## 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
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{_Obs._} & \multirow{2}{*}{\(i\)} & \multirow{2}{*}{\(\dfrac{\boldsymbol{t}}{\boldsymbol{t}}\)} & \multirow{2}{*}{\(\boldsymbol{HTC30\ d}\)} & \multirow{2}{*}{\(\boldsymbol{TCI\ d}\)} & \multirow{2}{*}{\(\boldsymbol{TCI\ d}\)} & \multirow{2}{*}{\(\boldsymbol{TCI\ d}\)-\(\boldsymbol{1}\)} & \multirow{2}{*}{\(\boldsymbol{TCI\ d}\)-\(\boldsymbol{2}\)} \\ \cline{5-6}  & & & & & & & & \\ \hline
1 & 1 & 2001 & 112 & HTC(82..112) & TCI 112 & TCI 94 & TCI 86 \\
2 & 1 & 2001 & 120 & HTC(90..120) & TCI 120 & TCI 112 & TCI 104 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
418 & 1 & 2019 & 280 & HTC(250..280) & TCI 280 & TCI 272 & TCI 264 \\
419 & 2 & 2001 & 112 & HTC(82..112) & TCI 112 & TCI 94 & TCI 86 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
836 & 2 & 2019 & 280 & HTC(250..280) & TCI 280 & TCI 272 & TCI 264 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
97 394 & 233 & 2019 & 280 & HTC(250..280) & TCI 280 & TCI 272 & TCI 264 \\ \hline \end{tabular}
\end{table} 
Table 1: Example of panel data for the fusion of meteorological and satellite indices. The goodness-of-fit on individual meteorological stations is largely balanced for all variants of the model. The correlation coefficients change in the range presented in the last column of Table 2, from R = 0.48 to 0.74, depending on station. The results included in Table 2 highlight the feasibility of using the model that fuses meteorological and satellite data to detect drought in different climatic regions. The exponential transformation of Equation (6) and the moving of MedHTC30 to the right side by multiplication affords an equation with HTC30 on the left side. The variability of the right side of the equation is based on TCI, while MedHTC30 is a fixed element characterizing the climatic aspect of a specific area. For the whole territory of Poland, the MedHTC30 image with a resolution of 1 km was prepared using interpolated meteorological data (Figure 6). Equation (6) was adopted as the Drought Information Satellite System (DISS) index:  
\[DISS_{t}\ =\ MedHTC30*\exp(a+b*TCI_{t}+c*TCI_{t-1}+d*TCI_{t-2}) \tag{7}\]  
where \(t\) is the eight-day time point in the growing season; and a is negative while b, c, and d are positive. Figure 11 illustrates the simulation of the differences between the values of DISS for climatic areas represented by various MedHTC. For the same values of the satellite indices in (7), the drought index DISS is higher for areas characterized by a more humid climate and lower for drier areas than for the average hydrothermal conditions. This allows us to compare the results from various regions using the uniform scale. It also allows us to maintain the characteristics of the area, regarding the frequency of the occurrence of drought, compatible with HTC distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**TCI Lag**} & **HTC Accumulation** & **Pearson’s R** & **Standard** & **Number of** & **Range of R for** \\  & **Period in Days** & **Coefficient** & **Error** & **Observations** & **Meteo Stations** \\ \hline
3 & 30 & 0.63 & 0.34 & 8492 & 0.48–0.74 \\
4 & 40 & 0.63 & 0.30 & 7702 & 0.49–0.72 \\
5 & 50 & 0.64 & 0.26 & 7692 & 0.50–0.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of multivariate regression (Equation (4)) in three versions of HTC accumulation (i.e., 30, 40, and 50 days) for agricultural area. Figure 10: Histogram of dependent variable \(YHTC30=\log(HTC30)-\log(MedHTC30)\). The number of time steps in the moving average phase of (7) is flexible, depending on the period of accumulation of the precipitation and temperature in HTC equation (1). The more the preceding satellite indices are aggregated, the longer the horizon of agricultural moisture conditions that we are interested in.