The depth extent, density, and dynamical role of apparent subcontinental keels are investigated by using constraints provided by the very long-wavelength representations of several geophysical fields. We first consider local cross correlations between the nonhydrostatic free-air gravity field, the map of the ice sheets at Last Glacial Maximum, and the surface expression of continental cratons. We initially confine our analyses to North America and observe that equally good local correlations exist between the gravity field and either of the others. The case of Eurasia is also considered. We observe that correlation analyses of this type cannot be employed to unambiguously infer the cause of long-wavelength continental gravity anomalies; therefore we revert to explicit postglacial rebound and tomography-based viscous flow modeling of the gravity field. For viscosity profiles that optimally reconcile relative sea level constraints from the Laurentide platform, we find that the rebound process accounts for only 10% of the observed free-air gravity low over Hudson Bay. We consider mantle convection as the more likely source of this gravity anomaly and alternatively investigate the implications of assuming that seismically fast, deep structure imaged tomographically beneath the continent represents either negatively, neutrally, or positively buoyant material. In addition to the gravity constraint we introduce the independent constraint of continental dynamic surface topography. We infer this new datum by using the Crust 5.1 global model of crustal structure. Remarkably, continents are found to systematically reside in topographic depressions of the order of 1--2 km. Within the context of our modeling assumptions we find that optimal model descriptions of the joint gravity and dynamic surface topography constraints over the continents require deep and dense subcontinental undercurrents. ¿ 2000 American Geophysical Union