Qualifying uncertainty of precipitation projections over China: mitigating uncertainty with emergent constraints

Predicting future mean precipitation poses significant challenges due to uncertainties among climate models, complicating water resource management. In this study, we introduce a novel methodology to mitigate uncertainty in future mean precipitation projections over China on a grid-by-grid basis. By constraining precipitation parameters of the Gamma distribution, we establish emergent constraints on parameters, revealing significant correlations between historical and future simulations. Our analysis spans the periods 2040–2069 and 2070–2099 under low-to-moderate and high emission scenarios. We observe reductions in uncertainty across most regions of China, with constrained mean precipitation indicating increases in monsoon regions and decreases in non-monsoon zones relative to raw projections. Notably, the observed 30%–40% increase in mean precipitation for the whole of China underscores the efficacy of our methodology. These observationally constrained results provide valuable insights into current precipitation projections, offering actionable information for water resource planning and climate adaptation strategies amidst future uncertainties.


Introduction
Quantifying the future response of precipitation to global warming in China, the world's most populous country (Piao et al 2010), is crucial for effective water resource management.However, accurate prediction of future precipitation changes based solely on historical temperature and precipitation data is challenging due to various constraints such as location, seasonality, and the magnitude of warming (Shaw et al 2011, Wasko and Sharma 2015, Wang et al 2017).Moreover, climate model projections are inherently uncertain, posing significant challenges in understanding future precipitation changes and formulating adaptation strategies.The uncertainty mainly arises from variations in future scenarios, diverse responses to identical atmospheric forcing due to disparities in model parameters and structure, and internal variability within the climate system (Hawkins and Sutton 2009, Tebaldi and Knutti 2007, Dai et al 2018, Dai and Bloecker 2019).Notably, for mean precipitation over the global terrestrial monsoon area, model uncertainty is the primary contributor, accounting for more than 90% of the uncertainty beyond 2040 (Zhou et al 2020).
One common approach to mitigate uncertainty is to select the optimal ensemble of climate models based on their performance in representing recent observations (Lee and Wang 2014, Chen et al 2023a, 2023b).However, climate models exhibit nonlinear responses to small changes in forcing, indicating that future precipitation changes may not strongly correlate with recent performance (Chen et al 2016, Monerie et al 2017).Therefore, there is an urgent need to explore the application of novel methods to constrain precipitation projection results, thereby reducing the uncertainty associated with precipitation predictions (IPCC 2021).
To reduce the spread of climate change projections and improve the reliability for informing decisionmaking, a methodology known as 'emergent constraints' has been developed.This approach leverages statistical and/or physical relationships between the intermodel spread of an observable variable and the spread of responses to projected changes in future precipitation (Klein and Hall 2015, Hall et al 2019, Brient 2020).Using emergent constraints on precipitation, Thackeray et al (2022) projected a global increase of approximately 32% ± 8% in the occurrence of precipitation extremes by the end of the century under a moderate emissions pathway, based on observed frequencies of precipitation extremes.Zhang et al (2022) reduced the uncertainty of predicted precipitation extremes by 20%-40% in temperate regions through consideration of current precipitation variability.For mean precipitation projections, Chai et al (2022) adjusted future precipitation growth rates for Asia by accounting for historical simulated temperature growth rates.Zhou et al (2023) found that the future global mean precipitation is above the raw projection and reduced the projection uncertainty by 27% based on the observed pattern of the climatological cloud effect.However, there remains a gap in research providing grid-by-grid constraints on future mean precipitation for China.
Recent studies on emergent constraints for climate attribution suggest that anthropogenic greenhouse gas emissions are the primary drivers of rising temperatures and increasing precipitation in China from the 1970s to the end of the 21st century (Zhang et al 2019, Wang et al 2023, Xu et al 2024).This implies that the mean precipitation in the mid-to-late 21st century may be constrained by precipitation patterns from the 1970s to the early 21st century, given consistent driving factors during these periods.
The Gamma distribution, widely adopted as a foundational probability distribution model for precipitation (Groisman et al 1999, New et al 2002), has been shown to be the best fit for modeling monthly precipitation data (Ding 1994).Its parameters, the shape parameter and scale parameter, are determined based on the mean and standard deviation of precipitation data (Choi and Wette 1969).Moreover, precipitation distributions in different regions may exhibit different skewness characteristics, and the gamma distribution makes it possible to adapt to these regional variations through its shape and scale parameters.These parameters find application across various domains, including analysis and prediction of precipitation extremes, as well as the refinement of projected precipitation outcomes in China (Wang et al 2008, Li et al 2019, Zhao et al 2020).Since the mean of monthly precipitation for a given period can be represented by the product of the shape and scale parameters of the Gamma distribution of monthly precipitation during that period, constraining the shape and scale parameters separately can reduce the uncertainty of the monthly precipitation mean.Here, we extend constraints on the future mean precipitation based on the recent past mean precipitation to emerging constraints on the parameters of the future precipitation distribution based on the parameters of the recent past precipitation distribution, further refining the mean value of constrained future precipitation based on constrained precipitation parameters.We show that climate models underestimate monthly mean precipitation in monsoon regions and overestimate it in non-monsoon regions.The constraint reduces the model uncertainty in monthly precipitation projections at most grid points, with particularly notable reductions observed in mountainous areas.

Data and methods
The CN05.1 dataset, covering the period from 1961 to 2022 with a spatial resolution of 0.25°× 0.25° (Xu et al 2009, Wu and Gao 2013, Wu et al 2017), is widely recognized as a pivotal tool for evaluating reanalysis data and climate model outputs over China (Yang et al 2017, Zhu andYang 2020).Given that sufficient constraints by observations can reduce the uncertainty of a climate model's sensitivity (Jackson et al 2008), this dataset was chosen as the observational benchmark to constrain future precipitation distribution parameters.
Model data from 22 CMIP6 models were compiled to constrain the future mean precipitation (table S1).The all-forcing historical simulations from 1985 to 2014 served as the historical data for emergent constraints, while future simulations from 2040 to 2069 and 2070 to 2099, under low-to-moderate (SSP2-4.5)and high (SSP5-8.5)emission scenarios (O'Neill et al 2017), were selected as separate constrained targets in this study.Additionally, 30 years of data from a pre-industrial control (piControl) run of each model were used as the noise for the emergent constraints.To facilitate standardized comparison, we tried various interpolation methods, including bilinear interpolation, nearest neighbor interpolation, cubic interpolation, and spline interpolation, and the results obtained were basically the same.Therefore, all precipitation datasets were interpolated to a unified resolution of 1°× 1°using bilinear interpolation method, the most commonly used method for processing meteorological grid data.
For granular analysis and regional contrasts in precipitation dynamics, China was divided into three distinct sub-regions (figure 2 Raw parameters (shape and scale) of the precipitation distribution for the 22 models were estimated by fitting the Gamma distribution of monthly precipitation over the 30-year period for each model.As shown in figure 1, the observationally constrained ranges of future precipitation distribution parameters were calculated using the hierarchical emergent constraints framework based on historical and observed parameters, assuming a Gaussian distribution of parameters (Bowman et al 2018).Here, z and x are the future parameters and recent parameters of 22 models, respectively, while y denotes the observations.μ z and δ z stand for the expectation and standard deviation of z, respectively, and μ x and δ x denote the expectation and standard deviation of x, respectively.δ n represents the standard deviation of noise.The correlation between z and x is denoted by ρ.We estimate the mean ( | ) E z y and standard deviation ( | ) var z y of the constrained future parameters by the following equations: 3. Results

Emergent constraints on parameters of the Gamma distribution
To establish emergent constraints on future parameters (shape and scale) of the Gamma distribution, it is essential to identify parameters from past periods that exhibit strong correlations with those in the future.We find a statistically significant correlation (r 0.50) between the future parameters (from 2040 to 2069 and from 2070 to 2099) and those from recent decades (from 1985 to 2014) in most areas (figure 2).However, regions such as the Tibetan Plateau and Tarim Basin exhibit fewer grid points conforming to a Gaussian distribution due to the concentration of monthly precipitation values in smaller magnitudes (Li et al 2021, Zhang et al 2021a, 2021b), resulting in shape parameters below 1.Consequently, disparities in monthly precipitation  simulation among different models may result in a significant gap in shape parameters.Despite this, given a strong correlation (r 0.8) between historical and future simulations in this region, we proceed with the assumption of a Gaussian distribution.We apply the hierarchical emergent constraints framework for emergent constraints.
The shape parameter controls the form of the monthly precipitation distribution.When the shape parameter is small, the distribution exhibits a higher peak, indicating a higher frequency of events with smaller monthly precipitation amounts.Conversely, a larger shape parameter results in a smoother distribution, indicating a higher frequency of events with larger monthly precipitation amounts.The scale parameter determines the spread of the monthly precipitation distribution.A larger scale parameter widens the distribution, implying that monthly precipitation data are distributed over a larger range, while a smaller scale parameter narrows the distribution, indicating that monthly precipitation data are concentrated within a smaller range (Wilks 2011).
The variance of the raw simulated shape parameter exceeds 1.8 in regions north of the Tianshan Mountains, along the southern edge of the Tarim Basin, and within the Sichuan Basin (refer to figures 3(a), (d), (g), (j)), indicating a notable discrepancy among models in simulating the peaks of monthly precipitation in these areas.This discrepancy is potentially due to the fact that these topographically complex regions exhibit strong and intricate atmospheric systems, coupled with sparse distributions of gauge stations, leading to more pronounced variance among climate models compared to flatter regions (Zhao and Yatagai 2014, Li et al 2020, Zhang et al 2021aZhang et al , 2021b)).
Additionally, the variance of the raw simulated scale parameter surpasses 1600 in the Himalayas, Hengduan Mountains, and Southeastern Hills under the SSP2-4.5 scenario (figures 4(a), (d)).Similarly, it exceeds 1600 in these regions and the Southeastern Hills under the SSP5-8.5 scenario (figures 4(g), (j)).This indicates that the range of monthly precipitation distributions simulated by different models differ significantly in these areas.Given that these areas are primarily concentrated in regions where CMIP6 significantly overestimate annual precipitation (Zhang et al 2024), it remains a challenge to improve the ability of climate models to simulate precipitation accurately in these regions.
By comparing simulated past parameters with observations from CN05.1, we observe a reduction in the high-value area of variance for the future shape parameter from 1.8 to 1.1 (figures 3(b), (e), (h), (k)), and a corresponding decrease in the high-value area of variance for the future scale parameter from 2000 to 1400 (figures 4(b), (e), (h), (k)).Nationally, the variance of both the shape and scale parameter decreases after constraint, relative to raw variance.The maximum relative reduction in variance for the future shape parameter ranges from 50% to 80% in North China, while the minimum reduction ranges from 0% to 30% in the lower and middle reaches of the Yangtze River (figures 3(c), (f), (i), (l)).Similarly, the maximum relative reduction in variance for the future scale parameter varies from 30% to 50% in Southwest China and central and eastern North China, with the minimum reduction ranging from 0% to 25% in the lower and middle reaches of the Yangtze River (figures 4(c), (f), (i), (l)).

Future projections of constrained precipitation
After implementing the constraint, the variances of both parameters have decreased.Consequently, we utilize the parameters before and after the constraints, based on the gamma distribution, to calculate the respective uncertainties in future mean monthly precipitation.We assess the efficacy of the constraints on precipitation by quantifying the uncertainty (in mm/month), defined as the difference between the precipitation calculated according to the 90th and 10th percentiles of the shape and scale parameters across 22 models, where larger uncertainty implies a broader range of mean precipitation.
Figure 4 illustrates that the region with the largest precipitation uncertainty lies in the eastern Himalayas, Hengduan Mountains, the Sichuan Basin, and the Southeastern Hills.Following the application of constraints, uncertainty in these regions has decreased (figures 5(c), (f), (i), (l)).Notably, in the middle and lower reaches of the Yangtze River, uncertainty has increased by 0%-10%, which can be attributed to the relatively low correlation between historical and future parameters in this region, the constraints rely more on observational data, resulting in low effectiveness for constraining future monthly precipitation in this area.Therefore, in the subsequent analysis of the constrained results for monthly mean precipitation (figure 6), caution should be exercised when interpreting results for the middle and lower reaches of the Yangtze River.
Due to the substantial decrease in the uncertainty of monthly average precipitation (figure 3), our subsequent analysis focuses on the mean and its changes in raw mean precipitation and constrained mean precipitation.As can be seen in figure 4, constrained mean precipitation (mm/month) decreases in the Tarim Basin, Qaidam Basin, and Hengduan Mountains compared with raw mean precipitation, with reductions primarily ranging from 0% to 30%.Conversely, constrained mean precipitation increases in the monsoon region, with values mainly falling within the range of 30%-70%.
For better comparison of the constraints on future monthly mean precipitation in various regions of China, we divided China into three regions (figure 7).Among the three regions, Southeast China experiences the most notable increase in monthly mean precipitation under the two scenarios, with a growth of approximately 40% to 50%.In contrast, Southwest China shows the least noticeable change in monthly precipitation after constraint, ranging from approximately 6% to 10%.This is due to the decrease in modeled precipitation in Xinjiang (figures 6(c), (f), (i), (l)), resulting in a mean increase in monthly precipitation for Southwest China that is lower compared to North China and Southeast China.Meanwhile, North China has an increase ranging from about 37% to 43% (table 1).

Summary and discussion
This study employed observational constraints to reduce the uncertainty in the parameters of the Gamma distribution of monthly mean precipitation for the periods 2040-2069 and 2070-2099, leveraging strong correlations between historical and future parameters in our analysis.Regions with a correlation below 0.7 are mainly concentrated in areas with higher annual precipitation (China's wet region).These regions are influenced by a combination of climatic factors, including the East Asian summer monsoon, oceanic climate, and local topography (Dai et al 2013, Zhou et al 2018).To obtain better constraint effects in this area, it may be beneficial to further improve the accuracy and reliability of simulations by optimizing existing climate models and adding new constraints.
Notably, the variance of both the shape and scale parameters is significantly reduced in mountainous regions with complex terrain, resulting in decreased uncertainty associated with future monthly precipitation distribution in these areas.However, it is worth mentioning that in the middle and lower reaches of the Yangtze River, the effectiveness of the constraints is not as significant as in other areas due to the weaker correlations upon which the constraints rely.
Emergent constraints based on parameters of the Gamma distribution demonstrate good performance in estimating monthly mean precipitation under both scenarios.Utilizing constrained parameters, the uncertainty of future monthly mean precipitation across 22 models was calculated.Except for certain areas in the middle and lower reaches of the Yangtze River, the uncertainty in mean precipitation decreases in all regions, with particularly notable reductions observed in mountainous areas.Compared to the raw mean precipitation, the constrained mean precipitation increases primarily in the monsoon regions, with the increases ranging from 30% to 70%, while decreases in the mean precipitation are mainly located in the non-monsoon zone, with reductions ranging from 0% to 30%.
For regional analysis, we divided China into three regions, with the change in mean precipitation following emergent constraints relative to raw mean precipitation for each region under two scenarios being 40%-50% (Southeast China), 37%-43% (North China), and 6%-10% (Southwest China).Across China as a whole, the change in mean monthly precipitation following emergent constraints during the periods 2040-2069 and 2070-2099 ranges from 30% to 40% relative to the raw values, indicating an underestimation of mean precipitation during the two periods by the multimodel ensemble results.
It is noteworthy that, by applying the constraints, the average future temperature in China is expected to decrease compared to the unconstrained scenario (Chen et al 2023, Dong et al 2024), while the atmospheric precipitable water over China is projected to increase (Zhang et al 2019).These results underscore the intricate nature of the relationship between future precipitation and temperature changes, which cannot be adequately captured by the Clausius-Clapeyron equation.
Findings from this study are expected to enhance the precision of constrained forecasts of future precipitation, thereby advancing our comprehension of the trajectory of future climate change and furnishing policymakers with a robust scientific foundation.However, further research is warranted to refine and expand upon these findings, incorporating additional factors and improving modeling techniques to enhance the accuracy of future climate projections.
(a)) based on climatic attributes and topographical nuances (Zhao et al 2015, Zhang and Zhao 2019) : Southeast China, North China, and Southwest China.

Figure 1 .
Figure 1.Schematic outline of the overall methodology to get the observationally constrained ranges of future monthly precipitation distribution.

Figure 2 .
Figure 2. Correlation coefficients of the shape parameters (left panels) and scale parameters (right panels) of the historical monthly precipitation distributions with that of future scenarios for 22 models over China.Dots indicate data adhering to the Gaussian distribution.Note that all the correlation coefficients are statistically significant at the 95% confidence level.Panel (a) shows the three divisions of China used in this study: Southeast China (SEC), North China (NC), and Southwest China (SWC).

Figure 4 .
Figure 4.As in figure 3, but for scale parameters of variance.

Table 1 .
Raw monthly mean precipitation and constrained monthly mean precipitation in different regions under two scenarios.