Forecast models often depend on unknown parameters, such as model initial and boundary conditions, or other tunable parameters not necessarily having any physical meaning. Calibration of these parameters to minimize errors between forecasted and observed states is called data assimilation. A common approach in this context are variational methods, of which four dimensional data variation (4D-VAR) is studied in this thesis. In 4D-VAR, a cost function is defined that penalizes misfits between observations and the corresponding numerical model results, obtained by running the model with the chosen configuration. Performing optimization with regard to this cost function yields an improved initial parameter set. Associated with this type of methods, however, are difficulties in connection with programming the adjoint model, which is needed to compute the exact gradient of the cost function. Additionally, having to integrate the adjoint model backwards in time adds significantly to the computational cost of the data assimilation process. To avoid manual implementation of adjoint code and to reduce computational complexity, approximation of the gradient calculation is considered through the use of proper orthogonal decomposition (POD), a flexible data-driven order reduction method. To facilitate this, a finite element model of the shallow water equations is tested with both the full adjoint 4D-VAR method and two different POD-reduced approaches. Twin experiments are performed and comparisons are made in terms of accuracy, computational complexity and sensitivity to perturbation and number of observation points.