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

2011 ◽  
Vol 139 (11) ◽  
pp. 3389-3404 ◽  
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.

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

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 ◽  
Error Covariance

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.

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

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 ◽  
Error Covariance

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.

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

2016 ◽  
Vol 145 (1) ◽  
pp. 199-213 ◽  
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.

Including the spatial observation error correlation in data assimilation of AMSU-A radiances

2021 ◽  
Author(s):  
Koji Terasaki ◽  
Takemasa Miyoshi
Keyword(s):  
Data Assimilation ◽  
Covariance Matrix ◽  
Weather Prediction ◽  
Computation Time ◽  
Horizontal Resolution ◽  
Observation Error ◽  
Correlated Errors ◽  
Error Covariance Matrix ◽  
Error Covariance ◽  
Error Correlation

<p>Recent developments in sensing technology increased the number of observations both in space and time. It is essential to effectively utilize the information from observations to improve numerical weather prediction (NWP). It is known to have correlated errors in observations measured with a single instrument, such as satellite radiances. The observations with the horizontal error correlation are usually thinned to compensate for neglecting the error correlation in data assimilation. This study explores to explicitly include the horizontal observation error correlation of Advanced Microwave Sounding Unit-A (AMSU-A) radiances using a global atmospheric data assimilation system NICAM-LETKF, which comprises the Nonhydrostatic ICosahedral Atmospheric Model (NICAM) and the Local Ensemble Transform Kalman Filter (LETKF). This study performs the data assimilation experiments at 112-km horizontal resolution and 38 vertical layers up to 40 km and with 32 ensemble members.</p><p>In this study, we estimate the horizontal observation error correlation of AMSU-A radiances using innovation statistics. The computation cost of inverting the observation error covariance matrix will increase when non-zero off-diagonal terms are included. In this study, we assume uncorrelated observation errors between different instruments and observation variables, so that the observation error covariance matrix becomes block diagonal with only horizontal error correlations included. The computation time of the entire LETKF analysis procedure is increased only by up to 10 % compared with the case using the diagonal observation error covariance matrix. The analyses and forecasts of temperature and zonal wind in the mid- and upper-troposphere are improved by including the horizontal error correlations. We will present the most recent results at the workshop.</p>

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

2013 ◽  
Vol 30 (5) ◽  
pp. 1303-1312 ◽  
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 ◽  
Error Covariance

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

2010 ◽  
Vol 138 (3) ◽  
pp. 932-950 ◽  
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.

scholarly journals Ozone-sensitive channel selection over IASI full spectrum with correlated observation errors for NWP

2019 ◽  
Author(s):  
Olivier Coopmann ◽  
Vincent Guidard ◽  
Béatrice Josse ◽  
Virginie Marécal ◽  
Nadia Fourrié
Keyword(s):  
Covariance Matrix ◽  
Weather Prediction ◽  
Channel Selection ◽  
Spectral Interval ◽  
Observation Error ◽  
Full Spectrum ◽  
Error Covariance Matrix ◽  
Error Covariance ◽  
One Year ◽  
Selection Of

Abstract. The Infrared Atmospheric Sounding Interferometer (IASI) onboard the Metop satellites provides 8461 channels in the infrared spectrum, covering the spectral interval 645–2760 cm−1 at a resolution of 0.5 cm−1. The high volume of data observation resulting from IASI presents many challenges. In current Numerical Weather Prediction (NWP) models, assimilating all channels is not feasible, due to data transmission, data storage and significant computational costs. One of the methods for reducing the data volume is the channel selection. Many NWP centres use a subset of 314 IASI channels including 15 ozone-sensitive channels. However, this channel selection has been carried out assuming uncorrelated observation errors. In addition, these ozone-sensitive channels have been selected only for ozone information. The objective of this study is to carry out a new selection of IASI ozone-sensitive channels from the full spectrum over a spectral range of 1000–1070 cm−1, in a direct radiance assimilation framework. This selection is done with a full observation error covariance matrix to take into account cross-channel error correlations. A sensitivity method based on the channel spectral sensitivity to variables and a statistical approach based on the Degrees of Freedom for Signal (DFS) have been chosen. To be representative of atmospheric variability, 345 profiles from around the world over a one-year period were selected. The new selection, is evaluated in a One-Dimensional Variational (1D-Var) analyses framework. This selection highlights a new set of 15 IASI ozone-sensitive channels. The results are very encouraging since by adding these 15 channels to 122 operational channels, temperature and humidity analyses are improved by 13.8 % and 20.9 % respectively. Obviously, these 15 channels significantly improve ozone analyses. In addition to considering inter-channel observation error correlations, the channel selection method uses a robust background error covariance matrix that takes into account temperature, humidity and ozone errors using a lagged forecast method over a one-year period. The new selection of IASI ozone-sensitive channels will be soon used in the global 4D-Var ARPEGE (Action de Recherche Petite Echelle Grande Echelle) data assimilation system.

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.

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