We investigate the convergence of incremental four-dimensional variational data assimilation (4D-Var) when an approximation to the tangent linear model is used within the inner loop. Using a semi-implicit semi-Lagrangian model of the one-dimensional shallow water equations, we perform data assimilation experiments using an exact tangent linear model and using an inexact linear model (a perturbation forecast model). We find that the two assimilations converge at a similar rate and the analyses are also similar, with the difference between them dependent on the amount of noise in the observations. To understand the numerical results, we present the incremental 4D-Var algorithm as a Gauss–Newton iteration for solving a least-squares problem and consider its fixed points. Copyright © 2005 Royal Meteorological Society.