## 2 Materials and Methods
### Methods
#### 2.3.3 CDHI  
Copula functions, with their flexible construction and strong adaptability, are widely used to construct bivariate and multivariate joint distributions [56]. There are many types of Copula functions. Among them, the commonly used ones include elliptical Copulas and Archimedean Copulas (Table 1). Elliptical Copulas, such as the Gaussian Copula and T Copula, can describe both positive and negative multivariate relationships. They are suitable for modeling multivariate positive and negative correlations. Archimedean Copulas, such as the Clayton Copula, Frank Copula, and Gumbel Copula, are primarily used to depict the dependence relationships between bivariate variables. Among them, the Frank Copula can describe both positively and negatively correlated variables. However, other types mainly describe positively correlated variables [57]. \begin{table}
\begin{tabular}{c c c} \hline
**Copula** & **Display Formula** & **Parameter \(\mathbf{\alpha}\)** \\ \hline Gumbel & \(C(u,v)=\exp\biggl{\{}-\bigl{[}(-\ln u)^{a}+(\ln v)^{a}\bigr{]}^{\frac{1}{2}} \biggr{\}}\) & [1,\(\infty\)] \\ [PERSON] & \(C(u,v)=\max\biggl{[}(u^{-a}+v^{-a}-1)^{-\frac{1}{2}},0\biggr{]}\) & \((-1,0)\cup(0,\infty)\) \\ [PERSON] & \(C(u,v)=-\frac{1}{a}\ln\bigl{[}1+\frac{(e^{-a}-1)(e^{-a}-1)}{e^{-a}-1}\bigr{]}\) & \((-\infty,0)\cup(0,\infty)\) \\ [PERSON] & \(C(u,v)=\int_{-\infty}^{\Phi^{-1}(u)}\int_{-\infty}^{\Phi^{-1}(v)}\frac{1}{2 \pi\sqrt{1-a^{2}}}\exp\bigl{[}\frac{x^{2}-2 ost+x^{2}}{2(1-a^{2})}\bigr{]}dsdt\) & \((-1,1)\) \\ T & \(C(u,v)=\int_{-\infty}^{\Phi^{-1}(u)}\int_{-\infty}^{\Phi^{-1}(u)}\frac{1}{2 \pi\sqrt{1-a^{2}}}\bigl{[}1+\frac{2-2 ost+x^{2}}{k(1-a^{2})}\bigr{]}^{-\frac{k +2}{2}}dsdt\) & \((-1,1)\), k\(\
eq 0\) \\ \hline \end{tabular} Note: In all Copula expressions, \(u\) and \(v\) represent the cumulative distribution function (CDF) values of the two standardized marginal variables, i.e., \(u=F_{x}(x)\) and \(v=F_{y}(y)\), ranging from 0 to 1. \(\alpha\) is the dependence parameter of the Copula, which reflects the strength and structure of the dependence. \(\Phi^{-1}\) refers to the inverse of the standard normal CDF, \(s\) and \(t\) are integration variables used in Gaussian and T-Copulas. \(k\) is the degrees of freedom in the T Copula, controlling the heaviness of the tails.  
\end{table} 
Table 1: Expressions for 5 Copula functions. In order to construct the CDHI, we first conducted Pearson correlation analysis to assess the correlations between the SPEI, SPI, SRHI, SSMI, and STI-1 at different time scales (1, 3, 6, 9, and 12 months), where stronger negative correlations indicate a higher co-occurrence of droughs and heatwaves, suggesting a greater likelihood of CDHEs. Then, we selected the drought index with the strongest negative correlation with STI-1 as the drought indicator for CDHEs. After selecting the drought and heatwave indicators, we designated the Standardized Drought Index (SDI) and heatwave index as SDI and STI, respectively. Then, we defined CDHEs as instances where the variable SDI is less than or equal to a threshold (_sdi_), while simultaneously, the variable STI is greater than or equal to another threshold (_sti_). Based on this definition, we calculated the joint probability of CDHEs. The specific calculation formula is shown in Equation (3):  
\[ \begin{array}{l}P=P(\mathrm{SDI}\leq sdi,\mathrm{STI}\geq sti)\\ =P(\mathrm{SDI}\leq sdi)-P(\mathrm{SDI}\leq sdi,\mathrm{STI}\leq sti)\\ =F(sdi)-C(F(sdi),F(sit))\end{array}  \tag{3}\]  
where \(F\) is the marginal distribution function, and \(C\) is the Copula function. Unlike marginal probabilities in univariate cases, joint probabilities may not be uniformly distributed. Therefore, we fitted the Gringorten empirical frequency distribution to the joint probability [58]. Meanwhile, in order to facilitate the expression and quantification of the severity of CDHEs, we standardized \(P\) to obtain the CDHI. In this study, we adopted the same classification standards as those for drought indices such as the SPEI to define the levels of the CDHI.