Unbiased fully nonlinear data assimilation. Part 2: The stochastic particle flow smoother

Building on the stochastic particle flow filter (SPFF) introduced in Part 1, this study develops a stochastic particle flow smoother (SPFS) for fully nonlinear data assimilation in high‐dimensional dynamical systems. Whereas the SPFF successfully provides unbiased ensemble spread and handles non‐Gaussian distributions in sequential filtering, many applications require historical state estimates over an extended assimilation window. The SPFS is formulated as a strong‐constraint smoother in which observation information is propagated backward in time to optimize the initial ensemble. In this study, two backward‐propagation strategies are examined. One uses the exact adjoint model, and the other uses an ensemble‐averaged sensitivity matrix approximation. Experiments with the 1000‐dimensional Lorenz 96 system demonstrate that the SPFS with an adjoint model achieves superior trajectory fits compared with the filter, particularly in strongly nonlinear regimes, by allowing particles to follow individual gradients and represent multimodal posteriors. In contrast, the ensemble‐averaged sensitivity matrix approximation, though computationally appealing, degrades in highly nonlinear settings and requires shorter assimilation windows. These results establish the SPFS as an effective solution to the fully nonlinear smoothing problem in high‐dimensional systems.