The standard practice in filtering with model errors is to augment the dynamical equations for the physical process with ad-hocly chosen dynamical equations for the parameters (Friedland 1969, 1982). On the other hand, Fredericson and O'Kane (2009) showed high filtering skill of atmospheric blocking event with a more elaborate stochastic model; they parameterized damping and forcing with global nonlinear interaction across scales.
Motivated by these works, we (Gershgorin, Harlim, and Majda 2010a, b) introduced a novel strategy "Stochastic Parameterized Extended Kalman Filter" (SPEKF) as a simple nonlinear stochastic model with exactly solvable statistics that includes both the forcing and damping corrections ``on-the-fly". The exactly solvable feature is important because it suggests that no linear tangent approximation is required in SPEKF, as opposed to the standard Extended Kalman Filter. In this talk, I will briefly review the underlying theory and present numerical results with an SPDE that mimics the barotropic Rossby waves as well as recent results on the two-layer quasi-geostrophic model with baroclinic instabilities (Harlim and Majda 2010).
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