  Data assimilation algorithms combine information from observations and prior
model information to obtain the most likely state of a dynamical system. The
linearised weak-constraint four-dimensional variational assimilation problem
can be reformulated as a saddle point problem, which admits more scope for
preconditioners than the primal form. In this paper we design new terms which
can be used within existing preconditioners, such as block diagonal and
constraint-type preconditioners. Our novel preconditioning approaches: (i)
incorporate model information, and (ii) are designed to target correlated
observation error covariance matrices. To our knowledge (i) has not previously
been considered for data assimilation problems. We develop new theory
demonstrating the effectiveness of the new preconditioners within Krylov
subspace methods. Linear and non-linear numerical experiments reveal that our
new approach leads to faster convergence than existing state-of-the-art
preconditioners for a broader range of problems than indicated by the theory
alone. We present a range of numerical experiments performed in serial.
