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:

For6     (6)

where R is the cumulative deviation from the arithmetic mean of the time series, S is the standard deviation of the time series, and N is the length of the time series. Coefficient h is the Hurst coefficient. It has been demonstrated that if h=0.5, then the time series has no memory and is called random walk (Salas et al., 1988). Low values of the Hurst coefficient, h<0.5, represent time series with a short memory, while high values, h>0.5, indicate that the process has a long memory. The widespread R/S analysis (Blasco et al., 1996) is used to determine the Hurst coefficient; it is based on the different time interval of the time series used to define the ratio (R/S).

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 f. It differs from the discrete spectrum, which determines periodic intensities at accurately defined frequencies. It is therefore a more precise tool for defining hidden periodicity. The spectral function S(f) is expressed as (Meko, 2011):

For7     (7)

where Rx(t) is the covariance function at time step 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 Rx. 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.


In order to generate the spectrum applying the B-T method, the covariate function Rx needs to be smoothed over a certain number of steps (Meko, 2011). This method does not make use of all 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):

For8     (8)

where λk is the weight of the covariate function Rx, while M(<N) is the length of the truncated covariate function. The value of coefficient λk is computed from:

For9     (9)

The weight λk varies from 1 in the zero step to 0 in the final step of window width 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 χ2 distribution. The number of degrees of freedom was defined as ν=2.67N/M and the confidence of the spectrum was determined as follows (Meko, 2011):

For10     (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 bw and length M (Meko, 2011) was:

For11     (11)

According to Eq. (11), the length of the truncated covariate function M needed to be increased so that the frequency bandwidth bw would be as small as possible, to better discern the spectral properties. A general recommendation concerning the value M is that it should fall within the interval (1/20)N<M<(1/3)N (Chatfield, 2004).