The impact basins of Mars reveal important insights on Martian tectonic evolution. They involve strongly disrupted, depressed regions of crust with likely enhanced porosity and permeability that may locally concentrate water and other crustal fluids. Hence the crustal properties of impact basins can also be important in the hunt for water and related clues for life on Mars. We assess the crustal details of impact basins by separating the Mars Global Surveyor free-air anomalies into terrain-correlated and terrain-decorrelated components. The separation is based on the correlation spectrum between the free-air anomalies and the gravity effects evaluated from the topography mapped by the Mars Orbital Laser Altimeter (MOLA). For topographically visible multiring basins like Isidis, striking circular patterns of alternating terrain-correlated free-air maxima and minima mark the uncompensated components of the central mantle plug and surrounding rings. The first vertical derivatives of these anomalies effectively estimate the basin ring locations and a transient cavity depth-to-diameter ratio of 0.09 that is consistent with the ratio observed for lunar nearside multiring basins. For the Isidis Basin we obtain an excavation depth of roughly 66 km that reflects considerably disrupted crust for concentrating local fluids since the Noachian impact. Furthermore, the related crustal terrain-decorrelated free-air anomalies suggest up to 2 km of high-density basin fill may cap the central basin. Subtle quasi-circular depressions in the relatively featureless MOLA terrain of the northern hemisphere have identified potentially buried impact basins <Frey et al., 2002>. An altimetry depression in Acidalia Planitia and another in Utopia are also associated with ringed patterns of terrain-decorrelated free-air anomalies that may mark the uncompensated mass effects of buried impact basins. The gravity-derived transient excavation depths for these inferred basins are roughly 41 and 20 km, respectively, while the related ring diameters (D) follow the ubiquitous $sqrt{2}$D rule of planetary impact basins. The crust of these buried basins is likely to contain water at higher levels than the crust of the equatorial basins that was substantially dewatered with the development of the great northern basin. Hence analyzing surface fractures, faults, and other crustal discontinuities that can tap the more saturated crust of these inferred buried basins may facilitate the hunt for water on Mars.