Coupling Ensemble Kalman Filter with Four-dimensional Variational Data Assimilation

This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.

Testing a Four-Dimensional Variational Data Assimilation Method Using an Improved Intermediate Coupled Model for ENSO Analysis and Prediction

A four-dimensional variational （4D-Var） data assimilation method is implemented in an improved intermediate coupled model （ICM） of the tropical Pacific. A twin experiment is designed to evaluate the impact of the 4D-Var data assimilation algorithm on ENSO analysis and prediction based on the ICM. The model error is assumed to arise only from the parameter uncertainty. The ＂observation＂ of the SST anomaly, which is sampled from a ＂truth＂ model simulation that takes default parameter values and has Gaussian noise added, is directly assimilated into the assimilation model with its parameters set erroneously. Results show that 4D-Var effectively reduces the error of ENSO analysis and therefore improves the prediction skill of ENSO events compared with the non-assimilation case. These results provide a promising way for the ICM to achieve better real-time ENSO prediction.

A theoretical study of the multigrid three-dimensional variational data assimilation scheme using a simple bilinear interpolation algorithm

In order to solve the so-called ＂bull-eye＂ problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational （3DVAR） data assimilation scheme, a smoothing term, equivalent to a penalty term, is introduced into the cost function to serve as a means of troubleshooting. A theoretical analysis is first performed to figure out what on earth results in the issue of ＂bull-eye＂, and then the meaning of such smoothing term is elucidated and the uniqueness of solution of the multigrid 3DVAR with the smoothing term added is discussed through the theoretical deduction for one-dimensional （1D） case, and two idealized data assimilation experiments （one- and two-dimensional （2D） cases）. By exploring the relationship between the smoothing term and the recursive filter theoretically and practically, it is revealed why satisfied analysis results can be achieved by using such proposed solution for the issue of the multigrid 3DVAR.

Rainfall Assimilation Using a New Four-Dimensional Variational Method:A Single-Point Observation Experiment

Accurate forecast of rainstorms associated with the mei-yu front has been an important issue for the Chinese economy and society. In July 1998 a heavy rainstorm hit the Yangzi River valley and received widespread attention from the public because it caused catastrophic damage in China. Several numerical studies have shown that many forecast models, including Pennsylvania State University National Center for Atmospheric Research’s fifth-generation mesoscale model （MM5）, failed to simulate the heavy precipitation over the Yangzi River valley. This study demonstrates that with the optimal initial conditions from the dimension-reduced projection four-dimensional variational data assimilation （DRP-4DVar） system, MM5 can successfully reproduce these observed rainfall amounts and can capture many important mesoscale features, including the southwestward shear line and the low-level jet stream. The study also indicates that the failure of previous forecasts can be mainly attributed to the lack of mesoscale details in the initial conditions of the models.

Study and application of an improved four-dimensional variational assimilation system based on the physical-space statistical analysis for the South China Sea

The four-dimensional variational assimilation(4D-Var)has been widely used in meteorological and oceanographic data assimilation.This method is usually implemented in the model space,known as primal approach(P4D-Var).Alternatively,physical space analysis system(4D-PSAS)is proposed to reduce the computation cost,in which the 4D-Var problem is solved in physical space(i.e.,observation space).In this study,the conjugate gradient(CG)algorithm,implemented in the 4D-PSAS system is evaluated and it is found that the non-monotonic change of the gradient norm of 4D-PSAS cost function causes artificial oscillations of cost function in the iteration process.The reason of non-monotonic variation of gradient norm in 4D-PSAS is then analyzed.In order to overcome the non-monotonic variation of gradient norm,a new algorithm,Minimum Residual(MINRES)algorithm,is implemented in the process of assimilation iteration in this study.Our experimental results show that the improved 4D-PSAS with the MINRES algorithm guarantees the monotonic reduction of gradient norm of cost function,greatly improves the convergence properties of 4D-PSAS as well,and significantly restrains the numerical noises associated with the traditional 4D-PSAS system.

Evaluation of Two Momentum Control Variable Schemes and Their Impact on the Variational Assimilation of Radar Wind Data:Case Study of a Squall Line

Different choices of control variables in variational assimilation can bring about different influences on the analyzed atmospheric state. Based on the WRF model＇s three-dimensional variational assimilation system, this study compares the be- havior of two momentum control variable options-streamfunction velocity potential （ψ-χ） and horizontal wind components （U-V）-in radar wind data assimilation for a squall line case that occurred in Jiangsu Province on 24 August 2014. The wind increment from the single observation test shows that the ψ-χ control variable scheme produces negative increments in the neighborhood around the observation point because streamfunction and velocity potential preserve integrals of velocity. On the contrary, the U-V control variable scheme objectively reflects the information of the observation itself. Furthermore, radial velocity data from 17 Doppler radars in eastern China are assimilated. As compared to the impact of conventional observation, the assimilation of radar radial velocity based on the U-V control variable scheme significantly improves the mesoscale dynamic field in the initial condition. The enhanced low-level jet stream, water vapor convergence and low-level wind shear result in better squall line forecasting. However, the ψ-χ control variable scheme generates a discontinuous wind field and unrealistic convergence/divergence in the analyzed field, which lead to a degraded precipitation forecast.

Idealized Experiments for Optimizing Model Parameters Using a 4D-Variational Method in an Intermediate Coupled Model of ENSO

Large biases exist in real-time ENSO prediction, which can be attributed to uncertainties in initial conditions and model parameters. Previously, a 4D variational （4D-Vat） data assimilation system was developed for an intermediate coupled model （ICM） and used to improve ENSO modeling through optimized initial conditions. In this paper, this system is further applied to optimize model parameters. In the ICM used, one important process for ENSO is related to the anomalous temperature of subsurface water entrained into the mixed layer （Te）, which is empirically and explicitly related to sea level （SL） variation. The strength of the thermocline effect on SST （referred to simply as ＂the thermocline effect＂） is represented by an introduced parameter, （l＇Te. A numerical procedure is developed to optimize this model parameter through the 4D-Var assimilation of SST data in a twin experiment context with an idealized setting. Experiments having their initial condition optimized only, and having their initial condition plus this additional model parameter optimized, are compared. It is shown that ENSO evolution can be more effectively recovered by including the additional optimization of this parameter in ENSO modeling. The demonstrated feasibility of optimizing model parameters and initial conditions together through the 4D-Var method provides a modeling platform for ENSO studies. Further applications of the 4D-Vat data assimilation system implemented in the ICM are also discussed.

The Regularized WSM6 Microphysical Scheme and Its Validation in WRF 4D-Var

A cold cloud assimilation scheme was developed that fully considers the water substances,i.e.,water vapor,cloud water,rain,ice,snow,and graupel,based on the single-moment WSM6 microphysical scheme and four-dimensional variational(4D-Var)data assimilation in the Weather Research and Forecasting data assimilation(WRFDA)system.The verification of the regularized WSM6 and its tangent linearity model(TLM)and adjoint mode model(ADM)was proven successful.Two groups of single observation and real sounding data assimilation experiments were set up to further verify the correctness of the assimilation scheme.The results showed that the consideration of ice,snow,and graupel in the assimilation system of the 4D-Var,as opposed to their omission in the warm rain Kessler scheme,allowed the water substances to be reasonably updated,further improving the forecast.Before it can be further applied in the assimilation of observational data,radar reflectivities,and satellite radiances,the cold cloud assimilation scheme needs additional verification,including using conventional ground and sounding observations in the 4D-Var assimilation system.

Impacts of XBT,TAO,Altimetry and ARGO Observations on the Tropical Pacific Ocean Data Assimilation

This study aims at assessing the relative impacts of four major components of the tropical Pacific Ocean observing system on assimilation of temperature and salinity fields. Observations were collected over a period between January 2001 through June 2003 including temperature data from the expendable bathythermographs （XBT）, thermistor data from the Tropical Ocean Global Atmosphere Tropical Atmosphere-Ocean （TOGA-TAO） mooring array, sea level anomalies from the Topex/Poseidon and Jason-1 altimetry （T/P-J）, and temperature and salinity profiles from the Array for Real-time Geostrophic Oceanography （ARGO） floats. An efficient three-dimensional variational analysis-based method was introduced to assimilate the above data into the tropical-Pacific circulation model. To evaluate the impact of the individual component of the observing system, four observation system experiments were carried out. The experiment that assimilated all four components of the observing system was taken as the reference. The other three experiments were implemented by withholding one of the four components. Results show that the spatial distribution of the data influences its relative contribution. XBT observations produce the most distinguished effects on temperature analyses in the off-equatorial region due to the large amount of measurements and high quality. Similarly, the impact of TAO is dominant in the equatorial region due to the focus of the spatial distribution. The Topex/Poseidon-Jason-1 can be highly complementary where the XBT and TAO observations are sparse. The contribution of XBT or TAO on the assimilated salinity is made by the model dynamics because no salinity observations from them are assimilated. Therefore, T/P-J, as a main source for providing salinity data, has been shown to have greater impacts than either XBT or TAO on the salinity analysis. Although ARGO includes the subsurface observations, the relatively smaller number of observation makes it have the smallest contribution to the assimilation system.

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.

An explicit four-dimensional variational data assimilation method

A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are ex-pressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost func-tion with respect to the control variables, is no longer needed. The new technique significantly simpli-fies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method.

Forming proper ensemble forecast initial members with four-dimensional variational data assimilation method

A method has been presented to improve ensemble forecast by utilizing these initial members generated by four-dimensional variational data assimilation (4-D VDA), to conquer limitation of those initial members generated by Monte Carlo forecast (MCF) or lagged average forecast (LAF). This method possesses significant statistical characteristic of MCF, and by virtue of LAF that contains multi-time information and its initial members are harmonic with

Assimilation of Total Lightning Data Using the Three-Dimensional Variational Method at Convection-Allowing Resolution

A large number of observational analyses have shown that lightning data can be used to indicate areas of deep convection. It is important to assimilate observed lightning data into numerical models, so that more small-scale information can be incorporated to improve the quality of the initial condition and the subsequent forecasts. In this study, the empirical relationship between flash rate, water vapor mixing ratio, and graupel mixing ratio was used to adjust the model relative humidity, which was then assimilated by using the three-dimensional variational data assimilation system of the Weather Research and Forecasting model in cycling mode at 10-min intervals. To find the appropriate assimilation time-window length that yielded significant improvement in both the initial conditions and subsequent forecasts, four experiments with different assimilation time-window lengths were conducted for a squall line case that occurred on 10 July 2007 in North China. It was found that 60 min was the appropriate assimilation time-window length for this case, and longer assimilation window length was unnecessary since no further improvement was present. Forecasts of 1-h accumulated precipitation during the assimilation period and the subsequent 3-h accumulated precipitation were significantly improved compared with the control experiment without lightning data assimilation. The simulated reflectivity was optimal after 30 min of the forecast, it remained optimal during the following 42 min, and the positive effect from lightning data assimilation began to diminish after 72 min of the forecast. Overall,the improvement from lightning data assimilation can be maintained for about 3 h.

闪电资料同化对河南郑州“7.20”特大暴雨预报的影响

研究了闪电资料同化对2021年7月20日河南郑州特大暴雨短临预报的影响。利用天气研究与预报(WRF)模式的三维变分(3DVAR)数据同化系统(WRFDA),开展了两组循环同化试验:(1)同化地面和探空常规观测资料(包括风速、风向、温度和气压)的试验(CONV);(2)同化常规观测资料和由闪电资料反演的伪相对湿度的试验(LGDA),并与无资料同化的试验(NoDA)进行对比。结果表明,CONV的分析场和NoDA都未能模拟出强对流系统的回波结构,但由于LGDA增加了对整层大气的湿度场的调整,其分析场在闪电发生位置处的水凝物增量较大,相对湿度和反射率的分布情况与中国全球大气再分析资料(CRA)及雷达反射率观测值最接近。降水预报方面,LGDA显著提高了大暴雨雨带(6 h累积降水量≥50 mm)和强降水中心(6 h累积降水量≥200 mm)位置和强度的预报效果,对本次强降水过程的预报起到了积极作用,尤其是对前3 h的降水预报。

基于NLS-4DVar方法的雷达资料同化及其在暴雨预报中的应用

在基于本征正交分解POD(Proper Orthogonal Decomposition)的集合四维变分同化方法(POD4DEn Var)建立的雷达资料同化系统(PRAS)的基础上,本文利用非线性最小二乘法的集合四维变分同化方法(NLS-4DVar)对PRAS进行改进,解决PRAS在高度非线性情况下的适应性问题,建立了新的雷达资料同化系统(NRAS)。通过观测系统模拟试验OSSEs(Observing System Simulation Experiments)和两次实际暴雨同化试验(2010年7月8日,中国中部地区;2014年3月30日,中国华南地区)对NRAS进行检验,并与PRAS的同化结果进行了对比。结果表明:无论是OSSEs还是实际雷达资料的同化,相对于PRAS,NRAS能够进一步提高同化效果。通过增加迭代的次数,NRAS能够有效地调整初始场的风场和水汽场,进一步提高了降水强度和位置的预报精度。但随着迭代次数的增加,对初始场的调整变小,进而对降水预报效果的改进也减小。试验结果表明NRAS能够有效解决PRAS在高度非线性情况下的应用问题,通过有限次数的迭代,即可得到近似收敛的结果。因而NRAS有望在数值预报中更有效地同化雷达资料,提高中小尺度天气的预报水平。

在模式参数有误差下使用四维变分同化的LAF法

为了有效利用各种观测资料改善观测误差来提高集合预报的效果,提出了一种新的预报方法,即拟在模式参数有误差的情况下,利用四维变分同化方法获取的初始场进行滞后时间集合预报(LAF)。通过比较在不同的观测误差方差、不同观测个数以及不同观测间隔的条件下,新方法与传统的滞后时间集合预报方法和单考虑最优初始场的四维同化确定性预报的预报效果的差异。实验结果表明:使用四维变分同化分析场的滞后时间集合预报在短期内的预报效果较好,但随着预报时长的增加,预报效果与单滞后时间集合预报的预报效果相接近。

DRP-4DVar方法同化AIRS反演资料在一次江淮流域暴雨中的应用

利用经济省时的降维投影四维变分同化方法（DRP-4DVar）,在2009年7月22~23日江淮流域的一次大暴雨过程中同化晴空条件下高光谱大气红外探测仪（AIRS）反演温度、湿度廓线,改进此次强降水过程的模拟。试验结果分析显示,同化AIRS反演的温度及湿度场后,基于四维变分同化系统的模式约束,能够改进湿度场、高度场、高低层散度场。从累积降水量偏差图及同化试验增量图可以看到,正降水量偏差对应于正湿度增量、负位势高度增量及低层负散度高层正散度增量,负降水量偏差则与之相反。同化试验较参照试验可更好地模拟出暴雨的天气形势、对暴雨的落区及强度有更好的反映。此外,从单次同化与连续同化的试验对比结果看出,连续同化试验结果较单次同化结果有进一步的改进,说明不断加入新的观测资料可以更好地模拟强降水过程。

三维变分数值气象预报在弹道解算中应用研究

利用数值天气预报快速获取高空气象参数是当前战时远程火炮射击诸元解算重点研究内容之一,但是数值天气预报初始场误差大,直接影响了气象参数预报精度。采用三维变分同化法,同化NCEP ADP全球高空观测天气数据,结合背景场气象资料使用WRF进行气象预报,并对其应用于有风状态下的6自由度弹道的解算效果进行了分析。结果表明:同化试验对风速预报改善作用显著,应用循环同化后的预报结果较未同化数值预报结果,可使弹道计算射程X方向射程的预测相对误差减小了约0.14%;Z方向侧偏的预测相对误差减小了约3.89%。

改进三维变分同化模型及应用:动力场数据同化

在动力场中,如何消除观测数据受观测条件以及观测仪器误差对仿真物理模型的影响,以获得高精度、高质量的数据至关重要。三维变分作为连续型变分数据同化算法的主流,在融合同化来自动力场的模式和观测值时,其代价函数中构造的背景误差协方差矩阵和观测误差协方差矩阵不可逆,是造成无法使用梯度降低的方式求出其最优估计量问题的原因。对于动力场的数据同化方法,提出了两种改进的三维变分同化模型,不同于优化算法、变量变换、维度分解等其他数学求解方法,所提出的粒度化数据同化和改进三维变分代价函数方法解决了矩阵不可逆导致的算法无法计算的问题。在实证分析中,对于维度差异较高的实验数据和模型仿真数据,这两种方法都具有很好的算法鲁棒性,而且展现出较高的算法精度。

UNIFICATION AND APPLICATIONS OF MODERN OCEANIC/ATMOSPHERIC DATA ASSIMILATION ALGORITHMS

The key mathematics and applications of various modern atmospheric/oceanicdata assimilation methods including Optimal Interpolation (OI), 4-dimensional variational approach(4D-Var) and filters were systematically reviewed and classified. Based on the data assimilationphilosophy, i. e. , using model dynamics to extract the observational information, the commoncharacter of the problem, such as the probabilistic nature of the evolution of theatmospheric/oceanic system, noisy and irregularly spaced observations, and the advantages anddisadvantages of these data assimilation algorithms, were discussed. In the filtering framework, allmodern data assimilation algorithms were unified: OI/3D-Var is a stationary filter, 4D-Var is alinear (Kalman) filter and an ensemble of Kalman filters is able to construct a nonlinear filter.The nonlinear filter such as the Ensemble Kalman Filter (EN-KF), Ensemble Adjustment Kalman Filter(EAKF) and Ensemble Transformation Kalman Filter (ETKF) can, to some extent, account for thenon-Gaussian information of the prior distribution from the model. The flow-dependent covarianceestimated by an ensemble filter may be introduced to OI and 4D-Var to improve these traditionalalgorithms. In practice, the performance of algorithms may depend on the specific numerical modeland the choice of algorithm may depend on the specific problem. However, the unification ofalgorithms allows us to establish a unified test system to evaluate these algorithms, which providesmore insights into data assimilation philosophies and helps improve data assimilation techniques.