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

Application of ensemble transform data assimilation methods for parameter estimation in reservoir modelling

Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable methods of MCMC are computationally expensive. Sequential ensemble methods such as ensemble Kalman filers and particle filters provide a favourable alternative. However, Ensemble Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle Filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimation in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup and observe that for a small number of uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands a knowledge at which mode to truncate

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

Statistical relevance of vorticity conservation with the Hamiltonian particle-mesh method

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with the theoretical models, as well as with the continuum statistical mechanical theory for ideal fluid flow developed by Ellis et al. (2002). In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean

Comparison of regularized ensemble Kalman filter and tempered ensemble transform particle filter for an elliptic inverse problem with uncertain boundary conditions

In this paper, we focus on parameter estimation for an elliptic inverse problem. We consider a 2D steady-state single- phase Darcy flow model, where permeability and boundary conditions are uncertain. Permeability is parameterized by the Karhunen-Loeve expansion and thus assumed to be Gaussian distributed. We employ two ensemble-based data assimilation methods: ensemble Kalman filter and ensemble transf

Statistical mechanics of Arakawa`s discretizations

The results of statistical analysis of simulation data obtained from long-time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. Statistical mechanical theories are constructed for three discretizations of the quasi-geostrophic model due to Arakawa (1966), each having different conservation properties. It is shown that the three statistical theories accurately explain the differences observed in statistics derived from the discretizations. The effect of the time discretization is also considered, and it is shown that projection is insufficient if the underlying spatial discretization is not conservative

Parameter estimation for subsurface flow using ensemble data assimilation

Over the years, different data assimilation methods have been implemented to acquire improved estimations of model parameters by adjusting the uncertain parameter values in such a way that the mathematical model approximates the observed data as closely and consistently as possible. However, most of these methods are developed on the assumption of Gaussianity, e.g. Ensemble Kalman Filters, whic

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

Impact of the initialisation on the predictability of the Southern Ocean sea ice at interannual to multi-decadal timescales

In this study, we assess systematically the impact of different initialisation procedures on the predictability of the sea ice in the Southern Ocean. These initialisation strategies are based on three data assimilation methods: the nudging, the particle filter with sequential importance resampling and the nudging proposal particle filter. An Earth system model of intermediate complexity is used to perform hindcast simulations in a perfect model approach. The predictability of the Antarctic sea ice at interannual to multi-decadal timescales is estimated through two aspects: the spread of the hindcast ensemble, indicating the uncertainty of the ensemble, and the correlation between the ensemble mean and the pseudo-observations, used to assess the accuracy of the prediction. Our results show that at decadal timescales more sophisticated data assimilation methods as well as denser pseudo-observations used to initialise the hindcasts decrease the spread of the ensemble. However, our experiments did not clearly demonstrate that one of the initialisation methods systematically provides with a more accurate prediction of the sea ice in the Southern Ocean than the others. Overall, the predictability at interannual timescales is limited to 3 years ahead at most. At multi-decadal timescales, the trends in sea ice extent computed over the time period just after the initialisation are clearly better correlated between the hindcasts and the pseudo-observations if the initialisation takes into account the pseudo-observations. The correlation reaches values larger than 0.5 in winter. This high correlation has likely its origin in the slow evolution of the ocean ensured by its strong thermal inertia, showing the importance of the quality of the initialisation below the sea ice