Cycling the representer algorithm for variational data assimilation with a nonlinear reduced gravity ocean model

The representer method was used by [Ngodock, H.E., Jacobs, G.A., Chen, M., 2006. The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: a comparison study using a nonlinear reduced gravity ocean model. Ocean Modelling 12, 378–400] in a comparison study with the ensemble Kalman filter and smoother involving a 1.5 nonlinear reduced gravity idealized ocean model simulating the Loop Current (LC) and the Loop Current eddies (LCE) in the Gulf of Mexico. It was reported that the representer method was more accurate than its ensemble counterparts, yet it had difficulties fitting the data in the last month of the 4-month assimilation window when the data density was significantly decreased. The authors attributed this failure to increased advective nonlinearities in the presence of an eddy shedding causing the tangent linear model (TLM) to become inaccurate. In a separate study [Ngodock, H.E., Smith, S.R., Jacobs, G.A., 2007. Cycling the representer algorithm for variational data assimilation with the Lorenz attractor. Monthly Weather Review 135 (2), 373–386] applied the cycling representer algorithm to the Lorenz attractor and demonstrated that the cycling solution was able to accurately fit the data within each cycle and beyond the range of accuracy of the TLM, once adjustments were made in the early cycles, thus overcoming the difficulties of the non-cycling solution. The cycling algorithm is used here in assimilation experiments with the nonlinear reduced gravity model. It is shown that the cycling solution overcomes the difficulties encountered by the non-cycling solution due to a limited time range of accuracy of the TLM. Thus, for variational assimilation applications where the TLM accuracy is limited in time, the cycling representer becomes a very powerful and attractive alternative, given that its computational cost is significantly lower than that of the non-cycling algorithm.     

The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: A comparison study using a nonlinear reduced gravity ocean model

This paper compares contending advanced data assimilation algorithms using the same dynamical model and measurements. Assimilation experiments use the ensemble Kalman filter (EnKF), the ensemble Kalman smoother (EnKS) and the representer method involving a nonlinear model and synthetic measurements of a mesoscale eddy. Twin model experiments provide the “truth” and assimilated state. The difference between truth and assimilation state is a mispositioning of an eddy in the initial state affected by a temporal shift. The systems are constructed to represent the dynamics, error covariances and data density as similarly as possible, though because of the differing assumptions in the system derivations subtle differences do occur. The results reflect some of these differences in the tangent linear assumption made in the representer adjoint and the temporal covariance of the EnKF, which does not correct initial condition errors. These differences are assessed through the accuracy of each method as a function of measurement density. Results indicate that these methods are comparably accurate for sufficiently dense measurement networks; and each is able to correct the position of a purposefully misplaced mesoscale eddy. As measurement density is decreased, the EnKS and the representer method retain accuracy longer than the EnKF. While the representer method is more accurate than the sequential methods within the time period covered by the observations (particularly during the first part of the assimilation time), the representer method is less accurate during later times and during the forecast time period for sparse networks as the tangent linear assumption becomes less accurate. Furthermore, the representer method proves to be significantly more costly (2–4 times) than the EnKS and EnKF even with only a few outer iterations of the iterated indirect representer method.     

Data assimilation with a barotropically unstable shallow water system using representer algorithms

The cycling representer algorithm of Xu and Daley (2000) is a weak constraint four-dimensional variational data assimilation algorithm. It was successfully applied to a one-dimensional transport problem and was able to successfully extract the signal from noisy and sparse observations. The algorithm, however, has not previously been applied to a multivariate, multidimensional system with dynamic instability. The algorithm is also very computationally demanding and awaits considerable enhancement in computer power before being practical for operational forecast models. We have two objectives in this paper. The first is to apply the cycling representer algorithm to a two-dimensional, multivariate barotropically unstable linear shallow water system. The second objective is to formulate and test an accelerated representer algorithm that is much more computationally tractable than the cycling representer algorithm itself. A linear shallow water system with a barotropically unstable basic state was used as a test bed to conduct data assimilation experiments. The evolution of a 'neutral' eastward-propagating singular vector was selected as the 'truth', against which all data assimilation experiments were to be evaluated. The results indicated that the cycling representer algorithm was capable of providing satisfying state estimates for a multivariate, multidimensional system. The results from the accelerated representer algorithm were very encouraging because it is sufficiently computationally tractable to be used on present day multi-processor machines for operational applications.     

Asynchronous data assimilation with the EnKF

This study revisits the problem of assimilation of asynchronous observations, or four-dimensional data assimilation, with the ensemble Kalman filter (EnKF). We show that for a system with perfect model and linear dynamics the ensemble Kalman smoother (EnKS) provides a simple and efficient solution for the problem: one just needs to use the ensemble observations (that is, the forecast observations for each ensemble member) from the time of observation during the update, for each assimilated observation. This recipe can be used for assimilating both past and future data; in the context of assimilating generic asynchronous observations we refer to it as the asynchronous EnKF. The asynchronous EnKF is essentially equivalent to the four-dimensional variational data assimilation (4D-Var). It requires only one forward integration of the system to obtain and store the data necessary for the analysis, and therefore is feasible for large-scale applications. Unlike 4D-Var, the asynchronous EnKF requires no tangent linear or adjoint model.     

Towards a true 4‐dimensional data assimilation algorithm: application of a cycling representer algorithm to a simple transport problem

3‐dimensional variational algorithms are widely used for atmospheric data assimilation at the present time, particularly on the synoptic and global scales. However, mesoscale and convective scale phenomena are considerably more chaotic and intermittent and it is clear that true 4‐dimensional data assimilation algorithms will be required to properly analyze these phenomena. In its most general form, the data assimilation problem can be posed as the minimization of a 4‐dimensional cost function with the forecast model as a weak constraint. This is a much more difficult problem than the widely discussed 4DVAR algorithm where the model is a strong constraint. Bennett and collaborators have considered a method of solution to the weak constraint problem, based on representer theory. However, their method is not suitable for the numerical weather prediction problem, because it does not cycle in time. In this paper, the representer method is modified to permit cycling in time, in a manner which is entirely internally consistent. The method was applied to a simple 1‐dimensional constituent transport problem where the signal was sampled (perfectly and imperfectly) with various sparse observation network configurations. The cycling representer algorithm discussed here successfully extracted the signal from the noisy, sparse observations     

EnKF和SIR-PF在贝叶斯滤波框架下的比较和结合

贝叶斯估计理论为非线性、非高斯系统的数据同化提供了一个统一的框架。在本文中,我们利用著名的洛伦茨吸引子(Lorenz'63)模式对两种基于贝叶斯滤波理论的数据同化方法——集合卡尔曼滤波器(EnKF)和重取样粒子滤波器(SIR-PF)——进行了较为全面的比较。比较的结果揭示了两种方法的优缺点:即当集合成员数目较多时,SIR-PF的同化效果优于EnKF;反之,则EnKF的表现较好。进一步地,我们使用统计方法分析了两者表现的差异和原因。最近提出的一种集合卡尔曼粒子滤波器(EnKPF)通过使用一个可控的参数整合EnKF和SIR-PF的分析格式,可以结合两者的优点。本文在充分比较两种方法的前提下,重新阐释并改进了原有的EnKPF算法,使之适用于非线性的观测算子。通过使用相同的洛伦茨模式实验,我们揭示了EnKPF实质上提供了关于EnKF和SIR-PF的连续插值,使得后两者可以视为其特殊情况。并且,在集合成员数目有限的前提下,EnKPF可以在一定程度上避免滤波退化的发生,取得优于EnKF和SIR-PF的同化效果。     

Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): Development and application for a baroclinic coastal upwelling system

We describe the development and preliminary application of the inverse Regional Ocean Modeling System (ROMS), a four dimensional variational (4DVAR) data assimilation system for high-resolution basin-wide and coastal oceanic flows. Inverse ROMS makes use of the recently developed perturbation tangent linear (TL), representer tangent linear (RP) and adjoint (AD) models to implement an indirect representer-based generalized inverse modeling system. This modeling framework is modular. The TL, RP and AD models are used as stand-alone sub-models within the Inverse Ocean Modeling (IOM) system described in [Chua, B.S., Bennett, A.F., 2001. An inverse ocean modeling system. Ocean Modell. 3, 137–165.]. The system allows the assimilation of a wide range of observation types and uses an iterative algorithm to solve nonlinear assimilation problems. The assimilation is performed either under the perfect model assumption (strong constraint) or by also allowing for errors in the model dynamics (weak constraints). For the weak constraint case the TL and RP models are modified to include additional forcing terms on the right hand side of the model equations. These terms are needed to account for errors in the model dynamics.Inverse ROMS is tested in a realistic 3D baroclinic upwelling system with complex bottom topography, characterized by strong mesoscale eddy variability. We assimilate synthetic data for upper ocean (0–450 m) temperatures and currents over a period of 10 days using both a high resolution and a spatially and temporally aliased sampling array. During the assimilation period the flow field undergoes substantial changes from the initial state. This allows the inverse solution to extract the dynamically active information from the synthetic observations and improve the trajectory of the model state beyond the assimilation window. Both the strong and weak constraint assimilation experiments show forecast skill greater than persistence and climatology during the 10–20 days after the last observation is assimilated.Further investigation in the functional form of the model error covariance and in the use of the representer tangent linear model may lead to improvement in the forecast skill.     

Cycling the Representer Method for 4D-variational data assimilation with the Navy Coastal Ocean Model

The Cycling Representer Method, which is a technique for solving 4D-variational data assimilation problems, has been demonstrated to improve the assimilation accuracy with simpler nonlinear models. In this paper, the Cycling Representer Method will be used to assimilate an array of ADCP velocity observations with the Navy Coastal Ocean Model (NCOM). Experiments are performed in a high-resolution Mississippi Bight domain for the entire month of June, 2004 and demonstrate the usefulness of this assimilation technique in a realistic application.The Representer Method is solved by minimizing a cost function containing the weighted squared errors of velocity measurements, initial conditions, boundary conditions, and model dynamics. NCOM, however, is a highly nonlinear model and in order to converge towards the global minimum of this cost function, NCOM is linearized about a background state using tangent linearization. The stability of this tangent linearized model (TLM) is a very sensitive function of the background state, the level of nonlinearity of the model, open boundary conditions, and the complexity of the bathymetry and flow field. For the Mississippi Bight domain, the TLM is stable for only about a day. Due to this short TLM stability time period, the Representer Method is cycled by splitting the time period of the assimilation problem into short intervals. The interval time period needs to be such that it is short enough for the TLM to be stable, but long enough to minimize the loss of information due to reducing the temporal correlation of the dynamics and data. For each new cycle, a background is created as a nonlinear forecast from the previous cycle’s assimilated solution. This background, along with the data that falls within this new cycle, is then used to calculate a new assimilated solution. The experiments presented in this paper demonstrate the improvement of the assimilated solution as the time window of the cycles is reduced to 1 day. The 1-day cycling, however, was only optimal for the first half of the experiment. This is because there was a strong wind event near the middle of June that significantly reduced the stability of the 1-day cycling and caused substantial errors in the assimilation. Therefore, the 12-h cycling worked best for the second half of the experiment. This paper also demonstrates that the forecast skill is improved as the assimilation system progresses through the cycles.     

Picard iterations for a finite element shallow water equation model

ADCIRC, a finite element circulation model for shelves, coasts and estuaries, will be used for variational data assimilation. The nonlinear Euler–Lagrange (EL) problem will be solved using the iterated indirect representer algorithm. This algorithm makes such large, nonlinear but functionally smooth optimization problems feasible by iterating on linear approximations of the nonlinear problem (Picard iterations) and by making preconditioned searches in the “data subspace” at each iterate. Before solving the nonlinear EL using such Picard iterations, it essential that the iteration scheme be carefully examined within the framework of the nonassimilative or forward problem.The purpose of this paper is (1) to detail a Picard iteration procedure for ADCIRC, including the problematic bottom friction term; (2) to examine the ability of the iteration scheme to recover the nonlinear forward solution from deficient background fields; and (3) to present a study of different interpolation methods for reducing the memory/disk requirements of the iteration scheme. The iteration scheme is shown to be quite robust in its ability to recover the nonlinear solution from a variety of deficient background fields. A new cubic Hermitian interpolation method is shown to be a more effective alternative to standard linear interpolation for reducing memory/disk requirements, especially for high frequency overtides.     

基于海气耦合模式的均权重粒子滤波与集合卡尔曼滤波比较研究

在大气和海洋环境研究中,粒子滤波(PF)由于在非线性数据同化方面突出的优势,逐渐成为研究热点。最近改进的均权重粒子滤波(EWPF)为粒子滤波的进一步发展指明了新方向。集合卡尔曼滤波方法 (EAKF)作为当前主要应用的数据同化方法,使用高斯假设和线性假设来解决非线性问题,然而对均权重粒子滤波方法和卡尔曼滤波方法在非线性模式下的同化结果和特点还缺少系统详细的比较研究。本文在非线性耦合气候模式下,比较研究两种同化方法,采用均方根误差(RMSE)作为评价比较标准。实验结果表明,在非线性低频观测耦合模式中EWPF结果均优于EAKF。同时根据RMSE的结果得出,EWPF的同化结果更接近观察结果,而EAKF的同化结果更接近模式真值。     

集合滤波和三维变分混合数据同化方法研究

发展了一种新的混合数据同化方法——基于集合滤波和三维变分的混合数据同化方法。该方法将集合调整卡尔曼滤波(ensemble adjustment Kalman filter,EAKF)得到的集合样本扰动通过一个转换矩阵的形式直接作用到背景场上,利用顺序滤波的思想得到分析场的一个扰动;然后在三维变分(three dimensional variational analysis,3D-Var)的框架下与观测数据进行拟合,从而给出分析场的最优估计。文中以Lorenz63模型为例,开展了理想数据同化试验,结果表明,相比于集合调整卡尔曼滤波,这种新的混合同化方法可以给出更好的同化结果。     

集合变换卡尔曼滤波数据同化方法中的协方差局地化

在集合数据同化中,协方差局地化（covariance localization,CL）方法的使用存在限制。集合转换卡尔曼滤波（ensemble transform Kalman filter,ETKF）作为集合平方根滤波的变种方法,是一种应用较广、计算高效的数据同化方法。本文分析了CL方法应用于ETKF方法的困难,从而改进CL方法使其可以适用于ETKF方法。另外,结合浅水方程,利用Askey函数作为多元局地化函数,提出了一种适用于多元数值模型的CL方法。通过具体实验验证,得到了较好的分析结果。     

现代海洋/大气资料同化方法的统一性及其应用进展

海洋／大气资料同化的理论基础是用数值模式作为动力学强迫对观测信息进行提炼，或者说，从包含观测误差(噪声)的空间分布不均匀的实测资料中依据动力系统自身的演化规律(动力学方程或模式)来确定海洋／大气系统状态的最优估计。本文对主要的现代海洋／大气资料同化方法，包括最优插值(()ptimal Interpolation，简称()Ⅰ)、变分方法(3—Dimensional Variational和4—Dimensional Variational，分别简称3DVAR和4DVAR)和滤波方法(Filtering)的原理、算法设计和实际应用进行系统地回顾，并对这些资料同化方法的优缺点进行分析和讨论。在滤波框架下，所有的现代资料同化方法都被统一了：()Ⅰ和3DVAR是不随时间变化的滤波器，4DVAR和卡曼滤波是线性滤波器，即非线性滤波的退化情形；而集合滤波能构建非线性的滤波器，因为集合在某种程度上体现了系统的非高斯信息。一个非线性滤波器的主要优点是能计算和应用随时间变化的各阶误差统计距，如误差协方差矩阵。将非线性滤波器计算的随时间变化的误差协方差矩阵引入到（）Ⅰ或4DVAR中，也许能实质性地改进这些传统方法。在实际应用中，方法的优劣可能取决于所选用的数值模式和可获得的计算资源，因此需针对不同的问题选取不同的资料同化方法。由于各种资料同化方法具有统一性，因此可建立测试系统来评价这些方法，从而对各种方法获得更深入的理解，改进现有的资料同化技术，并提高人们对海洋／大气环境的预测能力。     

解析四维集合变分参数优化方法研究

传统的四维变分数据同化方法在同化观测资料的同时可以对数值模式参数进行优化,然而传统的四维变分方法需要针对不同的数值模式编写特有的伴随模式,因此算法的可移植性差,同时计算时耗费大量资源。本文提出了一种新的基于解析四维集合变分的参数优化方法,该方法以迭代搜索得到的模式参数为基准展开扰动并构建样本集合,由此显式地计算协方差矩阵,并得到代价函数极小值的解析解,从而避免了伴随模式的使用。基于Lorenz-63模型对该方法进行单参数和多参数数值试验和优化效果检验,并在不同的同化时间窗口长度和观测采样间隔情况下,采用传统四维变分方法与之进行对比,结果显示,新方法表现出与传统四维变分相同的优化性能,都能有效收敛到真值,而新方法不需要计算伴随模式,可移植性好。本文还测试了不同的集合成员个数和模式参数真值的情况下新方法的同化效果,结果表明,新方法对集合样本个数及模型参数真值不敏感,采用较少的集合样本即可完成数据同化。     

MASNUM-WAM海浪模式集合Kalman滤波同化研究——Ⅱ.集合样本对同化效果的影响

背景误差相关结构的确定是影响海浪同化效果的关键因素之一。集合Kalman滤波是一种较为成熟的同化方法,其可以对背景误差进行实时更新和动态估计,现已广泛应用于海洋和大气领域的研究。本文基于MASNUM-WAM海浪模式,分别采用静态样本集合Kalman滤波和EAKF方法,针对2014年全球海域开展海浪数据同化实验,同化资料为Jason-2卫星高度计数据,利用Saral卫星高度计资料对同化实验结果进行检验。结果表明,两组同化方案均有效提高了海浪模式的模拟水平,EAKF方案在风场变化较大的西风带区域表现显著优于静态样本集合Kalman滤波方案,但总体上两者相差不大。综合考虑计算成本和同化效果,静态样本集合Kalman滤波方案更适用于海浪业务化预报。     

EnKF集合同化下黄海海雾数值确定性预报初始场构造方法的探究

在黄海海雾的数值模拟中,EnKF（ensemble Kalman filter）是一种优于3DVAR（three-dimensional variational）的数据同化方法。研究发现,对EnKF初始场集合体采取常用的集合平均所产生的确定性预报初始场,会出现初始场中海雾在预报开始后就迅速消失以及接下来海雾难以生成的异常现象。通过详细的海雾个例研究,清晰地揭示并解释了此现象,指出这是集合平均造成初始场中云水与温度湿度之间存在不协调关系所导致的后果,并提出了一种择优加权平均方法来取代常用的集合平均。研究结果表明,海雾确定性预报采用择优加权平均所构建的初始场,可以消除上述异常现象,显著改进海雾模拟效果。     

Assimilation of surface currents into a regionalmodel over Qingdao coastal waters of China

Surface currents measured by high frequency （HF） radar arrays are assimilated into a regional ocean model over Qingdao coastal waters based on Kalman filter method. A series of numerical experiments are per- formed to evaluate the performance of the data assimilation schemes. In order to optimize the analysis pro- cedure in the traditional ensemble Kalman filter （ENKF）, a different analysis scheme called quasiensemble Kaman filter （QENKF） is proposed. The comparisons between the ENKF and the QENKF suggest that both them can improve the simulated error and the spatial structure. The estimations of the background error covariance （BEC） are also assessed by comparing three different methods： Monte Carlo method; Canadian quick covariance （CQC） method and data uncertainty engine （DUE） method. A significant reduction of the root-mean-square （RMS） errors between model results and the observations shows that the CQC method is able to better reproduce the error statistics for this coastal ocean model and the corresponding external forcing. In addition, the sensibility of the data assimilation system to the ensemble size is also analyzed by means of different scales of the ensemble size used in the experiments. It is found that given the balance of the computational cost and the forecasting accuracy, the ensemble size of 50 will be an appropriate choice in the Qingdao coastal waters.     

Observation bias correction with an ensemble Kalman filter

This paper considers the use of an ensemble Kalman filter to correct satellite radiance observations for state dependent biases. Our approach is to use state-space augmentation to estimate satellite biases as part of the ensemble data assimilation procedure. We illustrate our approach by applying it to a particular ensemble scheme—the local ensemble transform Kalman filter (LETKF)—to assimilate simulated biased atmospheric infrared sounder brightness temperature observations from 15 channels on the simplified parameterizations, primitive-equation dynamics (SPEEDY) model. The scheme we present successfully reduces both the observation bias and analysis error in perfect-model simulations.