A Bayesian data assimilation scheme is formulated for
advection-dominated or hyperbolic evolutionary problems, and
observations. It uses the physics to dynamically update the likelihood
in order to extend the impact of the likelihood on the posterior, a
strategy that would be particularly useful when the the observation
network is sparse in space and time and the associated measurement
uncertainties are low. 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. By comparing to the exact
estimate the dynamic likelihood filter is shown to be superior to model
outcomes and to 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.