Cluster Sampling Filters for Non-Gaussian Data Assimilation

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 $C\ell \mathrm{MCMC}$ , $C\ell \mathrm{HMC}$ , MC- $C\ell \mathrm{HMC}$ , and MC- $C\ell \mathrm{HMC}$ are presented. $C\ell \mathrm{MCMC}$ uses a Gaussian proposal density to sample the posterior, and $C\ell \mathrm{HMC}$ is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC- $C\ell \mathrm{MCMC}$ and MC- $C\ell \mathrm{HMC}$ are multi-chain versions of the cluster sampling filters $C\ell \mathrm{MCMC}$ and $C\ell \mathrm{HMC}$ 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.