Improving the ensemble transform Kalman filter using a second-order Taylor approximation of the nonlinear observation operator

Abstract

Abstract. The ensemble transform Kalman filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the ensemble Kalman filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF still has some limitations when the observation operator is strongly nonlinear. One problem in the minimization of a nonlinear objective function similar to 4D-Var is that the nonlinear operator and its tangent-linear operator have to be calculated iteratively if the Hessian is not preconditioned or if the Hessian has to be calculated several times. This may be computationally expensive. Another problem is that it uses the tangent-linear approximation of the observation operator to estimate the multiplicative inflation factor of the forecast errors, which may not be sufficiently accurate.This study attempts to solve these problems. First, we apply the second-order Taylor approximation to the nonlinear observation operator in which the operator, its tangentlinear operator and Hessian are calculated only once. The related computational cost is also discussed. Second, we propose a scheme to estimate the inflation factor when the observation operator is strongly nonlinear. Experimentation with the Lorenz 96 model shows that using the secondorder Taylor approximation of the nonlinear observation operator leads to a reduction in the analysis error compared with the traditional linear approximation method. Furthermore, the proposed inflation scheme leads to a reduction in the analysis error compared with the procedure using the traditional inflation scheme.

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

“…This finding indicates that the filter is over reliant on the model forecasts and excludes the observations. It can eventually result in the divergence of the filter (Anderson and Anderson, 1999;Constantinescu et al, 2007;Wu et al, 2014).…”

“…The covariance inflation technique is used to mitigate filter divergence by inflating the empirical covariance in EnKF, and it can increase the weight of the observations in the analysis state (Xu et al, 2013). In reality, this method will perturb the subspace spanned by the ensemble vectors and better capture the sub-growing directions that may not have been captured by the original ensemble (Yang et al, 2015).…”

Linear fitting Kalman filter

“…Other non‐linear filters, such as ensemble Kalman filter and ensemble transform Kalman filter, use similar concept as PF, but more efficient than the PF in implementation [11–13]. These filters are widely employed in the problems of weather prediction and geophysical systems, where the sizes of the state and measurement vectors are large.…”