A Non-Gaussian Ensemble Filter Update for Data Assimilation

2010 ◽  
Vol 138 (11) ◽  
pp. 4186-4198 ◽  
Author(s):  
Jeffrey L. Anderson
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Sampling Error ◽  
Weather Prediction ◽  
Large Ensembles ◽  
Stochastic Filter ◽  
Pathological Behavior ◽  
Non Gaussian ◽  
Observation Space

Abstract A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large ensembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability distribution in a fashion analogous to the use of rank histograms for ensemble forecast evaluation. This rank histogram filter can represent non-Gaussian observation space priors and posteriors and is shown to be competitive with existing filters for problems as large as global numerical weather prediction. The ability to represent non-Gaussian distributions is useful for a variety of applications such as convective-scale assimilation and assimilation of bounded quantities such as relative humidity.

scholarly journals Convective-Scale Data Assimilation for the Weather Research and Forecasting Model Using the Local Particle Filter

2017 ◽  
Vol 145 (5) ◽  
pp. 1897-1918 ◽  
Author(s):  
Jonathan Poterjoy ◽  
Ryan A. Sobash ◽  
Jeffrey L. Anderson
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Squall Line ◽  
Monte Carlo Data ◽  
Mixing Ratios ◽  
Convective Scale ◽  
Non Gaussian ◽  
Mean Square Errors

Abstract Particle filters (PFs) are Monte Carlo data assimilation techniques that operate with no parametric assumptions for prior and posterior errors. A data assimilation method introduced recently, called the local PF, approximates the PF solution within neighborhoods of observations, thus allowing for its use in high-dimensional systems. The current study explores the potential of the local PF for atmospheric data assimilation through cloud-permitting numerical experiments performed for an idealized squall line. Using only 100 ensemble members, experiments using the local PF to assimilate simulated radar measurements demonstrate that the method provides accurate analyses at a cost comparable to ensemble filters currently used in weather models. Comparisons between the local PF and an ensemble Kalman filter demonstrate benefits of the local PF for producing probabilistic analyses of non-Gaussian variables, such as hydrometeor mixing ratios. The local PF also provides more accurate forecasts than the ensemble Kalman filter, despite yielding higher posterior root-mean-square errors. A major advantage of the local PF comes from its ability to produce more physically consistent posterior members than the ensemble Kalman filter, which leads to fewer spurious model adjustments during forecasts. This manuscript presents the first successful application of the local PF in a weather prediction model and discusses implications for real applications where nonlinear measurement operators and nonlinear model processes limit the effectiveness of current Gaussian data assimilation techniques.

scholarly journals Scale-Dependent Representation of the Information Content of Observations in the Global Ensemble Kalman Filter Data Assimilation

2016 ◽  
Vol 144 (8) ◽  
pp. 2927-2945
Author(s):  
Nedjeljka Žagar ◽  
Jeffrey Anderson ◽  
Nancy Collins ◽  
Timothy Hoar ◽  
Kevin Raeder ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Information Content ◽  
Ensemble Kalman Filter ◽  
Large Scale ◽  
Variance Reduction ◽  
Weather Prediction ◽  
Current Data ◽  
Model Framework ◽  
The Tropics

Abstract Global data assimilation systems for numerical weather prediction (NWP) are characterized by significant uncertainties in tropical analysis fields. Furthermore, the largest spread of global ensemble forecasts in the short range on all scales is in the tropics. The presented results suggest that these properties hold even in the perfect-model framework and the ensemble Kalman filter data assimilation with a globally homogeneous network of wind and temperature profiles. The reasons for this are discussed by using the modal analysis, which provides information about the scale dependency of analysis and forecast uncertainties and information about the efficiency of data assimilation to reduce the prior uncertainties in the balanced and inertio-gravity dynamics. The scale-dependent representation of variance reduction of the prior ensemble by the data assimilation shows that the peak efficiency of data assimilation is on the synoptic scales in the midlatitudes that are associated with quasigeostrophic dynamics. In contrast, the variance associated with the inertia–gravity modes is less successfully reduced on all scales. A smaller information content of observations on planetary scales with respect to the synoptic scales is discussed in relation to the large-scale tropical uncertainties that current data assimilation methodologies do not address successfully. In addition, it is shown that a smaller reduction of the large-scale uncertainties in the prior state for NWP in the tropics than in the midlatitudes is influenced by the applied radius for the covariance localization.

scholarly journals Impact of surface data assimilation using an ensemble Kalman filter technique on the improvement of weather prediction

2019 ◽  
Vol 1127 ◽  
pp. 012041
Author(s):  
N J Trilaksono ◽  
M Taqiyya ◽  
N Dewani ◽  
I D G A Junnaedhi ◽  
E Riawan ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Filter Technique ◽  
Surface Data

E4DVar: Coupling an Ensemble Kalman Filter with Four-Dimensional Variational Data Assimilation in a Limited-Area Weather Prediction Model

2012 ◽  
Vol 140 (2) ◽  
pp. 587-600 ◽  
Author(s):  
Meng Zhang ◽  
Fuqing Zhang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Prediction Model ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Coupled System ◽  
Limited Area ◽  
Ensemble Forecasts ◽  
Background Error ◽  
Hybrid Data Assimilation

A hybrid data assimilation approach that couples the ensemble Kalman filter (EnKF) and four-dimensional variational (4DVar) methods is implemented for the first time in a limited-area weather prediction model. In this coupled system, denoted E4DVar, the EnKF and 4DVar systems run in parallel while feeding into each other. The multivariate, flow-dependent background error covariance estimated from the EnKF ensemble is used in the 4DVar minimization and the ensemble mean in the EnKF analysis is replaced by the 4DVar analysis, while updating the analysis perturbations for the next cycle of ensemble forecasts with the EnKF. Therefore, the E4DVar can obtain flow-dependent information from both the explicit covariance matrix derived from ensemble forecasts, as well as implicitly from the 4DVar trajectory. The performance of an E4DVar system is compared with the uncoupled 4DVar and EnKF for a limited-area model by assimilating various conventional observations over the contiguous United States for June 2003. After verifying the forecasts from each analysis against standard sounding observations, it is found that the E4DVar substantially outperforms both the EnKF and 4DVar during this active summer month, which featured several episodes of severe convective weather. On average, the forecasts produced from E4DVar analyses have considerably smaller errors than both of the stand-alone EnKF and 4DVar systems for forecast lead times up to 60 h.

scholarly journals E3DVar: Coupling an Ensemble Kalman Filter with Three-Dimensional Variational Data Assimilation in a Limited-Area Weather Prediction Model and Comparison to E4DVar

2013 ◽  
Vol 141 (3) ◽  
pp. 900-917 ◽  
Author(s):  
Fuqing Zhang ◽  
Meng Zhang ◽  
Jonathan Poterjoy
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Three Dimensional ◽  
Variational Data Assimilation ◽  
Lead Times ◽  
Limited Area ◽  
Model Errors ◽  
Mean Square Errors

Abstract This study examines the performance of a hybrid ensemble-variational data assimilation system (E3DVar) that couples an ensemble Kalman filter (EnKF) with the three-dimensional variational data assimilation (3DVar) system for the Weather Research and Forecasting (WRF) Model. The performance of E3DVar and the component EnKF and 3DVar systems are compared over the eastern United States for June 2003. Conventional sounding and surface observations as well as data from wind profilers, aircraft and ships, and cloud-tracked winds from satellites, are assimilated every 6 h during the experiments, and forecasts are verified using standard sounding observations. Forecasts with 12- to 72-h lead times are found to have noticeably smaller root-mean-square errors when initialized with the E3DVar system, as opposed to the EnKF, especially for the 12-h wind and moisture fields. The E3DVar system demonstrates similar performance as an EnKF, while using less than half the number of ensemble members, and is less sensitive to the use of a multiphysics ensemble to account for model errors. The E3DVar system is also compared with a similar hybrid method that replaces the 3DVar component with the WRF four-dimensional variational data assimilation (4DVar) method (denoted E4DVar). The E4DVar method demonstrated considerable improvements over E3DVar for nearly all model levels and variables at the shorter forecast lead times (12–48 h), but the forecast accuracies of all three ensemble-based methods (EnKF, E3DVar, and E4DVar) converge to similar results at longer lead times (60–72 h). Nevertheless, all methods that used ensemble information produced considerably better forecasts than the two methods that relied solely on static background error covariance (i.e., 3DVar and 4DVar).

scholarly journals Pseudo-Orbit Data Assimilation. Part I: The Perfect Model Scenario

2014 ◽  
Vol 71 (2) ◽  
pp. 469-482 ◽  
Author(s):  
Hailiang Du ◽  
Leonard A. Smith
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
State Estimation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Weather Prediction ◽  
Attractive Alternative ◽  
Perfect Model ◽  
Improved Performance ◽  
Orbit Data

Abstract State estimation lies at the heart of many meteorological tasks. Pseudo-orbit-based data assimilation provides an attractive alternative approach to data assimilation in nonlinear systems such as weather forecasting models. In the perfect model scenario, noisy observations prevent a precise estimate of the current state. In this setting, ensemble Kalman filter approaches are hampered by their foundational assumptions of dynamical linearity, while variational approaches may fail in practice owing to local minima in their cost function. The pseudo-orbit data assimilation approach improves state estimation by enhancing the balance between the information derived from the dynamic equations and that derived from the observations. The potential use of this approach for numerical weather prediction is explored in the perfect model scenario within two deterministic chaotic systems: the two-dimensional Ikeda map and 18-dimensional Lorenz96 flow. Empirical results demonstrate improved performance over that of the two most common traditional approaches of data assimilation (ensemble Kalman filter and four-dimensional variational assimilation).

Application of the Ensemble Kalman Filter to a One-Dimensional Linear Advection Model

2015 ◽  
Vol 804 ◽  
pp. 287-290
Author(s):  
Somsiri Payakkarak ◽  
Dusadee Sukawat
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Atmospheric Model ◽  
Ensemble Member ◽  
Observation Data ◽  
State Estimate ◽  
One Dimensional ◽  
Weather Forecasts

Data assimilation is used in numerical weather prediction to improve weather forecasts by incorporating observation data into the model forecast. The Ensemble Kalman Filter (EnKF) is a method of data assimilation which updates an ensemble of states to provide a state estimate and associated error at each step. The atmospheric model that is used in this research is a one-dimensional linear advection model. This model describes the motion of a scalar field as it is advected by a known speed field. The result shows that by selecting appropriate initial ensemble, model noise and measurement perturbations, it is possible to achieve a significant improvement in the EnKF results. The accuracy of the EnKF increases when the number of ensemble member grows. That is, the larger ensemble sizes perform better than those of smaller sizes.

An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation

SPE Journal ◽  
2007 ◽  
Vol 12 (04) ◽  
pp. 438-446 ◽  
Author(s):  
Yaqing Gu ◽  
Dean S. Oliver
Keyword(s):  
Porous Media ◽  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Weather Prediction ◽  
Flow In Porous Media ◽  
Petroleum Reservoir ◽  
State Variables ◽  
Highly Nonlinear

Summary The dynamical equations for multiphase flow in porous media are highly nonlinear and the number of variables required to characterize the medium is usually large, often two or more variables per simulator gridblock. Neither the extended Kalman filter nor the ensemble Kalman filter is suitable for assimilating data or for characterizing uncertainty for this type of problem. Although the ensemble Kalman filter handles the nonlinear dynamics correctly during the forecast step, it sometimes fails badly in the analysis (or updating) of saturations. This paper focuses on the use of an iterative ensemble Kalman filter for data assimilation in nonlinear problems, especially of the type related to multiphase ow in porous media. Two issues are key:iteration to enforce constraints andensuring that the resulting ensemble is representative of the conditional pdf (i.e., that the uncertainty quantification is correct). The new algorithm is compared to the ensemble Kalman filter on several highly nonlinear example problems, and shown to be superior in the prediction of uncertainty. Introduction For linear problems, the Kalman filter is optimal for assimilating measurements to continuously update the estimate of state variables. Kalman filters have occasionally been applied to the problem of estimating values of petroleum reservoir variables (Eisenmann et al. 1994; Corser et al. 2000), but they are most appropriate when the problems are characterized by a small number of variables and when the variables to be estimated are linearly related to the observations. Most data assimilation problems in petroleum reservoir engineering are highly nonlinear and are characterized by many variables, often two or more variables per simulator gridblock. The problem of weather forecasting is in many respects similar to the problem of predicting future petroleum reservoir performance. The economic impact of inaccurate predictions is substantial in both cases, as is the difficulty of assimilating very large data sets and updating very large numerical models. One method that has been recently developed for assimilating data in weather forecasting is ensemble Kalman filtering (Evensen 1994; Houtekamer and Mitchell 1998; Anderson and Anderson 1999; Hamill et al. 2000; Houtekamer and Mitchell 2001; Evensen 2003). It has been demonstrated to be useful for weather prediction over the North Atlantic. The method is now beginning to be applied for data assimilation in groundwater hydrology (Reichle et al. 2002; Chen and Zhang 2006) and in petroleum engineering (Nævdal et al. 2002, 2005; Gu and Oliver 2005; Liu and Oliver 2005a; Wen and Chen 2006, 2007; Zafari and Reynolds 2007; Gao et al. 2006; Lorentzen et al. 2005; Skjervheim et al. 2007; Dong et al. 2006), but the applications to state variables whose density functions are bimodal has proved problematic (Gu and Oliver 2006). For applications to nonlinear assimilation problems, it is useful to think of the ensemble Kalman filter as a least squares method that obtains an averaged gradient for minimization not from a variational approach but from an empirical correlation between model variables (Anderson 2003; Zafari et al. 2006). In addition to providing a mean estimate of the variables, a Monte Carlo estimate of uncertainty can be obtained directly from the variability in the ensemble.

scholarly journals Balance and Ensemble Kalman Filter Localization Techniques

2011 ◽  
Vol 139 (2) ◽  
pp. 511-522 ◽  
Author(s):  
Steven J. Greybush ◽  
Eugenia Kalnay ◽  
Takemasa Miyoshi ◽  
Kayo Ide ◽  
Brian R. Hunt
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Covariance Matrix ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Length Scale ◽  
Observation Error ◽  
Long Distance ◽  
Error Covariance Matrix ◽  
Error Covariance

Abstract In ensemble Kalman filter (EnKF) data assimilation, localization modifies the error covariance matrices to suppress the influence of distant observations, removing spurious long-distance correlations. In addition to allowing efficient parallel implementation, this takes advantage of the atmosphere’s lower dimensionality in local regions. There are two primary methods for localization. In B localization, the background error covariance matrix elements are reduced by a Schur product so that correlations between grid points that are far apart are removed. In R localization, the observation error covariance matrix is multiplied by a distance-dependent function, so that far away observations are considered to have infinite error. Successful numerical weather prediction depends upon well-balanced initial conditions to avoid spurious propagation of inertial-gravity waves. Previous studies note that B localization can disrupt the relationship between the height gradient and the wind speed of the analysis increments, resulting in an analysis that can be significantly ageostrophic. This study begins with a comparison of the accuracy and geostrophic balance of EnKF analyses using no localization, B localization, and R localization with simple one-dimensional balanced waves derived from the shallow-water equations, indicating that the optimal length scale for R localization is shorter than for B localization, and that for the same length scale R localization is more balanced. The comparison of localization techniques is then expanded to the Simplified Parameterizations, Primitive Equation Dynamics (SPEEDY) global atmospheric model. Here, natural imbalance of the slow manifold must be contrasted with undesired imbalance introduced by data assimilation. Performance of the two techniques is comparable, also with a shorter optimal localization distance for R localization than for B localization.

scholarly journals Comparison of ensemble Kalman filter and variational approaches for CO<sub>2</sub> data assimilation

2013 ◽  
Vol 13 (5) ◽  
pp. 12825-12865
Author(s):  
A. Chatterjee ◽  
A. M. Michalak
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Prediction ◽  
Co2 Flux ◽  
Estimation Problem ◽  
Advantages And Disadvantages ◽  
Flux Estimation ◽  
Operational Constraints ◽  
The Impact

Abstract. Data assimilation (DA) approaches, such as the variational and the ensemble Kalman filter, provide a computationally efficient framework for solving the CO2 source-sink estimation problem. Unlike DA applications for weather prediction and constituent assimilation, however, the advantages and disadvantages of alternative DA approaches for CO2 flux estimation have not been extensively explored. In this study, we compare and assess estimates from two advanced DA methods (an ensemble square root filter and a variational technique) using a simple 1-dimensional advection-diffusion inverse problem that has been designed to capture the nuances of a real CO2 flux estimation problem. Experiments are specifically designed to identify the impact of the observational density, heterogeneity, and uncertainty, as well as operational constraints (i.e., ensemble size, number of descent iterations) in order to isolate the degradation in the DA estimates relative to the estimates from a batch inverse modeling scheme. No dynamical model is explicitly specified for the DA methods to keep the problem setup analogous to a real CO2 flux estimation problem. Results demonstrate that the performance of the DA approaches depends on a complex interplay between the measurement network and the operational constraints imposed to make the DA algorithms practically feasible. The overall advantages/disadvantages of the two examined DA approaches are complementary and highlight that, specifically for CO2 applications, selection of one method over the other should be dictated by the carbon science questions being asked, and the inversion conditions under which the approaches are being applied.

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