  A Bayesian data assimilation scheme is formulated for advection-dominated
advective and diffusive evolutionary problems, based upon the Dynamic
Likelihood (DLF) approach to filtering. The DLF was developed specifically for
hyperbolic problems -waves-, and in this paper, it is extended via a split step
formulation, to handle advection-diffusion problems. In the dynamic likelihood
approach, observations and their statistics are used to propagate probabilities
along characteristics, evolving the likelihood in time. The estimate posterior
thus inherits phase information. For advection-diffusion the advective part of
the time evolution is handled on the basis of observations alone, while the
diffusive part is informed through the model as well as observations. We
expect, and indeed show here, that in advection-dominated problems, the DLF
approach produces better estimates than other assimilation approaches,
particularly when the observations are sparse and have low uncertainty. The
added computational expense of the method is cubic in the total number of
observations over time, which is on the same order of magnitude as a standard
Kalman filter and can be mitigated by bounding the number of forward propagated
observations, discarding the least informative data.
