Spectral Analysis of Flood Waves in Open Watercourse

Transcription

1 International Symposium on Water Management and Hydraulic Engineering Ohrid/Macedonia, 1-5 September 2009 Paper: A65 Spectral Analysis of Flood Waves in Open Watercourse Neven Kuspilić, Kristina Novak, Eva Ocvirk University of Zagreb, Faculty of Civil Engineering, Zagreb, Croatia, Abstract. The Sava river is one of three longest rivers in Croatia, and therefore very important for water resources management in the Republic of Croatia. The length of the Sava River is 945 km, and its catchment area is 95,720 sq. km. Because of its importance, the Sava is the subject of continuous hydrological research. This paper contributes to this body of research by outlining the application of spectral analyses. Generally, large floods do not result from a single flood wave, but from a number of successive flood waves. This paper is based on the study whose aim was to carry out spectral analysis of successive flood waves on the basis of measured water levels. By Fourier transformation, spectral density functions were developed from measured water levels. They reflect the hydrological character of the studied watercourse and its catchment area. From stable water wave spectra it is possible to derive synthetic level graphs, which, in turn provide the basis for hydraulic analyses of propagation of successive synthetic water waves. Keywords: Spectral analysis, water levels, flood, synthetic flood wave 1 Introduction Hydrological regime of a river is a part of complex hydrological processes and systems in the nature. This paper deals with the hydrological regime of water levels in the Sava River by describing the dominant periods by spectral analysis. Unlike the traditional analyses in the time domain, application of spectral analysis in hydrological practice is infrequent. Moreover, relevant literature tends to provide only theoretical accounts, and lacks reviews of how particular methods are applied [3]. This paper presents an example of application of spectral analysis on hydrological data. The first part briefly describes how the hydrological series was selected and formed. It is followed by a short theoretical background, and it finishes with the discussion related to the obtained results. Our central analyses dealt with power spectral density (PSD)

2 686 obtained by the periodogram method based on FFT algorithm (Fast Fourier Transformation). The periodogram method was used, which gave satisfactory results. FFT algorithm is currently the most widespread in available computer programs, from generally accepted MS Excel to more advanced mathematical programs such as Matlab (MathWorks). 2 Data The analysis involved hydrological data in the form of measured water level graphs from six gauging stations on the Sava River. These are, going upstream, the gauging stations (GS): Slavonski Kobaš, Davor pumping station, Mačkovac weir, Stara Gradiška, Jasenovac, and Crnac. The average distance between the six selected stations is 37 km, on the Sava river reach of km. While selecting the gauging stations we were governed by the idea to make the distance between the first and the last GS not too long, assuming that it would be easier to notice similarities on a shorter river reach, and in the part of the course with approximately similar conditions of sediment discharge and channel behavior. The analysis covered the middle reaches of the Sava River, and therefore the comparison did not include gauging stations upstream from GS Rugvica. The spectral analysis of water levels has been conducted on the data from the past 25 years. However, due to the war in the nineties, the process of recording water levels in the gauging stations was interrupted, which left us with incomplete records. GS Kobaš has continuous records for the entire period The largest problem was GS Stara Gradiška where the gauging records are incomplete in the period from 1991 to 1999 (Table 1). Table 1. List of gauging stations on the Sava, their mutual distances and distance from the river mouth, and the years of missing data (due to war) Gauging station (GS) Years of missing data Distance from river mouth Distance from previous GS [1] [1] Slavonski Kobaš Davor pumping station Mačkovac weir Stara Gradiška Jasenovac Crnac Average 36.9 Hydrological data of GS are given as daily water level observations at hours over the period from 1982 to The data are segmented in the series of 1048 days

3 687 (i.e observations, because observations were taken once a day), each series starting on January 1 of the initial year of the series. Hourly observations were taken because our intention was to analyze the actual state of the watercourse, i.e. the idea was not to analyze the data which had already been statistically processed and/or analyzed. We made sure to make the total number of observations as a power of number 2 (2 11 =2048) because this is the precondition for maximum efficiency of the Fourier transformation in the form of FFT algorithm [2]. In addition to the above condition related to the number of observations, it was also necessary that the number of daily observations (length of the series) describes seasonal and annual behavior of the Sava River. The total length of the analyzed data covers the period of 25 years ( ), which is statistically significant, and the sub-series of 5.6 years (2048 days) within the record cover both seasonal and annual behaviors of the Sava River. 3 Method Algorithms for transformation of discrete data from the time domain into the frequency domain, and vice versa, belong to the group of DFT (Discrete Fourier Transform) algorithms. Several authors have independently worked out fast DFT algorithms, and contributed a great deal to their further development. The work by Tukey and Cooley from 1965 [2] is considered the starting point of modern use of FFT (Fast Fourier Transform), and is nowadays a conventional method widely used in electrical engineering, biomedicine, geophysics, astrology, oceanography, etc. [1], [4],[5], [6]. The forward Fourier transform takes a time-domain signal g(t) (i.e. the observed hydrological time series) and transforms it into a frequency-domain signal, G (f), where f is frequency, and t time (i.e., the time elapsed from the start of the data record analysed). The forward Fourier transform is generally defined as 2πift G(f ) = g(t)e dt (1) wherei = 1, and all other values as previously defined. Also, Fourier transform may be explained as decomposition of time series into sine waves of varying amplitudes, phases and periods. Summing up sine waves of the mentioned characteristics by Fourier analysis, results in the original time series (signal). This physical interpretation by decomposition into sine waves is complicated by the fact that Fourier transform gives complex values. It is necessary to mention that, unlike the simpler Fourier series, the Fourier transform can be efficiently applied to non-periodic functions. Inverse Fourier transform, which transforms the frequency-domain signal back to the time domain, is defined as: 2πift g(t) = G(f )e df (2) From among many equivalent expressions for the above forward-inverse Fourier transform pair, equations (1) and (2) were chosen in the given form as the most prac-

4 688 tical for hydrological applications. The reason is that frequency units are equal to the inversed value of time units of the input signal, g(t), thus, if t is expressed in days, f is in days -1 The equations for Fourier transform pair, generally prevailing in literature, are the following: and where iωt G( ω ) = g(t)e dt (3) 1 g(t) = ω π G( )e 2 i ω t d and ω is angular frequency in radians per unit of time. ω (4) iωt e = cos ωt isin ωt (5) In this analysis, only the one-dimensional form of Fourier transform was used, which is appropriate for hydrological time series because input and output signals are each function of only one independent variable, i.e. time. For the input signal g(t) which assumes real values (as in our case), Fourier transform G(t) is complexly-conjugated symmetrical, and these values can be presented either in complex notation: G (f ) = R(f ) + ii(f ) (6) wherein R(f) and I(f) are real (or cosine) and imaginary (or sine) parts of G(f), respectively, or in polar notation: G(f ) = A(f )e iφ(f ) ) (7) wherein A(f) and Φ(f) are the amplitude and phase of the signal spectrum, respectively. These two notations are related by the following equations: and 2 2 = R(f ) I(f ) (8) A (f ) + Φ(f ) = tan 1 I(f ) R(f ) Graphic presentation of the distribution of Fourier coefficients (given in the forms R(f) and I(f) in the complex plane is difficult to interpret, and there is a need for other presentations, such as periodogram [3]. Periodogram method belongs to nonparametric methods where PSD is determined directly from the signal. The fact that the periodogram is directly proportional to the square of the spectrum amplitude of (9)

5 689 the process enables easy calculation of the periodogram in any mathematical program (e.g. Matlab or MathCAD). Identification of frequency components in the data requires only the first half of the frequency range (from 0 to Nyquist frequency) because the other half is only the reflection of the first. Nyquist critical frequency (f c ) is the highest frequency that can be detected in the data by Fourier transform. If there are frequencies higher than Nyquist s, overlapping occurs and a portion of the signal-related data is lost, and the basic signal can no more be recovered from the sampled signal [3]. 4 Spectral Analyses of Flood Waves The number of signal data is N=2048 which means that the minimum frequency interval is 1/2048= 0, day -1 ; the time interval is Δt=1 day, and Nyquist frequency is 0,5 day -1. Almost all dominant frequencies (peaks) are in the narrow frequency range between 0 and 0,02 day -1 (which corresponds to the period of 50 days). The graphic presentation is given for this range (Figs. 1 and 2). Frequencies higher than 0,1 day -1 (10 days) do not contain any more information about periods. The fundamental idea of spectral analysis of water level graphs was to examine time and spatial stability of the spectrum in 6 GS for the entire period of 25 years. For each day of the analyzed 2048, the signal was obtained by calculating the deviation from the hourly water level at hours from the mean value for the observed series (signal). Thus, the mean value becomes zero around which the water level varies, and we obtain a cyclic phenomenon for which spectral density of power is determined. Time stability was tested by forming first 11 signals for GS Kobaš with time shift of 2 years, taking the first datum with the beginning of the calendar year (e.g. January 1, 1982). These signals form a particular time window of 2048 days, which shifts through the series of 25 years. By overlapping of 11 spectra, two expressedly predominating frequencies are noted: one of 0,0034 day -1 (293 days) and 0,0059 day -1 (171 days). It should be noted that for the time period , the predominant frequency 0,0030 day -1 (341 day) also occurs. The same dominant frequencies were noticed also on the remaining 5 GS (Fig. 1), where the analyses involved from 6 in GS Stara Gradiška (where formation of more spectra was not possible due to the longest pause in measurements), to 10 signals in GS Davor, Mačkovac and Jasenovac (Fig. 1). The same dominant frequencies serve as evidence of the time stability of the spectrum, and physically these dominant periods would represent annual (341 and 293 days) and seasonal (171 days) fluctuations of extreme water levels in the Sava. Somewhat higher spectrum instability was noticed in GS Crnac, which may be explained by the influence of the Kupa River backwater. In spite of that, the same dominant periods may be noticed in this GS as well.

6 690 Fig. 1. Analysis of time stability of spectra in GS Kobaš (a), Davor (b); Mačkovac (c), St.Gradiška (d), Jasenovac (e) and Crnac (f) Spatial stability of the spectrum is proved by comparison of signals in all 6 GS for the same time periods. The comparison was conducted for only 4 time signals, but even this analysis confirmed the same dominant frequencies. A deviation is noticed for GS Crnac for the time sequence , which may also be explained by the influence of the Kupa River backwater, whereas for other spectra a good overlapping may be seen in the shape of spectra. This makes sense in physical terms because the same flood wave passes through all GS in a few days (i.e. very short time period for our analysis (Fig. 2)). GS Stara Gradiška was omitted from analysis for time se-

7 691 quence because of a lack of measured water level data in first 45 days of the year Fig. 2. Analysis of spatial stability of spectrum for time sequences (a), (b), (c), and (d) 5 Conclusions This paper delineates, in a descriptive manner and with the help of spectral analysis, the behavior of water levels in the Sava River. The analysis proves spatial and time stability of flood wave spectra on a km stretch in the middle reaches of the Sava River. This opens the possibility for further research of these spectra towards, which may lead to developing a single synthetic spectrum, as well as forming appropriate synthetic water level graphs. References 1. Alpar B, Yüce H. Sea-level Variations and their Interactions Between the Black Sea and the Aegean Sea. Estuarine, Coastal and Shelf Science. 1998;46(5):

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