一种求解条件非线性最优扰动的快速算法及其在台风目标观测中的初步检验

条件非线性最优扰动(CNOP)是Mu等2003年提出的一个新的理论方法,它是线性奇异向量在非线性情形的推广,克服了线性奇异向量不能代表非线性系统最快发展扰动的缺陷,成为非线性系统可预报性和敏感性等研究新的有效工具.然而,由于以往CNOP的求解需要采用伴随技术,计算量相当巨大,限制了该方法的推广应用.为了克服这一困难,本文基于经验正交分解(EOF),提出了一种求解CNOP的快速算法,利用GRAPES区域业务预报模式实现了CNOP快速计算,并在台风"麦莎"的目标观测研究中得到初步检验,通过观测系统模拟实验(OSSE)检验了该方法确定敏感性区域(瞄准区)的有效性和可行性.试验结果表明,用快速算法求解的CNOP,其净能量随时间快速地发展,而且发展呈非线性.在台风"麦莎"个例的目标观测试验中,用快速算法得到的预报时间为24 h的CNOP可以有效地识别瞄准区,并通过瞄准区内初值的改善,可明显减少目标区域(检验区)内24 h累计降水预报误差.尤其,累计降水预报的这种改进效果能够延伸到更长时间(如72 h),尽管检验时间是设在第24小时.进一步分析发现,24 h累计降水预报误差的减少是通过利用瞄准区内改善的初值改进初始时刻台风暖心结构、高空相对涡度以及水汽条件等而得以实现的.     

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.     

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.     

Applications of conditional nonlinear optimal perturbation to the study of the stability and sensitivity of the Jovian atmosphere

A two-layer quasi-geostrophic model is used to study the stability and sensitivity of motions on smallscale vortices in Jupiter’s atmosphere. Conditional nonlinear optimal perturbations (CNOPs) and linear singular vectors (LSVs) are both obtained numerically and compared in this paper. The results show that CNOPs can capture the nonlinear characteristics of motions in small-scale vortices in Jupiter’s atmosphere and show great difference from LSVs under the condition that the initial constraint condition is large or the optimization time is not very short or both. Besides, in some basic states, local CNOPs are found. The pattern of LSV is more similar to local CNOP than global CNOP in some cases. The elementary application of the method of CNOP to the Jovian atmosphere helps us to explore the stability of variousscale motions of Jupiter’s atmosphere and to compare the stability of motions in Jupiter’s atmosphere and Earth’s atmosphere further.     

伴随敏感性方法、第一奇异向量方法以及条件非线性最优扰动方法在台风目标观测敏感区识别中的比较研究

本文通过深入分析伴随敏感性（ADS）方法、第一奇异向量（LSV）方法、以及条件非线性最优扰动（CNOP）方法在目标观测敏感区识别方面的原理,提出了非线性程度的概念和计算方法,考察了转向型和直线型台风的非线性程度,分析了上述三种方法在不同非线性程度下识别的敏感区的异同,同时对比了转向型和直线型台风的敏感区的差异,并通过敏感性试验探讨了在不同非线性程度下以及在转向型与直线型台风中,预报对敏感区内初值的敏感性程度,进而探讨台风目标观测在不同情况下的有效性。结果表明,转向型台风的非线性程度差别比较大,或者特别强,或者特别弱;而直线型台风非线性程度居中,不同台风个例之间的非线性程度差别较小。对于非线性较弱的台风,三种方法识别的敏感区较为相似,而对于非线性较强的台风,LSV方法与ADS方法识别的敏感区较为相似,但是与CNOP方法识别的敏感区具有较大的差别。对于转向型台风,敏感区主要位于行进路径的右前方,而对于直线型台风,敏感区主要位于初始台风位置的后方。敏感性试验表明,不论台风非线性强弱,转向还是直行,CNOP敏感区内的随机扰动发展最大,而LSV敏感区内叠加的随机扰动发展次之,ADS敏感区内叠加的扰动发展最小;此外,非线性弱的台风,扰动的发展大于非线性强的台风的扰动的发展,表明非线性弱的台风预报受初值影响更大,目标观测的效果可能会更明显。     

What Kind of Initial Errors Cause the Severest Prediction Uncertainty of EI Nino in Zebiak-Cane Model

With the Zebiak-Cane (ZC) model, the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation (CNOP). The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model. By analyzing the behavior of CNOP- type errors, we find that for the normal states and the relatively weak EI Nino events in the ZC model, the predictions tend to yield false alarms due to the uncertainties caused by CNOP. For the relatively strong EI Nino events, the ZC model largely underestimates their intensities. Also, our results suggest that the error growth of EI Nino in the ZC model depends on the phases of both the annual cycle and ENSO. The condition during northern spring and summer is most favorable for the error growth. The ENSO prediction bestriding these two seasons may be the most difficult. A linear singular vector (LSV) approach is also used to estimate the error growth of ENSO, but it underestimates the prediction uncertainties of ENSO in the ZC model. This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes. CNOP yields the severest prediction uncertainty. That is to say, the prediction skill of ENSO is closely related to the types of initial error. This finding illustrates a theoretical basis of data assimilation. It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.     

What Kind of Initial Errors Cause the Severest Prediction Uncertainty of El Nino in Zebiak-Cane Model

With the Zebiak-Cane （ZC） model, the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation （CNOP）. The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model. By analyzing the behavior of CNOPtype errors, we find that for the normal states and the relatively weak E1 Nifio events in the ZC model, the predictions tend to yield false alarms due to the uncertainties caused by CNOP. For the relatively strong E1 Nino events, the ZC model largely underestimates their intensities. Also, our results suggest that the error growth of E1 Nifio in the ZC model depends on the phases of both the annual cycle and ENSO. The condition during northern spring and summer is most favorable for the error growth. The ENSO prediction bestriding these two seasons may be the most difficult. A linear singular vector （LSV） approach is also used to estimate the error growth of ENSO, but it underestimates the prediction uncertainties of ENSO in the ZC model. This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes. CNOP yields the severest prediction uncertainty. That is to say, the prediction skill of ENSO is closely related to the types of initial error. This finding illustrates a theoretical basis of data assimilation. It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.     

A New Approach for Parameter Optimization in Land Surface Model

In this study,a new parameter optimization method was used to investigate the expansion of conditional nonlinear optimal perturbation (CNOP) in a land surface model (LSM) using long-term enhanced field observations at Tongyu station in Jilin Province,China,combined with a sophisticated LSM (common land model,CoLM).Tongyu station is a reference site of the international Coordinated Energy and Water Cycle Observations Project (CEOP) that has studied semiarid regions that have undergone desertification,salination,and degradation since late 1960s.In this study,three key land-surface parameters,namely,soil color,proportion of sand or clay in soil,and leaf-area index were chosen as parameters to be optimized.Our study comprised three experiments:First,a single-parameter optimization was performed,while the second and third experiments performed triple-and six-parameter optimizations,respectively.Notable improvements in simulating sensible heat flux (SH),latent heat flux (LH),soil temperature (TS),and moisture (MS) at shallow layers were achieved using the optimized parameters.The multiple-parameter optimization experiments performed better than the single-parameter experminent.All results demonstrate that the CNOP method can be used to optimize expanded parameters in an LSM.Moreover,clear mathematical meaning,simple design structure,and rapid computability give this method great potential for further application to parameter optimization in LSMs.     

A comparison study of the methods of conditional nonlinear optimal perturbations and singular vectors in ensemble prediction

The authors apply the technique of conditional nonlinear optimal perturbations (CNOPs) as a means of providing initial perturbations for ensemble forecasting by using a barotropic quasi-geostrophic (QG) model in a perfect-model scenario. Ensemble forecasts for the medium range (14 days) are made from the initial states perturbed by CNOPs and singular vectors (SVs). 13 different cases have been chosen when analysis error is a kind of fast growing error. Our experiments show that the introduction of CNOP provides better forecast skill than the SV method. Moreover, the spread-skill relationship reveals that the ensemble samples in which the first SV is replaced by CNOP appear superior to those obtained by SVs from day 6 to day 14. Rank diagrams are adopted to compare the new method with the SV approach. The results illustrate that the introduction of CNOP has higher reliability for medium-range ensemble forecasts.     

Optimal Initial Error Growth in the Prediction of the Kuroshio Large Meander Based on a High-resolution Regional Ocean Model

Based on the high-resolution Regional Ocean Modeling System(ROMS) and the conditional nonlinear optimal perturbation(CNOP) method, this study explored the effects of optimal initial errors on the prediction of the Kuroshio large meander(LM) path, and the growth mechanism of optimal initial errors was revealed. For each LM event, two types of initial error(denoted as CNOP1 and CNOP2) were obtained. Their large amplitudes were found located mainly in the upper 2500 m in the upstream region of the LM, i.e., southeast of Kyushu. Furthermore, we analyzed the patterns and nonlinear evolution of the two types of CNOP. We found CNOP1 tends to strengthen the LM path through southwestward extension. Conversely,CNOP2 has almost the opposite pattern to CNOP1, and it tends to weaken the LM path through northeastward contraction.The growth mechanism of optimal initial errors was clarified through eddy-energetics analysis. The results indicated that energy from the background field is transferred to the error field because of barotropic and baroclinic instabilities. Thus, it is inferred that both barotropic and baroclinic processes play important roles in the growth of CNOP-type optimal initial errors.     

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。     

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.     

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.     

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。     

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...     

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.     

An Adjoint-Free CNOP–4DVar Hybrid Method for Identifying Sensitive Areas in Targeted Observations: Method Formulation and Preliminary Evaluation

This paper proposes a hybrid method, called CNOP–4 DVar, for the identification of sensitive areas in targeted observations, which takes the advantages of both the conditional nonlinear optimal perturbation(CNOP) and four-dimensional variational assimilation(4 DVar) methods. The proposed CNOP–4 DVar method is capable of capturing the most sensitive initial perturbation(IP), which causes the greatest perturbation growth at the time of verification; it can also identify sensitive areas by evaluating their assimilation effects for eliminating the most sensitive IP. To alleviate the dependence of the CNOP–4 DVar method on the adjoint model, which is inherited from the adjoint-based approach, we utilized two adjointfree methods, NLS-CNOP and NLS-4 DVar, to solve the CNOP and 4 DVar sub-problems, respectively. A comprehensive performance evaluation for the proposed CNOP–4 DVar method and its comparison with the CNOP and CNOP–ensemble transform Kalman filter(ETKF) methods based on 10 000 observing system simulation experiments on the shallow-water equation model are also provided. The experimental results show that the proposed CNOP–4 DVar method performs better than the CNOP–ETKF method and substantially better than the CNOP method.     

条件非线性最优扰动方法在黑潮目标观测研究中的应用

对近年来利用条件非线性最优扰动（Conditional Nonlinear Optimal Perturbation,CNOP）方法开展的黑潮目标观测研究进行了总结,主要包括日本南部黑潮路径变异的目标观测研究、黑潮延伸体模态转变的目标观测研究和源区黑潮流量变化的目标观测研究。通过计算这些事件的CNOP型扰动,发现这些事件的CNOP型扰动具有局地特征,可以作为实施目标观测的敏感区。理想回报试验结果表明,如果在由CNOP方法识别的敏感区内实施目标观测,则会大幅度提高上述事件的预报技巧。     

Exploring the phase-strength asymmetry of the North Atlantic Oscillation using conditional nonlinear optimal perturbation

Negative-phase North Atlantic Oscillation(NAO) events are generally stronger than positive-phase ones, i.e., there is a phase-strength asymmetry of the NAO. In this work, we explore this asymmetry of the NAO using the conditional nonlinear optimal perturbation(CNOP) method with a three-level global quasi-geostrophic spectral model. It is shown that, with winter climatological flow forcing, the CNOP method identifies the perturbations triggering the strongest NAO event under a given initial constraint. Meanwhile, the phase-strength asymmetry characteristics of the NAO can be revealed. By comparing with linear results, we find that the process of perturbation self-interaction promotes the onset of negative NAO events, which is much stronger than during positive NAO onset. Results are obtained separately using the climatological and zonal-mean flows in boreal winter(December–February) 1979–2006 as the initial basic state. We conclude, based on the fact that NAO onset is a nonlinear initial-value problem, that phase-strength asymmetry is an intrinsic characteristic of the NAO.     

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.