Abstract
This paper presents a fully non-Gaussian filter for sequential data assimilation. The filter is named the “cluster sampling filter”, and works by directly sampling the posterior distribution following a Markov Chain Monte-Carlo (MCMC) approach, while the prior distribution is approximated using a Gaussian Mixture Model (GMM). Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a GMM to the prior ensemble. Using the data likelihood function, the posterior density is then formulated as a mixture density, and is sampled following an MCMC approach. Four versions of the proposed filter, namely C ℓ MCMC , C ℓ HMC , MC- C ℓ HMC , and MC- C ℓ HMC are presented. C ℓ MCMC uses a Gaussian proposal density to sample the posterior, and C ℓ HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC- C ℓ MCMC and MC- C ℓ HMC are multi-chain versions of the cluster sampling filters C ℓ MCMC and C ℓ HMC respectively. The multi-chain versions are proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. The new methodologies are tested using a simple one-dimensional example, and a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption especially in the HMC filtering paradigm.
Highlights
Data assimilation (DA) is a complex process that involves combining information from different sources in order to produce accurate estimates of the true state of a physical system such as the atmosphere
The data assimilation testing suite (DATeS) [38,39] is used to carry out the numerical experiments presented in this work
The results below suggest that relaxing the Gaussian-prior using an ensemble-based Gaussian Mixture Model (GMM) estimate could be dangerous, unless the sampler is guaranteed to cover all posterior probability modes
Summary
Data assimilation (DA) is a complex process that involves combining information from different sources in order to produce accurate estimates of the true state of a physical system such as the atmosphere. IEnKF, assumes that the underlying probability distributions are Gaussian and the analysis state is best estimated by the posterior mode These families of filters can generally be tuned (e.g., using inflation and localization) for optimal performance on the problem at hand. Experiments with linear settings aim to compare the performance of the proposed algorithms, in the presence of benchmark results produced by EnKF This has the benefit of demonstrating the advantage of replacing the Gaussian prior with a GMM, for MCMC sampling, even in the simplified linear settings. Numerical results with nonlinear settings, where EnKF fails, suggest that the proposed relaxation of the Gaussian-prior assumption is beneficial, especially for the sequential application of MCMC sampling filters in the presence of nonlinearities.
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