Reply to “Comment on ‘Ensemble Kalman filter with the unscented transform”’

This is a reply to the comment of Dr. Sakov on the work “Ensemble Kalman filter with the unscented transform” of Luo and Moroz (2009) [2].     

Scaled unscented transform Gaussian sum filter: Theory and application

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. We introduce a framework, called the scaled unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas: the scaled unscented Kalman filter (SUKF) based on the concept of scaled unscented transform (SUT) (Julier and Uhlmann (2004) [16]), and the Gaussian mixture model (GMM). The SUT is used to approximate the mean and covariance of a Gaussian random variable which is transformed by a nonlinear function, while the GMM is adopted to approximate the probability density function (pdf) of a random variable through a set of Gaussian distributions. With these two tools, a framework can be set up to assimilate nonlinear systems in a recursive way. Within this framework, one can treat a nonlinear stochastic system as a mixture model of a set of sub-systems, each of which takes the form of a nonlinear system driven by a known Gaussian random process. Then, for each sub-system, one applies the SUKF to estimate the mean and covariance of the underlying Gaussian random variable transformed by the nonlinear governing equations of the sub-system. Incorporating the estimations of the sub-systems into the GMM gives an explicit (approximate) form of the pdf, which can be regarded as a “complete” solution to the state estimation problem, as all of the statistical information of interest can be obtained from the explicit form of the pdf (Arulampalam et al. (2002) [7]).In applications, a potential problem of a Gaussian sum filter is that the number of Gaussian distributions may increase very rapidly. To this end, we also propose an auxiliary algorithm to conduct pdf re-approximation so that the number of Gaussian distributions can be reduced. With the auxiliary algorithm, in principle the SUT-GSF can achieve almost the same computational speed as the SUKF if the SUT-GSF is implemented in parallel.As an example, we will use the SUT-GSF to assimilate a 40-dimensional system due to Lorenz and Emanuel (1998) [27]. We will present the details of implementing the SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF.     

History matching of facies distribution with the EnKF and level set parameterization

In this work, we develop a methodology to combine the Ensemble Kalman filter (EnKF) and the level set parameterization for history matching of facies distribution. With given prior knowledge about the facies of the reservoir geology, initial realizations are generated by commonly used software as the prior guesses of the unknown field. Furthermore, level set functions are used to reparameterize these initial realizations. In the reparameterization process, a representing node system is set up, on which the values of level set functions are assigned using Gaussian random numbers. The mean and the standard deviation of the Gaussian random numbers are designed according to the facies proportion, and the sign of the random numbers depends on the facies type at the representing nodes. The values of the level set functions at the other grid nodes are obtained by linear interpolation. The level set functions on the representing nodes are the model parameters of the EnKF state vector and are updated in the data assimilation process. On the basis of our numerical examples for two-dimensional reservoirs with two or three facies, the proposed method is demonstrated to be able to capture the main features of the reference facies distributions.     

Capturing eddy shedding in the Gulf of Mexico from Lagrangian observations

The nonlinear process of eddy shedding is studied in the context of the Gulf of Mexico. We show that model runs which do not include eddy detachment can reproduce such an event with the assimilation of suitable data obtained from a control run with eddy detachment. This works surprisingly well and with small amounts of data provided the data originates from instruments that are carried by the flow, i.e. Lagrangian. This is compared with analogous assimilation of data from fixed stations which capture the eddy poorly. The remarkable efficacy of Lagrangian data assimilation in this context is explained by considering the structure of the correlation functions and their associated regions of influence.     

On a nonlinear Kalman filter with simplified divided difference approximation

We present a new ensemble-based approach that handles nonlinearity based on a simplified divided difference approximation through Stirling’s interpolation formula, which is hence called the simplified divided difference filter (sDDF). The sDDF uses Stirling’s interpolation formula to evaluate the statistics of the background ensemble during the prediction step, while at the filtering step the sDDF employs the formulae in an ensemble square root filter (EnSRF) to update the background to the analysis. In this sense, the sDDF is a hybrid of Stirling’s interpolation formula and the EnSRF method, while the computational cost of the sDDF is less than that of the EnSRF. Numerical comparison between the sDDF and the EnSRF, with the ensemble transform Kalman filter (ETKF) as the representative, is conducted. The experiment results suggest that the sDDF outperforms the ETKF with a relatively large ensemble size, and thus is a good candidate for data assimilation in systems with moderate dimensions.     

A Bayesian tutorial for data assimilation

Data assimilation is the process by which observational data are fused with scientific information. The Bayesian paradigm provides a coherent probabilistic approach for combining information, and thus is an appropriate framework for data assimilation. Viewing data assimilation as a problem in Bayesian statistics is not new. However, the field of Bayesian statistics is rapidly evolving and new approaches for model construction and sampling have been utilized recently in a wide variety of disciplines to combine information. This article includes a brief introduction to Bayesian methods. Paying particular attention to data assimilation, we review linkages to optimal interpolation, kriging, Kalman filtering, smoothing, and variational analysis. Discussion is provided concerning Monte Carlo methods for implementing Bayesian analysis, including importance sampling, particle filtering, ensemble Kalman filtering, and Markov chain Monte Carlo sampling. Finally, hierarchical Bayesian modeling is reviewed. We indicate how this approach can be used to incorporate significant physically based prior information into statistical models, thereby accounting for uncertainty. The approach is illustrated in a simplified advection–diffusion model.     

Unscented Kalman filter and its nonlinear application for tracking a moving target

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 40 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. To overcome these limitations, this paper proposes the unscented Kalman filter (UKF). And the algorithms of the FEKF, SEKF and UKF are given. Furthermore, the state models and measurement models of a target are set up. For comparison purpose, the three algorithms is simulated for the target tracking, and the algorithm performance is analyzed and compared by the simulation results of FEKF, SEKF and UKF. Numerical results demonstrate that FEKF and UKF give almost identical results while the estimates of SEKF are clearly worse. The UKF is easier to implement, avoiding Jacobian and Hessian matrices computation.     

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

State and parameter estimation in stochastic dynamical models

This paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model. In contrast to previous studies, the change in background distribution due to changes in model parameters is taken into account. An ensemble approach to solving the maximum likelihood estimates is proposed. An exact solution for the ensemble update based on a square root Kalman Filter is derived. This solution involves a two step procedure in which an ensemble is first produced by a standard ensemble Kalman Filter, and then “corrected” to account for parameter estimation, thereby allowing a user to take advantage of an existing ensemble filter. The solution is illustrated with simple, low-dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters.     

遥感反演时间序列叶面积指数的集合卡尔曼平滑算法

基于MODIS LAI产品数据集(MOD15A2)构建经验性的LAI动态模型,以LAI作为连接参数,将LAI动态模型与植被辐射传输模型MCRM2相耦合,提出了将耦合模型与时间序列MODIS反射率观测数据集(MOD09A1)同化进行LAI反演的方案.将集合卡尔曼平滑(EnKS)方法引入到LAI同化反演中,为更好地评价该算...     

基于粒子滤波的一种改进的资料同化方法

针对在粒子数较少时传统的集合卡尔曼滤波和粒子滤波方法不能有效表征后验概率密度函数(PDF)的问题, 提出了一种改进的粒子滤波方法. 主要思想是在预测步之后引入更新步, 并且将观测时刻与非观测时刻的同化分析进行区别处理. 对典型的低维和高维混沌系统的仿真结果表明:改进粒子滤波方法是一种非常有效的估计非线性非高斯随机系统状态的方法.     

四旋翼飞行器姿态解算与滤波

四旋翼飞行器的运动控制关键在于对飞行过程中的实时姿态角控制。目前实时姿态角信息还不能直接测量出来。为了能利用已有的传感器数据解算出更准确的姿态角,通过物理实验详细分析了四旋翼飞行器姿态角的解算和滤波算法。首先,通过联立欧拉方向余弦矩阵与四元数矩阵,得到用四元数表达的姿态角表达式。然后,结合加速度计和磁强计实时测量的数据,分别采用互补滤波和卡尔曼滤波两种方法来补偿四元数结果,分别分析如何选取最佳参数,并对比分析了两种滤波方式的优缺点。在一定精度要求范围内,这两种滤波方式都能获得更加准确的姿态角,但是互补滤波相对卡尔曼滤波有一定的解算时延。因此在精度要求一般的系统中,这两种滤波方式都可以用来求解姿态角,卡尔曼滤波方法则更适于对实时性要求更高的系统。     

Sampling strong tracking nonlinear unscented Kalman filter and its application in eye tracking

The unscented Kalman filter is a developed well-known method for nonlinear motion estimation and tracking. However, the standard unscented Kalman filter has the inherent drawbacks, such as numerical instability and much more time spent on calculation in practical applications. In this paper, we present a novel sampling strong tracking nonlinear unscented Kalman filter, aiming to overcome the difficulty in nonlinear eye tracking. In the above proposed filter, the simplified unscented transform sampling strategy with n+2 sigma points leads to the computational efficiency, and suboptimal fading factor of strong tracking filtering is introduced to improve robustness and accuracy of eye tracking. Compared with the related unscented Kalman filter for eye tracking, the proposed filter has potential advantages in robustness, convergence speed, and tracking accuracy. The final experimental results show the validity of our method for eye tracking under realistic conditions.     

Adaptive Two-Step Bearing-Only Underwater Uncooperative Target Tracking with Uncertain Underwater Disturbances

The bearing-only tracking of an underwater uncooperative target can protect maritime territories and allows for the utilization of sea resources. Considering the influences of an unknown underwater environment, this work aimed to estimate 2-D locations and velocities of an underwater target with uncertain underwater disturbances. In this paper, an adaptive two-step bearing-only underwater uncooperative target tracking filter (ATSF) for uncertain underwater disturbances is proposed. Considering the nonlinearities of the target’s kinematics and the bearing-only measurements, in addition to the uncertain noise caused by an unknown underwater environment, the proposed ATSF consists of two major components, namely, an online noise estimator and a robust extended two-step filter. First, using a modified Sage-Husa online noise estimator, the uncertain process and measurement noise are estimated at each tracking step. Then, by adopting an extended state and by using a robust negative matrix-correcting method in conjunction with a regularized Newton-Gauss iteration scheme, the current state of the underwater uncooperative target is estimated. Finally, the proposed ATSF was tested via simulations of a 2-D underwater uncooperative target tracking scenario. The Monte Carlo simulation results demonstrated the reliability and accuracy of the proposed ATSF in bearing-only underwater uncooperative tracking missions.     

扩展卡尔曼滤波和不敏卡尔曼滤波在实时雷达回波反演大气波导中的应用

为了改善雷达回波反演大气波导(RFC)方面存在的单时次、单方位角反演的问题,提出利用扩展卡尔曼滤波和不敏卡尔曼滤波的反演算法对大气波导结构的多方位角实时跟踪反演. 在卡尔曼滤波方法中分别给出大气波导结构的参数化方程、观测方程、滤波算法的状态转移方程,最后导出滤波反演算法的迭代求解流程. 在大气波导结构不随时间变化和随时间变化的两种条件下,对扩展卡尔曼滤波和不敏卡尔曼滤波算法进行数值实验. 实验结果表明,不敏卡尔曼滤波更适用于RFC这高度非线性反演问题,它可能今后为大气波导结构多方位角实时跟踪反演的业务化运行提供理论基础与技术保证. 关键词： 大气波导 雷达回波 扩展卡尔曼滤波 不敏卡尔曼滤波     

BESⅢ track fitting algorithm

A track fitting algorithm based on the Kalman filter method has been developed for BESⅢ of BEPCⅡ.The effects of multiple scattering and energy loss when the charged particles go through the detector,non-uniformity of magnetic field (NUMF) and wire sag, etc., have been carefully handled.This algorithm works well and the performance satisfies the physical requirements tested by the simulation data.     

一种多滤波器组合导航数据融合方法

针对组合导航系统中使用单天线GPS接收机时导致姿态角不易收敛的问题,提出了一种互补滤波器和卡尔曼滤波器相结合的数据融合算法。该方法首先通过MEMS惯性传感器与磁强计设计了一种互补滤波算法。针对载体在变速运动过程中加速度计的倾角测量值有较大误差,影响互补滤波器输出精度的问题,通过GPS接收机和加速度计设计了卡尔曼滤波模型,将卡尔曼滤波器输出速度的微分量反馈给互补滤波器,实现了对互补滤波器中载体运动加速度的补偿。基于以上解算方法,以FPGA为核心处理器设计了组合导航系统并进行了车载实验。实验中,该方法有效补偿了汽车变速过程中的倾角测量误差,证明了该方法的有效性。     

基于Huber的高阶容积卡尔曼跟踪算法

为改善高阶容积卡尔曼滤波算法的滤波精度和鲁棒性, 提出了一种新的基于Huber的高阶容积卡尔曼滤波算法. 在采用统计线性回归模型近似非线性量测模型的基础上, 利用Huber M 估计算法实现状态的量测更新. 进一步结合高阶球面-径向容积准则的状态预测模块构成基于 Huber的高阶容积卡尔曼跟踪算法. 重点分析了Huber代价函数的调节因子对算法跟踪性能的影响. 通过对纯方位目标跟踪和再入飞行器跟踪两个实例验证了所提算法的跟踪性能优于传统高阶容积卡尔曼滤波算法.     

基于CamShift和Kalman组合的改进目标跟踪算法

针对应用CamShift算法进行目标跟踪过程中,当目标被严重遮挡、目标被与目标颜色相近的背景干扰时易丢失跟踪目标的问题,提出了一种基于CamShift和Kalman滤波组合的改进跟踪算法;为克服目标因严重遮挡而丢失的缺陷,利用自适应算法改进了传统的CamShift算法,扩大了搜索窗口,使运动目标位于搜索窗口内;为解决目标因颜色相近背景干扰而丢失的问题,改善跟踪准确率,利用卡尔曼滤波预测目标运动空间位置,作为下一帧搜索窗口的质心坐标;基于上述改进,利用C++语言,研发了改进的CamShift目标跟踪软件模块,给出了该模块的算法流程;实验结果表明,改进后的目标跟踪算法能有效地克服传统CamShift算法的缺陷,大大提高运动目标跟踪的准确性;所提的算法可以应用于运动小车跟踪,人脸识别等领域。     

自动引导车电池SOC估算方法研究

自动引导车(AGV车)工况特殊,电流积分法估算电池剩余容量(SOC)误差较大,而且存在累积误差。为了提高AGV车电池剩余容量估算的准确度,对扩展卡尔曼滤波法估算AGV车电池剩余容量进行了研究,分析了AGV车特殊工况,提出将扩展卡尔曼滤波法的滤波增益改进为动态调整滤波增益,有效提高扩展卡尔曼滤波法的跟踪效果。实验表明使用扩展卡尔曼滤波法估算AGV车电池剩余容量精度较高,采用动态校正的滤波增益提高了估算过程的跟踪效果,解决了AGV车电池剩余容量估算不准确的问题。