Analysis of erosional valleys, geologic materials and features, and topography through time in the Thaumasia region of Mars using co-registered digital spatial data sets reveals significant associations that relate to valley origin. Valleys tend to originate (1) on Noachian to Early Hesperian (stages 1 and 2) large volcanoes, (2) within 50--100 km of stages 1 and 2 rift systems, and (3) within 100 km of Noachian (stage 1) impact craters >50 km in diameter. These geologic preferences explain observations of higher valley-source densities (VSDs) in areas of higher elevations and regional slopes (>1¿) because the volcanoes, rifts, and craters form high, steep topography or occur in terrain of high relief. Other stage 1 and stage 2 high, steep terrains, however, do not show high VSDs. The tendency for valleys to concentrate near geologic features and the overall low drainage densities in Thaumasia compared to terrestrial surfaces rule out widespread precipitation as a major factor in valley formation (as is proposed in warm, wet climate scenarios) except perhaps during the Early Noachian, for which much of the geologic record has been obliterated. Instead, volcanoes and rifts may indicate the presence of shallow crustal intrusions that could lead to local hydrothermal circulation, melting of ground ice and snow, and groundwater sapping. However, impact-crater melt would provide a heat source at the surface that might drive away water, forming valleys in the process. Post-stage 1 craters mostly have low nearby VSDs, which, for valleys incised in older rocks, suggests burial by ejecta and, for younger valleys, may indicate desiccation of near-surface water and deepening of the cryosphere. Later Hesperian and Amazonian (stages 3 and 4) valleys originate within 100--200 km of three young, large impact craters and near rifts systems at Warrego Valles and the southern part of Coprates rise. These valleys likely developed when the cryosphere was a couple kilometers or more thick, inhibiting valley development by hydrothermal circulation. However, eruption of groundwater may have occurred from impact-induced fracturing and lateral and perhaps minor upward transport of water due to seismic pumping. The two smaller craters formed along the plateau margin where the highest potential hydraulic head would occur in aquifers beneath the plateau. In the case of the larger crater (Lowell, 200 km in diameter), potential aquifers would likely be at depths of kilometers below the cryosphere. Seismic energy generated by the Lowell impactor would have been much greater, pumping both groundwater and perhaps fluidized slurry to the surface from beneath the cryosphere to form the young valleys and flow deposit. Along the margin of Thaumasia, tectonic pressurization of groundwater also may have contributed to valley formation. Dissection of rim materials of the Argyre impact may relate to tectonic activity and the unconsolidated state of basin ejecta.