4.3.2. Estimation of F<sup>b</sup>

*F*<sup>b</sup> is a concept corresponding to *F*v. It includes the reflectance and temperature characteristics of the object surface and is not a supplement to *F*v. *F*<sup>b</sup> enriches the research index of grassland ecosystem on the basis of remote sensing technology. In this study, we chose Wang's estimation formula for *F*<sup>b</sup> [54], which is expressed as follows:

$$F\_{\rm b} = \frac{NDII - NDII\_{\rm min}}{NDII\_{\rm max} - NDII\_{\rm min}} \times 100\% \tag{2}$$

$$NDII = \frac{\lambda\_R - \lambda\_T}{\lambda\_R + \lambda\_T} \tag{3}$$

where *F*<sup>b</sup> delineates the surface bareness fractions, *NDII* is the normalized difference impervious index, *NDII*min refers to the minimum value of *NDII* (high grassland coverage and low-temperature pixel), and *NDII*max represents the maximum value of *NDII* (high temperature and reflectivity). *λ<sup>R</sup>* is red band 1 from MOD13C2, and *λ<sup>T</sup>* refers to thermal infrared band 32 from MOD11C3.

### 4.3.3. Estimation of NPP

*NPP* was extracted from the dataset at a spatial resolution of 1 km and calculated on the basis of the BIOME-BGC model [55]. The specific formula is as follows:

$$NPP = \sum\_{\mathbf{t}}^{365} PSNet - \left(\mathbf{R}\_{\mathbf{m}} + \mathbf{R}\_{\mathbf{g}}\right) \tag{4}$$

$$PSNet = GPP - \mathbb{R}\_{lr} \tag{5}$$

where *NPP* represents the actual *NPP* (g cm−<sup>2</sup> year−<sup>1</sup> ), and *PSNet* refers to the net photosynthesis. R*<sup>m</sup>* is the annual maintenance respiration of live cells in woody tissue, and R*<sup>g</sup>* delineates the annual growth respiration. *GPP* is the gross primary productivity from MOD17A2 datasets, and R*lr* is the daily leaf and fine root maintenance respiration.

#### 4.3.4. Grassland Dynamic Analysis

The grassland vegetation dynamic analysis is a significant ecological process of grassland health condition. We can assess grassland degradation or restoration by using *F*v, *F*b, and NPP as fundamental indicators. The slope was determined by using ordinary least squares regression, which is expressed as follows:

$$Slope\_A = \frac{n \times \sum\_{i=1}^{n} i \times (A)\_i - (\sum\_{i=1}^{n} i)(\sum\_{i=1}^{n} (A)\_i)}{n \times \sum\_{i=1}^{n} i^2 - \left(\sum\_{i=1}^{n} i\right)^2} \tag{6}$$

where *A* refers to grassland *F*v, *F*b, and NPP; *i* is the sequence number of the year (in this study, 1 is for the year 2000, 2 is for the year 2001, and so on); *n* represents the number of years, which is 14 in this study. A negative slope value shows a degradation trend, whereas a positive slope value shows a restoration trend. In this study, combined analysis of slopes (Table 3) was conducted to quantitatively evaluate the grassland response to drought.


**Table 3.** Scenarios to assess the role of Fv, F<sup>b</sup> and NPP responding to scPDSI in the MP.

The significance of the variation tendency was determined in terms of *F*-test to represent the confidence level of variation. The calculation for statistics is expressed as follows:

$$F = \mathcal{U} \times \frac{n-2}{\mathcal{Q}} \tag{7}$$

$$\mathcal{U} = \sum\_{i=1}^{n} (\mathcal{Y}\_i - \overline{\mathcal{Y}})^2 \tag{8}$$

$$Q = \sum\_{i=1}^{n} (y\_i - \mathfrak{H}\_i)^2 \tag{9}$$

$$
\mathfrak{H}\_{\mathrm{i}} = \mathrm{Slope} \times \mathrm{i} + \mathrm{b} \tag{10}
$$

$$b = \overline{y} - \text{Slope} \times \overline{\overline{i}} \tag{11}$$

where *U* represents the residual sum of the squares; *Q* is the regression sum; *y*ˆ*<sup>i</sup>* refers to the regression value; *y<sup>i</sup>* delineates the average data of year *i*; *y* is the mean value of *F*v, *F*<sup>b</sup> or NPP over *n* years; *b* refers to the intercept of the regression formula.

We classified the variation tendency into the following six levels on the basis of the *F*-test results: extremely significant decrease (ESD, slope < 0, *p* < 0.01); significant decrease (SD, slope < 0, 0.01 < *p* < 0.05); no significant change (NSC, slope = 0, *p* > 0.05); Significant Increase (SI, Slope > 0, 0.01 < *p* < 0.05); extremely significant increase (ESI, slope > 0, *p* < 0.01).

### 4.3.5. Correlation Analysis

The Pearson correlation coefficient was used to reflect the long-term dynamic of two variables in a given time *n*. The specific calculation formula is as follows:

$$r = \frac{n \times \sum\_{i=1}^{n} \left(\mathbf{x}\_{i} \times y\_{i}\right) - \left(\sum\_{i=1}^{n} \mathbf{x}\_{i}\right) \left(\sum\_{i=1}^{n} y\_{i}\right)}{\sqrt{n \times \left(\sum\_{i=1}^{n} \mathbf{x}\_{i}^{2}\right) - \left(\sum\_{i=1}^{n} \mathbf{x}\_{i}\right)^{2}} \sqrt{n \times \left(\sum\_{i=1}^{n} y\_{i}^{2}\right) - \left(\sum\_{i=1}^{n} y\_{i}\right)^{2}}} \tag{12}$$

where *r* is the correlation coefficient, *n* refers to the sequential year, which is 14 in this study; *x<sup>i</sup>* and *y<sup>i</sup>* represent *F*v, *F*<sup>b</sup> or NPP and climatic factors, respectively. *r* refers to a description of linear correlation degrees between the two variables. The value of *r* ranges from −1 to 1. −1 and 1 are completely related, whereas 0 indicates irrelevant. The greater the absolute value of *r*, the stronger the correlation, but no causal relationship is found.

#### **5. Conclusions**

This study investigated the grassland vegetation dynamic on the basis of multi-index and its response to droughts in the MP during 2000–2013 in terms of the relations between Fv, Fb, NPP, and scPDSI. The spatial distribution of grassland F<sup>v</sup> and NPP decreases from

northeast to southwest, showing an increasing trend of 0.18 and 0.43, respectively. On the contrary, F<sup>b</sup> increases from northeast to southwest, presenting a decreasing trend, with the value of −0.16. The grassland degradation condition of the MP shows a restoration trend during the study period. F<sup>v</sup> and NPP shows a positive relationship with scPDSI, whereas F<sup>b</sup> is exactly on the country. The areas with a positive correlation between Fv, NPP, and scPDSI are 84.08% and 93.88%. The grassland increase regions with control and counter response to drought account for 6.06% and 6.87%. However, the distribution of grassland decrease regions with control and counter response (0.44% vs. 0.29%) is minimal. The regions of grassland increase from control response mainly distribute in central MG, whereas the grassland increase regions with counter response are in the eastern and western MG and northeast IM. Such detailed analysis of grassland-related indexes and its responses to drought are useful to clarify the grassland condition, potential effect of drought, and is beneficial to help policymakers for develop proper measures for grassland protection.

**Author Contributions:** Conceptualization, Y.Z. and J.L.; Methodology, Z.W.; Software, Q.W.; Validation, Y.Z., Y.Y. and Z.W.; Formal Analysis, J.L. and Y.B.; Data Curation, Y.Z. and W.X.; Writing— Original Draft Preparation, Y.Z.; Visualization, Y.Z.; Supervision, J.L. and W.X.; Project Administration, J.L. and W.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (42067069), the Technology Innovation Leading Program of Shaanxi Province (2021KJXX-53), the National Key Research and Development project (2018YFD0800201), the Natural Science Foundation of China (32060279) and the National Research Project of the State Ethnic Affairs Commission of China (2019-GMD-034). We are grateful to the editor and anonymous reviewers. We also appreciate the Oak Ridge National Laboratory Distributed Active Archive Center (ORNL DAAC) for sharing a series of original remote sensing dataset and Climatic Research Unit in University of East Anglia for sharing climate dataset.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

