Brief communication: An innovation-based estimation method for model error covariance in Kalman filters

Abstract. We present a simple innovation-based model error covariance estimation method for Kalman filters. The method is based on Berry and Sauer (2013) and the simplification results from assuming known observation error covariance. We carry out experiments with a prescribed model error covariance using a Lorenz (1996) model and ensemble Kalman filter. The prescribed error covariance matrix is recovered with high accuracy.

Assimilation of Stratospheric Temperature and Ozone with an Ensemble Kalman Filter in a Chemistry–Climate Model

10.1175/2011mwr3540.1 ◽

2011 ◽

Vol 139 (11) ◽

pp. 3389-3404 ◽

Cited By ~ 17

Author(s):

Thomas Milewski ◽

Michel S. Bourqui

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Covariance Matrix ◽

Ensemble Kalman Filter ◽

Climate Model ◽

Observation Error ◽

Background Error ◽

Error Covariance Matrix ◽

Error Covariance ◽

Analysis Error

Abstract A new stratospheric chemical–dynamical data assimilation system was developed, based upon an ensemble Kalman filter coupled with a Chemistry–Climate Model [i.e., the intermediate-complexity general circulation model Fast Stratospheric Ozone Chemistry (IGCM-FASTOC)], with the aim to explore the potential of chemical–dynamical coupling in stratospheric data assimilation. The system is introduced here in a context of a perfect-model, Observing System Simulation Experiment. The system is found to be sensitive to localization parameters, and in the case of temperature (ozone), assimilation yields its best performance with horizontal and vertical decorrelation lengths of 14 000 km (5600 km) and 70 km (14 km). With these localization parameters, the observation space background-error covariance matrix is underinflated by only 5.9% (overinflated by 2.1%) and the observation-error covariance matrix by only 1.6% (0.5%), which makes artificial inflation unnecessary. Using optimal localization parameters, the skills of the system in constraining the ensemble-average analysis error with respect to the true state is tested when assimilating synthetic Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) retrievals of temperature alone and ozone alone. It is found that in most cases background-error covariances produced from ensemble statistics are able to usefully propagate information from the observed variable to other ones. Chemical–dynamical covariances, and in particular ozone–wind covariances, are essential in constraining the dynamical fields when assimilating ozone only, as the radiation in the stratosphere is too slow to transfer ozone analysis increments to the temperature field over the 24-h forecast window. Conversely, when assimilating temperature, the chemical–dynamical covariances are also found to help constrain the ozone field, though to a much lower extent. The uncertainty in forecast/analysis, as defined by the variability in the ensemble, is large compared to the analysis error, which likely indicates some amount of noise in the covariance terms, while also reducing the risk of filter divergence.

Data assimilation with multiple types of observation boreholes via the ensemble Kalman filter embedded within stochastic moment equations

Hydrology and Earth System Sciences ◽

10.5194/hess-25-1689-2021 ◽

2021 ◽

Vol 25 (4) ◽

pp. 1689-1709

Author(s):

Chuan-An Xia ◽

Xiaodong Luo ◽

Bill X. Hu ◽

Monica Riva ◽

Alberto Guadagnini

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Ensemble Kalman Filter ◽

Type A ◽

Observation Error ◽

Type B ◽

Moment Equations ◽

Error Covariance Matrix ◽

Monitoring Wells ◽

Abstract. We employ an approach based on the ensemble Kalman filter coupled with stochastic moment equations (MEs-EnKF) of groundwater flow to explore the dependence of conductivity estimates on the type of available information about hydraulic heads in a three-dimensional randomly heterogeneous field where convergent flow driven by a pumping well takes place. To this end, we consider three types of observation devices corresponding to (i) multi-node monitoring wells equipped with packers (Type A) and (ii) partially (Type B) and (iii) fully (Type C) screened wells. We ground our analysis on a variety of synthetic test cases associated with various configurations of these observation wells. Moment equations are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of conductivity) and are solved by an efficient transient numerical scheme proposed in this study. The use of an inflation factor imposed to the observation error covariance matrix is also analyzed to assess the extent at which this can strengthen the ability of the MEs-EnKF to yield appropriate conductivity estimates in the presence of a simplified modeling strategy where flux exchanges between monitoring wells and aquifer are neglected. Our results show that (i) the configuration associated with Type A monitoring wells leads to conductivity estimates with the (overall) best quality, (ii) conductivity estimates anchored on information from Type B and C wells are of similar quality, (iii) inflation of the measurement-error covariance matrix can improve conductivity estimates when a simplified flow model is adopted, and (iv) when compared with the standard Monte Carlo-based EnKF method, the MEs-EnKF can efficiently and accurately estimate conductivity and head fields.

Data assimilation with multiple types of observation boreholes via ensemble Kalman filter embedded within stochastic moment equations

10.5194/hess-2020-588 ◽

2020 ◽

Author(s):

Chuan-An Xia ◽

Xiaodong Luo ◽

Bill X. Hu ◽

Monica Riva ◽

Alberto Guadagnini

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Ensemble Kalman Filter ◽

Type A ◽

Observation Error ◽

Type B ◽

Moment Equations ◽

Error Covariance Matrix ◽

Monitoring Wells ◽

Abstract. We employ an approach based on ensemble Kalman filter coupled with stochastic moment equations (MEs-EnKF) of groundwater flow to explore the dependence of conductivity estimates on the type of available information about hydraulic heads in a three-dimensional randomly heterogeneous field where convergent flow driven by a pumping well takes place. To this end, we consider three types of observation devices, corresponding to (i) multi-node monitoring wells equipped with packers (Type A), (ii) partially (Type B) and (iii) fully (Type C) screened wells. We ground our analysis on a variety of synthetic test cases associated with various configurations of these observation wells. Moment equations are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of conductivity) and are solved by an efficient transient numerical scheme proposed in this study. The use of an inflation factor imposed to the observation error covariance matrix is also analyzed to assess the extent at which this can strengthen the ability of the MEs-EnKF to yield appropriate conductivity estimates in the presence of a simplified modeling strategy where flux exchanges between monitoring wells and aquifer are neglected. Our results show that (i) the configuration associated with Type A monitoring wells leads to conductivity estimates with the (overall) best quality; (ii) conductivity estimates anchored on information from Type B and C wells are of similar quality; (iii) inflation of the measurement-error covariance matrix can improve conductivity estimates when an incomplete/simplified flow model is adopted; and (iv) when compared with the standard Monte Carlo -based EnKF method, the MEs-EnKF can efficiently and accurately estimate conductivity and head fields.

Balance and Ensemble Kalman Filter Localization Techniques

10.1175/2010mwr3328.1 ◽

2011 ◽

Vol 139 (2) ◽

pp. 511-522 ◽

Cited By ~ 136

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 ◽

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.

An Extension of the Ensemble Kalman Filter for Estimating the Observation Error Covariance Matrix Based on the Variational Bayes’s Method

10.1175/mwr-d-16-0139.1 ◽

2016 ◽

Vol 145 (1) ◽

pp. 199-213 ◽

Cited By ~ 5

Author(s):

Akio Nakabayashi ◽

Genta Ueno

Keyword(s):

Kalman Filter ◽

Maximum Likelihood ◽

Covariance Matrix ◽

Ensemble Kalman Filter ◽

Numerical Experiments ◽

Observation Error ◽

Error Covariance Matrix ◽

Error Covariance ◽

Long Run ◽

The Stability

Abstract This paper presents an extension of the ensemble Kalman filter (EnKF) that can simultaneously estimate the state vector and the observation error covariance matrix by using the variational Bayes’s (VB) method. In numerical experiments, this capability is examined for a time-variant observation error covariance matrix, and it is noteworthy that this method works well even when the true observation error covariance matrix is nondiagonal. In addition, two complementary studies are presented. First, the stability of a long-run assimilation is demonstrated when there are unmodeled disturbances. Second, a maximum-likelihood (ML) method is derived and demonstrated for optimizing the hyperparameters used in this method.

Estimating model error covariance matrix parameters in extended Kalman filtering

Nonlinear Processes in Geophysics ◽

10.5194/npg-21-919-2014 ◽

2014 ◽

Vol 21 (5) ◽

pp. 919-927 ◽

Cited By ~ 11

Author(s):

A. Solonen ◽

J. Hakkarainen ◽

A. Ilin ◽

M. Abbas ◽

A. Bibov

Keyword(s):

Covariance Matrix ◽

Weather Prediction ◽

Estimation Method ◽

Model Error ◽

Tuning Parameter ◽

Extended Kalman Filtering ◽

Error Covariance Matrix ◽

Nonlinear Dynamical ◽

Abstract. The extended Kalman filter (EKF) is a popular state estimation method for nonlinear dynamical models. The model error covariance matrix is often seen as a tuning parameter in EKF, which is often simply postulated by the user. In this paper, we study the filter likelihood technique for estimating the parameters of the model error covariance matrix. The approach is based on computing the likelihood of the covariance matrix parameters using the filtering output. We show that (a) the importance of the model error covariance matrix calibration depends on the quality of the observations, and that (b) the estimation approach yields a well-tuned EKF in terms of the accuracy of the state estimates and model predictions. For our numerical experiments, we use the two-layer quasi-geostrophic model that is often used as a benchmark model for numerical weather prediction.

Using analysis state to construct a forecast error covariance matrix in ensemble Kalman filter assimilation

Advances in Atmospheric Sciences ◽

10.1007/s00376-012-2133-5 ◽

2013 ◽

Vol 30 (5) ◽

pp. 1303-1312 ◽

Cited By ~ 7

Author(s):

Xiaogu Zheng ◽

Guocan Wu ◽

Shupeng Zhang ◽

Xiao Liang ◽

Yongjiu Dai ◽

...

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Ensemble Kalman Filter ◽

Forecast Error ◽

Error Covariance Matrix ◽

Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals

10.1175/2009mwr3085.1 ◽

2010 ◽

Vol 138 (3) ◽

pp. 932-950 ◽

Cited By ~ 22

Author(s):

Jean-Michel Brankart ◽

Emmanuel Cosme ◽

Charles-Emmanuel Testut ◽

Pierre Brasseur ◽

Jacques Verron

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Error Estimates ◽

Forecast Error ◽

Observation Error ◽

Square Root ◽

Error Statistics ◽

Error Covariance Matrix ◽

Error Covariance ◽

Optimal Estimates

Abstract In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a scaling factor for the observation error correlation length scale. As an application, the resulting adaptive filter is used to estimate the time evolution of ocean mesoscale signals using observations of the ocean dynamic topography. To check the behavior of the adaptive mechanism, this is done in the context of idealized experiments, in which model error and observation error statistics are known. This ideal framework is particularly appropriate to explore the ill-conditioned situations (inadequate prior assumptions or uncontrollability of the parameters) in which adaptivity can be misleading. Overall, the experiments show that, if used correctly, the efficient optimal adaptive algorithm proposed in this paper introduces useful supplementary degrees of freedom in the estimation problem, and that the direct control of these statistical parameters by the observations increases the robustness of the error estimates and thus the optimality of the resulting Kalman filter.

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

Nonlinear Processes in Geophysics ◽

10.5194/npg-24-329-2017 ◽

2017 ◽

Vol 24 (3) ◽

pp. 329-341 ◽

Cited By ~ 1

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.

A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective

Nonlinear Processes in Geophysics ◽

10.5194/npg-28-1-2021 ◽

2021 ◽

Vol 28 (1) ◽

pp. 1-22

Author(s):

Olivier Pannekoucke ◽

Richard Ménard ◽

Mohammad El Aabaribaoune ◽

Matthieu Plu

Keyword(s):

Kalman Filter ◽

Numerical Model ◽

Covariance Matrix ◽

Model Error ◽

Filter Method ◽

Advection Equation ◽

Parametric Modelling ◽

Test Bed ◽

Error Covariance Matrix ◽

Abstract. This contribution addresses the characterization of the model-error covariance matrix from the new theoretical perspective provided by the parametric Kalman filter method which approximates the covariance dynamics from the parametric evolution of a covariance model. The classical approach to obtain the modified equation of a dynamics is revisited to formulate a parametric modelling of the model-error covariance matrix which applies when the numerical model is dissipative compared with the true dynamics. As an illustration, the particular case of the advection equation is considered as a simple test bed. After the theoretical derivation of the predictability-error covariance matrices of both the nature and the numerical model, a numerical simulation is proposed which illustrates the properties of the resulting model-error covariance matrix.