Bivariate copulas and PMF in ESMs. The bivariate copulas are commonly used to describe the dependence between two random variables and to calculate the joint probability of an event. The joint probability distribution function FX,Y(x, y) of random variables X and Y can be expressed asEmbedded Image(1)where P is the joint cumulative probability. The marginal cumulative distribution functions are given by FX(x) = P(Xx) and FY(y) = P(Yy). Here, we used a copula function C to describe a bivariate distribution function, so the joint probability distribution of X and Y can be written asEmbedded Image(2)where the two marginal cumulative distribution functions are transformed into two uniformly distributed random variables u and v (i.e., the normalized ranks of x and y). As an example, the probability of an event, such as the variables exceed or are below given thresholds, can be expressed asEmbedded Image(3)

In this study, we focused on the joint probability of extreme VPD (u, above its 90th percentile) and SM (v, below its 10th percentile). So, the probability of this compound extreme event is given byEmbedded Image(4)

The commonly used bivariate copula families include Gaussian copula, Student’s t copula, and Archimedean copulas, which could be used to describe a wide range of possible dependence structures of two variables (18). With these copulas, the joint probability of an event can be easily calculated.

We first transformed the marginal distributions of VPD and SM into normalized ranks ranging from 0 to 1 [from FX(x), FY(y) to u, v]. To select an appropriate bivariate copula for each grid cell, we fit all possible bivariate copula families (40 in total, listed in table S3) and selected the best one according to the Bayesian information criterion in the R package, VineCopula (function “BiCopSelect”) (42). With the copula of the best fit, we then calculated the joint probability of extreme high VPD (above its 90th percentile) and low SM (below its 10th percentile) using the function “BiCopCDF” in VineCopula.

We defined the PMF as the ratio of the joint probability with the copula and that assuming independent distributions (P = 0.01). In other words, PMF quantifies the increase in frequency due to the covariations between VPD and SM, a value of one meaning that there is no change in frequency. To further evaluate the quality of the copula retrieval, we also directly estimated the joint probability by counting the joint occurrence rate of extreme high VPD and low SM based on model simulations and calculated the PMF using that method. Minimal differences were observed, both in historical and future simulations, so that the reliability of the copula method was confirmed (fig. S17). The consistent PMF derived using the two methods also demonstrated that the thresholds of VPD and SM extremes determined based on model simulations were reliable from the perspective of bivariate probability distribution. This was important for assessing the responses of carbon uptake to extreme high VPD and low SM, which were sensitive to the definition (or thresholds) of VPD and SM extremes.

Compound extreme events and their impacts. The VPD-SM coupling was evaluated by assessing the bivariate distributions of VPD and SM and calculating the Spearman correlation coefficient between them. We sorted observed daily VPD and SM from the flux tower sites into 10 × 10 percentile bins in each site and calculated the mean probability of each percentile bin of VPD and SM across the 66 sites. PMF was calculated as the ratio of the probability in the top left bin in Fig. 1A and the assumed probability (P = 0.01) for independent VPD and SM extremes.

We calculated mean anomalies of GPP, TER, and NEP in the 10 × 10 percentile bins across the 66 sites to assess the observed mean responses of these variables to VPD and SM, especially the responses to extreme high VPD and low SM. In ESMs, mean anomalies of GPP, TER, and NEP due to compound VPD and SM extremes were calculated in each grid cell in historical and future simulations individually. Historical VPD and SM extremes were determined according to extreme VPD (above its 90th percentile) and SM (below its 10th percentile) in historical simulations. For all evaluations of future projections, we defined the high VPD and low SM thresholds in two ways: based on (i) the historical period and (ii) the future period. These two thresholds were used to calculate future PMF and mean anomalies of GPP, TER, and NEP due to compound VPD and SM extremes. A comparison of the thresholds between the two periods reflects changes in the intensity of compound extreme events. PMF changes based on the same thresholds reflect changes in the frequency of compound extreme events.

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