  In this paper we propose to numerically assess the performance of standard
Gaussian approximations to probe the posterior distribution that arises from
Bayesian data assimilation in petroleum reservoirs. In particular we assess the
performance of (i) the linearization around the maximum a posterior estimate,
(ii) the randomized maximum likelihood and (iii) standard ensemble Kalman
filter-type methods. In order to fully resolve the posterior distribution we
implement a state-of-the art MCMC method that scales well with respect to the
dimension of the parameter space. Our implementation of the MCMC method
provides the gold standard against which to assess the aforementioned Gaussian
approximations. We present numerical synthetic experiments where we quantify
the capability of each of the {\em ad hoc} Gaussian approximation in
reproducing the mean and the variance of the posterior distribution
(characterized via MCMC) associated to a data assimilation problem. The main
objective of our controlled experiments is to exhibit the substantial
discrepancies of the approximation properties of standard {\em ad hoc} Gaussian
approximations. Numerical investigations of the type we present here will lead
to greater understanding of the cost-efficient, but {\em ad hoc}, Bayesian
techniques used for data assimilation in petroleum reservoirs, and hence
ultimately to improved techniques with more accurate uncertainty
quantification.
