Inflation Adjustment on Error Covariance Matrices for Ensemble Kalman Filter Assimilation

This paper mainly discusses some details when applying data assimilation method via an ensemble Kalman filter (EnKF) to improve prediction of adsorptive solute Cr(VI) transfer from soil into runoff. Based on this work, we could make better use of our theoretical model to predict adsorptive solute transfer from soil into surface runoff in practice. The results show that the ensemble number of 100 is reasonable, considering assimilation effect and efficiency after selecting its number from 25 to 225 at an interval of 25. While the initial ensemble value makes little difference to data assimilation (DA) results. Besides, DA results could be improved by multiplying an amplification factor to forecast error covariance matrix due to underestimation of forecast error.

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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 ◽

Error Covariance

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

Monthly Weather Review ◽

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.

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Evaluating Ensemble Kalman Filter Analyses of Severe Hailstorms on 8 May 2017 in Colorado: Effects of State Variable Updating and Multimoment Microphysics Schemes on State Variable Cross Covariances

Monthly Weather Review ◽

10.1175/mwr-d-19-0300.1 ◽

2020 ◽

Vol 148 (6) ◽

pp. 2365-2389

Author(s):

Jonathan Labriola ◽

Nathan Snook ◽

Youngsun Jung ◽

Ming Xue

Keyword(s):

Kalman Filter ◽

Ensemble Kalman Filter ◽

Forecast Error ◽

Variable Cross ◽

Background Error ◽

Mixing Ratios ◽

State Variable ◽

Error Covariance ◽

Thermodynamic Variables ◽

The Impact

Abstract Ensemble Kalman filter (EnKF) analyses of the storms associated with the 8 May 2017 Colorado severe hail event using either the Milbrandt and Yau (MY) or the NSSL double-moment bulk microphysics scheme in the forecast model are evaluated. With each scheme, two experiments are conducted in which the reflectivity (Z) observations update in addition to dynamic and thermodynamic variables: 1) only the hydrometeor mixing ratios or 2) all microphysical variables. With fewer microphysical variables directly constrained by the Z observations, only updating hydrometeor mixing ratios causes the forecast error covariance structure to become unreliable, and results in larger errors in the analysis. Experiments that update all microphysical variables produce analyses with the lowest Z root-mean-square innovations; however, comparing the estimated hail size against hydrometeor classification algorithm output suggests that further constraint from observations is needed to more accurately estimate surface hail size. Ensemble correlation analyses are performed to determine the impact of hail growth assumptions in the MY and NSSL schemes on the forecast error covariance between microphysical and thermodynamic variables. In the MY scheme, Z is negatively correlated with updraft intensity because the strong updrafts produce abundant small hail aloft. The NSSL scheme predicts the growth of large hail aloft; consequently, Z is positively correlated with storm updraft intensity and hail state variables. Hail production processes are also shown to alter the background error covariance for liquid and frozen hydrometeor species. Results in this study suggest that EnKF analyses are sensitive to the choice of MP scheme (e.g., the treatment of hail growth processes).

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Including Uncertainties of Sea Surface Temperature in an Ensemble Kalman Filter: A Case Study of Typhoon Sinlaku (2008)

Weather and Forecasting ◽

10.1175/waf-d-11-00136.1 ◽

2012 ◽

Vol 27 (6) ◽

pp. 1586-1597 ◽

Cited By ~ 16

Author(s):

Masaru Kunii ◽

Takemasa Miyoshi

Keyword(s):

Kalman Filter ◽

Surface Temperature ◽

Sea Surface Temperature ◽

Ensemble Kalman Filter ◽

Forecast Error ◽

Ocean Model ◽

Sea Surface ◽

Ensemble Spread ◽

Error Covariance ◽

The Impact

Abstract Sea surface temperature (SST) plays an important role in tropical cyclone (TC) life cycle evolution, but often the uncertainties in SST estimates are not considered in the ensemble Kalman filter (EnKF). The lack of uncertainties in SST generally results in the lack of ensemble spread in the atmospheric states near the sea surface, particularly for temperature and moisture. In this study, the uncertainties of SST are included by adding ensemble perturbations to the SST field, and the impact of the SST perturbations is investigated using the local ensemble transform Kalman filter (LETKF) with the Weather Research and Forecasting Model (WRF) in the case of Typhoon Sinlaku (2008). In addition to the experiment with the perturbed SST, another experiment with manually inflated ensemble perturbations near the sea surface is performed for comparison. The results indicate that the SST perturbations within EnKF generally improve analyses and their subsequent forecasts, although manually inflating the ensemble spread instead of perturbing SST does not help. Investigations of the ensemble-based forecast error covariance indicate larger scales for low-level temperature and moisture from the SST perturbations, although manual inflation of ensemble spread does not produce such structural effects on the forecast error covariance. This study suggests the importance of considering SST perturbations within ensemble-based data assimilation and promotes further studies with more sophisticated methods of perturbing SST fields such as using a fully coupled atmosphere–ocean model.

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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.

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On the Kalman Filter error covariance collapse into the unstable subspace

Nonlinear Processes in Geophysics ◽

10.5194/npg-18-243-2011 ◽

2011 ◽

Vol 18 (2) ◽

pp. 243-250 ◽

Cited By ~ 31

Author(s):

A. Trevisan ◽

L. Palatella

Keyword(s):

Kalman Filter ◽

Lyapunov Exponents ◽

Extended Kalman Filter ◽

Covariance Matrices ◽

Reduced Form ◽

Asymptotic State ◽

Numerical Verification ◽

Error Covariance ◽

Neutral Subspace ◽

Unstable Subspace

Abstract. When the Extended Kalman Filter is applied to a chaotic system, the rank of the error covariance matrices, after a sufficiently large number of iterations, reduces to N+ + N0 where N+ and N0 are the number of positive and null Lyapunov exponents. This is due to the collapse into the unstable and neutral tangent subspace of the solution of the full Extended Kalman Filter. Therefore the solution is the same as the solution obtained by confining the assimilation to the space spanned by the Lyapunov vectors with non-negative Lyapunov exponents. Theoretical arguments and numerical verification are provided to show that the asymptotic state and covariance estimates of the full EKF and of its reduced form, with assimilation in the unstable and neutral subspace (EKF-AUS) are the same. The consequences of these findings on applications of Kalman type Filters to chaotic models are discussed.

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Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain

ISRN Applied Mathematics ◽

10.1155/2014/417623 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Nicholas Assimakis ◽

Maria Adam

Keyword(s):

Kalman Filter ◽

Steady State ◽

Prediction Error ◽

Covariance Matrices ◽

Asymptotically Stable ◽

System Parameters ◽

Estimation And Prediction ◽

Error Covariance ◽

Algebraic Algorithms ◽

Time Invariant

The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain.

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On the Propagation of Information and the Use of Localization in Ensemble Kalman Filtering

Journal of the Atmospheric Sciences ◽

10.1175/2010jas3452.1 ◽

2010 ◽

Vol 67 (12) ◽

pp. 3823-3834 ◽

Cited By ~ 8

Author(s):

Young-noh Yoon ◽

Edward Ott ◽

Istvan Szunyogh

Keyword(s):

Correlation Function ◽

Kalman Filter ◽

Ensemble Kalman Filter ◽

Atmospheric Model ◽

Spatial Correlation Function ◽

Wave Dynamics ◽

Error Covariance ◽

Principal Tool ◽

Ensemble Kalman Filtering ◽

Made In

Abstract Several localized versions of the ensemble Kalman filter have been proposed. Although tests applying such schemes have proven them to be extremely promising, a full basic understanding of the rationale and limitations of localization is currently lacking. It is one of the goals of this paper to contribute toward addressing this issue. The second goal is to elucidate the role played by chaotic wave dynamics in the propagation of information and the resulting impact on forecasts. To accomplish these goals, the principal tool used here will be analysis and interpretation of numerical experiments on a toy atmospheric model introduced by Lorenz in 2005. Propagation of the wave packets of this model is shown. It is found that, when an ensemble Kalman filter scheme is employed, the spatial correlation function obtained at each forecast cycle by averaging over the background ensemble members is short ranged, and this is in strong contrast to the much longer range correlation function obtained by averaging over states from free evolution of the model. Propagation of the effects of observations made in one region on forecasts in other regions is studied. The error covariance matrices from the analyses with localization and without localization are compared. From this study, major characteristics of the localization process and information propagation are extracted and summarized.

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