GRAPES的新初始化方案

四维变分同化由于引入预报模式作为一项约束,理论上它的分析场已经具有较好的平衡性,但实施时还会有诸多因重力波导致的高频振荡过程,因此,四维变分同化(4DVar)分析仍需要初始化。文中描述了GRAPES全球四维变分同化系统(GRAPES-4DVar)的新初始化方案的科学设计、公式演绎以及试验结果。GRAPES-4DVar的新初始化方案采用数字滤波方案作为代价函数的一项约束控制重力波引发的不平衡结构,约束强加在分析增量上与极小化迭代过程同步进行。新的初始化方案是变分同化系统的一部分,数字滤波的积分时间与4DVar的同化时间窗一致,不会对4DVar产生额外的计算资源消耗;并能适应长时间窗的同化,不会因为时间窗的延长而削弱慢波过程。新初始化方案中,模式轨迹的光滑程度可在变分同化中通过重力波控制项的权重系数方便控制。GRAPES全球四维变分同化的理想和循环同化批量试验都表明,在四维变分同化中,重力波的控制依然非常重要,具有初始化的GRAPES试验,无论分析还是预报技巧都较无初始化的有明显优势。与以前分析和滤波独立实施的旧初始化方案相比,新方案的分析和预报效果略优,同时有效地节省循环同化系统的运行时间,这对四维变分同化来说非常重要。     

线性化物理过程对GRAPES 4DVAR同化的影响

线性化物理过程能够改善四维变分同化中极小化收敛的稳定性和增加极小化过程中对大气物理过程和动力更加精确的描述,它是四维变分同化中非常重要的一部分。通过在GRAPES全球模式中研究线性化物理过程,尤其是两个湿线性化物理过程,改善切线性模式预报精度,来提高GRAPES全球四维变分同化的分析和预报效果。线性化物理过程的开发首先需要简化原非线性化物理过程中的强非线性项,然后对线性化物理过程进行规约化,以抑制切线性扰动的异常增长。目前GRAEPS全球模式中的线性化物理过程主要包括次网格尺度地形参数化、垂直扩散、积云深对流和大尺度凝结。线性化物理过程预报精度的检验方法是通过选择合适大小的初始扰动（同化分析增量）,来比较非线性模式和切线性模式中的扰动演化的纬向平均误差。然后以绝热版本的切线性模式为基础,通过冬、夏两个个例试验来分别检验4个线性化物理过程的12 h预报效果。试验结果表明,通过添加次网格地形参数化和垂直扩散两个干线性化物理过程方案,可以有效抑制住绝热版本切线性模式低层扰动的异常增长,大幅度改善切线性模式预报效果。通过添加积云深对流和大尺度凝结两个湿线性化物理过程,可以在热带区域和中、高纬度地区提高切线性模式中湿变量和温度变量的近似精度,提高GRAPES全球四维变分同化的分析和预报效果。      

GRAPES全球三维变分同化业务系统性能

近年来,GRAPES全球三维变分同化系统分析性能和稳定性有了长足进步。该文简要介绍了近两年GRAPES全球:三维变分同化技术的发展与改进情况,包括同化框架技术、资料同化应用技术与系统稳定性等方面。分析诊断了两年的同化循环试验结果,以探空资料作为参考,对ERA-Interim再分析场、NCEP FNL分析场和GRAPES全球三维变分分析场的统计特征进行了比较;以ERA-Interim再分析场作为参考,对NCEP FNL分析场、T639分析场和GRAPES全球三维变分分析场进行比较。结果表明:GRAPES分析场的质量明显优于T639分析场,性能上达到了业务化的要求,但相比NCEP FNL分析场还有一定差距,特别是对流层内湿度分析场的误差还比较大。     

MPAS动力框架和伴随模式简介以及未来扩展和应用

跨尺度大气预报模式(MPAS-A)的动力学框架是非结构化的球面质心Voronoi网格,它代表了过去几十年来数值天气预报最重要进展之一。MPAS-A具有开放性、计算机程序和文件规范性、科学性及先进性等特点,因此可选其作为未来全球四维变分资料同化系统模式,目前已经发展了MPAS-A的切线线性模式和伴随模式。选择MPAS-A为全球四维变分资料同化系统模式,不仅可以避免MPAS-A预报模式网格与资料同化网格之间的来回插值所产生的误差,减少每次极小化迭代时MPAS-A模式变量与资料同化分析变量之间的转换所需计算时间,同时也为非结构化球面质心网格的跨尺度全球资料同化研究提供新机遇。     

基于扰动模式的四维变分资料同化系统框架的设计完善和数值试验

为了建立一个应用于区域数值预报的四维变分资料同化（4DVar）系统,在近期开发的扰动预报模式GRAPES_PF基础上,开发完善增量四维变分同化系统框架。该框架中暂不包含物理过程（长短波辐射、边界层过程、对流参数化和云微物理等）。对比业务使用的GRAPES 3DVar系统,增加了温度控制变量。将无量纲Exner气压与流函数的线性风压平衡方程直接在地形追随垂直坐标面上求解,且通过广义共轭余差法（GCR）求解扰动亥姆霍兹（Helmholtz）伴随方程。利用人造“探空”资料对2015年10月台风“彩虹”进行了理想数值试验。试验结果表明,所开发的扰动四维变分同化框架得到了预期的结果,即同化更多资料并反复受到模式约束的四维变分同化系统能有效改善初值质量,进而改善区域数值预报。建立的区域四维变分同化框架合理可行,为进一步发展包含完整物理过程的区域四维变分同化系统奠定了研究基础。      

GRAPES集合卡尔曼滤波资料同化系统Ⅱ：区域分析及集合预报

GRAPES集合卡尔曼滤波资料同化方法能够分批同化常规观测资料,GRAPES集合卡尔曼滤波同化系统的设计及其与GRAPES三维变分同化系统的对比试验结果表明,GRAPES集合卡尔曼滤波系统能够得到合理的分析,并且具有实际运行能力。在此基础上,进行集合卡尔曼滤波区域同化分析及集合预报试验,对比区域模式面三维变分同化分析预报结果,研究表明,集合卡尔曼滤波分析比三维变分分析具有一定优势,降水预报更接近实况。考察了预报误差特征随天气形势的变化情况,表明预报误差相关场和均方差的分布随着天气形式不同而变化。     

COSMIC资料在GRAPES全球三维变分同化系统的初步研究

利用COSMIC获得的GPS无线电掩星数据所反演得到的大气温湿度廓线资料,在GRAPES全球三维变分同化系统中做连续循环同化试验,研究了加入COSMIC资料后得到的GRAPES分析场及其5天中期预报是否有所改善。研究结果表明：通过加入COSMIC资料连续循环同化得到的分析场,相对原来没有加COSMIC资料连续同化出来的GRAPES分析场有明显改善,由此GRAPES全球模式全球5天中期预报水平有明显提高,其中加入没有稀疏化后COSMIC资料对南半球GRAPES分析场的改善以及GRAPES全球模式5天中期预报水平的提高都比较显著;加入稀疏化后的COSMIC资料对北半球GRAPES分析场的改善以及GRAPES全球模式5天中期预报水平的提高都有比较显著的效果。     

GRAPES全球三维变分同化中卫星微波温度计亮温的背景误差及在质量控制中的应用

模式变量背景误差在观测空间的投影,也即观测变量的背景误差包含了变分同化系统的重要信息,其在诊断和分析变分同化系统中资料的影响等方面具有重要作用,特别是在背景场检查质量控制中。在GRAPES全球三维变分同化(3DVar)系统中仅给定了控制变量的背景误差,并未直接给定观测变量的背景误差。为了能够对GRAPES全球3DVar进行全面的诊断和分析,改进卫星微波温度计资料的质量控制,推导出GRAPES全球3DVar同化系统控制变量随机扰动方法估计观测变量的背景误差的公式,为分析和改进GRAPES全球3DVar提供了一个有力工具,并进而估计了AMSU-A亮温的背景误差,分析了AMSU-A不同通道亮温的背景误差特征,将其应用于GRAPES全球3DVar的AMSU-A亮温的背景场检查质量控制中。结果表明,控制变量随机扰动方法估计的GRAPES全球3DVar同化系统AMSU-A亮温的背景误差正确合理。同化循环预报试验结果表明,亮温的背景误差在背景场检查中的应用显著提高了GRAPES全球3DVar同化的亮温资料的数量,显著提高了GRAPES南半球对流层中高层位势高度场的预报技巧。在GRAPES全球3DVar同化系统中推导和实现的控制变量扰动方法为诊断和分析GRAPES全球3DVar观测资料同化效果提供了有力工具。      

极轨卫星高层通道在GRAPES同化系统中的应用

针对GRAPES(Global/Regional Assimilation and Prediction System)模式三维变分系统高层背景场温湿廓线外推方案的局限性,提出以气候垂直廓线重新构造高层温湿垂直结构,以减小外推方案的偏差。首先采用一维变分同化系统,展开模拟实验:分析目前模式中使用的外推方案误差及其对反演结果的影响,利用高层大气气候廓线构造垂直结构并分析同化偏差。最后,运用GRAPES全球分析预报系统进行同化实验并分析改进程度。结果显示:模拟研究表明采用高层背景场温湿廓线外推方案与实际观测相比最大偏差在1 h Pa附近可达数十度以上,不仅影响平流层,而且对对流层也有影响;用气候温度数据修正GRAPES高层温度数据,可以减少50%以上的偏差,证明了用气候值高层数据优化现行GRAPES模式中同化系统高层插值方案的可行性。全球GRAPES三维变分同化试验结果显示,改进方案不仅显著的改善平流层分析质量,对对流层中高层也有改进。     

集合预报误差在GRAPES全球四维变分同化中的应用研究Ⅱ：参数选取与数值试验

研究的第一部分讨论了如何有效应用集合预报误差的科学方案,确定了集合预报误差在GRAPES（Global Regional Assimilation and PrEdiction System）全球4DVar（four dimensional variational data assimilation）中应用的分析框架。在此基础上研究了针对集合预报误差实际应用于GRAPES全球4DVar,解决接近或超过100个集合样本数时高效生成的计算效率问题,以及与GRAPES全球4DVar匹配的同化关键参数确定问题。选择基于4DVar的集合资料同化方法生成集合样本,通过将第1个样本极小化迭代过程中产生的预调节信息用于其他样本极小化做预调节,将计算效率提高了2倍。通过时间错位扰动方法增加集合样本数,实现集合样本增加到3倍。对集合方差进行膨胀,并选择水平局地化相关尺度为流函数背景误差水平相关的1.4倍。通过批量数值试验方法确定背景误差与集合预报误差的权重系数,对60个集合样本当集合预报误差权重为0.7时预报效果最好。对北半球夏、冬两季各52 d的批量试验表明,对于南、北半球En4DVar （ensemble 4DVar）较4DVar的改进在冬季主要集中在700—30 hPa,而在夏季主要集中在400—150 hPa。赤道地区受季节影响较小,En4DVar对位势高度、风场与温度的改进都较为明显,且经向风场的改进最为显著。文中研发的集合预报误差在GRAPES全球4DVar中应用的方法合理可行。      

Conjugate Gradient Algorithm in the Four-Dimensional Variational Data Assimilation System in GRAPES

Minimization algorithms are singular components in four-dimensional variational data assimilation (4DVar). In this paper, the convergence and application of the conjugate gradient algorithm (CGA), which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework, are investigated in the 4DVar minimization. First, the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied. The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization, while the orthogonalized Lanczos algorithm converges stably. Second, the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System (GRAPES). The CGA is 40% more computationally efficient than the quasi-Newton method, although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method. Thus, the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.     

GRAPES全球切线性和伴随模式的调优

伴随技术是四维变分同化（4DVar）系统中计算代价函数梯度的最佳办法,切线性和伴随模式的效果和效率直接影响着4DVar系统的发展。基于GRAPES（Global and Regional Assimilation PrEdiction System）全球切线性和伴随模式1.0版本,利用GRAPES全球模式2.0版本在并行框架和性能等方面的改善,重新优化和设计了GRAPES全球切线性伴随模式2.0版本,提高了GRAPES全球切线性和伴随模式的效果和效率,优化了切线性模式程序结构,使其计算时间最优可控制在非线性模式的1.2倍以内;采用在切线性模式中保存基态的方法,重构了伴随模式的程序结构,使其计算时间最优控制在非线性模式的1.5倍以内;在GRAPES全球切线性物理过程的设计中,将线性物理过程的轨迹基态计算和切线性扰动计算解耦,提高了GRAPES全球切线性和伴随模式的计算效果和效率。     

The GRAPES Variational Bias Correction Scheme and Associated Preliminary Experiments

The variational assimilation theory is generally based on unbiased observations. In practice, however, almost all observations suffer from biases arising from observational instruments, radiative transfer operator, precondition of data, and so on. Therefore, a bias correction scheme is indispensable. The current scheme for radiance bias correction in the GRAPES 3DVar system is an offline scheme. It is actually a static correction for the radiance bias before the process of cost function minimization. In consideration of its effects on forecast results, this kind of scheme has some shortcomings. Thus, this study provides a variational bias correction (VarBC) scheme for the GRAPES 3DVar system following Dee’s idea. In the VarBC scheme, the observation operator is modified and a new control variable is defined by taking the predictor coefficients as the control parameters. According to the feature of the GRAPES-3DVAR, an incremental formulation is applied and the original bias correction scheme is maintained in the actual process of observations. The VarBC is designed to co-exist with the original scheme, because it is a dynamic revision to the observational operator on the basis of the old method, i.e., it adjusts the model state vector along with the control parameters to an unbiased state in the process of minimization and the assimilation system remains consistent with available information automatically. Preliminary experimental results show that the mean departures of background-minus-observation and analysis-minus-observation are reduced as expected. In a case study of the heavy rainfall that happened in South China on 11–13 June 2008, the 500-hPa geopotential height is better simulated using the analyzed field from the VarBC as the initial condition.     

集合预报误差在GRAPES全球四维变分同化中的应用研究Ⅰ：局地化方案设计与动力平衡特征分析

通过引入流依赖的集合预报误差,使得同化分析与天气形势紧密相关,是改善初值分析质量的重要途径。文中在GRAPES（Global Regional Assimilation and PrEdiction System）全球四维变分资料同化（4DVar）中研究了如何有效应用集合预报误差,包括增加扩展控制变量时如何降低其计算消耗以及如何在局地化过程中保持不同变量之间的动力平衡。利用高斯分布的谱滤波实现水平局地化,利用垂直正交经验函数分解实现垂直局地化,并采用前8个主导特征模态来限制控制变量空间维数增加。引入20至180个集合样本,在水平二维局地化情形下,控制变量总数的增长可以限制在1.1—1.8倍,而在三维局地化情形下,控制变量总数的增长限制在1.7—7.1倍。对60个集合样本和1°水平分辨率内循环,4DVar引入扩展控制变量后墙钟时间增加了约30%。进一步,通过采用在非平衡分析变量上进行水平局地化,然后再将风压地转平衡关系重新叠加到非平衡分析变量上,使得分析更好地保持了风压平衡关系,初始场地面气压倾向变化减小。此外,虽然垂直局地化对分析平衡影响较大,但依靠目标函数中的数字滤波弱约束,分析变量之间仍能较好满足动力平衡关系。结果表明,GRAPES全球4DVar中发展的增加扩展控制变量、谱滤波实现水平局地化、非平衡分析变量进行水平局地化等有效应用集合预报误差的方法,适合集合样本数超过100个的情况,在分析质量改善的同时,4DVar系统的计算和存储消耗没有显著增加。      

The second order adjoint analysis: Theory and applications

Summary The adjoint method application in variational data assimilation provides a way of obtaining the exact gradient of the cost functionj with respect to the control variables. Additional information may be obtained by using second order information. This paper presents a second order adjoint model (SOA) for a shallow-water equation model on a limited-area domain. One integration of such a model yields a value of the Hessian (the matrix of second partial derivatives, ² J) multiplied by a vector or a column of the Hessian of the cost function with respect to the initial conditions. The SOA model was then used to conduct a sensitivity analysis of the cost function with respect to distributed observations and to study the evolution of the condition number (the ratio of the largest to smallest eigenvalues) of the Hessian during the course of the minimization. The condition number is strongly related to the convergence rate of the minimization. It is proved that the Hessian is positive definite during the process of the minimization, which in turn proves the uniqueness of the optimal solution for the test problem. Numerical results show that the sensitivity of the response increases with time and that the sensitivity to the geopotential field is larger by an order of magnitude than that to theu andv components of the velocity field. Experiments using data from an ECMWF analysis of the First Global Geophysical Experiment (FGGE) show that the cost functionJ is more sensitive to observations at points where meteorologically intensive events occur. Using the second order adjoint shows that most changes in the value of the condition number of the Hessian occur during the first few iterations of the minimization and are strongly correlated to major large-scale changes in the reconstructed initial conditions fields.With 17 Figures     

An Extension of the Dimension-Reduced Projection 4DVar

This paper extends the dimension-reduced pro- jection four-dimensional variational assimilation method （DRP-4DVar） by adding a nonlinear correction process, thereby forming the DRP-4DVar with a nonlinear correction, which shall hereafter be referred to as the NC-DRP- 4DVar. A preliminary test is conducted using the Lorenz-96 model in one single-window experiment and several multiple-window experiments. The results of the single-window experiment show that compared with the adjoint-based traditional 4DVar, the final convergence of the cost function for the NC-DRP-4DVar is almost the same as that using the traditional 4DVar, but with much less computation. Furthermore, the 30-window assimilation experiments demonstrate that the NC-DRP-4DVar can alleviate the linearity approximation error and reduce the root mean square error significantly.     

IMPACT OF DIABATIC PROCESSES IN AGCM ON 4-DIMENSIONAL VARIATIONAL DATA ASSIMILATION*

The impact of diabatic processes on 4-dimensional variational data assimilation (4D-Var) was studied using the 1995 version of NCEP's global spectral model with and without full physics.The adjoint was coded manually.A cost function measuring spectral errors of 6-hour forecasts to "observation" (the NCEP reanalysis data) was minimized using the L-BFGS (the limited memory quasi-Newton algorithm developed by Broyden,Fletcher,Goldfard and Shanno) for optimizing parameters and initial conditions.Minimization of the cost function constrained by an adiabatic version of the NCEP global model converged to a minimum with a significant amount of decrease in the value of the cost function.Minimization of the cost function using the diabatic model, however,failed after a few iterations due to discontinuities introduced by physical parameterizations.Examination of the convergence of the cost function in different spectral domains reveals that the large-scale flow is adjusted during the first 10 iterations,in which discontinuous diabatic parameterizations play very little role.The adjustment produced by the minimization gradually moves to relatively smaller scales between 10-20th iterations.During this transition period,discontinuities in the cost function produced by "on-off" switches in the physical parameterizations caused the cost function to stay in a shallow local minimum instead of continuously decreasing toward a deeper minimum. Next,a mixed 4D-Var scheme is tested in which large-scale flows are first adiabatically adjusted to a sufficient level,followed by a diabatic adjustment introduced after 10 to 20 iterations. The mixed 4D-Var produced a closer fit of analysis to observations,with 38% and 41% more decrease in the values of the cost function and the norm of gradient,respectively,than the standard diabatic 4D-Var,while the CPU time is reduced by 21%.The resulting optimal initial conditions improve the short-range forecast skills of 48-hour statistics.The detrimental effect of parameterization discontinuities on minimization was also reduced.