  This paper provides a detailed theoretical analysis of methods to approximate
the solutions of high-dimensional (>10^6) linear Bayesian problems. An optimal
low-rank projection that maximizes the information content of the Bayesian
inversion is proposed and efficiently constructed using a scalable randomized
SVD algorithm. Useful optimality results are established for the associated
posterior error covariance matrix and posterior mean approximations, which are
further investigated in a numerical experiment consisting of a large-scale
atmospheric tracer transport source-inversion problem. This method proves to be
a robust and efficient approach to dimension reduction, as well as a natural
framework to analyze the information content of the inversion. Possible
extensions of this approach to the non-linear framework in the context of
operational numerical weather forecast data assimilation systems based on the
incremental 4D-Var technique are also discussed, and a detailed implementation
of a new Randomized Incremental Optimal Technique (RIOT) for 4D-Var algorithms
leveraging our theoretical results is proposed.
