AN ENSEMBLE KALMAN FILTER USING THE CONJUGATE GRADIENT SAMPLER

Abstract

The ensemble Kalman filter (EnKF) is a technique for dynamic stateestimation. EnKF approximates the standard extended Kalman filter (EKF)by creating an ensemble of model states whose mean and empiricalcovariance are then used within the EKF formulas. The technique has anumber of advantages for large-scale, nonlinear problems. First,large-scale covariance matrices required within EKF are replaced bylow-rank and low-storage approximations, making implementation of EnKFmore efficient. Moreover, for a nonlinear state space model,implementation of EKF requires the associated tangent linear and adjointcodes, while implementation of EnKF does not. However, for EnKF to beeffective, the choice of the ensemble members is extremely important. Inthis paper, we show how to use the conjugate gradient (CG) method, andthe recently introduced CG sampler, to create the ensemble members ateach filtering step. This requires the use of a variational formulationof EKF. The effectiveness of the method is demonstrated on both alarge-scale linear, and a small-scale, nonlinear, chaotic problem. Inour examples, the CG-EnKF performs better than the standard EnKF,especially when the ensemble size is small.