Analysis of the 3DVAR filter for thepartially observed Lorenz'63 model

The problem of effectively combining data with amathematical model constitutes a major challengein applied mathematics. It is particularchallenging for high-dimensional dynamical systemswhere data is received sequentially in time andthe objective is to estimate the system statein an on-line fashion; this situation arises, forexample, in weather forecasting.The sequential particle filteris then impractical and ad hoc filters, whichemploy some form of Gaussian approximation, arewidely used. Prototypical of these ad hoc filtersis the 3DVAR method. The goal of this paper isto analyze the 3DVARmethod, using the Lorenz '63 model to exemplify thekey ideas. The situation where the data is partialand noisy is studied, and both discrete time andcontinuous time data streams are considered. Thetheory demonstrates how the widely used techniqueof variance inflation acts to stabilize the filter,and hence leads to asymptotic accuracy.