High-dimensional state estimation and numerical weather prediction

Daily weather forecasts follow in large part from numerical weather prediction, in which large numerical models simulate the atmospheric flow beginning from initial conditions (a state estimate) based on the most recent atmospheric observations.  The state estimation problem poses computational and mathematical challenges because it is "big" -- the dimension of the model state is large and there are many observations.  I will review some of the challenges of high-dimensional state estimation and address several questions:  How "big" is state estimation for numerical weather prediction?  What measures of problem size are most useful as guidance for computational difficulty, especially for Monte-Carlo approaches to state estimation?  And how do these depend on the underlying dynamics of atmospheric flows and on the details of the observational network?