A nonlinear test model for filtering turbulent signals from partial
observations of nonlinear slow-fast systems with multiple time scales is
developed here. This model is a nonlinear stochastic real triad model with
one slow mode, two fast modes, and catalytic nonlinear interaction of the
fast modes depending on the slow mode. Despite the nonlinear and
non-Gaussian features of the model, exact solution formulas are developed
here for the mean and covariance. These formulas are utilized to develop a
suite of statistically exact extended Kalman filters for the slow-fast
system. Important practical issues such as filter performance with partial
observations, which mix the slow and fast modes, model errors through linear
filters for the fast modes, and the role of observation frequency and
observational noise strength are assessed in unambiguous fashion in the test
model by utilizing these exact nonlinear statistics.