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Lagrangian data assimilation exploits the trajectories of moving tracers asobservations to recover the underlying flow field. One major challenge inLagrangian data assimilation is the intrinsic nonlinearity that impedes usingexact Bayesian formulae for the state estimation of high-dimensional systems.In this paper, an analytically tractable mathematical framework forcontinuous-in-time Lagrangian data assimilation is developed. It preserves thenonlinearity in the observational processes while approximating the forecastmodel of the underlying flow field using linear stochastic models (LSMs). Acritical feature of the framework is that closed analytic formulae areavailable for solving the posterior distribution, which facilitatesmathematical analysis and numerical simulations. First, an efficient iterativealgorithm is developed in light of the analytically tractable statistics. Itaccurately estimates the parameters in the LSMs using only a small number ofthe observed tracer trajectories. Next, the framework facilitates thedevelopment of several computationally efficient approximate filters and thequantification of the associated uncertainties. A cheap approximate filter witha diagonal posterior covariance derived from the asymptotic analysis of theposterior estimate is shown to be skillful in recovering incompressible flows.It is also demonstrated that randomly selecting a small number of tracers ateach time step as observations can reduce the computational cost whileretaining the data assimilation accuracy. Finally, based on a prototype modelin geophysics, the framework with LSMs is shown to be skillful in filteringnonlinear turbulent flow fields with strong non-Gaussian features.

Uncertainty Quantification of Nonlinear Lagrangian Data Assimilation Using Linear Stochastic Forecast Models