## 3 Materials and Methods
### Dmc  
The DMC method is often applied to verify the consistency of hydro-meteorological data. This approach compares data from a single station with a pattern generated using data from several other stations across a given area. We utilize the DMC method in this study in order to elucidate the relationships between different ecosystem services. However, as the elements we incorporated into our comparative analysis require high degrees of correlation, we performed an appropriate test for each pair of ecosystem services before constructing a graph. In the case of two highly correlated variables, \(E_{1}\) and \(E_{2}\), which represent one pair of ecosystem services, the main procedure used to generate a DMC includes the initial calculation of ratios for each pair of functions based on raster map layers (\(E_{1}/E_{2}\)). This ensures that \(E_{2}\) decreases as values on the \(x\)-axis increase. Ratio grid cells on each map layer are then arranged based on their ascending values,and a scatter plot is generated by accumulating these values, which leads to a DMC for each pair of ecosystem services as in Equation (5), as follows:  
\[Y_{i}=\sum_{1}^{i}y_{i}\quad\text{and}\quad X_{i}=\sum_{1}^{i}x_{i}\quad(i=1, \cdots,k) \tag{5}\]  
In this expression, \(k\) denotes each ecosystem service observation, while the series \(y_{1}\),..., \(y_{k}\), denotes observations for \(E_{1}\), and the series \(x_{1}\),..., \(x_{k}\) includes observations for \(E_{2}\). A graph drawn using this method can assume one of three different shapes, including a straight line (\(OA\)), a convex (\(AO_{3}B\)) or concave curve (\(OO_{6}A\)) (Figure 5). In order to conveniently illustrate these differences, we selected seven points (\(O_{1}\), \(O_{2}\),..., \(O_{7}\)) on the curves; thus, points \(O_{1}\) and \(O_{2}\) are a straight line, while \(O_{3}\) to \(O_{7}\) are on the curves (Figure 5). Thus, for any point on the curve (i.e., \(O_{1,i}\), \(O_{2,i}\)), the coordinate difference to its preceding counterpart (i.e., \(O_{1,i-1}\), \(O_{2,i-1}\)) is its ecosystem service value (\(E_{1,i}\), \(E_{2,i}\)), as expressed by Equation (6), as follows:  
\[(E_{1,i},E_{2,i})=(y_{i-1}-y_{i},x_{i-1}-x_{i}) \tag{6}\]  
In the straight line case (\(OA\)), \(E_{2,i}\) increases in concert with \(E_{1,i}\), while the ratio of changes in \(E_{1}\) versus a related \(E_{2}\) will be fixed. This relationship is expressed by Equation (7), as follows:  
\[\frac{\Delta E_{1,i}}{\Delta E_{2,i}}=\frac{\Delta E_{1,1}}{\Delta E_{2,1}}=C \tag{7}\]  
In this expression, \(\Delta E\) denotes the change in \(E\) while \(C\) is constant. Compared to a straight line, however, the slope of a curve (such as \(AO_{3}B\)) is not fixed. This is expressed by Equation (8), as follows:  
\[\left\{ \begin{array}{l}\frac{\Delta E_{1,i}}{\Delta E_{2,i}}<\frac{\Delta E_ {1,i+1}}{\Delta E_{2,i+1}}\text{ if the point $i$ is on convex curve}\\ \frac{\Delta E_{1,i}}{\Delta E_{2,i}}>\frac{\Delta E_{1,i+1}}{\Delta E_{2,i+1} }\text{ if the point $i$ is on concave curve}\end{array} \right. \tag{8}\]  
Thus, if the point O lies on the convex curve (Figure 5b), then the slope of this curve will decrease with increasing \(x\)-axis values. This indicates that point \(E_{1}\) on the convex curve will decrease (OM4)  
Figure 5: Double mass curve (DMC) sketches (**a**) and comparisons between different slopes of DMC (**b**). more than if it falls on a straight line (OM\({}_{2}\)); similarly, if point O lies on the concave curve, then the slope in this case will also increase with increasing \(x\)-axis values. This indicates that point \(E_{1}\) on the concave curve has increased (M\({}_{2}\)M\({}_{3}\)) relative to its position on a straight line (OM\({}_{2}\)). Applying the definitions of synergies and trade-offs in ecosystem services, both a straight-line and a convex curve imply the simultaneous enhancement of two services that can be defined as synergistic.