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.     

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。     

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。     

Data assimilation in a system with two scales—combining two initialization techniques

An ensemble Kalman filter (EnKF) is used to assimilate data onto a non-linear chaotic model, coupling two kinds of variables. The first kind of variables of the system is characterized as large amplitude, slow, large scale, distributed in eight equally spaced locations around a circle. The second kind of variables are small amplitude, fast, and short scale, distributed in 256 equally spaced locations. Synthetic observations are obtained from the model and the observational error is proportional to their respective amplitudes. The performance of the EnKF is affected by differences in the spatial correlation scales of the variables being assimilated. This method allows the simultaneous assimilation of all the variables. The ensemble filter also allows assimilating only the large-scale variables, letting the small-scale variables to freely evolve. Assimilation of the large-scale variables together with a few small-scale variables significantly degrades the filter. These results are explained by the spurious correlations that arise from the sampled ensemble covariances. An alternative approach is to combine two different initialization techniques for the slow and fast variables. Here, the fast variables are initialized by restraining the evolution of the ensemble members, using a Newtonian relaxation toward the observed fast variables. Then, the usual ensemble analysis is used to assimilate the large-scale observations.     

Ensemble covariances adaptively localized with ECO-RAP. Part 2: a strategy for the atmosphere

Part 1's localization method, Ensemble COrrelations Raised to A Power (ECO-RAP), is incorporated into a Local Ensemble Transform Kalman Filter (LETKF). Because brute force incorporation would be too expensive, we demonstrate a factorization property for Part 1's Covariances Adaptively Localized with ECO-rap (CALECO) forecast error covariance matrix that, together with other simplifications, reduces the cost. The property inexpensively provides a large CALECO ensemble whose covariance is the CALECO matrix. Each member of the CALECO ensemble is an element-wise product between one raw ensemble member and one column of the square root of the ECO-RAP matrix. The LETKF is applied to the CALECO ensemble rather than the raw ensemble. The approach enables the update of large numbers of variables within each observation volume at little additional computational cost. Under plausible assumptions, this makes the CALECO and standard LETKF costs similar. The CALECO LETKF does not require artificial observation error inflation or vertically confined observation volumes both of which confound the assimilation of non-local observations such as satellite observations. Using a 27 member ensemble from a global Numerical Weather Prediction (NWP) system, we depict four-dimensional (4-D) flow-adaptive error covariance localization and test the ability of the CALECO LETKF to reduce analysis error.     

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.     

Comparison of EnOI and EnKF regional ocean reanalysis systems

This study compares two regional eddy resolving ocean reanalysis systems, based on the ensemble Kalman filter (EnKF) and ensemble optimal interpolation (EnOI), focusing on data assimilation aspects. Both systems are configured for the Tasman Sea using the same ocean model with 0.1° resolution and commonly available observations of satellite altimetry, sea surface temperature and subsurface temperature and salinity. The primary goals are to quantify the difference in performance of the EnKF and EnOI and investigate how important this difference might be from an oceanographic perspective. We find that both systems generally constrain mesoscale circulation in the region, with some exceptions for the East Australian Current separation region, the most energetic and chaotic part of the domain. Overall, the EnKF is found to consistently outperform the EnOI, producing on average 9–21% smaller innovations. The EnKF also has better forecast skill relative to the persisted analysis than the EnOI. For SST the EnKF forecast outperforms persisted analysis by about 17%, which indicates that the surface circulation is mainly constrained. The EnKF and EnOI are shown to produce qualitatively different increments of unobserved or sparsely observed variables; however, we find only moderate improvements of the EnKF over EnOI in subsurface temperature fields when compared against withheld XBT observations. We attribute this lack of a major improvement in subsurface reconstruction to the inability of the EnKF to linearly constrain the system due to initialisation shock, model error caused by open boundaries, and possibly insufficient observations.     

对地球系统模式FIO-ESM同化实验中北极海冰模拟的评估

本文评估了地球系统模式FIO-ESM(First Institute of Oceanography-Earth System Model)基于集合调整Kalman滤波同化实验对1992-2013年北极海冰的模拟能力。结果显示:尽管同化资料只包括了全球海表温度和全球海面高度异常两类数据,而并没有对海冰进行同化,但实验结果能很好地模拟出与观测相符的北极海冰基本态和长期变化趋势,卫星观测和FIO-ESM同化实验所得的北极海冰覆盖范围在1992-2013年间的线性变化趋势分别为-7.06×105和-6.44×105 km2/(10a),同化所得的逐月海冰覆盖范围异常和卫星观测之间的相关系数为0.78。与FIO-ESM参加CMIP5(Coupled Model Intercomparison Project Phase 5)实验结果相比,该同化结果所模拟的北极海冰覆盖范围的长期变化趋势和海冰密集度的空间变化趋势均与卫星观测更加吻合,这说明该同化可为利用FIO-ESM开展北极短期气候预测提供较好的预测初始场。     

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。     

Wave Data Assimilation to Modify Wind Forcing Using an Ensemble Kalman Filter

In order to improve the predictability of winter storm waves in the East Sea, this article explores the use of the ensemble Kalman filter technique for dat     

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。     

基于地球系统模式的局地化粒子滤波器与集合卡尔曼滤波器同化实验

粒子滤波器(PF)是一种非常具有应用前景的非线性资料同化方法。但由于其算法本身存在的粒子退化问题,目前尚未被广泛地应用于大型地球物理模式。目前主流的集合同化系统仍然倾向于使用集合卡尔曼滤波器(EnKF)及其衍生方法。一种新近被提出的局地化粒子滤波器(LPF)在经典的粒子滤波器算法中引入局地化技术,可以使用较小的计算成本有效地避免退化问题,具有非常大的业务应用潜力。本文在全耦合的通用地球系统模式中开展了LPF和EnKF的同化实验,同化资料为模拟的卫星海表温度资料。着重考察了不同局地化参数对两种方法的不同影响,对比了局地化粒子滤波器与集合卡尔曼滤波器的同化效果差异。比较的结果表明,LPF的同化效果对于局地化参数的选择非常敏感,在使用最优局地化参数的条件下,LPF能达到与EnKF相当甚至优于后者的同化效果,并具有较大的改进空间。     

中国近海现场海洋观测系统设计评估

中国科学院正在发展一个在中国近海(包括黄海、东海和南海)现场海洋观测系统。观测系统包括3个沿岸观测站点、4个近海离岸浮标和由观测船只按固定航线做的船舶观测断面。观测站点、浮标和断面的位置已经预先确定,这个计划在2008-2011实施。利用基于卡尔曼理论的样本集合方法对这样一个能够监测大尺度的季节和年季变率的观测系统设计进行了评估。根据卡尔曼滤波理论,用集合样本的方法能够给出经过同化这个观测系统位置的观测资料后能够减少多少分析误差和分析场的不确定性。用2个来自不同模式、不同分辨率的模式的结果作为集合样本来计算静态的背景误差协方差,这2套样本分别是来自分辨率是0.5°×0.5°的模式同化结果和高分辨0.125°×0.125°的模式结果。由这2个不同资料得到的结果是一致的。发现来自3个近岸和4个离岸浮标得到的观测能够有效地减少SST在渤海、黄海、东海和南海中部的分析误差。然而在越南东部和台湾东部海域,分析误差减少的百分比相对要小。最后,给出了中国近海最优的观测位置序列设计。     

Parameter sensitivity of three Kalman filter schemes for assimilation of water levels in shelf sea models

In applications of data assimilation algorithms, a number of poorly known assimilation parameters usually need to be specified. Hence, the documented success of data assimilation methodologies must rely on a moderate sensitivity to these parameters. This contribution presents a parameter sensitivity study of three well known Kalman filter approaches for the assimilation of water levels in a three dimensional hydrodynamic modelling system. The filters considered are the ensemble Kalman filter (EnKF), the reduced rank square root Kalman filter (RRSQRT) and the steady Kalman filter. A sensitivity analysis of key parameters in the schemes is undertaken for a setup in an idealised bay. The sensitivity of the resulting root mean square error (RMSE) is shown to be low to moderate. Hence the schemes are robust within an acceptable range and their application even with misspecified parameters is to be encouraged in this perspective. However, the predicted uncertainty of the assimilation results are sensitive to the parameters and hence must be applied with care. The sensitivity study further demonstrates the effectiveness of the steady Kalman filter in the given system as well as the great impact of assimilating even very few measurements.     

集合调整卡尔曼滤波方法在M2分潮数值模拟中的水深估计研究

数据同化利用观测信息对模型状态场调整的同时也可以对数值模型中的不确定参数进行估计,从而改进数值模型,提高数值模拟的精度。本文基于集合调整卡尔曼滤波方法,采用广义坐标系统的美国普林斯顿大学海洋模式的外模式开展了渤海和部分黄海海域M₂分潮模拟中的水深估计研究。理想数据同化试验结果表明,集合调整卡尔曼滤波方法能很好地降低模式模拟的水位误差并反演出“真实”的水深参数。而在NAO.99Jb和验潮站数据的实际数据同化试验中,与验潮站数据相比较,水深参数估计后,模式模拟的M₂分潮振幅与迟角误差分别降低了40.27%和49.19%。     

融合法及其在数据同化中的应用研究

根据预报值具有最小方差这一要求,详细推导了融合法在观测数据为一维、多维和维数不同的情况下的具体同化表达形式,同时还给出了不同情况下与同化表达式相对应的预报误差公式.利用这些公式,可以用融合法处理常见的海洋观测数据的同化问题.在陆架海模式HAMSOM基础上,以4月份的渤海海表温度为例,我们验证了同化公式的正确性,并给出了同化后较好的同化结果。最后将融合法的同化结果与卡尔曼滤波同化结果进行了对比.比较表明,融合法使用起来更简单,且能有效地处理常见的海洋观测数据.     

Using the ensemble Kalman filter to estimate multiplicative model parameters

This paper proposes a simple approach to estimating multiplicative model parameters using the ensemble square root filter. The basic idea, following previous studies, is to augment the state vector by the model parameters. While some success with this approach has been reported if the model parameters enter as additive terms in the tendency equations, this approach is problematic if the model parameters are multiplied by the state variables. The reason for this difficulty is that multiplicative parameters change the dynamical properties of the model, and in particular can cause the model to become dynamically unstable. This paper shows that model instability can be avoided if the usual persistence model for parameter update is replaced by a temporally smoothed version of the update model. In addition, the augmentation approach can be interpreted as two simultaneously decoupled ensemble Kalman filters for the model state and the parameter state, respectively. Implementation of the parameter estimation does not require changing the data assimilation algorithm—it just has to be supplemented by a parameter estimation step that is similar to the state estimation step. Covariance localization is found to be necessary not only for the model state, but also for augmented model parameters, if they are spatially dependent. The new formulation is illustrated with the Lorenz-96 model and shown to be capable of estimating additive and multiplicative model parameters, as well as the state, under relatively challenging conditions (e.g. using 20 observations to estimate 120 unknown variables).