A comparison of sequential assimilation schemes for ocean prediction with the HYbrid Coordinate Ocean Model (HYCOM): Twin experiments with static forecast error covariances

We assess and compare four sequential data assimilation methods developed for HYCOM in an identical twin experiment framework. The methods considered are Multi-variate Optimal Interpolation (MVOI), Ensemble Optimal Interpolation (EnOI), the fixed basis version of the Singular Evolutive Extended Kalman Filter (SEEK) and the Ensemble Reduced Order Information Filter (EnROIF). All methods can be classified as statistical interpolation but differ mainly in how the forecast error covariances are modeled. Surface elevation and temperature data sampled from an 1/12° Gulf of Mexico HYCOM simulation designated as the truth are assimilated into an identical model starting from an erroneous initial state, and convergence of assimilative runs towards the truth is tracked. Sensitivity experiments are first performed to evaluate the impact of practical implementation choices such as the state vector structure, initialization procedures, correlation scales, covariance rank and details of handling multivariate datasets, and to identify an effective configuration for each assimilation method. The performance of the methods are then compared by examining the relative convergence of the assimilative runs towards the truth. All four methods show good skill and are able to enhance consistency between the assimilative and truth runs in both observed and unobserved model variables. Prediction errors in observed variables are typically less than the errors specified for the observations, and the differences between the assimilated products are small compared to the observation errors. For unobserved variables, RMS errors are reduced by 50% relative to a non-assimilative run and differ between schemes on average by about 5%. Dynamical consistency between the updated state space variables in the data assimilation algorithm, and the data adequately sampling significant dynamical features are the two crucial components for reliable predictions. The experiments presented here suggest that practical implementation details can have at least as much an impact on the accuracy of the assimilated product as the choice of assimilation technique itself. We also present a discussion of the numerical implementation and the computational requirements for the use of these methods in large scale applications.     

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.     

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.     

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。     

A state decoupling approach to estimate unobservable trackingsystems

If a system is unobservable, the error covariance associated with a Kalman filter will be nearly singular. As a consequence, an optimum estimation in the sense of minimum error covariance does not exist. In this paper, we show that this (unobservable) system can be transformed into a nonlinear system with a linear measurement equation. In addition to other useful features, this transformation also serves to decouple the state in such a way that an observable part can be extracted and estimated while no information can be gained and processed for the unobservable part     

黄海海雾短时临近预报中云水路径的EnKF同化研究

在海雾的短时临近预报中,初始场的水汽凝结状态扮演着重要角色。为了改进初始场的云水含量,本文提出直接同化雾体云水信息的思路。针对2011年5月一次大范围的黄海海雾,借助EnKF (Ensemble Kalman Filter)方法,尝试进行了极轨卫星反演云水路径数据的同化试验。结果表明:(1)通过利用EnKF将云水混合比增加到背景场和分析场的控制变量中,构建云水观测数据与背景场之间的关系,实现云水路径数据的直接同化是可行的;(2)同化云水路径可显著改善海面气温与湿度状态,大幅提高海雾预报效果;(3)EnKF能够基于集合体动态统计流依赖的背景误差协方差是其取得良好同化效果的主要原因。值得指出的是,受集合样本误差的影响,需要特别关注云水含量与风之间的相关关系。     

Low‐dimensional representation of error covariance

Ensemble and reduced‐rank approaches to prediction and assimilation rely on low‐dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time‐independent systems are used to identify factors that cause the steady‐state analysis error covariance to admit a low‐dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix , a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady‐state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time‐dependent systems. If much of the steady‐state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady‐state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting non‐modal transient growth. Failure to observe growing modes leads to increased steady‐state analysis error variances. Leading eigenvectors of the steady‐state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest‐order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady‐state analysis error covariance matrix.     

Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models

In atmospheric data assimilation (DA), observations over a 6–12-h time window are used to estimate the state. Non-adaptive moderation or localization functions are widely used in ensemble DA to reduce the amplitude of spurious ensemble correlations. These functions are inappropriate (1) if true error correlation functions move a comparable distance to the localization length scale over the time window and/or (2) if the widths of true error correlation functions are highly flow dependent. A method for generating localization functions that move with the true error correlation functions and that also adapt to the width of the true error correlation function is given. The method uses ensemble correlations raised to a power (ECO-RAP). A gallery of periodic one-dimensional error models is used to show how the method uses error propagation information and error correlation width information retained by powers of raw ensemble correlations to propagate and adaptively adjust the width of the localization function. It is found that ECO-RAP localization outperforms non-adaptive localization when the true errors are propagating or the error correlation length scale is varying and is as good as non-adaptive localization when such variations in error covariance structure are absent.     

Pseudo long base line navigation algorithm for underwater vehicles with inertial sensors and two acoustic range measurements

This paper presents an integrated navigational algorithm for unmanned underwater vehicles (UUV) using two acoustic range transducers and strap-down inertial measurement unit (SD-IMU). A range measurement model is derived for a UUV having one acoustic transducer and cruising around two reference transponders at sea floor or surface. The proposed algorithm, called pseudo long base line (PLBL), estimates the position of the vehicle integrating the SD-IMU signals corrected with the two range measurements. Extended Kalman filter was applied to propagate error covariance, to update measurement errors and to correct state equation whenever the external measurements are available. Simulations were conducted to illustrate the effectiveness of the PLBL using the 6-d.o.f. nonlinear numerical model of a UUV at current flow, excluding bottom-fixed DVL. This paper also shows the error convergence of the vehicle's initial position by the additional range measurements without velocity information.     

船舶动力定位系统的预测模糊控制

在船舶动力定位中采用预测模糊控制策略,即通过自校正滤波与Ｋａｌｍ ａｎ 滤波得到系统低频运动位置偏差与相应速度的预测值作为模糊控制器的输入,以实现对其在水平面内的运动控制。因为基于系统模型的滤波器输出最终是经模糊化后输入至模糊控制器的,于是可大大降低对系统建模的精度要求,控制器本身具有强的鲁棒性。仿真结果说明了该策略的可行性及良好的控制性能。     

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。     

Data assimilation considerations for improved ocean predictability during the Gulf of Mexico Grand Lagrangian Deployment (GLAD)

Ocean prediction systems rely on an array of assumptions to optimize their data assimilation schemes. Many of these remain untested, especially at smaller scales, because sufficiently dense observations are very rare. A set of 295 drifters deployed in July 2012 in the north-eastern Gulf of Mexico provides a unique opportunity to test these systems down to scales previously unobtainable. In this study, background error covariance assumptions in the 3DVar assimilation process are perturbed to understand the effect on the solution relative to the withheld dense drifter data. Results show that the amplitude of the background error covariance is an important factor as expected, and a proposed new formulation provides added skill. In addition, the background error covariance time correlation is important to allow satellite observations to affect the results over a period longer than one daily assimilation cycle. The results show the new background error covariance formulations provide more accurate placement of frontal positions, directions of currents and velocity magnitudes. These conclusions have implications for the implementation of 3DVar systems as well as the analysis interval of 4DVar systems.     

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.     

Simulation of an Inertial Acoustic Navigation System With Range Aiding for an Autonomous Underwater Vehicle

This paper presents an integrated navigation system for underwater vehicles to improve the performance of a conventional inertial acoustic navigation system by introducing range measurement. The integrated navigation system is based on a strapdown inertial navigation system (SDINS) accompanying range sensor, Doppler velocity log (DVL), magnetic compass, and depth sensor. Two measurement models of the range sensor are derived and augmented to the inertial acoustic navigation system, respectively. A multirate extended Kalman filter (EKF) is adopted to propagate the error covariance with the inertial sensors, where the filter updates the measurement errors and the error covariance and corrects the system states when the external measurements are available. This paper demonstrates the improvement on the robustness and convergence of the integrated navigation system with range aiding (RA). This paper used experimental data obtained from a rotating arm test with a fish model to simulate the navigational performance. Strong points of the navigation system are the elimination of initial position errors and the robustness on the dropout of acoustic signals. The convergence speed and conditions of the initial error removal are examined with Monte Carlo simulation. In addition, numerical simulations are conducted with the six-degrees-of-freedom (6-DOF) equations of motion of an autonomous underwater vehicle (AUV) in a boustrophedon survey mode to illustrate the effectiveness of the integrated navigation system.     

海洋资料同化对气候季节-年际预测技巧及初始场的影响试验

集合卡尔曼滤波(Ensemble Kalman filter, EnKF)是一种国内外广泛使用的海洋资料同化方案, 用集合成员的状态集合表征模式的背景误差协方差, 结合观测误差协方差, 计算卡尔曼增益矩阵, 有效地将观测信息添加到模式初始场中。由于季节、年际预测很大程度上受到初始场的影响, 因此资料同化可以提高模式的预测性能。本文在NUIST-CFS1.0预测系统逐日SST nudging的初始化方案上, 利用EnKF在每个月末将全场(full field)海表温度(sea surface temperature, SST)、温盐廓线(in-situ temperature and salinity profiles, T-S profiles)以及卫星观测海平面高度异常(sea level anomalies, SLA)观测资料同化到模式初始场中, 对比分析了无海洋资料同化以及加入同化后初始场的区别、加入海洋资料同化后模式提前1~24个月预测性能的差异以及对于厄尔尼诺-南方涛动(El Niño-southern oscillation, ENSO)预测技巧的影响。结果表明, 加入海洋资料同化能有效地改进初始场, 并且呈现随深度增加初始场改进越显著的特征。加入同化后, 对全球SST、次表层海水温度的平均预测技巧均有一定的提高, 也表现出随深度增加预测技巧改进越明显的特征。但加入海洋资料同化后, 模式对ENSO的预测技巧有所下降, 可能是由于模式误差的存在, 使得同化后的预测初始场从接近观测的状态又逐渐恢复到与模式动力相匹配的状态, 加剧了赤道太平洋冷舌偏西、中东部偏暖的气候平均态漂移。     

Preconditioned Optimizing Utility for Large-dimensional analyses (POpULar)

I present the derivation of the Preconditioned Optimizing Utility for Large-dimensional analyses (POpULar), which is developed for adopting a non-diagonal background error covariance matrix in nonlinear variational analyses (i.e., analyses employing a non-quadratic cost function). POpULar is based on the idea of a linear preconditioned conjugate gradient method widely adopted in ocean data assimilation systems. POpULar uses the background error covariance matrix as a preconditioner without any decomposition of the matrix. This preconditioning accelerates the convergence. Moreover, the inverse of the matrix is not required. POpULar therefore allows us easily to handle the correlations among deviations of control variables (i.e., the variables which will be analyzed) from their background in nonlinear problems. In order to demonstrate the usefulness of POpULar, we illustrate two effects which are often neglected in studies of ocean data assimilation before. One is the effect of correlations among the deviations of control variables in an adjoint analysis. The other is the nonlinear effect of sea surface dynamic height calculation required when sea surface height observation is employed in a three-dimensional ocean analysis. As the results, these effects are not so small to neglect.     

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.     

INS/GPS组合导航中的病态问题及其处理方法

在地球表面附近的组合导航中,一般以经、纬度表示水平位置,单位为弧度(rad),以高程表示竖直位置,单位为米(m)。1m的定位误差仅相当于约10-7rad,从而造成定位协方差矩阵的条件数达1013量级。Kalman滤波中要对协方差矩阵求逆,过高的条件数将引起严重的病态问题,从而造成很大的数值误差,影响滤波精度,甚至造成滤波发散。提出一种直接的解决方法,即构建一种避免病态问题的组合导航滤波模型。具体过程为:引入一种尺度因子(即平均地球半径)对组合导航系统的状态量、观测量、状态方程以及观测方程进行线性变换,从而对经、纬度误差进行适当的尺度化,明显降低协方差矩阵的条件数,有效避免了滤波过程中病态问题的出现。新方法在不明显增加计算量的前提下有效解决了病态问题,并保证了Kalman滤波的最优性质。数值仿真验证了该方法的有效性。     

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.     

Towed array shape estimation using Kalman filters-theoreticalmodels

The dynamical behavior of a thin flexible array towed through the water is described by the Paidoussis equation. By discretizing this equation in space and time a finite-dimensional state-space representation is obtained where the states are the transverse displacements of the array from linearity in either the horizontal or vertical plane. The form of the transition matrix in the state-space representation describes the propagation of transverse displacements down the array. The outputs of depth sensors and compasses located along the array are shown to be related in a simple, linear manner to the states. From this state-space representation a Kalman filter which recursively estimates the transverse displacements and hence the array shape is derived. It is shown how the properties of the Kalman filter reflect the physics of the propagation of motion down the array. Solutions of the Riccati equation are used to predict the mean square error of the Kalman filter estimates of the transverse displacements