Internal Stochastic Structure of Annual Discharge Time Series of Serbia’s Large Rivers - page 01


The detection of hidden periodicity has for ages been a developing field of statistical mathematics. Shuster (1897) proposed the first approach, which has been widely used and applied to long time series. Significant progress was made at the beginning of the 20th century in connection with the theory of probability and statistics. The contribution of Einstein (1914) is evident in the use of spectral smoothing techniques to better register the periodic component. In the mid-20th century statistical mathematics made further strides in the determination of time series spectra. Like Einstein (1914), Daniell (1946) concluded that a smoothed spectrum is a useful tool for determining the periodicity. Bartlett (1946), Kendall (1948), and Hamming and Tukey (1949) proposed a technique for determining the periodicity, which comprised three possibilities: periodogram smoothing, truncated and smoothed covariance functions, and smoothing of time series members. The ensuing period witnessed significant attention being devoted to spectral assessment, with major contributions coming from Grenier and Rosenblatt (1953, 1957), Parzen (1957a, 1957b) and Blackman and Tukey (1959). Blackman and Tukey (1959) proposed the B-T (Blackman-Tukey) method for truncating and smoothing covariance functions by means of a nonlinear filter. The next significant discovery was the Fourier transform, which speeded up spectral calculations (Cooley and Tukey, 1965). This discovery allowed the spectral theory to be implemented without extensive calculations, in two types of discrete and fast Fourier transforms. Singular spectral analysis was also used. It is a nonparametric method based on the principle of multivariate time series statistics (Broomhead and King, 1986). Further development of spectral assessment included parametric methods based on stochastic linear models (Beran, 1994). Given the fact that time series are non-stationary, the wavelet transform method became increasingly important as a useful tool for analyzing time series periodicity (Labat, 2006).


The present research considers the internal structure of the hydrologic process. The nature of the hydrologic process—stationary or non-stationary—is determined. The internal structure is defined by the average value trend and the periodicity of the hydrologic process. An annual time step of the time series is considered, leaving out the intra-annual seasonal cycle, in order to analyze long-term macroperiodicity. The internal structure analysis addresses the Danube River Basin in South East Europe.


Method and Results

The object of the present analysis is the Danube River, along with its major tributaries (the Sava, Tisa and Velika Morava). The analysis focuses on the territory of Serbia. The needed hydrologic time series were obtained from the National Hydrometeorological Service of Serbia and those from beyond Serbia were made available by the German Federal Institute of Hydrology from Koblenz. The studied gauging stations on the Danube included the station at Bogojevo, where transboundary waters from the Danube River Basin enter Serbia. The exit station was the station at Orsova in Romania, which reflects water contributions from the catchment areas of the Sava, Tisa and Velika Morava. The Sava River was analyzed at the station in Sremska Mitrovica and the Tisa at Senta. The Velika Morava is a national river, whose catchment area occupies some 42% of Serbia's territory. The hydrological analyses were conducted for a synchronous period at all stations from 1931 to 2012. The considered stations are identified in Table 1, along with information on catchment size F, long-term average discharge Q, runoff modulus q, standard deviation σ and coefficient of asymmetry Cs.


Table 1. Gauging stations in the Danube River Basin