  Lagrangian data assimilation of complex nonlinear turbulent flows is an
important but computationally challenging topic. In this article, an efficient
data-driven statistically accurate reduced-order modeling algorithm is
developed that significantly accelerates the computational efficiency of
Lagrangian data assimilation. The algorithm starts with a Fourier transform of
the high-dimensional flow field, which is followed by an effective model
reduction that retains only a small subset of the Fourier coefficients
corresponding to the energetic modes. Then a linear stochastic model is
developed to approximate the nonlinear dynamics of each Fourier coefficient.
Effective additive and multiplicative noise processes are incorporated to
characterize the modes that exhibit Gaussian and non-Gaussian statistics,
respectively. All the parameters in the reduced order system, including the
multiplicative noise coefficients, are determined systematically via closed
analytic formulae. These linear stochastic models succeed in forecasting the
uncertainty and facilitate an extremely rapid data assimilation scheme. The new
Lagrangian data assimilation is then applied to observations of sea ice floe
trajectories that are driven by atmospheric winds and turbulent ocean currents.
It is shown that observing only about $30$ non-interacting floes in a
$200$km$\times200$km domain is sufficient to recover the key multi-scale
features of the ocean currents. The additional observations of the floe angular
displacements are found to be suitable supplements to the center-of-mass
positions for improving the data assimilation skill. In addition, the observed
large and small floes are more useful in recovering the large- and small-scale
features of the ocean, respectively. The Fourier domain data assimilation also
succeeds in recovering the ocean features in the areas where cloud cover
obscures the observations.
