  Weak constraint four-dimensional variational data assimilation is an
important method for incorporating data (typically observations) into a model.
The linearised system arising within the minimisation process can be formulated
as a saddle point problem. A disadvantage of this formulation is the large
storage requirements involved in the linear system. In this paper, we present a
low-rank approach which exploits the structure of the saddle point system using
techniques and theory from solving large scale matrix equations. Numerical
experiments with the linear advection-diffusion equation, and the non-linear
Lorenz-95 model demonstrate the effectiveness of a low-rank Krylov subspace
solver when compared to a traditional solver.
