\(\begin{equation} \label{eq:3}
Y_t=X_t-r_1X_{t-1}
\end{equation}\)
Yt is the residual time series, Xt is the original sequence, and t represents the sequence rank ranging from 1 to n. If r1 is less than 0.1, the MK test was applied to the original time-series. Otherwise, the Mann-Kendall test was applied to the prewhitened series (x2-r1x1, x3-r1x2,...xn-r1n-1). The Trend-Free Pre-Whitening Mann-Kendall Test (TFPWMK) is a variant of the MK test that specifically addresses autocorrelation issues in time series data. The TFPWMK test first applies a pre-whitening step to remove autocorrelation and then applies the MK test to the pre-whitened data. This approach provides a more accurate assessment of the trend in time series data with autocorrelation compared to the classical MK test. The trend’s slope can be estimated using the TSA \cite{Sen_1968,henri1950} as follows: