We present a new method for estimating the three-dimensional field of geostrophic velocity from hydrographic station data. Very simply, we ask for the smoothest velocity field (in the sense of an arbitrarily defined norm) which is consistent with the data and with selected approximate dynamical constraints to within prescribed misfits, which, we will argue, should never be zero. The misfits represent errors in the data and in the approximate dynamical constraints. By varying the misfits relatively to one another, we explore the full envelope of physically plausible estimates of the average geostrophic flow. We illustrate the method by application to hydrographic measurements in the Labrador Sea.