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).     

Directional spread parameter at intermediate water depth

The characteristics of directional spread parameters at intermediate water depth are investigated based on a cosine power ‘2s' directional spreading model. This is based on wave measurements carried out using a Datawell directional waverider buoy in 23 m water depth. An empirical equation for the frequency dependent directional spreading parameter is presented. Directional spreading function estimated based on the Maximum Entropy Method is compared with those obtained using a cosine power ‘2s' parameter model. A set of empirical equations relating the directional spreading parameter corresponding to the peak of wave spectrum to other wave parameters like significant wave height and period are obtained. It shows that the wave directional spreading at peak wave frequency can be related to the non-linearity parameter, which allows estimation of directional spreading without reference to wind information.     

Spatially and temporally varying adaptive covariance inflation for ensemble filters

Ensemble filters are used in many data assimilation applications in geophysics. Basic implementations of ensemble filters are trivial but are susceptible to errors from many sources. Model error, sampling error and fundamental inconsistencies between the filter assumptions and reality combine to produce assimilations that are suboptimal or suffer from filter divergence. Several auxiliary algorithms have been developed to help filters tolerate these errors. For instance, covariance inflation combats the tendency of ensembles to have insufficient variance by increasing the variance during the assimilation. The amount of inflation is usually determined by trial and error. It is possible, however, to design Bayesian algorithms that determine the inflation adaptively. A spatially and temporally varying adaptive inflation algorithm is described. A normally distributed inflation random variable is associated with each element of the model state vector. Adaptive inflation is demonstrated in two low-order model experiments. In the first, the dominant error source is small ensemble sampling error. In the second, the model error is dominant. The adaptive inflation assimilations have better mean and variance estimates than other inflation methods.     

A stochastic operational forecasting system of the Black Sea: Technique and validation

In this article, we present the latest version of an ensemble forecasting system of the hydrodynamics of the Black Sea, based on the GHER model. The system includes the Weakly Constrained Ensembles algorithm to generate random, but physically balanced perturbations to initialize members of the ensemble. On top of initial conditions, the ensemble accounts also for uncertainty on the atmospheric forcing fields, and on some scalar parameters such as river flows or model diffusion coefficients. The forecasting system also includes the Ocean Assimilation Kit, a sequential data assimilation package implementing the SEEK and Ensemble Kalman filters. A novel aspect of the forecasting system is that not only our best estimate of the future ocean state is provided, but also the associated error estimated from the ensemble of models. The primary goal of this paper is to quantitatively show that the ensemble variability is a good estimation of the model error, regardless of the magnitude of the forecast errors themselves. In order for this estimation to be meaningful, the model itself should also be well validated. Therefore, we describe the model validation against general circulation patterns. Some particular aspects critical for the Black Sea circulation are validated as well: the mixed layer depth and the shelfopen sea exchanges. The model forecasts are also compared with observed sea surface temperature, and errors are compared to those of another operational model as well.     

Estimation of Transmission Loss in the Presence of Geoacoustic Inversion Uncertainty

A common problem in sonar system prediction is that the ocean environment is not well known. Utilizing probabilistic based results from geoacoustic inversions we characterize parameters relevant to sonar performance. This paper describes the estimation of transmission loss and its statistical properties based on posterior parameter probabilities obtained from inversion of ocean acoustic array data. This problem is solved by first finding an ensemble of relevant environmental model parameters and the associated posterior probability using a likelihood based inversion of the acoustic array data. In a second step, each realization of these model parameters is weighted with their posterior probability to map into the transmission loss domain. This approach is illustrated using vertical-array data from a recent benchmark data set and from data acquired during the Asian Seas International Acoustics Experiment (ASIAEX) 2001 in the East China Sea. The environmental parameters are first estimated using a probabilistic-based geoacoustic inversion technique. Based on the posterior probability that each of these environmental models fits the ocean acoustic array data, each model is mapped into transmission loss. This enables us to compute a full probability distribution for the transmission loss at selected frequencies, ranges, and depths, which potentially could be used for sonar performance prediction.     

基于Sobol法的海底反向散射模型参数敏感度分析

为了对海底声学参数反演中的估计精度做出预估,合理解释反演结果,本文基于Jackson海底声散射模型,利用Sobol算法,对该模型中的耗散系数、速度比等7个参数进行定量的敏感度分析。Sobol算法可以给出参数的一阶敏感度和参数间相互作用的敏感度,适用于分析散射强度的影响因子。仿真结果表明,所选择的声波频率对于参数的敏感度影响不大,模型各参数交互作用较为强烈,速度比的全局敏感度最大,而耗散系数敏感度很小。将参数划分为地声属性参数、粗糙度参数和非均匀性参数,地声属性参数敏感度最大。模型参数敏感度分析结果对于反演有一定的指导作用。     

转移矩阵未知时带乘性噪声系统的自适应滤波算法

针对带乘性噪声系统状态转移矩阵未知的情况,提出1种在线性最小方差意义下的系统参数和状态联合滤波算法。以迭代方式获得模型参数和系统状态的递推估计:首先,利用之前时刻的状态估计值,根据投影定理,对系统未知参数即系统状态转移矩阵作出估计;其次,利用已得到的系统参数估计值,获取当前时刻的状态滤波。计算机仿真结果表明了算法的有效性。     

南海中央海盆岩石圈纵向演化模拟

基于南海海盆的地质构造历史、地球物理和构造应力场资料建立准三维有限元模型,对南海中央海盆进行岩石圈纵向演化动力学模拟.本文采用弹塑性各向同性连续岩石介质模型,充分考虑拉伸速率、拉伸位移以及模型参数对模拟结果的影响,共建立了3个模型进行比较.从数值模拟结果可以看出中央海盆在被动拉张的地质构造背景下是单向生长的,动态模拟出南海中央海盆在形成过程中的陆缘裂离、海底扩张两个阶段的岩石圈纵向演化过程,并且分析了两个阶段岩石圈的动力学性质.提出水平的被动拉张力是南海岩石圈纵向演化前两个阶段形成的主要因素,并且在这两个阶段中,岩石圈的纵向演化时间主要集中在海底扩张阶段.     

CNOP-P-based parameter sensitivity for double-gyre variation in ROMS with simulated annealing algorithm

Reducing the error of sensitive parameters by studying the parameters sensitivity can reduce the uncertainty of the model,while simulating double-gyre variation in Regional Ocean Modeling System(ROMS).Conditional Nonlinear Optimal Perturbation related to Parameter(CNOP-P)is an effective method of studying the parameters sensitivity,which represents a type of parameter error with maximum nonlinear development at the prediction time.Intelligent algorithms have been widely applied to solving Conditional Nonlinear Optimal Perturbation(CNOP).In the paper,we proposed an improved simulated annealing(SA)algorithm to solve CNOP-P to get the optimal parameters error,studied the sensitivity of the single parameter and the combination of multiple parameters and verified the effect of reducing the error of sensitive parameters on reducing the uncertainty of model simulation.Specifically,we firstly found the non-period oscillation of kinetic energy time series of double gyre variation,then extracted two transition periods,which are respectively from high energy to low energy and from low energy to high energy.For every transition period,three parameters,respectively wind amplitude(WD),viscosity coefficient(VC)and linear bottom drag coefficient(RDRG),were studied by CNOP-P solved with SA algorithm.Finally,for sensitive parameters,their effect on model simulation is verified.Experiments results showed that the sensitivity order is WD>VC>>RDRG,the effect of the combination of multiple sensitive parameters is greater than that of single parameter superposition and the reduction of error of sensitive parameters can effectively reduce model prediction error which confirmed the importance of sensitive parameters analysis.     

Global parameter estimation of the Cochlodinium polykrikoides model using bioassay data

Cochlodinium polykrikoides is a notoriously harmful algal species that inflicts severe damage on the aquacultures of the coastal seas of Korea and Japan. Information on their expected movement tracks and boundaries of influence is very useful and important for the effective establishment of a reduction plan. In general, the information is supported by a red-tide(a.k.a algal bloom) model. The performance of the model is highly dependent on the accuracy of parameters, which are the coefficients of functions approximating the biological growth and loss patterns of the C. polykrikoides. These parameters have been estimated using the bioassay data composed of growth-limiting factor and net growth rate value pairs. In the case of the C. polykrikoides, the parameters are different from each other in accordance with the used data because the bioassay data are sufficient compared to the other algal species. The parameters estimated by one specific dataset can be viewed as locally-optimized because they are adjusted only by that dataset. In cases where the other one data set is used, the estimation error might be considerable. In this study, the parameters are estimated by all available data sets without the use of only one specific data set and thus can be considered globally optimized. The cost function for the optimization is defined as the integrated mean squared estimation error, i.e., the difference between the values of the experimental and estimated rates. Based on quantitative error analysis, the root-mean squared errors of the global parameters show smaller values, approximately 25%–50%, than the values of the local parameters. In addition, bias is removed completely in the case of the globally estimated parameters. The parameter sets can be used as the reference default values of a red-tide model because they are optimal and representative. However, additional tuning of the parameters using the in-situ monitoring data is highly required.As opposed to the bioassay data, it is necessary because the bioassay data have limitations in terms of the in-situ coastal conditions.     

Application of particle filter combined with extended Kalman filter in model identification of an autonomous underwater vehicle based on experimental data

This study proposes a method for identification of the nonlinear dynamic model of an AUV while some states are unmeasured; hence, it concentrates on a nonlinear “state and parameter estimation” issue. In this method, a local linearization is used for solving the nonlinear dynamics based on the extended Kalman filter (EKF), and a particle filter (PF) is used to minimize errors and variances of the nonlinear system. In other words, the PF is combined with the EKF in the form of the extended Kalman particle filter (EKPF). The EKPF method is independent of the initial values and satisfies the limits of the parameters and also the assumption that the hydrodynamic coefficients are constant. Hence, it is shown when the ranges or signs of some parameters are known, the EKPF is a more accurate estimator than the EKF. Moreover, a new simulation is done using the model estimated by the EKPF and the results are compared and validated with the measured data of a new experimental test. It is shown that the obtained model can predict the trajectory path with the total normalized root-mean-square error (NRMSE) of 14% and the surge mean speed with the NRMSE of 5%; and it describes the 6DOF motion of the AUV more accurate than the EKF model.     

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。     

A time-varying parameter model of a body oscillating in pitch

Among the assumptions upon which linear time-invariant models of floating bodies are based is that the body motions are so small that any change in the body’s angular position can be disregarded. However, it is often a major design requirement of a wave energy conversion device that the response amplitude is large, thereby invalidating one of the assumptions of the linear model. In particular, the immersed geometry of a body undergoes considerable variation when it is moved in pitch. With regard to this we investigate the difference in performance between a quasi-linear model in which the change of immersed surface is modelled by time-varying parameters and a basic linear model in which the immersed surface is time-invariant. The time-varying parameter model is realized by interpolation between the appropriate parameter values of a set of linear time-invariant (LTI) models derived for the different immersed surfaces that occur at discrete body displacements. It is shown that the responses predicted using the time-varying parameter model are closer to those measured experimentally than those of a standard frequency-domain model. Particular improvement occurs when the responses are large, such as at or near the resonance frequency. A problem which may limit the general use of the model is also discussed.     

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。     

岭估计在多项式曲面拟合GPS高程中的应用

探讨了岭估计方法在GPS高程拟合中的应用,并给出了岭估计中确定岭参数k值的一种新方法——方差扩大因子法。通过某地区GPS水准网数据进行实验验证,结果表明,与传统多项式曲面拟合模型比较,岭估计方法具有拟合精度高、适用复杂的地形等特点。     

An inverse problem methodology for multiple parameter estimation in bend stiffeners

The flexible riser top connection is a critical region for lifetime assessment due to large tension/curvature variations and modeling uncertainties. The bend stiffener polyurethane mechanical response not only presents a nonlinear loading rate and temperature dependency but is also subjected to weather ageing during operation, which may affect its mechanical behavior over time. The top tension, employed for riser local cross-section stress calculation, is usually obtained from global dynamic analyses performed under selected environmental conditions, if direct measurement is not available. As a consequence, both the bend stiffener effect on the curvature distribution and the top tension time series present inherent uncertainties for riser lifetime (re)assessment. In the present work, a proposed monitoring approach composed by gyrometers installed along flexible riser/bend stiffener top connection system length combined with an inverse problem methodology is numerically investigated to estimate the following parameters: (i) polyurethane hyperelastic response and (ii) effective top tension. The top connection system is modeled using a large deflection beam bending model and the parameters are estimated using a damped least-square minimization approach with the Levenberg–Marquardt algorithm. For the preliminary feasibility investigation, the gyrometer experimental data is numerically estimated through Monte Carlo simulations. A case study is carried out to investigate the influence that the number of sensors, sensors arrangement, loading conditions and top connection model have on the inverse parameters estimation. The results indicate that the proposed monitoring approach and inverse parameter estimation methodology may effectively reduce flexible riser lifetime calculation uncertainties.     

Comparison of climatic efficiency of the mechanisms of land-surface albedo changes caused by land use

Changes in ecosystem types, including situations when natural vegetation is replaced by agricultural lands, leads to surface albedo changes and the development of the corresponding short-wave radiative forcing (RF). This work analyzes ensemble numerical experiments with the climatic model (CM) of the Institute of Atmospheric Physics at the Russian Academy of Sciences (IAP RAS) for the 16th–21st centuries. The responses to changes in the contents of greenhouse gases and sulfate aerosols (tropospheric and stratospheric), in the solar constant, and in the land-surface albedo when natural vegetation is replaced by agricultural lands were modeled during these experiments. Different members of these ensemble experiments were obtained by varying the model parameters affecting the RF on the climate during land use: the albedo of agricultural lands was varied within the interval from 0.15 to 0.25 and the parameter controlling the efficiency of snow masking by tree vegetation was varied in the range from the absence of this effect to its maximally possible efficiency. It has been established that changes in surface albedo when natural vegetation is replaced by agricultural lands have the largest influence on the globally averaged annual mean RF at the top of the atmosphere, whereas the influence of snow masking on the RF is substantially less. This phenomenon is caused by the fact that snow masking by tree vegetation can take place only in winter in regions of temperate and high latitudes, when insolation is relatively low. A comparison of the spatial structure of the annual mean response of the surface temperature with the HadCRUT3v and GISS observational data makes it possible to narrow the admissible range of model parameter values. In particular, it can be inferred that the key parameter values which control the influence that land use has on the surface albedo in the IAP RAS CM are close to optimal. In addition, variations in these parameters do not lead to a significant influence of land use on climate change in the 21st century if the Land Use Harmonization (LUH) scenarios of changes in the area of agricultural lands are used: the uncertainty of the model response associated with the uncertainty of values of such controlling parameters in the 21st century does not exceed 0.1 K.     

Wave height parameter for damage description of rubble-mound breakwaters

In this paper it will be shown that the wave height parameter H₅₀, defined as the average wave height of the 50 highest waves reaching a rubble-mound breakwater in its useful life, can describe the effect of the wave height on the history of the armor damage caused by the wave climate during the structure's usable life.Using Thompson and Shuttler (Thompson, D.M., Shuttler, R.M., 1975. Riprap design for wind wave attack: A laboratory study on random waves. HRS Wallingford, Report 61, UK) data it will be shown that H₅₀ is the wave parameter that best represents the damage evolution with the number of waves in a sea state. Using this H₅₀ parameter, formulae as van der Meer (van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. PhD Thesis. Technical University of Delft) and Losada and Giménez-Curto (Losada, M.A., Gimenez–Curto, L.A., 1979. The joint effect of the wave height and period on the stability of rubble mound breakwaters using Iribarren's number. Coastal Engineering, 3, 77–96) are transformed into sea-state damage evolution formulae. Using these H₅₀-transformed formulae for regular and irregular sea states it will be shown how damage predictions are independent of the sea state wave height distribution.To check the capability of these H₅₀-formulae to predict damage evolution of succession of sea states with different wave height distributions, some stability tests with regular and irregular waves have been carried out. After analysing the experimental results, it will be shown how H₅₀-formulae can predict the observed damage independently of the sea state wave height distribution or the succession of sea states.     

Different approaches to model error formulation in 4D-Var: a study with high-resolution advection schemes

All numerical models are imperfect. Weak constraint variational data assimilation ( VDA ), which provides a treatment of the modelling errors, is studied; building on the approach of Vidard et al. (Tellus, 56 A, pp. 177–188, 2004). The evolution of model error ( ME ) is modelled using ordinary differential equations, which involve a scalar parameter. These approaches were tested using different high-resolution advection schemes. The first set of experiments were constructed to see if it is possible to account for (numerical) discretization error within such a framework. In other set of experiments, a systematic source of modelling error was introduced by deliberately specifying an incorrect value for the Coriolis parameter in the model. Results with observational state at half of the model state resolution, are also presented. We also discuss a method of estimating the scalar parameter in the ME through VDA . In all cases, the inclusion of ME provides reduction in forecasting errors. Also, our experiments indicate that different settings of the model (e.g. using different high-resolution advection schemes) would need different ME formulation. Results presented in this paper could be used to formulate sophisticated ME forms to account for systematic errors in higher dimensional models with complex advection schemes.     

渤、黄海生态环境模拟的参数敏感度空间差异分析

随着海洋生态系统模型的发展,生态变量增多,众多生物过程参数量值的确定成为制约生态环境模拟的瓶颈问题,生态系统结构区域性要求模型中的生态参数具有区域差异。为探究不同海区的关键参数及参数敏感度的空间差异,本研究在渤、黄海建立了ROMS-CoSiNE物理–生物耦合的高分辨率生态系统模型,并对13种生态参数的敏感度空间分布进行分析。结果表明:南黄海中部与渤海及近岸海域的敏感度差异较大。渤海敏感度最大的参数为决定光合速率的浮游植物P-I曲线初始斜率,其次为浮游动物捕食半饱和常数和浮游动物最大捕食率。而南黄海中部敏感度最大的参数为浮游动物最大捕食率,其次为浮游植物死亡率和浮游植物P-I曲线初始斜率。结合敏感度分布及浮游植物生物量收支得出,渤海水体透明度较南黄海偏低、浮游植物生长光限制较强,是引起浮游植物P-I曲线初始斜率敏感度在渤海高于黄海的主要原因。浮游动物最大捕食率及浮游植物死亡率的敏感度空间差异,受渤、黄海浮游植物生物量差异的影响,与生态系统中的高度非线性特征有关。