## 2 Materials and Methods
### Estimation and Projection Models for the Net Primary Productivity
Therefore, in this study, the difference in vegetation distribution was reflected using NDVI instead of LAI as a variable. Air pollutants, such as sulfur oxide, nitrogen oxide, and ozone, as well as the atmospheric carbon dioxide concentration, directly or indirectly affect the net primary productivity of forests [7; 54; 55]. In particular, a high ozone concentration directly destroys leaf tissue [56], and photosynthesis is reduced in plants that are exposed to ozone [57]. In this study, ozone was selected as the air pollution factor. Of these, the average concentration of ozone in the second and third quarters was utilized. The concentration of ozone remained high from early spring to late summer [58; 59] and the average concentration in the second and third quarters was used because the increase in forest productivity was the most dynamic. According to the literature review, correlation analysis, and expert interview (see Appendix A), we finally selected annual temperature range, precipitation, NDVI, and the previous year's NDVI to develop a model to predict future NDVI. Furthermore, solar radiation, elevation, NDVI, and the concentration of ozone in the 2 nd and 3 rd quarters were selected as variables for developing a model  
Figure 1: Framework of the research. for estimating future NPP. These variables were considered to be valid in previous studies and so were selected as major factors for this study. Input data for the development of the two models were obtained from the Meteorological Agency and the National Institute of Environmental Research. The ozone concentration was obtained from the National Institute of Environmental Research. A total of 480 observatories are distributed throughout Korea, among which, data from 10 forest stations and 52 stations that are located around 100 m from the edge of forests were used to obtain data from 2001 to 2010. The collected data from 2001 to 2010, including solar radiation and the weather data, were used by the Korea Meteorological Administration. The NDVI data from 2001 to 2010 were collected from satellite imagery provided by the United States Geological Survey (USGS). In addition, future temperature and precipitation data were also used as detailed in the RCP 8.5 scenario. 2.2 Statistical Models for Estimating the Normal Distribution Vegetation Index and the Net Primary Productivity  
In this study, a two-step statistical model was constructed. To reflect the difference of forests in the future, we first constructed a model to predict NDVI and developed a model to predict forest productivity. The characteristics of data that were used in this study are multidimensional data with temporal and spatial characteristics defined as data reflecting a period of 10 years from the 62 aforementioned stations. This can be viewed as panel data with multiple observations obtained over time from the same observatory [60; 61; 62]. Therefore, a prediction model using cross-section time series analysis was developed to estimate the NDVI and the net primary productivity of forests. We used two kinds of models for developing the prediction model. One type of model is the vector autoregressive model and another is the panel analysis model. In the vector autoregressive model, the dynamic response of variable changes to endogenous variables can be explained [63], and each variable can be expressed as a linear function of its past values, while the past values of other variables are expressed as errors [64]. The general structure of VAR is as follows. \[X_{i,\ t}=a+\sum_{t=1}^{T}\beta_{i,t-1}X_{i,\ t-1}+e_{i,t} \tag{1}\]  
where \(X_{i,t}\) is a dependent variable, \(e_{i,t}\) is idiosyncratic error. We estimated the future yearly NDVI from 2001 to 2100 as the first step ahead forecasting using VAR. However, the NDVI used the 10-year average of the target year because the climate data used a 10-year average to reduce variation [65]. Also, we used a panel analysis model to predict NPP.