集合变换卡尔曼滤波方法在集合预报和适应性观测中的初步应用
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摘要
近年来,为了最大程度地改进预报技能,一种利用预报系统信息来确定进行补充观测区域的方法迅速发展起来,我们称之为适应性观测方法或目标观测方法。确定的观测区域称之为敏感区域,在敏感区内改善分析质量对后续的预报技能产生最大的预期影响。目前适应性观测研究已经成为THORPEX计划中的一个子计划。集合变换卡尔曼滤波(theensemble transform Kalman filer简称ETKF)方法最早是作为一种适应性观测算法提出的,现又被用于集合预报初始扰动的生成。ETKF方法是一种次优的卡尔曼滤波方案,像其它卡尔曼滤波方案一样,它不仅可以同化观测资料而且可以估计出观测对预报误差协方差的影响。它与其它集合卡尔曼滤波方案不同在于它利用集合变换和无量纲的思想求解与观测有关的误差协方差矩阵,可以快速估计出不同附加观测所造成的预报误差的减少量。现有的研究成果表明,ETKF方法无论是作为集合预报初始扰动的生成方法还是作为一种适应性观测算法都具有业务应用的潜力。目前国内在集合变换卡尔曼滤波方面的研究开展地比较少,因此对集合变换卡尔曼滤波方法的研究具有重要的科学价值和现实意义。
本文在前人的理论研究基础之上,构建ETKF系统平台,利用了实际大气预报模式进行数值试验,对集合变换卡尔曼滤波理论及其在集合预报和适应性观测两方面的应用进行了研究。下面是本文的研究结果和结论:
(1)集合变换卡尔曼滤波方法作为一种集合预报初始扰动的生成方法时,在假设系统是线性和不随时间变化、观测网的地理分布不变、观测误差固定、并且初始集合扰动代表误差增长的方向时,所得到的分析扰动在彼此正交化的观测空间中的概率分布是一样的。本文选取了一次暴雨过程进行试验,根据试验结果统计分析发现在真实大气模式的试验设置下,利用ETKF方法得到的分析扰动在集合子空间基本上也是等概率分布的。
(2)在集合预报试验当中,由于集合数的限制会造成分析误差协方差矩阵的低估,需要引入扩大因子(inflation factor)调整集合预报扰动的幅度。本文利用两种不同的扩大因子计算方案进行试验,试验1选用Wang和Bishop(2003)文中介绍的方法进行试验,试验2是采用类似Breeding方法调整集合扰动的大小。试验1得到的扰动的幅度是先增大后减小,最后趋于稳定;试验2的扰动是缓慢的增长,增长到一定大小保持不变。两组试验得到的扰动场最终都趋于稳定,扰动维持在观测误差大小附近,说明两种方法得到的扰动增长是合理变化的。
(3)考察了集合数的变化对扩大因子的影响,试验表明随着集合数的增加一倍扩大因子的数值减少了将近一半。说明在集合数增大的过程中扩大因子的作用在逐渐变小。
(4)集合变换卡尔曼滤波方法作为一种适应性观测算法,可以直接估计出加入观测后预报误差的减少量。本文用ETKF方法得到的预报误差减少量的变化与深层平均风(DLM)的结果进行了对比,结果表明利用ETKF方法估计得到的信号方差是合理的。
Recently strategies were developed that use forecast-system information to identifylocations where additional observations would provide maximal improvements in the expectedskill of forecast. We refer to these as adaptive observation, or targeted observation, commonlycalled targeting. Targeting identifies localized areas, referred to as sensitive region, in whichthe quality of the analysis has the greatest expected influence on the subsequent skill of theforecast. Now the adaptive observation becomes a sub-program of THORPEX. The EnsembleTransform Kalman filter is initially proposed as an adaptive observation method, than it is usedinto the ensemble forecast. The ETKF is a sub-optimal Kalman filter scheme. Like otherKalman filter, it provides a framework for assimilating observations and also for estimating theeffect of observations on the forecast error covariance. It differs from other ensemble Kalmanfilter in that it uses ensemble transformation and a normalization to rapidly obtain theprediction error covariance matrix associated with a particular deployment of observationresources. This rapidity enables it to quickly assess the ability of a large number of futurefeasible sequences of observational networks to reduce forecast error variance. All the resultsindicate that the ETKF could be put into operational environment. It can be used into theadaptive observation, and also can be used as an ensemble forecast method. Now the domesticscientists do not pay much attention to the ETKF method.So it is necessary to implement themethod on this field.
This paper is based on the ETKF theory established by Bishop and Wang Xu guang et al.The characteristics of ETKF are studied in this paper by virtue of the model GRAPES. Themajor results and conclusions are summarized as follows:
(1) In a system with fixed and perfect linear dynamics and fixed observation distributionand error statistics, provided that the initial ensemble perturbation span the vector subspace ofthe linear dynamics operator, the ETKF ensemble would eventually maintain error variance inall amplifying normal modes. And the experiment results indicate that the spectrum of theETKF eigenvalues is flat in the experiment setup.
(2) When the number of ensemble perturbations is much smaller than the number ofdirections to which the forecast error variance projects, the ETKF ensembles wouldunderestimate total analysis error variance because it lacks contribution from important parts ofthe error space. To avoid this problem, we introduce the inflation factor. In the paper I use twomethods to calculate the inflation factor. One is the method introduced by Wang and Bishop(2003), and the other one is similar to the simple Breeding method. The results indicate that theWB method enlarges the ensembles at the beginning, and then it resizes them. The ensemblesin the simple Breeding method grow up slowly. But in the end the two methods nearly get the
same size.(3) The inflation factor's size depends on the ensemble number. The results show that theinflation factor will decrease to the half of the original one if you increased the ensemblenumber from 15 to 30. It indicates that as the ensemble number increases the inflation factorwill have less effect on ensembles.(4) The ETKF method can quickly assess the ability of a large number of future feasiblesequences of observational networks to reduce forecast error variance. In this paper the ETKFand the Deep layer mean wind variance (DLM) method is compared. Results indicate that thesignal variance calculated by the ETKF method is reasonable.
本文在前人的理论研究基础之上,构建ETKF系统平台,利用了实际大气预报模式进行数值试验,对集合变换卡尔曼滤波理论及其在集合预报和适应性观测两方面的应用进行了研究。下面是本文的研究结果和结论:
(1)集合变换卡尔曼滤波方法作为一种集合预报初始扰动的生成方法时,在假设系统是线性和不随时间变化、观测网的地理分布不变、观测误差固定、并且初始集合扰动代表误差增长的方向时,所得到的分析扰动在彼此正交化的观测空间中的概率分布是一样的。本文选取了一次暴雨过程进行试验,根据试验结果统计分析发现在真实大气模式的试验设置下,利用ETKF方法得到的分析扰动在集合子空间基本上也是等概率分布的。
(2)在集合预报试验当中,由于集合数的限制会造成分析误差协方差矩阵的低估,需要引入扩大因子(inflation factor)调整集合预报扰动的幅度。本文利用两种不同的扩大因子计算方案进行试验,试验1选用Wang和Bishop(2003)文中介绍的方法进行试验,试验2是采用类似Breeding方法调整集合扰动的大小。试验1得到的扰动的幅度是先增大后减小,最后趋于稳定;试验2的扰动是缓慢的增长,增长到一定大小保持不变。两组试验得到的扰动场最终都趋于稳定,扰动维持在观测误差大小附近,说明两种方法得到的扰动增长是合理变化的。
(3)考察了集合数的变化对扩大因子的影响,试验表明随着集合数的增加一倍扩大因子的数值减少了将近一半。说明在集合数增大的过程中扩大因子的作用在逐渐变小。
(4)集合变换卡尔曼滤波方法作为一种适应性观测算法,可以直接估计出加入观测后预报误差的减少量。本文用ETKF方法得到的预报误差减少量的变化与深层平均风(DLM)的结果进行了对比,结果表明利用ETKF方法估计得到的信号方差是合理的。
Recently strategies were developed that use forecast-system information to identifylocations where additional observations would provide maximal improvements in the expectedskill of forecast. We refer to these as adaptive observation, or targeted observation, commonlycalled targeting. Targeting identifies localized areas, referred to as sensitive region, in whichthe quality of the analysis has the greatest expected influence on the subsequent skill of theforecast. Now the adaptive observation becomes a sub-program of THORPEX. The EnsembleTransform Kalman filter is initially proposed as an adaptive observation method, than it is usedinto the ensemble forecast. The ETKF is a sub-optimal Kalman filter scheme. Like otherKalman filter, it provides a framework for assimilating observations and also for estimating theeffect of observations on the forecast error covariance. It differs from other ensemble Kalmanfilter in that it uses ensemble transformation and a normalization to rapidly obtain theprediction error covariance matrix associated with a particular deployment of observationresources. This rapidity enables it to quickly assess the ability of a large number of futurefeasible sequences of observational networks to reduce forecast error variance. All the resultsindicate that the ETKF could be put into operational environment. It can be used into theadaptive observation, and also can be used as an ensemble forecast method. Now the domesticscientists do not pay much attention to the ETKF method.So it is necessary to implement themethod on this field.
This paper is based on the ETKF theory established by Bishop and Wang Xu guang et al.The characteristics of ETKF are studied in this paper by virtue of the model GRAPES. Themajor results and conclusions are summarized as follows:
(1) In a system with fixed and perfect linear dynamics and fixed observation distributionand error statistics, provided that the initial ensemble perturbation span the vector subspace ofthe linear dynamics operator, the ETKF ensemble would eventually maintain error variance inall amplifying normal modes. And the experiment results indicate that the spectrum of theETKF eigenvalues is flat in the experiment setup.
(2) When the number of ensemble perturbations is much smaller than the number ofdirections to which the forecast error variance projects, the ETKF ensembles wouldunderestimate total analysis error variance because it lacks contribution from important parts ofthe error space. To avoid this problem, we introduce the inflation factor. In the paper I use twomethods to calculate the inflation factor. One is the method introduced by Wang and Bishop(2003), and the other one is similar to the simple Breeding method. The results indicate that theWB method enlarges the ensembles at the beginning, and then it resizes them. The ensemblesin the simple Breeding method grow up slowly. But in the end the two methods nearly get the
same size.(3) The inflation factor's size depends on the ensemble number. The results show that theinflation factor will decrease to the half of the original one if you increased the ensemblenumber from 15 to 30. It indicates that as the ensemble number increases the inflation factorwill have less effect on ensembles.(4) The ETKF method can quickly assess the ability of a large number of future feasiblesequences of observational networks to reduce forecast error variance. In this paper the ETKFand the Deep layer mean wind variance (DLM) method is compared. Results indicate that thesignal variance calculated by the ETKF method is reasonable.
引文
[1] Panel on model-assimilated data sets (D R Johnson, J T Bates, G. P Brasseur, M.Ghil, A.Hollingsworth, R L jenne, K Miyakoda, E.Rasmusson, E S Sarachikand, and T T Warner). Four-Dimentioanal Model Assimilation of Data: A Strategy for the Earth System Science. National Academy Press, Washington D C, 1991, 78pp
[2] 王跃山.客观分析和四维同化(II)客观分析的主要方法(1)[J].气象科技.2001,1:1 -9
[3] Panofsky H . Objective weather-map analysis [J]. J Appl meteor, 1949, 6:386-392
[4] Gilchrist B and Cressman G.. An experiment in objective analysis [J]. Tellus, 1954, 6:309-318
[5] Bergthorsson P and Doos B. Numerical weather map analysis [J]. Tellus, 1955, 7:329-340
[6] Cressman G. An operational objective analysis system [J].Mon Wea Rev.1959, 87:367-374
[7] 高山红,吴增茂. Kalman 滤波在气象数据同化中的发展与应用[J].地球科学进展,2000,5(4):571-575.
[8] Bishop C. H., Etherton B. J. and Majumdar S. J. Adaptive Sampling with the Ensemble Transform Kalman Filter. PartΙ : Theoretical Aspects [J]. Mon Wea Rev,2001,129:420-436
[9] THORPEX 中国委员会秘书处.THORPEX 国际科学计划中译本.2005
[10] Bishop C. H. and Toth Z. Ensemble Transform and Adaptive Observations [J]. J Atmos Sci, 1999,56:1748-1765
[11] Kalman R E .A new approach to linear filtering and prediction problems. Transaction of the ASME-Journal of Basic Engineering, 1960:35-55
[12] 刘成思 .集合卡尔曼滤波资料同化方案的设计和研究[硕士学位论文].北京,中国气象科学研究院,2005
[13] Wang X, and Bishop C H. A comparison between ETKF and breeding ensemble forecast schemes [J]. J .Atmos .sci, 2003, 60 :1140-1158
[14] Majumdar S J, Bishop C H, Etherton B J et al. Can an Ensemble Transform Kalman Filter predict the reduction in forecast error variance produced by targeted observations? [J] Q J R Meteorol Soc, 2001,127: 2803-2820
[15] C. H. Bishop. Estimation Theory and Adaptive Observing Techniques [J]. 2000. submitted to Q J R Meteorol Soc
[16] S. J. Majumdar, C. H. Bishop, B. J. Etherton and Z. Toth, 2002, Adaptive Sampling with the Ensemble Transform Kalman Filter. Part II: Field Program Implementation [J]. Mon Wea Rev. 2002, 130:1357-1369
[17] Majumdar S J ,Bishop C H, Buizza,R et al .A Comparison of Ensemble Transform Kalman Filter Targeting Guidance with ECMWF and NRL Total Energy Singular Vector Guidance [J]. Q J R Meteorol Soc.2002, 128:2527-2549
[18] Wang X, et al. Which is better, an ensemble of positive-negative pairs or centered spherical simplex ensemble?[J] Mon. Wea.Rev.2004,132:1590-1605
[19] Wei Mo zheng et al. Ensemble Transform Kalman Filter-based ensemble perturbation in an operational global prediction system at NCEP [J].Tellus, 2006, 58A:28-44
[20] Jone Harlim and Brian R.Hunt .Local Ensemble Transform Kalman Filter: An Efficient Scheme For Assimilating Atmospheric Data. 2005,from http://www.atmos.umd.edu/~ekalnay/harlim_hunt05.pdf
[21] 薛纪善,陈德辉等.新一代数值预报系统(GRAPES)研究与应用(内部材料).北京.2005
[22] 谭晓伟,观测系统影响试验与可预报性研究[硕士学位论文].北京.中国气象科学研究院,2005
[23] Tanguay M et al. A semi-implicit semi-Lagrangian fully compressible regional forecast model [J]. Mon Wea Rev.,1990,118:1970—1980.
[24] Ikawa M. Comparison of some schemes for non-hydrostatic models with orography [J]. J Meteor Soc Japan,1988,66: 753—776
[25] Bates J R.,Moorthi S,Higgins R.W. A global multilevel atmospheric model using a vector semi-Lagrangian finite difference scheme. Part I: Adiabatic formulation [J]. Mon Wea Rev.,1993,121:244-263.
[26] Staniforth A.,Cote J. Semi-Lagrandian integration scheme for atmospheric models – A review [J]. Mon Wea Rev,1991,119:2206-2223.
[27] Skamarock W. C.,Klemp J. B. The stability of time—split numerical methods for the hydrostatic and the non-hydrostatic elastic equations [J]. Mon Wea Rev,1992,120: 2109—2127
[28] 陈静.中尺度暴雨短期集合预报研究[博士学位论文].北京:中国气象科学研究院,2003
[29] Toth Z, and Kalnay E. Ensemble forecasting at NMC: The generation of perturbations. Bull Amer Meteor soc,1993,74:2317-2330
[30] 董佩明,张昕.目标观测设计与伴随敏感性分析[J].气象科技,2004,32(1):1-5
[31] Palmer, T. N. et al. Singular vectors, metrics, and adaptive observations [J]. J Atmos Sci., 1998, 55:633–653.
[32] Thierry Bergot and Gwenaelle Hello et al. Adaptive Observation: A feasibility Study [J]. Mon Wea Rev,1999,127:743-765
[33] Langland R H, and Rohaly G D, 1996: Adjoint-based targeting of observations for FASTEX cyclones [J]. Preprints, Seventh Conf on Mesoscale Processes, Phoenix, AZ .Amer Meteor Soc, 1996:369–371.
[34] Pu Z, and Kalnay E. Targeting observations with the quasilinear inverse and adjoint NCEP global models: Performance during FASTEX [J]. Quart J Roy Meteor Soc, 1999, 125:3329–3338.
[35] Lorenz E N, and Emanuel K A. Optimal sites for supplementary observation sites: Simulation with a small model [J]. J Atmos Sci , 1998, 55:399–414.
[36] Wu Chun-Chieh , Lin Po-Hsiung , Chen Jan-Huey et al. Targeted Observations of Tropical Cyclones Based on the Adjoint-Derived Sensitivity Steering Vector .from http://typhoon.as.ntu.edu.tw/wu_paper/ADSSV-GRL-Wu.pdf
[37] John Lewis, Andrew Van Tuyl, and Christopher Velden. A Dynamical Method for Building Continuity into the Deep-Layer Mean Wind. Mon Wea Rev, 1987, 115:885-893
[38] Berliner L M, Q Lu, and C Snyder. Statistical design for adaptive weather observations[J]. J Atmos Sci.1999, 56: 2536-2552.
[39] Sim D.Aberson.Targeted Observation to Improve Operational Tropical Cyclone Track Forecast Guidance [J]. Mon Wea Rev.2003,131:1613-1627
[2] 王跃山.客观分析和四维同化(II)客观分析的主要方法(1)[J].气象科技.2001,1:1 -9
[3] Panofsky H . Objective weather-map analysis [J]. J Appl meteor, 1949, 6:386-392
[4] Gilchrist B and Cressman G.. An experiment in objective analysis [J]. Tellus, 1954, 6:309-318
[5] Bergthorsson P and Doos B. Numerical weather map analysis [J]. Tellus, 1955, 7:329-340
[6] Cressman G. An operational objective analysis system [J].Mon Wea Rev.1959, 87:367-374
[7] 高山红,吴增茂. Kalman 滤波在气象数据同化中的发展与应用[J].地球科学进展,2000,5(4):571-575.
[8] Bishop C. H., Etherton B. J. and Majumdar S. J. Adaptive Sampling with the Ensemble Transform Kalman Filter. PartΙ : Theoretical Aspects [J]. Mon Wea Rev,2001,129:420-436
[9] THORPEX 中国委员会秘书处.THORPEX 国际科学计划中译本.2005
[10] Bishop C. H. and Toth Z. Ensemble Transform and Adaptive Observations [J]. J Atmos Sci, 1999,56:1748-1765
[11] Kalman R E .A new approach to linear filtering and prediction problems. Transaction of the ASME-Journal of Basic Engineering, 1960:35-55
[12] 刘成思 .集合卡尔曼滤波资料同化方案的设计和研究[硕士学位论文].北京,中国气象科学研究院,2005
[13] Wang X, and Bishop C H. A comparison between ETKF and breeding ensemble forecast schemes [J]. J .Atmos .sci, 2003, 60 :1140-1158
[14] Majumdar S J, Bishop C H, Etherton B J et al. Can an Ensemble Transform Kalman Filter predict the reduction in forecast error variance produced by targeted observations? [J] Q J R Meteorol Soc, 2001,127: 2803-2820
[15] C. H. Bishop. Estimation Theory and Adaptive Observing Techniques [J]. 2000. submitted to Q J R Meteorol Soc
[16] S. J. Majumdar, C. H. Bishop, B. J. Etherton and Z. Toth, 2002, Adaptive Sampling with the Ensemble Transform Kalman Filter. Part II: Field Program Implementation [J]. Mon Wea Rev. 2002, 130:1357-1369
[17] Majumdar S J ,Bishop C H, Buizza,R et al .A Comparison of Ensemble Transform Kalman Filter Targeting Guidance with ECMWF and NRL Total Energy Singular Vector Guidance [J]. Q J R Meteorol Soc.2002, 128:2527-2549
[18] Wang X, et al. Which is better, an ensemble of positive-negative pairs or centered spherical simplex ensemble?[J] Mon. Wea.Rev.2004,132:1590-1605
[19] Wei Mo zheng et al. Ensemble Transform Kalman Filter-based ensemble perturbation in an operational global prediction system at NCEP [J].Tellus, 2006, 58A:28-44
[20] Jone Harlim and Brian R.Hunt .Local Ensemble Transform Kalman Filter: An Efficient Scheme For Assimilating Atmospheric Data. 2005,from http://www.atmos.umd.edu/~ekalnay/harlim_hunt05.pdf
[21] 薛纪善,陈德辉等.新一代数值预报系统(GRAPES)研究与应用(内部材料).北京.2005
[22] 谭晓伟,观测系统影响试验与可预报性研究[硕士学位论文].北京.中国气象科学研究院,2005
[23] Tanguay M et al. A semi-implicit semi-Lagrangian fully compressible regional forecast model [J]. Mon Wea Rev.,1990,118:1970—1980.
[24] Ikawa M. Comparison of some schemes for non-hydrostatic models with orography [J]. J Meteor Soc Japan,1988,66: 753—776
[25] Bates J R.,Moorthi S,Higgins R.W. A global multilevel atmospheric model using a vector semi-Lagrangian finite difference scheme. Part I: Adiabatic formulation [J]. Mon Wea Rev.,1993,121:244-263.
[26] Staniforth A.,Cote J. Semi-Lagrandian integration scheme for atmospheric models – A review [J]. Mon Wea Rev,1991,119:2206-2223.
[27] Skamarock W. C.,Klemp J. B. The stability of time—split numerical methods for the hydrostatic and the non-hydrostatic elastic equations [J]. Mon Wea Rev,1992,120: 2109—2127
[28] 陈静.中尺度暴雨短期集合预报研究[博士学位论文].北京:中国气象科学研究院,2003
[29] Toth Z, and Kalnay E. Ensemble forecasting at NMC: The generation of perturbations. Bull Amer Meteor soc,1993,74:2317-2330
[30] 董佩明,张昕.目标观测设计与伴随敏感性分析[J].气象科技,2004,32(1):1-5
[31] Palmer, T. N. et al. Singular vectors, metrics, and adaptive observations [J]. J Atmos Sci., 1998, 55:633–653.
[32] Thierry Bergot and Gwenaelle Hello et al. Adaptive Observation: A feasibility Study [J]. Mon Wea Rev,1999,127:743-765
[33] Langland R H, and Rohaly G D, 1996: Adjoint-based targeting of observations for FASTEX cyclones [J]. Preprints, Seventh Conf on Mesoscale Processes, Phoenix, AZ .Amer Meteor Soc, 1996:369–371.
[34] Pu Z, and Kalnay E. Targeting observations with the quasilinear inverse and adjoint NCEP global models: Performance during FASTEX [J]. Quart J Roy Meteor Soc, 1999, 125:3329–3338.
[35] Lorenz E N, and Emanuel K A. Optimal sites for supplementary observation sites: Simulation with a small model [J]. J Atmos Sci , 1998, 55:399–414.
[36] Wu Chun-Chieh , Lin Po-Hsiung , Chen Jan-Huey et al. Targeted Observations of Tropical Cyclones Based on the Adjoint-Derived Sensitivity Steering Vector .from http://typhoon.as.ntu.edu.tw/wu_paper/ADSSV-GRL-Wu.pdf
[37] John Lewis, Andrew Van Tuyl, and Christopher Velden. A Dynamical Method for Building Continuity into the Deep-Layer Mean Wind. Mon Wea Rev, 1987, 115:885-893
[38] Berliner L M, Q Lu, and C Snyder. Statistical design for adaptive weather observations[J]. J Atmos Sci.1999, 56: 2536-2552.
[39] Sim D.Aberson.Targeted Observation to Improve Operational Tropical Cyclone Track Forecast Guidance [J]. Mon Wea Rev.2003,131:1613-1627