An application of Ensemble Kalman Filter in integral-balance subsurface modeling

Data assimilation method provides a framework to decrease the uncertainty of hydrological modeling by sequentially incorporating observations into numerical model. Such a process involves estimating statistical moments of different order based on the evolution of conditional probability distribution function. Because of the nonlinearity of many hydrological dynamics, explicit and analytical solutions for moments of state distribution are often impossible. Evensen [J Geophys Res 99(c5): 10143–10162 (1994)] introduced Ensemble Kalman Filtering (EnKF) method to address such problems. We test and evaluate the performance of EnKF in fusing model predictions and observations for a saturated–unsaturated integral-balance subsurface model. We find EnKF improve the model predictions, and also we conclude a good estimate of state variance is essential for the success of EnKF.     

On ensemble representation of the observation-error covariance in the Ensemble Kalman Filter

Evensen (2003) presents a modification of the Ensemble Kalman Filter (EnKF), in which the observation-error and background-error covariance matrices are both represented by ensembles, in contrast to the usual practice, where only the background error is so represented. It is shown that this modification can cause the ensemble to collapse to a single member, in the common situation where the number of observations is more than twice the number of ensemble members, and to be rank-deficient when the number of observations is greater than or equal to the ensemble size. It is also shown that some further modifications to the scheme, presented by Evensen as offering numerical efficiencies, can prevent this collapse. However, these latter modifications are shown in some simple numerical examples to require tuning to produce acceptable results, which are nevertheless inferior to those of the standard EnKF.Acknowledgements The author acknowledges useful discussions with Peter Steinle, and other participants at the EnKF workshop held in BMRC in November, 2003.     

The Ensemble Kalman Filter: theoretical formulation and practical implementation

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.Responsible Editor: Jörg-Olaf Wolff     

Calibration of a Land Subsidence Model Using InSAR Data via the Ensemble Kalman Filter

The application of interferometric synthetic aperture radar (InSAR) has been increasingly used to improve capabilities to model land subsidence in hydrogeologic studies. A number of investigations over the last decade show how spatially detailed time‐lapse images of ground displacements could be utilized to advance our understanding for better predictions. In this work, we use simulated land subsidences as observed measurements, mimicking InSAR data to inversely infer inelastic specific storage in a stochastic framework. The inelastic specific storage is assumed as a random variable and modeled using a geostatistical method such that the detailed variations in space could be represented and also that the uncertainties of both characterization of specific storage and prediction of land subsidence can be assessed. The ensemble Kalman filter (EnKF), a real‐time data assimilation algorithm, is used to inversely calibrate a land subsidence model by matching simulated subsidences with InSAR data. The performance of the EnKF is demonstrated in a synthetic example in which simulated surface deformations using a reference field are assumed as InSAR data for inverse modeling. The results indicate: (1) the EnKF can be used successfully to calibrate a land subsidence model with InSAR data; the estimation of inelastic specific storage is improved, and uncertainty of prediction is reduced, when all the data are accounted for; and (2) if the same ensemble is used to estimate Kalman gain, the analysis errors could cause filter divergence; thus, it is essential to include localization in the EnKF for InSAR data assimilation.     

The Ensemble Kalman Filter for Groundwater Plume Characterization: A Case Study

The Kalman filter is an efficient data assimilation tool to refine an estimate of a state variable using measured data and the variable's correlations in space and/or time. The ensemble Kalman filter (EnKF) (Evensen 2004, 2009) is a Kalman filter variant that employs Monte Carlo analysis to define the correlations that help to refine the updated state. While use of EnKF in hydrology is somewhat limited, it has been successfully applied in other fields of engineering (e.g., oil reservoir modeling, weather forecasting). Here, EnKF is used to refine a simulated groundwater tetrachloroethylene (TCE) plume that underlies the Tooele Army Depot‐North (TEAD‐N) in Utah, based on observations of TCE in the aquifer. The resulting EnKF‐based assimilated plume is simulated forward in time to predict future plume migration. The correlations that underpin EnKF updating implicitly contain information about how the plume developed over time under the influence of complex site hydrology and variable source history, as they are predicated on multiple realizations of a well‐calibrated numerical groundwater flow and transport model. The EnKF methodology is compared to an ordinary kriging‐based assimilation method with respect to the accurate representation of plume concentrations in order to determine the relative efficacy of EnKF for water quality data assimilation.     

The Local Ensemble Transform Kalman Filter with the Weather Research and Forecasting Model: Experiments with Real Observations

The local ensemble transform Kalman filter (LETKF) is implemented with the Weather Research and Forecasting (WRF) model, and real observations are assimilated to assess the newly-developed WRF-LETKF system. The WRF model is a widely-used mesoscale numerical weather prediction model, and the LETKF is an ensemble Kalman filter (EnKF) algorithm particularly efficient in parallel computer architecture. This study aims to provide the basis of future research on mesoscale data assimilation using the WRF-LETKF system, an additional testbed to the existing EnKF systems with the WRF model used in the previous studies. The particular LETKF system adopted in this study is based on the system initially developed in 2004 and has been continuously improved through theoretical studies and wide applications to many kinds of dynamical models including realistic geophysical models. Most recent and important improvements include an adaptive covariance inflation scheme which considers the spatial and temporal inhomogeneity of inflation parameters. Experiments show that the LETKF successfully assimilates real observations and that adaptive inflation is advantageous. Additional experiments with various ensemble sizes show that using more ensemble members improves the analyses consistently.     

Comparison of Ensemble Kalman Filter groundwater-data assimilation methods based on stochastic moment equations and Monte Carlo simulation

Traditional Ensemble Kalman Filter (EnKF) data assimilation requires computationally intensive Monte Carlo (MC) sampling, which suffers from filter inbreeding unless the number of simulations is large. Recently we proposed an alternative EnKF groundwater-data assimilation method that obviates the need for sampling and is free of inbreeding issues. In our new approach, theoretical ensemble moments are approximated directly by solving a system of corresponding stochastic groundwater flow equations. Like MC-based EnKF, our moment equations (ME) approach allows Bayesian updating of system states and parameters in real-time as new data become available. Here we compare the performances and accuracies of the two approaches on two-dimensional transient groundwater flow toward a well pumping water in a synthetic, randomly heterogeneous confined aquifer subject to prescribed head and flux boundary conditions.     

Global assessments of the NCEP Ensemble Forecast System using altimeter data

Ocean Dynamics - Forecasts of 10-m wind (U10) and significant wave height (Hs) from the National Centers for Environmental Prediction (NCEP) Ensemble Forecast System are evaluated using altimeter...     

State analysis using the Local Ensemble Transform Kalman Filter (LETKF) and the three-layer circulation structure of the Luzon Strait and the South China Sea

A new circulation model of the western North Pacific Ocean based on the parallelized version of the Princeton Ocean Model and incorporating the Local Ensemble Transform Kalman Filter (LETKF) data assimilation scheme has been developed. The new model assimilates satellite data and is tested for the period January 1 to April 3, 2012 initialized from a 24-year simulation to estimate the ocean state focusing in the South China Sea (SCS). Model results are compared against estimates based on the optimum interpolation (OI) assimilation scheme and are validated against independent Argo float and transport data to assess model skills. LETKF provides improved estimates of the western North Pacific Ocean state including transports through various straits in the SCS. In the Luzon Strait, the model confirms, for the first time, the three-layer transport structure previously deduced in the literature from sparse observations: westward in the upper and lower layers and eastward in the middle layer. This structure is shown to be robust, and the related dynamics are analyzed using the results of a long-term (18 years) unassimilated North Pacific Ocean model. Potential vorticity and mass conservations suggest a basin-wide cyclonic circulation in the upper layer of the SCS (z?>??570 m), an anticyclonic circulation in the middle layer (?570 m?≥?z?>??2,000 m), and, in the abyssal basin (<?2,000 m), the circulation is cyclonic in the north and anticyclonic in the south. The cyclone–anticyclone abyssal circulation is confirmed and explained using a deep-layer reduced-gravity model as being caused by overflow over the deep sill of the Luzon Strait, coupled with intense, localized upwelling west of the strait.     

Application of the Kalman filter to the Nash model

The Nash model was used for application of the Kalman filter. The state vector of the rainfall–runoff system was constituted by the IUH (instantaneous unit hydrograph) estimated by the Nash model and the runoff estimated by the Nash model using the Kalman filter. The initial values of the state vector were assumed as the average of 10% of the IUH peak values and the initial runoff estimated from the average IUH. The Nash model using the Kalman filter with a recursive algorithm accurately predicted runoff from a basin in Korea. The filter allowed the IUH to vary in time, increased the accuracy of the Nash model and reduced physical uncertainty of the rainfall–runoff process in the river basin. © 1998 John Wiley & Sons, Ltd.     

电离层TEC卡尔曼滤波成像研究

随着太空探测技术的进步,对TEC(Total Electron Content,简称TEC)探测精度要求越来越高.本文利用COSMOS 2414卫星数据资料获得观测TEC,在电离层NeQuick模型下,得到电离层电子密度,并使用卡尔曼滤波算法反演电子密度,最后结合电离层测高仪数据对实验结果进行判定.结果发现利用卡尔曼滤波反演信标资料算法,可以获得可靠的二维电子密度场.     

Calibration/Data Assimilation Approach for Integrating GRACE Data into the WaterGAP Global Hydrology Model (WGHM) Using an Ensemble Kalman Filter: First Results

We introduce a new ensemble-based Kalman filter approach to assimilate GRACE satellite gravity data into the WaterGAP Global Hydrology Model. The approach (1) enables the use of the spatial resolution provided by GRACE by including the satellite observations as a gridded data product, (2) accounts for the complex spatial GRACE error correlation pattern by rigorous error propagation from the monthly GRACE solutions, and (3) allows us to integrate model parameter calibration and data assimilation within a unified framework. We investigate the formal contribution of GRACE observations to the Kalman filter update by analysis of the Kalman gain matrix. We then present first model runs, calibrated via data assimilation, for two different experiments: the first one assimilates GRACE basin averages of total water storage and the second one introduces gridded GRACE data at \(5^\circ\) resolution into the assimilation. We finally validate the assimilated model by running it in free mode (i.e., without adding any further GRACE information) for a period of 3 years following the assimilation phase and comparing the results to the GRACE observations available for this period.     

基于集合卡尔曼滤波的湖泊富营养化模型Delft3D-BLOOM数据同化

富营养化模型是进行湖泊水环境质量预测和管理的重要工具,然而模型客观存在的误差一直是应用者关心的重要问题.数据同化作为连接观测数据与数值模型的重要方法,可以有效提高模型的准确性.集合卡尔曼滤波(En KF)是众多数据同化算法中应用最为广泛的一种,可进行非线性系统的数据同化,并能有效降低数据同化的计算量.本研究以太湖作为具体实例,选择Delft3D-BLOOM作为富营养化模型,在数值实验确定En KF集合数为100、观测误差方差为1%、模拟误差方差为10%的基础上分别进行模型状态变量同化以及状态变量与关键参数同步同化.结果显示,仅同化状态变量时,模型预测精度有所增加;同时同化状态变量和关键参数时,可显著提升模型在湖泊水环境质量预测中的精度.该研究为应用集合卡尔曼滤波以提高复杂的湖库富营养化模型模拟精度提供了有效的方法.     

基于无迹卡尔曼滤波的新安江模型实时校正方法

通过利用实时水文观测数据对洪水预报模型进行校正,可增加流域洪水预报的实时性和精确度.本文讨论了水文模型状态变量选取对滤波效果的影响,并给出了状态变量选取原则.在集总式新安江模型的基础上,结合状态变量选取原则,应用无迹卡尔曼滤波技术构建了新安江模型的实时校正方法.方法应用于闽江邵武流域洪水预报的计算结果表明,采用无迹卡尔曼滤波方法后,不仅能够直接校正模型状态,同时也能有效地提高模型预报精度,适合应用于实际流域洪水预报作业中.     

Assessment of the Long-Term Hydrological Forecast Skill Evolution across Lead-Times within the Ensemble Streamflow Prediction Framework

Long-term or seasonal forecasting is crucial for the management of large water systems. Advances in catchment hydrology, such as mathematical models of catchment processes, are proven to be capable of creating reliable streamflow forecasting systems. In this study, the limits of predictability of streamflow in a snowmelt-dominated river basin are examined and a new illustration of the forecast efficiency across different issue dates and lead times—the so-called “forecastability map”—is demonstrated.     

Noise Reduction in Radon Monitoring Data Using Kalman Filter and Application of Results in Earthquake Precursory Process Research

Monitoring the concentration of radon gas is an established method for geophysical analyses and research, particularly in earthquake studies. A continuous radon monitoring station was implemented in Jooshan hotspring, Kerman province, south east Iran. The location was carefully chosen as a widely reported earthquake-prone zone. A common issue during monitoring of radon gas concentration is the possibility of noise disturbance by different environmental and instrumental parameters. A systematic mathematical analysis aiming at reducing such noises from data is reported here; for the first time, the Kalman filter (KF) has been used for radon gas concentration monitoring. The filtering is incorporated based on several seismic parameters of the area under study. A novel anomaly defined as “radon concentration spike crossing” is also introduced and successfully used in the study. Furthermore, for the first time, a mathematical pattern of a relationship between the radius of potential precursory phenomena and the distance between epicenter and the monitoring station is reported and statistically analyzed.     

HPPPR模型及其在地震综合预测研究中的初步应用

在综合考虑经典投影寻踪算法特点的基础上,针对投影寻踪计算中存在的一些不利因素,给出相应的解决思路.利用数值仿真技术进行基于粒子群优化算法与厄密特多项式构建的投影寻踪回归模型建模能力与计算精度的检验,再将其应用于多维地震时间序列综合建模预测中.计算结果和进一步分析表明,基于粒子群优化算法与厄密特多项式构建的投影寻踪回归投影寻踪模型具有简单、快速、有效的特点,在实际地震综合预测建模中取得了满意的效果,可作为地震综合预测的一种回归分析方法.     

An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering

The ensemble Kalman filter (EnKF) is a commonly used real-time data assimilation algorithm in various disciplines. Here, the EnKF is applied, in a hydrogeological context, to condition log-conductivity realizations on log-conductivity and transient piezometric head data. In this case, the state vector is made up of log-conductivities and piezometric heads over a discretized aquifer domain, the forecast model is a groundwater flow numerical model, and the transient piezometric head data are sequentially assimilated to update the state vector. It is well known that all Kalman filters perform optimally for linear forecast models and a multiGaussian-distributed state vector. Of the different Kalman filters, the EnKF provides a robust solution to address non-linearities; however, it does not handle well non-Gaussian state-vector distributions. In the standard EnKF, as time passes and more state observations are assimilated, the distributions become closer to Gaussian, even if the initial ones are clearly non-Gaussian. A new method is proposed that transforms the original state vector into a new vector that is univariate Gaussian at all times. Back transforming the vector after the filtering ensures that the initial non-Gaussian univariate distributions of the state-vector components are preserved throughout. The proposed method is based in normal-score transforming each variable for all locations and all time steps. This new method, termed the normal-score ensemble Kalman filter (NS-EnKF), is demonstrated in a synthetic bimodal aquifer resembling a fluvial deposit, and it is compared to the standard EnKF. The proposed method performs better than the standard EnKF in all aspects analyzed (log-conductivity characterization and flow and transport predictions).     

应用FIR滤波器消除天津数字地震记录中的干扰

数字地震记录中包含大量的干扰信号,影响数据质量,需要加以排除。针对天津数字地震记录中存在的干扰信号,介绍了FIR数字滤波器的原理与方法,并详细阐述了利用汉宁窗函数的带阻滤波器、高通滤波器滤除天津数字地震记录中的干扰,从而达到突出地震信号特征、消除干扰的目的。实践表明:这种方法设计简单、方便、实用性强、具有良好的效果。