for the very large nonlinear systems that arise in meteorology and oceanography, the available observations are not sufficient to initiate a numerical forecasting model. data assimilation is a technique for combining the measured observations with the model predictions in order to generate accurate estimates of the expected system states - both current and future. four-dimensional variational assimilation techniques (4d-var) are attractive because they deliver the best statistically linear unbiased estimate of the model solution given the available observations and their error covariances. the optimal estimates minimize an objective function that measures the mismatch between the model predictions and the observed system states, weighted by the inverse of the covariance matrices. the model equations are treated as strong constraints. \vskip .1in \noindent gradient methods are used, typically, to solve the large-scale constrained optimization problem. currently popular is the ``incremental 4d-var" procedure, in which a sequence of linearly constrained, convex, quadratic cost functions are minimized. we show here that this procedure approximates a gauss-newton method for treating nonlinear least squares problems. we review the known convergence theory for this method and then investigate the effects of approximations on the convergence of the procedure. specifically we consider the effects of truncating the inner iterations and of using approximate linear constraints in the inner loop. to illustrate the behaviour of the method, we apply incremental 4d-var schemes to a discrete numerical model of the one-dimensional nonlinear shallow water equations.