Sequential Markov chain Monte Carlo for Lagrangian data assimilation with applications to unknown data locations

We consider a class of high‐dimensional spatial filtering problems, where the spatial locations of observations are unknown and driven by the partially observed hidden signal. This problem is exceptionally challenging, as not only is it high‐dimensional, but the model for the signal yields longer‐range time dependences through the observation locations. Motivated by this model, we revisit a lesser‐known and provably convergent computational methodology from Berzuini et al. (1997, Journal of the American Statistical Association, 92, 1403–1412); Centanniand Minozzo (2006, Journal of the American Statistical Association, 101, 1582–1597); Martin et al. (2013, Annals of the Institute of Statistical Mathematics, 65, 413–437) that uses sequential Markov Chain Monte Carlo (MCMC) chains. We extend this methodology for data filtering problems with unknown observation locations. We benchmark our algorithms on linear Gaussian state‐space models against competing ensemble methods and demonstrate a significant improvement in both execution speed and accuracy. Finally, we implement a realistic case study on a high‐dimensional rotating shallow‐water model (of about – dimensions) with real and synthetic data. The data are provided by the National Oceanic and Atmospheric Administration (NOAA) and contain observations from ocean drifters in a domain of the Atlantic Ocean restricted to the longitude and latitude intervals , , respectively.