Parameter and state estimation with ensemble Kalman filter based algorithms for convective‐scale applications

Abstract

Representation of clouds in convection‐permitting models is sensitive to numerical weather prediction (NWP) model parameters that are often very crudely known (for example roughness length). Our goal is to allow for uncertainty in these parameters and estimate them from data using the ensemble Kalman filter (EnKF) approach. However, to deal with difficulties associated with convective‐scale applications, such as non‐Gaussianity and constraints on state and parameter values, modifications to classical EnKF are necessary. In this article, we evaluate and extend several recently developed EnKF‐based algorithms that either incorporate constraints such as mass conservation and positivity of precipitation explicitly or introduce higher order moments on the joint state and parameter estimation problem. We compare their results with the localized EnKF for a common idealized test case. The test case uses perfect model experiments with the one‐dimensional modified shallow‐water model, which was designed to mimic important properties of convection. We use a stochastic dynamical model for parameters, in order to prevent underdispersion in parameter space. To deal with localization for estimation of parameters, we introduce a method called global updating, which is a computationally cheap modification of spatial updating and was proven successful in this context. The sensitivity of the results to the number of ensemble members and localization, as well as observation coverage and frequency, is shown. Although all algorithms are capable of reducing the initial state and parameter errors, it is concluded that mass conservation is important when the localization radius is small and/or the observations are sparse. In addition, accounting for higher order moments in the joint space and parameter estimation problem is beneficial when the ensemble size is large enough or when applied to parameter estimation only.

Online learning of both state and dynamics using ensemble Kalman filters

“…Their authors have proposed to make the global parameters local in the DL update step, followed by a spatial averaging of the local updated parameters to form the global parameters and be able to propagate the ensemble using these updated global parameters. By contrast, CL was chosen in [38,50], and localization was not applied in the global parameter space. Indeed, it is difficult to choose a priori any correlation structure for the global parameters among themselves.…”

“…Nonetheless, a tapering coefficient could be used for the state/parameter covariances. Ruckstuhl and Janjić [50] empirically chose N x −1 as tapering coefficient, and argue that this could make the localization matrix positive semi-definite (and hence a genuine correlation matrix). Hereafter, we choose to work with the latter approach where there is no localization in parameter space but a tapering can be applied to the state/parameter covariances.…”

“…In this section, arguments are given for the use of a tapering coefficient of the state/parameter covariances. We follow the discussion of [50] in their Section 3.1, and slightly generalize their derivation. The main argument of [50] is that the localization matrix C should be positive definite.…”

“…We follow the discussion of [50] in their Section 3.1, and slightly generalize their derivation. The main argument of [50] is that the localization matrix C should be positive definite. We note, however, that this is a sufficient condition for the regularized B to be positive definite, not a necessary condition.…”

“…We note, however, that this is a sufficient condition for the regularized B to be positive definite, not a necessary condition. In contrast to [50], all parameters are considered here. We have C xx = ρ, the localization matrix in state space, which is assumed to be positive definite.…”