Estimating unobserved quantities such as velocities and transport indirectly from temperature and salinity measurements is one of the best studied problems in physical oceanography. It is ill posed. The ill posedness is traditionally removed by adding prior information such as a level of no motion or a set of conservation principles. We propose to use the deep pressure gradient taken from a prognostic integration of an ocean general circulation model in primitive equations as a weak constraint when using the stationary inverse of the same model to derive transports. First results for the western boundary currents in the North Atlantic compare well with previous estimates from several studies including direct measurements. Copyright 2006 by the American Geophysical Union.