Evolution Law Analysis on Monthly Streamflow Series in Tuwei River Watershed, China

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1 International Journal of Applied Environmental Sciences ISSN Volume 11, Number 3 (016), pp Research India Publications Evolution Law Analysis on Monthly Streamflow Series in Tuwei River Watershed, China Zhang Yinin 1, Yang Huilong, Wang Lishu 1, Du xinyan 1, Cheng Congmin 1 and Ju Juanli 3, * 1 College of Water Conservancy and Hydropower, Hebei University of Engineering, 178 S. Zhonghua Street, Handan, Hebei , China. Sichuan Water Resources and Hydroelectric Investigation & Design Institute, \99 Changjiang West Road, Deyang, Sichuan , China. 3 College of Water Resources and Architectural Engineering, Northwest A&F University, 3 Weihui Road, Yangling, Shaanxi 71100, China. Abstract Fractal theory was utilized to analyze long memory characteristic and evolution rule of streamflow time series considering its highly nonlinear characteristics. Some basic characteristic values were computed according to monthly streamflow series on Gaojiachuan ( ) and Gaojiabu ( ) stations in Tuwei River watershed. After that, evolution law of monthly streamflow series was analyzed by employing V/S and MF-DFA methods. Results indicated that the largest streamflow occurred in July and August and it occupied 10.6% and 7.65% within annual total streamflow, respectively. M N values were all above during the V/S analysis, which demonstrated that streamflow time series had long memory in the study watershed and this evolution trend in the future would be opposite compared with the past instead. At the same time scales, the corresponding generalized Hurst indexes of the original and random seuences were a kind diminishing function, however, H'() values were generally larger than that of H(). Overall, monthly streamflow series in Tuwei River watershed had long memory and also were multifractal process. Keywords: Monthly streamflow series, V/S analysis, MF-DFA method, generalized Hurst index.

2 73 Zhang Yinin et al 1. INTRODUCTION Hydrologic system is a serious non-linear system. Nonlinear methods and new methods and theories of math science need to be applied to describe future streamflow reasonably (Huang et al. 008). For time series studies, Giraitis et al. (003) constructed V/S (rescaled variance) method diagnosing long memories of series. V/S statistics are stable while testing long memories of series from Monte Carol perspective. After that, Sun et al. (011) used V/S analysis method to study long memories of annual streamflow series. Although V/S analysis method was much more stable and effective compared with R/S analysis method, both of them were based on Hurst index. Wen (010) attempted to test V/S analysis method from statistic perspective and good results were obtained. The scale of time series should be unchanged if V/S analysis method was used to test it. However, series usually have trend fluctuation, which lead to wrong test for V/S analysis method. Meanwhile, Hurst index is a constant if single fractal was applied to describe the variation of series. Multi-fractal Detrended Fluctuation Analysis (MF-DFA) method was introduced by Kantelhardt et al. (00), in which Hurst index was added based on DFA. This model can not only overcome noise disturb due to random factors in time series and avoid the wrong judgment to long range relevance, but also accurately excavate variation characteristics of sub-series with different waves. It provided an objective judgment standard for the validity of judgment and also expressed optimal in the application of multi-fractal analysis for non-stable time series (Huang et al. 008). Thus, MF-DFA method had been used widely in hydrologic field (Kantelhardt et al. 006; Movahed and Hermanis 008; Zhang et al. 009; Rego et al. 013; Li et al. 015). Tuwei River is originated from Gongbohaizi of Jinjie town, Shenmu county, Shannxi Province, China and passes through Jinjie, Gaojiabu, Qiaochatan, et al. and finally converges Yellow River in Hekoucha village in Wanzhen town. Streamflow in Tuwei River watershed is mainly originated from precipitation. Interannual variability of streamflow is similar to precipitation. Since 1980s, annual streamflow decreased dramatically. Inflow of Tuwei River is relatively concentrated. And 78.5 percentege of streamflow is from Gaojiabu of upstream (Gao et al. 006). On the basis of calculating essential characteristics of monthly streamflow in Gaojiachuan ( ) and Gaojiabu ( ) hydrologic stations, V/S analysis method and MF-DFA method were used to describe the evolution of streamflow series in Tuwei River watershed in this study.. METHODS.1 Statistic characteristics Streamflow intra-annual distribution non-uniform coefficient (C v ) and complete adjustment coefficient (C r ) were selected to describe the non-uniform of streamflow

3 Evolution law analysis on monthly streamflow series in Tuwei River watershed 733 intra-annual distribution in Tuwei River watershed. The corresponding computational formula were expressed as (Wang et al. 008): C / R R R / R (1) V i i 1 i 1 1 i C t R t R / R( t ) r i 1 i 1 () t 0, R( t) 1, R( t) where, R i is monthly streamflow; R is the average monthly streamflow; The larger C v value was, the more non-uniform interannual distribution of streamflow was. To calculate the concentration ratio (C d ) and concentration period (D), monthly streamflow was considered as a vector. Values of monthly streamflow was the vector length, and the corresponding month was the vector direction. Position angle of every month from January to December was 0, 30, 60,, 360, respectively. Meanwhile, monthly streamflow was divided as components on x and y directions. The corresponding composite vector was represented as (Wang et al. 008): R R () 1 R R( i )cos x i 1 i 1 R R( i )sin (3) y i 1 i Then, the combination of streamflow was expressed as Thus, C d and D were defined as: R R R. x y 1 C R / R( i ) D arctan( R / R ) (4) d i 1 y x Where, D clearly shows the total effects of the combined monthly streamflow. That is, the time of the largest monthly streamflow occurs. C d reflects the percentage of streamflow in D occupying the total annual streamflow.. Analysis on the long memory of monthly streamflow As for streamflow time series x( t) ( t 1,,, N), statistics of V/ S analysis method was expressed as (Giraitis et al. 003; Wen 010): N k N k 1 1 M ( x x ) ( x x ) ˆ N ( ) j N j N sn, N k 1 j 1 N k 1 j 1 (5) where 1 sˆ ( x x ) ( ) N ˆ N, j N j j N j 1 j 1 (6)

4 734 Zhang Yinin et al j ( ) 1 j, 1 N j 1 ˆ ( x x )( x x ), 0 j N. j i N i j N N i 1 (7) For the optimal value of window width (), there is no good solutions. Giraitis et al. (003) suggested 0, 1,, 5, 10, 0 and 30 to, and indicated that values were correlated with the length of time series. According to V/S analysis method, V n ( V / S) n n.3 MF-DFA method V n statistics was defined as: (8) As for streamflow seuence x( t) ( t 1,,, N),the calculation steps of MF-DFA method were concluded in the following (Huang et al. 008; Wan et al. 01; Chen et al. 011): (1) Calculate the accumulated deviation of seuence x( t) ( t 1,,, N), i y( i) ( x( t) x)( i 1,,, n) (9) t 1 () yi () and its inverted seuence were partitioned subinterval with the same length. Every sub-range had s observed values (6 s N/6), that is, m mutually disjoint sub-range could be derived. s observed values within every sub-range v( v 1,,, m) were fitted using m order polynomial. The corresponding polynomial was expressed as: k yˆ () i a a i a i a i ( i 1,,, s; k 1,,3, ) (10) v 0 1 k (3) Mean suare error F v, s was calculated by the following euations: s 1 F v s y v s i yˆ v i v m s t 1, ( 1) ( ) ( 1,,, ) s 1 F v s y n v m s i yˆ v i v m m m s i 1, ( ) ( ) ( 1,,, ) (4) order wave function of the whole seuence was represented as: (11)

5 Evolution law analysis on monthly streamflow series in Tuwei River watershed 735 m 1/ 1 / F s F v, s m v 1 1 F0 s F v s 4m m exp ln, 0) v 1 (1) For the larger s value, F () s was increasing as the form of power-law. Different values described the effect of different levels of volatility on F () s. Specifically, when =, multiple fractal would be degenerated into a single fractal; When <0, F () s value mainly depended on the small fluctuation of F v, s ; When >0, F () s value mainly depended on the large fluctuation of F v, s. (5) F () s ~ s H( ) was expressed as () H( ) F s As. Natural logarithm was conducted on both sides of the former euation. Then linear regression was made using Least Suare Method. The derived estimate slope was the order generalized Hurst index (H()). When H() was not rely on, indicating that seuence a single fractal process; On the contrary, the series was a multi-fractal process. x i was 3 RESULTS 3.1 Statistic characteristic of monthly streamflow Statistic characteristic values of Gaojiachuan and Gaojiabu stations in Tuwei River watershed were shown in Table 1. Skewness of streamflow series of both stations were positive and the Kurtosis values were greater than 3, indicating that the distribution of streamflow series of both stations is the shape of peak and rear on the whole. They were the proper characteristics of fraction distribution (Zhang et al. 010). The calculation results of C v and C r showed that interannual streamflow of Gaojiachuan station was much more non-uniform compared with that of Gaojiabu station. The mean values of C d at the time scales for Gaojiachuan and Gaojiabu stations were and 765, respectively. Meanwhile, the corresponding D existed in July and August.

6 736 Zhang Yinin et al Table 1: Statistic characteristic values of streamflow time series in Tuwei River watershed Hydrologic Mean Standard Skewness Kurtosis Min Max stations m 3 /s deviation m 3 /s m 3 C v C r C d /s Gaojiachuan Gaojiabu V/S analysis results Considered the above streamflow series were comply with the fraction distribution, V/S analysis method (Giraitis et al. 003; Wen 010) was used to study the long memory of the monthly streamflow since self-similarity was a typical feature of the fraction and its scale was certainly unchanged. In this study, was assigned within the length of time series. The computation results of statistics M N () was shown in Table. If M N was larger than , the time series would have obvious long memory at 5% significant level (Zhang et al. 010). The analysis results by V/S method showed that M N values of monthly streamflow series were all greater than even though was assigned as different values, showing that monthly streamflow series of both stations had remarkable long memories. With the increase of values, variations of M N () values appeared a trend of decreasing and then increasing. Table : Analysis results of V/ S analysis method of streamflow series in Tuwei River watershed Gaojiachuan M N () Gaojiabu M N () V n test of V/S analysis V n values of monthly streamflow series in Gaojiachuan and Gaojiabu stations were calculated by Euation (8). And then the relationship between V n and logn was shown in Figure 1. Vn statistics curve was downward on the whole, indicating that the evolution trend of monthly streamflow in the future would be opposite with the trend in the past. However, Vn statistics curve of Gaojiachuan station in recent 50 years and Gaojiabu station in recent 39 years had two and one prominent breakpoint, respectively.

7 Evolution law analysis on monthly streamflow series in Tuwei River watershed Gao jiachuan 30 5 Gao jiabu 16 0 Vn 1 Vn logn logn Figure 1: watershed Vn statistics curve of Gaojiachuan and Gaojiabu stations in Tuwei River 3.4 Analysis on monthly streamflow time series by MF-DFA method According to (1)~(5) steps, m=1,, 3, 4 was adopted to fit polynomial at every sub-interval for monthly streamflow in Gaojiachuan and Gaojiabu gaging station, respectively. Herein, was assigned as integral number between -10 and 10. Finally, H() was derived by linear regression using Least Suare Method. The relationship between H() and for different steps m were shown in Figure. H() Gaojiachuan m=1 m= Gaojiabu m=3 m= H() Figure : The relationship between H() and with different m values in Tuwei River watershed As shown in Figure, H() values of Gaojiachuan and Gaojiabu stations presented decrease trend with the increase of steps. Thus, the original time series were multifractal process. At the same time scales, the larger fitting step was, the larger

8 738 Zhang Yinin et al fluctuation intervals of H() was;at the same steps, H() values of monthly streamflow series for both stations were similar except for one step. Then, the original time series were disturbed randomly. The corresponding generalized Hurst index was noted as H (). Taking m= and m=3 for examples, the relationship between H () and were shown in Figure 3. Compared with the original time series, H () values was greater than that of H() as a whole, although the relationship between H () and was also decreasing. Generalized Hurst Inddex Original seuence Random seuence Generalized Hurst Inddex Original seuence Random seuence Gaojiachan (m=) Gaojiachan (m=3) Generalized Hurst Inddex Original seuence Random seuence Generalized Hurst Inddex Original seuence Random seuence Gaojiabao (m=) Gaojiabao (m=3) Figure 3: The relationship between generalized Hurst index and CONCLUSIONS (1) The basic statistic characteristics of monthly streamflow time series in Gaojiachuan and Gaojiabu stations indicated that the original streamflow series had fractal characteristics. By contrast, the distribution of interannual streamflow in Gaojiachuan station was much more non-uniform. The maximum monthly streamflow appeared in July and August for Gaojiachuan and Gaojiabu stations, respectively. The corresponding streamflow occupied 10. 6% and 7.65% of the total annual streamflow.

9 Evolution law analysis on monthly streamflow series in Tuwei River watershed 739 () For different values, M N statistics calculated by V/S analysis method were both greater than , indicating that monthly streamflow series of Gaojiachuan and Gaojiabu gaging station existed significant long memories. Furthermore, M N () appeared trend decrease firstly and then increase trend with the increase of values. The future evolution trend of monthly streamflow series for both stations would be opposite with the past trend. (3) The results of MF-DFA with different steps showed that the larger fitting step m was, the larger fluctuation intervals of H() was at the same time scales. At the same steps, H() values of monthly streamflow series for both stations were similar except for one step. H () of the original time series disturbed randomly was also a decrease function. But H () values was greater than that of H() as a whole. Overall, streamflow time series in Tuwei River watershed is the multi-fractal process. ACKNOWLEDGMENTS This study was financially supported by Doctoral Degree Construction Project in Hebei University of Engineering ( D). REFERENCES [1] Chen Y, Yu J, Cui C. Efficient market hypothesis and predictability of the international oil price fluctuations: reassessment based on MF-DFA model. Finance & Trade Economics, 011(5):19-135, 3 [] Gao Y, He X, X J. Analysis of the characters of runoff and sediment in Tuweihe River Basin. Journal of Water Resources and Water Engineering, 006(3): (in Chinese) [3] Giraitis L, Kokoszkab P, Leipusc R, et a1. Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics, 003, 11: [4] Huang Q, Zhao X, et a1. Analysis and prediction on runoff time series: theory and method [M]. Zhengzhou: The Yellow River Water Conservancy Press, (in Chinese) [5] Kantelhardt J W, Zschiegner S A, Koscielny B E, et al. Multifractal detrended fluctuation analysis of nonstationary time series.physical A, 00, 316(1-4): [6] Kantelhardt J W, Koscielny-Bunde E, Rybski D, et al. Long-term persistence and multifractality of precipitation and river runoff records. Journal of Geophysical Research, 006 (111).

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