Internal Stochastic Structure of Annual Discharge Time Series of Serbia’s Large Rivers

Milan Stojković1, Stevan Prohaska1, Jasna Plavšić2




1 Jaroslav Černi Institute for the Development of Water Resources, Jaroslava Černog 80, Belgrade, Serbia, E-mail: This e-mail address is being protected from spambots. You need JavaScript enabled to view it

2 Faculty of Civil Engineering, University of Belgrade, Belgrade, Bulevar Kralja Aleksandra 83, Serbia




The internal structure of hydrologic processes is discussed in the paper. Time series of mean annual river discharges are assessed to determine their stationarity, trend and periodicity. The time series of annual discharges are determined to be non-stationary. The trend varies in different time segments, with changing trend intensity and direction. Hurst analysis shows that the time series of annual discharges have a long memory. Long-term periodicity of annual time series is considered, while macroperiods are determined and classified into four groups of mean values: 9.5, 13.5, 22.5 and 31.5 years. Secondary microperiodicity is found to exist in the interval from 3.6 to 4.9 years.

Keywords: multi-temporal trend analysis, Mann-Kendall trend test, macroperiodicity, Blackman-Tukey method




Assessment of the internal structure of hydrologic time series is an important contributor to the definition of the rules that govern the development of a process, which is the basis for decision-making in water management. The question whether this internal structure has changed over the past decades, or whether climate change is having an impact on the development of the hydrologic process, is being asked with increasing frequency.

Yevjevich proposed the parametric TIPS method (Yevjevich, 1984) for defining the stochastic structure of daily hydrologic time series. This method is used to represent a time series via its tendency, intermittency, periodicity and stochasticity (TIPS). The intermittent component characterizes time series of daily precipitation, where there are frequent discontinuities. Intermittency is not characteristic of hydrologic time series of large rivers, but trends and periodicity are. The periodicity is comprised of intra-annual seasonal periodicity and long-term macroperiodicity.

Parametric and nonparametric tests are applied in trend analyses of time series. The statistical significance of a parametric linear trend is assessed by means of a test proposed by Kendall and Stuart (1966). This test was later modified (Varga and Prohaska, 1976) in order to standardize it for several time series of different lengths. Many trend analyses are based on the nonparametric Mann-Kendall trend test (Kendall, 1962; Douglas et al., 2000).

According to that test, members of a time series are ranked and each data point in the time series is compared with the data points that follow in time. A trend analysis may be misleading if long-term discharge fluctuations are disregarded; to assess the trend, several full cycles of the time series need to be considered. Not addressing spatial correlation may pose an additional problem, leading to a dramatic reduction in the amount of data for trend analysis (Douglas et al., 2000). Techniques like pre-whitening, variance correction and block boot-strap (Khaliq et al., 2009) are used to remove the effect of correlation between hydrologic time series. A trend analysis requires trend assessment using data from the first to the last year of the time series. The multi-temporal approach is an alternative trend assessment method, which considers a combination of subseries, from the first to the last member of the time series. A subseries constitutes a set of time series members in a continuous multiple-year period. McCabe and Wolock (2002) were the first to propose visualizing a time series trend by combining subseries; they used the Kendall-tau nonparametric correlation statistic to assess the trend. This approach was followed in trend analyses of maximum annual discharges in Germany (Petrow et al., 2009) and Switzerland (Schmocker-Fackel and Naef, 2010). Hannaford et al. (2013) analyzed time series of mean annual and maximum discharges using the multi-temporal approach to assess a large number of small catchments in Europe. They grouped the discharge time series by catchment and region, and standardized the time series members within each group. Then they smoothed the average standardized discharges by the LOESS method. Using this approach, they demonstrated that annual discharge fluctuations have a significant effect on the time series trend. They applied the Mann-Kendall test to determine the significance of the trend.