  In this paper we apply the 4D-Var data assimilation scheme to the
initialization problem for a family of quasilinear evolution equations. The
resulting variational problem is non-convex, so it need not have a unique
minimizer. We comment on the implications of non-uniqueness in numerical
applications, then prove uniqueness results in the following situations: 1) the
observational times are all sufficiently small; 2) the prior covariance is
sufficiently small. We also give an example of a data set where the cost
functional has a critical point of arbitrarily large Morse index.
