THE EFFECTIVENESS OF GENETIC ALGORITHM IN CAPTURING CONDITIONAL NONLINEAR OPTIMAL PERTURBATION WITH PARAMETERIZATION “ON-OFF” SWITCHES INCLUDED BY A MODEL

In the typhoon adaptive observation based on conditional nonlinear optimal perturbation (CNOP), the ‘on-off’ switch caused by moist physical parameterization in prediction models prevents the conventional adjoint method from providing correct gradient during the optimization process. To address this problem, the capture of CNOP, when the “on-off” switches are included in models, is treated as non-smooth optimization in this study, and the genetic algorithm (GA) is introduced. After detailed algorithm procedures are formulated using an idealized model with parameterization “on-off” switches in the forcing term, the impacts of “on-off” switches on the capture of CNOP are analyzed, and three numerical experiments are conducted to check the effectiveness of GA in capturing CNOP and to analyze the impacts of different initial populations on the optimization result. The result shows that GA is competent for the capture of CNOP in the context of the idealized model with parameterization ‘on-off’ switches in this study. Finally, the advantages and disadvantages of GA in capturing CNOP are analyzed in detail.     

THE EFFECTIVENESS OF GENETIC ALGORITHM IN CAPTURING CONDITIONAL NONLINEAR OPTIMAL PERTURBATION WITH PARAMETERIZATION "ON-OFF" SWITCHES INCLUDED BY A MODEL

In the typhoon adaptive observation based on conditional nonlinear optimal perturbation (CNOP), the ‘on-off’ switch caused by moist physical parameterization in prediction models prevents the conventional adjoint method from providing correct gradient during the optimization process. To address this problem, the capture of CNOP, when the on-off switches are included in models, is treated as non-smooth optimization in this study, and the genetic algorithm (GA) is introduced. After detailed algorithm procedur...     

A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations简

A variant constrained genetic algorithm （VCGA） for effective tracking of conditional nonlinear optimal perturbations （CNOPs） is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite pararneters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has ＂on off＂ switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method （ADJ）, and a GA using tournament selection operation and the niching technique （GA-DEB） were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of ~CGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.     

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.     

INTELLIGENT ALGORITHMS FOR SOLVING CNOP AND THEIR APPLICATIONS IN ENSO PREDICTABILITY AND TROPICAL CYCLONE ADAPTIVE OBSERVATIONS

Some intelligent algorithms (IAs) proposed by us, including swarm IAs and single individual IAs, have been applied to the Zebiak-Cane (ZC) model to solve conditional nonlinear optimal perturbation (CNOP) for studying El Ni?o – Southern Oscillation (ENSO) predictability. Compared to the adjoint-based method (the ADJ-method), which is referred to as a benchmark, these IAs can achieve approximate CNOP results in terms of magnitudes and patterns. Using IAs to solve CNOP can avoid the use of an adjoint model and widen the application of CNOP in numerical climate and weather modeling. Of the proposed swarm IAs, PCA-based particle swarm optimization (PPSO) obtains CNOPs with the best patterns and the best stability. Of the proposed single individual IAs, continuous tabu search algorithm with sine maps and staged strategy (CTS-SS) has the highest efficiency. In this paper, we compare the validity, stability and efficiency of parallel PPSO and CTS-SS using these two IAs to solve CNOP in the ZC model for studying ENSO predictability. The experimental results show that CTS-SS outperforms parallel PPSO except with respect to stability. At the same time, we are also concerned with whether these two IAs can effectively solve CNOP when applied to more complicated models. Taking the sensitive areas identification of tropical cyclone adaptive observations as an example and using the fifth-generation mesoscale model (MM5), we design some experiments. The experimental results demonstrate that each of these two IAs can effectively solve CNOP and that parallel PPSO has a higher efficiency than CTS-SS. We also provide some suggestions on how to choose a suitable IA to solve CNOP for different models.     

On the Chaotic Behavior and Predictability of the Real Atmosphere

In this paper the concept of Chaos and its applications to the study of predictability theory is introduced. The author's attempt is to give a general overview of ideas and methods involved in this problem to scientists,who are interested in the problem of predictability but not familiar with the theory of chaos. The problem is discussed in 4 sections. In the first section, the concept of chaos and the study methods are outlined briefly; in the second section, the methods of quantitatively measuring the main characteristics of chaos which are the basis for the predictability theory are introduced; the third section discusses the time series analysis for directly studying chaotic phenomena in practical problems; and the last section presents some research results on the chaotic characteristics and the predictability of the real atmosphere.     

Mesoscale predictability of moist baroclinic waves: Variable and scale-dependent error growth

This study seeks to quantify the predictability of different forecast variables at various scales through spectral analysis of the difference between perturbed and unperturbed cloud-permitting simulations of idealized moist baroclinic waves amplify- ing in a conditionally unstable atmosphere. The error growth of a forecast variable is found to be strongly associated with its reference-state （unperturbed） power spectrum and slope, which differ significantly from variable to variable. The shallower the reference state spectrum, the more spectral energy resides at smaller scales, and thus the less predictable the variable since the error grows faster at smaller scales before it saturates. In general, the variables with more small-scale components （such as vertical velocity） are less predictable, and vice versa （such as pressure）. In higher-resolution simulations in which more rigorous small-scale instabilities become better resolved, the error grows faster at smaller scales and spreads to larger scales more quickly before the error saturates at those small scales during the first few hours of the forecast. Based on the reference power spectrum, an index on the degree of lack （or loss） of predictability （LPI） is further defined to quantify the predictive time scale of each forecast variable. Future studies are needed to investigate the scale- and variable-dependent predictability under different background reference flows, including real case studies through ensemble experiments.     

STABILITY AND POTENTIAL PREDICTABILITY OF ANNUAL PERCIPITATION OVER CHINA

The precipitation is a primary element which directly affects the agricultural production of thecountry with one fifth of the world population.With the economic development the water resourcestress is getting greater.In this paper,based on the data at 162 stations selected evenly over Chinafrom 1960 to 1991 the stability and potential predictability of annual precipitation have been stud-ied.The eastern and southern parts of the country having abundant precipitation enjoy more stableprecipitation.The north and northwest parts of the country where the precipitations are deficienthave unstable precipitations.The potential predictability approximates to the ratio of the estimatedinterannual variance to the climatic noise.Generally the annual precipitation over China is poten-tially predictable.In the area between the Huanghe River and Changjiang River and the east ofnortheastern China the potential predictability is the lowest in the country.In the north and north-west of the country the potential predictability is greater.The southeastern coast has relatively lowvalues of potential predictability.Also,the method of estimating climatic noise of annual precipita-tion has been discussed from the idea of Yamamoto et al.(1985)in order to estimate the potentialpredictability.     

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states.Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system,the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.     

A Fast Algorithm for Solving CNOP and Associated Target Observation Tests

Conditional Nonlinear Optimal Perturbation （CNOP） is a new method proposed by Mu et al. in 2003, which generalizes the linear singular vector （LSV） to include nonlinearity. It has become a powerful tool for studying predictability and sensitivity among other issues in nonlinear systems. This is because the CNOP is able to represent, while the LSV is unable to deal with, the fastest developing perturbation in a nonlinear system. The wide application of this new method, however, has been limited due to its large computational cost related to the use of an adjoint technique. In order to greatly reduce the computational cost, we hereby propose a fast algorithm for solving the CNOP based on the empirical orthogonal function （EOF）. The algorithm is tested in target observation experiments of Typhoon Matsa using the Global/Regional Assimilation and PrEdiction System （GRAPES）, an operational regional forecast model of China. The effectivity and feasibility of the algorithm to determine the sensitivity （target） area is evaluated through two observing system simulation experiments （OSSEs）. The results, as expected, show that the energy of the CNOP solved by the new algorithm develops quickly and nonlinearly. The sensitivity area is effectively identified with the CNOP from the new algorithm, using 24 h as the prediction time window. The 24-h accumulated rainfall prediction errors （ARPEs） in the verification region are reduced significantly compared with the ＂true state,＂ when the initial conditions （ICs） in the sensitivity area are replaced with the ＂observations.＂ The decrease of the ARPEs can be achieved for even longer prediction times （e.g., 72 h）. Further analyses reveal that the decrease of the 24-h ARPEs in the verification region is attributable to improved simulations of the typhoon＇s initial warm-core, upper level relative vorticity, water vapor conditions, etc., as a result of the updated ICs in the sensitivity area.     

遗传算法求解最大预报误差的可行性研究

利用条件非线性最优扰动（conditional nonlinear optimal perturbation,CNOP）可以实现最大预报误差的上界估计。CNOP通常由基于梯度信息的约束优化算法进行求解,且其中的梯度信息由伴随模式提供。然而当非线性模式中含不连续＂开关＂时,传统伴随方法不能为优化过程提供正确的梯度方向,从而导致优化失败。为此,采用自适应变异和混合交叉的遗传算法,联赛选择机制和小生境技术的约束处理方法来求解最大预报误差上界。为检验新方法的有效性,以修改的Lorenz模型作为预报模式,对3个初始态分别用新方法和传统伴随方法进行比较,数值试验结果显示新方法求解出的最大预报误差的上界更加精确。     

A New Strategy for Solving a Class of Constrained Nonlinear Optimization Problems Related to Weather and Climate Predictability

There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems: the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly efficient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly efficient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally efficient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.     

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation (CNOP), which is a nonlinear generalization of the linear singular vector (LSV), is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming (SQP) algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, and the spectral projected gradients (SPG2) algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.     

A Preliminary Application of the Differential Evolution Algorithm to Calculate the CNOP

A projected skill is adopted by use of the differential evolution （DE） algorithm to calculate a conditional nonlinear optimal perturbation （CNOP）. The CNOP is the maximal value of a constrained optimization problem with a constraint condition, such as a ball constraint. The success of the DE algorithm lies in its ability to handle a non-differentiable and nonlinear cost function. In this study, the DE algorithm and the traditional optimization algorithms used to obtain the CNOPs are compared by analyzing a theoretical grassland ecosystem model and a dynamic global vegetation model. This study shows that the CNOPs generated by the DE algorithm are similar to those by the sequential quadratic programming （SQP） algorithm and the spectral projected gradients （SPG2） algorithm. If the cost function is non-differentiable, the CNOPs could also be caught with the DE algorithm. The numerical results suggest the DE algorithm can be employed to calculate the CNOP, especially when the cost function is non-differentiable.     

Applications of conditional nonlinear optimal perturbation in predictability study and sensitivity analysis of weather and climate

Considering the limitation of the linear theory of singular vector (SV), the authors and their collaborators proposed conditional nonlinear optimal perturbation (CNOP) and then applied it in the predictability study and the sensitivity analysis of weather and climate system. To celebrate the 20th anniversary of Chinese National Committee for World Climate Research Programme (WCRP), this paper is devoted to reviewing the main results of these studies. First, CNOP represents the initial perturbation that has largest nonlinear evolution at prediction time, which is different from linear singular vector (LSV) for the large magnitude of initial perturbation or/and the long optimization time interval. Second, CNOP, rather than linear singular vector (LSV), represents the initial anomaly that evolves into ENSO events most probably. It is also the CNOP that induces the most prominent seasonal variation of error growth for ENSO predictability; furthermore, CNOP was applied to investigate the decadal variability of ENSO asymmetry. It is demonstrated that the changing nonlinearity causes the change of ENSO asymmetry. Third, in the studies of the sensitivity and stability of ocean’s thermohaline circulation (THC), the nonlinear asymmetric response of THC to finite amplitude of initial perturbations was revealed by CNOP. Through this approach the passive mechanism of decadal variation of THC was demonstrated; Also the authors studies the instability and sensitivity analysis of grassland ecosystem by using CNOP and show the mechanism of the transitions between the grassland and desert states. Finally, a detailed discussion on the results obtained by CNOP suggests the applicability of CNOP in predictability studies and sensitivity analysis.     

关于线性奇异向量和条件非线性最优扰动差别的一个注记

奇异向量(singular vectors,SVs)和条件非线性最优扰动(conditional nonlinear optimal perturbation,CNOP)已广泛应用于研究大气—海洋系统的不稳定性以及与其相关的可预报性、集合预报和目标观测问题研究。本文首先回顾了SVs和CNOP的发展历史,并简单描述了它们的基本原理;然后针对二维正压准地转模式,使用不同的范数组合,分析了第一线性奇异向量(first singular vector,FSV)和CNOP之间的异同。结果表明,当优化时间较短时,度量SVs和CNOP大小的范数不同也将导致FSV和CNOP相差很大,而当度量SVs和CNOP大小的范数相同时,FSV和CNOP之间的差别则主要是由非线性物理过程作用的结果。因此,针对不同的物理问题,应该选取合适的度量范数研究FSV和CNOP以及其所引起的大气或海洋动力学的异同,从而揭示非线性物理过程的影响机理。