A Local Implementation of the POD-Based Ensemble 4DVar with R-Localization

The purpose of this paper is to provide a robust and flexible implementation of a proper orthogonal decomposition-based ensemble four-dimensional variational assimilation method(PODEn4DVar) through Rlocalization.With R-localization,the implementation of the local PODEn4DVar analysis can be coded for parallelization with enhanced assimilation precision.The feasibility and effectiveness of the PODEn4DVar local implementation with R-localization are demonstrated in a two-dimensional shallow-water equation model with simulated observations(OSSEs) in comparison with the original version of the PODEn4DVar with B-localization and that without localization.The performance of the PODEn4DVar with localization shows a significant improvement over the scheme with no localization,particularly under the imperfect model scenario.Moreover,the R-localization scheme is capable of outperforming the Blocalization case to a certain extent.Further,the assimilation experiments also demonstrate that PODEn4DVar with R-localization is most efficient due to its easy parallel implementation.     

Doppler Radar Data Assimilation with a Local SVD-En3DVar Method

An observation localization scheme is introduced into an ensemble-based three-dimensional variational (3DVar) assimilation method based on the singular value decomposition technique (SVD-En3DVar) to improve assimilation skill. A point-by-point analysis technique is adopted in which the weight of each observation decreases with increasing distance between the analysis point and the observation point. A set of numerical experiments, in which simulated Doppler radar data are assimilated into the Weather Research and Forecasting (WRF) model, is designed to test the scheme. The results are compared with those obtained using the original global and local patch schemes in SVD-En3DVar, neither of which includes this type of observation localization. The observation localization scheme not only eliminates spurious analysis increments in areas of missing data, but also avoids the discontinuous analysis fields that arise from the local patch scheme. The new scheme provides better analysis fields and a more reasonable short-range rainfall forecast than the original schemes. Additional forecast experiments that assimilate real data from 10 radars indicate that the short-term precipitation forecast skill can be improved by assimilating radar data and the observation localization scheme provides a better forecast than the other two schemes.     

基于历史样本投影的四维变分陆面数据同化方法及其初步应用

法面临着计算量上的挑战。本研究将一种历史样本投影的四维变分同化方法(Historical-Sample-Projection4DVar,简写为HSP-4DVar)应用于陆面数据同化,建立起CoLM陆面模型的HSP-4DVar系统。相比其他四维变分同化方法,HSP-4DVar的分析值是显式求解,不需要编写和使用伴随模式,从而大大节省了计算量,是一种易于实现的同化方案。通过同化56个月的土壤湿度观测数据表明,新的陆面同化系统不仅省时,而且能够有效吸取观测信息,使得同化后的均方根误差显著降低,各层土壤湿度模拟都有所改善,陆表1000mm层的改善最为明显。     

一种新的资料同化方法

为寻求一种快速有效的四维变分资料同化(英文缩写4DVar)作了有意义的尝试,提出了映射观测的新概念和反向四维变分资料同化的新思路,并以此为基础建立了三维变分映射资料同化(英文缩写为3DVM:3-DimensionalVariational data assimilation of Mapped observation)。该方法与传统的四维变分资料同化一样,不仅考虑了模式的动力和物理约束,使得同化后的初值与模式协调,而且通过模式方程对同化窗口中不同时刻的观测资料作了最佳拟合。与传统四维变分同化方法不同的是,由3DVM得到的初值不在同化窗口的始端,而在窗口的末端。正是所求初值时刻的改变,使得该方法的计算代价大大减少,几乎与三维变分资料同化(英文缩写3DVar)相当,这实际上是用3DVar的代价实现了4DVar的功能。同时,由于3DVM不再需要切线性和伴随近似来计算代价函数的梯度也提高了同化的精度。对具体的台风个例(Dan)用AMSU-A反演的温度场进行变分同化模拟试验,发现3DVM能比传统4DVar产生更好的初值,而且所花计算时间只需4DVar的1/7。     

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。      

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。      

An Economical Approach to Four-dimensional Variational Data Assimilation

Four-dimensional variational data assimilation (4DVar) is one of the most promising methods to provide optimal analysis for numerical weather prediction (NWP). Five national NWP centers in the world have successfully applied 4DVar methods in their global NWPs, thanks to the increment method and adjoint technique. However, the application of 4DVar is still limited by the computer resources available at many NWP centers and research institutes. It is essential, therefore, to further reduce the computational cost of 4DVar. Here, an economical approach to implement 4DVar is proposed, using the technique of dimension-reduced projection (DRP), which is called ``DRP-4DVar." The proposed approach is based on dimension reduction using an ensemble of historical samples to define a subspace. It directly obtains an optimal solution in the reduced space by fitting observations with historical time series generated by the model to form consistent forecast states, and therefore does not require implementation of the adjoint of tangent linear approximation. To evaluate the performance of the DRP-4DVar on assimilating different types of mesoscale observations, some observing system simulation experiments are conducted using MM5 and a comparison is made between adjoint-based 4DVar and DRP-4DVar using a 6-hour assimilation window.     

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。     

Numerical experiments for Typhoon Dan incorporating AMSU-A retrieved data with 3DVM

Two sets of assimilation experiments on a landfalling typhoon—Typhoon Dan(1999)over the western North Pacific were designed to compare the performances of two kinds of variational data assimilation schemes that are the 3-Dimensional Variational data assimilation of Mapped observation(3DVM)and the 4-dimensional variational data assimilation(4DVar).Results show that:(1)both the 3DVM and 4DVar successfully improved the simulations of typhoon intensity and track incorporating the satellite AMSU-A retrieved temperature and wind data into the initial conditions,and the 3DVM more significantly due to the flow-dependent of background error covariance matrix and observation error covariance matrix like 3-dimensional variational data assimilation(3DVar)circle;(2)inclusions of extra model integration iterations at each observation time in the 3DVM make it more consistent with prediction model;(3)the 3DVM is much more time-saving due to the exclusion of the adjoint technique in it.     

Assimilation of Doppler Radar Data with an Ensemble 3DEnVar Approach to Improve Convective Forecasting

An ensemble three-dimensional ensemble-variational(3DEnVar)data assimilation(E3DA)system was developed within the Weather Research and Forecasting model’s 3DVar framework to assimilate radar data to improve convective forecasting.In this system,ensemble perturbations are updated by an ensemble of 3DEnVar and the ensemble forecasts are used to generate the flow-dependent background error covariance.The performance of the E3DA system was first evaluated against one experiment without radar DA and one radar DA experiment with 3DVar,using a severe storm case over southeastern China on 5 June 2009.Results indicated that E3DA improved the quantitative forecast skills of reflectivity and precipitation,as well as their spatial distributions in terms of both intensity and coverage over 3DVar.The root-mean-square error of radial velocity from 3DVar was reduced by E3DA,with stronger low-level wind closer to observation.It was also found that E3DA improved the wind,temperature and water vapor mixing ratio,with the lowest errors at the surface and upper levels.3DVar showed moderate improvements in comparison with forecasts without radar DA.A diagnosis of the analysis revealed that E3DA increased vertical velocity,temperature,and humidity corresponding to the added reflectivity,while 3DVar failed to produce these adjustments,because of the lack of reasonable cross-variable correlations.The performance of E3DA was further verified using two convective cases over southern and southeastern China,and the reflectivity forecast skill was also improved over 3DVar.     

A New Approach to Data Assimilation

A significant attempt to design a timesaving and efficient four-dimensional variational data assimilation (4DVar) has been made in this paper, and a new approach to data assimilation, which is noted as 'three-dimensional variational data assimilation of mapped observation (3DVM)' is proposed, based on the new concept of mapped observation and the new idea of backward 4DVar. Like the available 4DVar, 3DVM produces an optimal initial condition (IC) that is consistent with the prediction model due to the inclusion of model constraints and best fits the observations in the assimilation window through the model solution trajectory. Different from the 4DVar, the IC derived from 3DVM is located at the end of the assimilation window rather than at the beginning conventionally. This change greatly reduces the computing cost for the new approach, which is almost the same as that of the three-dimensional variational data assimilation (3DVar). Especially, such a change is able to improve assimilation accuracy because it does not need the tangential linear and adjoint approximations to calculate the gradient of cost function. Therefore, in numerical test, the new approach produces better IC than 4DVar does for 72-h simulation of TY9914 (Dan), by assimilating the three-dimensional fields of temperature and wind retrieved from the Advanced Microwave Sounding Unit-A (AMSU-A) observations. Meanwhile, it takes only 1/7 of the computing cost that the 4DVar requires for the same initialization with the same retrieved data.     

The Structure of Background-error Covariance in a Four-dimensional Variational Data Assimilation System: Single-point Experiment

A four dimensional variational data assimilation (4DVar) based on a dimension-reduced projection (DRP-4DVar) has been developed as a hybrid of the 4DVar and Ensemble Kalman filter (EnKF) concepts. Its good flow-dependent features are demonstrated in single-point experiments through comparisons with adjoint-based 4DVar and three-dimensional variational data (3DVar) assimilations using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model (MM5). The results reveal that DRP-4DVar can reasonably generate a background error covariance matrix (simply B-matrix) during the assimilation window from an initial estimation using a number of initial condition dependent historical forecast samples. In contrast, flow-dependence in the B-matrix of MM5 4DVar is barely detectable. It is argued that use of diagonal estimation in the B-matrix of the MM5 4DVar method at the initial time leads to this failure. The experiments also show that the increments produced by DRP-4DVar are anisotropic and no longer symmetric with respect to observation location due to the effects of the weather trends captured in its B-matrix. This differs from the MM5 3DVar which does not consider the influence of heterogeneous forcing on the correlation structure of the B-matrix, a condition that is realistic for many situations. Thus, the MM5 3DVar assimilation could only present an isotropic and homogeneous structure in its increments.     

物理滤波初始化四维变分在临近预报中的应用

运用WRF模式（Weather Research and Forecasting Model,天气研究和预报模式）和WRFDA同化（WRF Data Assimilation,WRF资料同化）系统,探究采用物理滤波初始化四维变分同化方法提高数值预报在临近预报时效的预报能力的可能性。通过采用12 min同化窗,在不显著增加计算量的情况下,得到更协调的模式初始场,从而提高模式预报能力。选取2018年8月华北地区17个降水个例进行研究,结果表明:采用物理滤波初始化四维变分同化技术能够明显改进模式短时临近降水预报能力,明显提高对大量级降水预报的ETS评分,6 h累积降水大于25.0 mm量级的ETS评分由0.125提高到0.190,且6 h累积降水大于60.0 mm量级的ETS评分由0.016提高到0.081。研究还表明:同化雷达风场通过改进初始动力场使次网格尺度降水过程（积云参数化）快速响应,可提高短时临近时段的降水预报能力。     

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 Analysis-time Tuning on the Performance of the DRP-4DVar Approach

In this study we extend the dimension-reduced projection-four dimensional variational data assimilation (DRP-4DVar) approach to allow the analysis time to be tunable, so that the intervals between analysis time and observation times can be shortened. Due to the limits of the perfect-model assumption and the tangentlinear hypothesis, the analysis-time tuning is expected to have the potential to further improve analyses and forecasts. Various sensitivity experiments using the Lorenz-96 model are conducted to test the impact of analysistime tuning on the performance of the new approach under perfect and imperfect model scenarios, respectively. Comparing three DRP-4DVar schemes having the analysis time at the start, middle, and end of the assimilation window, respectively, it is found that the scheme with the analysis time in the middle of the window outperforms the others, on the whole. Moreover, the advantage of this scheme is more pronounced when a longer assimilation window is adopted or more observations are assimilated.     

华南暴雨区域集合预报中不同同化方案的影响试验研究

基于全球集合预报系统(GEFS)资料,利用WRF中尺度模式及GEFS动力降尺度获取区域集合预报初值场,通过对同化后的分析场进行模式积分实现华南前汛期区域集合预报。对2019年6月10日的一次华南前汛期暴雨过程进行不同同化方案的试验:混合同化(Hybrid)、三维变分(3Dvar)、集合卡尔曼滤波(EnKF)和对比试验(Ctrl)四组试验的对比分析,探讨具有不同背景误差协方差矩阵的同化方案对区域集合预报集合扰动和集合离散随时间演变特征的影响,评估不同试验的降水模拟效果。(1) Hybrid对模式初始场有较好的改善作用,而3DVar和EnKF对初始场的改善作用不明显。(2) 对风场、温度场和湿度场,在前期预报中Hybrid的预报误差小于3DVar和EnKF,在中后期的预报中,3DVar和EnKF的预报误差得到改善,且好于Hybrid。同样,集合扰动能量,Hybrid和Ctrl在前期预报发展好于3DVar和EnKF,而在中后期的预报3DVar和EnKF好于Hybrid和Ctrl。(3) 从24 h累积降水评分中,整体上同化试验好于Ctrl,3DVar和EnKF好于Hybrid,且3DVar对大中雨级别的降水评分较好,而EnKF对暴雨以上级别的降水评分较好。(4) 对于集合统计检验分析,同化试验的AUC值都大于Ctrl的AUC值,24 h累积降水量阈值在10~100 mm的AUC值,3DVar最好;而125 mm阈值的AUC值,EnKF最好。      

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全球四维变分同化系统极小化算法预调节

在进行多次外循环更新的增量分析框架下,前一次极小化迭代过程中产生的信息可提供给下一次极小化做预调节。该文在GRAPES全球四维变分同化系统中对极小化算法——L-BFGS算法实施了这种预调节,通过全观测的个例试验和批量试验进行评估,发现进行预调节后L-BFGS算法的收敛效率得到明显提高,而且在1个月的循环试验中表现十分稳定。该工作可以帮助GRAPES全球四维变分同化系统有效减少极小化的迭代次数,有利于满足业务化运行的时效要求。另外,间隔6 h和间隔24 h的两次4DVar分析对应的海森矩阵变化不大,因此,前一时刻极小化过程产生的信息提供给后一时刻的极小化进行预调节也有一定效果。     

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。