At its core, this dissertation aims to formalize and explain—through a statistical lens—the empirical success of popular ensemble-based algorithms in the data assimilation literature. A key component of this effort is the derivation of non-asymptotic, dimension-free bounds for the estimation of covariance operators. To achieve this, we leverage existing techniques from high-dimensional probability while also developing new theoretical tools to analyze the behavior of a certain class of covariance estimators under structural assumptions. This dissertation rigorously establishes fundamental guarantees for these estimators, shedding light on the mechanisms that drive their effectiveness and providing a deeper understanding of their practical success.