Estimating unobserved quantities such as velocitiesand transports indirectly from temperature and salinity measurements is oneof the best studied problems in physical oceanography. It is illposed. The ill posedness is traditionally removed by adding priorinformation such as a level of no motion or a set of conservationprinciples. We propose to use the deep pressure gradient takenfrom a prognostic integration of an ocean general circulation modelin primitive equations as a weak constraint when using the stationaryinverse of the same model to derive transports. First results forthe western boundary currents in the North Atlantic compare well withprevious estimates from several studies including direct measurements.