Fig. 1 Spatial distribution of the relative change in summer precipitation over the Northwestern China predicted by CMIP6 models for the end of 21st century (2081-2100) compared to the historical period (1995-2014). (The data on the map represent the regional average changes, the bold black solid line in the figure represents the zero line of relative change in precipitation)

Fig. 2 The first empirical orthogonal function (EOF) of the inter-model decomposition of relative changes in summer precipitation over Northwestern China. (a) The first mode, (b) the corresponding contribution sequences of each model and the regional average sequences of each model

Fig. 3 Regression distribution of summer average precipitation over Northwestern China with contemporaneous sea surface temperatures in the tropical Indian Ocean (a) and the 700 hPa wind field (b), respectively. (The coloring area represent the 95% confidence level, the boxed area in (a) indicates the key factor region. The bold black lines represent areas passing the 95% confidence test, the boxed area in (b) indicates Northwestern China)

Fig. 4 Regression of summer average precipitation over Northwestern China with the East Asian subtropical 200 hPa zonal wind (a), and the zonal wind and meridional circulation along the Northwestern China region (74°-105°E) (b), respectively. (The dots represent areas passing the 95% confidence test, the boxed area indicates the key factor region. The red shading indicates west wind anomalies, while the blue shading indicates east wind anomalies)

Fig. 5 Scatter plot of the tropical Indian Ocean sea surface temperature (a) and the East Asian subtropical 200 hPa zonal wind (b) simulated by 25 CMIP6 models and the estimated changes in summer average precipitation over Northwestern China. (The red line represents the observed region mean sea surface temperature over the tropical Indian Ocean)

Fig. 6 Scatter plot of the summer average precipitation over Northwestern China, constrained by the tropical Indian Ocean sea surface temperature (a) and East Asian subtropical 200 hPa zonal wind (b), showing the estimates after constraints versus unconstrained estimates

Fig. 7 Optimal models obtained from the Pareto-optimal ensemble schemes. (a) The Pareto-optimal ensemble of summer precipitation in the Northwest region and tropical Indian Ocean sea surface temperature, (b) the Pareto-optimal ensemble of summer precipitation in the Northwest region and East Asian subtropical 200 hPa zonal wind, (c) the Pareto-optimal ensemble based on three variables, namely, summer precipitation in the Northwest region, tropical Indian Ocean sea surface temperature, and East Asian subtropical 200 hPa zonal wind. (Each point in the graph represents a model, with filled legends indicating the optimal models)

Fig. 8 Boxplots of future summer precipitation estimates for the Northwest region at the end of the 21st century based on different schemes. (The upper and lower boundaries of the box represent the 75th and 25th percentiles of the estimated results, while the upper and lower limits of the box represent the 90th and 10th percentiles, the black line inside the box represents the median)

Fig. 9 Relative change in summer precipitation in the Northwest region at the end of the 21st century (2081-2100) compared to the historical period (1995-2014) based on the three-variable Pareto optimal ensemble scheme (a), along with the difference from the unconstrained estimate (b)