Uncertainty reduction for precipitation prediction in North America

Large differences in projected future annual precipitation increases in North America exists across 27 CMIP6 models under four emission scenarios. These differences partly arise from weak representations of land-atmosphere interactions. Here we demonstrate an emergent constraint relationship between annual growth rates of future precipitation and growth rates of historical temperature. The original CMIP6 projections show 0.49% (SSP126), 0.98% (SSP245), 1.45% (SSP370) and 1.92% (SSP585) increases in precipitation per decade. Combining observed warming trends, the constrained results show that the best estimates of future precipitation increases are more likely to reach 0.40–0.48%, 0.83–0.93%, 1.29–1.45% and 1.70–1.87% respectively, implying an overestimated future precipitation increases across North America. The constrained results also are narrow the corresponding uncertainties (standard deviations) by 13.8–31.1%. The overestimated precipitation growth rates also reveal an overvalued annual growth rates in temperature (6.0–13.2% or 0.12–0.37°C) and in total evaporation (4.8–14.5%) by the original models’ predictions. These findings highlight the important role of temperature for accurate climate predictions, which is important as temperature from current climate models’ simulations often still have systematic errors.


Introduction
The climate of North America varies due to changes in latitude, and a range of geographic features (including mountains and deserts), ranging from the frost-free tropical of southernmost Florida to the perennial ice and snow of the northernmost islands of the Greenland.Generally, on the mainland, the climate of becomes warmer the further south one travels, and drier the further west, until one reaches the West Coast.The annual mean precipitation, temperature and evapotranspiration are 562 mm, -3.2˚C and 342 mm, respectively.The current population of Northern America is around 0.38 billion.However, North America has experienced severe droughts in recent decades [1,2], which has negatively influenced agriculture, energy production, food security, forestry, drinking water, and tourism [3][4][5].Typically, the main drivers of drought are below-average precipitation and/or above-average temperature and evaporation [6].Thereby, accurate predictions of future precipitation, temperature and total evaporation are crucial for mitigating the climate-driven drought risks [4].
Earth System models (ESMs) are widely used to investigate past climate variations and future climate predictions in response to various radiative forcings [7].Despite consideration of physical, chemical, and biological processes, ESMs often poorly predict the most basic quantities, such precipitation, temperature, and evaporation, as seen in a series of publications of the Fifth Phase of the Coupled Model Intercomparison Project (CMIP5) [8][9][10].For example, the spread of future increases in global mean surface temperatures during 2081-2100 across CMIP5 models is large, ranging from 0.3˚C to 1.7˚C (RCP2.6),1.1˚C to 2.6˚C (RCP4.5),1.4˚C to 3.1˚C (RCP6.0)and 2.6˚C to 4.8˚C (RCP8.5)[11].The future global mean precipitation feedbacks with temperature also exhibits substantial uncertainties (0.5-4% ˚C-1 ) [11].The new generation of ESMs (CMIP6) that has higher horizontal-vertical resolutions and more comprehensive experimental designs [12], still yields considerable uncertainties in predicting the basic climate variables [13][14][15].This inaccurate information makes planning climate mitigation and adaptation measures more challenging [16].
In recent years, the emergent constraint technique that is based on the significant statistical relationship between simulated changes of historical climate variable X and predicted changes of future climate variable Y across an ESM ensemble has emerged [17][18][19][20][21][22] (See methods).This empirical relationship, combined with observed changes of climate variable X, has successfully reduced uncertainties in predicted changes of future climate variable Y (e.g., permafrost melt [23,24], marine primary production [25], Arctic sea-ice albedo feedback [26] and precipitation extremes [27]).
The key of the emergent constraint technique is to explore the mechanisms that underpin the emergent constraint relationship [16].Thereby, we first explore the main driving factor (i.e.variable x) which dominates the large spread of predicted future annual precipitation growth rates (i.e.variable y) across the 27 CMIP6 models.The identified factor then is used to build the emergent constraint relationship with future annual precipitation growth rates across the CMIP6 models under scenarios SSP126, SSP245, SSP370 and SSP585.SSP126 represents the sustainable and "green" pathway with minimizing material resource and energy usage.SSP245 is the medium pathway that the world follows a path in which social, economic, and technological trends do not shift markedly from historical patterns.SSP370 has the high GHG emissions that a low international priority for addressing environmental concerns leads to strong environmental degradation in some regions.SSP585 has a very high GHG emissions (CO 2 emissions triple by 2075) that inequality is rising.Some regions suffer drastic environmental damage.By combing the observations, we aim to reduce the uncertainty of future annual precipitation in North America during 2015-2100.To verify the robustness of the emergent constraint relationships from CMIP6, 33 CMIP5 models under RCP45 and RCP85 are also used for cross-checking of the constraint relationship on a new model ensemble, using exactly the same constraint processes.Finally, the constrained future precipitation changes, with smaller uncertainties relative to the raw model predictions, are used to re-estimate growth rates of future temperature and evapotranspiration in North America.

Emergent constraint technique
In this study, we built the emergent constraint relationships between annual growth rates of simulated historical temperature and that of future precipitation across the CMIP6 and CMIP5 models (S1 and S2 Tables), by using the least-squares linear regression method (Eq 1).The prediction error of the regression (σ y ) is estimated by Eq 2, following the method from Cox et al [21] and Chai et al [22].Combining the observed temperature from four different data sets, the emergent relationship provides a tight constraint on future annual precipitation growth rates.
where y i (future predicted climate variable y, i.e., future annual precipitation growth rates) is the value given by x i (historical simulated climate variable x, i.e., simulated historical annual temperature growth rates); a and b are the slope and intercept values, respectively; ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where s is used for minimizing the least-squares error, calculating by Eq 3; N is the number of models.σ x is the variance of x i Eq 4; � x is the mean value; ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi

Calculation of probability density function (PDF)
For checking how significant the changes of future annual precipitation growth rates before and after applying the emergent constraint technique, we estimated the PDFs of the future annual precipitation growth rates before applying the technique (Eq 5).After the constraint, the PDFs for the constrained future annual precipitation growth rates (PDF(F)) is calculated by numerically integrating PDF(F/ob) and PDF(ob) (Eq 6).
where PDF(y/x) is the PDF around the best-fit linear regression, representing the PDF of y given x.
where PDF(F/ob) is the probability density for the "future climate projected variable" given the "historical observable variable"; and PDF(ob) is the observation-based PDF for "historical observable variable".
Where q s and T are the saturation specific humidity and the temperature, respectively.L v and R v are the latent heat of condensation at temperature T and the gas constant for water vapour (461.5 J kg −1 K −1 ), respectively.Here we assumed that Lv = 2.5 10 6 J kg −1 and the total pressure is much larger than the water vapour pressure.

Thermodynamic equations indicating a linear relationship between precipitation and temperature
Held and Soden [29] proposed a thermodynamic scaling where precipitation (Pre) can be approximated as a product of convective mass flux (M f ) and specific humidity (q), near the global land surface (Eq 9).Thus, combining Eqs 8 and 9, we can further obtain the Eq 10 [28].Under a unchanged atmospheric circulation (ΔM f = 0), Eq 10 shows a linear positive relationship between precipitation and temperature, i.e. per increase in temperature will lead to 7% increase in precipitation.

Global land hydrological budget equation
The changes in precipitation (ΔP) are normally accompanied by coinciding changes in land surface runoff (ΔR), total evaporation (ΔET), soil water storage (ΔSW).These closely connected processes make up the land water cycle, which is described by Eq 11 [30][31][32].
Ɛ is the other minor components in land water cycle (e.g., snow melting and human water uses).

Multiple regression technique
To estimate the relative contribution of each potential driving factor (i.e., temperature, land surface runoff, soil water content and total evaporation) on the future precipitation changes, we used the multiple regression method that has been widely applied by previous studies [33][34][35][36].After building the multiple regression relationships (Eq 12) between future precipitation changes and the physically relevant observed driving factors, we can obtain the regression coefficient of each driving factor (Eq 12).Then, the regression coefficients are used to estimate the standardized coefficient of each potential driving factor (Eq 13).
Where P, R, ET, SW and T are the precipitation, land surface runoff, total evaporation, soil water content and temperature, respectively.Ɛ is the residual error term.β 1 , β 2, β 3 and β 4 are the regression coefficient of each driving factor.Stc_R is the standardized coefficient of runoff (Eq 13).Similarly, we can obtain the standardized coefficients of total evaporation (Stc_ET), soil water content (Stc_SW) and temperature (Stc_T).The higher the value of standardized coefficient of a driving factor it is, the more large the effects of this factor will be [36].Thereby, we used the Eq 14 to estimate the relative contribution of runoff (C_R) on the future precipitation changes.Similarly, we can obtain the contributions of total evaporation (C_ET), soil water content (C_SW) and temperature (C_T).

Definition and calculation of annual extreme light rain days
Extreme light rainfall days here are defined as the days with rainfall (including days without rainfall) lower than the long-term 10th percentile.Based on the outputs of the daily precipitation during 2015-2100 from 12 CMIP6 models, we estimated the annual extreme light rainfall days in each grid.The mean value of the annual extreme light rainfall days in all terrestrial grids is regarded as the average number of annual drought days in North America.

Mechanisms of emergent constraint technique
The basic idea of emergent constraint technique is to identify a variable x of the observable climate that both varies significantly across an ESM ensemble and that exhibits a statistically significant relationship with variations in some relevant variable y (i.e., Future annual precipitation growth rates in this study) describing the ESM's future simulated climate state [37] (See schematic diagram in S1 Fig) .The observed variable x normally has smaller uncertainties compared to the range of simulated values (S1 Fig) [16].Thereby, by projecting the observed estimate of the variable x with its uncertainty onto the y-axis using an empirical linear relationship, we can obtain a more reliable and accurate variable y with smaller difference across models (See details in the caption of S1 Fig).
In North America, although the 27 CMIP6 models and the 33 CMIP5 models all indicate a trend of increasing precipitation during 2015-2100 (Figs 1a, 1b and S2), the predicted trend values have large differences across the models.We estimated that the future annual precipitation growth rates (i.e., future climate variable y in S1 Fig) from CMIP6 models are predicted to reach 0.3538 ± 0.1676 mm.year -1 (SSP126), 0.7043 ± 0.2182 mm.year -1 (SSP245), 1.0752 ± 0.3470 mm.year -1 (SSP370), and 1.4364 ± 0.4165 mm.year -1 (SSP585).Accordingly, the ranges across the CMIP5 models are also large (0.5248 ± 0.1825 mm.year -1 under RCP45 and 1.0139 ± 0.2393 mm.year -1 under RCP85).Before using the emergent constraint technique to reduce the uncertainties, we need to explore the dominant factor (i.e., Climate veriable x in S1 Fig) that drives the large inter-model spread of future precipitation changes.
The Clausius-Clapeyron equation (C-C, Eqs 7 and 8) indicates that a warming climate can have an increase of about 7% K -1 in the atmospheric water-holding capacity.Thereby, increases in the water vapor holding capacity of the atmosphere due to global warming largely attributed to the future increases in precipitation [39,40].Assuming an unchanged atmospheric circulation, the thermodynamic equations (Eqs 9 and 10) from Kjellsson [28] also present a similar linear relationship between precipitation and temperature.That is, a degrees K increase in temperature will lead to 7% increase in precipitation [28].Certainly, by considering radiative balance, positive sensitivity of precipitation to temperature will be also constrained by radiative cooling [41], and thus the increase rates in precipitation will be weakened by 4-6% K -1 [41,42], but generally precipitation still present a positive feedback behavior to temperature (1-3% K -1 ).
Supporting by this mechanism, the warming temperature in North America is identified as an important driving factor on the local precipitation increases, which is also verified by tight observational relationships (r = 0.38, p-value <0.001) between temperature and precipitation anomalies during 1901-2014 (Figs 1c and S3).The predictions from CMIP6 models also indicate a positive precipitation feedback to temperature changes under all the four emission scenarios (Fig 1d , r�0.92, p-value <0.001), with the strong correlations (p-value <0.001) over most of North America's land surface (covering 68.7-83.6% of the totals), which also confirms a strong constraint of temperature on local precipitation change.
However, the North American climate is also affected by other climate factors (e.g., monsoon, El Niño Southern Oscillation and the Arctic Oscillation) that can lead to extreme temperatures and precipitation [43][44][45].These abnormal values can non-linearly affect trends in the climate of North America and possible cause us to obtain spurious precipitation-temperature relations.To exclude the interference of extreme climates, we smooth out extreme fluctuations using a series of moving windows with different lengths (5-10 years).This improves the reliability of the identified positive feedback behavior of precipitation to temperature [19,22] and indicates even tighter positive relationships between precipitation and temperature both in observations and simulations with the correlation coefficient increasing from 0.38 to 0.6-0.65 (S4 Fig) .Furthermore, positive feedback behavior has been enhanced due to the increased sensitivity of precipitation to temperature (i.e., slope values in S4 Fig) .This indicates that the dominant role of temperature in driving the local precipitation changes is robust, and becomes even more distinct after accounting for the disturbance other climate conditions.
The land water balance (Eq 11) indicates that precipitation changes may also be affected by other potential factors such as changes in total evaporation, soil moisture and land surface runoff [46][47][48].Using a multiple regression (See Methods), we estimated the relative contributions of these drivers on the future precipitation changes.The results still indicate a primary role of temperature in affecting future precipitation changes persists under all the four SSPs, with the contributions of 47.6-80.8%(Figs 1f and S5, depends on the SSPs).By contrast, total evaporation (12.1-28.3%),soil moisture (3.4-20.8%)and land surface runoff (3.3-16.4%)all have much smaller contributions.
The linear relationships between annual growths in precipitation and in temperature (Figs 1e, S6 and S7, each circle represents a CMIP6/CMIP5 model) is tight for all historical periods or all future period under all the emission scenarios (0.60�r�0.88, p-value<0.001).These positive relationships indicate that the precipitation increases proportionally to the warming trends in North America.In other words, a warmer model (higher temperature) tens to exhibit a stronger precipitation increase, while a model with less warming tends to has a less strong precipitation increases.Thus, as long as the temperature increases, a stronger feedback implies more precipitation in both the past and the future.Thereby, due to the key role of warming on precipitation changes in North America, simulated historical annual temperature growth rate, we consider it the variable x for building the emergent constraint relationship with future annual precipitation growth rates (S1 Fig).

Emergent constraint on future annual precipitation growth rates
During 1970-2014, the multi-model CMIP6 ensemble mean estimate of the annual precipitation growth rate is 0.58 ± 0.042 mm.year -1 per year.Based on the four observational datasets of HadCRUT4, NOAA, GISS and GHCN, the observed precipitation growth rate is 0.11 ± 0.17 mm.year -1 per year.We conclude that the CMIP6 models largely overestimate the historical precipitation increase trend in North America.
By building linear relationships between annual growth of simulated historical temperature and that of future precipitation across the 27 CMIP6 models (Fig 2, p value<0.001), we found the strong correlations between them under all emission scenarios (i.e., SSP126, SSP245, SSP370 and SSP585).After cross-checking using the 33 CMIP5 models' simulations, the tight constraint relationships still hold under both the emission scenarios of RCP45 and RCP85 (S8   (c), and (d) are the emergent constraint relationships between the simulated historical annual temperature growth rates and the predicted future annual precipitation growth rates across the CMIP6 models under the emission scenarios of SSP126, SSP245, SSP370, and SSP585, respectively.Green shading is the 90% prediction error of the linear regression.Each dot represents a model.Four vertical shadings are the observed annual temperature growth rates (Mean ± one standard deviation) from HadCRUT4 (0.0357 ± 0.0050 ˚C year -1 ), NOAA (0.0346 ± 0.0048 ˚C year -1 ), GISS (0.0394 ± 0.0056 ˚C year -1 ) and GHCN (0.0365 ± 0.0050 ˚C year -1 ) data sets, respectively.Black curves and colorized curves are the PDFs of the future annual precipitation growth rates before and after the emergent constraint, respectively.Gray histograms and colorized histograms in the right panels of (SSP126), 0.7043 mm.year -1 (SSP245), 1.0752 mm.year -1 (SSP370), and 1.4364 mm.year -1 (SSP585).The constrained results indicate that the best estimates are more likely to reach 0.2903-0.3455mm.year -1 (Depends on observational data sets), 0.6012-0.6748mm.year -1 , 0.9569-1.0742mm.year -1 , and 1.2691-1.3926mm.year -1 respectively (Colorized histograms in right panels of Fig 2 and S6 Table).This shifting presents a largely overestimated future annual precipitation growth rates in North America by the original CMIP6 models' predictions.The overestimated percentages of future water availability are 2.3-17.9%(SSP126), 4.2-14.6%(SSP245), 0.1-11.0%(SSP370) and 3.0-11.6%(SSP585).
In contrast, the raw 33 CMIP5 models' predictions have underestimated the future terrestrial water supply from precipitation.After the constraint, the future annual precipitation growth rates have been increased from the raw predictions of 0.5248 mm.year -1 (RCP45) and 1.0139 mm.year -1 (RCP85) to the constrained values of 0.5445-0.5959mm.year -1 and 1.0661-1.1533mm.year -1 (Depends on observational data sets), respectively (S8 Fig and S7 Table).Thereby, the CMIP5 models have underestimated future precipitation increases (3.8-13.7%),which is contrary to the constrained results of the CMIP6 models [49].The main reason of this difference might be due to a stronger positive cloud feedback from decreasing extratropical low cloud coverage and albedo in CMIP6 models that tends to enhance the responses of temperature to increasing atmospheric CO 2 concentration and then leads to an overestimated predictions of future precipitation increases.
No matter which observed data set we use, the PDFs of the constrained results (Colorized curves in Figs 2 and S8) all have been shifted tightly for each emission scenario in comparison with the raw CMIP5 and CMIP6 predictions (Black curves).The original standard deviations of future annual precipitation growth rates from CMIP6 models (Gray histograms in right panels of Fig 2) are 0.1676 mm.year -1 (SSP126), 0.2182 mm.year -1 (SSP126), 0.3470 mm.year -1 (SSP126) and 0.4165 mm.year -1 (SSP126).After the constraint, the standard deviations (Colorized histograms in right panels of Fig 2) have been successfully decreased to 0.1155-0.1303mm.year -1 , 0.1548-0.1712mm.year -1 , 0.2742-0.2869mm.year -1 and 0.3326-0.3591mm.year -1 respectively.Accordingly, the percentages of the reduced uncertainties are up to 13.8-31.1% after the constraint, indicating a more reliable and accurate estimate of future annual precipitation growth rates in relative the raw CMIP6 predictions.Consistently, the uncertainties of predictions from the 33 CMIP5 models are also reduced by 10.9-24.1% after constraining by the observed warming trends (S8 Fig).
Due to complex atmosphere-vegetation-soil interactions, positive feedback behavior of precipitation to temperature (Fig 1e and 1f) are unexpectedly nonexistent in the southern regions of North America (10.6-14.4% of total areas of North America with negative correlations), i.e., an increased temperature but with a decreased precipitation.In these regions, the mechanisms underpin the emergent constraint relationships might be unrealistic, due to dominant role of other factors in driving the local precipitation changes, instead of warming temperature.After excluding these regions, the emergent constraint relationships (red fitting lines in S8    The constrained results reveal an overestimated future warming trends in North America.The raw CMIP6 outputs (Gray histograms in Fig 3b) predict an increasing temperature of 0.0158 ± 0.0086 ˚C year -1 (SSP126), 0.0395 ± 0.0105 ˚C year -1 (SSP245), 0.0671 ± 0.0171 ˚C year -1 (SSP370) and 0.0897 ± 0.0210 ˚C year -1 (SSP585).After the constraint (Colorized histograms in Fig 3b and S8 Table ), we estimated that the rates of annual temperature increases have been decreased to 0.0132-0.0155± 0.0057-0.0065˚C year -1 ,0.0356-0.0383± 0.0078-0.0082˚Cyear -1 , 0.0619-0.0670± 0.0130-0.0138˚C year -1 and 0.0830-0.0879± 0.0175-0.0185˚C year -1 , respectively.This shifting indicates that the raw CMIP6 predictions overestimated the future temperature increases in North America under all the four SSPs, with the overvalued percentages up to 0.1 ± 16.5%.On the contrary, the future warming trends have been underestimated by the CMIP5 models (2.8 ± 11.9%, S10 Fig and S9 Table).

Implications on future warming trends
To verify the reliability of the constrained results of future warming trends in Fig 3b, we introduced a new emergent constraint that relies on the significant statistical relationship between annual growth rates in historical temperature and that of future temperature (Fig 1c).The plausible mechanisms underpin the relationships are that the responses of temperature are proportional to the radiative forcing, and thus, as long as the forcing increases, stronger feedbacks imply higher temperature in both the past and the future [50,51].After constraining by the observed temperature from the four different data sets, we found that the future warming trends in North America have been overestimated by the raw CMIP6 predictions under all the SSPs (overvalued by 0.  3c and  3d) is highly in agreement with the conclusions that are obtained by applying the constrained future precipitation growth rates (Fig 3a and 3b).In particular, the uncertainties of the temperature predictions have been successfully reduced by 31.6-46.0%(CMIP6) and 13.0-24.1% (CMIP5) due to the narrowed standard deviations (Fig 3d), implying a more reliable and accurate estimate of future temperature changes.
"Precipitation minus evapotranspiration" here is used to metric the water availability over land surface [54][55][56].Based on the constrained CMIP6 predictions of precipitation and total evaporation (Colorized histograms in Figs 2 and 3f), we estimated the constrained future water availability changes (S14 Table ).In relative to the raw CMIP6 outputs, the future water availability growth rates have been largely overestimated by 2.4-28.2%(S14 Table) due to the largely overvalued precipitation increases (Fig 2).This is also supported by investigating the negative relationships (S12 Table) between the future annual precipitation growth rates and the future yearly changes in average annual extreme light rainfall days (See Method).These negative relationships indicate that a model with a fewer precipitation normally has a higher frequency of average annual extreme light rainfall days.Thus, a potential overestimated future annual precipitation growth rates in CMIP6 models represents an underestimated frequency in future average annual extreme light days.

Conclusions
In this study, we successfully captured the emergent constraint relationships between annual growth rates of simulated historical temperature and that of future precipitation in North America, across 27 CMIP6 models (underSSP126, SSP245, SSP370 and SSP585) and 33 CMIP5 models (under RCP45 and RCP85).Combining the observed temperature from four different data sets, the emergent constraint relationships successfully reduced the uncertainties of precipitation predictions by 13.8-31.1%,and also reveal an overestimation of future precipitation growth rates under all the four SSPs by the raw CMIP6 predictions.Furthermore, the constrained future precipitation changes are further applied to capture the more accurate future temperature trends, which indicate that the future annual temperature growth rates have been overestimated by the CMIP6 models (overestimated by 0.1 ± 16.5% by the originally predicted), while the CMIP5 models underestimated the future warming trends by around 2.8 ± 11.9%.

Fig 1 .
Fig 1. Future annual precipitation growth rate and its potential driving factors in North America.(a) presents the simulated annual average daily precipitation (mm day -1 ) from 27 CMIP6 models (1970-2100) and the observed precipitation from HadCRUT4 data set (1970-2014) (b) presents the large difference of predicted future annual precipitation growth rates (mm year -1 ) across the 27 CMIP6 models (See name of each model in S1 Table) under the emission scenarios of SSP126, SSP245, SSP370 and SSP585 and across the 33 CMIP5 models (See name of each model in S2 Table) under the emission scenarios of RCP45 and RCP85.(c) is the linear regression relationships between the observational temperature anomaly and the observed precipitation anomaly for the period of 1970-2014 from the HadCRUT4 data set, following the least-squares criterion.Each circle represents a year.(d) presents the linear relationships between future temperature and future annual precipitation for the CMIP6 models under the emission scenarios of SSP126, SSP245, SSP370, and SSP585 during 2015-2100.Each circle represents a year.(e) presents the linear relationships between the future annual growth rates in precipitation and in temperature for the CMIP6 models under SSP585 during 2015-2100.Each dot represents a model.(f) presents the relative contributions of temperature, total evaporation, soil water content and land surface runoff on the future precipitation changes in North America for the CMIP6 ensemble under SSP585 during 2015-2100.The geographical distribution of study area can be referred to Roque-Malo et al [38].https://doi.org/10.1371/journal.pone.0301759.g001 Fig), implying the reliability of our introduced emergent constraint.These empirical relationships can reduce the uncertainties of the predicted future annual precipitation changes in North America by combining it with observations.Due to the large discrepancy of the observations across different data sets, the observed temperature from the four data sets (i.e., Had-CRUT4, NOAA, GISS and GHCN, See vertical shadings in left panels of Fig 2) are collected for the same constraint processes for obtaining the reliable constrained results.After the observed constraint, we find that the probability density functions (PDFs) of the constrained future annual precipitation growth rates have been shifted to the lower values under all the four SSPs (Colorized curves in Fig 2), in relative to the raw CMIP6 predictions (Black curves).The CMIP6 models originally predicted that the best estimates of future annual precipitation growth rates (Gray histograms in right panels of Fig 2) are 0.3538 mm.year -1

Fig 2 .
Fig 2. Emergent constraint (EC) on the future annual precipitation growth rates (2015-2100) in North America combining the observed annual temperature growth rates (1970-2014) for the CMIP6 projections.(a), (b), (c),and (d) are the emergent constraint relationships between the simulated historical annual temperature growth rates and the predicted future annual precipitation growth rates across the CMIP6 models under the emission scenarios of SSP126, SSP245, SSP370, and SSP585, respectively.Green shading is the 90% prediction error of the linear regression.Each dot represents a model.Four vertical shadings are the observed annual temperature growth rates (Mean ± one standard deviation) from HadCRUT4 (0.0357 ± 0.0050 ˚C year -1 ), NOAA (0.0346 ± 0.0048 ˚C year -1 ), GISS (0.0394 ± 0.0056 ˚C year -1 ) and GHCN (0.0365 ± 0.0050 ˚C year -1 ) data sets, respectively.Black curves and colorized curves are the PDFs of the future annual precipitation growth rates before and after the emergent constraint, respectively.Gray histograms and colorized histograms in the right panels of Fig 2 the constrained and unconstrained future annual precipitation growth rates (Mean ± one standard deviation), respectively.
Fig 2. Emergent constraint (EC) on the future annual precipitation growth rates (2015-2100) in North America combining the observed annual temperature growth rates (1970-2014) for the CMIP6 projections.(a), (b), (c),and (d) are the emergent constraint relationships between the simulated historical annual temperature growth rates and the predicted future annual precipitation growth rates across the CMIP6 models under the emission scenarios of SSP126, SSP245, SSP370, and SSP585, respectively.Green shading is the 90% prediction error of the linear regression.Each dot represents a model.Four vertical shadings are the observed annual temperature growth rates (Mean ± one standard deviation) from HadCRUT4 (0.0357 ± 0.0050 ˚C year -1 ), NOAA (0.0346 ± 0.0048 ˚C year -1 ), GISS (0.0394 ± 0.0056 ˚C year -1 ) and GHCN (0.0365 ± 0.0050 ˚C year -1 ) data sets, respectively.Black curves and colorized curves are the PDFs of the future annual precipitation growth rates before and after the emergent constraint, respectively.Gray histograms and colorized histograms in the right panels of Fig 2 the constrained and unconstrained future annual precipitation growth rates (Mean ± one standard deviation), respectively.https://doi.org/10.1371/journal.pone.0301759.g002 Fig) basically remained the same under all the SSPs in comparison with the constraint relationships that include the southern North America (Black lines in S9 Fig and Gray lines in Fig 2), indicating a slight influence of such regions on the area-averaged emergent constraint relationships.

Future
annual growth rates of temperature and that of precipitation exist a great positive linear relationship between them across the 27 CMIP6 models under all the four SSPs (Fig 3a, Each circle represents a model), i.e., a model with a less precipitation normally associates with a colder temperature.In Fig 2, we obtained the more reliable and accurate estimates of future annual precipitation growth rates after the constraint, which may bring some implications on future temperature changes in North America.By projecting the constrained future annual precipitation growth rates (Obtaining from Fig 2) into y-axis through the significant statistical relationships of Fig 3a, then we can obtain the constrained future warming trends.

Fig 3 .
Fig 3. Emergent constraints on the future annual growth rates of temperature and that of total evaporation (ET) based on the CMIP6 predictions during 2015-2100.(a) presents the linear relationships between future annual growth rates of precipitation and that of temperature.(b) presents the constrained and the unconstrained future annual temperature growth rates by applying the constrained future annual precipitation growth rates and the relationships in Fig 3a.(c) presents the linear relationships between annual growth rates of historical temperature (1970-2014) and that of future precipitation (2015-2100).(d) presents the constrained and the unconstrained future annual temperature growth rates by applying the observed warming trends and the relationships in Fig 3c.(e) presents the linear relationships between future annual growth rates of precipitation and that of total evaporation.(f) presents the constrained and the unconstrained future annual total evaporation growth rates by applying the constrained future annual precipitation growth rates and the relationships in Fig 3e. https://doi.org/10.1371/journal.pone.0301759.g003

Fig 2 )
into y-axis through the relationships in Fig 3e, we estimated that the future annual total evaporation growth rates (Colorized histograms in Fig 3f and S12 Table 1 ± 26.9%, Fig 3d and S10 Table), while the original CMIP5 outputs underestimated future temperature increases (3.3 ± 15.6%, S10 Fig).This result (Fig