*3.3. Statistical Analyses*

We applied different statistical tools and tests to analyze the relationship between *T<sup>s</sup>* and *Ta*. Firstly, the coefficient of correlation (*r*), coefficient of determination (R<sup>2</sup> ), standard error of regression (SE), and root mean square difference (RMSD) between *T<sup>s</sup>* and *T<sup>a</sup>* for all the stations was calculated. The SE is the standard deviation of the difference between two datasets while RMSD is the square root of mean of squared difference between two datasets. The R<sup>2</sup> explains the efficiency of the regression model. In other words, it is the degree to which the independent variable will be able to explain the dependent variable. During the analysis, the *T<sup>a</sup>* was considered to be the dependent variable (y) and *T<sup>s</sup>* was considered as the independent variable (x). The *p*-values for all the analyses were <0.01 at 99% confidence level. Additionally, we estimated these statistical parameters for all available data for different climate zones namely monsoon-dominated, transition, westerlies-dominated, and precipitation shadow and for all the stations. The value of modified R<sup>2</sup> which is adjusted for the number of predictors was observed to be around unadjusted R<sup>2</sup> and therefore was not shown in the table. The *p*-value for each of the analyses was found to be less than 0.01 at 99% confidence level showing the effect of predictors. In addition, we analyzed the variation in the magnitude of the coefficient of the difference between *T<sup>s</sup>* and *T<sup>a</sup>* observed after the multiple regression taking January as the base month. We also plotted the box and whisker plots for the daily difference between *T<sup>s</sup>* and *T<sup>a</sup>* to graphically represent the overall range of the data, median of the data, and distribution of the data in different quartiles.

#### **4. Results**

### *4.1. Ts vs. Ta Relationship*

We performed different statistical analyses to derive several first-hand conclusions regarding the relationship between *Ts* and *Ta* in the Himalayan region. The results show a strong relationship between observed daily mean *Ta* and its respective daily mean *Ts* in general for all the stations (R<sup>2</sup> = 0.77, RMSD = 5.9 ◦C, SE = 4.76, *n* = 11,101, *p*-value <0.01 at 99% confidence level) with variations corresponding to the altitudinal locations of the stations (Figure 2 and Table 3). The strongest relationship between *T<sup>a</sup>* and *T<sup>s</sup>* at daily scale was observed for Shimla (R<sup>2</sup> = 0.94; RMSD = 1.5 ◦C, SE = 1.2 ◦C, *n* = 304, *p*-value <0.01 at 99% confidence level) and Mukteshwar stations (R<sup>2</sup> = 0.94; RMSD = 1.6 ◦C, SE = 1.2 ◦C, *n* = 355, *p*-value <0.01 at 99% confidence level) which are located on the southern slopes in monsoon-dominated precipitation regime. The coefficient of determination is considerable for all the stations (R<sup>2</sup> > 0.69, *p*-value <0.01 at 99% confidence level) at daily scale.

**Table 3.** Summary of all the statistical tests used for analysis between *Ts* and *Ta* for all the stations, climate regimes and for all observations at daily and 8-day scale. (R<sup>2</sup> = Coefficient of determination; SE = Standard Error of Regression; RMSD = Root mean square difference).


**Figure 2.** The scatter plot between daily *Ts* (x-axis) and *Ta* (y-axis) for all the stations and overall observations with respective coefficient of determination (R<sup>2</sup> ) and root mean square difference (RMSD) in ◦C.

The R<sup>2</sup> and RMSD for all the stations show slight improvement for 8-day average (Figure 3 and Table 3). The relationship between *T<sup>a</sup>* and *T<sup>s</sup>* for 8-day average was also found to be strongest for Shimla (R<sup>2</sup> = 0.97; RMSD = 1.4 ◦C, SE = 0.96 ◦C, *n* = 55, *p*-value <0.01 at 99% confidence level) and Mukteshwar stations (R<sup>2</sup> = 0.96; RMSD = 1.2 ◦C, SE = 1.05, *n* = 63, *p*-value <0.01 at 99% confidence level). Overall, the *T<sup>a</sup>* and *T<sup>s</sup>* relationship was found to be stronger (R<sup>2</sup> = 0.96; RMSD = 5.7 ◦C, SE = 4.5, *n* = 3552, *p*-value <0.01 at 99% confidence level) at 8-day scale for all the stations as well. The regression equation for all the analyses was also given which can be used for estimating *T<sup>a</sup>* for different climate regimes with continuity over large spatiotemporal domain using *Ts* (Table 3) at both daily and 8-day scales.

**Figure 3.** The scatter plot between 8-day *Ts* (x-axis) and *Ta* (y-axis) for all the stations and overall observations with respective coefficient of determination (R<sup>2</sup> ) and RMSD in ◦C.

The number of data points available for 8-day analysis is significantly less in comparison to the daily analysis (Table 1). Additionally, the use of 8-day data gives spatiotemporal continuity due to correction of cloud contaminated pixels but poses a restriction on the frequency of comparisons. We decided to represent the analysis of the relationship between *Ta* and *Ts* at daily scale henceforth because there was very small improvement in the results observed after the use of 8-day data and the daily observation are crucial for different geophysical models as explained in the Introduction section. We noticed certain spatial patterns in the daily differences between *T<sup>s</sup>* and *T<sup>a</sup>* which are discussed in detail in the following paragraphs.

#### *4.2. Altitudinal Relationship*

First, we present the relationship between *T<sup>s</sup>* − *T<sup>a</sup>* by considering altitudinal positions of the stations. The variation in altitude affects the *T<sup>a</sup>* due to difference in density of air which causes a reduced green-house effect in the higher reaches. The RMSD between *T<sup>s</sup>* and *T<sup>a</sup>* for stations has a direct correlation with the elevation of the station (Figures 2–4). Although, the RMSD increases systematically with increase in elevation in general, small variation in this trend is observed for northernmost stations (Skardu and Srinagar). The annual mean RMSD is strongly correlated to the elevation (R<sup>2</sup> = 0.74) in general (Figure 5a) except for two stations (Skardu and Srinagar), which even when located at comparatively low elevations show higher magnitude of RMSD (Figure 4). The R<sup>2</sup> is stronger for monsoon season (Figure 5b) in comparison to annual (Figure 5a) and summer season values (Figure 5c). The observed *T<sup>a</sup>* was unavailable for Skardu for winter months (Figure 5d). Therefore, the R<sup>2</sup> is highest for winter when compared to monsoon, summer, and annual analysis. Furthermore, the magnitude of *T<sup>s</sup>* − *T<sup>a</sup>* is observed to be higher in summer months in comparison to the winter months for all the stations in general (Figure 4).

**Figure 4.** Graph showing the monthly mean of RMSD between daily *T<sup>s</sup>* and *T<sup>a</sup>* for different stations with respective elevation. The RMSD is higher in summer months and increases with increase in elevation.

**Figure 5.** The graph showing the relationship between average of RMSD between daily *Ts* and *Ta* for each station and its corresponding elevation for (**a**) the entire study period, (**b**) monsoon (JASO), (**c**) summer (MAMJ), and (**d**) winter (NDJF). The map shows the precipitation intensity (mm/h) data from Tropical Rainfall Measuring Mission (GES DISC, 2016) for the period 1 January 1999 to 31 December 2017 plotted through GIOVAANI (https://giovanni.gsfc.nasa.gov/giovanni/) [41].
