四维变分同化关键技术研究与并行计算
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摘要
由于初始场在数值天气预报中的重要性以及被精确确定的困难性,先进的气象资料同化方法已经成为提高数值天气预报效果的核心技术之一。本文以解决四维变分同化(4D-Var)业务化关键技术为目标,综合运用大气科学、计算数学和计算机应用技术等方面的知识,对制约四维变分同化效果和计算性能的关键算法和并行计算技术进行了研究,并实现了一个基于增量公式的多分辨率四维变分同化试验系统YH4DVAR。论文完成的主要工作和创新成果如下:
一、研究了四维变分同化基本原理和实现技术,对四维变分同化算法进行了理论推导和分析。针对增量方法中内循环最优化计算量大,容易丢失中小尺度信息等缺点,提出了多分辨率多增量四维变分同化实现方法。多增量方法可以加快内循环最优化迭代的收敛速度,减少计算量,同时保留不同尺度的大气信息,从而能够获得更高分辨率的分析增量和更准确的分析场。针对一般控制变量变换中背景误差协方差水平和垂直相关相分离的简化处理,构造了基于球面小波变换的控制变量转换新算子,通过小波尺度实现了水平和垂直相关的依赖。
二、对背景误差协方差的统计方法进行了研究。针对传统的NMC统计方法只能估计出气候意义上的或静止的背景误差统计的不足,提出了基于集合预报的NMC方法,实现了具有流依赖(flow-dependent)特征的背景误差协方差统计。针对谱方法估计背景误差统计量的缺点,提出了基于球面小波的全球背景误差协方差模拟方法。理论研究和试验结果表明,基于球面小波的背景误差协方差模型能够模拟出既依赖于不同尺度又依赖于地理空间位置的局地垂直相关矩阵,这对于提高全球范围的中尺度预报效果具有重要意义。
三、针对非高斯型的观测误差导致非二次型目标函数的问题,将变分原理应用于观测资料质量控制过程中,设计了基于多分辨率迭代的变分质量控制算法。实现了观测资料质量控制和四维变分同化分析过程相结合的技术,其中考虑了显著误差的非高斯性质,在迭代中对目标函数的观测项作了修正。变分质量控制改善了资料的利用情况,对于前一次迭代中被拒绝的观测资料,如果周围的观测与之相一致,则可以在下一次迭代中被重新评估和利用,并最终影响分析场。试验结果表明,通过变分质量控制,资料的利用率和同化效果得到了明显改进。
四、研究了大规模最优化算法原理和实现技术。为了加快四维变分同化最优化计算迭代收敛速度,在分析共轭梯度方法和有限存储LBFGS方法的基础上,基于预条件和混合迭代步思想,并考虑目标函数的局部特性,提出了迭代时利用线性共轭梯度法进行预条件的改进LBFGS方法。与传统LBFGS方法比较,该方法能够保证更快地收敛。数值实验表明,当目标函数不是严格的二次型时,计算性能的提升更加明显,这对于采用复杂处理而导致目标函数非严格二次型的四维变分同化具有很好的应用前景。
五、针对四维变分同化系统的计算性能瓶颈,研究了高分辨率谱模式和变分同化框架的大规模可扩展并行算法。针对谱模式的两类空间计算(格点空间和谱空间)和两类变换(傅立叶变换和勒让德变换)算法特点,提出了格点空间的二维非规则区域分解算法,解决了一维数据划分导致的计算负载不平衡和通讯效率低的问题。针对四维变分同化的计算特点,提出了基于类型的观测资料混合数据划分算法,有效地解决了由于观测资料的随机分布导致的观测空间计算负载不平衡。实现了基于二维数据剖分和三维数据转置的谱模式并行计算和四维变分同化框架的多阶段并行。实际分析场数据的预报试验表明,谱模式512个进程并行计算能够在15分钟以内完成5天预报,完全满足业务预报的时效性要求。
六、研究了四维变分系统设计和实现技术。针对四维变分同化多任务、复杂数据流、计算量大的特点,基于多分辨率多增量算法设计了四维变分同化计算流程,建立了业务时序图。针对四维变分同化分析效果和计算性能瓶颈问题,采用论文提出的基于球面小波的背景误差协方差模型、变分质量控制、基于小波的控制变量变换、迭代时预条件LBFGS算法,和高分辨率谱模式、变分同化框架的并行计算技术实现了一个四维变分同化试验系统YH4DVAR。通过在我军某气象中心的试运行表明,YH4DVAR四维变分同化系统可以同化12小时时间窗口的气象观测资料,得到与模式协调一致的分析场,与模式一起组成的同化预报系统可以取得较好的预报效果。
Because of the importance of initial fields to the numerical weather prediction (NWP) and the difficulty of them being precisely determined, the advanced data assimilation techniques have become one of the most pivotal technologies to improve the quality of NWP. This thesis concentrates on the goal to resolve the key technologies to establish an operational four-dimensional variational assimilation (4D-Var) system. With the comprehensive utilization of various knowledge from meteorology, computational mathematics and computer application technology, this thesis researches the key algorithms and the parallel computing techniques which are crucial to improve assimilation effect and computing efficiency, and an experimental multi-resolutional 4D-Var system YH4DVAR based on incremental formulation is implemented. The main contents and innovations are as follows:
(1) The basic principle and implementation technologies of 4D-Var are systematically investigated and the algorithms of it are theoretically deduced and analyzed as well. Against the defects of large computing cost and the information loss of medium and small scales in the inner loop optimization, a multi-resolutional and multi-incremental implementation of 4D-Var is proposed. The multi-incremental method can speed up the convergence of inner loop optimization iteration, reduce the computation cost and maintain the meteoric information of various scales. Therefore, we can obtain the incremental analysis with higher resolution and the analysis fields with more accuracy. Aimed at the simplification of which vertical and horizontal background error covariances are computed separately in the procedure of control variable transformation, a new control variable transformation operator is constructed based on the spherical wavelet transformation. The dependence between horizontal and vertical covariances is managed in the wavelet scale.
(2) The statistical methods of background error covariances are investigated. The traditional NMC methods only estimate the background error theoretically or statically, so a NMC method based on the collective forecast is proposed, which can estimate the background error statistics flow-dependently. Against the defects while estimating the background error statistics with spectral method, a new method for simulating the global background error covariances is proposed basing on the spherical wavelet. It is proved theoretically and experimentally that the background error covariances model based on the spherical wavelet can get the local vertical matrix which depends on various scales and positions in the geographic space, which is quite important to improve the quality of global mesoscale forecast.
(3) Considering the situation that the non-Gaussian observation error makes the cost function non-quadratic, a variational quality control algorithm based on the multi-resolutional iteration is proposed, in which the variation principle is applied to the quality control of observations. The quality control of observations and the procedure of 4D-Var are implemented integratedly and the observation term of cost function is modified in the iteration. The non-Gaussian property of obvious errors is also taken into consideration. The variational quality control can improve the utilization of various data. The observations, which are refused in the latest iteration, can be estimated and used again in the next iteration if they are consistent with the observations around. Therefore, they can affect the analysis fields all the same. It is indicated from the experiment results that the utilization of observations and quality of data assimilation make a significant improvement by the variational quality control.
(4) The principle and implementation methods of massive optimal algorithms are investigated. In order to speed up the convergence of optimization iteration in the 4D-Var, based on the analysis of the conjugate gradient method and the LBFGS method, a new preconditioned LBFGS method is proposed, which uses the linear conjugate gradient method to perform the preconditioning. The new method builds upon the preconditioning and hybrid iteration step theories and takes the locality of cost function into consideration as well. Compared to the traditional LBFGS method, the new method has a sooner convergence. Numerical experiments show that the performance improvement is even more dramatic when the cost function is not strictly quadric. Therefore, the new method is quite adaptive to the 4D-Var of which the cost function is usually not strictly quadric because of the complex processing.
(5) Against the computational bottlenecks of 4D-Var system, the massive scalable parallel algorithm for the spectral model with high resolution and variation framework is investigated. Considering the properties of the two space computation (grid space and spectrum space) algorithms and the two transformation (Fourier transformation and Legendre transformation) algorithms, a two-dimensional irregular domain decomposition algorithm is proposed, which eliminates the load imbalance and the low communication efficiency problems appearing in the one-dimensional data partition mode. Considering the computational properties of 4D-Var, an observation hybrid data partition algorithm is proposed according to the observation types, which effectively solves the load imbalance while calculating the observation space resulting from the randomly-distributed observations. This thesis parallelizes the spectrum model and 4D-Var framework according to the two-dimensional data partition and three-dimensional data transform. The forecasting experiment with real analysis fields indicates that it can dramatically satisfy the real-time requirements of the operational forecast, since the five days' forecast can be done in fifteen minutes by spectrum model with 512 processes.
(6) The design and implementation of 4D-Var system is investigated. As 4D-Var system has the features of multitasking, complex data flow and huge computation, the computing flow is carefully designed and the operational time sequence is set up for the 4D-Var system based on the multi-resolutional multi-incremental algorithm. Considering the analysis effect and computational bottleneck, an experimental 4D-Var system YH4DVAR is implemented to which many new technologies above are applied, including background error covariance model based on the spherical wavelet, variational quality control, control variables transformation based on the wavelet, LBFGS algorithm with iteration preconditioning and the parallel computing of spectrum model with high resolution and the variational data assimilation framework. Being experimentally deployed in one of the PLA weather forecasting center, YH4DVAR can assimilate the observations within 12-hour assimilation window and get the analysis fields consistent with the model. YH4DVAR can achieve wonderful forecasting quality when it is combined with the model.
一、研究了四维变分同化基本原理和实现技术,对四维变分同化算法进行了理论推导和分析。针对增量方法中内循环最优化计算量大,容易丢失中小尺度信息等缺点,提出了多分辨率多增量四维变分同化实现方法。多增量方法可以加快内循环最优化迭代的收敛速度,减少计算量,同时保留不同尺度的大气信息,从而能够获得更高分辨率的分析增量和更准确的分析场。针对一般控制变量变换中背景误差协方差水平和垂直相关相分离的简化处理,构造了基于球面小波变换的控制变量转换新算子,通过小波尺度实现了水平和垂直相关的依赖。
二、对背景误差协方差的统计方法进行了研究。针对传统的NMC统计方法只能估计出气候意义上的或静止的背景误差统计的不足,提出了基于集合预报的NMC方法,实现了具有流依赖(flow-dependent)特征的背景误差协方差统计。针对谱方法估计背景误差统计量的缺点,提出了基于球面小波的全球背景误差协方差模拟方法。理论研究和试验结果表明,基于球面小波的背景误差协方差模型能够模拟出既依赖于不同尺度又依赖于地理空间位置的局地垂直相关矩阵,这对于提高全球范围的中尺度预报效果具有重要意义。
三、针对非高斯型的观测误差导致非二次型目标函数的问题,将变分原理应用于观测资料质量控制过程中,设计了基于多分辨率迭代的变分质量控制算法。实现了观测资料质量控制和四维变分同化分析过程相结合的技术,其中考虑了显著误差的非高斯性质,在迭代中对目标函数的观测项作了修正。变分质量控制改善了资料的利用情况,对于前一次迭代中被拒绝的观测资料,如果周围的观测与之相一致,则可以在下一次迭代中被重新评估和利用,并最终影响分析场。试验结果表明,通过变分质量控制,资料的利用率和同化效果得到了明显改进。
四、研究了大规模最优化算法原理和实现技术。为了加快四维变分同化最优化计算迭代收敛速度,在分析共轭梯度方法和有限存储LBFGS方法的基础上,基于预条件和混合迭代步思想,并考虑目标函数的局部特性,提出了迭代时利用线性共轭梯度法进行预条件的改进LBFGS方法。与传统LBFGS方法比较,该方法能够保证更快地收敛。数值实验表明,当目标函数不是严格的二次型时,计算性能的提升更加明显,这对于采用复杂处理而导致目标函数非严格二次型的四维变分同化具有很好的应用前景。
五、针对四维变分同化系统的计算性能瓶颈,研究了高分辨率谱模式和变分同化框架的大规模可扩展并行算法。针对谱模式的两类空间计算(格点空间和谱空间)和两类变换(傅立叶变换和勒让德变换)算法特点,提出了格点空间的二维非规则区域分解算法,解决了一维数据划分导致的计算负载不平衡和通讯效率低的问题。针对四维变分同化的计算特点,提出了基于类型的观测资料混合数据划分算法,有效地解决了由于观测资料的随机分布导致的观测空间计算负载不平衡。实现了基于二维数据剖分和三维数据转置的谱模式并行计算和四维变分同化框架的多阶段并行。实际分析场数据的预报试验表明,谱模式512个进程并行计算能够在15分钟以内完成5天预报,完全满足业务预报的时效性要求。
六、研究了四维变分系统设计和实现技术。针对四维变分同化多任务、复杂数据流、计算量大的特点,基于多分辨率多增量算法设计了四维变分同化计算流程,建立了业务时序图。针对四维变分同化分析效果和计算性能瓶颈问题,采用论文提出的基于球面小波的背景误差协方差模型、变分质量控制、基于小波的控制变量变换、迭代时预条件LBFGS算法,和高分辨率谱模式、变分同化框架的并行计算技术实现了一个四维变分同化试验系统YH4DVAR。通过在我军某气象中心的试运行表明,YH4DVAR四维变分同化系统可以同化12小时时间窗口的气象观测资料,得到与模式协调一致的分析场,与模式一起组成的同化预报系统可以取得较好的预报效果。
Because of the importance of initial fields to the numerical weather prediction (NWP) and the difficulty of them being precisely determined, the advanced data assimilation techniques have become one of the most pivotal technologies to improve the quality of NWP. This thesis concentrates on the goal to resolve the key technologies to establish an operational four-dimensional variational assimilation (4D-Var) system. With the comprehensive utilization of various knowledge from meteorology, computational mathematics and computer application technology, this thesis researches the key algorithms and the parallel computing techniques which are crucial to improve assimilation effect and computing efficiency, and an experimental multi-resolutional 4D-Var system YH4DVAR based on incremental formulation is implemented. The main contents and innovations are as follows:
(1) The basic principle and implementation technologies of 4D-Var are systematically investigated and the algorithms of it are theoretically deduced and analyzed as well. Against the defects of large computing cost and the information loss of medium and small scales in the inner loop optimization, a multi-resolutional and multi-incremental implementation of 4D-Var is proposed. The multi-incremental method can speed up the convergence of inner loop optimization iteration, reduce the computation cost and maintain the meteoric information of various scales. Therefore, we can obtain the incremental analysis with higher resolution and the analysis fields with more accuracy. Aimed at the simplification of which vertical and horizontal background error covariances are computed separately in the procedure of control variable transformation, a new control variable transformation operator is constructed based on the spherical wavelet transformation. The dependence between horizontal and vertical covariances is managed in the wavelet scale.
(2) The statistical methods of background error covariances are investigated. The traditional NMC methods only estimate the background error theoretically or statically, so a NMC method based on the collective forecast is proposed, which can estimate the background error statistics flow-dependently. Against the defects while estimating the background error statistics with spectral method, a new method for simulating the global background error covariances is proposed basing on the spherical wavelet. It is proved theoretically and experimentally that the background error covariances model based on the spherical wavelet can get the local vertical matrix which depends on various scales and positions in the geographic space, which is quite important to improve the quality of global mesoscale forecast.
(3) Considering the situation that the non-Gaussian observation error makes the cost function non-quadratic, a variational quality control algorithm based on the multi-resolutional iteration is proposed, in which the variation principle is applied to the quality control of observations. The quality control of observations and the procedure of 4D-Var are implemented integratedly and the observation term of cost function is modified in the iteration. The non-Gaussian property of obvious errors is also taken into consideration. The variational quality control can improve the utilization of various data. The observations, which are refused in the latest iteration, can be estimated and used again in the next iteration if they are consistent with the observations around. Therefore, they can affect the analysis fields all the same. It is indicated from the experiment results that the utilization of observations and quality of data assimilation make a significant improvement by the variational quality control.
(4) The principle and implementation methods of massive optimal algorithms are investigated. In order to speed up the convergence of optimization iteration in the 4D-Var, based on the analysis of the conjugate gradient method and the LBFGS method, a new preconditioned LBFGS method is proposed, which uses the linear conjugate gradient method to perform the preconditioning. The new method builds upon the preconditioning and hybrid iteration step theories and takes the locality of cost function into consideration as well. Compared to the traditional LBFGS method, the new method has a sooner convergence. Numerical experiments show that the performance improvement is even more dramatic when the cost function is not strictly quadric. Therefore, the new method is quite adaptive to the 4D-Var of which the cost function is usually not strictly quadric because of the complex processing.
(5) Against the computational bottlenecks of 4D-Var system, the massive scalable parallel algorithm for the spectral model with high resolution and variation framework is investigated. Considering the properties of the two space computation (grid space and spectrum space) algorithms and the two transformation (Fourier transformation and Legendre transformation) algorithms, a two-dimensional irregular domain decomposition algorithm is proposed, which eliminates the load imbalance and the low communication efficiency problems appearing in the one-dimensional data partition mode. Considering the computational properties of 4D-Var, an observation hybrid data partition algorithm is proposed according to the observation types, which effectively solves the load imbalance while calculating the observation space resulting from the randomly-distributed observations. This thesis parallelizes the spectrum model and 4D-Var framework according to the two-dimensional data partition and three-dimensional data transform. The forecasting experiment with real analysis fields indicates that it can dramatically satisfy the real-time requirements of the operational forecast, since the five days' forecast can be done in fifteen minutes by spectrum model with 512 processes.
(6) The design and implementation of 4D-Var system is investigated. As 4D-Var system has the features of multitasking, complex data flow and huge computation, the computing flow is carefully designed and the operational time sequence is set up for the 4D-Var system based on the multi-resolutional multi-incremental algorithm. Considering the analysis effect and computational bottleneck, an experimental 4D-Var system YH4DVAR is implemented to which many new technologies above are applied, including background error covariance model based on the spherical wavelet, variational quality control, control variables transformation based on the wavelet, LBFGS algorithm with iteration preconditioning and the parallel computing of spectrum model with high resolution and the variational data assimilation framework. Being experimentally deployed in one of the PLA weather forecasting center, YH4DVAR can assimilate the observations within 12-hour assimilation window and get the analysis fields consistent with the model. YH4DVAR can achieve wonderful forecasting quality when it is combined with the model.
引文
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