Fitting oceanographic general circulation models to data is a difficult computational undertaking. This paper addresses difficulties stemming from the rigid lid approximation. Adjusting the barotropic stream function to achieve the fit, in effect, requires the solution of a hidden elliptical equation, the resolution-dependent difficulty of the elliptical problem manifest in the optimization problem. Adjusting the independent (subsurface) baroclinic velocity variables also contributes to the ill-conditioning of the least squares computation, but Lagrange multipliers can be used to adjust all baroclinic velocity variables (surface and subsurface) symmetrically. Similarly, Lagrange multipliers can be used to adjust the total velocity variables, thereby circumventing the use of barotropic stream function and baroclinic velocities; determining the multipliers at each optimization iteraction requires the solution of an elliptical partial differential equation, the solution of which can be thought of as a preconditioning transformation. ¿ American Geophysical Union 1994