Traditional data assimilation methods like ensemble Kalman filters (EnKF) face intrinsic challenges with system nonlinearity and high-dimensional dynamics. We propose a novel approach based on reproducing kernel Hilbert spaces (RKHS), embedding dynamical system observables into time-evolving RKHSs. This framework leads to reconstruction of trajectories with time-invariant linear combinations of ensemble members, which enables aggregating observations over time and assimilate them during a same analysis step. The aim of the present paper is to construct an ensemble data assimilation method formulated in this time-evolving RKHS. We assess the approach using a multilayer quasi-geostrophic ocean model with synthetic sea surface height (SSH) observations derived from satellite tracks. Results show significantly improved reconstruction accuracy and robustness compared to a classical ensemble filter. This work summarizes a more detailed study (Jaouen et al. 2025).