Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems

Abstract

Performance of an advanced, derivativeless, sigma-point Kalman filter (SPKF) data assimilation scheme in a strongly nonlinear dynamical model is investigated. The SPKF data assimilation scheme is compared against standard Kalman filters such as the extended Kalman filter (EKF) and ensemble Kalman filter (EnKF) schemes. Three particular cases—namely, the state, parameter, and joint estimation of states and parameters from a set of discontinuous noisy observations—are studied. The problems associated with the use of tangent linear model (TLM) or Jacobian when using standard Kalman filters are eliminated when using SPKF data assimilation algorithms. Further, the constraints and issues of SPKF data assimilation in real ocean or atmospheric models are emphasized. A reduced sigma-point subspace model is proposed and investigated for higher-dimensional systems. A low-dimensional Lorenz 1963 model and a higher-dimensional Lorenz 1995 model are used as the test beds for data assimilation experiments. The results of SPKF data assimilation schemes are compared with those of standard EKF and EnKF, in which a highly nonlinear chaotic case is studied. It is shown that the SPKF is capable of estimating the model state and parameters with better accuracy than EKF and EnKF. Numerical experiments showed that in all cases the SPKF can give consistent results with better assimilation skills than EnKF and EKF and can overcome the drawbacks associated with the use of EKF and EnKF.

Evaluation of two modified Kalman gain algorithms for radar data assimilation in the WRF model

“…It was developed from the Extended Kalman Filter (EKF) to avoid the linearisation of non-linear models, and the prediction (background) error covariance is approximated by using an ensemble of model forecasts. The underlying conceptualisation of the EnKF is that if the dynamical model is expressed as a stochastic differential equation, the prediction error statistics, which are described by the FokkerÁPlank equation, can be estimated by using ensemble integrations (Evensen, 1994(Evensen, , 1997Ambadan and Tang, 2009). Therefore, the error covariance matrix can be calculated by integrating the ensemble of model states.…”

“…One is the potential problem of the Kalman gain algorithm related to the non-linear measurement operator. Ambadan and Tang (2009) and Tang and Ambandan (2009) argued that the non-linear measurement treatment in the classical EnKF contains an implicit assumption that the forecast of the measurement function is unbiased or the mean of the forecast equals the forecast of the mean. Recently, Tang et al (2014) further analysed the current EnKF algorithm in a statistically rigorous sense and developed two modified algorithms of the Kalman gain.…”

Adaptive Filter for High Dimensional Inverse Engineering Problems: From Theory to Practical Implementation

“…That is 10 times greater than the dimension of model state. As reported in [15], the EnKF with 19 ensembles produces the estimation error of 3 times higher than that based on 1000 ensembles. In contrast, even with , the performance of the PEF is comparable with that of the EnKF-100.…”

“…The variation becomes less important with increase in the ensemble size. As reported in [15], the EnKF can yield an accurate performance if the ECM is estimated on the basis of the ensemble of 1000 samples.…”

Technical Note: Propagating correlations in atmospheric inversions using different Kalman update smoothers

“…The fixed-lag sigma-point Kalman smoother deterministically samples a group of points in the state variable space for the ensemble simulations (Ambadan and Tang, 2009;Van der Merwe, 2004). It is derivative free in assimilating the measurements (Nørgaard et al, 1998).…”