  Atmospheric motion vectors (AMVs) extracted from satellite imagery are the
only wind observations with good global coverage. They are important features
for feeding numerical weather prediction (NWP) models. Several Bayesian models
have been proposed to estimate AMVs. Although critical for correct assimilation
into NWP models, very few methods provide a thorough characterization of the
estimation errors. The difficulty of estimating errors stems from the
specificity of the posterior distribution, which is both very high dimensional,
and highly ill-conditioned due to a singular likelihood. Motivated by this
difficult inverse problem, this work studies the evaluation of the (expected)
estimation errors using gradient-based Markov Chain Monte Carlo (MCMC)
algorithms. The main contribution is to propose a general strategy, called here
chilling, which amounts to sampling a local approximation of the posterior
distribution in the neighborhood of a point estimate. From a theoretical point
of view, we show that under regularity assumptions, the family of chilled
posterior distributions converges in distribution as temperature decreases to
an optimal Gaussian approximation at a point estimate given by the Maximum A
Posteriori, also known as the Laplace approximation. Chilled sampling therefore
provides access to this approximation generally out of reach in such
high-dimensional nonlinear contexts. From an empirical perspective, we evaluate
the proposed approach based on some quantitative Bayesian criteria. Our
numerical simulations are performed on synthetic and real meteorological data.
They reveal that not only the proposed chilling exhibits a significant gain in
terms of accuracy of the point estimates and of their associated expected
errors, but also a substantial acceleration in the convergence speed of the
MCMC algorithms.
