# Spatio-Temporal Variability of Water Temperature of Arctic Rivers in Russia over the Past 60 Years
## 2 Materials and Methods
### Statistical Analysis  
An important task in assessing the magnitude of long-term changes in hydrological characteristics is choosing the year of division of the series of observations into two compared periods. In this paper, we implement two approaches to series separation simultaneously. First, we use the classical approach, comparing data for the periods 1961-1990 and 1991-2020 for each month from May to October at each hydrological post. Second, for each month from May to October, we identify a change point year at each hydrological post, with respect to which subsequent comparisons are carried out. This approach, previously used to study long-term changes in water flow characteristics [51], and thermal runoff and water temperature in the lower reaches of large rivers [41], has not been used for datasets with significant volume and spatial coverage before. The change point years in this study were determined using two well-known methods: the Pettit and the Buishand tests [52; 53]. The results of these tests are relatively close, and they show similar patterns for the identified change point years of hydrometeorological characteristics [54]. The analysis also showed that these tests produced similar results for the water temperature series. In the paper, we discussed the separation of observations using the Buishand test because it is more sensitive to local extremes in the data, which is a positive aspect given the relatively low inter-annual variability in water temperature. The lengths of the records after the change point year are long enough for statistical analysis (n > 10). Due to the different length of the time series, the value of the standard error of the mean (SEM) was calculated in addition to the mean values and compared for each month. Changes were only considered significant if their magnitudes exceed the SEM for the selected period (1961-1990 or 1991-2021 or before change point year or after change point year) and for the entire observation period. In this paper, we only discuss those values of change that are greater than the SEM for both the period 1961-2021 and the periods before and after 1991, or the change point year. Additionally, only these values were used for spatial zoning. The direction of change was assessed by the presence or absence of trends identified by the Mann-Kendall test, both for the entire study period and for selected sub-periods within. The Mann-Kendall test was chosen for several reasons. Firstly, it is a non-parametric test. Secondly, it is quite sensitive to changes in the data. Disruptions of homogeneity of series were assessed by the parametric [PERSON]'s \(t\)-test and [PERSON]'s test and the non-parametric Mann-Whitney test (U-test). Note that the results of the Student's and [PERSON] tests were identical, so this article will only discuss the results of Student's test. The Mann-Whitney test was selected among non-parametric tests because of its high level of robustness and its ability to detect uniformity violations, even in short datasets, for example, in the case of a relatively early change point year. Thus, the work simultaneously assesses the statistical significance of deviations from both the mean and variance, and changes in the shape of the distribution in a large number of series under consideration. Normality of data distribution within the analyzed time series was determined by Pearson and Kolmogorov-Smirnov tests, which are the most robust normality tests. The significance level in all statistical tests was set at 5%. With such a wide range of statistical analysis techniques, taking into account the extensive spatial and temporal coverage of river water temperature data, we can identify patterns of long-term change with high accuracy and assess the impact of various factors on these changes. Data analysis was performed using Python 3.0, specifically the numpy, pandas, SciPy, pymannkendal, pyhomogenity, xarray, and seaborn libraries. The gaps in the data were filled in using numpy.NaN (IEEE 754 floating point representation of Not a Number).