  There is growing awareness that errors in the model equations cannot be
ignored in data assimilation methods such as four-dimensional variational
assimilation (4D-Var). If allowed for, more information can be extracted from
observations, longer time windows are possible, and the minimisation process is
easier, at least in principle. Weak constraint 4D-Var estimates the model error
and minimises a series of linear least-squares cost functions using the
conjugate gradient (CG) method; minimising each cost function is called an
inner loop. CG needs preconditioning to improve its performance. In previous
work, limited memory preconditioners (LMPs) have been constructed using
approximations of the eigenvalues and eigenvectors of the Hessian in the
previous inner loop. If the Hessian changes significantly in consecutive inner
loops, the LMP may be of limited usefulness. To circumvent this, we propose
using randomised methods for low rank eigenvalue decomposition and use these
approximations to cheaply construct LMPs using information from the current
inner loop. Three randomised methods are compared. Numerical experiments in
idealized systems show that the resulting LMPs perform better than the existing
LMPs. Using these methods may allow more efficient and robust implementations
of incremental weak constraint 4D-Var.
