## Internal Stochastic Structure of Annual Discharge Time Series of Serbia’s Large Rivers - page 03

When the multi-temporal approach is used in trend analysis, additional parameters are introduced, such as the length of the time series and the position of the analyzed segment of the time series in the time domain. The goal of the analysis is to indicate whether periodic multiple-year alternating dry and wet periods in the river basin affect trend direction and intensity.
Long-term periodicity of time series Hydrological processes tend to group within multiple-year wet and dry periods. This feature of the time series is called the Hurst phenomenon (Hurst, 1951). The equation that describes the Hurst phenomenon is: (6) where A continuous spectrum has been used to assess the macroperiodicity of hydrologic annual time series, while a discrete spectrum served as an alternative tool to produce a relative cumulative periodogram for determining significant periods (Stojković et al., 2013). The continuous spectrum is the answer to the question about the nature of change in periodogram intensity at continuous values of frequency (7) where t, f is the frequency and N is the number of members in the time series. It is apparent from Eq. (7) that the spectrum represents the transformation of covariance function R. Bartlett (1946) and Deniel (1946) showed that it was necessary to correct the spectrum derived from Eq. (7), or to correct the values of the covariate function, to facilitate interpretation._{x} |
In order to generate the spectrum applying the N steps, but only the first steps (high values of the covariate function—M values), as demonstrated by the following formula (Blackman and Tukey, 1959):(8) where λ R, while _{x}M(<N) is the length of the truncated covariate function. The value of coefficient λk is computed from:(9) The weight λ M. The average weight is 0.5. If a low value of M is selected, a softer spectrum than the basic spectrum is obtained, according to Eq. (7). If M is high, the deviation from the basic spectrum is smaller. Once the spectrum was established from Eq. 8, the next step was to determine significant periods. Fisher's test was applied to determine the significance of macroperiods (Yevjevich, 1972; Pekarova et al., 2003), with a confidence threshold of α=0.05.The goodness of spectral density adjustment and of the approximate spectrum using the Tukey window was tested by means of ν=2.67N/M and the confidence of the spectrum was determined as follows (Meko, 2011):(10) The frequency bandwidth bw represented the limits within which spectral properties could be recognized. If this bandwidth is too high, close peaks might not be differentiated. For the Tukey window, the correlation between the frequency bandwidth (11) According to Eq. (11), the length of the truncated covariate function |