We present a new Lagrangian diagnostic for identifying the sources of water vapor for precipitation. Unlike previous studies, the method allows for a quantitative demarcation of evaporative moisture sources. This is achieved by taking into account the temporal sequence of evaporation into and precipitation from an air parcel during transport, as well as information on its proximity to the boundary layer. The moisture source region diagnostic was applied to trace the origin of water vapor for winter precipitation over the Greenland ice sheet for 30 selected months with pronounced positive, negative, and neutral North Atlantic Oscillation (NAO) index, using the European Centre for Medium-Range Weather Forecasts' ERA-40 reanalysis data. The North Atlantic and the Nordic seas proved to be the by far dominant moisture sources for Greenland. The location of the identified moisture sources in the North Atlantic basin strongly varied with the NAO phase. More specifically, the method diagnosed a shift from sources north of Iceland during NAO positive months to a maximum in the southeastern North Atlantic for NAO negative months, qualitatively consistent with changes in the concurrent large-scale mean flow. More long-range moisture transport was identified during the NAO negative phase, leading to the advection of moisture from more southerly locations. Different regions of the Greenland ice sheet experience differing changes in the average moisture source locations; variability was largest in the north and west of Greenland. The strong moisture source variability for Greenland winter precipitation with the NAO found here can have a large impact on the stable isotope composition of Greenland precipitation and hence can be important for the interpretation of stable isotope data from ice cores. In a companion paper, the implications of the present results are further explored in that respect.