GRAPES变分框架下背景误差协方差矩阵的优化
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摘要
在变分资料同化中,背景误差协方差矩阵的结构一般通过物理变换和空间变换隐式表达。其中,物理变换抽取了变量间的动力平衡约束,反映的是背景误差协方差矩阵中的误差交叉协相关关系。这些交叉协相关关系控制着观测信息在不同变量之间的传播,并能抑制同化分析场中的高频重力波,对于同化分析十分重要。
GRAPES模式面三维变分同化框架下原有的物理变换方案采用线性平衡方程来约束风场和质量场之间的分析,并引入了一组参考等压面解决平衡方程在高度地形追随模式面上求解困难的问题。该方案在操作中涉及反复垂直插值,并且线性平衡方程的适用范围也有一定的限制。此外,原方案只考虑旋转风和质量之间的动力平衡,这使得背景误差协方差矩阵中的交叉协相关结构在边界层等特殊区域略嫌简单。
为解决原有方案存在的问题,本研究在这一同化框架中引入了一个新的物理变换方案,以优化变分框架中背景误差协方差矩阵的结构。新方案采用模式面上统计得到的流函数和无量纲气压之间的平衡算子N代替原方案中的线性平衡方程,来表达旋转风和质量之间的平衡关系；采用模式面上统计得到的流函数和势函数之间的平衡算子M,补充表达了原方案中所没有的旋转风和散度风之间的平衡关系,统计样本采用NMC方法获得。N算子的统计采用单层相关和多层相关两种模型进行比较,并最终选用了单层相关结果进入到新的物理变换方案；M算子的统计直接采用的是单层相关模型。新的物理变换方案可以直接在模式面上实施,避免了额外插值。
统计结果表明,流函数和无量纲气压之间的动力平衡约束在赤道外对流层顶以下最强,在赤道地区很弱。而流函数与势函数的动力平衡约束相对要小很多,且大值区主要位于两个半球中高纬边界层地区。通过随机扰动试验和单点试验可以发现,在地转关系成立较好的区域,新方案的性能和原方案十分接近；而在地转关系不适用的区域,新方案可以有效减小旋转风和质量的耦合程度,使得这些地区的风场和质量场的分析更加独立。单点试验结果还表明,由于新方案中添加了旋转风和散度风之间的动力平衡约束,边界层的风场分析结构更加完善。同化预报循环试验表明,在同化常规观测资料的情况下,新的物理变换方案对于提高南北半球的同化分析场以及东亚地区短期预报场的质量有正效果。
In variational data assimilation, the background error covariance is defined implicitly by the physical variable transform operator and the spatial transform operator. The physical variable transform operator defines the error cross-correlations between analysis variables by extracting the dynamic balance constraints of them. These cross-correlations are very important to variational assimilation, because they control the observation information spreading between the variables and can suppress noise caused by gravity waves.
The existing GRAPES three-dimensional variational data assimilation system, which defined on sigma coordinates, uses linear balance equation to ensure that mass and wind analysis increments to be geostrophically coupled. In this formulation, to deal with the difficulties in solving the balance equation at sigma levels, analysis variables need to be interpolated to a series of auxiliary isobaric surface to calculate balanced components, In addition to the repeated interpolation problem, another disadvantage of the formulation is that the linear balance equation is not appropriate in some spatial regions.
To solve the problems in the existing program and optimize the structure of the background error covariance, a new physical variable transform formulation has been developed in the GRAPES variation assimilation system. In the new scheme, dynamic balance operator (N) obtained by statistical methods between stream function and dimensionless pressure (Exner function) is used to describe the balance relationship between rotational wind and mass field. In addition, the balance relationship between rotational wind and divergent wind is similarly described by dynamic balance operator (M) between stream function and velocity potential which is also obtained by statistical methods. In the calculation of the operator N, single-layer-related model and multilayer-related model were compared, and ultimately chose the former into the new formulation. Single-layer-related model was also chose to calculate operator M. Compared to the original scheme, the new formulation can avoid repeated interpolations along the vertical direction.
Statistical results show that, the explained variance of dimensionless pressure is primarily in the extratropics with the variance best explained below100hPa in this new formulation. And the explained velocity potential ratio has a maximum in the middle-and high-latitude near the surface. Results of randomization and single-observation experiments indicate that, in regions where geostrophic balance is appropriate, the new formulation behaves similarly to the old scheme. However, in regions where geostrophic balance is not appropriate, the new formulation could allow for a smooth decoupling of stream function and dimensionless pressure, while the old scheme cannot. Moreover, by adding the balance relationship between rotational wind and divergent wind, the new formulation could derive a more reasonable wind field in boundary layer. The results of analysis forecast cycle experiment show that, the new formulation could help improve the quality of the analysis fields of both the northern and southern hemispheres and could help improve short-term forecasting results of the East Asian region.
GRAPES模式面三维变分同化框架下原有的物理变换方案采用线性平衡方程来约束风场和质量场之间的分析,并引入了一组参考等压面解决平衡方程在高度地形追随模式面上求解困难的问题。该方案在操作中涉及反复垂直插值,并且线性平衡方程的适用范围也有一定的限制。此外,原方案只考虑旋转风和质量之间的动力平衡,这使得背景误差协方差矩阵中的交叉协相关结构在边界层等特殊区域略嫌简单。
为解决原有方案存在的问题,本研究在这一同化框架中引入了一个新的物理变换方案,以优化变分框架中背景误差协方差矩阵的结构。新方案采用模式面上统计得到的流函数和无量纲气压之间的平衡算子N代替原方案中的线性平衡方程,来表达旋转风和质量之间的平衡关系；采用模式面上统计得到的流函数和势函数之间的平衡算子M,补充表达了原方案中所没有的旋转风和散度风之间的平衡关系,统计样本采用NMC方法获得。N算子的统计采用单层相关和多层相关两种模型进行比较,并最终选用了单层相关结果进入到新的物理变换方案；M算子的统计直接采用的是单层相关模型。新的物理变换方案可以直接在模式面上实施,避免了额外插值。
统计结果表明,流函数和无量纲气压之间的动力平衡约束在赤道外对流层顶以下最强,在赤道地区很弱。而流函数与势函数的动力平衡约束相对要小很多,且大值区主要位于两个半球中高纬边界层地区。通过随机扰动试验和单点试验可以发现,在地转关系成立较好的区域,新方案的性能和原方案十分接近；而在地转关系不适用的区域,新方案可以有效减小旋转风和质量的耦合程度,使得这些地区的风场和质量场的分析更加独立。单点试验结果还表明,由于新方案中添加了旋转风和散度风之间的动力平衡约束,边界层的风场分析结构更加完善。同化预报循环试验表明,在同化常规观测资料的情况下,新的物理变换方案对于提高南北半球的同化分析场以及东亚地区短期预报场的质量有正效果。
In variational data assimilation, the background error covariance is defined implicitly by the physical variable transform operator and the spatial transform operator. The physical variable transform operator defines the error cross-correlations between analysis variables by extracting the dynamic balance constraints of them. These cross-correlations are very important to variational assimilation, because they control the observation information spreading between the variables and can suppress noise caused by gravity waves.
The existing GRAPES three-dimensional variational data assimilation system, which defined on sigma coordinates, uses linear balance equation to ensure that mass and wind analysis increments to be geostrophically coupled. In this formulation, to deal with the difficulties in solving the balance equation at sigma levels, analysis variables need to be interpolated to a series of auxiliary isobaric surface to calculate balanced components, In addition to the repeated interpolation problem, another disadvantage of the formulation is that the linear balance equation is not appropriate in some spatial regions.
To solve the problems in the existing program and optimize the structure of the background error covariance, a new physical variable transform formulation has been developed in the GRAPES variation assimilation system. In the new scheme, dynamic balance operator (N) obtained by statistical methods between stream function and dimensionless pressure (Exner function) is used to describe the balance relationship between rotational wind and mass field. In addition, the balance relationship between rotational wind and divergent wind is similarly described by dynamic balance operator (M) between stream function and velocity potential which is also obtained by statistical methods. In the calculation of the operator N, single-layer-related model and multilayer-related model were compared, and ultimately chose the former into the new formulation. Single-layer-related model was also chose to calculate operator M. Compared to the original scheme, the new formulation can avoid repeated interpolations along the vertical direction.
Statistical results show that, the explained variance of dimensionless pressure is primarily in the extratropics with the variance best explained below100hPa in this new formulation. And the explained velocity potential ratio has a maximum in the middle-and high-latitude near the surface. Results of randomization and single-observation experiments indicate that, in regions where geostrophic balance is appropriate, the new formulation behaves similarly to the old scheme. However, in regions where geostrophic balance is not appropriate, the new formulation could allow for a smooth decoupling of stream function and dimensionless pressure, while the old scheme cannot. Moreover, by adding the balance relationship between rotational wind and divergent wind, the new formulation could derive a more reasonable wind field in boundary layer. The results of analysis forecast cycle experiment show that, the new formulation could help improve the quality of the analysis fields of both the northern and southern hemispheres and could help improve short-term forecasting results of the East Asian region.
引文
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[3]Talagrand O. Assimilation of observations, an introduction. Met. Soc. Japan, 1997, Special Issue 75, 1B:191-209.
[4]Barker M D, W Huang, Y -R Guo, A Bourgeois,2003:A three-dimensional (3DVAR) data assimilation system for use with MM5. NCAR Tech. Note.2pp. [Obtained from the web site http://www.mmm.ucar.edu/3dvar/].
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[7]薛纪善,陈德辉,等.数值预报系统GRAPES的科学设计与应用.北京：科学出版社,2008,6：1-64.
[8]Panofsky H. Objective weather-map analysis. J. Appl. Meteor.1949,6:386-392.
[9]Kalnay E. Atmospheric modeling, data assimilation and predictability. Cambridge University Press.2003:136-204
[10]Bergthorsson P, B Doos, S Frykland, et al. Routine forecasting with the barotropics model. Tellus,1955,7:329-340.
[11]王跃山.客观分析和四维同化Ⅱ：客观分析的主要方法(1).气象科技,2011,1：1-11
[12]Lorenc A C. A global three-dimensional multivariate statistical interpolation scheme. Mon. Wea. Rev.,1981,109:701-721
[13]杨晓霞,沈桐立,徐文金,等.最优插值客观分析方法.南京气象学院学报,1991,14(4)：566-574.
[14]Parrish D F, J C Derber. The National Meteorological Center's Spectral Statistical Interpolation analysis system. Mon. Wea. Rev.,1992,120:1747-1763.
[15]Rabier F. Overview of global data assimilation developments in numerical weather prediction centres. Q. J. R. Meteorol. Soc.,2005,131:3215-3233.
[16]Lorenc A. Analysis methods for numerical weather prediction. Q. J. R. Meteorol. Soc.,1986,112:1177-1194.
[17]龚建东,邱崇践.区域四维变分资料同化的数值试验气象学报1999,57(2)：131-142.
[18]杨毅,邱崇践,龚建东,等.三维变分和物理初始化方法相结合同化多普勒雷达资料的试验研究.气象学报,2008,66(4)：479-488.
[19]薛谌彬.FY-2E云导风的定高误差及对同化的影响.南京信息工程大学硕士论文2010.
[20]Bannister R N. A review of forecast error covariance statistics in atmospheric variational data assimilation. I:Characteristics and measurements of forecast error covariances. Q. J. R. Meteorol Soc.,2008,134:1951-1970.
[21]Evensen G. Data Assimilation-the Ensemble Kalman Filter. Springer,2009: 38-44.
[22]Sakov P, G Evensen, Bertino L. Asynchronous data assimilation with the EnKF. Tellus,2010,62A:24-29
[23]Evensen G. Data Assimilation the ensemble kalman filter. Second Edition. Springer Dordrecht Heidelberg London New York.2009:27-45.
[24]Lorenc A. The potential of the ensemble Kalman filter for NWP:A comparison with 4D-Var. Q. J. R. Meteorol. Soc.,2003,129:3183-3203.
[25]Wang X, D Barker, C Snyder, et al. A hybrid ETKF-3DVAR data assimilation scheme for the WRF model. Part I:observing system simulation experiment. Mon. Wea. Rev.,2008,136:5116-5131.
[26]Fisher M. Background Error Covariance Modeling. ECMWF Seminar on "Recent Developments in Data Assimilation for Atmosphere and Ocean",2003: 45-63
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