*2.3. Methods*

PUE was identified as the ratio of NPP to P [11]:

$$PUE = \frac{NPP}{P} \tag{1}$$

where NPP is the annual net primary productivity (unit: <sup>g</sup>·m<sup>−</sup>2) and P is the annual precipitation (unit: mm). NPP is obtained through the Thornthwaite memorial model [64]:

$$NPP = 3000(1 - e^{-0.0009695(ET\_a - 20)})\tag{2}$$

where ETa is the actual evapotranspiration. It can be obtained as Zhou and Zhang [65]. Although there are many methods to estimate ETa, such as the water balance method, surface energy balance method, remote sensing analysis method, etc., they all have limitations, such as complicated parameters and difficulty to determine [66,67]. Zhou and Zhang's method fully reflects the limiting effect of energy and water on evapotranspiration, with its few parameters and clear physical significance making it high practicability. The method is as follows:

$$ET\_a = \frac{P \times R \times \left(P^2 + R^2 + P \times R\right)}{\left(P + R\right) \times \left(P^2 + R^2\right)}\tag{3}$$

Here, P is the annual precipitation, and R is the annual net radiation correction factor, which can be obtained by Equation (4):

$$R = (ET\_0 \times P)^{0.5} \times (0.369 + 0.598 \times \left(ET\_0 / P\right)^{0.5} \tag{4}$$

where ET0 is the potential evapotranspiration calculated using the FAO Penman-Monteith method. See Allen et al. [68] for details.

The sensitivity coefficient and the relative change were used to measure the contribution of environmental factors to the PUE change:

$$\text{Con}\_X = \text{RC}\_X \times \text{S}\_X \tag{5}$$

$$RC\_X = \frac{58 \times Trend\_X}{|ave\_X|} \times 100\% \tag{6}$$

where *ConX* is the contribution of a factor to the PUE change, *RCX* is the relative change rate of the factor, *TrendX* and *aveX* are the change rate and average of the factor, respectively, and *Sx* is the sensitivity coefficient of the PUE with respect to environmental factor X. *Sx* can be obtained as McCuen [69]:

$$S\_x = \lim\_{\Delta X \to 0} \left( \frac{\Delta PUE/PUE}{\Delta X/X} \right) = \frac{\partial PUE}{\partial X} \times \frac{X}{PUE} \tag{7}$$
