## 2 Materials and Methods
### Approach for Drought Assessment Using Satellite and Meteorological Data
#### 2.5.2 Design of the Drought Index Based on the TCI under Different Climatic Areas  
The data from one meteorological station were taken as the entity in Equation (3). HTC30 coefficients were calculated for 30 days back for the time points with an eight-day step coupled with the days of the year in the MODIS image database. The exemplary cases in the system of equations (3) for stations \(i\) 1..233, years 2001..2019, and eight-day time steps \(d\) 112..280 for the left and right side variables are presented in Table 1.  
A logarithmic operation was carried out on HTC30 to ensure a normal distribution. The expected value of it is log(MedHTC30). The number of lagged TCI variables in the model expression is determined during the estimation process. The value equal to 0.5 as the expected value of the TCI for each eight-day step of MODIS data was taken. It was confirmed by the Student's \(t\)-test that there is no significant difference between the value 0.5 and the multiyear averages of the TCI for most entity areas and time points. Thus, the formulation of the model, combining spatiotemporal satellite and meteorological data according to Equation (4) for 30 days of HTC accumulation, becomes  
\[ \begin{array}{ll}\log(HTC30_{i,t})-\log(MedHTC30_{i})\\ =&\beta_{0}(TCI_{i,t}-0.5)+\beta_{1}(TCI_{i,t-1}-0.5)+\beta_{2}(TCI_{i,t-2}-0.5 )+\varepsilon_{i,t}\end{array}  \tag{5}\]  
where  
\(HTC30_{i,t}\)--HTC30 coefficient of station \(i\) at time point \(t\) (eight-day step in the growing season);  
\(MedHTC30_{i}\)--median of all HTC30 observations at station \(i\);  
\(TCI_{i,t}\)--TCI for the area close to station \(i\) at time point \(t\);  
0.5--the expected mean of the TCI variable for every station at every time point;  
\(\varepsilon_{i,t}\)--the error for station \(i\) at time point \(t\);  
\(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\)--the parameters to estimate. The dependent variable in Equation (5) has the form \(YHTC30_{i,t}=\log(HTC30_{i,t})-\log(MedHTC30_{i})\). The mean subtraction results in the dependent variable becoming free from the atmospheric characteristics represented by the median MedHTC30 of each station. The learning sample was limited to such cases where the number of agricultural pixels within 10 km of the meteorological station exceeded 200. Such a choice provides more averaging of area in the agricultural cover class, which is advantageous for modeling. TCI data come from the aggregation of 1 km\({}^{2}\) pixel values for the sample area. A histogram of the dependent variable, called YHTC30, presented in Figure 10, shows the normal distribution necessary for using the LMSQ regression method. The equation, being the result of regression after recalculating the estimated parameters (slope and intercept), can be presented in the form  
\[\log(HTC30_{t})-\log(MedHTC30)\ =\ a+b*TCI_{t}+c*TCI_{t-1}+d*TCI_{t-2}+\ \varepsilon_{t} \tag{6}\]  
where \(t\) is the eight-day time point of the growing season; and \(a\), \(b\), \(c\), and \(d\) are the estimated parameters. The length of the period for HTC accumulation was set to 30, 40, and 50 previous days. This allowed us to fit it to 3, 4, or 5 time steps back of aggregating TCI, respectively, according to Equation (4). The Pearson's R coefficient as a goodness-of-fit measure and the standard errors for the models using the TCI in the 3, 4, and 5 time steps are presented in Table 2. The standard errors decrease as the TCI lag increases; thus, more intensive agricultural drought is predicted with more precision in relation to atmospheric conditions.