Abstract: Wavelet analysis is a useful tool for analyzing multi-time scale and heteroscedastic time series. In this paper，the multi-time scale cross-wavelet transformation is used to study runoff and climate factors as well as their relationships in the upstream of the Heihe River. The runoff time series（AAR）is constituted of annual average runoff at Yingluoxia Station from 1944 to 2010，and the climate factors consist of: annual index of Arctic Oscillation （AOI）from 1950-2010，annual average temperature（AAT）at Yeniugou Station（1959-2010）and Qilian Station （1957-2010），annual precipitation （AP）at Yeniugou Station（1959-2010）and Qilian Station（1957-2010）. Firstly，continuous wavelet transformation was used to analyze runoff（AAR）and meteorological time series（i.e. AOI，AAT and AP），and then we used cross-wavelet transformation and wavelet coherence to study relationships between AAR and meteorological time series（i.e. AOI，AAT and AP）respectively. Besides，Monte Carlo methods were used to assess the statistical significance against red noise backgrounds because the addition of statistical significance tests will improve the quantitative nature of wavelet analysis. Results of continuous wavelet transformation show that AOI has significant 3-5 a periods，AAT has a significant 3 a period，AP has relatively significant 3 a and 4-6 a periods，and AAR has three relatively significant periods which are 3 a，2.5-4 a and 5 a. Besides，the fact that the high-energy zone of AAR covers most of high-energy areas of AAT and AP indicates that AAR variability in the upstream of the Heihe River has a positive response to variability of AAT and AP，and the runoff increase is mainly affected by warm and humid climate. Moreover，results of cross wavelet power spectrum and wavelet coherence show that AAR is inversely related to AOI with a 3-a resonant period，and AAR is almost negatively related to AAT with 3-4 a resonant period; the fact that AAR has significant 2-7 a resonant periods with AP demonstrates that the precipitation has a great influence on runoff and it is the main supply of the runoff; Influenced by AOI，precipitation and temperature individually，the runoff exhibits a 3-a periodic variation of high flow and low flow in the mutations year 1987，1986，1974 and 1996，respectively. In addition，the results imply that precipitation and temperature are the dominant factors that influence the variation of runoff. The cross wavelet analysis is capable to reveal the correlation between the variation of hydrological element（runoff）and the variability of meteorological elements（temperature and precipitation）in the upstream of the Heihe River. Forecasts of the future evolutions of water resources in the upstream of the Heihe River is feasible based on the results. Moreover，it is hoped that the analysis presented here will be proved useful in studies of nonstationary time series.

Key words: cross wavelet analysis, climate change, runoff variability, nonstationary time series, the Heihe River