The problem of effectively combining data with a mathematical model
constitutes a major challenge in applied mathematics. It is particular
challenging for high-dimensional dynamical systems where data is received
sequentially in time and the objective is to estimate the system state in an
on-line fashion; this situation arises, for example, in weather forecasting.
The sequential particle filter is then impractical and ad hoc filters, which
employ some form of Gaussian approximation, are widely used. Prototypical of
these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze
the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The
situation where the data is partial and noisy is studied, and both discrete
time and continuous time data streams are considered. The theory demonstrates
how the widely used technique of variance inflation acts to stabilize the
filter, and hence leads to asymptotic accuracy.