  Lagrangian trajectories are widely used as observations for recovering the
underlying flow field via Lagrangian data assimilation (DA). However, the
strong nonlinearity in the observational process and the high dimensionality of
the problems often cause challenges in applying standard Lagrangian DA. In this
paper, a Lagrangian-Eulerian multiscale DA (LEMDA) framework is developed. It
starts with exploiting the Boltzmann kinetic description of the particle
dynamics to derive a set of continuum equations, which characterize the
statistical quantities of particle motions at fixed grids and serve as Eulerian
observations. Despite the nonlinearity in the continuum equations and the
processes of Lagrangian observations, the time evolutions of the posterior
distribution from LEMDA can be written down using closed analytic formulae.
This offers an exact and efficient way of carrying out DA, which avoids using
ensemble approximations and the associated tunings. The analytically solvable
properties also facilitate the derivation of an effective reduced-order
Lagrangian DA scheme that further enhances computational efficiency. The
Lagrangian DA within the framework has advantages when a moderate number of
particles is used, while the Eulerian DA can effectively save computational
costs when the number of particle observations becomes large. The Eulerian DA
is also valuable when particles collide, such as using sea ice floe
trajectories as observations. LEMDA naturally applies to multiscale turbulent
flow fields, where the Eulerian DA recovers the large-scale structures, and the
Lagrangian DA efficiently resolves the small-scale features in each grid cell
via parallel computing. Numerical experiments demonstrate the skilful results
of LEMDA and its two components.
