  Data assimilation leads naturally to a Bayesian formulation in which the
posterior probability distribution of the system state, given the observations,
plays a central conceptual role. The aim of this paper is to use this Bayesian
posterior probability distribution as a gold standard against which to evaluate
various commonly used data assimilation algorithms.
  A key aspect of geophysical data assimilation is the high dimensionality and
low predictability of the computational model. With this in mind, yet with the
goal of allowing an explicit and accurate computation of the posterior
distribution, we study the 2D Navier-Stokes equations in a periodic geometry.
We compute the posterior probability distribution by state-of-the-art
statistical sampling techniques. The commonly used algorithms that we evaluate
against this accurate gold standard, as quantified by comparing the relative
error in reproducing its moments, are 4DVAR and a variety of sequential
filtering approximations based on 3DVAR and on extended and ensemble Kalman
filters.
  The primary conclusions are that: (i) with appropriate parameter choices,
approximate filters can perform well in reproducing the mean of the desired
probability distribution; (ii) however they typically perform poorly when
attempting to reproduce the covariance; (iii) this poor performance is
compounded by the need to modify the covariance, in order to induce stability.
Thus, whilst filters can be a useful tool in predicting mean behavior, they
should be viewed with caution as predictors of uncertainty. These conclusions
are intrinsic to the algorithms and will not change if the model complexity is
increased, for example by employing a smaller viscosity, or by using a detailed
NWP model.
