We compare estimates of atmospheric precipitation during the Martian Noachian-Hesperian boundary 3.8 Gyr ago as calculated in a radiative-convective column model of the atmosphere with runoff values estimated from a geomorphological analysis of dendritic valley network discharge rates. In the atmospheric model, we assume CO2-H2O-N2 atmospheres with surface pressures varying from 20 mb to 3 bar with input solar luminosity reduced to 75% the modern value. Results from the valley network analysis are of the order of a few mm d-1 liquid water precipitation (1.5-10.6 mm d-1, with a median of 3.1 mm d-1). Atmospheric model results are much lower, from about 0.001-1 mm d-1 of snowfall (depending on CO2 partial pressure). Hence, the atmospheric model predicts a significantly lower amount of precipitated water than estimated from the geomorphological analysis. Furthermore, global mean surface temperatures are below freezing, i.e. runoff is most likely not directly linked to precipitation. Therefore, our results strongly favor a cold early Mars with episodic snowmelt as a source for runoff. Our approach is challenged by mostly unconstrained parameters, e.g. greenhouse gas abundance, global meteorology (for example, clouds) and planetary parameters such as obliquity- which affect the atmospheric result - as as well as by inherent problems in estimating discharge and runoff on ancient Mars, such as a lack of knowledge on infiltration and evaporation rates and on flooding timescales, which affect the geomorphological data. Nevertheless, our work represents a first step in combining and interpreting quantitative tools applied in early Mars atmospheric and geomorphological studies.Read less <