Effects of Misspecified Time-correlated Model Error in the (Ensemble) Kalman Smoother

Data assimilation (DA) is often performed under the perfect model assumption (strong-constraint). There is an increasing number of works accounting for model errors (weak-constraint), but often with different degrees of approximation or simplification without knowing their impact on the DA results. We investigate the effect of an inaccurate specification of the model error -- in particular an inaccurate time correlation-- on the quality of the DA results using the Kalman Smoother (KS) and the Ensemble Kalman Smoother (EnKS).

We use a simple linear model in which the true evolution of the system contains model error with correlation time-scale ω_r. The forecast model contains model error with a guessed time-scale ω_g which is used in the data assimilation process. The performance of the KS solution is measured using the mean squared error of the analysis mean and its ratio with the analysis spread. Analytical results are obtained for a 1-dimensional system, and their generality confirmed through numerical experiments with an EnKS on higher dimensional systems. The results are highly dependent on the model error time scale, with longer guess time scales allowing for more influence of observations over the time window, but also on the observation frequency. For instance, with a single observation the posterior variance attains a maximum at a certain ω_g value which decreases over time. When we increase the number of observations, the posterior variance becomes a monotonic decreasing function of ω_g. The posterior MSE with a single observation increases with both ω_r and the mismatch between ω_g and ω_r. However increasing the number of observations the impact of the mismatch between two decorrelation time-scales reduces, and the MSE mainly depends on time and size of the model error time scale.

Finally we use state augmentation to estimate the model error correlation time-scale. We find that when the observation density is high we are able to obtain converging results. However, with only one observation in a time window, the problem becomes too nonlinear and the estimation process is slow and often does not converge.