Using the ensemble Kalman filter to estimate multiplicative model parameters

This paper proposes a simple approach to estimating multiplicative model parameters using the ensemble square root filter. The basic idea, following previous studies, is to augment the state vector by the model parameters. While some success with this approach has been reported if the model parameters enter as additive terms in the tendency equations, this approach is problematic if the model parameters are multiplied by the state variables. The reason for this difficulty is that multiplicative parameters change the dynamical properties of the model, and in particular can cause the model to become dynamically unstable. This paper shows that model instability can be avoided if the usual persistence model for parameter update is replaced by a temporally smoothed version of the update model. In addition, the augmentation approach can be interpreted as two simultaneously decoupled ensemble Kalman filters for the model state and the parameter state, respectively. Implementation of the parameter estimation does not require changing the data assimilation algorithm-it just has to be supplemented by a parameter estimation step that is similar to the state estimation step. Covariance localization is found to be necessary not only for the model state, but also for augmented model parameters, if they are spatially dependent. The new formulation is illustrated with the Lorenz-96 model and shown to be capable of estimating additive and multiplicative model parameters, as well as the state, under relatively challenging conditions (e.g. using 20 observations to estimate 120 unknown variables).