Evaluating Data Assimilation Algorithms

3 Data assimilation leads naturally to a Bayesian formulation in which the posterior probability 4 distribution of the system state, given all the observations on a time window of interest, 5 plays a central conceptual role. The aim of this paper is to use this Bayesian posterior 6 probability distribution as a gold standard against which to evaluate various commonly used 7 data assimilation algorithms. 8 A key aspect of geophysical data assimilation is the high dimensionality and limited 9 predictability of the computational model. We study the 2D Navier-Stokes equations in a 10 periodic geometry, which has these features and yet is tractable for explicit and accurate com11 putation of the posterior distribution by state-of-the-art statistical sampling techniques. The 12 commonly used algorithms that we evaluate, as quantified by the relative error in reproduc13 ing moments of the posterior, are 4DVAR and a variety of sequential filtering approximations 14 based on 3DVAR and on extended and ensemble Kalman filters. 15 The primary conclusions are that under the assumption of a well-defined posterior prob16 ability distribution: (i) with appropriate parameter choices, approximate filters can perform 17 well in reproducing the mean of the desired probability distribution; (ii) however they do 18 not perform as well in reproducing the covariance; (iii) the error is compounded by the need 19 to modify the covariance, in order to induce stability. Thus, filters can be a useful tool in 20 predicting mean behavior, but should be viewed with caution as predictors of uncertainty. 21 These conclusions are intrinsic to the algorithms when assumptions underlying them are not 22 valid and will not change if the model complexity is increased. 23