ARTICLE IN PRESS. Testing for nonlinearity of streamflow processes at different timescales. Accepted 8 February 2005

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1 Testing for nonlinearity of streamflow processes at different timescales Wen Wang a,b, *, J.K. Vrijling b, Pieter H.A.J.M. Van Gelder b, Jun Ma c a Faculty of Water Resources and Environment, Hohai University, Nanjing 198, China b Faculty of Civil Engineering and Geosciences, Section of Hydraulic Engineering, Delft University of Technology. P.O. Box 58, GA Delft, Netherlands c Yellow River Conservancy Commission, Hydrology Bureau, Zhengzhou 5, China Abstract Accepted 8 February 5 Streamflow processes are commonly accepted as nonlinear. However, it is not clear what kind of nonlinearity is acting underlying the streamflow processes and how strong the nonlinearity is for the streamflow processes at different timescales. Streamflow data of four rivers are investigated in order to study the character and type of nonlinearity that are present in the streamflow dynamics. The analysis focuses on four characteristic time scales (i.e. one year, one month, 1/3 month and one day), with BDS test to detect for the existence of general nonlinearity and the correlation exponent analysis method to test for the existence of a special case of nonlinearity, i.e. low dimensional chaos. At the same time, the power of the BDS test as well as the importance of removing seasonality from data for testing nonlinearity are discussed. It is found that there are stronger and more complicated nonlinear mechanisms acting at small timescales than at larger timescales. As the timescale increases from a day to a year, the nonlinearity weakens, and the nonlinearity of some 1/3-monthly and monthly streamflow series may be dominated by the effects of seasonal variance. While nonlinear behaviour seemed to be present with different intensity at the various time scales, the dynamics would not seem to be associable to the presence of low dimensional chaos. q 5 Published by Elsevier B.V. Keywords: Nonlinearity; BDS test; Stationarity; KPSS test; Chaos detection; Correlation dimension 1. Introduction A major concern in many scientific disciplines is whether a given process should be modeled as linear or as nonlinear. It is currently well accepted that many natural systems are nonlinear with feedbacks over Journal of Hydrology xx (xxxx) 1 many space and timescales. However, certain aspects of these systems may be less nonlinear than others and the nature of nonlinearity may not be always clear (Tsonis, 1). As an example of natural systems, streamflow processes are also commonly perceived as nonlinear. They could be governed by various nonlinear mechanisms acting on different temporal and spatial scales. Investigations on nonlinearity and applications of nonlinear models to streamflow series have received much attention in the past two decades * Corresponding author. Tel.: C address: w.wang@1.com (W. Wang). -19/$ - see front matter q 5 Published by Elsevier B.V. doi:1.11/j.jhydrol HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

2 W. Wang et al. / Journal of Hydrology xx (xxxx) (e.g. Rogers, 198, 198; Rogers and Zia, 198; Rao and Yu, 199; Chen and Rao, 3). Rogers (198, 198) and Rogers and Zia (198) developed a heuristic method to quantify the degree of nonlinearity of drainage basins by using rainfall-runoff data. Rao and Yu (199) used Hinich bispectrum test (198) to investigate the linearity and nongaussian characteristics of annual streamflow and daily rainfall and temperature series. They detected nonlinearity in daily meteorological series, but not in annual streamflow series. Chen and Rao (3) investigated nonlinearity in monthly hydrologic time series with the Hinich test. The results indicate that all of the stationary segments of standardized monthly temperature and precipitation series are either Gaussian or linear, and some of the standardized monthly streamflow are nonlinear. As a special case of nonlinearity, chaos is widely concerned in the last two decades, and chaotic mechanism of streamflows has been increasingly gaining interests of the hydrology community (e.g. Wilcox et al., 1991; Jayawardena and Lai, 199; Porporato and Ridolfi, 199; Sivakumar et al., 1999; Elshorbagy et al., ). Most of the research in literature confirms the presence of chaos in the hydrologic time series. Nonetheless, the existence of low-dimensional chaos has been a topic in wide dispute (e.g. Ghilardi and Rosso, 199; Koutsoyiannis and Pachakis, 199; Pasternack, 1999; Schertzer et al., ). In spite of all the advances in the research on the nonlinear characteristics of streamflow processes, further investigation is still desirable, because on one hand, there is no common knowledge about what type of nonlinearity exists in the streamflow process, and on the other hand, it is not clear how the character and intensity of nonlinearity of streamflow processes changes as the timescale changes. More insights into the nature of nonlinearity would allow one to decide whether a specific process should be modeled with a linear or a nonlinear model. It is hard to explore different types of nonlinearity one by one which may possibly act underlying streamflow processes. We here want to investigate the existence of general nonlinearity in the streamflow process from a univariate time series data based quantitative point of view. However, there is no direct general measure of nonlinearity so far, therefore, testing for nonlinearity basically is carried out by testing for linearity as an alternative. There are a wide variety of methods available presently to test for linearity or nonlinearity, which may be divided into two categories: portmanteau tests, which test for departure from linear models without specifying alternative models, and the tests designed for some specific alternatives. Patterson and Ashley () applied portmanteau test methods to 8 artificially generated nonlinear series of different types, and found that the BDS test is the best and clearly stands out in terms of overall power against a variety of alternatives. The power of BDS test and some nonparametric tests have also recently been compared and applied to residual analysis of fitted models for monthly rainfalls by Kim et al. (3), and the results also indicate the effectiveness of BDS test. As for the test for the existence of chaos, there are many methods available nowdays, among which the correlation exponent method (e.g. Grassberger and Procaccia, 1983a), the Lyapunov exponent method (e.g. Wolf et al., 1985), the Kolmogorov entropy method (e.g. Grassberger and Procaccia, 1983b), the nonlinear prediction method (e.g. Farmer and Sidorowich, 1987; Sugihara and May, 199), and the surrogate data method (e.g. Theiler et al., 199; Schreiber and Schmitz, 199) are commonly used. In this paper, two issues are addressed. First, in Section, streamflow series of different timescales, namely, one year, one month, 1/3-month and one day, of four streamflow processes in different climate regions are studied to investigate the existence and intensity of general nonlinearity with the BDS test. Second, in Section, correlation exponent method will be applied to test for the presence of chaos in the streamflow series of four rivers. Correlation exponent method is the most important method for detecting chaos, and it is used by almost all the researchers for detecting chaos in hydrological processes (e.g. Jayawardena and Lai, 199; Porporato and Ridolfi, 1997; Pasternack, 1999; Bordignon and Lisi, ; Elshorbagy et al., ). The analysis based on the correlation exponent method is done with software TISEAN (Hegger et al., 1999). In addition, in Section, the datasets which are used for this study are described, followed by an introduction to the BDS test in Section 3, and the power analysis of BDS test in HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

3 W. Wang et al. / Journal of Hydrology xx (xxxx) Section 5. Finally, the paper ends with a discussion and conclusion in Sections 7 and 8, respectively.. Data used Streamflow series of four rivers, i.e. the Yellow River in China, the Rhine River in Europe, the Umpqua River and the Ocmulgee River in the United States, are analyzed in this study. The first streamflow process is the streamflow of the Yellow River at Tangnaihai. The gauging station Tangnaihai has a 133,5 km drainage basin in the northeastern Tibet Plateau, including an permanently snow-covered area of 19 km. The length of main channel in this watershed is over 15 km. Most of the watershed is 3 m above sea level. Snowmelt water composes about 5% of total runoff. Because the watershed is partly permanently snowcovered and sparsely populated, without any largescale hydraulic works, the streamflow process is fairly pristine. The second one is the streamflow of the Rhine River at Lobith, the Netherlands. The Rhine is one of Europe s best-known and most important rivers. Its length is 13 km. The catchment area is about 17, km. The gauging station Lobith is located at the lower reaches of the Rhine, near German- Dutch border. Due to favorable distribution of precipitation over the catchment area, the Rhine has a rather equal discharge. The data are provided by the Global Runoff Data Centre (GRDC) in Germany ( The third one is the streamflow of the Umpqua River near Elkton, Oregon in the United states. The drainage area is 9535 km. The datum of the gauge is 9. feet above sea level. The record started from October 195. Regulation by powerplants on North Umpqua River ordinarily does not affect discharge at this station. There are diversions for irrigation upstream from the station. The fourth one is the streamflow of the upper Ocumlgee River at Macon, Georgia. The station Macon has a drainage area of 5799 km. Its gauge datum is 9.8 feet above sea level. The headwaters of the Ocmulgee River begin in the highly urbanized Atlanta metropolitan area, and downstream its watershed is dominated by agriculture and forested areas. The daily discharge data of both the Umpqua River and the Ocmulgee River are available from the USGS (United States Geological Survey) website Monthly series are obtained from daily data by taking average of daily discharges in every month. For the 1/3-monthly series, the 1st and rd 1/3-month streamflows are the averages of the first and the second 1-days daily discharges, and the 3rd 1/3- month discharge is the average of the last 8 11 days daily discharges of a month depending on the length of the month. All the daily data series used here start from January 1, and end on December 31. The statistical characteristics of the streamflow series at different timescales are summarized in Table 1. The plots of mean daily discharges and standard deviations of these streamflow series are shown in Fig BDS test The BDS test (Brock et al., 199) is a nonparametric method for testing for serial independence and nonlinear structure in a time series based on the correlation integral of the series. As stated by the authors, the BDS statistic has its origins in the work on deterministic nonlinear dynamics and chaos theory, it is not only useful in detecting deterministic chaos, but also serves as a residual diagnostic tool that can be used to test the goodness-of-fit of an estimated model. The null hypothesis is that the time series sample comes from an independent identically distributed (i.i.d.) process. The alternative hypothesis is not specified. In this section, the theoretical aspects of BDS test are presented. Embed a scalar time series {x t } of length N into a m-dimensional space, and generate a new series {X t }, X t Z(x t, x tkt,.,x tk(mk1)t ), X t R m. Then, calculate the correlation integral C m,m (r) given by (Grassberger and Procaccia, 1983a): C m;m ðrþ Z M! K1 X Hðr KjjX i KX j jjþ; (1) 1%i!j%M where MZN-(mK1) t is the number of embedded points in m-dimensional space; r the radius of a sphere centered on X i ; H(u) is the Heaviside function, with HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

4 W. Wang et al. / Journal of Hydrology xx (xxxx) Table 1 Statistical characteristics of streamflow series River (station) Yellow (Tangnaihai) H(u)Z1 for uo, and H(u)Z for u%; k(k denotes the sup-norm. C m,m (r) counts up the number of points in the m-dimensional space that lie within a hypercube of radius r. Brock et al. (199) exploit the asymptotic normality of C m,m (r) under the null hypothesis (a) Discharge (m 3 /s) (c) 3 Discharge (ft 3 /s) Period of record Timescale Mean (m 3 /s) Standard deviation (m 3 /s) Yellow River at Tangnaihai Mean SD Day Umpqua River near Elkton that {x t } is an i.i.d. process to obtain a test statistic which asymptotically converges to a unit normal. If the series is generated by a strictly stationary stochastic process that is absolutely regular, then the limit C m ðrþzlim N/N C m;m ðrþ exists. In this case the Day Mean SD (b) Discharge (m 3 /s) (d) Discharge (ft 3 /s) Skewness coefficient Rhine River at Lobith Day Day Mean SD Ocmulgee River at Macon Fig. 1. Variation in daily mean and standard deviation of streamflow processes. Kurtosis coefficient Mean SD ACF(1) 195 Daily /3-monthly Monthly Annual 1.88 K.7.31 Rhine (Lobith) Daily /3-monthly Monthly Annual K.135 K.57.1 Umpqua 19 1 Daily (Elkton) 1/3-monthly Monthly Annual K Ocmulgee Daily (Macon) 1/3-monthly Monthly Annual HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

5 W. Wang et al. / Journal of Hydrology xx (xxxx) limit is C m ðrþ Z ðð Hðr KjjX KYjjÞdF m ðxþdf m ðyþ; () where F m denote the distribution function of embedded time series {X t }. When the process is independent, and since HðrKjjX i KY j jjþz Q m kz1 HðrKjX i;k KY j;k jþ, Eq. () implies that C m ðrþzc1 m ðrþ. Also C m ðrþkc1 m ðrþ has asymptotic normal distribution, with zero mean and variance given by 1 s m;mðrþ Z mðm KÞC mk ðk KC Þ CK m KC m C XmK1 jz1 ½C j ðk mkj KC mkj Þ KmC mk ðk KC ÞŠ: (3) The constants C and K in Eq. (3) can be estimated by C M ðrþ Z 1 X M X M M Hðr KjjX i KX j jjþ; and iz1 K M ðrþ Z 1 M 3 X M iz1 jz1 X M Hðr KjjX j KX k jjþ: X M jz1 kz1 Hðr KjjX i KX j jjþ Under the null hypothesis that {x t } is an i.i.d. process, the BDS statistic for mo1 is defined as p BDS m;m ðrþ Z ffiffiffiffi C M m ðrþ KC1 m ðrþ : () s m;m ðrþ It asymptotically converges to a unit normal as M/N. This convergence requires large samples for values of embedding dimension m much larger than, so m is usually restricted to the range from to 5. Brock et al. (1991) recommend that r is set to between half and three halves the standard deviation s of the data. We find that if r is set as half s, there would be too few or even no nearest neighbors for many points in the embedded m-dimensional space when m is large (e.g. mz5), especially for series of short size (e.g. less than 1); on the other hand, when r is set as three halves s, there would be too many nearest neighbors for many points in the embedded m-dimensional space when m is small (e.g. mz). Such kind of 33 shortage of neighbors or excess of neighbors will 3 probably bias the calculation of C m,m (r). Therefore, 35 we only consider r equal to the standard deviation of 3 the data in this study Test results for streamflow processes 1.1. Stationarity test 3 Because usually linearity/nonlinearity tests (e.g. BDS test) assume the series of interest is stationary, it 5 is necessary to test the stationarity before taking nonlinearity test. The stationarity test is carried out 7 with two methods, one is augmented Dickey-Fuller 8 (ADF) unit root test proposed by Dickey and Fuller 9 (1979), which tests for the presence of unit root in the 5 series (difference stationarity); another is KPSS test 51 proposed by Kwiatkowski et al. (199), which tests 5 for the stationarity around a deterministic trend (trend 53 stationarity) and the stationarity around a fixed level 5 (level stationarity). To achieve stationarity, if a 55 process is difference stationary with unit roots, the 5 appropriate treatment is to difference the series; if not 57 level stationary but trend stationary, which indicates 58 that there is a deterministic trend, then we should 59 remove the trend component from the series. Because on one hand both ADF test and KPSS test 1 are based on linear regression, which has normal distribution assumption; on the other hand, logarithmization can convert exponential trend possibly 3 present in the data into linear trend, therefore, it is 5 common to take logs of the data before applying ADF test and KPSS test (e.g. Gimeno et al., 1999). In this 7 study, the streamflow data are also logarithmized 8 before applying stationarity tests. An important 9 practical issue for the implementation of the ADF 7 test as well as the KPSS test is the specification of the 71 lag length l. Following Schwert (1989); Kwiatkowski 7 et al. (199), the number of lag length in this study is 73 chosen as lzint½xðt=1þ 1= Š, with xz, 1. 7 The stationarity test results are given in Table. 75 All the monthly and 1/3-monthly series appear to be 7 stationary, since we cannot accept the unit root 77 hypothesis with ADF test at 1% significance level 78 and cannot reject the trend stationarity hypothesis and 79 level stationarity hypothesis with KPSS test at 8 HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

6 W. Wang et al. / Journal of Hydrology xx (xxxx) Table Stationarity test results for streamflow series Station Series KPSS level stationary test KPSS trend stationary test ADF unit roots test Lag Results p-value Lag Results p-value Lag Results p-value Yellow (Tangnaihai) Rhine (Lobith) Umpqua (Elkton) Ocmulgee (Macon) Daily 1.3 O.5 1.3!.1 1 K !1 K.138 O O.5 K !1 K 1/ O O.1 8 K !1 K7 montly.113 O O.1 K !1 K7 Monthly.8 O.1.8 O.1 K !1 K O O.1 18 K !1 K9 Annual 3.15 O O.1 3 K.89.53!1 K3 9.1 O O.5 9 K.89.53!1 K3 Daily O !.1 17 K1.8.!1 K O O.1 51 K1.8.!1 K3 1/ O O.1 9 K !1 K5 montly 9.7 O O.1 9 K !1 K5 Monthly 7.88 O O.1 7 K1. 1.3!1 K3. O.1.59 O.1 K1. 1.3!1 K3 Annual 3.75 O O.1 3 K !1 K O O.1 11 K !1 K1 Daily 17.5 O.1 17.!.1 17 K !1 K O O.1 51 K !1 K 1/3-9.1 O O.1 9 K !1 K montly 9.13 O O.1 9 K !1 K Monthly 7.79 O O.1 7 K !1 K8.133 O.1.13 O.5 K !1 K8 Annual 3.11 O O.1 3 K !1 K O O.1 11 K !1 K7 Daily 1.53 O.1 1.8!.1 1 K3.3 1.!1 K O O.1 8 K3.3 1.!1 K115 1/ O O.1 9 K18..33!1 K57 montly 7.11 O O.1 7 K18..33!1 K57 Monthly.97 O.1.8 O.1 K !1 K9.81 O.1.7 O.1 K !1 K9 Annual 3.5 O O.1 3 K.311.7!1 K O O.1 11 K.311.7!1 K Note: Critical value of KPSS distribution for level stationarity hypothesis: 1% w.37; 5% w.3; 1% w.739; Critical value of KPSS distribution for trend stationarity hypothesis: 1% w.119; 5% w.1; 1% w.1. the 1% level. All the series pass KPSS level stationary test, which means that all the series are stationary around a fixed level and there is no significant change in mean. But some daily series cannot pass trend stationary test when the lag is small. This is probably partly because of the influence of serial dependence at short-lags, and partly because the trend fitted to the daily series in trend stationary test could be over-affected by some outlier data, thus make the whole series not stationary around such a biased trend. However, for large lags, all the daily series pass the trend-stationary test, although some pass at comparatively low significance level (O.1). Therefore, all the series are basically stationary, and no differencing or de-trending operation is needed... Nonlinearity test BDS test needs the extraction of linear structure from the original series by the use of an estimated linear filter. Therefore, the first step for the test is fitting linear models to the streamflow series. Because streamflow processes (except annual series) usually exhibit strong seasonality, to analysis HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

7 W. Wang et al. / Journal of Hydrology xx (xxxx) Table 3 Order of AR models fitted to streamflow series Timescale Yellow Rhine Umpqua Ocmulgee Raw Log Log-DS Raw Log Log-DS Raw Log Log-DS Raw Log Log-DS Daily / monthly Monthly Annual 1 1 the role of the seasonality played in nonlinearity test, the streamflow series are pre-processed in two ways, logarithmization and deseasonalization. Correspondingly, the pre-processed series are referred to as Log series and Log-DS series respectively. The Log-DS series is obtained with two steps. Firstly, logarithmize the flow series. Then deseasonalize them by subtracting the seasonal (e.g. daily or monthly) mean values and dividing by the seasonal standard deviations of the logarithmized series. To alleviate the stochastic fluctuations of the daily means and standard deviations, we smooth them with first 8 Fourier harmonics before using them for standardization. Annual series is analyzed without any transformation. All series are pre-whitened with AR models. The autoregressive orders of the AR models are selected according to AIC, shown in Table 3. Residuals are obtained from these models, then the BDS test is applied to the residual series. Test results are shown in Table. It is shown in Table that all the annual series pass the BDS test, indicating that annual flow series are linear. This result is in agreement with that of Rao and Yue (199). Among the monthly series, Log series of Ocmulgee and Log-DS series of Rhine pass the BDS test, while Log-DS series of Ocmulgee narrowly pass the test at significance level.5. But all the other series cannot pass BDS test at.1 significance level. It is noted that, with the increase of the timescale, the nonlinearity decreases. Among the flow series at four characteristic time scales, the strongest nonlinearity exists in daily series and the least nonlinearity exists in annual series. Except for the daily and monthly streamflow series of Ocmulgee, and daily flow of Umpqua, there is a general feature that the test statistics of Log-DS series are smaller than those of 35 the Log series, which implies that deseasonalization 3 may more or less alleviates the nonlinearity. 37 With a close inspection of the residual series, we 38 find that although the residuals are serially uncorrelated, there is seasonality in the variance of the 39 residual series. Therefore, it is worthwhile to have a 1 look at the residuals after removing such kind of season-dependent variance. Table 5 shows the BDS 3 test results for the residual series after being standardized with seasonal variance. 5 Comparing Tables and 5, we can find that, the BDS test statistics of all the series are generally 7 smaller than those of the series before standardization. 8 Especially, 1/3-montly and monthly Log-DS series of 9 the Yellow River, and the monthly Log series of the 5 Rhine River, which are nonlinear before standardization, pass the BDS test at.5 significance level after 5 51 standardization. Therefore, the seasonal variation in 53 variance in the residuals is probably a dominant 5 source of nonlinearity in the 1/3-montly and monthly 55 Log-DS series of the Yellow River, and the monthly 5 Log series of the Rhine River. But all the daily series, 57 most 1/3-monthly series and some monthly series still 58 exhibit nonlinearity even after standardization. That 59 indicates that the seasonal variance composes only a small, even negligible, fraction of the nonlinearity 1 underlying these processes, especially daily streamflow processes. 3 The above analysis indicates that there are stronger and more complicated nonlinearity mechanisms 5 acting at small timescales than at large timescales. As the timescale increases, the nonlinearity weakens, 7 and the effects of seasonal variance dominate the 8 nonlinearity of some 1/3-monthly and monthly 9 streamflow series. 7 Although most monthly flow series and some 71 1/3-monthly series are diagnosed as linear with BDS HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

8 8 W. Wang et al. / Journal of Hydrology xx (xxxx) Table BDS test results for pre-whitened streamflow series Series Rhine (Lobith) Umpqua (Elkton) Yellow (Tangnaihai) Ocmulgee (Macon) Transform Timescale mz mz3 mz mz5 test after, or even before, being standardized by seasonal variance, this does not exclude the possibility that there exists some weak nonlinearity in these series. For example, some studies indicate that monthly streamflow could be modeled by TAR model or PAR model (e.g. Thompstone et al., 1985). TAR model is a well-acknowledged nonlinear model. PAR model is also a nonlinear model, which differs from TAR model in that TAR model uses observed values as threshold whereas PAR model uses season as threshold. Passing BDS test does not mean that there is no nonlinearity such as TAR or PAR mechanism present in the time series. It is possible that BDS test is not powerful enough to detect weak nonlinearity. We will make an analysis on this issue in the next section. Statistic p-value Statistic p-value Statistic p-value Statistic p-value Log Daily /3-montly Monthly Log-DS Daily /3-montly Monthly Raw Annual K K ! 3.91.! 1 K 1 K 1 K Analysis of the power of BDS test We will analyze the power of BDS test with some 755 simulated series. Considering one AR model, two 75 TAR models, two bilinear models and Henon map 757 series of the following form: (1) Autoregressive: x t Z:7x tk1 C3 t ( 7 x t Z:9x tk1 C3 t for x tk1!1 71 () TARK1 : x t Z:3x tk1 C3 t for x tk1 R1 7 ( 73 x t Z:9x tk1 C3 t for x tk1!1 7 (3) TARK : x t ZK:3x tk1 C3 t for x tk1 R1 75 Log Daily /3-montly Monthly Log-DS Daily /3-montly Monthly Raw Annual K K K.8.8 Log Daily /3-montly Monthly Log-DS Daily /3-montly Monthly 3.91.! 1 K ! Raw Annual K.5.83 K K K Log Daily /3-montly Monthly Log-DS Daily /3-montly Monthly Raw Annual K K () Bilinear-1: x t Z:9x tk1 C:1x tk1!3 tk1 C3 t (5) Bilinear-: x t Z:x tk1 C:8x tk1!3 tk1 C3 t HYDROL //5 :5 SHYLAJA XML MODEL 3 pp. 1

9 HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1 Table 5 BDS test results for standardized pre-whitened streamflow series Series Transform Timescale mz mz3 mz mz5 Statistic p-value Statistic p-value Statistic p-value Statistic p-value Yellow (Tangnaihai) Rhine (Lobith) Umpqua (Elkton) Ocmulgee (Macon) Log Daily /3-montly !1 K !1 K Monthly !1 K !1 K !1 K !1 K Log-DS Daily /3-montly Monthly Log Daily /3-montly Monthly.9.9 K K K Log-DS Daily /3-montly Monthly K K K Log Daily /3-montly Monthly !1 K !1 K !1 K Log-DS Daily /3-montly Monthly Log Daily /3-montly Monthly Log-DS Daily /3-montly Monthly W. Wang et al. / Journal of Hydrology xx (xxxx) 1 9 DTD

10 1 W. Wang et al. / Journal of Hydrology xx (xxxx) () Henon map x series : tc1 Z1Kax t Cby t ; y tc1 Zx t az1:; bz:3 In all the above models, {x t } (or {y t }) is time series, and {3 t } is independent standard normal error. Obviously, among the above models, model TAR-1 and Bilinear-1 have weak nonlinearity while model TAR- and Bilinear- have stronger nonlinearity, because TAR- has a larger parameter difference and Bilinear- has a more significant bilinear item. Henon map series is a typical chaotic series (Henon, 197). For model (1) to (5), 1 simulations are generated, and each simulation has 5 points. For Henon series, one simulation with 5 points is generated (referred to as clean- Henon in Table ). Then the Henon series is divided into 1 segments, and each segment has 5 points. To evaluate the influence of noise on BDS test, noise is added to the simulated Henon series (referred to as noise-henon in Table ). The noise is normally distributed with zero mean, and its standard deviation is 5% of the standard deviation of the Henon series. Then we use BDS test to detect the presence of nonlinearity in the simulated series. All the series are pre-whitened with AR models. The test results are shown in Table. It is shown that the hypothesis of linearity for Henon series (pure or with noise) are firmly rejected, which indicates that BDS test is very powerful for detecting such kind of strong nonlinearity. In most cases, BDS test correctly rejects the hypothesis that TAR- and Table Rates of accepting linearity with BDS test based on 1 replications at significance level.5 Bilinear- processes are linear, but wrongly accepts TAR-1 and Blinear-1 processes as linear. That means that although BDS test is considered very powerful for testing nonlinearity, but not powerful enough for detecting weak nonlinearity in TAR-1 and Bilinear-1, whereas such kinds of weak nonlinearity probably present in the streamflow series, because it is impossible that streamflow processes are driven by the mechanism like TAR-, which switches between dramatically different regimes. Therefore, BDS test results tell us that there is strong nonlinearity present in daily streamflow series as well as most 1/3-monthly series, even after taking away the effects of seasonal variance, but there is no strong nonlinearity presents in most monthly streamflow series and some 1/3-monthly series after removing the effects of seasonal variance. However, we cannot say there is no nonlinearity present in those 1/3-monthly and monthly streamflow series even if they pass BDS test, because BDS test is not powerful enough for detecting weak nonlinearity. In addition, comparing the BDS test results for chaotic Henon series with those for streamflow series, while it is not clear whether most 1/3-monthly series and all the daily series have chaotic properties, it seems that all monthly series may not be chaotic because the BDS test p-values for monthly flow series are far much higher than those for chaotic Henon series. We would further detect the existence of chaos with correlation exponent method in the next section. Series mz mz3 mz mz5 p-value Accepted p-value Accepted p-value Accepted p-value Accepted AR(1) TAR TAR Bilinear Bilinear-.79!1 K 5.977!1 K 3.!1 K7 3.17!1 K7 Clean-Henon 7.35!1 K5 3.!1 K8 3.9!1 K115.5!1 K1 Noise-Henon 3.11!1 K !1 K8 5.89!1 K11 5.9!1 K138 Note: p-value in the table is the median value for each group of 1 replications HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

11 W. Wang et al. / Journal of Hydrology xx (xxxx) (a) x (t+1) (c) x (t+1) (a) 1.5 ACF / MI Lag (c) Lag ACF / MI x (t ) x (t ) (b) x (t+7) (d) x (t+) x (t ) x (t ) Fig. 3. x t -x tct state-space maps of daily streamflow series of the Yellow River at Tangnaihai with (a) tz1; (b) tz7; (c) tz1; (d) tz. (b) ACF / MI Lag ACF MI Fig.. ACF and MI of (a) daily, (b) 1/3-monthly and (c) monthly river flow of the Yellow River HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

12 1 W. Wang et al. / Journal of Hydrology xx (xxxx) Test for chaos in streamflow processes with correlation exponent method When testing for general nonlinearity, it is common to filter the data to remove linear correlations (prewhitening) (e.g. Brock et al., 199), because linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as correlation dimension and Lyapunov exponents. But it has been observed that in numerical practice prewhitening may severely impairs the underlying deterministic nonlinear structure of low-dimensional chaotic time series (e.g. Theiler and Eubank, 1993; Sauer and Yorke, 1993). Therefore, mostly chaos analyses are based on original series, and the same in our analysis. Correlation exponent method is most frequently employed to investigate the existence of chaos. The basis of this method is multi-dimension state space reconstruction. The most commonly used method for reconstructing the state space is the time-delay coordinate method proposed by Packard et al. (198); Takens (1981). In the time delay coordinate method, a scalar time series {x 1, x,.,x N } is converted to state vectors X t Z(x t,x t-t,.,x t-(mk1)t ) after determining two state space parameters: the embedding dimension m and delay time t. To check whether chaos exists, the correlation exponent values are calculated against the corresponding embedding dimension values. If the correlation exponent leads to a finite value as embedding dimension increasing, then the process under investigation is thought of as being dominated by deterministic dynamics. Otherwise, the process is considered as stochastic. To calculate the correlation exponent, the delay time t should be determined first. Therefore, the selection of delay time is discussed first in the following section, followed by the estimation of correlation dimension..1. Selection of delay time The delay time T is commonly selected by using the autocorrelation function (ACF) method where ACF first attains zeros or below a small value (e.g.. or.1), or the mutual information (MI) method (Fraser and Swinney, 198) where the MI first attains a minimum. We first take the streamflow of the Yellow River at Tangnaihai as an example to analyze the choice of T. We calculate ACF and MI of daily, 1/3-monthly and monthlyflowseriesoftheyellowriver, shown infig.. Because of strong seasonality, ACF first attains zeros at the lag time of about 1/ period, namely, 91, 9 and 3 for daily, 1/3-monthly and monthly series respectively. The MI method gives similar estimates for T to the ACF method, about approximately 1/ annual period. In practice, the estimate of t is usually application and author dependent nonetheless in practice. For instance, for daily flow series, some authors take the delay time as 1 day (Porporato and Ridolfi, 1997), days (Jayawardena and Lai, 199), 7 days (Islam and Sivakumar, ), 1 days (Elshorbagy et al., ), days (Wilcox et al., 1991) and 1 days (Pasternack, 1999). These differences may arise from different ACF structure. To compare the influence of different T on the reconstruction of state space, we can plot x t wx tct statespace maps for the streamflow series with different T. The best T value should make the state space best unfolded. For the streamflow series of the Yellow River, the x t wx tct state-space maps with small T values (i.e. 1, 7, 1, and ) are displayed in Fig. 3, and the - and 3-dimensional x t wx tct state-space maps with t taken as 1/ of the annual period are displayed in Fig.. Obviously, especially clearly in the 3-D maps, state spaces for daily, 1/3-monthly and monthly streamflow series are best unfolded when delay time TZ91, 9, 3 respectively. We therefore select TZ91, 9, 3 for estimating correlation dimension for the streamflow series of the Yellow River. Similar results are obtained for the sreamflow processes of the Umpqua River and the Ocmulgee River (to save space, the plots are not displayed here). But for the Rhine River, the seasonality is not that obvious. The ACF and MI of daily, 1/3- monthly and monthly flow series of the Rhine River are shown in Fig. 5. If we determine the delay time according to the lags where ACF attains or MI attains its minimum for the Rhine River, the lags would be about days which seems to be too large, which would possibly make the successive elements of the state vectors in the embedded multi-dimensional state space almost independent. Therefore we select the delay time equal to the lags before ACF attains.1, namely, TZ9, 9, 3 for daily, 1/3-monthly and monthly streamflow series, respectively HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

13 W. Wang et al. / Journal of Hydrology xx (xxxx) (a) x (t+91) (c) x (t+9) (e) x (t+3) x (t) x (t) x (t).. Estimation of correlation dimension The most commonly used algorithm for computing correlation dimension is Grassberger - Procaccia algorithm (Grassberger and Procaccia, 1983a), modified by Theiler (198). For a m-dimension phasespace, the modified correlation integral C(r) is defined (b) x (t+18) (d) (f) x (t+18) x (t+) x (t+91) x (t+9) x (t+3) Fig.. -D and 3-D state space maps of (a), (b) daily; (c), (d) 1/3-monthly; and (e), (f) monthly streamflow of the Yellow River at Tangnaihai with delay time tz91, 9 and 3. by (Theiler, 198) CðrÞ Z ðm C1 KwÞðM KwÞ X M X MKi iz1 jzicwc1 x (t) x (t) x (t) Hðr KjjX i KX j jjþ; (5) (d) HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

14 1 W. Wang et al. / Journal of Hydrology xx (xxxx) (a) ACF / MI (c) ACF / MI (a) lnc (r ) (c) LnC (r ).8... (b) ACF / MI Lag Lag ACF Lag Fig. 5. ACF and MI of (a) daily, (b) 1/3-monthly and (c) monthly river flow of the Rhine River. Yellow River (d) Fig.. ln C(r) versus ln r plot for daily streamflow processes. 15 Umpqua River (b) lnc (r ) lnc (r ) MI Rhine River Ocmulgee River HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

15 W. Wang et al. / Journal of Hydrology xx (xxxx) where M, r, H have the same meaning as in Eq. (1), w (R1) is the Theiler window to exclude those points which are temporally correlated. In this study, w is set as about half a year, namely 18, 18, and for daily, 1/3-monthly and monthly series respectively. For a finite dataset, there is a radius r below which there are no pairs of points, whereas at the other extreme, when the radius approaches the diameter of the cloud of points, the number of pairs will increase no further as the radius increases (saturation). The scaling region would be found somewhere between depopulation and saturation. When ln C(r) versus ln r is plotted for a given embedding dimension m, the range of ln r where the slope of the curve is approximately constant is the scaling region where fractal geometry is indicated. In this region C(r) increase as a power of r, with the scaling exponent being the correlation dimension D. If the scaling region vanishes as m increases, then finite value of correlation dimension cannot be obtained, and the system under investigation is considered as stochastic. (a) lnc (r ) (c) lnc (r ) Yellow River Umpqua River Local slopes of ln C(r) versus ln r plot can show scaling region clearly when it exists. Because the local slopes of ln C(r) versus ln r plot often fluctuate dramatically, to identify the scaling region more clearly, we can use Takens Theiler estimator or smooth Gaussian kernel estimator to estimate correlation dimension (Hegger et al., 1999). The ln C(r) versus ln r plots of daily, 1/3-monthly and monthly streamflow series of the four rivers are displayed in Figs. 8 respectively, and the Takens Theiler estimates ( ) of correlation dimension are displayed in Figs We cannot find any obvious scaling region from the Figs Take the Yellow River for instance, an ambiguous ln r region could be identified as scaling region is around ln rz7 7.5 for the three flow series of different timescales. But in this region, as shown in Fig. 1, the increases with the increment of the embedding dimension, which indicates that the system under investigation is stochastic. Rhine River Ocmulgee River (d) (b) lnc (r ) lnc (r ) Fig. 7. ln C(r) versus ln r plots for 1/3-monthly streamflow processes HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

16 1 W. Wang et al. / Journal of Hydrology xx (xxxx) (a) lnc (r ) 7. Discussion Yellow River (c) (d) lnc (r ) 7.1. On the estimation of correlation dimension Three issues regarding the estimation of correlation dimension should be noticed. First, about the minimum data size for estimating correlation dimension. Some authors claim that the size of 1 A (Procaccia, 1988) or1 (C.m) (Nerenberg and Essex, 199; Tsonis et al., 1993), where A is the greatest integer smaller than correlation dimension and m is the embedding dimension, is needed for estimating correlation dimension with an error less than 5%. Whereas some other researchers found that smaller data size is needed. For instance, the minimum data points for reliable correlation dimension pffiffiffiffiffiffiffiffiffi D is 1 D/ (Eckmann and Ruelle, 199), or! p ffiffiffi D 7:5 (Hong and Hong, 199), or 5 m to keep the edge effect error in correlation dimension estimation below 5% (Theiler, 199), and empirical results of dimension calculations are not substantially altered by going from 3 or points to subsets of 5 Umpqua River (b) lnc (r ) lnc (r ) Rhine River points (Abraham et al., 198). In our study, data length is long enough for estimating correlation dimension for daily flow, but the data size used for monthly streamflow analysis seems short, especially the size of 5 points of monthly flow series of the Yellow River. However, as shown in Figs. 11, there is no significant difference among the behavior of correlation integrals of the flow series with different sampling frequency. The agreement among the behavior of correlation integrals for daily, 1/3- monthly and monthly flow series indicates that the dimension calculations are very close to each other, therefore it is possible to make basically reliable correlation dimension calculation with a series of size as short as 5, which is consistent with the empirical result of Abraham et al. (198) and satisfying the theoretical minimum size of Hong and Hong (199) if the dimension is less than Second, about scaling region. Some authors do not provide scaling plot when investigating the existence of chaos (e.g. Jayawardena and Lai, 199; Sivakumar et al., 1999; Elshorbagy et al., ), whereas some Ocmulgee River Fig. 8. ln C(r) versus ln r plots for monthly streamflow processes HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

17 W. Wang et al. / Journal of Hydrology xx (xxxx) Yellow River Rhine River (a) (b) (c) Umpqua River (d) Ocmulgee River Fig. 9. Takens Theiler estimates of correlation dimension for daily streamflow processes. Yellow River (a) (c) Umpqua River Rhine River (b) (d) Ocmulgee River Fig. 1. Takens Theiler estimates of correlation dimension for 1/3-monthly streamflow processes HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1

18 18 W. Wang et al. / Journal of Hydrology xx (xxxx) other authors provide scaling plot, but give no obvious scaling region (e.g. Porporato and Ridolfi 199). However, a clearly discernible scaling region is crucial to make a convincing and reliable estimate of correlation dimension (Kantz and Schreiber, 1997). Third, about temporally related points for computing C(r). To exclude temporally related points from the computation of C(r), the Theiler window as in Eq. (5) is indispensable. Grassberger (199) remarked that when estimating the dimension of an attractor from a time sequence, one has to make sure that there exist no dynamical correlations between data points, so that all correlations are due to the geometry of the attractor rather than due to short-time correlations. He urged the reader to be very generous with the Theiler window parameter. Because streamflow series is highly temporally related, especially for daily flow, therefore, without setting Theiler window w, we would find a spurious scaling region between ln r Z 5 7 in the plot of versus ln r which gives an incorrect estimate of correlation dimension. (a) (c) Yellow River Umpqua River This problem has been pointed out by Wilcox et al. (1991) a decade ago, however, some authors ignored this (e.g. Elshorbagy et al., ), and some others take a very small Theiler window, which is maybe not large enough to exclude temporal correlations between the points (for example, Porporato and Ridolfi (199) take wz5 for daily flow series). Fig. 13 shows the Takens Theiler s estimate for daily streamflow series of the four rivers with w set to be. It is clear that with wz, we would find spurious scaling regions in all these plots. Furthermore, comparing the plots for the daily streamflow of the Rhine river with different values of w, namely, Figs. 13(b), 1(a) and (b), we can further find that the smaller the value of w, the lower the estimated correlation dimension. According to these plots, when wz, the correlation dimension D is less than ; when wz5, D is less than 8, and when wz15, D is less than 1. Therefore, the dimension estimate could be seriously too low if temporal coherence in the time series is mistaken for geometrical structure (Kantz and Schreiber, 1997). Rhine River (b) (d) Ocmulgee River Fig. 11. Takens Theiler estimates of correlation dimension for monthly streamflow processes HYDROL //5 : SHYLAJA XML MODEL 3 pp. 1