  We show, using idealized models, that numerical data assimilation can be
successful only if an effective dimension of the problem is not excessive. This
effective dimension depends on the noise in the model and the data, and in
physically reasonable problems it can be moderate even when the number of
variables is huge. We then analyze several data assimilation algorithms,
including particle filters and variational methods. We show that well-designed
particle filters can solve most of those data assimilation problems that can be
solved in principle, and compare the conditions under which variational methods
can succeed to the conditions required of particle filters. We also discuss the
limitations of our analysis.
