Ensemble member generation for sequential data assimilation

Using an ensemble of model forecasts to describe forecast error covariance extends linear sequential data assimilation schemes to nonlinear applications. This approach forms the basis of the Ensemble Kalman Filter and derivative filters such as the Ensemble Square Root Filter. While ensemble data assimilation approaches are commonly reported in the scientific literature, clear guidelines for effective ensemble member generation remain scarce. As the efficiency of the filter is reliant on the accurate determination of forecast error covariance from the ensemble, this paper describes an approach for the systematic determination of random error. Forecast error results from three factors: errors in initial condition, forcing data and model equations. The method outlined in this paper explicitly acknowledges each of these sources in the generation of an ensemble. The initial condition perturbation approach presented optimally spans the dynamic range of the model states and allows an appropriate ensemble size to be determined. The forcing data perturbation approach treats forcing observations differently according to their nature. While error from model physics is not dealt with in detail, discussion of some commonly used approaches and their limitations is provided. The paper concludes with an example application for a synthetic coastal hydrodynamic experiment assimilating sea surface temperature (SST) data, which shows better prediction capability when contrasted with standard approaches in the literature.     

Experiments of one-dimensional soil moisture assimilation system based on ensemble Kalman filter

Ensemble Kalman filter is a new sequential data assimilation algorithm which was originally developed for atmospheric and oceanographic data assimilation. It can be applied to calculate error covariance matrix through Monte-Carlo simulation. This approach is able to resolve the nonlinearity and discontinuity existed within model operator and observation operator. When observation data are assimilated at each time step, error covariances are estimated from the phase-space distribution of an ensemble of model states. The error statistics is then used to calculate Kalman gain matrix and analysis increments. In this study, we develop a one-dimensional soil moisture data assimilation system based on ensemble Kalman filter, the Simple Biosphere Model (SiB2) and microwave radiation transfer model (AIEM, advanced integration equation model). We conduct numerical experiments to assimilate in situ soil surface moisture measurements and low-frequency passive microwave remote sensing data into a land surface model, respectively. The results indicate that data assimilation can significantly improve the soil surface moisture estimation. The improvement in root zone is related to the model bias errors at surface layer and root zone. The soil moisture does not vary significantly in deep layer. Additionally, the ensemble Kalman filter is predominant in dealing with the nonlinearity of model operator and observation operator. It is practical and effective for assimilating observations in situ and remotely sensed data into land surface models.     

基于DEnKF的背景误差协方差局地化和协方差膨胀研究

尽管DEnKF同化不会引入观测采样误差,但小集合仍会造成背景误差协方差矩阵存在伪相关,出现滤波发散。为了减少小集合对数据同化结果的影响,结合Lorenz96模型和DEnKF同化方案分析了协方差局地化和协方差膨胀方法对背景误差协方差矩阵、增益矩阵及同化结果的影响。实验表明:协方差局地化方法能消除背景误差协方差矩阵和增益矩阵中的伪相关,增大背景误差协方差矩阵的秩,有助于滤波算法收敛到真实解;而协方差膨胀方法不能消除背景误差协方差矩阵和增益矩阵中的伪相关,只能改善在每个同化周期内背景误差协方差系统性被低估的现象;同化过程中采用合适的局地化半径和方差膨胀因子能够较好地改善同化结果的精度。     

观测误差协方差估计下的集合鲁棒滤波数据同化方法

在数据同化方法中,观测误差协方差矩阵是相关的,且与时间和状态有一定的依赖性。针对这种相关特性,将鲁棒滤波方法与观测误差协方差估计方法相结合,得到随状态时间变化的观测误差协方差,提出一种带有观测误差估计的鲁棒数据同化新方法,更新观测误差协方差,改善估计效果。从分析误差协方差,转移矩阵特征值放大等角度优化同化方法。利用非线性Lorenz-96混沌系统,对三种不同优化角度下带有观测误差估计的鲁棒滤波和原鲁棒滤波方法的鲁棒性和同化精度进行评估,并比较分析了两种方法在模型误差、观测数目和性能水平系数变化时的性能。结果表明：观测误差估计技术能够提高状态估计的精确性,带有观测误差估计的鲁棒滤波对系统参数变化具有较好的鲁棒性。     

A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation

The background error covariance matrix, B, is often used in variational data assimilation for numerical weather prediction as a static and hence poor approximation to the fully dynamic forecast error covariance matrix, P^f. In this paper the concept of an Ensemble Reduced Rank Kalman Filter (EnRRKF) is outlined. In the EnRRKF the forecast error statistics in a subspace defined by an ensemble of states forecast by the dynamic model are found. These statistics are merged in a formal way with the static statistics, which apply in the remainder of the space. The combined statistics may then be used in a variational data assimilation setting. It is hoped that the nonlinear error growth of small-scale weather systems will be accurately captured by the EnRRKF, to produce accurate analyses and ultimately improved forecasts of extreme events.     

Assimilating canopy reflectance data into an ecosystem model with an Ensemble Kalman Filter

An Ensemble Kalman Filter (EnKF) is used to assimilate canopy reflectance data into an ecosystem model. We demonstrate the use of an augmented state vector approach to enable a canopy reflectance model to be used as a non-linear observation operator. A key feature of data assimilation (DA) schemes, such as the EnKF, is that they incorporate information on uncertainty in both the model and the observations to provide a best estimate of the true state of a system. In addition, estimates of uncertainty in the model outputs (given the observed data) are calculated, which is crucial in assessing the utility of model predictions.Results are compared against eddy-covariance observations of CO₂ fluxes collected over three years at a pine forest site. The assimilation of 500 m spatial resolution MODIS reflectance data significantly improves estimates of Gross Primary Production (GPP) and Net Ecosystem Productivity (NEP) from the model, with clear reduction in the resulting uncertainty of estimated fluxes. However, foliar biomass tends to be over-estimated compared with measurements. Issues regarding this over-estimate, as well as the various assumptions underlying the assimilation of reflectance data are discussed.     

Objectively determined 10-day African rainfall estimates created for famine early warning systems

A method of estimation of accumulated precipitation which incorporates numerical model analyses, satellite and surface data has been developed for the African continent. An estimate for accumulated convective cloud precipitation is computed from cold cloud top temperature duration with a bias removal made from the use of rain-gauge data. Orographic precipitation from relatively warm cloud sources is estimated using a combination of surface and satellite data, orography, and numerical model analyses of relative humidity and wind. The results of a comparison of these precipitation estimates with independent rainfall data show this method produces skilful analyses of estimated accumulated precipitation for the Sahel region of Africa.     

Assimilating synthetic GOES-R radiances in cloudy conditions using an ensemble-based method

The weather research and forecasting (WRF) model and the maximum likelihood ensemble filter (MLEF) data assimilation approach are used to examine the potential impact of observations from the future Geostationary Operational Environmental Satellite, generation R (GOES-R) on improving our knowledge about clouds. Synthetic radiances are assimilated from the 10.35 μm channel of the GOES-R advanced baseline imager (ABI) employing a ‘non-identical twins’ experimental setup. The experimental results are examined for an extratropical cyclone named Kyrill that produced unusually strong winds, widespread damage and fatalities in Western Europe in January 2007. The data assimilation problem is especially challenging for this case, as there is a large error in the model-simulated radiances resulting from incorrect cloud location. Although this problem is difficult to eliminate, data assimilation results indicate the potential of GOES-R data to significantly reduce these errors.     

Using coarse remote sensing radar observations to control the trajectory of a simple Sahelian land surface model

In the Sahel, land surface processes are significantly interacting with climate dynamics. In this paper, we present an original method to control a simple Sahelian land surface model coupled to a radiative transfer model (RTM) on the basis of ERS wind scatterometer (WSC) observations. In a first step, a sensitivity study is implemented to identify those parameters of the land surface model that can be estimated through the assimilation of WSC data. The assimilation scheme relies on evolution strategies (ES) algorithm that aims at solving the parameter evaluation problem. These algorithms are particularly well suited for complex (nonlinear) inverse problems. The assimilation scheme is applied to several study sites located in the Sahelian mesoscale site of the African Monsoon Multidisciplinary Analysis Project (Gourma region, Mali). The results are compared with ground observations of herbaceous mass. After the WSC data assimilation, the simulated herbaceous mass curves compare well with observations [187 kilogram of dry matter per hectare (kg DM/ha) of average error]. The simulated water fluxes exhibit a behaviour in agreement with ground measurements performed over similar ecosystems during the Hapex Sahel experiment. The accuracy of estimated herbaceous mass and water fluxes resulting from uncertainties on climatic forcing variable is evaluated using a stochastic approach. The average error on the herbaceous mass values mainly depends on the rainfall estimate accuracy and ranges from 139 to 268 kg DM/ha that compares well with a previous study based on the sole inversion of the radiative transfer model. Finally, this study underlines the need for a multispectral assimilation approach to get a better constraint on water fluxes estimation.     

Estimation of regional terrestrial water cycle using multi-sensor remote sensing observations and data assimilation

An integrated data assimilation system is implemented over the Red-Arkansas river basin to estimate the regional scale terrestrial water cycle driven by multiple satellite remote sensing data. These satellite products include the Tropical Rainfall Measurement Mission (TRMM), TRMM Microwave Imager (TMI), and Moderate Resolution Imaging Spectroradiometer (MODIS). Also, a number of previously developed assimilation techniques, including the ensemble Kalman filter (EnKF), the particle filter (PF), the water balance constrainer, and the copula error model, and as well as physically based models, including the Variable Infiltration Capacity (VIC), the Land Surface Microwave Emission Model (LSMEM), and the Surface Energy Balance System (SEBS), are tested in the water budget estimation experiments. This remote sensing based water budget estimation study is evaluated using ground observations driven model simulations. It is found that the land surface model driven by the bias-corrected TRMM rainfall produces reasonable water cycle states and fluxes, and the estimates are moderately improved by assimilating TMI 10.67 GHz microwave brightness temperature measurements that provides information on the surface soil moisture state, while it remains challenging to improve the results by assimilating evapotranspiration estimated from satellite-based measurements.     

Integrating hydrological modelling,data assimilation and cloud computing for real-time management of water resources

Online data acquisition, data assimilation and integrated hydrological modelling have become more and more important in hydrological science. In this study, we explore cloud computing for integrating field data acquisition and stochastic, physically-based hydrological modelling in a data assimilation and optimisation framework as a service to water resources management. For this purpose, we developed an ensemble Kalman filter-based data assimilation system for the fully-coupled, physically-based hydrological model HydroGeoSphere, which is able to run in a cloud computing environment. A synthetic data assimilation experiment based on the widely used tilted V-catchment problem showed that the computational overhead for the application of the data assimilation platform in a cloud computing environment is minimal, which makes it well-suited for practical water management problems. Advantages of the cloud-based implementation comprise the independence from computational infrastructure and the straightforward integration of cloud-based observation databases with the modelling and data assimilation platform.     

基于Lorenz-96模型的顺序数据同化方法比较研究

顺序数据同化方法在数据同化系统中得到了广泛的应用,其性能各有优缺。选择3种典型的顺序数据同化算法,即集合Kalman滤波,集合转换Kalman滤波和确定性Kalman滤波,使用经典的Lorenz-96模型进行敏感性实验,研究不同的关键参数变化,如集合数目变化、观测数变化、误差放大因子变化和定位半径变化时对同化效果的影响。实验表明：集合数目和观测数目的多少直接影响3种方法的同化效果;协方差放大因子和定位半径的选择会提高同化精度。综合比较,确定性集合Kalman滤波算法是一种具有较强鲁棒性的滤波算法,能够在集合数较小的情况下达到较好的同化效果。     

迭代集合卡尔曼滤波方法的性能比较研究

针对数据同化过程中模型的非线性问题,通过分析对比得出了一种适合强非线性系统的迭代集合Kalman滤波(IEnKF)。在Lorenz\|63模型的框架内,比较分析集合Kalman滤波(EnKF)、迭代集合Kalman滤波(IEnKF)和迭代扩展卡Kalman滤波(IEKF)在集合数、观测误差方差、放大因子和模型步长不同时同化性能差异,由此探讨这3种方法的优劣。研究结果表明:随着集合数的增加,3种算法的同化性能都得到了一定的改善;放大因子的增大,使其同化性能变差且EnKF呈现出多重波峰波谷的现象;3种方法的均方误差(RMSE）随观测误差方差和模型步长的增大而增大,其同化精度都变差;而IEnKF同化性能最优,更具有鲁棒性。     

Data assimilation framework: Linking an open data assimilation library (OpenDA) to a widely adopted model interface (OpenMI)

Data assimilation optimally merges model forecasts with observations taking into account both model and observational uncertainty. This paper presents a new data assimilation framework that enables the many Open Model Interface (OpenMI) 2.0 .NET compliant hydrological models already available, access to a robust data assimilation library. OpenMI is an open standard that allows models to exchange data during runtime, thus transforming a complex numerical model to a ‘plug and play’ like component. OpenDA is an open interface standard for a set of tools, filters, and numerical techniques to quickly implement data assimilation. The OpenDA–OpenMI framework is presented and tested on a synthetic case that highlights the potential of this new framework. MIKE SHE, a distributed and integrated hydrological model is used to assimilate hydraulic head in a catchment in Denmark. The simulated head over the entire domain were significantly improved by using an ensemble based Kalman filter.     

基于双EnKF的土壤水分与土壤属性参数同时估计

为提高土壤水分数据同化结果的精度,将基于双集合卡尔曼滤波(Dual Ensemble Kalman Filter,DEnKF)的状态-参数估计方案与简单生物圈模型(simple biosphere model 2,SiB2)相结合,同时更新土壤水分和优化模型参数(土壤属性参数)。选用2008年6月1日~10月29日黑河上游阿柔冻融观测站为参考站,开展了同化表层土壤水分观测数据的实验。研究结果表明:DEnKF可同时优化土壤属性参数和改进土壤水分估计,该方法对表层土壤水分估计的精度0.04高于EnKF算法的精度0.05。当观测数据稀少时,DEnKF算法仍然可以得到较高精度的土壤水分估计,3层土壤水分的估计精度在0.02~0.05之间。     

Regularization techniques for ill-posed inverse problems in data assimilation

Optimal state estimation from given observations of a dynamical system by data assimilation is generally an ill-posed inverse problem. In order to solve the problem, a standard Tikhonov, or L₂, regularization is used, based on certain statistical assumptions on the errors in the data. The regularization term constrains the estimate of the state to remain close to a prior estimate. In the presence of model error, this approach does not capture the initial state of the system accurately, as the initial state estimate is derived by minimizing the average error between the model predictions and the observations over a time window. Here we examine an alternative L₁ regularization technique that has proved valuable in image processing. We show that for examples of flow with sharp fronts and shocks, the L₁ regularization technique performs more accurately than standard L₂ regularization.     

Modeling and assimilation of root zone soil moisture using remote sensing observations in Walnut Gulch Watershed during SMEX04

Soil moisture status in the root zone is an important component of the water cycle at all spatial scales (e.g., point, field, catchment, watershed, and region). In this study, the spatio-temporal evolution of root zone soil moisture of the Walnut Gulch Experimental Watershed (WGEW) in Arizona was investigated during the Soil Moisture Experiment 2004 (SMEX04). Root zone soil moisture was estimated via assimilation of aircraft-based remotely sensed surface soil moisture into a distributed Soil-Water-Atmosphere-Plant (SWAP) model. An ensemble square root filter (EnSRF) based on a Kalman filtering scheme was used for assimilating the aircraft-based soil moisture observations at a spatial resolution of 800 m × 800 m. The SWAP model inputs were derived from the SSURGO soil database, LAI (Leaf Area Index) data from SMEX04 database, and data from meteorological stations/rain gauges at the WGEW. Model predictions are presented in terms of temporal evolution of soil moisture probability density function at various depths across the WGEW. The assimilation of the remotely sensed surface soil moisture observations had limited influence on the profile soil moisture. More specifically, root zone soil moisture depended mostly on the soil type. Modeled soil moisture profile estimates were compared to field measurements made periodically during the experiment at the ground based soil moisture stations in the watershed. Comparisons showed that the ground-based soil moisture observations at various depths were within ± 1 standard deviation of the modeled profile soil moisture. Density plots of root zone soil moisture at various depths in the WGEW exhibited multi-modal variations due to the uneven distribution of precipitation and the heterogeneity of soil types and soil layers across the watershed.     

An error bound for sensor fusion with application to Doppler frequency based emitter location

This note provides an error bound on an estimate resulting from a simple fusion scheme combining individual estimates based on measurements of different sensing mechanisms. An important benefit of the bound is that it requires only knowledge of the individual sensor covariances rather than the complete inter-sensor covariance structure. An example drawn from a previous paper is presented to assess the benefit to the accuracy of emitter location using a Doppler sensor for which the error bound is applied to the estimate of a set of local states entering the Doppler sensor model as parameters.     

An evaluation of the nonlinear/non-Gaussian filters for the sequential data assimilation

This paper aims to investigate several new nonlinear/non-Gaussian filters in the context of the sequential data assimilation. The unscented Kalman filter (UKF), the ensemble Kalman filter (EnKF), the sampling importance resampling particle filter (SIR-PF) and the unscented particle filter (UPF) are described in the state-space model framework in the Bayesian filtering background. We first evaluated those methods with a simple highly nonlinear Lorenz model and a scalar nonlinear non-Gaussian model to investigate the filter stability and the error sensitivity, and then their abilities in the one-dimensional estimation of the soil moisture content with the synthetic microwave brightness temperature assimilation experiment in the land surface model VIC-3L. All the results are compared with the EnKF. The advantages and disadvantages of each filter are discussed.The results in the Lorenz model showed that the particle filters are suitable for the large measurement interval assimilation and that the Kalman filters were suitable for the frequent measurement assimilation as well as small measurement uncertainties. The EnKF also showed its feasibility for the non-Gaussian noise. The performance of the SIR-PF was actually not as good as that of the UKF or the EnKF regarding a very small observation noise level compared with the uncertainties in the system. In the one-dimensional brightness temperature assimilation experiment, the UKF, the EnKF and the SIR-PF all proved to be flexible and reliable nonlinear filter algorithms for the low dimensional sequential land data assimilation application. For the high dimensional land surface system that takes the horizontal error correlations into account, the UKF is restricted by its computational demand in the covariance propagation; we must use the EnKF, the SIR-PF and other covariance reduction algorithms. The large computational cost prevents the UPF from being applied in practice.     

On Parameter Estimation Using Nonparametric Noise Models

Fitting multidimensional parametric models in frequency domain using nonparametric noise models is considered in this paper. A nonparametric estimate of the noise statistics is obtained from a finite number of independent data sets. The estimated noise model is then substituted for the the true noise covariance matrix in the maximum likelihood loss function to obtain suboptimal parameter estimates. The goal here is to present an analysis of the resulting estimates. Sufficient conditions for consistency are derived, and an asymptotic accuracy analysis is carried out. The first- and second-order statistics of the cost function at the global minimum point are also explored, which can be used for model validation. The analytical findings are validated using numerical simulation results.