集合变分资料同化关键技术及其并行算法研究
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摘要
地球大气系统是一个随时间演变的非线性混沌系统，其数值预报模式对初始条件非常敏感，这就要求准确的数值预报结果必须以高质量的模式初值为前提。近年来，能否获得满足质量要求的初始条件已经成为制约当前数值天气预报发展的瓶颈之一，因此为数值预报模式提供初始场的资料同化技术成为数值天气预报领域的关键研究问题。
资料同化系统是数值天气预报系统的一个重要组成部分，它通过数值模式将当前的观测信息与过去的观测信息融合起来，为预报模式提供尽可能精确的初始场。资料同化技术从二十世纪五六十年代开始发展起来，相继经历了逐步订正和最优插值客观分析、三维变分同化等阶段，目前国际上已有发达国家的气象业务中心实现了四维变分同化和集合滤波等方案。
当前的资料同化技术主要集中在变分同化与集合滤波方面。变分同化方法具有可方便增添附加动力约束条件、直接同化非常规观测资料等优点，而集合滤波方法则具有动态生成背景误差协方差、可获得分析扰动不确定性统计量等优点，因此将两类方法进行结合是今后一段时间内同化技术的重要研究方向。
本文针对变分同化与集合滤波的各自优缺点，对混合资料同化方法进行了重点探索和研究，以下是本文的主要工作和研究内容。
（1）本文首先针对粒子滤波的特点，提出了一种改进的粒子滤波资料同化方法。集合滤波中一类重要的方法是粒子滤波。粒子滤波不需要假定高斯分布、线性系统等条件，因此更能符合实际天气系统的的非高斯、非线性特性。研究粒子滤波方法，对于发展数值天气预报技术具有重要意义。考虑到模式系统多步积分之后才有观测资料的情况，该方法将同化过程分为非观测时刻同化与观测时刻同化。通过在非观测时刻引入“伪”观测、在观测时刻进行预重采样，该方法能较好地解决非线性系统的初值估计问题。应用到低维和高维混沌系统的实验结果显示，该方法使用极少的粒子数就能有效追踪系统状态的演化轨迹，且对模式误差和观测误差的适应范围比标准粒子滤波和标准集合卡尔曼滤波更为广泛。
（2）针对三维变分资料同化和粒子滤波方法各自的优点，提出了一种混合三维变分粒子滤波方法。通过迭代求解最小化代价函数，可以生成一个更好的状态后验分布。该方法将粒子集合划分为若干个子集合，每个子集合独立计算各自的“中心粒子”（包括分析期望值及其权重），然后通过加权平均得出全局分析期望值与分析误差协方差等统计量。为了避免标准粒子滤波中的随机采样步骤，提出了一种确定性采样策略，使用该策略生成新粒子集可以避免随机重采样方法导致的粒子多样性匮乏等问题。通过仿真实验发现，新的混合同化方法的性能优于标准集合卡尔曼滤波和标准粒子滤波，尤其是应用于高度非线性物理系统的资料同化时效果更为明显。
（3）针对目前三种主要的基于集合的三维变分资料同化方法进行了对比研究，证明了它们在理论上都等价于集合转换卡尔曼滤波（ETKF）。集合卡尔曼滤波是另一类重要的集合滤波方法，该方法近年来吸引了众多研究者的关注。三维变分资料同化通过迭代方法求解控制变量，易于添加非线性等物理约束，而集合卡尔曼滤波方法为三维变分资料同化提供流依赖（flow-dependent）的预报误差协方差，因此将三维变分资料同化与集合卡尔曼滤波进行结合是资料同化领域中的一项重要研究内容。文中讨论的几种主要的基于集合的三维变分资料同化方法均使用预报误差协方差替代原三维变分资料同化方法中静态的背景误差协方差，虽然在计算形式上各不相同，但理论分析表明，这几种方法得出的分析状态向量和分析集合扰动矩阵与ETKF是一致的。另外，基于PSAS算法，本文提出了一种区别于ETKF的新的集合变分混合同化方法EnPSAS，该方法在观测空间求解最小化代价函数，具有Hessian矩阵条件数较小和易于空间局部化等优点。
（4）针对四维变分资料同化和集合卡尔曼滤波的特点，本文提出了一种新的混合资料同化方法，称为四维集合转换局部卡尔曼滤波方法。基于集合的滤波方法具有一个共同的缺点，就是有限的集合成员数目必将引入额外的样本误差，而通过空间局部化操作和引入膨胀系数等方式则可以去除协方差矩阵中的伪相关和方差低估问题。该方法将空间局部化直接作用到预报集合扰动上，而非通常情况下完整的预报误差协方差矩阵，这样既扩展了分析增量的生成空间，同时又避免了基于集合的预报误差协方差矩阵的严重退化性。另外，该方法设计了生成分析集合的确定性策略，可以获得比随机扰动生成方式更好的后验分布。
（5）研究了四维集合转换卡尔曼滤波方法对于各种参数的敏感性。将该方法应用到Lorenz96系统，并与标准集合卡尔曼滤波进行了对比研究，测试和验证了该方法的性能。考察了不同参数对该方法性能的影响，包括集合数目、膨胀因子、相关距离尺度、模式误差方差、观测误差方差、观测时间间隔、观测数量等。实验结果表明，该方法具有较强的鲁棒性，性能明显优于标准集合卡尔曼滤波，尤其是在集合规模较小、相关距离尺度较大、观测数量较少时，其先进性更为明显。
（6）将四维集合转换卡尔曼滤波方法简化为三维集合转换卡尔曼滤波，研究其并行性并设计了相关并行算法。由于三维集合转换卡尔曼滤波是基于集合设计的资料同化方法，因此保留了集合滤波易于并行的特点。每个集合成员可以对应一个独立计算节点，且可通过将扩展集合扰动矩阵及其在观测空间上的投影矩阵分解成若干个小矩阵的方式，将每个小矩阵在独立计算节点上进行相关计算和操作，这样可以较好地实现负载均衡。根据三维集合转换卡尔曼滤波的算法特性，重点针对预报集合扰动矩阵、观测集合扰动矩阵、分析集合扰动矩阵的计算以及最小化迭代过程进行了并行化设计。将并行化后的三维集合转换卡尔曼滤波方法应用到48个独立CPU的并行计算机上进行测试，结果显示，该并行方法具有较高的并行加速比和良好的并行效率。
The atmosphere system is a nonlinear and chaos system which evolves over timeand its numerical prediction model is sensitive to the initial condition, therefore, ahigh-quality initial condition is needed for an accurate prediction. Recently, how toobtain a good-enough initial state is a bottleneck problem in Numerical WeatherPrediction (NWP), thus, the data assimilation (DA) technique which supplies the initialcondition for NWP becomes a key tool.
DA is an important component of NWP, since it provides the initial values for theforecast system as accurately as possible. The idea of DA is to combine the informationfrom past observations (as encapsulated in a short term model forecasts) along withcurrent observations in an optimal way. Since the1950-1960s, data assimilation hasexperienced several stages, such as the method of successive corrections, the optimalinterpolation, three-dimensional variational data assimilation (3D-Var), etc. Nowadays,four-dimensional variational data assimilation (4D-Var) and ensemble filtering havebeen already implemented in some operational NWP centers.
There are two major methods for current data assimilation techniques, such asvariational data assimilation and ensemble filtering. One of the merits of the variationaldata assimilation is that additional constraints can be introduced easily, while theensemble filtering has the advantages of dynamically generating the background errorcovariance and naturally obtaining the uncertainty of the analysis. Thus, combiningthese two methods is an important research direction of data assimilation at present andin the future.
In this paper, hybrid data assimilation methods are explored and studied, accordingto their respective advantages and disadvantages. The main work in this paper is shownas follows.
In the first part, an improved particle filter DA method is proposed. Particle filter isone kind of the ensemble filtering methods. Without assumptions of Gaussiandistribution and linearity, particle filtering can approximate the features of the realweather system better than other filtering methods. It is significant to study particlefiltering methods for the next generation of data assimilation techniques. Consideringthe situation that observations are obtained after several model integrations, this methoddivided the data assimilation process into two stages, data assimilation atnon-observation times and data assimilation at observation times. By introducing―pseudo‖observations at non-observation times and resampling in advance atobservation times, this method can solve the initial value problem of nonlinear systemswell. After being applied to a low-dimension system and a high-dimension system, theresults demonstrates that the system state can be well traced by using only a few particles, and the range for the model error and the observation error where the methodis credible is wider than the standard particle filter and the standard Kalman filter.
For the respective merits of the3D-Var and the particle filtering, a hybridthree-dimensional variational particle filtering was presented. By solving a minimumcost function iteratively, this method can obtain a good posterior distribution. Theparticle ensemble is divided into several sub-ensembles, and an expected analysis andits weight are computed for each sub-ensemble. Afterwards, the global analysis and itserror covariance are obtained by using a weighted average. A deterministic resamplingstrategy was proposed, which can void the problem of lack of diversity resulting fromstochastically resampling. Simulation results show that the new hybrid method performsbetter than the standard ensemble Kalman filtering and the standard particle filtering,especially applied to highly nonlinear systems.
Another important ensemble filtering method is the so-called ensemble Kalmanfiltering (EnKF), which attracts much attention in recent years. The3D-Var seeks theanalysis of the control variables in an iterative way, and can introduce extra nonlinearconstraints expediently. Ensemble Kalman filtering can supply a flow-dependentforecast error covariance for the3D-Var; therefore, combining the3D-Var and theEnKF is one of the significant research directions. This paper compared a fewensemble-based3D-Var methods, and proved that they are all theoretically equivalent tothe ensemble transform Kalman filtering (ETKF). Furthermore, a new method calledensemble physical space analysis system (EnPSAS) was proposed, where the Hessianmatrix of the cost function has a small condition number and the space localization canbe introduced easily.
There is a common disadvantage among the ensemble filters that limited ensemblesize will introduce extra sampling errors. However, through applying spatial localizationor introducing an inflation factor, the spurious correlations in the error covarianceproduced from samples can be reduced. Therefore, a new hybrid4D-Var and EnKFmethod was presented, which applied the spatial localization directly into the forecastensemble anomalies, not the full forecast error covariance. In this way, the generatingspace of the analysis increment is expanded; meanwhile, the severe degeneration of theensemble-based forecast error covariance matrix can be avoided. Moreover, adeterministic strategy was designed, which can obtain a better posterior distribution thanthe stochastic perturbation way.
The previous hybrid method (called4DETLKF) was applied to the Lorenz96system, and its performance was investigated in comparison of the standard EnKF. Thesensitivity to different parameters was investigated, such as ensemble size, inflationfactor, correlation length scale, observation error variance, observation time interval,and observation number, etc. The experimental results demonstrate that, this newmethod is robust, and performs significantly better than the standard EnKF, especially with a relatively small ensemble size, a large correlation length scale or a smallobservation number.
The3DETLKF, the reduced version of the4DETLKF, is applied to the ShallowWater model, its parallel characteristics are studied and the associated parallel algorithmis presented. Since the3DETLKF is ensemble-based, it can be parallelized easier thanthe primal variational methods. Each ensemble member can correspond to an individualcomputing node, and a good load balancing can be achieved through dividing theextended ensemble perturbation matrix into a number of sub-matrices and dealing withthem on the individual computing nodes. The most work is focused on theparallelization of the forecast ensemble perturbation matrix, the observation ensembleperturbation matrix, the analysis ensemble perturbation matrix and the minimization ofthe cost function. By applying the parallelized3DETLKF method to a parallel computerwith48independent CPUs, the result shows that a high parallel speedup ratio and agood parallel efficiency can be achieved.
资料同化系统是数值天气预报系统的一个重要组成部分，它通过数值模式将当前的观测信息与过去的观测信息融合起来，为预报模式提供尽可能精确的初始场。资料同化技术从二十世纪五六十年代开始发展起来，相继经历了逐步订正和最优插值客观分析、三维变分同化等阶段，目前国际上已有发达国家的气象业务中心实现了四维变分同化和集合滤波等方案。
当前的资料同化技术主要集中在变分同化与集合滤波方面。变分同化方法具有可方便增添附加动力约束条件、直接同化非常规观测资料等优点，而集合滤波方法则具有动态生成背景误差协方差、可获得分析扰动不确定性统计量等优点，因此将两类方法进行结合是今后一段时间内同化技术的重要研究方向。
本文针对变分同化与集合滤波的各自优缺点，对混合资料同化方法进行了重点探索和研究，以下是本文的主要工作和研究内容。
（1）本文首先针对粒子滤波的特点，提出了一种改进的粒子滤波资料同化方法。集合滤波中一类重要的方法是粒子滤波。粒子滤波不需要假定高斯分布、线性系统等条件，因此更能符合实际天气系统的的非高斯、非线性特性。研究粒子滤波方法，对于发展数值天气预报技术具有重要意义。考虑到模式系统多步积分之后才有观测资料的情况，该方法将同化过程分为非观测时刻同化与观测时刻同化。通过在非观测时刻引入“伪”观测、在观测时刻进行预重采样，该方法能较好地解决非线性系统的初值估计问题。应用到低维和高维混沌系统的实验结果显示，该方法使用极少的粒子数就能有效追踪系统状态的演化轨迹，且对模式误差和观测误差的适应范围比标准粒子滤波和标准集合卡尔曼滤波更为广泛。
（2）针对三维变分资料同化和粒子滤波方法各自的优点，提出了一种混合三维变分粒子滤波方法。通过迭代求解最小化代价函数，可以生成一个更好的状态后验分布。该方法将粒子集合划分为若干个子集合，每个子集合独立计算各自的“中心粒子”（包括分析期望值及其权重），然后通过加权平均得出全局分析期望值与分析误差协方差等统计量。为了避免标准粒子滤波中的随机采样步骤，提出了一种确定性采样策略，使用该策略生成新粒子集可以避免随机重采样方法导致的粒子多样性匮乏等问题。通过仿真实验发现，新的混合同化方法的性能优于标准集合卡尔曼滤波和标准粒子滤波，尤其是应用于高度非线性物理系统的资料同化时效果更为明显。
（3）针对目前三种主要的基于集合的三维变分资料同化方法进行了对比研究，证明了它们在理论上都等价于集合转换卡尔曼滤波（ETKF）。集合卡尔曼滤波是另一类重要的集合滤波方法，该方法近年来吸引了众多研究者的关注。三维变分资料同化通过迭代方法求解控制变量，易于添加非线性等物理约束，而集合卡尔曼滤波方法为三维变分资料同化提供流依赖（flow-dependent）的预报误差协方差，因此将三维变分资料同化与集合卡尔曼滤波进行结合是资料同化领域中的一项重要研究内容。文中讨论的几种主要的基于集合的三维变分资料同化方法均使用预报误差协方差替代原三维变分资料同化方法中静态的背景误差协方差，虽然在计算形式上各不相同，但理论分析表明，这几种方法得出的分析状态向量和分析集合扰动矩阵与ETKF是一致的。另外，基于PSAS算法，本文提出了一种区别于ETKF的新的集合变分混合同化方法EnPSAS，该方法在观测空间求解最小化代价函数，具有Hessian矩阵条件数较小和易于空间局部化等优点。
（4）针对四维变分资料同化和集合卡尔曼滤波的特点，本文提出了一种新的混合资料同化方法，称为四维集合转换局部卡尔曼滤波方法。基于集合的滤波方法具有一个共同的缺点，就是有限的集合成员数目必将引入额外的样本误差，而通过空间局部化操作和引入膨胀系数等方式则可以去除协方差矩阵中的伪相关和方差低估问题。该方法将空间局部化直接作用到预报集合扰动上，而非通常情况下完整的预报误差协方差矩阵，这样既扩展了分析增量的生成空间，同时又避免了基于集合的预报误差协方差矩阵的严重退化性。另外，该方法设计了生成分析集合的确定性策略，可以获得比随机扰动生成方式更好的后验分布。
（5）研究了四维集合转换卡尔曼滤波方法对于各种参数的敏感性。将该方法应用到Lorenz96系统，并与标准集合卡尔曼滤波进行了对比研究，测试和验证了该方法的性能。考察了不同参数对该方法性能的影响，包括集合数目、膨胀因子、相关距离尺度、模式误差方差、观测误差方差、观测时间间隔、观测数量等。实验结果表明，该方法具有较强的鲁棒性，性能明显优于标准集合卡尔曼滤波，尤其是在集合规模较小、相关距离尺度较大、观测数量较少时，其先进性更为明显。
（6）将四维集合转换卡尔曼滤波方法简化为三维集合转换卡尔曼滤波，研究其并行性并设计了相关并行算法。由于三维集合转换卡尔曼滤波是基于集合设计的资料同化方法，因此保留了集合滤波易于并行的特点。每个集合成员可以对应一个独立计算节点，且可通过将扩展集合扰动矩阵及其在观测空间上的投影矩阵分解成若干个小矩阵的方式，将每个小矩阵在独立计算节点上进行相关计算和操作，这样可以较好地实现负载均衡。根据三维集合转换卡尔曼滤波的算法特性，重点针对预报集合扰动矩阵、观测集合扰动矩阵、分析集合扰动矩阵的计算以及最小化迭代过程进行了并行化设计。将并行化后的三维集合转换卡尔曼滤波方法应用到48个独立CPU的并行计算机上进行测试，结果显示，该并行方法具有较高的并行加速比和良好的并行效率。
The atmosphere system is a nonlinear and chaos system which evolves over timeand its numerical prediction model is sensitive to the initial condition, therefore, ahigh-quality initial condition is needed for an accurate prediction. Recently, how toobtain a good-enough initial state is a bottleneck problem in Numerical WeatherPrediction (NWP), thus, the data assimilation (DA) technique which supplies the initialcondition for NWP becomes a key tool.
DA is an important component of NWP, since it provides the initial values for theforecast system as accurately as possible. The idea of DA is to combine the informationfrom past observations (as encapsulated in a short term model forecasts) along withcurrent observations in an optimal way. Since the1950-1960s, data assimilation hasexperienced several stages, such as the method of successive corrections, the optimalinterpolation, three-dimensional variational data assimilation (3D-Var), etc. Nowadays,four-dimensional variational data assimilation (4D-Var) and ensemble filtering havebeen already implemented in some operational NWP centers.
There are two major methods for current data assimilation techniques, such asvariational data assimilation and ensemble filtering. One of the merits of the variationaldata assimilation is that additional constraints can be introduced easily, while theensemble filtering has the advantages of dynamically generating the background errorcovariance and naturally obtaining the uncertainty of the analysis. Thus, combiningthese two methods is an important research direction of data assimilation at present andin the future.
In this paper, hybrid data assimilation methods are explored and studied, accordingto their respective advantages and disadvantages. The main work in this paper is shownas follows.
In the first part, an improved particle filter DA method is proposed. Particle filter isone kind of the ensemble filtering methods. Without assumptions of Gaussiandistribution and linearity, particle filtering can approximate the features of the realweather system better than other filtering methods. It is significant to study particlefiltering methods for the next generation of data assimilation techniques. Consideringthe situation that observations are obtained after several model integrations, this methoddivided the data assimilation process into two stages, data assimilation atnon-observation times and data assimilation at observation times. By introducing―pseudo‖observations at non-observation times and resampling in advance atobservation times, this method can solve the initial value problem of nonlinear systemswell. After being applied to a low-dimension system and a high-dimension system, theresults demonstrates that the system state can be well traced by using only a few particles, and the range for the model error and the observation error where the methodis credible is wider than the standard particle filter and the standard Kalman filter.
For the respective merits of the3D-Var and the particle filtering, a hybridthree-dimensional variational particle filtering was presented. By solving a minimumcost function iteratively, this method can obtain a good posterior distribution. Theparticle ensemble is divided into several sub-ensembles, and an expected analysis andits weight are computed for each sub-ensemble. Afterwards, the global analysis and itserror covariance are obtained by using a weighted average. A deterministic resamplingstrategy was proposed, which can void the problem of lack of diversity resulting fromstochastically resampling. Simulation results show that the new hybrid method performsbetter than the standard ensemble Kalman filtering and the standard particle filtering,especially applied to highly nonlinear systems.
Another important ensemble filtering method is the so-called ensemble Kalmanfiltering (EnKF), which attracts much attention in recent years. The3D-Var seeks theanalysis of the control variables in an iterative way, and can introduce extra nonlinearconstraints expediently. Ensemble Kalman filtering can supply a flow-dependentforecast error covariance for the3D-Var; therefore, combining the3D-Var and theEnKF is one of the significant research directions. This paper compared a fewensemble-based3D-Var methods, and proved that they are all theoretically equivalent tothe ensemble transform Kalman filtering (ETKF). Furthermore, a new method calledensemble physical space analysis system (EnPSAS) was proposed, where the Hessianmatrix of the cost function has a small condition number and the space localization canbe introduced easily.
There is a common disadvantage among the ensemble filters that limited ensemblesize will introduce extra sampling errors. However, through applying spatial localizationor introducing an inflation factor, the spurious correlations in the error covarianceproduced from samples can be reduced. Therefore, a new hybrid4D-Var and EnKFmethod was presented, which applied the spatial localization directly into the forecastensemble anomalies, not the full forecast error covariance. In this way, the generatingspace of the analysis increment is expanded; meanwhile, the severe degeneration of theensemble-based forecast error covariance matrix can be avoided. Moreover, adeterministic strategy was designed, which can obtain a better posterior distribution thanthe stochastic perturbation way.
The previous hybrid method (called4DETLKF) was applied to the Lorenz96system, and its performance was investigated in comparison of the standard EnKF. Thesensitivity to different parameters was investigated, such as ensemble size, inflationfactor, correlation length scale, observation error variance, observation time interval,and observation number, etc. The experimental results demonstrate that, this newmethod is robust, and performs significantly better than the standard EnKF, especially with a relatively small ensemble size, a large correlation length scale or a smallobservation number.
The3DETLKF, the reduced version of the4DETLKF, is applied to the ShallowWater model, its parallel characteristics are studied and the associated parallel algorithmis presented. Since the3DETLKF is ensemble-based, it can be parallelized easier thanthe primal variational methods. Each ensemble member can correspond to an individualcomputing node, and a good load balancing can be achieved through dividing theextended ensemble perturbation matrix into a number of sub-matrices and dealing withthem on the individual computing nodes. The most work is focused on theparallelization of the forecast ensemble perturbation matrix, the observation ensembleperturbation matrix, the analysis ensemble perturbation matrix and the minimization ofthe cost function. By applying the parallelized3DETLKF method to a parallel computerwith48independent CPUs, the result shows that a high parallel speedup ratio and agood parallel efficiency can be achieved.
引文
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