## 2 Materials and Methods
### Study Area
Standard error (SE):  
\[SE=\sqrt{\frac{1}{n-2}\left\{\sum_{i=1}^{n}\bigl{(}M_{i}-\overline{M}\bigr{)}^ {2}-\frac{\left[\sum_{i=1}^{n}\bigl{(}P_{i}-\overline{P}\bigr{)}\bigl{(}M_{i} -\overline{M}\bigr{)}\right]^{2}}{\sum_{i=1}^{n}\bigl{(}P_{i}-\overline{P} \bigr{)}^{2}}\right\}} \tag{1}\]  
Maximum error (ME):  
\[ME=max|P_{i}-M_{i}|_{i=1}^{n} \tag{2}\]  
Root mean square error (RMSE):  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}\bigl{(}P_{i}-M_{i}\bigr{)}^{2}}{n}}\frac{100}{ \overline{M}} \tag{3}\]  
Mean absolute percentage error (MAPE):  
\[MAPE=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{M_{i}-P_{i}}{M_{i}}\right| \tag{4}\]  
Coefficient of determination (R\({}^{2}\)):  
\[R^{2}=\frac{\sum_{i=1}^{n}\bigl{(}M_{i}-\overline{M}\bigr{)}^{2}}{\sum_{i=1}^ {n}\bigl{(}P_{i}-\overline{M}\bigr{)}^{2}} \tag{5}\]  
Nash-Sutcliffe model efficiency coefficient (NSE):  
\[NSE=1-\frac{\sum_{i=1}^{n}\bigl{(}P_{i}-M_{i}\bigr{)}^{2}}{\sum_{i=1}^{n} \bigl{(}M_{i}-\overline{M}\bigr{)}^{2}} \tag{6}\]  
The above statistical metrics are often used when validating models [27; 35; 36]. Moreover, for the purpose of the estimation of errors related to the reconstruction, the application of a linear regression equation was proposed in the form of P = aM + b, describing the dependency between measured (M) and forecasted water temperatures (P). The values of the equation coefficients can be calculated from the following formulas:  
\[a=\frac{n\sum_{i=1}^{n}M_{i}P_{i}-\sum_{i=1}^{n}M_{i}\sum_{i=1}^{n}P_{i}}{n\left( \sum_{i=1}^{n}M_{i}^{2}\right)-\left(\sum_{i=1}^{n}M_{i}\right)^{2}} \tag{7}\]  
Figure 1.0  
The analysis of the location of the regression line P = aM + b in reference to the regression line in the form of P = M (full compatibility of results P and M) permits drawing detailed conclusions regarding the prediction (reconstruction) results. Solving the system of equations (Equation (9)) allows for the determination of the values of M\({}_{0}\) and P\({}_{0}\) (i.e., point of line intersection). \[\left\{ \begin{aligned} P=aM+b\\ P=M\end{aligned} \right. \tag{9}\]  
From the point of view of the assessment of prediction results themselves, it is possible to underestimate or overestimate the values throughout the variation range, or to under- or overestimate the values in certain intervals (Table 2). At the final stage of model validation, directions and trendline slope values for the courses of river water temperatures in the period from 1987 to 2020 were also obtained, based on original measurement data and based on data with partial reconstruction performed for the years 2015-2020. A Mann-Kendall test [37] was used to determine the direction of changes in river water temperatures. The extent of river water temperature changes was determined using a non-parametric Sen test [38]. In order to remove the auto-correlation from the data series, a trend-free prewhitening procedure was used [39]. The adopted procedure is the most frequently used when analysing hydrological data [40; 41]. This is because the method based on the Mann-Kendall and Sen test is less sensitive to extreme values compared to the linear regression method and Student's \(t\)-test. A modified [PERSON] and Spearman's Rho Trend Tests package developed by [PERSON] and [PERSON] [42] was used to analyse the direction and extent of river water temperature changes.