A consistent interpretation of the stochastic version of the Ensemble Kalman Filter - Leeuwen, Peter Jan (Quarterly journal of the Royal Meteorological Society, 2020)
Ensemble Kalman Filters are used extensively in all geoscience areas. Often a stochastic variant is used, in which each ensemble member is updated via the Kalman Filter equation with an extra perturbation in the innovation. These perturbations are essential for the correct ensemble spread in a stochastic Ensemble Kalman Filter, and are applied either to the observations or to the modelled observations. This paper investigates if there is a preference for either of these two perturbation methods. Both versions lead to the same posterior mean and covariance when the prior and the likelihood are Gaussian in the state. However, ensemble verification methods, Bayes' Theorem and the Best Linear Unbiased Estimate (BLUE) suggest that one should perturb the modelled observations. Furthermore, it is known that in non‐Gaussian settings the perturbed modelled observation scheme is preferred, illustrated here for a skewed likelihood. Existing reasons for the perturbed observation scheme are shown to be incorrect, and no new arguments in favour of that scheme have been found. Finally, a new and consistent derivation and interpretation of the stochastic version of the EnKF equations is derived based on perturbing modelled observations. It is argued that these results have direct consequences for (iterative) Ensemble Kalman Filters and Smoothers, including “perturbed observation” 3D‐ and 4D‐Vars, both in terms of internal consistency and implementation. The result of an Ensemble Kalman Filter on a skewed likelihood for the perturbed modelled observation version (left) and the perturbed observation version (right). The prior is denoted as the broad Gaussian with the black line, the true posterior as the blue line, and the stochastic EnKF posterior as the red line. Note that perturbed modelled observation EnKF has the correct positive skewness, while the perturbed observation variant has a skewness of the wrong sign compared to the posterior.