集合降维变分方法及其在全球谱模式T106中的应用研究
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摘要
为了解决资料同化中“观测量小于模式变量数”这一欠定性问题,Qiu和Chou从在大气模式吸引子上求解资料同化问题的思想出发,提出了一种基于奇异值分解(SVD)的四维资料同化方法。这一方法建议利用奇异值分解技术从一个模式输出的样本集合中产生支撑起模式大气吸引子的正交基向量,将观测投影到这些基向量张成的空间,通过变分方法得到分析场。这一方法经过不断发展完善形成了一种基于集合的降维变分同化方法(Eensemble-based ReducedDimension Variational Assimilation Method,简称ERDVar)。ERDVar方法把同化问题从高维空间变换到低维空间来求解,这种方法和四维变分(4DVar)一样可以在一个同化循环里使用多个观测,又不需要计算切线性模式和伴随模式,能够减小程序工作量;它不像三维变分(3DVar)、四维变分(4DVar)那样采用固定不变、各向均匀同性的预报误差协方差,而是通过预报样本来估计预报误差协方差;但是和一般的集合卡尔曼滤波不同,ERDVar是在变分框架下的同化方法,更容易加入其他的物理约束,更适合同化遥感资料。另外,和集合卡尔曼滤波不同的是,ERDVar每次循环都要由新的初始扰动场出发积分模式产生样本,由这样的样本得到的误差协方差的流依赖性介于EnKF和3DVar之间。
本文对ERDVar方法作了进一步的研究,特别集中研究了影响该方法同化效果的两个基本问题:(1)如何产生好的初始扰动场;(2)如何减少用来生成大量预报样本花费的计算时间。本文首次将ERDVar方法应用到全球谱模式T106上,利用模式资料进行了同化探空资料和卫星资料的试验;本文还发展了一种分区同化的ERDVar方案,一定程度上解决了全球同化中的局地化问题。
通过这些研究得到以下结果:
(1)运用浅水方程模式和模式资料进行的同化试验表明,初始扰动场的生成方式对预报结果有重要的影响。通过对3种不同扰动方法的比较看到,Evensen给出的方法得到的同化效果最好。通过比较不同的特征尺度的扰动场的同化效果得到结论:在所有模式变量都有观测时,同化精度和扰动特征尺度的关系较小,但是观测变量不完全时同化精度和扰动特征尺度的关系明显,选择和预报误差特征尺度相近的扰动尺度有利于得到好的同化效果。采用独立的初始扰动和满足地转关系的初始扰动得到的同化效果大体相当。
(2)利用多次预报/同化循环产生的预报时间序列,统计分析预报误差的统计特征,并将它们和由扰动预报集合估计的预报误差的统计特征比较,得到结论:预报误差协方差随位置不同有所不同,随初始扰动尺度不同也有所不同,但是变量h的自相关结构都接近于高斯函数分布,变量u和v的自相关结构都接近于按地转关系得到的相关结构。不同变量间的相关也大体符合按地转关系得到的相关结构,但是由扰动预报样本得到的相关结构和由预报时间序列统计的相关结构相差较大,不同点和不同初始扰动得到的结果相差也较大。
(3)对不同尺度的初始扰动产生的预报样本分析发现,初始扰动尺度过小得到的预报扰动不容易发展,尺度和实际的预报误差尺度不匹配,这应该是造成同化效果不好的原因。
(4)通过同化试验说明,ERDVar方法的同化效果对样本容量是敏感的。样本数量对同化的影响来自于两方面:过少的样本得到的预报误差协方差不可靠;如果样本过少,在将变量按照基向量展开时有较大的截断误差。通过对预报样本的统计分析得出,截断误差是造成同化效果对样本容量敏感的主要原因。
(5)通过对预报样本的统计分析看到,ERDVar方法所用的预报集合估计的误差协方差的流依赖性比一般的EnKF的误差协方差弱,随时间变化较慢,为减少计算时间提供了可能。基于这一研究,提出了几种减少计算时间的方案,这些方案的基本思想是尽量利用较大容量的预报集合,但是减少预报样本更新频率,或者采取固定样本和动态样本相结合的方式。试验表明,这样的方法可以大大减少计算时间,而且可以提高预报精度,这对将ERDVar方法用于业务是很有意义的。
(6)设计了ERDVar方法与T106全球模式相结合的同化流程,用于同化探空和卫星资料;对初始扰动产生方法进行了分析,比较了通过扰动Fourier展开系数得到随机扰动场的方法和NMC扰动方法,分析表明,NMC方法得到的扰动场和模式动力协调,在预报中不容易衰减,随机扰动场的离散度则过小,不利于同化;应用这两种扰动方法分别进行了敏感性试验,研究样本个数、截断阶数、观测间隔、观测误差对同化结果的影响。
(7)针对全球同化的特点,提出了分区同化的RERDVar方案,进行了同化探空资料的分区试验,初步试验表明分区同化大部分区域的效果比全球同化要好。说明具备局地化功能的RERDVar方案在T106模式同化中有很大优势。试验还表明,同时加入探空资料和卫星资料比只同化卫星资料效果好。
ERDVar (Ensemble-based reduced dimension Variational assimilation method) , a new and promising data assimilation method, which was initially proposed by Qiu and Chou, has continuously developed and completed. The method tries to seek a solution in the attractor of model atmosphere to reduce the under-determination of the data assimilation. In this method, the SVD was used to extract the leading singular vectors from an ensemble of 4D perturbation fields produced by the model and the analysis is obtained by projecting actual observation data into a space spanned by the base vectors, as getting the best expanding coefficients through minimizing a cost function. As 4DVar does, the ERDVar is able to assimilate the observations at multi-time levels in an analysis cycle but avoid the heavy labor on coding the tangent and adjoint model in 4DVAR. As conventional EnKF does, the ERDVar estimates the forecast error covariance matrix from the forecast ensemble but it is belong to a variational assimilation method. The method is especially more suitable than conventional EnKF for assimilating various remote sense data by considering additional physical restricts in the cost function. Besides, the ERDVar requests to produce the forecast ensemble by integrating model from a set of initial disturbed fields in every analysis cycle, which is different from EnKF. The "flow- dependent" characteristic of the background error covariance estimated by such ensemble is intermediate between EnKF and 3DVar, which maybe is a problem but also can be a advantage for the weak dependent on model overcoming the systematic errors as the model with it.
Based on the previous studies on ERDVar method, we find that there are still at least two questions must be solved before its application for real operation: the first one is the technique of producing initial perturbation, which is the common headache for ensemble based method, the other one is making a balance between improving the analysis accuracy of the assimilation and the computational efficiency by reducing the frequency of perturbation updating. In this dissertation we make a research on the above problems first. Then we put ERDVar into an application with the global spectral model T106, absorbing the simulative radiosonde data and satellite data. Until recently it is unusual in assimilating satellite data in global models with ensemble method. We develop a regional assimilation method of ERDVar to resolve the localization of background error covariance in global assimilation.
The first part in this dissertation is about the perturbation research: (1) A series of experiments on ERDVar were performed with a shallow water model with three different perturbation methods (A. Monte Carlo method, B. Monte Carlo method improved by Shao, C. Evensen perturbation method). Perturbations produced by A and C are statistically uniform and isotropic with a gauss probability distribution function, while perturbation produced by B just appear approximately to that. The result of assimilation analysis shows the C method as the best one. (2) Then we make assimilation analysis using perturbation C with different de-correlation length. It is shown that the analysis accuracy is sensitive to de-correlation length when the observation types is incomplete (only height and wind), which is opposite with the complete observation types. It is benefit for assimilation by choosing proper de-correlation length of perturbation which is similar to that of forecast errors; (3) A conclusion based on the model forecast errors time-series shows that the forecast error covariance changes as location or the initial de-correlation length varies, the variance of h (or u, v) is similar to gauss function distribution (or the correlation structure derived from the geostrophic relationship). And the correlation of different variables fit the correlation structure derived from the geostrophic relationship. But the correlation structure of the perturbation forecast samples is quite different from that of forecast time series, and greatly varies with respect to different location and different initial perturbation. (4) We discover that from experiments the bad assimilation analysis comes with too small de-correlation length of the initial perturbation compared with that of the real forecast errors, also without an easily developing perturbation. The probable reason is the filter effect of the model on the high-frequency noise.
Secondly, we make a research on the possibility of reducing the updating frequency of perturbation. (1) In every assimilation cycle of ERDVar, forecast perturbation sample need to be produced from integrating model. Usually an ensemble with too small size cause a bad analysis, maybe because the unreliable forecast error covariance or the truncation errors when the observation innovations are expand on the base vectors with too small size. We verify the truncation error as the leading reason through the experiments. (2) We verify that the error covariance based on forecast ensemble in ERDVar, changing slowly as time varies, shows less 'flow-dependent' than that of conventional EnKF methods, which makes it possible to reduce the updating frequency of perturbation. (3) Based on the research above, we propose several methods to improve the computational efficiency. The main idea is using forecast ensemble with large size as possible as we can, but reducing the updating frequency, or using fixed ensemble mixed with dynamic ensemble. The results of these experiments are effective on both reducing computational cost greatly and improving the analysis accuracy, which is very meaningful to apply ERDVar to real operation.
Thirdly, a frame of ERDVar global assimilation with T106, the global medium-range numerical weather prediction spectral model, is set up and applied to absorb radiosonde data and satellite data. We also make a research on ensemble perturbation methods, here for Fourier random perturbation and NMC perturbation in global assimilation. The first leading singular vector of the Fourier random perturbation forecast ensemble can cover the 99% cumulative variance contribution; but is terribly filtered especially for high-order perturbation. While the NMC ensemble perturbation forecast is quite consisitent with the model dynamics, and there is no filter attenuation by the model in forecast process. Then we perform ERDVar Observing System Simulation Experiments (OSSE) with these perturbation methods, to make clear the impact of ensemble size, truncated order, observation interval, observation error on the assimilation analysis. Then in order to realize the covariance localization in ERDVar global assimilation, we develop a regional ERDVar method, and the preliminary study of OSSE with radiosonde data shows its advantage over full-area ERDVar. Then we design the OSSE with satellite data and radiosonde data, the result shows that assimilation of both the two types of data is better than that of satellite data alone.
本文对ERDVar方法作了进一步的研究,特别集中研究了影响该方法同化效果的两个基本问题:(1)如何产生好的初始扰动场;(2)如何减少用来生成大量预报样本花费的计算时间。本文首次将ERDVar方法应用到全球谱模式T106上,利用模式资料进行了同化探空资料和卫星资料的试验;本文还发展了一种分区同化的ERDVar方案,一定程度上解决了全球同化中的局地化问题。
通过这些研究得到以下结果:
(1)运用浅水方程模式和模式资料进行的同化试验表明,初始扰动场的生成方式对预报结果有重要的影响。通过对3种不同扰动方法的比较看到,Evensen给出的方法得到的同化效果最好。通过比较不同的特征尺度的扰动场的同化效果得到结论:在所有模式变量都有观测时,同化精度和扰动特征尺度的关系较小,但是观测变量不完全时同化精度和扰动特征尺度的关系明显,选择和预报误差特征尺度相近的扰动尺度有利于得到好的同化效果。采用独立的初始扰动和满足地转关系的初始扰动得到的同化效果大体相当。
(2)利用多次预报/同化循环产生的预报时间序列,统计分析预报误差的统计特征,并将它们和由扰动预报集合估计的预报误差的统计特征比较,得到结论:预报误差协方差随位置不同有所不同,随初始扰动尺度不同也有所不同,但是变量h的自相关结构都接近于高斯函数分布,变量u和v的自相关结构都接近于按地转关系得到的相关结构。不同变量间的相关也大体符合按地转关系得到的相关结构,但是由扰动预报样本得到的相关结构和由预报时间序列统计的相关结构相差较大,不同点和不同初始扰动得到的结果相差也较大。
(3)对不同尺度的初始扰动产生的预报样本分析发现,初始扰动尺度过小得到的预报扰动不容易发展,尺度和实际的预报误差尺度不匹配,这应该是造成同化效果不好的原因。
(4)通过同化试验说明,ERDVar方法的同化效果对样本容量是敏感的。样本数量对同化的影响来自于两方面:过少的样本得到的预报误差协方差不可靠;如果样本过少,在将变量按照基向量展开时有较大的截断误差。通过对预报样本的统计分析得出,截断误差是造成同化效果对样本容量敏感的主要原因。
(5)通过对预报样本的统计分析看到,ERDVar方法所用的预报集合估计的误差协方差的流依赖性比一般的EnKF的误差协方差弱,随时间变化较慢,为减少计算时间提供了可能。基于这一研究,提出了几种减少计算时间的方案,这些方案的基本思想是尽量利用较大容量的预报集合,但是减少预报样本更新频率,或者采取固定样本和动态样本相结合的方式。试验表明,这样的方法可以大大减少计算时间,而且可以提高预报精度,这对将ERDVar方法用于业务是很有意义的。
(6)设计了ERDVar方法与T106全球模式相结合的同化流程,用于同化探空和卫星资料;对初始扰动产生方法进行了分析,比较了通过扰动Fourier展开系数得到随机扰动场的方法和NMC扰动方法,分析表明,NMC方法得到的扰动场和模式动力协调,在预报中不容易衰减,随机扰动场的离散度则过小,不利于同化;应用这两种扰动方法分别进行了敏感性试验,研究样本个数、截断阶数、观测间隔、观测误差对同化结果的影响。
(7)针对全球同化的特点,提出了分区同化的RERDVar方案,进行了同化探空资料的分区试验,初步试验表明分区同化大部分区域的效果比全球同化要好。说明具备局地化功能的RERDVar方案在T106模式同化中有很大优势。试验还表明,同时加入探空资料和卫星资料比只同化卫星资料效果好。
ERDVar (Ensemble-based reduced dimension Variational assimilation method) , a new and promising data assimilation method, which was initially proposed by Qiu and Chou, has continuously developed and completed. The method tries to seek a solution in the attractor of model atmosphere to reduce the under-determination of the data assimilation. In this method, the SVD was used to extract the leading singular vectors from an ensemble of 4D perturbation fields produced by the model and the analysis is obtained by projecting actual observation data into a space spanned by the base vectors, as getting the best expanding coefficients through minimizing a cost function. As 4DVar does, the ERDVar is able to assimilate the observations at multi-time levels in an analysis cycle but avoid the heavy labor on coding the tangent and adjoint model in 4DVAR. As conventional EnKF does, the ERDVar estimates the forecast error covariance matrix from the forecast ensemble but it is belong to a variational assimilation method. The method is especially more suitable than conventional EnKF for assimilating various remote sense data by considering additional physical restricts in the cost function. Besides, the ERDVar requests to produce the forecast ensemble by integrating model from a set of initial disturbed fields in every analysis cycle, which is different from EnKF. The "flow- dependent" characteristic of the background error covariance estimated by such ensemble is intermediate between EnKF and 3DVar, which maybe is a problem but also can be a advantage for the weak dependent on model overcoming the systematic errors as the model with it.
Based on the previous studies on ERDVar method, we find that there are still at least two questions must be solved before its application for real operation: the first one is the technique of producing initial perturbation, which is the common headache for ensemble based method, the other one is making a balance between improving the analysis accuracy of the assimilation and the computational efficiency by reducing the frequency of perturbation updating. In this dissertation we make a research on the above problems first. Then we put ERDVar into an application with the global spectral model T106, absorbing the simulative radiosonde data and satellite data. Until recently it is unusual in assimilating satellite data in global models with ensemble method. We develop a regional assimilation method of ERDVar to resolve the localization of background error covariance in global assimilation.
The first part in this dissertation is about the perturbation research: (1) A series of experiments on ERDVar were performed with a shallow water model with three different perturbation methods (A. Monte Carlo method, B. Monte Carlo method improved by Shao, C. Evensen perturbation method). Perturbations produced by A and C are statistically uniform and isotropic with a gauss probability distribution function, while perturbation produced by B just appear approximately to that. The result of assimilation analysis shows the C method as the best one. (2) Then we make assimilation analysis using perturbation C with different de-correlation length. It is shown that the analysis accuracy is sensitive to de-correlation length when the observation types is incomplete (only height and wind), which is opposite with the complete observation types. It is benefit for assimilation by choosing proper de-correlation length of perturbation which is similar to that of forecast errors; (3) A conclusion based on the model forecast errors time-series shows that the forecast error covariance changes as location or the initial de-correlation length varies, the variance of h (or u, v) is similar to gauss function distribution (or the correlation structure derived from the geostrophic relationship). And the correlation of different variables fit the correlation structure derived from the geostrophic relationship. But the correlation structure of the perturbation forecast samples is quite different from that of forecast time series, and greatly varies with respect to different location and different initial perturbation. (4) We discover that from experiments the bad assimilation analysis comes with too small de-correlation length of the initial perturbation compared with that of the real forecast errors, also without an easily developing perturbation. The probable reason is the filter effect of the model on the high-frequency noise.
Secondly, we make a research on the possibility of reducing the updating frequency of perturbation. (1) In every assimilation cycle of ERDVar, forecast perturbation sample need to be produced from integrating model. Usually an ensemble with too small size cause a bad analysis, maybe because the unreliable forecast error covariance or the truncation errors when the observation innovations are expand on the base vectors with too small size. We verify the truncation error as the leading reason through the experiments. (2) We verify that the error covariance based on forecast ensemble in ERDVar, changing slowly as time varies, shows less 'flow-dependent' than that of conventional EnKF methods, which makes it possible to reduce the updating frequency of perturbation. (3) Based on the research above, we propose several methods to improve the computational efficiency. The main idea is using forecast ensemble with large size as possible as we can, but reducing the updating frequency, or using fixed ensemble mixed with dynamic ensemble. The results of these experiments are effective on both reducing computational cost greatly and improving the analysis accuracy, which is very meaningful to apply ERDVar to real operation.
Thirdly, a frame of ERDVar global assimilation with T106, the global medium-range numerical weather prediction spectral model, is set up and applied to absorb radiosonde data and satellite data. We also make a research on ensemble perturbation methods, here for Fourier random perturbation and NMC perturbation in global assimilation. The first leading singular vector of the Fourier random perturbation forecast ensemble can cover the 99% cumulative variance contribution; but is terribly filtered especially for high-order perturbation. While the NMC ensemble perturbation forecast is quite consisitent with the model dynamics, and there is no filter attenuation by the model in forecast process. Then we perform ERDVar Observing System Simulation Experiments (OSSE) with these perturbation methods, to make clear the impact of ensemble size, truncated order, observation interval, observation error on the assimilation analysis. Then in order to realize the covariance localization in ERDVar global assimilation, we develop a regional ERDVar method, and the preliminary study of OSSE with radiosonde data shows its advantage over full-area ERDVar. Then we design the OSSE with satellite data and radiosonde data, the result shows that assimilation of both the two types of data is better than that of satellite data alone.
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