Our estimates of the VN volume and water volume are based on the following assumptions: (1) the valley shoulder elevations did not change significantly since their formation; (2) the amount of sediments carried into the valleys from elsewhere (for example, from hillslope by sheet flow) was negligible. In addition, the post-formation infill by other processes such as eolian process or mass wasting were deemed minimum and not considered. Thus, the estimated water volume is the minimum cumulative volume of water required to carve the VNs on Mars globally. Yet, this minimum volume of water is larger than the volume of the hypothesized northern ocean (ranging from 156 to 548 m GEL10,14), which suggests that the water must have cycled through the VN system many times (that is, implying an active hydrologic cycle) and the early climate was likely to be wetter than suggested by some previous studies19,24. We realize that not all VNs on Mars drain to the northern ocean. If we only include those VNs that drain to the northern lowlands (based on current topography), our estimate of cumulative water needed to carve them would be 3.32 km GEL (based on the topographically derived VNs), still larger than previous cumulative water volume estimates and the volume of hypothesized ocean, suggesting an active hydrologic cycle. As on Earth, the great amount of water recycling needed to carve VNs would probably require a large open water body (ocean) on Mars contemporaneous with the VN formation, and a warm and wet climate to support the active hydrologic cycle. Without an ocean-sized open body of water, it would be hard to imagine the high rate of water cycling suggested by our new estimates. Given the large inventory of water from the perspective of global VNs, and that the age of most of the Martian VNs is more than three billion years old (carved into the ancient Noachian highlands), the active hydrologic cycle, the warm and wet climate, and the existence of an ocean must have happened early in Martian history.

There is no ground truth to assess the real accuracy of our estimation. However, we can establish the confidence of our estimates by comparing them with those of previous studies at the same locations using similar data, for example, Hoke et al.28 and Matsubara et al.32 (Table 2). As shown in Table 2, our estimates are generally consistent with these previous studies to within one order of magnitude with the ratio of our estimate to theirs ranging from 0.15 to 1.9. The differences are due to different methodologies used in calculating the volume and how the valley areas were defined. Matsubara et al.32 used the 75th percentile elevation within a search radius along the valley as the shoulder elevation and minimum elevation within that search window as the valley bottom elevation to estimate the eroded volume. Their estimates are larger than ours, as their method generally results in valley areas larger than ours. If we convert our VN volume estimate of the Evros and Samara, Parana and Loire Valles (first and last rows of Table 2) into volume of sediments, our numbers would be 2.28 × 1012 and 9.79 × 1012 m3, respectively, much closer to their estimates. Hoke et al.28 estimated the VN volumes based on manually drawn boundary along the outer walls of the visible valleys and measurements of width and length and estimates of depth. The manually drawn VN boundary is likely different from that derived from PBTH method and may be the primary reason for the differences in volume estimates. Applying the Hoke et al.28 method to the same simulated landscape with the same VN boundary29 resulted in a relative accuracy of ∼85% (see Table 3, calculation courtesy of Brian Hynek and William Nelson, University of Colorado, Boulder, personal communication). They have recently improved their methodology by using the Reimman sums and the Simpson rule. Again, using the same simulated landscape and same VN boundary, the improved method resulted in a slightly better relative accuracy than ours (by about 2.7%, see Table 3, calculation courtesy of Brian Hynek and William Nelson, personal communication). However, their new method requires considerably more human intervention and the user still has to provide a valley area boundary, which BPTH method can automatically generate. The small gain in relative accuracy would not make any significant difference in global estimate of the VN volumes.

Table 2 Comparison of VN volume estimates with previous studies. Full size table

Table 3 Comparison of volume estimates based on the same simulated landscape29. Full size table

Thus, we are confident about PBTH method as a robust, accurate and efficient method for estimating the global VN volume. The PBTH method objectively and consistently delineates the VN boundaries and estimates valley depth at pixel level. Because of the automated procedure, we have included all the VNs mapped at the global scale to date. Our results provided an independent source of estimate of global water inventory on Mars. Our result is consistent with a warm and wet early Mars climate and the existence of an ancient northern ocean. If erosion of the VNs required significant chemical or physical weathering to produce transportable sediment, fluvial abrasion of channel beds33 or transport of appreciable quantities of gravel34, the required volume of water may have been many times our conservative estimate.

Existing climate models have not been able to reproduce an early Mars climate sufficient to promote an active hydrological cycle35,36,37 (see also a recent review by Wordworth38 and references therein); this has led to hypotheses suggesting accumulation of thick cold-based ice on the equatorial highlands with VN formation during short-lived episodes of top–down melting39. If true, this argues against an unfrozen ocean. However, an equivalent amount of erosion and equivalent total runoff would still need to be accounted, which may be challenging to achieve under cold climate scenario36. The gap between the geomorphic evidence such as this study, which suggests a warm and wet climate, and the climate models that struggle to get temperature high enough for early Mars36 still requires further study.