  A Bayesian data assimilation scheme is formulated for advection-dominated or
hyperbolic evolutionary problems, and observations. The method is referred to
as the dynamic likelihood filter because it exploits the model physics to
dynamically update the likelihood with the aim of making better use of low
uncertainty sparse observations.
  The filter is applied to a problem with linear dynamics and Gaussian
statistics, and compared to the exact estimate, a model outcome, and the Kalman
filter estimate. Its estimates are shown to be superior to the model outcomes
and the Kalman estimate, when the observation system is sparse. The added
computational expense of the method is linear in the number of observations and
thus computationally efficient, suggesting that the method is practical even if
the space dimensions of the physical problem are large.
