## 2 Materials and Methods
### Method of Data Analysis
#### 2.3.1 Trend and Variability Analysis for Climate Data  
To analyze the trends and variability of rainfall and temperature in the study area, the study was employed the Mann-Kendall statistical test, Sen's slope estimator, and the coefficient of variation. The Mann-Kendall test, formulated by [44], is a non-parametric method for detecting trends, i.e., it is used for any distribution, not requiring the normality of the data. The test analyzes differences in signs between earlier and later data points, because if a trend is present, the sign values will tend to increase constantly or decrease constantly. The distribution of the test statistic was later provided by [45], specifically for testing non-linear trends and identifying turning points. The test produces a simple statistic (Z) and \(p\)-value, making interpretation straightforward. The Mann-Kendall statistic is calculated as follows [46]:  
\[S=\sum_{j=1}^{n-1}\sum_{i=j+1}^{n}sign(Zi-Zj) \tag{3}\]  
where n is the length series (i.e., number of years), j = 1, 2, 3.........n, and Zj is the variable to be analyzed (e.g., annual rainfall in year j), and for every i > j,  
\[Sign(Zi-Zj)= \begin{cases}+1&if\ (Zi-Zj)>0\\ 0&if\ (Zi-Zj)=0\\ -1&if\ (Zi-Zj)<0\end{cases}  \tag{4}\]  
[PERSON] [45] utilized a normal-approximation test for large datasets comprising over 10 values. This method utilizes a normal distribution with mean and variance to address non-monotonic trends in the data. The computation considers if there are tied groups in the data. A tied group is a collection of sample data with the same value. If the Zj series contain K tied groups (i.e., the K of the values occurs repeatedly), then t\({}_{\text{k}}\) is the value of the kth tied group (k = 1... K). The expected value of S is E(S) = 0, and the variance of S is calculated as:  
\[Var\ (S)=\frac{(n-1)(2n+5)-\sum_{k=1}^{K}t_{k}(t_{k}-1)(2t_{k}+5)}{18} \tag{5}\]  
If there are no tied groups, this summary process can be ignored [30]. In cases where the sample size is >10, the standard normal test statistic Z is computed as follows:  
\[Z= \begin{cases}\frac{S-1}{\sqrt{Var(s)}},&if\ S>0\\ 0\,&if\ S=0\\ \frac{S+1}{\sqrt{Var(s)}},&if\ S<0\end{cases}  \tag{6}\]  
Positive Z values indicate increasing trends, whereas negative Z values show decreasing trends. The \(p\)-value determines the statistical significance of the trend. The lower the \(p\)-value, the more statistically significant the observed difference. The null hypothesis H0 implies no trend (the data are independent and randomly ordered). In contrast, the alternative hypothesis H1 indicates the existence of a trend. [PERSON]'s slope estimator is a non-parametric test used to discover trends in univariate time series drawn from unknown distributions, thus it is an alternative to the least-squares regression line [47]. If Sen's slope is positive, then the trend in the time series is increasing; if it is negative, then the trend is decreasing. [PERSON]'s slope estimator is calculated in the following way. First calculate the slope (Ri) of all the data pairs as in Equation (7):  
\[R_{i}=\frac{Zi-Zj}{i-j}\text{ for }i,j=1,2,3\ldots\ldots\ldots n,\text{ i }>\text{j}, \tag{7}\]  
Thus, we arrive at N = n (n - 1)/2 values of R\({}_{i}\). Sen's slope estimator is the median of these \(N\) values of \(R_{i}\), ranked from smallest to largest, computed as in Equation (8):  
\[R_{med=}\left\{ \begin{array}{l}R_{k}\text{ {\it where }}k=\frac{(N+1)}{2}\text{ {\it if }}N\text{ {\it is odd}}\\ \frac{R_{i}+R_{k+1}}{2}\text{ {\it where }}k=\frac{N}{2}\text{ {\it if }}N\text{ {\it is even}}\end{array} \right. \tag{8}\]  
The coefficient of variation (CV) is a measure of statistical significance that evaluates the variability of a set of values, independent of a unit of measurement. It is also used to compare the distributions within various units [48]. In this study, the coefficient of variation was used to quantify the consistency and variability of rainfall and temperature across time in the study area.