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The problem of effectively combining data with a mathematical modelconstitutes a major challenge in applied mathematics. It is particularchallenging for high-dimensional dynamical systems where data is receivedsequentially in time and the objective is to estimate the system state in anon-line fashion; this situation arises, for example, in weather forecasting.The sequential particle filter is then impractical and ad hoc filters, whichemploy some form of Gaussian approximation, are widely used. Prototypical ofthese ad hoc filters is the 3DVAR method. The goal of this paper is to analyzethe 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. Thesituation where the data is partial and noisy is studied, and both discretetime and continuous time data streams are considered. The theory demonstrateshow the widely used technique of variance inflation acts to stabilize thefilter, and hence leads to asymptotic accuracy.

Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model