We will now introduce an ensemble representation or approximation for all the covariances appearing in Eqs. ( 4.2 ) and ( 4.5 ). This approximation leads to ensemble methods like EnKF and EnRML, which compute the updates using low-rank representations of the covariance matrices. We will see that the space spanned by the prior ensemble also contains the updated ensemble solution. Accordingly, we can simplify the derivation and formulation of the assimilation update equations by searching for the solution in the ensemble subspace defined by the prior ensemble.