,Manfred M.,Riefel,Christopher A.1967:Formulation and testing of a program for the objective assembly of meteorological satellite cloud observations.Monthly Weather Review:Vol.95,No.4,pp.171-187.
[5]Danard,Maurice B.,Holl,Manfred M,Clark,James R.1968:Fields by correlation assembly- A numerical analysis technique.Monthly Weather Review:"Vol.96,No.3,pp.141-149.
[6]Thompson,Philip D.1969:Reduction of Analysis Error Through Constraints of Dynamical Consistency.Journal of Apptied Meteorology:Vol.8,No.5,pp.738-742.
[7]Lorenz,E.N.,1963:Deterministic nonperiodic flow.Journal of Atmospheric Soienoe:Vol.20,pp.130-14.
[8]Ehrendorfer,Martin.1992:Four-dimensional data assimilation - Comparison of variational and sequential algorithms.Royal Meteorological Societ,[Quarterly Journal].Vol.118,No.506,pp.673-713.
[9]Evensen,G.,1992:Using the extended Kalman filter with a multiplayer quasi-geostrophic ocean model[J].Geophs.Res.,97,17 905-17 924.
[10]Panofsky,H.1949:Objective weather-map analysis [J].Journal of Applied Meteorology:Vol.6,No.3,pp.86-92.
[11]Cressman.1959:An operational objective analysis system.Monthly Weather Review:Vol.87, No. 10, pp. 367-374.
[12] Gandin, L, 1963: Objective analysis of meteorological fields (Leningrad: Gridromet). English translation (Jerusalem: Isracl Problem for Scientific Translation). 1965
[13] Kalman, R E. 1960: A new approach to linear filtering and prediction problem [J]. Translation of the ASME, Journal of Basic Engineering, vol. 82D, pp. 34-45.
[14] Epstein, E S. 1969: Stochastic dynamic prediction (Stochastic dynamic prediction assuming deterministic laws for atmospheric behavior while seeking solutions corresponding to probabilistic statements of initial conditions) .Tellus: Vol. 21,No. 6, pp. 739-759.
[15] Kushner, H J. 1967: Dynamical equations for optimal nonlinear filtering (Dynamical equations for optimal nonlinear filtering and representation of conditional expectation as solution to stochastic differential equation) Journal of Differential Equations: Vol. 3, pp. 179-190.
[16] Budgell. W .P., 1986: Nonlinear data assimilation for shallow water equations in branched channels[J]. G-eophys.Res., 91C(9), 10, 633-10,644.
[17] Budgell, W. P., 1987: Stochastic filtering of linear shallow water wave process ,SIMA J. Sci. Stat.Com-put., 8(2), 152-170.
[18] Miller, R.,1986: Toward the application of the kalman filter to regional open ocean modeling.J. Phys. Oceanogr.,16.72-86.
[19] Bennett, A. F., and W. P. Budgell, 1989:The Kalman smoother for a linear quasigeo strophic model of ocean circulation. Dyn. Atmos. Oceans, 13 (3-4), 219-267.
[20] Bennett, A. F., 1992: Inverse Methods in Physical Oceanography. Cambridge University Press.
[21] Ghil, M., 1989: Meteorological data assimilation for oceanpgraphers, Part. I, Description and theoretical framework. Dyn.Atmos. Oceans, 13(3-4), 171-218.
[22] Ghil, S. E. Cohn. J. Tavantzis, K. Bube, and E. Isaacson, 1981: Application of estimation theory to numerical weather prediction. Dynamic Meteorology: Data Assimilation Method s, L.Bengtsson,M.Ghil and E.Kallen,Eds.,Springer-Verlag,139-224.
[23]Kushner,H.,1967:Approximation to optimal nonlinear filter.IEEE Tran Autom Control,AC.,Vol.12,pp:546-556.
[24]Bouttier,F.,1994:A dynamical estimation of forecast error covariances in an assimilation system.Monthly Weather Review:Vol.122,No.10,pp:2376-2390.
[25]Miller,R.N.,M.Ghil,and F.Gauthier.,1994:Advanced data assimilation in strongly nonlinear dynamical systems.J.Atmos.Sci.,51,1037-1056.
[26]Evensen,G.,1994a:Inverse Methods and data assimilation in nonlinear ocean models. Physica.77 D,108-I29.
[27]Evensen,G.,1994b:Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics.J.Geophys.Res.,99C(5),10,143-10,162.
[28]Evensen,G.,.and P.van Leeuwen,1996:Assimilation of GEOSAT altimeter data for the Agulhas Current using the ensemble kalman filter with a quasi-geostrophic model.Monthly Weather Review: Vol.124,No.1,pp:85-96.
[29]Evensen,G.,1997:Advanced data assimilation for strongly nonlinear dynamics.Monthly Weather Review:Vol.125,No.6,pp:1342-1354.
[30]Burgers,G.,van Leeuwen,P J,Evensen,G.,1998:Analysis scheme in the ensemble Kalman Filter.Monthly Weather Review:Vol.126,No.6,pp:1719-1724.
[31]Houtekamer,P L.,Mitchell,H L.,1998:Data Assimilation using an Ensemble Kalman Filter Technique. Monthlv Weather Review:Vol.126,No.3,pp:796-811.
[32]Anderson J L,Anderson S L.,1999:A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts.Monthly Weather Review:Vol.127,No.12,pp:2741-2758.
[33]Evensen,G.,van Leeuwen,P J,2000:An Ensemble Kalman Smoother for Nonlinear Dynamics.Monthly Weather Review:Vol.128 No.6,pp:1852-1867.
[34]Mitchell H L,Houtekamer P L.2000:An Adaptive Ensemble Kalman Filter.Monthly Weather Reviev:Vol.128,No.2,pp:416-433
[35]Hamill,TM.,Snyder,C.,2000:A hybrid Ensemble Kalman Filter-3D variational analysis scheme.Monthly Weather Review:Vol.128,No.8,pp:2905-2919.
[36]Hansen,JA.,Smith,LA.,2001:Probabilistic noise reduction. Tellus,Ser.,53 A:585-598.
[37]Mitchel,HL.,Houtekamer,PL.,Pellerin,G.,2002:Ensemble size and model-error representation in an Ensemble Kalman Filter.Monthly Weather Review.Vol.130,No.11,pp:2791-2808.
[38]Snyder,C.,and F.Zhang,2003:Assimilation of simulated Doppler radar observations with an ensemble Kalman filter.Monthly Weather Review:Vol.131,No.8,pp:1663-1677.
[39]M.Tong,M.Xue,2005:Ensemble Kalman Filter Assimilation of Doppler Radar Data with a Compressible Nonhydrostatic Model:OSS Experiments.Monthly Weather Review:Vol.133,No.7,pp:1789-1807.
[40]F.Zhang,Z.Meng,A.Aksoy,2006:Test of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation. Partl: Perfect Model Experiments.Monthly Weather Review:Vol.134,No.2,pp:722-736.
[41]Z.Meng,F.Zhang,2007:Test of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation.Part2:Perfect Model Experiments.Monthly Weather Review:Vol.135,pp:1403-1423.
[2]Daley.1991:Atmospheric data analysis.Cambridge Univ.Press
[3]Jones,Robert W.1964:On Improving initial Data for Numerical Forecasts of Hurricane Trajectories by the Steering Method.Journal of Applied Meteorology:Vol.3,No.3,pp.277-284.
[4]Nagle,Roland E.,Clare,James R.,Holl,Manfred M.,Riefel,Christopher A.1967:Formulation and testing of a program for the objective assembly of meteorological satellite cloud observations.Monthly Weather Review:Vol.95,No.4,pp.171-187.
[5]Danard,Maurice B.,Holl,Manfred M,Clark,James R.1968:Fields by correlation assembly- A numerical analysis technique.Monthly Weather Review:"Vol.96,No.3,pp.141-149.
[6]Thompson,Philip D.1969:Reduction of Analysis Error Through Constraints of Dynamical Consistency.Journal of Apptied Meteorology:Vol.8,No.5,pp.738-742.
[7]Lorenz,E.N.,1963:Deterministic nonperiodic flow.Journal of Atmospheric Soienoe:Vol.20,pp.130-14.
[8]Ehrendorfer,Martin.1992:Four-dimensional data assimilation - Comparison of variational and sequential algorithms.Royal Meteorological Societ,[Quarterly Journal].Vol.118,No.506,pp.673-713.
[9]Evensen,G.,1992:Using the extended Kalman filter with a multiplayer quasi-geostrophic ocean model[J].Geophs.Res.,97,17 905-17 924.
[10]Panofsky,H.1949:Objective weather-map analysis [J].Journal of Applied Meteorology:Vol.6,No.3,pp.86-92.
[11]Cressman.1959:An operational objective analysis system.Monthly Weather Review:Vol.87, No. 10, pp. 367-374.
[12] Gandin, L, 1963: Objective analysis of meteorological fields (Leningrad: Gridromet). English translation (Jerusalem: Isracl Problem for Scientific Translation). 1965
[13] Kalman, R E. 1960: A new approach to linear filtering and prediction problem [J]. Translation of the ASME, Journal of Basic Engineering, vol. 82D, pp. 34-45.
[14] Epstein, E S. 1969: Stochastic dynamic prediction (Stochastic dynamic prediction assuming deterministic laws for atmospheric behavior while seeking solutions corresponding to probabilistic statements of initial conditions) .Tellus: Vol. 21,No. 6, pp. 739-759.
[15] Kushner, H J. 1967: Dynamical equations for optimal nonlinear filtering (Dynamical equations for optimal nonlinear filtering and representation of conditional expectation as solution to stochastic differential equation) Journal of Differential Equations: Vol. 3, pp. 179-190.
[16] Budgell. W .P., 1986: Nonlinear data assimilation for shallow water equations in branched channels[J]. G-eophys.Res., 91C(9), 10, 633-10,644.
[17] Budgell, W. P., 1987: Stochastic filtering of linear shallow water wave process ,SIMA J. Sci. Stat.Com-put., 8(2), 152-170.
[18] Miller, R.,1986: Toward the application of the kalman filter to regional open ocean modeling.J. Phys. Oceanogr.,16.72-86.
[19] Bennett, A. F., and W. P. Budgell, 1989:The Kalman smoother for a linear quasigeo strophic model of ocean circulation. Dyn. Atmos. Oceans, 13 (3-4), 219-267.
[20] Bennett, A. F., 1992: Inverse Methods in Physical Oceanography. Cambridge University Press.
[21] Ghil, M., 1989: Meteorological data assimilation for oceanpgraphers, Part. I, Description and theoretical framework. Dyn.Atmos. Oceans, 13(3-4), 171-218.
[22] Ghil, S. E. Cohn. J. Tavantzis, K. Bube, and E. Isaacson, 1981: Application of estimation theory to numerical weather prediction. Dynamic Meteorology: Data Assimilation Method s, L.Bengtsson,M.Ghil and E.Kallen,Eds.,Springer-Verlag,139-224.
[23]Kushner,H.,1967:Approximation to optimal nonlinear filter.IEEE Tran Autom Control,AC.,Vol.12,pp:546-556.
[24]Bouttier,F.,1994:A dynamical estimation of forecast error covariances in an assimilation system.Monthly Weather Review:Vol.122,No.10,pp:2376-2390.
[25]Miller,R.N.,M.Ghil,and F.Gauthier.,1994:Advanced data assimilation in strongly nonlinear dynamical systems.J.Atmos.Sci.,51,1037-1056.
[26]Evensen,G.,1994a:Inverse Methods and data assimilation in nonlinear ocean models. Physica.77 D,108-I29.
[27]Evensen,G.,1994b:Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics.J.Geophys.Res.,99C(5),10,143-10,162.
[28]Evensen,G.,.and P.van Leeuwen,1996:Assimilation of GEOSAT altimeter data for the Agulhas Current using the ensemble kalman filter with a quasi-geostrophic model.Monthly Weather Review: Vol.124,No.1,pp:85-96.
[29]Evensen,G.,1997:Advanced data assimilation for strongly nonlinear dynamics.Monthly Weather Review:Vol.125,No.6,pp:1342-1354.
[30]Burgers,G.,van Leeuwen,P J,Evensen,G.,1998:Analysis scheme in the ensemble Kalman Filter.Monthly Weather Review:Vol.126,No.6,pp:1719-1724.
[31]Houtekamer,P L.,Mitchell,H L.,1998:Data Assimilation using an Ensemble Kalman Filter Technique. Monthlv Weather Review:Vol.126,No.3,pp:796-811.
[32]Anderson J L,Anderson S L.,1999:A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts.Monthly Weather Review:Vol.127,No.12,pp:2741-2758.
[33]Evensen,G.,van Leeuwen,P J,2000:An Ensemble Kalman Smoother for Nonlinear Dynamics.Monthly Weather Review:Vol.128 No.6,pp:1852-1867.
[34]Mitchell H L,Houtekamer P L.2000:An Adaptive Ensemble Kalman Filter.Monthly Weather Reviev:Vol.128,No.2,pp:416-433
[35]Hamill,TM.,Snyder,C.,2000:A hybrid Ensemble Kalman Filter-3D variational analysis scheme.Monthly Weather Review:Vol.128,No.8,pp:2905-2919.
[36]Hansen,JA.,Smith,LA.,2001:Probabilistic noise reduction. Tellus,Ser.,53 A:585-598.
[37]Mitchel,HL.,Houtekamer,PL.,Pellerin,G.,2002:Ensemble size and model-error representation in an Ensemble Kalman Filter.Monthly Weather Review.Vol.130,No.11,pp:2791-2808.
[38]Snyder,C.,and F.Zhang,2003:Assimilation of simulated Doppler radar observations with an ensemble Kalman filter.Monthly Weather Review:Vol.131,No.8,pp:1663-1677.
[39]M.Tong,M.Xue,2005:Ensemble Kalman Filter Assimilation of Doppler Radar Data with a Compressible Nonhydrostatic Model:OSS Experiments.Monthly Weather Review:Vol.133,No.7,pp:1789-1807.
[40]F.Zhang,Z.Meng,A.Aksoy,2006:Test of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation. Partl: Perfect Model Experiments.Monthly Weather Review:Vol.134,No.2,pp:722-736.
[41]Z.Meng,F.Zhang,2007:Test of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation.Part2:Perfect Model Experiments.Monthly Weather Review:Vol.135,pp:1403-1423.