Filtering with One-Step-Ahead Smoothing for Efficient Data Assimilation

Online estimation of colored observation‐noise parameters within an ensemble Kalman filtering framework

“…This is consistent with the results of Raboudi et al . (2018), who argued that the two‐stage update step constrains the ensemble sampling with information from “future” data, which helps to mitigate issues related to undersampling and strong nonlinearities (Ait‐El‐Fquih and Hoteit, 2022a). On average, OSA smoothing improves the filtering results by about $$$ 20\% $$$ in both scenarios, with the corresponding state estimates clearly being less sensitive to the choice of inflation and localization values.…”

“…Filtering with one‐step‐ahead (OSA) smoothing is an alternative filtering strategy (Ait‐El‐Fquih and Hoteit, 2022a). It involves an extra OSA smoothing step with the “future” observation between two successive analyses, within a Bayesian framework (Desbouvries et al ., 2011).…”

A variational Bayesian approach for ensemble filtering of stochastically parametrized systems

“…In light of the conditional independence properties of the augmented system of Equation (i.e., Equations – ), this system is a hidden Markov‐chain model endowed with a transition pdf and a likelihood, respectively given as (Ait‐El‐Fquih et al ., 2016; Ait‐El‐Fquih and Hoteit, 2022) $$$$ p\left({\mathbf{z}}_n|{\mathbf{z}}_{n-1},\boldsymbol{\vartheta}, q\right)\kern0.5em =p\left({\mathbf{x}}_n|{\mathbf{x}}_{n-1},{\boldsymbol{\epsilon}}_{n-1},\boldsymbol{\theta}, \phi \right)p\left({\boldsymbol{\epsilon}}_n|{\boldsymbol{\epsilon}}_{n-1},\phi, q\right), $$$$ $$$$ p\left({\mathbf{y}}_n|{\mathbf{z}}_n\right)\kern0.5em =p\left({\mathbf{y}}_n|{\mathbf{x}}_n\right). $$$$ These are key ingredients that guarantee the recursivity of the proposed filtering algorithms (Ait‐El‐Fquih and Hoteit, 2022).…”

“…Although the OSAS formulation of the state‐parameter filtering problem uses the observation three times during one assimilation cycle, this does not in any way affect the Bayesian character of the resulting algorithms, as this (same) observation is not used to update the same quantity (the current state), but three different quantities: the previous state, the parameter vector, and the current state.This has already been discussed in detail in Raboudi et al . (2018) (see also Desbouvries et al ., 2011; Ait‐El‐Fquih and Hoteit, 2022) in the context of state estimation, emphasizing in particular that, in linear Gaussian state‐space systems, KF and OSAS‐based KF provide exactly the same (theoretical) forecast and analysis estimates.…”

“…In view of the heuristic nature of the dual EnKF, the one‐step‐ahead smoothing (OSAS) filtering approach has been adopted more recently to set up a Bayesian consistent filtering framework for this approach (Desbouvries et al ., 2011; Gharamti et al ., 2015; Ait‐El‐Fquih et al ., 2016; Ait‐El‐Fquih and Hoteit, 2022). The OSAS filtering approach reverses the classical (time update then observation update) order within the assimilation cycles, which results in three EnKF‐like updates based on the same (current) observation: the first to smooth the previous state, the second to update the parameters, and the third to analyse the current state conditionally on the resulting smoothed state and updated parameters (e.g., Raboudi et al ., 2018; Ait‐El‐Fquih and Hoteit, 2022). Compared with the dual EnKF, the one‐step‐ahead smoothing‐based ensemble Kalman filter (EnKF‐OSAS) involves a threefold, instead of twofold, use of the same observation in the state analysis, and this was shown to provide more information for enhanced state and parameter estimates (Gharamti et al ., 2015; Ait‐El‐Fquih et al ., 2016; Khaki et al ., 2020).…”