2.2 Methods
Modeling the nonlinear relationship between soil moisture and vegetation growth
First, we removed the impact of temperature on vegetation growth by fitting a linear regression between temperature and NDVI in each grid and then using the fitted regression to remove the temperature-driven component from the NDVI values. Then, we established relationships between NDVI anomalies (after removing the temperature impacts) and soil moisture in each grid based on the following nonlinear equation (Eq. 1) 31 32:
$$NDVI\left(sm\right)=NDVImax -{b\left(SM-SMpoint\right)}^{2}$$
1
where NDVI(sm) is the temperature-free NDVI anomaly at a given soil moisture (SM), and SMpoint is the soil moisture optima determined by the vertex of the unimodal curve. NDVImax represents the local maximum vegetation growth potential under the SMpoint in each grid; b is a parameter describing the spread of the parabola.
Defining soil water use efficiency by vegetation
We defined vegetation soil water use efficiency as the amount of vegetation growth change due to soil moisture consumption per unit (every 0.01 cm3/cm3 decrease in soil moisture). As we used Eq. (1) to describe the soil moisture (SM) – vegetation growth (NDVI) relation for each pixel, we derived the vegetation use efficiency of soil water (\(\frac{{d}_{NDVI}}{{d}_{SM}}\)) at a given soil moisture using the following equation:
$$\frac{{d}_{NDVI}}{{d}_{SM}}=2*b(SM-SMpoint)$$
2
Quantifying the impacts of soil properties on vegetation soil water use
In this step, we quantified the impacts of soil properties on vegetation growth and identified the important soil properties for vegetation soil water use. We considered soil texture (clay, silt, sand), absolute soil depth, soil bulk density, soil organic matter content, soil pH, and soil CEC, as they have been reported to have global impacts on the efficiency of water storage and release and thus vegetation growth in previous studies 33 34.
Next, we linked soil properties to the three key parameters describing the nonlinear relationships between soil moisture and vegetation growth (NDVImax, SMpoint and b). We used a multiple regression model to quantify the impacts of soil properties on these three parameters (Equations (3)-(5)), and the corresponding parameters (α, β and γ) were estimated by fitting the line after deleting the strongly collinear variables by minimizing the Akaike information criterion.
$$NDVImax=\alpha \text{*}soil+{NDVImax}_{-soil}$$
3
$$SMpoint=\beta \text{*}soil+{SMpoint}_{-soil}$$
4
$$b=\gamma \text{*}soil+{b}_{-soil}$$
5
where soil represents all selected soil properties by minimizing the Akaike information criterion and α, β and γ are the corresponding parameters from each fitted equation. NDVI− soil, SMpoint− soil and b− soil are the intercepts of the linear equations, which means that the effect is independent of soil properties.
We then substituted the equations above (Equations (3)-(5)) into Equations (1) and (2) and obtained the following equation:
$${NDVI\left(sm\right)}_{-\text{s}\text{o}\text{i}\text{l}}={NDVImax}_{-soil} -{{b}_{-soil}\left(SM-{SMpoint}_{-soil}\right)}^{2}$$
6
$$\frac{{d}_{NDVI}}{{d}_{SM-soil}}=2*{b}_{-soil}(SM-{SMopt}_{-soil})$$
7
where NDVI (sm)−soil is the NDVI change not caused by soil properties and \(\frac{{d}_{NDVI}}{{d}_{SM-soil}}\) is the vegetation soil water use efficiency independent of the soil property impacts.
We then quantified the impacts of soil properties on soil water use efficiency and vegetation growth as the difference between Eq. (2) and Eq. (7) and the difference between Eq. (1) and Eq. (6), respectively, using the following equations:
\({\frac{{d}_{NDVI}}{{d}_{SM}}}_{soil}=2*b\left(SM-SMpoint\right)\) \(-\) \(2*{b}_{-soil}\left(SM-{SMpoint}_{-soil}\right)\) (8)
\(NDVI\left(sm\right)-{NDVI\left(sm\right)}_{-soil}=[{b}_{-soil}{\left(SM-{SMpoint}_{-soil}\right)}^{2}-b{\left(SM-SMpoint\right)}^{2} ]\) \(+\) \(\left[NDVImax -{NDVImax}_{-soil}\right]\) (9)
In Eq. (8), \({\frac{{d}_{NDVI}}{{d}_{SM}}}_{soil}\) is the impact of soil properties on vegetation soil water use efficiency. In Eq. (9), \([{b}_{-soil}{\left(SM-{SMpoint}_{-soil}\right)}^{2}-b{\left(SM-SMpoint\right)}^{2} ]\) is the impact of soil properties on vegetation growth by adjusting soil water use, while \(\left[NDVImax-{NDVImax}_{-soil}\right]\) represents the baseline effect of soil properties on vegetation growth (not by adjusting soil water use). In our study, we focused on the impact of soil properties on vegetation growth by adjusting soil water use, and its theoretical effect is depicted in Fig. 1.