Deterministic Mean-field Ensemble Kalman Filtering

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $\kappa$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2\kappa$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI

Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error

In this paper, we trivially extend Tempered (Localized) Ensemble Transform Particle Filter---T(L)ETPF---to account for model error. We examine T(L)ETPF performance for non-additive model error in a low-dimensional and a high-dimensional test problem. The former one is a nonlinear toy model, where uncertain parameters are non-Gaussian distributed but model error is Gaussian distributed. The latter one is a steady-state single-phase Darcy flow model, where uncertain parameters are Gaussian distributed but model error is non-Gaussian distributed. The source of model error in the Darcy flow problem is uncertain boundary conditions. We comapare T(L)ETPF to a Regularized (Localized) Ensemble Kalman Filter---R(L)EnKF. We show that T(L)ETPF outperforms R(L)EnKF for both the low-dimensional and the high-dimensional problem. This holds even when ensemble size of TLETPF is 100 while ensemble size of R(L)EnKF is greater than 6000. As a side note, we show that TLETPF takes less iterations than TETPF, which decreases computational costs; while RLEnKF takes more iterations than REnKF, which incerases computational costs. This is due to an influence of localization on a tempering and a regularizing parameter

Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Data assimilation in slow-fast systems using homogenized climate models

A deterministic multiscale toy model is studied in which a chaotic fast subsystem triggers rare transitions between slow regimes, akin to weather or climate regimes. Using homogenization techniques, a reduced stochastic parametrization model is derived for the slow dynamics. The reliability of this reduced climate model in reproducing the statistics of the slow dynamics of the full deterministic model for finite values of the time scale separation is numerically established. The statistics however is sensitive to uncertainties in the parameters of the stochastic model. It is investigated whether the stochastic climate model can be beneficial as a forecast model in an ensemble data assimilation setting, in particular in the realistic setting when observations are only available for the slow variables. The main result is that reduced stochastic models can indeed improve the analysis skill, when used as forecast models instead of the perfect full deterministic model. The stochastic climate model is far superior at detecting transitions between regimes. The observation intervals for which skill improvement can be obtained are related to the characteristic time scales involved. The reason why stochastic climate models are capable of producing superior skill in an ensemble setting is due to the finite ensemble size; ensembles obtained from the perfect deterministic forecast model lacks sufficient spread even for moderate ensemble sizes. Stochastic climate models provide a natural way to provide sufficient ensemble spread to detect transitions between regimes. This is corroborated with numerical simulations. The conclusion is that stochastic parametrizations are attractive for data assimilation despite their sensitivity to uncertainties in the parameters.Comment: Accepted for publication in Journal of the Atmospheric Science