多精度计算和区间分析方法在科学计算中的应用

介绍了多精度计算和区间分析的概念、方法及应用,使用多精度计算研究Lorenz非线性动力系统计算结果对计算精度、步长的敏感依赖关系和不确定现象,研究Lorenz系统中长时间数值解的计算方法和基于多精度计算的最优步长、最大有效计算时间搜索方法,讨论了区间分析方法在计算机辅助证明Lorenz混沌吸引子存在性问题中的应用。     

浅水波方程在全球上的变时间步长积分

本文讨论了浅水波方程在全球上的变时间步长(以下简称变步长)积分。并讨论了两步Lax-Wendroff的变时间步长(以下简称TLW变步长)积分,发现积分结果对平滑系数κ很敏感,必须选择合适。井将常数时间步长(以下简称定步长)的积分结果和两种变步长的积分结果进行比较。发现用实际资料作预报,变步长法的预报均方根误差比定步长法的小。还发现变步长的24小时预报对近极地系统移动偏慢的问题比定步长的有较大的改善。变步长积分的计算稳定性也优于定步长。     

中国南海台风模式(TRAMS-v2.0)技术特点及其预报性能

中国南海台风模式(TRAMS)是基于GRAPES的非静力中尺度模式,采用半隐半拉格朗日时间差分方案,借助Helmholtz方程进行隐式求解,并在原模式的基础上,采用三维静力参考大气、非线性项分步计算、物理过程倾向隐式处理及与动力过程耦合等技术,形成新的模式动力过程计算方案。模式物理过程主要包括:长短波辐射、云微物理、湍流和深浅对流和海陆面等下垫面参数化方案,新版南海台风模式重点研发了海陆面参数化方案(SMS方案),改进了积云参数化方案(NSAS方案),并且引入地形重力波拖曳参数化方案(KA95方案)。预报模式的覆盖范围:81~161°E,0~51°N。水平格距为0.18°,垂直方向分65层,时间积分步长为100 s。2015年批量试验结果表明,新版南海台风模式预报性能稳定,误差较小,与EC全球模式同样本比较,发现短时效(如0~24 h)两模式台风路径预报误差水平基本相当,而较长时效(如48~72 h),南海台风模式的预报误差小于EC全球模式,具备较好的业务应用价值。     

多隐层BP神经网络在模式预报中的简化应用

EC细网格预报效果好,基本满足业务需要,在工作中被广泛应用。为进一步提升预报准确性,做好迪士尼园区的气象服务保障,选取2016年7月至2017年6月1年的2 m温度预报场,24 h预报时效的时间分辨率为3 h,72 h预报时效的时间分辨率为24 h,分别用回归分析法、S型和简化Line型BP神经网络法进行模式释用,与迪士尼气象站观测数据对比。结果表明:阈值为1℃时,对模式结果释用后,均方根误差减少了0.5℃到1.0℃,3—9 h和21—72 h预报时效的准确率由原来的50%和30%分别上升到70%和50%。采用S型多隐层BP神经网络误差最小,不同预报时效释用稳定性最高,同时该释用方法对t_(min)的预报特征把握更精准,释用效果明显优于对t_(max)的预报释用,但迭代计算耗费时间大幅增多,与预报效果的提升不成正比。简化Line型的BP神经网络通过8个半月的数据量和简单的网络模式,捕获了EC预报的特征,不但减小了计算量,大幅缩短了计算时间,而且预报结果也有显著提升,预报稳定性较好,具有广泛的业务应用空间。     

9711号热带气旋路径预报的检验评估

1引言众所周知,热带气旋路径预报的精度直接关系灾害性天气的预报服务效果。近年来,随着数值预报和各种热带气旋路径预报方法的不断改进,热带气旋路径的预报水平也在不断的提高。然而,随着社会经济的发展,对于热带气旋路径和灾害性天气预报的精度和时效的要求越来越高。目前国内外比较先进的数值预报和热带气旋路径预报,随着预报时效的延长和路径的变化,预报误差亦明显增大。因此,分析热带气旋路径预报的误差及各种数值预报对于热带气旋路径的预报能力,便于业务预报的合理释用,减小预报误差,获取最佳预报服务效果,是业务预报服…     

一种新的时间序列预报方法

本文将车贝雪夫多项式应用于时间序列的研究,提出了一种新的时间序列预报方法.这种方法精度较高,简便易行,可以用计算机算,也可以手工计算.文中给出了预报公式,估计了预报误差,并给出了预报实例.     

ECMWF细网格模式探空在阿勒泰地区短时强降水预报中的统计检验

采用2013—2018年5—9月ECMWF细网格资料和阿勒泰地区36次短时强降水资料,用Micaps平台的模式探空模块计算T-log P图及其对流参数,运用统计学方法进行了误差检验。结果表明:模式探空T-log P图48 h预报时效内一致性较高(72%),尤其是24 h预报时效内(92%);72 h预报时效内总指数和干暖盖指数及垂直风切变、60 h预报时效内沙氏指数、48 h预报时效内风暴相对螺旋度和36 h预报时效内850与500 hPa温度差、对流温度、最大抬升指数、抬升指数以及24 h预报时效内K指数、700与850 hPa假相当位温差、大风指数等对流参数的3种误差均较小(3.5),相关系数较高(0.60),特别是沙氏指数、K指数、700与850 hPa假相当位温差、垂直风切变和风暴相对螺旋度的3种误差1.5。T-log P图的一致性随时效的延长而减小,对流参数的误差随预报时效的延长变化不一致,在强对流潜势预报业务中注意订正运用。     

ECMWF细网格模式在北疆降雪预报中的统计检验

采用ECMWF细网格模式产品,对发生在北疆2015年1月—2017年4月共20场降雪天气过程进行统计学检验。结果表明,48 h预报时效内ECMWF细网格模式对形势场、850h Pa比湿、对流层位涡及对流层低层u、v风场预报误差较小,精度较高;对流层中低层垂直速度和相对湿度及300 hPa u、v风场的系统性误差较小,随机误差相对较大,并建立了ECMWF细网格模式48 h预报时效内在北疆降雪天气预报中的应用模型;模式对新疆北部暖区降雪的各量级预报随时效的延长准确率并非减小,尤其是中雪;72 h预报时效内,模式对12 h累计降雪量为小雪和中雪的预报相对较稳定,强降雪的误差较大,随时效的延长并非呈增大的趋势;在预报业务中注意订正应用。     

ECMWF产品在东亚中纬度地区预报精度检验

利用求预报场与实况场绝对误差和相关系数,对ECMWF资料进行预报精度检验,结果表明 500 hPa高度,850 hPa温度和相对湿度,海平面气压等4个物理量场预报较好; 850 hPa相对湿度预报比700 hPa的误差小;东西风预报比南北风预报好;绝对误差值随预报时效延长而增大,相关系数则随预报时效延长而减小.     

Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

由于满足计算的不确定性原理,需适当选取时间步长以保证非线性常微分方程组数值解的可靠性,目前尚未见关于有效步长区间的理论结果。本文对于给定的误差限,将方法截断误差与机器舍入误差的相关曲线分别进行平移,从而得到一种确定有效步长近似区间的方法,并推导出近似区间相比于原区间的相对误差公式。另外,研究了有效步长区间随积分时间的变化规律,并对已有的数值结果给出解释。本文所得结论可为数值求解常微分方程组选取有效步长并得到可靠的数值解提供理论支持。     

Analysis and application of multiple-precision computation and round-off error for nonlinear dynamical systems

This research reveals the dependency of floating point computation in nonlinear dynamical systems on machine precision and step-size by applying a multiple-precision approach in the Lorenz nonlinear equations. The paper also demoastrates the procedures for obtaining a real numerical solution in the Lorenz system with long-time integration and a new multiple-precision-based approach used to identify the maximum effective computation time （MECT） and optimal step-size （OS）. In addition, the authors introduce how to analyze round-off error in a long-time integration in some typical cases of nonlinear systems and present its approximate estimate expression.     

GRAPES全球模式静力平衡奇异向量改进及应用试验

采用线性化物理过程方案的GRAPES全球模式奇异向量在进行非线性模式积分时会有部分奇异向量出现崩溃问题,这说明奇异向量结构可能存在扰动变量之间不协调之处,需要对奇异向量扰动的计算方法优化,进而改进基于奇异向量的集合预报初值扰动,提高GRAPES全球集合预报效果.基于原有的GRAEPS全球奇异向量计算方法,在求解奇异向量...     

A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems

Based on the Taylor series method and Li's spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme's computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers' equation.     

Recent Advances in Predictability Studies in China （1999-2002）

Since the last International Union of Geodesy and Geophysics (IUGG) General Assembly (1999), the predictability studies in China have made further progress during the period of 1999-2002. Firstly, three predictability sub-problems in numerical weather and climate prediction are classified, which are concerned with the maximum predictability time, the maximum prediction error, and the maximum allowable initial error, and then they are reduced into three nonlinear optimization problems. Secondly, the concepts of the nonlinear singular vector (NSV) and conditional nonlinear optimal perturbation (CNOP) are proposed,which have been utilized to study the predictability of numerical weather and climate prediction. The results suggest that the nonlinear characteristics of the motions of atmosphere and oceans can be revealedby NSV and CNOP. Thirdly, attention has also been paid to the relations between the predictability and spatial-temporal scale, and between the model predictability and the machine precision, of which the investigations disclose the importance of the spatial-temporal scale and machine precision in the study of predictability. Also the cell-to-cell mapping is adopted to analyze globally the predictability of climate,which could provide a new subject to the research workers. Furthermore, the predictability of the summer rainfall in China is investigated by using the method of correlation coefficients. The results demonstrate that the predictability of summer rainfall is different in different areas of China. Analysis of variance, which is one of the statistical methods applicable to the study of predictability, is also used to study the potential predictability of monthly mean temperature in China, of which the conclusion is that the monthly mean temperature over China is potentially predictable at a statistical significance level of 0.10. In addition,in the analysis of the predictability of the T106 objective analysis/forecasting field, the variance and the correlation coefficient are calculated to explore the distribution characteristics of the mean-square errors.Finally, the predictability of short-term climate prediction is investigated by using statistical methods or numerical simulation methods. It is demonstrated that the predictability of short-term climate in China depends not only on the region of China being investigated, but also on the time scale and the atmospheric internal dynamical process.     

攀西地区月降水时序非线性特性分析

在介绍相空间重构理论的基础上, 以攀西地区4个站点47 a的月降水时间序列为例, 研究了该地月降水时间序列的非线性特性。首先, 运用定量的G-P关联维方法, 探讨了非线性分析的主要定量指标, 具体而言有, 饱和关联维数D2和柯尔莫哥诺夫熵, 计算表明攀西月降水时间序列具有一定的非线性混沌特性。其次, 结合定性的功率谱分析方法, 进一步验证了攀西月降水时序具有非线性特性。除此之外还应用Cao方法检验, 从而排除了月降水时序为随机序列的可能性。本文为进一步研究月降水时序数据的复杂性及其演化规律奠定了基础。      

On the Application of a Genetic Algorithm to the Predictability Problems Involving "On-Off" Switches

The lower bound of maximum predictable time can be formulated into a constrained nonlinear opti- mization problem, and the traditional solutions to this problem are the filtering method and the conditional nonlinear optimal perturbation (CNOP) method. Usually, the CNOP method is implemented with the help of a gradient descent algorithm based on the adjoint method, which is named the ADJ-CNOP. However, with the increasing improvement of actual prediction models, more and more physical processes are taken into consideration in models in the form of parameterization, thus giving rise to the on-off switch problem, which tremendously affects the effectiveness of the conventional gradient descent algorithm based on the ad- joint method. In this study, we attempted to apply a genetic algorithm (GA) to the CNOP method, named GA-CNOP, to solve the predictability problems involving on-off switches. As the precision of the filtering method depends uniquely on the division of the constraint region, its results were taken as benchmarks, and a series of comparisons between the ADJ-CNOP and the GA-CNOP were performed for the modified Lorenz equation. Results show that the GA-CNOP can always determine the accurate lower bound of maximum predictable time, even in non-smooth cases, while the ADJ-CNOP, owing to the effect of on-off switches, often yields the incorrect lower bound of maximum predictable time. Therefore, in non-smooth cases, using GAs to solve predictability problems is more effective than using the conventional optimization algorithm based on gradients, as long as genetic operators in GAs are properly configured.     

The Time and Regime Dependencies of Sensitive Areas for Tropical Cyclone Prediction Using the CNOP Method

This study examines the time and regime dependencies of sensitive areas identified by the conditional nonlinear optimal perturbation(CNOP) method for forecasts of two typhoons.Typhoon Meari(2004) was weakly nonlinear and is herein referred to as the linear case,while Typhoon Matsa(2005) was strongly nonlinear and is herein referred to as the nonlinear case.In the linear case,the sensitive areas identified for special forecast times when the initial time was fixed resembled those identified for other forecast times.Targeted observations deployed to improve a special time forecast would thus also benefit forecasts at other times.In the nonlinear case,the similarities among the sensitive areas identified for different forecast times were more limited.The deployment of targeted observations in the nonlinear case would therefore need to be adapted to achieve large improvements for different targeted forecasts.For both cases,the closer the forecast time,the higher the similarities of the sensitive areas.When the forecast time was fixed,the sensitive areas in the linear case diverged continuously from the verification area as the forecast period lengthened,while those in the nonlinear case were always located around the initial cyclones.The deployment of targeted observations to improve a special forecast depends strongly on the time of deployment.An examination of the efficiency gained by reducing initial errors within the identified sensitive areas confirmed these results.In general,the greatest improvement in a special time forecast was obtained by identifying the sensitive areas for the corresponding forecast time period.     

集合卡尔曼滤波数据同化在一维波动方程中的应用

简要回顾了集合卡尔曼滤波(EnKF：Ensemble Kalman Filter)数据同化方法的发展历史,并介绍了EnKF数据同化方法的基本原理,利用一维非线性波动方程进行了数值试验。EnKF数据同化方法的实现过程简单可行。避免了EKF中协方差演变方程预报过程中出现的计算不准确和关于协方差矩阵的大量数据的存储问题,最主要的是EnKF可以有效控制模式变量估计误差方差的增长,改善预报效果。     

Nonlinearity and finite-time instability in a T21L3 quasigeostrophic model

In this paper, a nonlinear optimization method is used to explore the finite-time instability of the atmospheric circulation with a three-level quasigeostrophic model under the framework of the conditional nonlinear optimal perturbation (CNOP). As a natural generalization of linear singular vector (SV), CNOP is defined as an initial perturbation that makes the cost function the maximum at a prescribed forecast time under certain physical constraint conditions. Special attentions are paid to the different structures and energy evolutions of the optimal perturbations.