A weak constraint variational formulation is used for inverse calculations and parameter estimation in a one-dimensional Ekman model. When parameters in the model are allowed to contain errors, the inverse problem becomes nonlinear even if the model itself is linear. It is shown that a convergent iteration can be defined for the nonlinear system of Euler-Lagrange equations and that improved estimates of the poorly known parameters can be calculated by solving the inverse problem for each of the linear iterates using the representer method. The formulation of the variational problem and the solution methods are illustrated using a simple example. The use of a simple dynamical model makes it possible to give an instructive presentation of the representer method. The method is finally used in an example using real current meter data. It is shown that the weak constraint formulation results in smooth solutions in good agreement with the data all through the water column and that it is superior to the traditional strong constraint inverse estimate.