Toward a multivariate formulation of the PKF assimilation: application to a simplified chemical transport model

Abstract

Abstract. This contribution explores a new approach to forecast multivariate covariances for atmospheric chemistry through the use of the parametric Kalman filter (PKF). In the PKF formalism, the error covariance matrix is modelized by a covariance model relying on parameters, for which the dynamics is then computed. The PKF has been formulated in univariate cases, and a multivariate extension for chemical transport models is explored here. To do so, a simplified two-species chemical transport model over a 1D domain is introduced, based on the nonlinear Lotka-Volterra equations, which allows to propose a multivariate pseudo covariance model. Then, the multivariate PKF dynamics is formulated and its results are compared with a large ensemble Kalman filter (EnKF) in several numerical experiments. In these experiments, the PKF accurately reproduces the EnKF. Eventually, the PKF is formulated for a more complex chemical model composed of six chemical species (Generic Reaction Set). Again, the PKF succeeds at reproducing the multivariate covariances diagnosed on the large ensemble.

Boundary Conditions for the Parametric Kalman Filter Forecast

“…Note that we consider that the PKF performs well here, even with a difference of 20% on the length‐scale near the middle of the domain (see Figures 6f, 6d, and 7b), because the length‐scale predicted by the PKF has the appropriate order of magnitude, and that the variance is really well reproduced (see Figures 6c, 6d, and 7a). From our experience, an error in variance is more damaging than an error in span length during assimilation cycle (see Pannekoucke, 2021a, Figure 9 or Perrot et al., 2023, Figure 7). However, the question to know if whether or not the PKF performs well should be addressed by comparing the results of the two schemes in the context of DA with synthetic observations, in order to see how the difference of length‐scale impacts the root mean squared error of the analyzed state, and see if this difference is admissible or not, which is beyond the scope of the present contribution.…”

“…However in multivariate statistics, the numerical cost of the PKF scales as the square of the number of prognostic fields, which is a strong limitation. For example, a multivariate assimilation in air quality should consider hundreds of chemical species (Perrot et al., 2023), but in practice, only a few species are assimilated, without correction for other unobserved chemical species (the forecast error of the observed and the unobserved chemical species are then assumed as decorrelated), which makes the PKF interesting for these applications (in CAMS regional air quality production 2.40 (CAMS, 2022), the univariate 3DVar system of MOCAGE is used for the separated assimilation of ozone, nitrogen dioxide, sulfur dioxide, and fine particulate matter PM2.5 and PM10, following a configuration similar to the one used for the monitoring atmospheric composition and climate: interim implementation forecast system detailed by Marécal et al. (2015)).…”

“…However, the very large length‐scale values, larger than the domain size Λ, as diagnosed from the ensemble is subject to the limitation of the numerical computation of Equation for large correlations that can present a positive bias (Pannekoucke et al., 2008), but the magnitude of this positive bias should be low in the present experiment where a large ensemble of 6,400 members is used. Moreover, the large length‐scale of the EnKF can also be influenced by model error (Pannekoucke et al., 2021), while the PKF can be less sensitive to numerical error (Perrot et al., 2023, Appendix A). Because of these limitations, it is not certain that the EnKF reproduces the true dynamics of the uncertainty for these extreme values of the length‐scales, while it is considered as the reference.…”

“…Hence, for a time integration over a time window [0, T ], the dynamics Equation is written as $$$\chi (T)={\mathbf{M}}_{T{\leftarrow}0}\chi (0),$$$ where M T ←0 denotes the propagator associated with the time integration of Equation over [0, T ]. Compared to the classical formulation of the Kalman filter equations, we do not consider here the model error because we won't try to characterize it here, while it can lead to spurious time evolution of ensemble error statistics compared to the true dynamics, as it has been shown using the PKF in previous contributions (Ménard et al., 2021; Pannekoucke, 2021a; Pannekoucke et al., 2021; Perrot et al., 2023).…”

“…The PKF is interesting to provide a flow dependency of the background error covariance matrix in variational DA for example, by providing a proxy for the diffusion tensor of the day in the covariance model based on the diffusion equation (Pannekoucke et al., 2016); or by providing the variance of the day in presence of model error thanks to the PKF dynamics of the variance, as in the assimilation of GOSAT methane observations using the hemispherical community multiscale air quality model (Voshtani et al., 2022). While the PKF has been mainly developed in univariate statistics, an exploration has begun to develop a multivariate statistics formulation (Perrot et al., 2023). At a more theoretical level, the parametric formulation has been considered to describe the covariance dynamics of white noise error in a conservative equation (Gilpin et al., 2022).…”