1. The Budyko Framework

The water balance equation for a closed basin can be expressed as:

$$P = Q + E + \Delta S \tag{2}$$

where *P* is precipitation (mm), *Q* is runoff (mm), *E* is evaporation (mm), and ∆*S* is the variation of water storage in the basin. For a long period, ∆*S* is approximately 0 and can be neglected. Therefore, the multi-year water balance equation can be simplified as: *P* = *Q* + *E*.

The actual evapotranspiration of a watershed depends on the available water supply and available heat, and Budyko [6] proposed that on a multi-year time scale, the multiyear average evapotranspiration depends on the multi-year average rainfall (*P*) and the multi-year average potential evapotranspiration (*ET*), expressed in the formula as:

$$\frac{E}{P} = f\left(\frac{ET}{P}, n\right) \tag{3}$$

where *n* is the underlying surface parameter of the basin.

Based on the Budyko framework, many studies have derived many different analytic forms, the more commonly used of which is the Choudhury–Yang [17] equation, the expression as:

$$E = \frac{ETP}{\left(P^n + ET^n\right)^{\frac{1}{n}}} \tag{4}$$

where *n* is the watershed underlying surface parameter, which reflects the characteristics of the watershed underlying surface, related to topography, soil, vegetation, etc., and changes mainly by human activities. *n* can be obtained by back-calculating the multi-year average *Q*, *ET*, and *P* and considering *P*, *ET*, and *n* as mutually independent variables in the above equation [18].

#### 2. Climate Elasticity Analysis Method

The variation in runoff can be attributed to the combined effect of climatic and underlying surface factors, where climatic factors mainly include precipitation and potential evapotranspiration. Assuming that the factors are independent of each other, the following equation can be obtained according to the water balance equation [18]:

$$
\Delta Q \approx \frac{\partial Q}{\partial P} \Delta P + \frac{\partial Q}{\partial ET} \Delta ET + \frac{\partial Q}{\partial n} \Delta n \tag{5}
$$

where ∆*Q*, ∆*P*, ∆*ET*, and ∆*n* are the changes in the average runoff depth, rainfall, potential evapotranspiration, and underlying surface parameters at different time periods, respectively. *<sup>∂</sup><sup>Q</sup> ∂P* , *∂Q <sup>∂</sup>ET* , *∂Q ∂n* are the sensitivity coefficients of runoff depth to precipitation, potential evapotranspiration, and parameters of the underlying surface, respectively, and the partial derivatives are obtained by combining Equations (2) and (4):

$$\begin{cases} \begin{array}{c} \frac{\partial Q}{\partial \overline{P}} = 1 - \frac{1}{\left[ \left( \frac{P}{ET} \right)^{n} + 1 \right]^{\frac{1}{\overline{\pi}} + 1}}\\ \frac{\partial Q}{\partial ET} = -\frac{1}{\left[ \left( \frac{ET}{P} \right)^{n} + 1 \right]^{\frac{1}{\overline{\pi}} + 1}}\\ \frac{\partial Q}{\partial n} = ETP \left[ \frac{ET^{n}lnET + P^{n}lnP}{n(ET^{n} + P^{n})^{1 + \frac{1}{\overline{\pi}}}} - \frac{ln(ET^{n} + P^{n})}{n^{2}(ET^{n} + P^{n})^{\frac{1}{\overline{\pi}}}} \right] \end{array} \tag{6}$$

Using Equation (6), the elasticity coefficients and the contribution to the change in runoff can be calculated for each factor:

$$
\varepsilon\_X = \frac{\partial Q}{\partial X} \frac{\mathbf{x}}{Q} \tag{7}
$$

$$
\delta Q\_x = \frac{\frac{\partial Q}{\partial X} \Delta X}{\frac{\partial Q}{\partial P} \Delta P + \frac{\partial Q}{\partial ET} \Delta ET + \frac{\partial Q}{\partial n} \Delta n} \times 100\% \tag{8}
$$

where *ε<sup>x</sup>* is the elasticity coefficient of *X* factor, *δQ<sup>x</sup>* is the contribution of *X* factor to the change of runoff, *<sup>∂</sup><sup>Q</sup> ∂X* ∆*X* indicates the contribution of *X* factor to the change of runoff.

#### **4. Results**
