Symplectic structure of statistical variational data assimilation

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t),x(t)} and Hamiltonian canonical coordinates {x(t),p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

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