  Data assimilation combines prior (or background) information with
observations to estimate the initial state of a dynamical system over a given
time-window. A common application is in numerical weather prediction where a
previous forecast and atmospheric observations are used to obtain the initial
conditions for a numerical weather forecast. In four-dimensional variational
data assimilation (4D-Var), the problem is formulated as a nonlinear
least-squares problem, usually solved using a variant of the classical
Gauss-Newton (GN) method. However, we show that GN may not converge if poorly
initialised. In particular, we show that this may occur when there is greater
uncertainty in the background information compared to the observations, or when
a long time-window is used in 4D-Var allowing more observations. The
difficulties GN encounters may lead to inaccurate initial state conditions for
subsequent forecasts. To overcome this, we apply two convergent GN variants
(line search and regularisation) to the long time-window 4D-Var problem and
investigate the cases where they locate a more accurate estimate compared to GN
within a given budget of computational time and cost. We show that these
methods are able to improve the estimate of the initial state, which may lead
to a more accurate forecast.
