scholarly journals Covariance Inflation in the Ensemble Kalman Filter: A Residual Nudging Perspective and Some Implications

2013 ◽  
Vol 141 (10) ◽  
pp. 3360-3368 ◽  
Author(s):  
Xiaodong Luo ◽  
Ibrahim Hoteit
Keyword(s):  
Kalman Filter ◽  
Numerical Experiment ◽  
Ensemble Kalman Filter ◽  
Sufficient Conditions ◽  
The State ◽  
State Estimate ◽  
Inflation Factor

Abstract This article examines the influence of covariance inflation on the distance between the measured observation and the simulated (or predicted) observation with respect to the state estimate. In order for the aforementioned distance to be bounded in a certain interval, some sufficient conditions are derived, indicating that the covariance inflation factor should be bounded in a certain interval, and that the inflation bounds are related to the maximum and minimum eigenvalues of certain matrices. Implications of these analytic results are discussed, and a numerical experiment is presented to verify the validity of the analysis conducted.

scholarly journals Penalized ensemble Kalman filters for high dimensional non-linear systems

PLoS ONE ◽  
2021 ◽  
Vol 16 (3) ◽  
pp. e0248046
Author(s):  
Elizabeth Hou ◽  
Earl Lawrence ◽  
Alfred O. Hero
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Estimation Error ◽  
Covariance Structure ◽  
Full Rank ◽  
The State ◽  
High Dimensional ◽  
Ensemble Kalman Filters ◽  
Non Linear ◽  
Non Linear System

The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of models, updated with data, to track the time evolution of a usually non-linear system. It does so by using an empirical approximation to the well-known Kalman filter. However, its performance can suffer when the ensemble size is smaller than the state space, as is often necessary for computationally burdensome models. This scenario means that the empirical estimate of the state covariance is not full rank and possibly quite noisy. To solve this problem in this high dimensional regime, we propose a computationally fast and easy to implement algorithm called the penalized ensemble Kalman filter (PEnKF). Under certain conditions, it can be theoretically proven that the PEnKF will be accurate (the estimation error will converge to zero) despite having fewer ensemble members than state dimensions. Further, as contrasted to localization methods, the proposed approach learns the covariance structure associated with the dynamical system. These theoretical results are supported with simulations of several non-linear and high dimensional systems.

scholarly journals An estimate of the inflation factor and analysis sensitivity in the ensemble Kalman filter

2017 ◽  
Vol 24 (3) ◽  
pp. 329-341 ◽  
Author(s):  
Guocan Wu ◽  
Xiaogu Zheng
Keyword(s):  
Kalman Filter ◽  
Covariance Matrix ◽  
Ensemble Kalman Filter ◽  
Forecast Error ◽  
Model Errors ◽  
Error Covariance Matrix ◽  
Spatially Correlated ◽  
Error Covariance ◽  
Inflation Factor ◽  
Analysis Error

Abstract. The ensemble Kalman filter (EnKF) is a widely used ensemble-based assimilation method, which estimates the forecast error covariance matrix using a Monte Carlo approach that involves an ensemble of short-term forecasts. While the accuracy of the forecast error covariance matrix is crucial for achieving accurate forecasts, the estimate given by the EnKF needs to be improved using inflation techniques. Otherwise, the sampling covariance matrix of perturbed forecast states will underestimate the true forecast error covariance matrix because of the limited ensemble size and large model errors, which may eventually result in the divergence of the filter. In this study, the forecast error covariance inflation factor is estimated using a generalized cross-validation technique. The improved EnKF assimilation scheme is tested on the atmosphere-like Lorenz-96 model with spatially correlated observations, and is shown to reduce the analysis error and increase its sensitivity to the observations.

Ensemble Kalman Filter estimates shear stress and seismic slip rates in synthetic laboratory experiment

2021 ◽  
Author(s):  
Hamed Ali Diab-Montero ◽  
Meng Li ◽  
Ylona van Dinther ◽  
Femke C Vossepoel
Keyword(s):  
Shear Stress ◽  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Slip Rate ◽  
The State ◽  
Shear Stresses ◽  
Limited Information ◽  
Slip Rates ◽  
Laboratory Setup

<p>Our ability to forecast earthquake events is hampered by limited information of the state of stress and strength of faults and their governing parameters. Ensemble data assimilation methods provide a means to estimate these variables by combining physics-based models and observations taking into account their uncertainties. In this study, we estimate earthquake occurrences in synthetic experiments representing a meter-scale laboratory setup of a straight-fault governed by rate-and-state friction. We test an Ensemble Kalman Filter implemented in the Parallel Data Assimilation Framework, which is connected with a 1D forward model using the numerical library GARNET. A perfect-model test shows that the filter can estimate shear stresses, slip rates and state θ acting on the fault even when simulating slip rates up to m/s and can thus be used for estimating earthquake occurrences. We assimilate shear stress and slip-rate observations, representing measurements obtained from shear strain gauges and piezoelectric transducers sensors, and their uncertainties acquired at a small distance to the fault in the homogeneous elastic medium. In this study we evaluate how the Ensemble Kalman filter estimates the state and strength of the faults using these observations, and assess the relative influence of assimilating various observations. The results suggest that the data assimilation improves the estimated timing of the earthquake occurrences. The assimilation of the shear stress observed in the medium improves in particular the estimates of the state θ and the shear stress on the fault, while assimilating observations of velocity in the medium improves the slip-rate estimation.</p>

scholarly journals Simultaneous estimation of model state variables and observation and forecast biases using a two-stage hybrid Kalman filter

2013 ◽  
Vol 10 (4) ◽  
pp. 5169-5224 ◽  
Author(s):  
V. R. N. Pauwels ◽  
G. J. M. De Lannoy ◽  
H.-J. Hendricks Franssen ◽  
H. Vereecken
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
The State ◽  
Bias Error ◽  
Simultaneous Estimation ◽  
State Variables ◽  
Two Stage ◽  
Forecast Bias ◽  
Error Covariance ◽  
Observation Bias

Abstract. In this paper, we present a two-stage hybrid Kalman filter to estimate both observation and forecast bias in hydrologic models, in addition to state variables. The biases are estimated using the Discrete Kalman Filter, and the state variables using the Ensemble Kalman Filter. A key issue in this multi-component assimilation scheme is the exact partitioning of the difference between observation and forecasts into state, forecast bias and observation bias updates. Here, the error covariances of the forecast bias and the unbiased states are calculated as constant fractions of the biased state error covariance, and the observation bias error covariance is a function of the observation prediction error covariance. In a series of synthetic experiments, focusing on the assimilation of discharge into a rainfall-runoff model, it is shown that both static and dynamic observation and forecast biases can be successfully estimated. The results indicate a strong improvement in the estimation of the state variables and resulting discharge as opposed to the use of a bias-unaware Ensemble Kalman Filter. The results suggest that a better performance of data assimilation methods should be possible if both forecast and observation biases are taken into account.

scholarly journals An Estimate of Inflation Factor and Analysis Sensitivity in Ensemble Kalman Filter

2016 ◽  
Author(s):  
Guocan Wu
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Forecast Error ◽  
Estimation Accuracy ◽  
Error Matrix ◽  
Spatially Correlated ◽  
Inflation Factor ◽  
Analysis Error ◽  
Lorenz 96 ◽  
Assimilation Scheme

Abstract. The estimation accuracy of forecast error matrix is crucial to the assimilation result. Ensemble Kalman filter (EnKF) is a widely used ensemble based assimilation method, which initially estimate the forecast error matrix using a Monte Carlo method with the short-term ensemble forecast states. However, this estimate needs to be further improved using inflation technique. In this study, the forecast error inflation factor is estimated based on cross validation and the analysis sensitivity is also investigated. The improved EnKF assimilation scheme is validated by assimilating spatially correlated observations to the atmosphere-like Lorenz-96 model. The experiment results show that, the analysis error is reduced and the analysis sensitivity to observations is improved.

scholarly journals Ensemble Kalman Filter Assimilation of Fixed Screen-Height Observations in a Parameterized PBL

2005 ◽  
Vol 133 (11) ◽  
pp. 3260-3275 ◽  
Author(s):  
Joshua P. Hacker ◽  
Chris Snyder
Keyword(s):  
Kalman Filter ◽  
Great Plains ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Lower Boundary ◽  
The State ◽  
Convective Initiation ◽  
Rich Data ◽  
Pbl Model ◽  
Surface Observations

Abstract In situ surface layer observations are a rich data source that could be more effectively utilized in NWP applications. If properly assimilated, data from existing mesonets could improve initial conditions and lower boundary conditions, leading to the possibility of improved simulation and short-range forecasts of slope flows, sea breezes, convective initiation, and other PBL circulations. A variance–covariance climatology is constructed by extracting a representative column from real-time mesoscale forecasts over the Southern Great Plains, and used to explore the potential for estimating the state of the PBL by assimilating surface observations. A parameterized 1D PBL model and an ensemble Kalman filter (EnKF) approach to assimilation are used to test this potential. Analysis focuses on understanding how effectively the EnKF can spread the surface observations vertically to constrain the state of the PBL model. Results confirm that assimilating surface observations can substantially improve the state of a modeled PBL. Experiments to estimate the moisture availability parameter through the data assimilation system show that the EnKF is a viable tool for parameter estimation, and may help mitigate model error in forecasting and simulating the PBL.

scholarly journals DATA ASSIMILATION METHOD FOR THE TASK OF PASSIVE IMPURITY PROPAGATION IN THE ATMOSPHERE BASED ON A DYNAMIC-STOCHASTIC APPROACH

2019 ◽  
Vol 6 (1) ◽  
pp. 94-100
Author(s):  
Marina Platonova ◽  
Ekaterina Klimova
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Stochastic Approach ◽  
Passive Impurity ◽  
Dynamic Stochastic ◽  
Inflation Factor ◽  
Impurity Propagation

In this paper, we consider the method of data assimilation for the problem the propagation of the concentration a passive impurity in the atmosphere. Classical approaches to solving such problems are described, features of the application of algorithms, their minuses and pros. Two algorithms are considered: the ensemble Kalman filter and the ensemble Kalmans moother. Various ways to improve the convergence of these algorithms, such as localization and inflation factor, are considered.

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.

scholarly journals A Nonlinear Rank Regression Method for Ensemble Kalman Filter Data Assimilation

2019 ◽  
Vol 147 (8) ◽  
pp. 2847-2860 ◽  
Author(s):  
Jeffrey L. Anderson
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
The State ◽  
Single Observation ◽  
Rank Regression ◽  
State Variable ◽  
Rank Histogram ◽  
Observation Space ◽  
The Impact ◽  
Standard Regression

Abstract It is possible to describe many variants of ensemble Kalman filters without loss of generality as the impact of a single observation on a single state variable. For most ensemble algorithms commonly applied to Earth system models, the computation of increments for the observation variable ensemble can be treated as a separate step from computing increments for the state variable ensemble. The state variable increments are normally computed from the observation increments by linear regression using the prior bivariate ensemble of the state and observation variable. Here, a new method that replaces the standard regression with a regression using the bivariate rank statistics is described. This rank regression is expected to be most effective when the relation between a state variable and an observation is nonlinear. The performance of standard versus rank regression is compared for both linear and nonlinear forward operators (also known as observation operators) using a low-order model. Rank regression in combination with a rank histogram filter in observation space produces better analyses than standard regression for cases with nonlinear forward operators and relatively large analysis error. Standard regression, in combination with either a rank histogram filter or an ensemble Kalman filter in observation space, produces the best results in other situations.

scholarly journals Simultaneous estimation of model state variables and observation and forecast biases using a two-stage hybrid Kalman filter

2013 ◽  
Vol 17 (9) ◽  
pp. 3499-3521 ◽  
Author(s):  
V. R. N. Pauwels ◽  
G. J. M. De Lannoy ◽  
H.-J. Hendricks Franssen ◽  
H. Vereecken
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
The State ◽  
Simultaneous Estimation ◽  
State Variables ◽  
Two Stage ◽  
Forecast Bias ◽  
Error Covariance ◽  
Observation Bias

Abstract. In this paper, we present a two-stage hybrid Kalman filter to estimate both observation and forecast bias in hydrologic models, in addition to state variables. The biases are estimated using the discrete Kalman filter, and the state variables using the ensemble Kalman filter. A key issue in this multi-component assimilation scheme is the exact partitioning of the difference between observation and forecasts into state, forecast bias and observation bias updates. Here, the error covariances of the forecast bias and the unbiased states are calculated as constant fractions of the biased state error covariance, and the observation bias error covariance is a function of the observation prediction error covariance. In a series of synthetic experiments, focusing on the assimilation of discharge into a rainfall-runoff model, it is shown that both static and dynamic observation and forecast biases can be successfully estimated. The results indicate a strong improvement in the estimation of the state variables and resulting discharge as opposed to the use of a bias-unaware ensemble Kalman filter. Furthermore, minimal code modification in existing data assimilation software is needed to implement the method. The results suggest that a better performance of data assimilation methods should be possible if both forecast and observation biases are taken into account.

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