We consider a class of high-dimensional spatial filtering or data assimilation 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 high-dimensional, but the model for the signal yields longer-range time dependencies through the observation locations. Motivated by this model we revisit a lesser-known and $\textit{provably convergent}$ computational methodology from Berzuini et al. (1997), Centanni $\&$ Minozzo (2006a) and Martin et al. (2013) 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 $10^4-10^5$ dimensions) with real and synthetic data. The data is provided by the National Oceanic and Atmospheric Administration (NOAA) and contains observations from ocean drifters {in a domain of the Atlantic Ocean restricted to the longitude and latitude intervals $[-51^{\circ}, -41^{\circ}]$, $[17^{\circ}, 27^{\circ}]$ respectively.

翻译：本文研究一类高维空间滤波或数据同化问题，其中观测数据的空间位置未知且由部分可观测的隐藏信号驱动。该问题极具挑战性：不仅维度高，而且信号模型通过观测位置引入了更长的时间依赖关系。受此模型启发，我们重新审视了Berzuini等（1997）、Centanni与Minozzo（2006a）以及Martin等（2013）提出的一种鲜为人知但具有理论收敛保障的计算方法，该方法利用顺序马尔可夫链蒙特卡洛（MCMC）链。我们将该方法的适用范围扩展至观测位置未知的数据滤波问题。在具有线性高斯状态空间模型的标准框架下，我们将所提算法与竞争性集成方法进行基准测试，结果表明算法在执行速度和精度方面均有显著提升。最后，我们在一个高维旋转浅水模型（维度约10^4-10^5）上开展了基于真实与合成数据的实际案例研究。数据由美国国家海洋与大气管理局（NOAA）提供，包含大西洋某海域（经度范围[-51°, -41°]，纬度范围[17°, 27°]）的海洋漂流浮标观测数据。