# Shifting of Meteorological to Hydrological Drought Risk at Regional Scale
## 2 Materials and Methods
### Identification of Linkage between Meteorological and Hydrological Drought and Lag Time  
A regression analysis is a statistical technique for estimating the magnitude of the effect of any independent time series (hereafter the meteorological drought index) on any dependent time series (hereafter the hydrological drought index) [38]. In particular,regression analysis is used to better understand the relationships between independent and dependent time series. As a result, a linear regression was used to evaluate the relationship and temporal changes in drought. Coefficient of determination (R\({}^{2}\)), slope, and \(p\)-value were calculated for each relationship using the two-tailed t-test, and R2 was used to indicate the best relationship between SDI and SPI [38]. Equation (5) depicts the mathematical expression of regression between any independent and any dependent time. In the current study, several simple linear regression models were developed between SPI and SDI of the same time scale, e.g., SPI-3 versus SDI-3; SPI-6 versus SDI-6; and similarly for other time scales.  
\[Y=\beta_{0}+aX+\varepsilon \tag{5}\]  
where \(Y\) is the dependent time series, \(a\) is the regression coefficient, \(X\) is the independent time series, \(\beta_{0}\) is the intercept, and \(\varepsilon\) is an error term.  
Furthermore, in this study, moving average (running average) analysis was performed at multiple time scales to better assess the correlation between SPI and SDI. It is a quite simple statistical analysis tool used as the fluctuation, trend, or lagging indicator. For a given time series and a fixed size subset, the first value of the moving average is calculated by averaging the fixed subset of the given time series. After this, the subset is shifting forward by including the next number and excluding the first number in the subset. For example, Let \((X_{1},\,t_{1})\), \((X_{2},\,t_{2}),\,\ldots\,\,\ldots\,\,(X_{n},\,t_{n})\) represents SPI or SDI time series, \(X_{1},\,X_{2},\,\ldots\,\,\), \(X_{n}\) are the values of SPI or SDI; against time periods \(t_{1},\,t_{2},\,\ldots\,\,t_{n}\), respectively. The first two moving averages of order \(k\) (here 2) can be calculated as given in Equations (6) and (7). A similar procedure was adopted up to the nth value.  
\[\begin{array}{c}\text{Figure 3 illustrates the severity of meteorological drought based on the 12-month time scale for both hydrological sub-basins and the difference in their SPI-12 values during the study period (i.e., 1983-2015). Based on the analysis, it resulted that the years from 1998 to 2004 were the driest years of the selected time series. Moreover, the difference in meteorological drought (SPI-12) severity at both sub-basins represents the notable spatial variations across the Soan river basins. The Mann-Kendall test was also used on SPI-12 to figure out the trend in meteorological drought at sub-basin 1 and sub-basin 2. The analysis showed an increase in meteorological drought severity during the past 33 years, and a significant decreasing trend in SPI-12 values was observed downstream, i.e., sub-basin 2. The Z-value of the Mann-Kendall test was 1.78 at upstream sub-basin 1, and 1.71 at downstream sub-basin 2.