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

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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 tangent-linear 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 second-order 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.