Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): I. Mathematical Framework

This work presents the “nth-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “nth-FASAM-N”), which will be shown to be the most efficient methodology for computing exact expressions of sensitivities, of any order, of model responses with respect to features of model parameters and, subsequently, with respect to the model’s uncertain parameters, boundaries, and internal interfaces. The unparalleled efficiency and accuracy of the nth-FASAM-N methodology stems from the maximal reduction of the number of adjoint computations (which are considered to be “large-scale” computations) for computing high-order sensitivities. When applying the nth-FASAM-N methodology to compute the second- and higher-order sensitivities, the number of large-scale computations is proportional to the number of “model features” as opposed to being proportional to the number of model parameters (which are considerably more than the number of features).When a model has no “feature” functions of parameters, but only comprises primary parameters, the nth-FASAM-N methodology becomes identical to the extant nth CASAM-N (“nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the nth-FASAM-N and the nth-CASAM-N methodologies are formulated in linearly increasing higher-dimensional Hilbert spaces as opposed to exponentially increasing parameter-dimensional spaces thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. Both the nth-FASAM-N and the nth-CASAM-N are incomparably more efficient and more accurate than any other methods (statistical, finite differences, etc.) for computing exact expressions of response sensitivities of any order with respect to the model’s features and/or primary uncertain parameters, boundaries, and internal interfaces.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): II. Illustrative Example

This work highlights the unparalleled efficiency of the “nth-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The nth-FASAM-N methodology is compared to the extant “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) showing that: (i) the 1st-FASAM-N and the 1st-CASAM-N methodologies are equally efficient for computing the first-order sensitivities;each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1st-LASS);(ii) the 2nd-FASAM-N methodology is considerably more efficient than the 2nd-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters;specifically for the Nordheim-Fuchs model, the 2nd-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2nd-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3rd-FASAM-N methodology is even more efficient than the 3rd-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3rd-FASAM-N methodology, while the application of the 3rd-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the nth-FASAM-N and the nth-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.

Melt Pond Scheme Parameter Estimation Using an Adjoint Model

Melt ponds significantly affect Arctic sea ice thermodynamic processes.The melt pond parameterization scheme in the Los Alamos sea ice model(CICE6.0) can predict the volume,area fraction(the ratio between melt pond area to sea ice area in a model grid),and depth of melt ponds.However,this scheme has some uncertain parameters that affect melt pond simulations.These parameters could be determined through a conventional parameter estimation method,which requires a large number of sensitivity simulations.The adjoint model can calculate the parameter sensitivity efficiently.In the present research,an adjoint model was developed for the CESM(Community Earth System Model) melt pond scheme.A melt pond parameter estimation algorithm was then developed based on the CICE6.0 sea ice model,melt pond adjoint model,and L-BFGS(Limited-memory Broyden-Fletcher-Goldfard-Shanno) minimization algorithm.The parameter estimation algorithm was verified under idealized conditions.By using MODIS(Moderate Resolution Imaging Spectroradiometer)melt pond fraction observation as a constraint and the developed parameter estimation algorithm,the melt pond aspect ratio parameter in CESM scheme,which is defined as the ratio between pond depth and pond area fraction,was estimated every eight days during summertime for two different regions in the Arctic.One region was covered by multi-year ice(MYI) and the other by first-year ice(FYI).The estimated parameter was then used in simulations and the results show that:(1) the estimated parameter varies over time and is quite different for MYI and FYI;(2) the estimated parameter improved the simulation of the melt pond fraction.

Theoretical Aspect of Suitable Spatial Boundary Condition Specified for Adjoint Model on Limited Area

Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers. Key words Variational data assimilation - Adjoint model - Over-specified partial boundary condition This research work is sponsored by the National Key Programme for Developing Basic Sciences (G1998040907), the Project of Natural Science Foundation of Jiangsu Province (BK99020), the President Foundation of Nanjing University (985) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

The Improvement Made by a Modified TLM in 4DVAR with a Geophysical Boundary Layer Model

The strong nonlinearity of boundary layer parameterizations in atmospheric and oceanic models can cause difficulty for tangent linear models in approximating nonlinear perturbations when the time integration grows longer. Consequently, the related 4—D variational data assimilation problems could be difficult to solve. A modified tangent linear model is built on the Mellor-Yamada turbulent closure (level 2.5) for 4-D variational data assimilation. For oceanic mixed layer model settings, the modified tangent linear model produces better finite amplitude, nonlinear perturbation than the full and simplified tangent linear models when the integration time is longer than one day. The corresponding variational data assimilation performances based on the adjoint of the modified tangent linear model are also improved compared with those adjoints of the full and simplified tangent linear models.

中尺度数值模式的资料同化系统——(一)伴随模式的设计

以一个有限区域的中尺度模式为基础，采用伴随模式技术进行有限区域气象资料的同化。伴随模式的方法是以数值天气预报的动力模式作为约束条件的变分方法，比传统的变分方法有了很大的改进。本文初步探讨了伴随模式系统的设计方法，特别是伴随模式的构造问题；用共轭码的方法导出伴随模式；初步试验表明该系统有较强的同化能力。

伴随模式同化系统在修正模式地形中的应用

任何模式都只是实际大气的一种近似模拟,存在误差。鉴于传统的四维资料伴随模式同化系统都是假设模式完全正确仅对初始场进行修正。将伴随模式同化系统应用于修正模式地形误差,通过对不同初始地形的修正试验表明,MM5伴随模式同化系统能很好地对地形误差进行修正,能够反演出与初始气象要素场分辨率相匹配、与模式更协调的地形场,为伴随模式同化系统的广泛应用提供了一种新的思路。

中尺度模式的四维伴随同化系统及其同化试验

以中尺度模式MM4为基础,利用伴随码技术改进、完善了MM4伴随模式同化系统,并利用该系统进行了常规资料和非常规资料的伴随模式同化试验,例如加入了可降水量资料等。试验表明:在伴随模式同化系统中加入常规和非常规资料,可以改进初始场,从而改善预报场。

伴随同化系统中用云导风修正模式地形的试验

以2002年7月23日的天气过程为例,利用中尺度模式MM5伴随模式同化系统,进行了修正模式地形高度的试验研究。结果发现,MM5伴随模式同化系统可以对地形进行修正,并得到与资料更为协调一致的地形高度。另外,利用非常规云导风资料能够比仅使用常规资料更有效地修正模式地形高度。将修正后的地形场作为模式地形,所得的预报结果比未经修正的地形有所改进。

中尺度数值模式的资料同化系统——(二)伴随模式系统的检验与试验

在《中尺度数值模式的资料同化系统——（一）伴随模式的设计》一文的基础上，进一步对伴随模式码的检验问题做了讨论。表明采用共轭码方法是可行的。其导出的伴随模式与理论推导的共轭方程是相容的。试验发现该同化系统有消除随机干扰的能力，同化效果好。在同化过程中，系统是先同化大尺度信息，然后逐步同化中小尺度信息。

云导风资料同化在伴随模式同化系统中的应用

以一次发生在长江流域的暴雨过程为例,设计了几种数值试验方案,并进行模拟。结果表明,MM5伴随模式同化系统能有效改善初始场与模式的协调能力,提高模式对于降水场和其它要素场的预报;使用云导风资料修正初始场后直接模拟的效果比未使用时直接模拟的效果要好,对部分区域的强降水预报精度有一定程度的改善;使用伴随模式同化系统后,加入云导风资料的同化试验对其它要素的改善与直接同化常规资料的效果相比,改善优势不明显,但从各要素的误差来看,对于风场的改善最好。

伴随模式同化系统在修正模式误差中的应用研究

资料误差和模式误差都是制约数值预报结果准确性提高的关键因素,传统四维资料同化都是假设模式完全正确仅对初始资料进行修正,忽略了模式误差本身造成的预报误差。伴随模式同化系统不仅具有修正资料误差的能力,也可应用于修正模式误差方面的研究。本文将卫星中心导风系统提供的GMS5卫星风场与非常规温度资料用于MM5伴随同化系统修正模式地形误差进行试验性研究表明,该方法能够反演出一个既与初始气象要素场相匹配,又与模式更协调的地形场,得到比一般包络地形更好的效果,改善模式对强降水中心及降水区域的预报;数值试验结果还揭示模式误差对模式预报造成一定的影响,用伴随方法对观测资料进行修正的同时也对模式误差进行修正的方法是可行的。

卫星云导风资料应用于伴随同化系统的数值试验研究

以2002年6月14日00时至15日00时发生在江南、华南一带的暴雨过程为例,利用MM5伴随模式同化系统,同化常规观测资料以及卫星云导风资料,研究MM5伴随模式对降水预报的改善以及云导风资料对于提高暴雨预报的作用。设计了4种数值试验方案,试验结果表明:MM5伴随同化系统对降水预报有较好的改善作用;GMS-5卫星云导风资料的加入增加了高空风场资料,将其同化到伴随同化系统中能有效改善模式初始场,提高了降水预报质量,特别对暴雨强降水的预报起到了较好的改善作用。

变分同化方法在Lorenz系统中的简单应用研究

利用Lorenz模式作变分同化数值试验 ,通过对一个简单系统的讨论 ,介绍四维变分同化方法。对初值敏感性和观测点的个数及观测值作了对比试验 ,发现随着模式对初值敏感性的增加 ,同化效果会越来越差 ;观测点越少 ,观测值误差越大 ,这些都会影响同化效果 ,甚至导致同化失败。

REM模式伴随系统的建立及其四维变分资料同化初步试验

Regional-Eta-Coordinate-Model(REM)中尺度模式对中国区域性降水显示出公认的较高预报能力,建立其四维变分资料同化系统是完善该模式,进一步提高其预报效果的重要工作。本研究编写了REM模式的切线性模式和伴随模式,介绍了建立REM模式伴随系统的过程,并利用实际天气个例资料,分别对REM模式的切线性模式、伴随模式及定义的目标函数梯度进行了正确性检验,检验结果表明对REM模式的切线性模式及伴随模式编写是成功的。利用REM模式的伴随系统,对1998年06月08日00时到09日00时和2000年08月01日00时到02日00时两个实际天气个例进行了四维变分资料同化试验。从数值试验的结果分析可以看到,进行四维变分资料同化后,两个天气个例在预报结束时刻其预报结果对风场和湿度场的预报都有明显改善,对温度场和高度场的预报也有所改善。对于累积降水的预报,两个个例利用四维变分资料同化后得到的初始场进行的预报结果则有较大不同,在个例1中,变分同化后对降水中心的位置和降水强度的预报都有明显改善,预报结果更接近于观测场;个例2中,变分同化后对降水中心位置和强度的预报则没有改善,产生这种现象的原因可能是由于定义的目标函数中没有加进背景场项,也可能是由于采用的观测资料时次比较少,还需要进一步进行研究和试验。

换相重叠角的计算机仿真及辩识

为避免繁杂的等效电抗测量计算,在确定发电机无刷励磁系统换相重叠角时,将其作为系统辨识的一个待估参数.在估计出电枢电感参数的基础上,从二极管的广义牛顿拉夫逊迭代模型与储能元件的离散化伴随模型出发,对无刷励磁系统的三相不可控整流电路进行仿真,并与换相重叠角的近似解析公式比较.经验证,仿真结果与解析式相对误差均值在1%以内,说明模型在这两个参数变动范围内有良好的泛化与逼近能力,可作为进一步发电机评估建模基础.仿真程序可以代替实际的无刷励磁模型,参与换相重叠角的辩识及可行的重叠值范围估计.

伴随系统及非线性优化方法在REM模式可预报性研究中的实际个例应用

利用REM模式的伴随系统和非线性优化方法,通过三个实际天气个例,对REM模式的可预报性问题进行了研究。结果表明,REM模式在给定的实际应用中可接受的预报误差范围内,对三个天气个例都具有预报能力。对于个例一,利用现有的常规报文初始观测场,进行简单的插值处理(最优插值等),REM数值模式就可以得到比较满意的预报结果;对于个例二和个例三,对现有的报文初始观测场进行处理(如四维变分资料同化)后,REM模式在给定的误差允许范围内,对这两个天气个例仍得到满意的预报。研究结果不仅对改进数值模式具有一定的指导意义,而且对如何改进数值模式的初值问题,特别是在中尺度天气预报中如何改进具有一定的参考价值。

具有生长特征的L-系统模型研究

传统的L系统对于植物生长特征的表达是将表示分枝结构和枝节生长的产生式加以区别,迭代生成描述植物结构的文法系统并结合海龟解释实现植物的可视化模拟,很难真正体现植物生长过程中的"生命"特征。综合考虑L系统对植物生长的表达能力,将具有一定生理年龄和生长年龄的生长单元视为子结构,考虑植物生长的随机特性,植物的动态生长按照隐式马尔可夫过程进行状态转移,从而提出一种广义的L-系统模型,符合植物本身的生长机理。实例验证表明,提出的模型可快速、有效地实现虚拟植物的可视化模型的建立。

Sensitivity Experiments of an Eastward-Moving Southwest Vortex to Initial Perturbations

Whether the initial conditions contain pronounced mesoscale signals is important to the simulation of the southwest vortex. An eastward-moving southwest vortex is simulated using the PSU/NCAR MM5. A modest degree of success is achieved, but the most serious failure is that the formation and displacement of the simulated vortex in its early phase are about fourteen hours later than the observed vortex. Considering the relatively sparse data on the mesoscale vortex and in an attempt to understand the cause of the forecast failure, an adjoint model is used to examine the sensitivity of the southwest vortex to perturbations of initial conditions. The adjoint sensitivity indicates how small perturbations of model variables at the initial time in the model domain can influence the vortex. A large sensitivity for zonal wind is located under 400 hPa, a large sensitivity for meridional wind is located under 500 hPa, a large sensitivity for temperature is located between 500 and 900 hPa, and almost all of the large sensitivity areas are located in the southwestern area. Based on the adjoint sensitivity results, perturbations are added to initial conditions to improve the simulation of the southwest vortex. The results show that the initial conditions with perturbations can successfully simulate the formation and displacement of the vortex; the wind perturbations added to the initial conditions appear to be a cyclone circulation under the middle level of the atmosphere in the southwestern area with an anticyclone circulation to its southwest; a water vapor perturbation added to initial conditions can strengthen the vortex and the speed of its displacement.

Application and Characteristic Analysis of the Moist Singular Vector in GRAPES-GEPS

The singular vector(SV)initial perturbation method can capture the fastest-growing initial perturbation in a tangent linear model(TLM).Based on the global tangent linear and adjoint model of GRAPES-GEPS(Global/Regional Assimilation and Prediction System-Global Ensemble Prediction System),some experiments were carried out to analyze the structure of the moist SVs from the perspectives of the energy norm,energy spectrum,and vertical structure.The conclusions are as follows:The evolution of the SVs is synchronous with that of the atmospheric circulation,which is flowdependent.The moist and dry SVs are located in unstable regions at mid-to-high latitudes,but the moist SVs are wider,can contain more small-and medium-scale information,and have more energy than the dry SVs.From the energy spectrum analysis,the energy growth caused by the moist SVs is reflected in the relatively small-scale weather system.In addition,moist SVs can generate perturbations associated with large-scale condensation and precipitation,which is not true for dry SVs.For the ensemble forecasts,the average anomaly correlation coefficient of large-scale circulation is better for the forecast based on moist SVs in the Northern Hemisphere,and the low-level variables forecasted by the moist SVs are also improved,especially in the first 72 h.In addition,the moist SVs respond better to short-term precipitation according to statistical precipitation scores based on 10 cases.The inclusion of the large-scale condensation process in the calculation of SVs can improve the short-term weather prediction effectively.