Weak constraint four-dimensional variational data assimilation is an important method for incorporating observations into a (usually
imperfect) model. The resulting minimisation process takes place in very high dimensions. In this talk we present two approaches for reducing the dimension and thereby the computational cost and storage of this optimisation problem. The first approach formulates the linearised system as a saddle point problem. 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 and low-rank Krylov subspace methods. The second approach uses projection methods for reducing the system dimension. Numerical experiments with the linear advection-diffusion equation, and the nonlinear Lorenz-95 model demonstrate the effectiveness of both approaches.