集合卡尔曼滤波中模式偏差的线性订正及其在有限区域地面观测资料同化中的应用
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摘要
数据同化是提高数值天气预报质量的重要方面。作为新一代数据同化方法，以依赖环流的背景误差协方差为主要优势的集合卡尔曼滤波及同类方法的业务应用问题是近年来数据同化研究领域的热点。本文以有限区域地面观测资料数据同化为出发点，研究了模式偏差对集合卡尔曼滤波数据同化带来的不利影响，提出了线性模式偏差模型和模式偏差线性齐次订正方法。使用Lorenz96系统开展了实验，分析了模式偏差的影响机制和偏差订正的效果。自主建立了以WRF为预报模式的EnSRF数据同化系统，使用该系统，对2006年4月28日一次飑线过程进行了数据同化研究，分析了预报误差协方差分布特征，并进一步分析了不同地面观测变量的模式偏差对数据同化带来的不同影响，通过海平面气压数据同化和地面气温数据同化进一步验证了线性模式偏差和线性齐次订正方法的有效性，另外，本文分析了不同地面观测变量同化可能存在的问题，讨论了地面观测资料的同化技术。
本文对模式偏差影响集合卡尔曼滤波数据同化的机制、模式偏差订正方法及效果进行了研究分析，得出以下结论：
(1)在集合卡尔曼滤波中，模式偏差的对滤波的影响与在观测资料中加入偏差是等价的，非高斯分布和对背景场的错误订正，增大了分析误差，并可造成系统不稳定和系统崩溃。
(2)对于一定的模式偏差，对应的背景场中观测值(即Hx)的离散度越大，造成的分析误差越小，反之，模式偏差会造成较大的分析误差。在集合背景场中离散度较大的区域，模式偏差容易造成较大的分析误差。
(3)采用本研究提出的模式偏差线性齐次订正方法，当不存在系统误差时，不会对同化带来明显的影响，即副作用很小。当存在系统误差时，采取了订正步骤后，可明显降低分析误差，增加系统的稳定性。不同的模式误差形态，订正的效果不同，对于线性齐次偏差，订正后可以基本消除模式偏差带来的不利影响，而对于其他类型的模式偏差，订正后只有部分改进，即仅能消除模式误差中能够使用线性齐次关系描述的部分偏差带来的影响。
(4)在同化实际观测资料时，对于预报误差协相关尺度较大的观测变量(如海平面气压)，有限区域中观测资料在正负协相关区域的分布往往存在不平衡，此时，模式偏差将明显降低同化效果，模式偏差对背景场某气象要素造成的错误订正的空间尺度与该要素误差协相关的尺度一致，而且观测资料越多、错误波动的振幅就会越大，因而必须进行模式偏差订正。反之，对于误差协方差尺度较小的观测变量，实际观测资料分布往往较为平衡，此时，模式偏差带来的不利影响较小。
本文分析了在有限区域集合卡尔曼数据同化中背景场预报误差协方差的分布特征，对地面资料同化技术进行了探讨，主要得出以下结论：
(1)如果将集合预报各个成员与集合平均的差可以看作是误差场波动在不同时刻的位移，那么背景场误差协相关的尺度可以看作是误差波动的尺度，不同的气象要素，以及相同的气象要素在不同的层次，误差波动的尺度是不同的。本研究中，海平面气压误差场波动尺度较大，远远大于实际地面观测的平均距离，因此，在海平面气压数据同化中可能存在“饱和”问题。在本研究采用的模式空间分辨率下，地面气温误差波动尺度呈现次天气尺度和中尺度特征，地面风场的误差波动尺度更小，因此，同化地面气温和风场资料时，应选用加密观测网。在高空气象要素预报中，误差波动的尺度由大到小依次为扰动位势、气温和风场，探空站的密度应该能够满足位势高度和高空气温数据同化的需求，对于高空风场的数据同化，探空资料的密度有些不足。
(2)在集合预报启动之初，受大气环流影响，叠加在初始场上的误差协方差结构随模式积分将作出快速地调整，在基本适应大气环流之后，预报误差协方差的结构跟随大气环流变化的速度则慢得多。
(3)地面比湿预报误差的协相关分布显示出明显的昼夜之分，夜间相关系数较大，而白天相关系数较小，应当与边界层水汽的垂直交换有关。
海平面气压与高空等σ面气压的预报误差相关性较强，海平面气压信息将影响整个对流层，对各个高度层是同等重要的。对于风场、气温和比湿而言，随着层次的升高，地面与高空相同要素的予报误差相关性迅速下降，但在白天的边界层中，由于湍流混合，气温与比湿预报误差与地面相同要素预报误差的相关性随高度下降较为缓慢。
实验结果显示地面2米气温与高空扰动位势的预报误差相关性较强，相关系数甚至高于地面气温与高空气温预报误差的相关系数，显示出地面气温同化对于位势高度场具有重要作用。其他地面与高空不同类型气象要素之间的预报误差相关性较差。
(4)受边界层物理方案的影响，在边界层中，与比湿大小的垂直分布相反，由边界层顶至近地面比湿的集合离散度迅速减小，边界层方案对近地面气温的离散度也有一定的抑制作用。在地面气温和比湿同化中，较小的离散度，可能会造成对背景场较大的虚假订正。
(5)海平面气压和地面气温的模式偏差分布基本符合本研究提出的线性模式偏差模型，使用线性齐次模式偏差订正方法取得较好的效果。而地面风场观测值与模式预报值的相关性较差，说明本研究提出的线性齐次订正方法可能不适用于风场模式偏差订正。
Data assimilation is important to the improvement of numerical weather prediction.Characterized by flow-dependent background error covarianee,and as new-generationdata assimilation techniques.Ensemble Kalman Filter (EnKF) has gained particularpopularity for environmental state estimation.However,there are still some obstaclesfor its fully application in operational works.In this work,oriented around surfaceobservations data assimilation in regional numerical forecast,some key scientificproblems of EnKF application are studied.The impact of model bias on EnKF isstudied,and a linear model bias model and homogeneous linear bias correctionmethodology are put forward.The impact mechanism of model bias on EnKF and theefficiency of the homogeneous linear bias correction method are studied inexperiments using Lorenz96 system.In this paper,an EnSRF data assimilation systemusing WRF as forecast model is set up independently.With the help of this system,adata assimilation case study is carried out on a disastrous weather caused by squallline occurred on 28 April,2006.The structure of background error covariance isstudied,and the impacts of model bias on different surface data assimilation areanalyzed.In sea-level pressure and surface air teraperature data assimilation,thelinear bias model and the Homogeneous Linear bias correction method are testified tobe efficient.In addition,the potential problems of different surface observation dataassimilation are discussed.
On the impact mechanism of model bias on EnKF,the Homogeneous Linear biascorrection method and its efficiency,the following conclusions are drawn in thiswork:
(1) In EnKF,model bias plays a simila(?) role as observation bias.The non-Gaussian distribution and erroneous background updating increase the analysis error and lead tosystem instability and cracking.
(2) With certain model bias,the smaller the backgroand ensemble spread ofobservation,the larger analysis error generated in model state analysis.The modelstates in area of larger ensemble spread or with lower predictability are likely to getlarger analysis error.
(3) With the help of homogeneous linear bias correction method proposed in this work,substantial improvement in analysis quality and system stability are got.The biascorrection method has little negative impact when there is no model bias in dataassimilation system.The efficiency of the homogeneous linear bias correction methodis different for different bias pattern.While the homogeneous linear bias can be fullycorrected,only part of the bias,which can be represented by homogeneous linear biasmodel in other bias patterns,can be corrected.
(4) In EnKF data assimilation of real observations,the model bias on observationalvariables with spatially large scale error covariance (for example sea level pressure)should be corrected to avoid large erroneous updating to background state.Whenmodel bias exist in observations of large scale background err covariance,the morethe observations and the more imbalance in its distribution,the larger the erroneousupdating is generated in regional data assimilation.
On the background error covariance structure and surface observation dataassimilation,the following conclusions are drawn in this work:
(1) The spatial scales of covariance are different with different variable and ondifferent levels.The sea level pressure has larger spatial scale,which may lead toimprovement“saturation”in data assimilation with relative dense observations.Oncontrary,with the spatial resolution in this work,the er(?)or covariance scale of surface air temperature,wind and humidity is much smaller.The density of wind observationin troposphere should be increased from current density of the soundings.
(2) The error covariance originally added on initial conditions adjusts fast till it fit forflow.The sea level pressure has important impact on all model levels,while theeovariance of surface wind,tenperature and humidity with the same variables withintroposphere decrease quickly with height.Different variables have small covarianceexcept for surface air temperature and perturbed geopotential.
(3) Affected by physics par(?)eterization scheme,within planate boundary level,thespread of humidity decreases fast with height,and the spread of temperature are alsosuppressed.It should be a potential problem in surface air temperature and specifichumidity data assimilation,since it could bring large spurious increment analysisfield.
(4) The model bias of sea level pressure and surface air tempperature are in agree withthe linear model put forward in the work.EnKF with homogeneous linear biascorrection method can get substantial improvement compared with assimilationwithout bias correction.
本文对模式偏差影响集合卡尔曼滤波数据同化的机制、模式偏差订正方法及效果进行了研究分析，得出以下结论：
(1)在集合卡尔曼滤波中，模式偏差的对滤波的影响与在观测资料中加入偏差是等价的，非高斯分布和对背景场的错误订正，增大了分析误差，并可造成系统不稳定和系统崩溃。
(2)对于一定的模式偏差，对应的背景场中观测值(即Hx)的离散度越大，造成的分析误差越小，反之，模式偏差会造成较大的分析误差。在集合背景场中离散度较大的区域，模式偏差容易造成较大的分析误差。
(3)采用本研究提出的模式偏差线性齐次订正方法，当不存在系统误差时，不会对同化带来明显的影响，即副作用很小。当存在系统误差时，采取了订正步骤后，可明显降低分析误差，增加系统的稳定性。不同的模式误差形态，订正的效果不同，对于线性齐次偏差，订正后可以基本消除模式偏差带来的不利影响，而对于其他类型的模式偏差，订正后只有部分改进，即仅能消除模式误差中能够使用线性齐次关系描述的部分偏差带来的影响。
(4)在同化实际观测资料时，对于预报误差协相关尺度较大的观测变量(如海平面气压)，有限区域中观测资料在正负协相关区域的分布往往存在不平衡，此时，模式偏差将明显降低同化效果，模式偏差对背景场某气象要素造成的错误订正的空间尺度与该要素误差协相关的尺度一致，而且观测资料越多、错误波动的振幅就会越大，因而必须进行模式偏差订正。反之，对于误差协方差尺度较小的观测变量，实际观测资料分布往往较为平衡，此时，模式偏差带来的不利影响较小。
本文分析了在有限区域集合卡尔曼数据同化中背景场预报误差协方差的分布特征，对地面资料同化技术进行了探讨，主要得出以下结论：
(1)如果将集合预报各个成员与集合平均的差可以看作是误差场波动在不同时刻的位移，那么背景场误差协相关的尺度可以看作是误差波动的尺度，不同的气象要素，以及相同的气象要素在不同的层次，误差波动的尺度是不同的。本研究中，海平面气压误差场波动尺度较大，远远大于实际地面观测的平均距离，因此，在海平面气压数据同化中可能存在“饱和”问题。在本研究采用的模式空间分辨率下，地面气温误差波动尺度呈现次天气尺度和中尺度特征，地面风场的误差波动尺度更小，因此，同化地面气温和风场资料时，应选用加密观测网。在高空气象要素预报中，误差波动的尺度由大到小依次为扰动位势、气温和风场，探空站的密度应该能够满足位势高度和高空气温数据同化的需求，对于高空风场的数据同化，探空资料的密度有些不足。
(2)在集合预报启动之初，受大气环流影响，叠加在初始场上的误差协方差结构随模式积分将作出快速地调整，在基本适应大气环流之后，预报误差协方差的结构跟随大气环流变化的速度则慢得多。
(3)地面比湿预报误差的协相关分布显示出明显的昼夜之分，夜间相关系数较大，而白天相关系数较小，应当与边界层水汽的垂直交换有关。
海平面气压与高空等σ面气压的预报误差相关性较强，海平面气压信息将影响整个对流层，对各个高度层是同等重要的。对于风场、气温和比湿而言，随着层次的升高，地面与高空相同要素的予报误差相关性迅速下降，但在白天的边界层中，由于湍流混合，气温与比湿预报误差与地面相同要素预报误差的相关性随高度下降较为缓慢。
实验结果显示地面2米气温与高空扰动位势的预报误差相关性较强，相关系数甚至高于地面气温与高空气温预报误差的相关系数，显示出地面气温同化对于位势高度场具有重要作用。其他地面与高空不同类型气象要素之间的预报误差相关性较差。
(4)受边界层物理方案的影响，在边界层中，与比湿大小的垂直分布相反，由边界层顶至近地面比湿的集合离散度迅速减小，边界层方案对近地面气温的离散度也有一定的抑制作用。在地面气温和比湿同化中，较小的离散度，可能会造成对背景场较大的虚假订正。
(5)海平面气压和地面气温的模式偏差分布基本符合本研究提出的线性模式偏差模型，使用线性齐次模式偏差订正方法取得较好的效果。而地面风场观测值与模式预报值的相关性较差，说明本研究提出的线性齐次订正方法可能不适用于风场模式偏差订正。
Data assimilation is important to the improvement of numerical weather prediction.Characterized by flow-dependent background error covarianee,and as new-generationdata assimilation techniques.Ensemble Kalman Filter (EnKF) has gained particularpopularity for environmental state estimation.However,there are still some obstaclesfor its fully application in operational works.In this work,oriented around surfaceobservations data assimilation in regional numerical forecast,some key scientificproblems of EnKF application are studied.The impact of model bias on EnKF isstudied,and a linear model bias model and homogeneous linear bias correctionmethodology are put forward.The impact mechanism of model bias on EnKF and theefficiency of the homogeneous linear bias correction method are studied inexperiments using Lorenz96 system.In this paper,an EnSRF data assimilation systemusing WRF as forecast model is set up independently.With the help of this system,adata assimilation case study is carried out on a disastrous weather caused by squallline occurred on 28 April,2006.The structure of background error covariance isstudied,and the impacts of model bias on different surface data assimilation areanalyzed.In sea-level pressure and surface air teraperature data assimilation,thelinear bias model and the Homogeneous Linear bias correction method are testified tobe efficient.In addition,the potential problems of different surface observation dataassimilation are discussed.
On the impact mechanism of model bias on EnKF,the Homogeneous Linear biascorrection method and its efficiency,the following conclusions are drawn in thiswork:
(1) In EnKF,model bias plays a simila(?) role as observation bias.The non-Gaussian distribution and erroneous background updating increase the analysis error and lead tosystem instability and cracking.
(2) With certain model bias,the smaller the backgroand ensemble spread ofobservation,the larger analysis error generated in model state analysis.The modelstates in area of larger ensemble spread or with lower predictability are likely to getlarger analysis error.
(3) With the help of homogeneous linear bias correction method proposed in this work,substantial improvement in analysis quality and system stability are got.The biascorrection method has little negative impact when there is no model bias in dataassimilation system.The efficiency of the homogeneous linear bias correction methodis different for different bias pattern.While the homogeneous linear bias can be fullycorrected,only part of the bias,which can be represented by homogeneous linear biasmodel in other bias patterns,can be corrected.
(4) In EnKF data assimilation of real observations,the model bias on observationalvariables with spatially large scale error covariance (for example sea level pressure)should be corrected to avoid large erroneous updating to background state.Whenmodel bias exist in observations of large scale background err covariance,the morethe observations and the more imbalance in its distribution,the larger the erroneousupdating is generated in regional data assimilation.
On the background error covariance structure and surface observation dataassimilation,the following conclusions are drawn in this work:
(1) The spatial scales of covariance are different with different variable and ondifferent levels.The sea level pressure has larger spatial scale,which may lead toimprovement“saturation”in data assimilation with relative dense observations.Oncontrary,with the spatial resolution in this work,the er(?)or covariance scale of surface air temperature,wind and humidity is much smaller.The density of wind observationin troposphere should be increased from current density of the soundings.
(2) The error covariance originally added on initial conditions adjusts fast till it fit forflow.The sea level pressure has important impact on all model levels,while theeovariance of surface wind,tenperature and humidity with the same variables withintroposphere decrease quickly with height.Different variables have small covarianceexcept for surface air temperature and perturbed geopotential.
(3) Affected by physics par(?)eterization scheme,within planate boundary level,thespread of humidity decreases fast with height,and the spread of temperature are alsosuppressed.It should be a potential problem in surface air temperature and specifichumidity data assimilation,since it could bring large spurious increment analysisfield.
(4) The model bias of sea level pressure and surface air tempperature are in agree withthe linear model put forward in the work.EnKF with homogeneous linear biascorrection method can get substantial improvement compared with assimilationwithout bias correction.
引文
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[32]Etherton,B.J.and C.H.Bishop,Resilience of Hybrid Ensemble/3DVAR.Analysis Schemes to Model Error and Ensemble Covariance Error.Monthly Weather Review,2004.132(5):p.1065-1080.
[33]高山红，吴．，谢红琴，Kalman滤波在气象数据同化中的发展与应用．地球科学进展，2000．15(5)：p．571-575．
[34]刘成思and薛纪善，关于集合Kalman滤波的理论和方法的发展．热带气象学报，2005．21(6)：p．628-633．
[35]费剑锋，韩．，集合卡尔曼滤波数据同化在一维波动方程中的应用．气象科技，2005．33(2)：p．110-119．
[36]朱琳，集合卡尔曼滤波同化方法的研究．2007，南京信息工程大学．
[37]官元红，周．，陆维松，陈建萍，资料同化方法的理论发展及应用综述．气象与减灾研究，2007．30(4)．
[38]徐小永 and刘黎平；郑国光，集合卡尔曼滤波同化多普勒雷达资料的数值试验．大气科学，2006．30(4)：p．713-728．
[39]林彩燕，朱江，and陆春谷，集合KALMAN滤波和最优插值方法在不同观测分布的比较理想试验．气候与环境研究，2006．11(5)：p．553-564．
[40]庄照荣，集合卡尔曼滤波资料同化系统的设计及其在集合预报中的应用．2007，中国科学院研究生院．
[41]朱琳，寿．，彭加毅，张红华，集合卡尔曼滤波同化探空资料的数值试验．南京气象学院学报，2008．31(2)：p．265-271．
[42]Dirren,S.,et al.,A Data Assimilation Case Study Using a Limited-Area Ensemble Kalman Filter.Monthly Weather Review,2007.135(4):p.1455-1473.
[43]Meng,Z.and F.Zhang,Tests of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation.Part Ⅲ:Comparison with 3DVAR in a Real-Data Case Study.Monthly Weather Review,2008.136(2):p.522-540.
[44] J. F. Meirinkl, P. B., and M. C. Krol, Four-dimensional variational data assimilation for inverse modelling of atmospheric methane emissions: method and comparison with synthesis inversion. Atmospheric Chemistry and Physics, 2008(8):p. 6341 - 6353.
[45] Zhang Lin, N. Y., Four-dimensional variational data assimilation experiments fora heavy rain case during the 2002 IOP in China. Advances in Atmospheric Sciences,2005. 22(2): p. 300-312.
[46] Vukievi, T. and J.-W. Bao, The Effect of Linearization Errors on 4DVAR Data Assimilation. Monthly Weather Review, 1998. 126(6): p. 1695-1706.
[47] Charron, M., P. L. Houtekamer, and P. Bartello, Assimilation with an Ensemble Kalman Filter of Synthetic Radial Wind Data in Anisotropic Turbulence: Perfect Model Experiment's. Monthly Weather Review, 2006. 134(2): p. 618-637.
[48] Caya, A., J. Sun, and C. Snyder, A Comparison between the 4DVAR and the Ensemble Kalman Filter Techniques for Radar Data Assimilation. Monthly Weather Review, 2005.133(11): p. 3081-3094.
[49] Meng, Z. and F. Zhang, Tests of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation. Part II: Imperfect Model Experiments. Monthly Weather Review, 2007. 135(4): p. 1403-1423.
[50] Edward OTT, B. R. H., Istvan Szunyogh, Aleksey V. ZIMIN, ERIC J. KOSTELICH, MATTEO CORAZZA, EUGENIA KALNAY, D. J. PATIL, JAMES A. YORKE A local ensemble Kalman filter for atmospheric data assimilation Tellus A 2004. 56(5): p. 415-428
[51] Istvan Szunyogh, E. J. K., Gyorgyi Gyarmati, EUGENIA KALNAY, BRIAN R. HUNT, EDWARD 0TT, ELIZABETH SATTERFIELD, JAMES A. YORKE A local ensemble transform Kalman filter data assimilation system for the NCEP global model Tellus A 2008. 60(1):p. 113-130.
[52] Evensen, G., Sample strategies and square root analysis schemes for EnKF Ocean Dynamics, 2004. 54: p. 539-560.
[53] Whitaker, J.S., et al., Reanalysis without Radiosondes Using Ensemble Data Assimilation. Monthly Weather Review, 2004. 132(5): p. 1190-1200.
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[61] Fujita, T., D.J. Stensrud, and D. C. Dowell, Surface Data Assimilation Using an Ensemble Kalman Filter Approach with Initial Condition and Model Physics Uncertainties. Monthly Weather Review, 2007. 135(5): p. 1846-1868.
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