A Bayesian approach to data assimilation

Data assimilation is formulated in a Bayesian context. This leads to a sampling problem in the space of continuous time paths. By writing down a density in path space, and conditioning on observations, it is pos-sible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired distribution in path space, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are stochastic par-tial differential equations which are ergodic and whose invariant measure gives the desired distribution in path space. Two examples are given to show how data assimilation can be for-mulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Further-more the relationship between the Bayesian approach outlined here and the commonly used Kalman filter-based techniques, prevalent in practice, is discussed. A simple paedogogical example is studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined.