Research ArticleMultifractal Analysis of Hydrologic Data Using WaveletMethods and Fluctuation Analysis

Tongzhou Zhao12 LiangWu3 Dehua Li1 and Yiming Ding4

1School of Automation Huazhong University of Science amp Technology Wuhan 430074 China2Hubei Province Key Laboratory of Intelligent Robot Wuhan Institute of Technology Wuhan 430073 China3School of Statistics Southwestern University of Finance and Economics Chengdu 611130 China4School of Science Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Liang Wu wuliangshinegmailcom

Received 14 February 2017 Revised 26 August 2017 Accepted 25 September 2017 Published 31 October 2017

Academic Editor David Arroyo

Copyright copy 2017 Tongzhou Zhao et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study themultifractal properties of water level with a high-frequency andmassive time series usingwaveletmethods (estimationof Hurst exponents multiscale diagram and wavelet leaders for multifractal analysis (WLMF)) and multifractal detrendedfluctuation analysis (MF-DFA) The dataset contains more than two million records from 10 observation sites at a northern Chinariver The multiscale behaviour is observed in this time series which indicates the multifractality This multifractality is detectedvia multiscale diagram Then we focus on the multifractal analysis using MF-DFA and WLMF The two methods give the sameconclusion that at most sites the records satisfy the generalized binomial multifractal model which is robust for different times(morning afternoon and evening) The variation in the detailed characteristic parameters of the multifractal model indicates thatboth human activities and tributaries influence the multifractality Our work is useful for building simulation models of the waterlevel of local rivers with many observation sites

1 Introduction

Long-range dependence (LRD) and multifractality are theinherent characteristics of many natural phenomena Long-range dependence (LRD) was found in the flood process byHurst [1] when he was studying the flow of the Nile RiverFractional Brown motion (FBM) which is introduced byMandelbrot and Van Ness [2] is the commonly used modelfor studying long-range dependence Long-range depen-dence is characterized by the Hurst parameter whichdescribes the intensity of long memory phenomena Manystudies [3ndash6] for estimating the Hurst parameter (or Hurstexponent) 119867 to judge the correlation effect have been madeavailable in recent years These studies reveal that in manyareas of applied sciences such as climate change stockmarkets telecommunication network and river flow thephenomena appear as long-range behaviour and the estima-tion of Hurst has been widely used in decisions and predic-tions

The concept of multifractality was originally introducedby Mandelbrot [7] in the context of turbulence The formal-ismofmultifractalitywas first introduced using box-countingalgorithms for the complexity of measures [8ndash10] and fur-ther generalized to functions using the structure functionmethod [11] Muzy et al [12] first developed the multifractalformalism using wavelets in the discussion of turbulencedata Schertzer and Lovejoy [13] concluded that amultifractalfield could be regarded as the result of amultiplicative cascadeprocess Until now the multifractality and its formalism havebeen applied in hydrology and many other contexts [14ndash20]

Previous studies on the LRD and multifractality ofhydrology (see eg [14ndash20]) were based on relatively largesampling periods such as 1 day 1 month or even 1 year rarelyon high-frequency hydrological data Venugopal et al [21ndash23] studied the multifractality of high-resolution temporalrainfall data using wavelet methods We focus on studyingthe properties of high-frequency hydrological data andwouldlike to further explore the typical LRD phenomena and

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 3148257 18 pageshttpsdoiorg10115520173148257

2 Discrete Dynamics in Nature and Society

multifractal behaviours in hydrology Our records are thehigh-frequency water level records (every 6 minutes) ofone river (a branch of the Haihe River in China) at its 10observation sites from April 2011 to September 2013 Wehave not yet seen studies conducted for such short samplingintervals and such awide range of observation sites (along theriver from upstream to downstream)

It is known that the hydrologic data are often affectedby trends or other nonstationarities for example due to theseasonal cycle or a change in climate which may lead to anoverestimation of the Hurst exponent Furthermore underthe influence of the trends the uncorrelated data may behaveas long-term correlated data Thus the methods appliedin this paper are based on wavelet analysis (estimation ofHurst exponents multiscale diagram and wavelet leaders formultifractal analysis (WLMF)) and multifractal detrendedfluctuation analysis (MF-DFA) The wavelet method caneliminate some trends as a result of the vanishing momentproperty and MF-DFA can avoid spurious detection ofcorrelations that are artifacts of nonstationarity in the records[24] The wavelet-based estimator of the Hurst exponent wasproposed by Veitch and Abry [25] and has been improved inrecent years [26 27] This estimate was further extended tothe situation of random fields [28 29] In addition to the esti-mation the wavelet-based method can also be used to obtaina multiscale diagram for detecting the multifractal behaviour[8 9 26 30] The multifractality has been studied in severalmethods since it was proposed the primary and advancedmethods are wavelet analysis [10 31 32] and multifractaldetrended fluctuation analysis [24 33]TheMF-DFA is basedon the standard fluctuation analysis (FA) and is a multifractalgeneralization of the detrended fluctuation analysis (DFA)[24 33] Compared with FA the method can be used toconduct nonstationary data for its detrending function Inrecent years the well-established MF-DFA has been widelyused in hydrologic data (see Koscielny-Bunde et al [14 16])Lashermes et al introduced a new multifractal formalismbased on wavelet leaders [34] which allows the multifractalspectrum to be obtained over its entire range Since thenmany studies on multifractality and its applications havebeen continuously conducted [35] The analysis methodnamely wavelet leaders multifractal analysis (WLMF) isa discrete version of wavelet transform modulus maxima(WTMM) method [36 37] The DFA and WTMM methodsare compared in many literatures (see eg [16 24 38])Comparing with WTMM DFA is easy to implement and hassmaller MSE than WTMM for very short time series Butthe DFA is lack of deep theoretical concepts underlying themultifractal formalism [38]

In this paper we first apply the wavelet-based methodto estimate the Hurst exponent with different sampling andwavelet scales and observe both the LRD phenomenon (119867 gt05) and multiscale behaviour in the high-frequency waterlevel records We think that the multiscale behaviour mayinduce multifractality This multifractality is detected viamultiscale diagram Then we apply MF-DFA and WLMF tothe records to study the multifractality in detail The twomethods give the same conclusion that the water level recordsat most sites (except some abnormal sites where the water

levels are seriously affected by human activities) satisfy thegeneralized binomial multifractal model which is robust fordifferent times (morning afternoon and evening) Anotherinteresting result is that by usingMF-DFA the values of 119886+119887vary from approximately 1 to less than 09 we conclude thatthis phenomenon occurs because there are four tributariesthat flow into the river changing the multifractality of theriver As a technical improvement in the estimation of theHurst parameter 119867 we show that the sampling intervalsand choice of scales of the discrete wavelet transformationshould be matched Our work is useful for those who want tobuild simulation models of the water level of local rivers withmany observation sites because the LRD and multifractalproperties of the simulation data can be checked by themethods used in our paper

The logical structure and organization of the paper aredescribed in Figure 1

The remaining parts of the paper are organized as followsWe introduce the data source and data extraction method foranalysis in Section 2Themethods employed in this paper areintroduced briefly in Section 3 The detailed analysis and theresults for the records are presented in Section 4 In Section 5we compare the results obtained from Section 4 with theprevious results We conclude our work in Section 6

2 Data

The records that are collected in real time from one river(a branch of the Haihe River in China) represent the waterlevel The river is located in northern China which has asignificant continental monsoon climate It is cold and dryin the winter and warm and humid in the summer Fromupstream to downstream 10 artificial canals have been builtto control the water level Next to these canals 10 water levelobservation sites (Site 1ndashSite 10) have been set up along theriver (see Figure 2) Four tributaries are located downstreamof Site 7 that flow into this river The data were obtainedfrom these 10 sites and collected every 6 minutes over thepast two and a half years from April 2011 to September 2013Table 1 shows the sampling information at each site Figure 3shows the water level records at Site 1 denoted by119883(119905) and thecumulative sum of 119883(119905) minus 119883 119883 is the mean of 119883(119905) The MF-DFA and WLMF methods require that the records shouldbe a measure and its cumulative sum should be a functionThe cumulative sum in Figure 3 displays a function formThis confirms the use of the MF-DFA and WLMF meth-ods Partial records for validation are available at GitHubhttpsgithubcomtongzhouzhaowater-level-records

The water level of the river is always influenced bythe weather conditions and human activities Every yearduring the traditional rainy season the upstream canals arealways filled withmuch rainwater During the dry season theupstream canals discharge the stored water Both the storageand discharge of water should be adjusted at any time accord-ing to the growing condition of crops along the bank of theriver and the actual demand of human activities Accordingto our survey water storage and discharge at Site 2 and Site 7are conducted frequently for crop growth and human activi-ties

Discrete Dynamics in Nature and Society 3

Table 1 Sampling information

Site 1 Site 2 Site 3 Site 4 Site 5 Site 6 Site 7 Site 8 Site 9 Site 10Number of records 182855 152307 189967 190959 188074 185557 190840 190445 185822 111600Sampling period (days) 904 837 904 904 904 904 904 904 903 698Sampling frequency 6min 6min 6min 6min 6min 6min 6min 6min 6min 6minNote The records numbers are not equal in the same sampling period because of the missing data

Water levelrecords

Preprocessing(delete the outliers and extract the necessary data)

Detect LRD

Multiscale behavior aconsistent relationshipbetween wavelet scale andsampling period (Tables 4ndash6)

Wavelet-basedMultiscalediagram

MF-DFA WLMF

Detectmultifractality

Conrm the results of MF-DFA(i) Satisfy the generalized

binomial multifractalmodel (Figure 8)

can describe the strength of the multifratality (Table 8)

(ii) Parameter Δ in the model

not satisfy equation Multifractality does

(q) = Hq (Figure 4)

(i) Satisfy the generalized binomial multifractalmodel (Figure 8)

(ii) e multifractality is robust in dierenttime (Figure 9)

strength of the multifratality (Table 8)

(v) e results of this river are similar tothose of Wuumlrm river which have similarclimate and location

(iv) ℎ(2) ⩾ 08 indicates LRD (Table 8)

(iii) Parameter Δ in the model can describe the

(Tables 4ndash6)LRD (H ⩾ 05)

Figure 1 Data processing flow chart

The following three preprocessing steps were conductedbefore themultifractal analyses (1) removing the outliers and(2) extracting the appropriate dataThese preprocessing stepswere conducted to exclude outliers noise and artifacts in thedata Firstly we deleted the outliers caused by the complexenvironment where the data were acquired For instance at124200 7182013 the water level measured at Site 8 wasminus189 which is obviously an outlier Secondly we adjust theartifact data records The artificial control of the sluice maycause the water level to rise or fall in a short period of timeFor example at 131200 752013 the record (Site 8) droppedfrom 273 to 028 The phenomenon of large fluctuations ina short time may be misdiagnosed as there is noise in thedataset Thirdly we fill the missing data appropriately Thereis a small portion of datamissing during times when the data-collecting equipment did notworkwell For instance at Site 5we received one dayrsquos last record at 164200 4272011 whichshould have been followed by another record sixminutes laterat 164800 on the same day However the next record timein our dataset is 84800 4292011 which means that almostone-day records are missing A situation like this can occur

at other sites The missing record is directly relevant to thenearby records For avoiding the effect of time in one day weuse the records at the same time of adjacent days to fill themissing data The used method is linear interpolation Forexample at Site 5 the record at 161800 4282011 is missingAccording to linear interpolationmethod it can be filled withthe mean of the records at 161800 4272011 and 1618004292011 The preprocessing may ensure a uniformly spacedsampling interval

3 Methods

31 Wavelet-Based Estimation of the Hurst Exponent Veitchet al [25 26] have proposed the wavelet-based method formany years Recently they improved it for non-Gaussianlong-range dependent processes [27] We use this methodto estimate the Hurst exponent The wavelet-based methodcan detrend the records with polynomial trend because ofthe property of vanishing moments of wavelet can avoid thespurious result caused by trend and is suitable for water levelrecords

4 Discrete Dynamics in Nature and Society

Site 1Site 2

Site 3 Site 4

Site 5 Site 6Site 7 Site 9

Site 8

Site 10

Figure 2 Site 1ndashSite 10 locations

times10410 155

480

4805

481

4815

X(t)

482

4825

483

4835

(a)times104

10 155minus10000

minus8000

minus6000

minus4000

minus2000

0

2000

Cum

ulat

ive s

um

(b)

Figure 3 The water level records 119883(119905) at Site 1 (a) and the cumulative sum of 119883(119905) minus 119883 (b)

Let 119883(119905) 119905 isin [0 119899) denote a second-order stationaryprocess with a spectrum labelled Γ119883(]) which satisfies

Γ119883 (]) sim 119862119883 |]|minus(2119867minus1) |]| 997888rarr 0 05 lt 119867 lt 1 (1)

The process 119883(119905) is said to be long-range dependentLet 120595(119905) be a wavelet function with a vanishing moment119873 The wavelet coefficients of 119883(119905) are

119889119883 (119895 119896) = int119877

120595119895119896 (119905) 119883 (119905) 119889119905where 120595119895119896 (119905) = 2minus1198952120595 (2minus119895119905 minus 119896)

(2)

Because of the stationary property of 119883(119905) we have119864119889119883 (119895 119896)2 = int

119877Γ119883 (]) 10038161003816100381610038161003816Ψ (2119895])10038161003816100381610038161003816 119889] (3)

and considering the property of the spectrum density (3)becomes

119864119889119883 (119895 119896)2 sim 1198621198832119895(2119867minus1) 2119895 997888rarr infin (4)

As a result

log2119864119889119883 (119895 119896)2 sim 119895 (2119867 minus 1) + log2119862119883 2119895 997888rarr infin (5)

Put

119878 (119895) fl 1119899119895 sum 119889119883 (119895 119896)2 (6)

where 119899119895 is the number of 119889119883(119895 119896) actually available atscale 2119895 119864119889119883(119895 119896)2 can be estimated by 119878(119895) Then theestimation of the Hurst parameter can be realized with alinear regression

Because of the different variances of log2119878(119895) at differentvalues of 119895 the weighted least squares for this regressionmodel are needed The weight for this regression is thereciprocal of the variance of log2119878(119895) Moreover

var log2119878 (119895) asymp (log2119890)2 119862 (119895) (7)

where 119862(119895) = var1198892119883(119895 sdot)[119899119895(1198641198892119883(119895 sdot))2] The 119862(119895) term canbe estimated for each octave by using the sample momentestimators of the fourth and second moments of 119889119883(119895 sdot)

Abry et al [27] studied the statistical performance of thisestimator for non-Gaussian long-range dependent processesobtained from point transformations of Gaussian processesTheir results suggest that the vanishing moment shouldsatisfy 119873 ge 2 Furthermore the linear regression should beperformed over scales of 2119895 ranging from the finest to thecoarsest available To obtain the estimators at different scales

Discrete Dynamics in Nature and Society 5

we do not comply with this suggestion in our experiments Itis also shown that the reduction of the dependence range ofwavelet coefficients from long to short by increasing 119873 above2 as observed for Gaussian processes does not hold for non-Gaussian processes This may lead to the fact that increasing119873 above 2 does not result in any decrease in the variance ofthe estimator

32 Wavelet-Based Multiscale Diagram Wavelet proceduresprovide us with a way to estimate the Hurst parameterSometimes the Hurst index depends on time 119905 (ie multipleexponents) such as multifractional process and multifractalprocess For themultifractal process its sample path has localregularity

|119883 (120596 119905 + 120591) minus 119883 (120596 119905)| asymp |120591|119867(120596119905) 120591 997888rarr 0 (8)

where 120596 denotes an element of the probability space [26]Abry et al [26] introduced a wavelet-based multiscale dia-gram for detecting multifractality of process This methodabandons the idea of estimating 119867 that changes with 119905 andstudies the process statistically by Legendre multifractalspectrum which can be obtained by calculating the structurefunction (or partition functions) from the sample path

119878119902 (120591) = int |119883 (120596 119905 + 120591) minus 119883 (120596 119905)|119902 119889119905 (9)

and for the multifractal case 119878119902(120591) asymp |120591|120577(119902) 120591 rarr 0 119902 isin 119877[26 39] Moreover the Legendre transform of the function120577(119902) is the Legendre multifractal spectrum

Wavelet processes are an effective tool for obtaining 120577(119902)because of the fact that the increments of the sample pathcan be seen as simple examples of the wavelet coefficientsThe wavelet coefficients are used to replace the incrementsin the partition function [9 26] Furthermore the wavelet-based partition function exhibits power-law behaviour forsmall scales [8 26 30 39]

int 1003816100381610038161003816119879119883 (119886 119905)1003816100381610038161003816119902 119889119905 asymp 119886120577(119902)+1199022 119886 997888rarr 0where 119879119883 (119886 119905) = int 1radic119886 119883 (119906) 120595 ( 119906 minus 119905119886 ) 119889119906

(10)

According to the power-law the statistics 119878(119895 119902) fl1119899119895sum119896 |119889119883(119895 119896)|119902 asymp 2119895(120577(119902)+1199022) 119902 isin 119877 for small 119895 So 120577(119902)can be estimated using linear regression [26]

If 120577(119902) = 119902119867 (linear form) the process is often calledldquomonofractalrdquo whose multifractal spectrum is determinedby one 119867 such as self-similar process (eg FBM) and LRDprocess Conversely if 120577(119902) is not linear the process exhibitsmultiscale property The multifractality of a process can bedetected by measuring the deviation of 120577(119902) from the linearform [26] Let 120577(119902) be the estimation of 120577(119902) The graph of120577(119902) against 119902 together with the confidence intervals relatedto 120577(119902) is designated as ldquomultiscale diagram (MD)rdquo [26]The confidence intervals are obtained according to Delbekeand Van Assche [40] The detection of multifractality can besolved by examining whether there is a lack of alignment inthe multiscale diagram

33Multifractal Detrended FluctuationAnalysis Thewavelet-basedmultiscale diagram can detect whether the process119883(119905)ismultifractalWe study themultifractality by usingMF-DFA[24] which is based on the standard fluctuation analysis (FA)

In MF-DFA one investigates the series

119884 (119894) fl 119894sum119896=1

(119883 (119896) minus 119883) 119894 = 1 119871 (11)

where 119871 is the length of 119883(119905) and 119883 is the mean of 119883(119905)Let119873119904 fl int(119871119904)119884(119905) is divided into119873119904 nonoverlapping

segments with equal length 119904 The fluctuation 1198652(V 119904) iscalculated using

1198652 (V 119904) = 1119904119904sum119894=1

119884 [(V minus 1) 119904 + 119894] minus 119910V (119894)2 V = 1 119873119904

(12)

where 119910V(119894) is the best polynomial fit of 119884[(Vminus 1)119904 + 119894] in eachof the 119873119904 segments V

In FA the fluctuation 1198652(V 119904) = (119884V119904 minus 119884(Vminus1)119904)2 thetrend in the original series is not considered In MF-DFAthe time series is detrended by subtracting 119910V(119894)The 119898-orderpolynomial used in the fitting procedure can eliminate thepolynomial trend of order 119898 minus 1 in the original series Thisensures the fluctuation analysis available for data affected bytrends or other nonstationarities

For a multifractal description of records one must con-sider all moments 119865119902(119904)

119865119902 (119904) fl 1119873119904119873119904sumV=1

[1198652 (V 119904)]11990221119902

V = 1 119873119904 (13)

If 119883(119905) is long-range dependent 119865119902(119904) satisfies [24]119865119902 (119904) sim 119904ℎ(119902) for large 119904 (14)

For very large 119904 that is 119904 gt 1198734119865119902(119904) becomes inaccuratedue to the statistical errors produced by the small numberof segments 119873119904 So we will concentrate on the scales 119904 lowerthan 1198734 The minimum scale chosen is equal to 25 to avoidspurious results produced by 119904 lt 30ℎ(119902) can be obtained by analyzing log-log plots 119865119902(119904)versus 119904 If 119902 = 2 ℎ(2) is the fluctuation exponent 119867 that isrelated to the correlation exponent 120574 by119867 = 1minus1205742 [24] For astationary time seriesℎ(2) is thewell-knownHurst exponentThe function ℎ(119902) is called the generalized Hurst exponent[24]

It is known that [14 24] for 119902 gt 0 ℎ(119902) shows the scalingbehaviour of the segments with large fluctuation and for 119902 lt0 ℎ(119902) shows that with small fluctuation

One can also get theRenyi scaling exponent 120591(119902) by calcu-lating 120591(119902) = 119902ℎ(119902)minus1Theway to describe themultifractalityis the singularity spectrum 119891(120572) (or multifractal spectrum)that is the Legendre transform of 120591(119902) [14 16 24]

120572 = 1205911015840 (119902) 119891 (120572) = 119902120572 minus 120591 (119902) (15)

6 Discrete Dynamics in Nature and Society

where 120572 is singularity strength (or Holder exponent) Thestrength of the multifractality of records can be characterizedby Δ120572 = 120572max minus 120572min

If ℎ(119902) satisfies the following equationℎ (119902) = 1119902 minus ln (119886119902 + 119887119902)

119902 ln 2 (16)

which can be obtained from a generalized binomial multi-fractal model [16] 120591(119902) = minus ln(119886119902 + 119887119902) ln 2 Δ120572 = ℎ(minusinfin) minusℎ(infin) In this case

Δ120572 = ln 119887 minus ln 119886ln 2 (17)

The three parameters 119886 119887 and Δ120572 can then be regardedas multifractal finger prints for records [14]

34 Wavelet Leaders for Multifractal Analysis For compar-ison with MF-DFA we introduce the method of waveletleaders for multifractal analysis (WLMF) proposed byWendtet al (see [41])

Assume that 120595(119905) is a wavelet function with compact timesupport and 119873 vanishing moment Let 120582119895119896 = [1198962119895 (119896 + 1)2119895)be dyadic intervals and let 3120582119895119896 denote the intervals 3120582119895119896 fl120582119895119896minus1 cup 120582119895119896 cup 120582119895119896+1 The wavelet leaders are defined as

119871119883 (119895 119896) fl sup[119896101584021198951015840(1198961015840+1)2119895

1015840)sub3120582119895119896

10038161003816100381610038161003816119889119883 (1198951015840 1198961015840)10038161003816100381610038161003816 (18)

The structure function is calculated by the averages of119871119883(119895 119896)119878119871119902 (119895) = 1119899119895

119899119895sum119896=1

119871119883 (119895 119896)119902 (19)

It can been shown [41] that under a mild uniform Holderregularity condition on 119883(119905)

119878119871119902 (119895) sim 1198651199022119895120577(119902) 2119895 997888rarr 0 (20)

So the estimation for 120577(119902) can be realized by a linearregression in 119895 versus log2119878119871119902(119895)

120577 (119902) = 1198952sum119895=1198951

120596119895log2119878119871119902 (119895) (21)

The weights 120596119895 satisfy sum11989521198951 119895120596119895 equiv 1 and sum11989521198951 120596119895 equiv 0To estimate the multifractal spectrum Wendt et al [41]

suggest a parametric formulation that avoids the computationof the Legendre transform

119891 (119902) = 1198952sum1198951

120596119895119880119871 (119895 119902)

ℎ (119902) = 1198952sum1198951

120596119895119881119871 (119895 119902) (22)

where 119880119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119877119902119883(119895 119896) + log2119899119895119881119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119871119883(119895 119896) and 119877119902119883(119895 119896) =119871119883(119895 119896)119902 sum119899119895119896=1 119871119883(119895 119896)119902

We emphasize that the wavelet transform with 119873 van-ishing moments can eliminate the polynomial trend of order119873 minus 1 in the original series

4 Computational Results

41 Wavelet-Based Results This section describes thewavelet-based estimation of Hurst parameters for the waterlevel records of the 10 sites We estimate the Hurst exponentsat different scales and different sampling interval timesThe results indicate that there is an interesting connectionbetween the wavelet scale and the sampling period For com-parison we repeat the procedure for a synthetic exponentiallong-range dependent process (ELRD) with length of180000 which is close to the actual sampling data lengthThevanishing moment of the used Daubechies wavelet is 119873 = 3(db3) The simulation results for ELRD are listed in Tables 2and 3 The values in brackets are errors bars The errors barsare based on the weighted linear least square fit which is usedto estimate 119867 and reveal 95 confidence interval

The simulation results show that the simulations for both119867 = 06 and 119867 = 08 are stable under changes in the intervallength and estimation scale One can see that the wavelet-based estimation of ELRD for theHurst exponent is valid andreliable and the selection of wavelet scale and sampling cycleis independent of the estimation

Tables 4ndash6 display the results of the practical water levelThere exists an expression 119886119894119895 asymp 119886119894+1119895minus1 (119886119894119895 is the element ofthe tables) For example in Table 4 H = 082 (12min scales(1sim5)) and H = 084 (6min scales (2sim6)) are approximatelyequalH = 070 (24min scales (1sim5))H = 068 (12min scales(2sim6)) and H = 068 (6min scales (3sim7)) are almost equaland so forthThe results represent the consistent relationshipbetween the wavelet scale and the sampling period Wepresent a mathematical proof in the Appendix

We can also see that the estimations for the Hurst expo-nent are not stable under changes in the interval length andestimation scale while the estimations along the secondarydiagonal direction of the tables are still stable We concludethat the massive hydrologic data have a multiscale consistentproperty while the simulation data do notThis indicates thatthere may exist multifractality in the records

42 Multiscale Diagram for Hydrological Data In this sec-tion we use the multiscale diagram to test the multifractalitythat may exist in the records The multiscale diagram is thegraph of 120577(119902) against 119902 together with the confidence intervalsrelated to 120577(119902)Themultifractality can be detected bymeasur-ing the deviation of 120577(119902) from the linear form [26]

We use a linear multiscale diagram to examine thealignment in the multiscale diagram in which 119867119902 = 120577(119902)119902is drawn against 119902 The multifractality can be determinedby examining whether there is a horizontal alignment in thelinearmultiscale diagramThehorizontal alignment indicates

Discrete Dynamics in Nature and Society 7

Table 2 Simulation data analysis for H (H = 06 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 061 (0009) 061 (0013) 060 (0018) 059 (0025) 059 (0036)2 061 (0013) 061 (0018) 060 (0025) 059 (0036) 060 (0051)4 061 (0018) 061 (0025) 060 (0036) 059 (0051) 061 (0071)8 060 (0025) 060 (0036) 060 (0051) 061 (0071) 062 (0106)

Table 3 Simulation data analysis for H (H = 08 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 083 (0009) 082 (0013) 081 (0018) 080 (0025) 080 (0034)2 083 (0013) 082 (0018) 081 (0025) 080 (0034) 082 (0049)4 083 (0018) 082 (0025) 081 (0034) 081 (0049) 080 (0072)8 082 (0025) 082 (0034) 082 (0049) 080 (0072) 079 (0100)

Table 4 Multiscales and multisampling for H of Site 1 (data length = 182855)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 084 (0062) 084 (0047) 068 (0046) 053 (0054)12min 082 (0067) 068 (0054) 053 (0054) mdash24min 070 (0046) 055 (0056) mdash mdash48min 057 (0053) mdash mdash mdash

Table 5 Multiscales and multisampling for H of Site 2 (data length = 152307)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 080 (0101) 086 (0086) 081 (0077) 061 (0086)12min 089 (0082) 080 (0079) 061 (0086) 050 (0111)24min 081 (0079) 063 (0087) 050 (0112) mdash48min 065 (0085) 052 (0116) mdash mdash

Table 6 Multiscales and multisampling for H of Site 3 (data length = 189967)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 079 (0047) 097 (0041) 087 (0044) 069 (0048)12min 095 (0046) 089 (0041) 069 (0047) 050 (0059)24min 091 (0044) 071 (0048) 050 (0059) mdash48min 074 (0046) 050 (0060) mdash mdash

that the Hurst parameter 119867 can control all of the scalingproperty The value of 119867 can be received by estimating thelevel The linear multiscale diagram and multiscale diagramare statistically equivalent

To determine the alignment of hydrological data weimpose the synthesized FBM (119867 = 08) known as the ldquolinearformrdquo for comparison The regressions for estimating 120577(119902)for each of the 10 different 119902 are performed for small scales(1198951 1198952) = (2 6) according to the power-law behaviour of

119878(119895 119902) discussed in Section 32 For Site 1ndashSite 10 as well asthe FBM the multiscale diagrams are shown in Figure 4 andthe values of 120577(119902) for all 10 sites are also shown in Table 7 (thesampling interval is 6 minutes)We choose db3 as the waveletof the wavelet transform which is orthogonal to quadratictrends

In Figure 4 the multiscale diagram in (a) shows a lackof alignment for the hydrologic data compared to the clearalignment of FBM for its self-similarity which indicates that

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

4 Discrete Dynamics in Nature and Society

Site 1Site 2

Site 3 Site 4

Site 5 Site 6Site 7 Site 9

Site 8

Site 10

Figure 2 Site 1ndashSite 10 locations

times10410 155

480

4805

481

4815

X(t)

482

4825

483

4835

(a)times104

10 155minus10000

minus8000

minus6000

minus4000

minus2000

0

2000

Cum

ulat

ive s

um

(b)

Figure 3 The water level records 119883(119905) at Site 1 (a) and the cumulative sum of 119883(119905) minus 119883 (b)

Let 119883(119905) 119905 isin [0 119899) denote a second-order stationaryprocess with a spectrum labelled Γ119883(]) which satisfies

Γ119883 (]) sim 119862119883 |]|minus(2119867minus1) |]| 997888rarr 0 05 lt 119867 lt 1 (1)

The process 119883(119905) is said to be long-range dependentLet 120595(119905) be a wavelet function with a vanishing moment119873 The wavelet coefficients of 119883(119905) are

119889119883 (119895 119896) = int119877

120595119895119896 (119905) 119883 (119905) 119889119905where 120595119895119896 (119905) = 2minus1198952120595 (2minus119895119905 minus 119896)

(2)

Because of the stationary property of 119883(119905) we have119864119889119883 (119895 119896)2 = int

119877Γ119883 (]) 10038161003816100381610038161003816Ψ (2119895])10038161003816100381610038161003816 119889] (3)

and considering the property of the spectrum density (3)becomes

119864119889119883 (119895 119896)2 sim 1198621198832119895(2119867minus1) 2119895 997888rarr infin (4)

As a result

log2119864119889119883 (119895 119896)2 sim 119895 (2119867 minus 1) + log2119862119883 2119895 997888rarr infin (5)

Put

119878 (119895) fl 1119899119895 sum 119889119883 (119895 119896)2 (6)

where 119899119895 is the number of 119889119883(119895 119896) actually available atscale 2119895 119864119889119883(119895 119896)2 can be estimated by 119878(119895) Then theestimation of the Hurst parameter can be realized with alinear regression

Because of the different variances of log2119878(119895) at differentvalues of 119895 the weighted least squares for this regressionmodel are needed The weight for this regression is thereciprocal of the variance of log2119878(119895) Moreover

var log2119878 (119895) asymp (log2119890)2 119862 (119895) (7)

where 119862(119895) = var1198892119883(119895 sdot)[119899119895(1198641198892119883(119895 sdot))2] The 119862(119895) term canbe estimated for each octave by using the sample momentestimators of the fourth and second moments of 119889119883(119895 sdot)

Abry et al [27] studied the statistical performance of thisestimator for non-Gaussian long-range dependent processesobtained from point transformations of Gaussian processesTheir results suggest that the vanishing moment shouldsatisfy 119873 ge 2 Furthermore the linear regression should beperformed over scales of 2119895 ranging from the finest to thecoarsest available To obtain the estimators at different scales

Discrete Dynamics in Nature and Society 5

we do not comply with this suggestion in our experiments Itis also shown that the reduction of the dependence range ofwavelet coefficients from long to short by increasing 119873 above2 as observed for Gaussian processes does not hold for non-Gaussian processes This may lead to the fact that increasing119873 above 2 does not result in any decrease in the variance ofthe estimator

32 Wavelet-Based Multiscale Diagram Wavelet proceduresprovide us with a way to estimate the Hurst parameterSometimes the Hurst index depends on time 119905 (ie multipleexponents) such as multifractional process and multifractalprocess For themultifractal process its sample path has localregularity

|119883 (120596 119905 + 120591) minus 119883 (120596 119905)| asymp |120591|119867(120596119905) 120591 997888rarr 0 (8)

where 120596 denotes an element of the probability space [26]Abry et al [26] introduced a wavelet-based multiscale dia-gram for detecting multifractality of process This methodabandons the idea of estimating 119867 that changes with 119905 andstudies the process statistically by Legendre multifractalspectrum which can be obtained by calculating the structurefunction (or partition functions) from the sample path

119878119902 (120591) = int |119883 (120596 119905 + 120591) minus 119883 (120596 119905)|119902 119889119905 (9)

and for the multifractal case 119878119902(120591) asymp |120591|120577(119902) 120591 rarr 0 119902 isin 119877[26 39] Moreover the Legendre transform of the function120577(119902) is the Legendre multifractal spectrum

Wavelet processes are an effective tool for obtaining 120577(119902)because of the fact that the increments of the sample pathcan be seen as simple examples of the wavelet coefficientsThe wavelet coefficients are used to replace the incrementsin the partition function [9 26] Furthermore the wavelet-based partition function exhibits power-law behaviour forsmall scales [8 26 30 39]

int 1003816100381610038161003816119879119883 (119886 119905)1003816100381610038161003816119902 119889119905 asymp 119886120577(119902)+1199022 119886 997888rarr 0where 119879119883 (119886 119905) = int 1radic119886 119883 (119906) 120595 ( 119906 minus 119905119886 ) 119889119906

(10)

According to the power-law the statistics 119878(119895 119902) fl1119899119895sum119896 |119889119883(119895 119896)|119902 asymp 2119895(120577(119902)+1199022) 119902 isin 119877 for small 119895 So 120577(119902)can be estimated using linear regression [26]

If 120577(119902) = 119902119867 (linear form) the process is often calledldquomonofractalrdquo whose multifractal spectrum is determinedby one 119867 such as self-similar process (eg FBM) and LRDprocess Conversely if 120577(119902) is not linear the process exhibitsmultiscale property The multifractality of a process can bedetected by measuring the deviation of 120577(119902) from the linearform [26] Let 120577(119902) be the estimation of 120577(119902) The graph of120577(119902) against 119902 together with the confidence intervals relatedto 120577(119902) is designated as ldquomultiscale diagram (MD)rdquo [26]The confidence intervals are obtained according to Delbekeand Van Assche [40] The detection of multifractality can besolved by examining whether there is a lack of alignment inthe multiscale diagram

33Multifractal Detrended FluctuationAnalysis Thewavelet-basedmultiscale diagram can detect whether the process119883(119905)ismultifractalWe study themultifractality by usingMF-DFA[24] which is based on the standard fluctuation analysis (FA)

In MF-DFA one investigates the series

119884 (119894) fl 119894sum119896=1

(119883 (119896) minus 119883) 119894 = 1 119871 (11)

where 119871 is the length of 119883(119905) and 119883 is the mean of 119883(119905)Let119873119904 fl int(119871119904)119884(119905) is divided into119873119904 nonoverlapping

segments with equal length 119904 The fluctuation 1198652(V 119904) iscalculated using

1198652 (V 119904) = 1119904119904sum119894=1

119884 [(V minus 1) 119904 + 119894] minus 119910V (119894)2 V = 1 119873119904

(12)

where 119910V(119894) is the best polynomial fit of 119884[(Vminus 1)119904 + 119894] in eachof the 119873119904 segments V

In FA the fluctuation 1198652(V 119904) = (119884V119904 minus 119884(Vminus1)119904)2 thetrend in the original series is not considered In MF-DFAthe time series is detrended by subtracting 119910V(119894)The 119898-orderpolynomial used in the fitting procedure can eliminate thepolynomial trend of order 119898 minus 1 in the original series Thisensures the fluctuation analysis available for data affected bytrends or other nonstationarities

For a multifractal description of records one must con-sider all moments 119865119902(119904)

119865119902 (119904) fl 1119873119904119873119904sumV=1

[1198652 (V 119904)]11990221119902

V = 1 119873119904 (13)

If 119883(119905) is long-range dependent 119865119902(119904) satisfies [24]119865119902 (119904) sim 119904ℎ(119902) for large 119904 (14)

For very large 119904 that is 119904 gt 1198734119865119902(119904) becomes inaccuratedue to the statistical errors produced by the small numberof segments 119873119904 So we will concentrate on the scales 119904 lowerthan 1198734 The minimum scale chosen is equal to 25 to avoidspurious results produced by 119904 lt 30ℎ(119902) can be obtained by analyzing log-log plots 119865119902(119904)versus 119904 If 119902 = 2 ℎ(2) is the fluctuation exponent 119867 that isrelated to the correlation exponent 120574 by119867 = 1minus1205742 [24] For astationary time seriesℎ(2) is thewell-knownHurst exponentThe function ℎ(119902) is called the generalized Hurst exponent[24]

It is known that [14 24] for 119902 gt 0 ℎ(119902) shows the scalingbehaviour of the segments with large fluctuation and for 119902 lt0 ℎ(119902) shows that with small fluctuation

One can also get theRenyi scaling exponent 120591(119902) by calcu-lating 120591(119902) = 119902ℎ(119902)minus1Theway to describe themultifractalityis the singularity spectrum 119891(120572) (or multifractal spectrum)that is the Legendre transform of 120591(119902) [14 16 24]

120572 = 1205911015840 (119902) 119891 (120572) = 119902120572 minus 120591 (119902) (15)

6 Discrete Dynamics in Nature and Society

where 120572 is singularity strength (or Holder exponent) Thestrength of the multifractality of records can be characterizedby Δ120572 = 120572max minus 120572min

If ℎ(119902) satisfies the following equationℎ (119902) = 1119902 minus ln (119886119902 + 119887119902)

119902 ln 2 (16)

which can be obtained from a generalized binomial multi-fractal model [16] 120591(119902) = minus ln(119886119902 + 119887119902) ln 2 Δ120572 = ℎ(minusinfin) minusℎ(infin) In this case

Δ120572 = ln 119887 minus ln 119886ln 2 (17)

The three parameters 119886 119887 and Δ120572 can then be regardedas multifractal finger prints for records [14]

34 Wavelet Leaders for Multifractal Analysis For compar-ison with MF-DFA we introduce the method of waveletleaders for multifractal analysis (WLMF) proposed byWendtet al (see [41])

Assume that 120595(119905) is a wavelet function with compact timesupport and 119873 vanishing moment Let 120582119895119896 = [1198962119895 (119896 + 1)2119895)be dyadic intervals and let 3120582119895119896 denote the intervals 3120582119895119896 fl120582119895119896minus1 cup 120582119895119896 cup 120582119895119896+1 The wavelet leaders are defined as

119871119883 (119895 119896) fl sup[119896101584021198951015840(1198961015840+1)2119895

1015840)sub3120582119895119896

10038161003816100381610038161003816119889119883 (1198951015840 1198961015840)10038161003816100381610038161003816 (18)

The structure function is calculated by the averages of119871119883(119895 119896)119878119871119902 (119895) = 1119899119895

119899119895sum119896=1

119871119883 (119895 119896)119902 (19)

It can been shown [41] that under a mild uniform Holderregularity condition on 119883(119905)

119878119871119902 (119895) sim 1198651199022119895120577(119902) 2119895 997888rarr 0 (20)

So the estimation for 120577(119902) can be realized by a linearregression in 119895 versus log2119878119871119902(119895)

120577 (119902) = 1198952sum119895=1198951

120596119895log2119878119871119902 (119895) (21)

The weights 120596119895 satisfy sum11989521198951 119895120596119895 equiv 1 and sum11989521198951 120596119895 equiv 0To estimate the multifractal spectrum Wendt et al [41]

suggest a parametric formulation that avoids the computationof the Legendre transform

119891 (119902) = 1198952sum1198951

120596119895119880119871 (119895 119902)

ℎ (119902) = 1198952sum1198951

120596119895119881119871 (119895 119902) (22)

where 119880119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119877119902119883(119895 119896) + log2119899119895119881119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119871119883(119895 119896) and 119877119902119883(119895 119896) =119871119883(119895 119896)119902 sum119899119895119896=1 119871119883(119895 119896)119902

We emphasize that the wavelet transform with 119873 van-ishing moments can eliminate the polynomial trend of order119873 minus 1 in the original series

4 Computational Results

41 Wavelet-Based Results This section describes thewavelet-based estimation of Hurst parameters for the waterlevel records of the 10 sites We estimate the Hurst exponentsat different scales and different sampling interval timesThe results indicate that there is an interesting connectionbetween the wavelet scale and the sampling period For com-parison we repeat the procedure for a synthetic exponentiallong-range dependent process (ELRD) with length of180000 which is close to the actual sampling data lengthThevanishing moment of the used Daubechies wavelet is 119873 = 3(db3) The simulation results for ELRD are listed in Tables 2and 3 The values in brackets are errors bars The errors barsare based on the weighted linear least square fit which is usedto estimate 119867 and reveal 95 confidence interval

The simulation results show that the simulations for both119867 = 06 and 119867 = 08 are stable under changes in the intervallength and estimation scale One can see that the wavelet-based estimation of ELRD for theHurst exponent is valid andreliable and the selection of wavelet scale and sampling cycleis independent of the estimation

Tables 4ndash6 display the results of the practical water levelThere exists an expression 119886119894119895 asymp 119886119894+1119895minus1 (119886119894119895 is the element ofthe tables) For example in Table 4 H = 082 (12min scales(1sim5)) and H = 084 (6min scales (2sim6)) are approximatelyequalH = 070 (24min scales (1sim5))H = 068 (12min scales(2sim6)) and H = 068 (6min scales (3sim7)) are almost equaland so forthThe results represent the consistent relationshipbetween the wavelet scale and the sampling period Wepresent a mathematical proof in the Appendix

We can also see that the estimations for the Hurst expo-nent are not stable under changes in the interval length andestimation scale while the estimations along the secondarydiagonal direction of the tables are still stable We concludethat the massive hydrologic data have a multiscale consistentproperty while the simulation data do notThis indicates thatthere may exist multifractality in the records

42 Multiscale Diagram for Hydrological Data In this sec-tion we use the multiscale diagram to test the multifractalitythat may exist in the records The multiscale diagram is thegraph of 120577(119902) against 119902 together with the confidence intervalsrelated to 120577(119902)Themultifractality can be detected bymeasur-ing the deviation of 120577(119902) from the linear form [26]

We use a linear multiscale diagram to examine thealignment in the multiscale diagram in which 119867119902 = 120577(119902)119902is drawn against 119902 The multifractality can be determinedby examining whether there is a horizontal alignment in thelinearmultiscale diagramThehorizontal alignment indicates

Discrete Dynamics in Nature and Society 7

Table 2 Simulation data analysis for H (H = 06 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 061 (0009) 061 (0013) 060 (0018) 059 (0025) 059 (0036)2 061 (0013) 061 (0018) 060 (0025) 059 (0036) 060 (0051)4 061 (0018) 061 (0025) 060 (0036) 059 (0051) 061 (0071)8 060 (0025) 060 (0036) 060 (0051) 061 (0071) 062 (0106)

Table 3 Simulation data analysis for H (H = 08 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 083 (0009) 082 (0013) 081 (0018) 080 (0025) 080 (0034)2 083 (0013) 082 (0018) 081 (0025) 080 (0034) 082 (0049)4 083 (0018) 082 (0025) 081 (0034) 081 (0049) 080 (0072)8 082 (0025) 082 (0034) 082 (0049) 080 (0072) 079 (0100)

Table 4 Multiscales and multisampling for H of Site 1 (data length = 182855)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 084 (0062) 084 (0047) 068 (0046) 053 (0054)12min 082 (0067) 068 (0054) 053 (0054) mdash24min 070 (0046) 055 (0056) mdash mdash48min 057 (0053) mdash mdash mdash

Table 5 Multiscales and multisampling for H of Site 2 (data length = 152307)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 080 (0101) 086 (0086) 081 (0077) 061 (0086)12min 089 (0082) 080 (0079) 061 (0086) 050 (0111)24min 081 (0079) 063 (0087) 050 (0112) mdash48min 065 (0085) 052 (0116) mdash mdash

Table 6 Multiscales and multisampling for H of Site 3 (data length = 189967)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 079 (0047) 097 (0041) 087 (0044) 069 (0048)12min 095 (0046) 089 (0041) 069 (0047) 050 (0059)24min 091 (0044) 071 (0048) 050 (0059) mdash48min 074 (0046) 050 (0060) mdash mdash

that the Hurst parameter 119867 can control all of the scalingproperty The value of 119867 can be received by estimating thelevel The linear multiscale diagram and multiscale diagramare statistically equivalent

To determine the alignment of hydrological data weimpose the synthesized FBM (119867 = 08) known as the ldquolinearformrdquo for comparison The regressions for estimating 120577(119902)for each of the 10 different 119902 are performed for small scales(1198951 1198952) = (2 6) according to the power-law behaviour of

119878(119895 119902) discussed in Section 32 For Site 1ndashSite 10 as well asthe FBM the multiscale diagrams are shown in Figure 4 andthe values of 120577(119902) for all 10 sites are also shown in Table 7 (thesampling interval is 6 minutes)We choose db3 as the waveletof the wavelet transform which is orthogonal to quadratictrends

In Figure 4 the multiscale diagram in (a) shows a lackof alignment for the hydrologic data compared to the clearalignment of FBM for its self-similarity which indicates that

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 5

we do not comply with this suggestion in our experiments Itis also shown that the reduction of the dependence range ofwavelet coefficients from long to short by increasing 119873 above2 as observed for Gaussian processes does not hold for non-Gaussian processes This may lead to the fact that increasing119873 above 2 does not result in any decrease in the variance ofthe estimator

32 Wavelet-Based Multiscale Diagram Wavelet proceduresprovide us with a way to estimate the Hurst parameterSometimes the Hurst index depends on time 119905 (ie multipleexponents) such as multifractional process and multifractalprocess For themultifractal process its sample path has localregularity

|119883 (120596 119905 + 120591) minus 119883 (120596 119905)| asymp |120591|119867(120596119905) 120591 997888rarr 0 (8)

where 120596 denotes an element of the probability space [26]Abry et al [26] introduced a wavelet-based multiscale dia-gram for detecting multifractality of process This methodabandons the idea of estimating 119867 that changes with 119905 andstudies the process statistically by Legendre multifractalspectrum which can be obtained by calculating the structurefunction (or partition functions) from the sample path

119878119902 (120591) = int |119883 (120596 119905 + 120591) minus 119883 (120596 119905)|119902 119889119905 (9)

and for the multifractal case 119878119902(120591) asymp |120591|120577(119902) 120591 rarr 0 119902 isin 119877[26 39] Moreover the Legendre transform of the function120577(119902) is the Legendre multifractal spectrum

Wavelet processes are an effective tool for obtaining 120577(119902)because of the fact that the increments of the sample pathcan be seen as simple examples of the wavelet coefficientsThe wavelet coefficients are used to replace the incrementsin the partition function [9 26] Furthermore the wavelet-based partition function exhibits power-law behaviour forsmall scales [8 26 30 39]

int 1003816100381610038161003816119879119883 (119886 119905)1003816100381610038161003816119902 119889119905 asymp 119886120577(119902)+1199022 119886 997888rarr 0where 119879119883 (119886 119905) = int 1radic119886 119883 (119906) 120595 ( 119906 minus 119905119886 ) 119889119906

(10)

According to the power-law the statistics 119878(119895 119902) fl1119899119895sum119896 |119889119883(119895 119896)|119902 asymp 2119895(120577(119902)+1199022) 119902 isin 119877 for small 119895 So 120577(119902)can be estimated using linear regression [26]

If 120577(119902) = 119902119867 (linear form) the process is often calledldquomonofractalrdquo whose multifractal spectrum is determinedby one 119867 such as self-similar process (eg FBM) and LRDprocess Conversely if 120577(119902) is not linear the process exhibitsmultiscale property The multifractality of a process can bedetected by measuring the deviation of 120577(119902) from the linearform [26] Let 120577(119902) be the estimation of 120577(119902) The graph of120577(119902) against 119902 together with the confidence intervals relatedto 120577(119902) is designated as ldquomultiscale diagram (MD)rdquo [26]The confidence intervals are obtained according to Delbekeand Van Assche [40] The detection of multifractality can besolved by examining whether there is a lack of alignment inthe multiscale diagram

33Multifractal Detrended FluctuationAnalysis Thewavelet-basedmultiscale diagram can detect whether the process119883(119905)ismultifractalWe study themultifractality by usingMF-DFA[24] which is based on the standard fluctuation analysis (FA)

In MF-DFA one investigates the series

119884 (119894) fl 119894sum119896=1

(119883 (119896) minus 119883) 119894 = 1 119871 (11)

where 119871 is the length of 119883(119905) and 119883 is the mean of 119883(119905)Let119873119904 fl int(119871119904)119884(119905) is divided into119873119904 nonoverlapping

segments with equal length 119904 The fluctuation 1198652(V 119904) iscalculated using

1198652 (V 119904) = 1119904119904sum119894=1

119884 [(V minus 1) 119904 + 119894] minus 119910V (119894)2 V = 1 119873119904

(12)

where 119910V(119894) is the best polynomial fit of 119884[(Vminus 1)119904 + 119894] in eachof the 119873119904 segments V

In FA the fluctuation 1198652(V 119904) = (119884V119904 minus 119884(Vminus1)119904)2 thetrend in the original series is not considered In MF-DFAthe time series is detrended by subtracting 119910V(119894)The 119898-orderpolynomial used in the fitting procedure can eliminate thepolynomial trend of order 119898 minus 1 in the original series Thisensures the fluctuation analysis available for data affected bytrends or other nonstationarities

For a multifractal description of records one must con-sider all moments 119865119902(119904)

119865119902 (119904) fl 1119873119904119873119904sumV=1

[1198652 (V 119904)]11990221119902

V = 1 119873119904 (13)

If 119883(119905) is long-range dependent 119865119902(119904) satisfies [24]119865119902 (119904) sim 119904ℎ(119902) for large 119904 (14)

For very large 119904 that is 119904 gt 1198734119865119902(119904) becomes inaccuratedue to the statistical errors produced by the small numberof segments 119873119904 So we will concentrate on the scales 119904 lowerthan 1198734 The minimum scale chosen is equal to 25 to avoidspurious results produced by 119904 lt 30ℎ(119902) can be obtained by analyzing log-log plots 119865119902(119904)versus 119904 If 119902 = 2 ℎ(2) is the fluctuation exponent 119867 that isrelated to the correlation exponent 120574 by119867 = 1minus1205742 [24] For astationary time seriesℎ(2) is thewell-knownHurst exponentThe function ℎ(119902) is called the generalized Hurst exponent[24]

It is known that [14 24] for 119902 gt 0 ℎ(119902) shows the scalingbehaviour of the segments with large fluctuation and for 119902 lt0 ℎ(119902) shows that with small fluctuation

One can also get theRenyi scaling exponent 120591(119902) by calcu-lating 120591(119902) = 119902ℎ(119902)minus1Theway to describe themultifractalityis the singularity spectrum 119891(120572) (or multifractal spectrum)that is the Legendre transform of 120591(119902) [14 16 24]

120572 = 1205911015840 (119902) 119891 (120572) = 119902120572 minus 120591 (119902) (15)

6 Discrete Dynamics in Nature and Society

where 120572 is singularity strength (or Holder exponent) Thestrength of the multifractality of records can be characterizedby Δ120572 = 120572max minus 120572min

If ℎ(119902) satisfies the following equationℎ (119902) = 1119902 minus ln (119886119902 + 119887119902)

119902 ln 2 (16)

which can be obtained from a generalized binomial multi-fractal model [16] 120591(119902) = minus ln(119886119902 + 119887119902) ln 2 Δ120572 = ℎ(minusinfin) minusℎ(infin) In this case

Δ120572 = ln 119887 minus ln 119886ln 2 (17)

The three parameters 119886 119887 and Δ120572 can then be regardedas multifractal finger prints for records [14]

34 Wavelet Leaders for Multifractal Analysis For compar-ison with MF-DFA we introduce the method of waveletleaders for multifractal analysis (WLMF) proposed byWendtet al (see [41])

Assume that 120595(119905) is a wavelet function with compact timesupport and 119873 vanishing moment Let 120582119895119896 = [1198962119895 (119896 + 1)2119895)be dyadic intervals and let 3120582119895119896 denote the intervals 3120582119895119896 fl120582119895119896minus1 cup 120582119895119896 cup 120582119895119896+1 The wavelet leaders are defined as

119871119883 (119895 119896) fl sup[119896101584021198951015840(1198961015840+1)2119895

1015840)sub3120582119895119896

10038161003816100381610038161003816119889119883 (1198951015840 1198961015840)10038161003816100381610038161003816 (18)

The structure function is calculated by the averages of119871119883(119895 119896)119878119871119902 (119895) = 1119899119895

119899119895sum119896=1

119871119883 (119895 119896)119902 (19)

It can been shown [41] that under a mild uniform Holderregularity condition on 119883(119905)

119878119871119902 (119895) sim 1198651199022119895120577(119902) 2119895 997888rarr 0 (20)

So the estimation for 120577(119902) can be realized by a linearregression in 119895 versus log2119878119871119902(119895)

120577 (119902) = 1198952sum119895=1198951

120596119895log2119878119871119902 (119895) (21)

The weights 120596119895 satisfy sum11989521198951 119895120596119895 equiv 1 and sum11989521198951 120596119895 equiv 0To estimate the multifractal spectrum Wendt et al [41]

suggest a parametric formulation that avoids the computationof the Legendre transform

119891 (119902) = 1198952sum1198951

120596119895119880119871 (119895 119902)

ℎ (119902) = 1198952sum1198951

120596119895119881119871 (119895 119902) (22)

where 119880119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119877119902119883(119895 119896) + log2119899119895119881119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119871119883(119895 119896) and 119877119902119883(119895 119896) =119871119883(119895 119896)119902 sum119899119895119896=1 119871119883(119895 119896)119902

We emphasize that the wavelet transform with 119873 van-ishing moments can eliminate the polynomial trend of order119873 minus 1 in the original series

4 Computational Results

41 Wavelet-Based Results This section describes thewavelet-based estimation of Hurst parameters for the waterlevel records of the 10 sites We estimate the Hurst exponentsat different scales and different sampling interval timesThe results indicate that there is an interesting connectionbetween the wavelet scale and the sampling period For com-parison we repeat the procedure for a synthetic exponentiallong-range dependent process (ELRD) with length of180000 which is close to the actual sampling data lengthThevanishing moment of the used Daubechies wavelet is 119873 = 3(db3) The simulation results for ELRD are listed in Tables 2and 3 The values in brackets are errors bars The errors barsare based on the weighted linear least square fit which is usedto estimate 119867 and reveal 95 confidence interval

The simulation results show that the simulations for both119867 = 06 and 119867 = 08 are stable under changes in the intervallength and estimation scale One can see that the wavelet-based estimation of ELRD for theHurst exponent is valid andreliable and the selection of wavelet scale and sampling cycleis independent of the estimation

Tables 4ndash6 display the results of the practical water levelThere exists an expression 119886119894119895 asymp 119886119894+1119895minus1 (119886119894119895 is the element ofthe tables) For example in Table 4 H = 082 (12min scales(1sim5)) and H = 084 (6min scales (2sim6)) are approximatelyequalH = 070 (24min scales (1sim5))H = 068 (12min scales(2sim6)) and H = 068 (6min scales (3sim7)) are almost equaland so forthThe results represent the consistent relationshipbetween the wavelet scale and the sampling period Wepresent a mathematical proof in the Appendix

We can also see that the estimations for the Hurst expo-nent are not stable under changes in the interval length andestimation scale while the estimations along the secondarydiagonal direction of the tables are still stable We concludethat the massive hydrologic data have a multiscale consistentproperty while the simulation data do notThis indicates thatthere may exist multifractality in the records

42 Multiscale Diagram for Hydrological Data In this sec-tion we use the multiscale diagram to test the multifractalitythat may exist in the records The multiscale diagram is thegraph of 120577(119902) against 119902 together with the confidence intervalsrelated to 120577(119902)Themultifractality can be detected bymeasur-ing the deviation of 120577(119902) from the linear form [26]

We use a linear multiscale diagram to examine thealignment in the multiscale diagram in which 119867119902 = 120577(119902)119902is drawn against 119902 The multifractality can be determinedby examining whether there is a horizontal alignment in thelinearmultiscale diagramThehorizontal alignment indicates

Discrete Dynamics in Nature and Society 7

Table 2 Simulation data analysis for H (H = 06 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 061 (0009) 061 (0013) 060 (0018) 059 (0025) 059 (0036)2 061 (0013) 061 (0018) 060 (0025) 059 (0036) 060 (0051)4 061 (0018) 061 (0025) 060 (0036) 059 (0051) 061 (0071)8 060 (0025) 060 (0036) 060 (0051) 061 (0071) 062 (0106)

Table 3 Simulation data analysis for H (H = 08 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 083 (0009) 082 (0013) 081 (0018) 080 (0025) 080 (0034)2 083 (0013) 082 (0018) 081 (0025) 080 (0034) 082 (0049)4 083 (0018) 082 (0025) 081 (0034) 081 (0049) 080 (0072)8 082 (0025) 082 (0034) 082 (0049) 080 (0072) 079 (0100)

Table 4 Multiscales and multisampling for H of Site 1 (data length = 182855)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 084 (0062) 084 (0047) 068 (0046) 053 (0054)12min 082 (0067) 068 (0054) 053 (0054) mdash24min 070 (0046) 055 (0056) mdash mdash48min 057 (0053) mdash mdash mdash

Table 5 Multiscales and multisampling for H of Site 2 (data length = 152307)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 080 (0101) 086 (0086) 081 (0077) 061 (0086)12min 089 (0082) 080 (0079) 061 (0086) 050 (0111)24min 081 (0079) 063 (0087) 050 (0112) mdash48min 065 (0085) 052 (0116) mdash mdash

Table 6 Multiscales and multisampling for H of Site 3 (data length = 189967)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 079 (0047) 097 (0041) 087 (0044) 069 (0048)12min 095 (0046) 089 (0041) 069 (0047) 050 (0059)24min 091 (0044) 071 (0048) 050 (0059) mdash48min 074 (0046) 050 (0060) mdash mdash

that the Hurst parameter 119867 can control all of the scalingproperty The value of 119867 can be received by estimating thelevel The linear multiscale diagram and multiscale diagramare statistically equivalent

To determine the alignment of hydrological data weimpose the synthesized FBM (119867 = 08) known as the ldquolinearformrdquo for comparison The regressions for estimating 120577(119902)for each of the 10 different 119902 are performed for small scales(1198951 1198952) = (2 6) according to the power-law behaviour of

119878(119895 119902) discussed in Section 32 For Site 1ndashSite 10 as well asthe FBM the multiscale diagrams are shown in Figure 4 andthe values of 120577(119902) for all 10 sites are also shown in Table 7 (thesampling interval is 6 minutes)We choose db3 as the waveletof the wavelet transform which is orthogonal to quadratictrends

In Figure 4 the multiscale diagram in (a) shows a lackof alignment for the hydrologic data compared to the clearalignment of FBM for its self-similarity which indicates that

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

6 Discrete Dynamics in Nature and Society

where 120572 is singularity strength (or Holder exponent) Thestrength of the multifractality of records can be characterizedby Δ120572 = 120572max minus 120572min

If ℎ(119902) satisfies the following equationℎ (119902) = 1119902 minus ln (119886119902 + 119887119902)

119902 ln 2 (16)

which can be obtained from a generalized binomial multi-fractal model [16] 120591(119902) = minus ln(119886119902 + 119887119902) ln 2 Δ120572 = ℎ(minusinfin) minusℎ(infin) In this case

Δ120572 = ln 119887 minus ln 119886ln 2 (17)

The three parameters 119886 119887 and Δ120572 can then be regardedas multifractal finger prints for records [14]

34 Wavelet Leaders for Multifractal Analysis For compar-ison with MF-DFA we introduce the method of waveletleaders for multifractal analysis (WLMF) proposed byWendtet al (see [41])

Assume that 120595(119905) is a wavelet function with compact timesupport and 119873 vanishing moment Let 120582119895119896 = [1198962119895 (119896 + 1)2119895)be dyadic intervals and let 3120582119895119896 denote the intervals 3120582119895119896 fl120582119895119896minus1 cup 120582119895119896 cup 120582119895119896+1 The wavelet leaders are defined as

119871119883 (119895 119896) fl sup[119896101584021198951015840(1198961015840+1)2119895

1015840)sub3120582119895119896

10038161003816100381610038161003816119889119883 (1198951015840 1198961015840)10038161003816100381610038161003816 (18)

The structure function is calculated by the averages of119871119883(119895 119896)119878119871119902 (119895) = 1119899119895

119899119895sum119896=1

119871119883 (119895 119896)119902 (19)

It can been shown [41] that under a mild uniform Holderregularity condition on 119883(119905)

119878119871119902 (119895) sim 1198651199022119895120577(119902) 2119895 997888rarr 0 (20)

So the estimation for 120577(119902) can be realized by a linearregression in 119895 versus log2119878119871119902(119895)

120577 (119902) = 1198952sum119895=1198951

120596119895log2119878119871119902 (119895) (21)

The weights 120596119895 satisfy sum11989521198951 119895120596119895 equiv 1 and sum11989521198951 120596119895 equiv 0To estimate the multifractal spectrum Wendt et al [41]

suggest a parametric formulation that avoids the computationof the Legendre transform

119891 (119902) = 1198952sum1198951

120596119895119880119871 (119895 119902)

ℎ (119902) = 1198952sum1198951

120596119895119881119871 (119895 119902) (22)

where 119880119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119877119902119883(119895 119896) + log2119899119895119881119871(119895 119902) = sum119899119895119896=1 119877119902119883(119895 119896)log2119871119883(119895 119896) and 119877119902119883(119895 119896) =119871119883(119895 119896)119902 sum119899119895119896=1 119871119883(119895 119896)119902

We emphasize that the wavelet transform with 119873 van-ishing moments can eliminate the polynomial trend of order119873 minus 1 in the original series

4 Computational Results

41 Wavelet-Based Results This section describes thewavelet-based estimation of Hurst parameters for the waterlevel records of the 10 sites We estimate the Hurst exponentsat different scales and different sampling interval timesThe results indicate that there is an interesting connectionbetween the wavelet scale and the sampling period For com-parison we repeat the procedure for a synthetic exponentiallong-range dependent process (ELRD) with length of180000 which is close to the actual sampling data lengthThevanishing moment of the used Daubechies wavelet is 119873 = 3(db3) The simulation results for ELRD are listed in Tables 2and 3 The values in brackets are errors bars The errors barsare based on the weighted linear least square fit which is usedto estimate 119867 and reveal 95 confidence interval

The simulation results show that the simulations for both119867 = 06 and 119867 = 08 are stable under changes in the intervallength and estimation scale One can see that the wavelet-based estimation of ELRD for theHurst exponent is valid andreliable and the selection of wavelet scale and sampling cycleis independent of the estimation

Tables 4ndash6 display the results of the practical water levelThere exists an expression 119886119894119895 asymp 119886119894+1119895minus1 (119886119894119895 is the element ofthe tables) For example in Table 4 H = 082 (12min scales(1sim5)) and H = 084 (6min scales (2sim6)) are approximatelyequalH = 070 (24min scales (1sim5))H = 068 (12min scales(2sim6)) and H = 068 (6min scales (3sim7)) are almost equaland so forthThe results represent the consistent relationshipbetween the wavelet scale and the sampling period Wepresent a mathematical proof in the Appendix

We can also see that the estimations for the Hurst expo-nent are not stable under changes in the interval length andestimation scale while the estimations along the secondarydiagonal direction of the tables are still stable We concludethat the massive hydrologic data have a multiscale consistentproperty while the simulation data do notThis indicates thatthere may exist multifractality in the records

42 Multiscale Diagram for Hydrological Data In this sec-tion we use the multiscale diagram to test the multifractalitythat may exist in the records The multiscale diagram is thegraph of 120577(119902) against 119902 together with the confidence intervalsrelated to 120577(119902)Themultifractality can be detected bymeasur-ing the deviation of 120577(119902) from the linear form [26]

We use a linear multiscale diagram to examine thealignment in the multiscale diagram in which 119867119902 = 120577(119902)119902is drawn against 119902 The multifractality can be determinedby examining whether there is a horizontal alignment in thelinearmultiscale diagramThehorizontal alignment indicates

Discrete Dynamics in Nature and Society 7

Table 2 Simulation data analysis for H (H = 06 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 061 (0009) 061 (0013) 060 (0018) 059 (0025) 059 (0036)2 061 (0013) 061 (0018) 060 (0025) 059 (0036) 060 (0051)4 061 (0018) 061 (0025) 060 (0036) 059 (0051) 061 (0071)8 060 (0025) 060 (0036) 060 (0051) 061 (0071) 062 (0106)

Table 3 Simulation data analysis for H (H = 08 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 083 (0009) 082 (0013) 081 (0018) 080 (0025) 080 (0034)2 083 (0013) 082 (0018) 081 (0025) 080 (0034) 082 (0049)4 083 (0018) 082 (0025) 081 (0034) 081 (0049) 080 (0072)8 082 (0025) 082 (0034) 082 (0049) 080 (0072) 079 (0100)

Table 4 Multiscales and multisampling for H of Site 1 (data length = 182855)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 084 (0062) 084 (0047) 068 (0046) 053 (0054)12min 082 (0067) 068 (0054) 053 (0054) mdash24min 070 (0046) 055 (0056) mdash mdash48min 057 (0053) mdash mdash mdash

Table 5 Multiscales and multisampling for H of Site 2 (data length = 152307)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 080 (0101) 086 (0086) 081 (0077) 061 (0086)12min 089 (0082) 080 (0079) 061 (0086) 050 (0111)24min 081 (0079) 063 (0087) 050 (0112) mdash48min 065 (0085) 052 (0116) mdash mdash

Table 6 Multiscales and multisampling for H of Site 3 (data length = 189967)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 079 (0047) 097 (0041) 087 (0044) 069 (0048)12min 095 (0046) 089 (0041) 069 (0047) 050 (0059)24min 091 (0044) 071 (0048) 050 (0059) mdash48min 074 (0046) 050 (0060) mdash mdash

that the Hurst parameter 119867 can control all of the scalingproperty The value of 119867 can be received by estimating thelevel The linear multiscale diagram and multiscale diagramare statistically equivalent

To determine the alignment of hydrological data weimpose the synthesized FBM (119867 = 08) known as the ldquolinearformrdquo for comparison The regressions for estimating 120577(119902)for each of the 10 different 119902 are performed for small scales(1198951 1198952) = (2 6) according to the power-law behaviour of

119878(119895 119902) discussed in Section 32 For Site 1ndashSite 10 as well asthe FBM the multiscale diagrams are shown in Figure 4 andthe values of 120577(119902) for all 10 sites are also shown in Table 7 (thesampling interval is 6 minutes)We choose db3 as the waveletof the wavelet transform which is orthogonal to quadratictrends

In Figure 4 the multiscale diagram in (a) shows a lackof alignment for the hydrologic data compared to the clearalignment of FBM for its self-similarity which indicates that

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 7

Table 2 Simulation data analysis for H (H = 06 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 061 (0009) 061 (0013) 060 (0018) 059 (0025) 059 (0036)2 061 (0013) 061 (0018) 060 (0025) 059 (0036) 060 (0051)4 061 (0018) 061 (0025) 060 (0036) 059 (0051) 061 (0071)8 060 (0025) 060 (0036) 060 (0051) 061 (0071) 062 (0106)

Table 3 Simulation data analysis for H (H = 08 data length = 180000)

Interval 1198951sim5 2sim6 3sim7 4sim8 5sim9

1 083 (0009) 082 (0013) 081 (0018) 080 (0025) 080 (0034)2 083 (0013) 082 (0018) 081 (0025) 080 (0034) 082 (0049)4 083 (0018) 082 (0025) 081 (0034) 081 (0049) 080 (0072)8 082 (0025) 082 (0034) 082 (0049) 080 (0072) 079 (0100)

Table 4 Multiscales and multisampling for H of Site 1 (data length = 182855)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 084 (0062) 084 (0047) 068 (0046) 053 (0054)12min 082 (0067) 068 (0054) 053 (0054) mdash24min 070 (0046) 055 (0056) mdash mdash48min 057 (0053) mdash mdash mdash

Table 5 Multiscales and multisampling for H of Site 2 (data length = 152307)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 080 (0101) 086 (0086) 081 (0077) 061 (0086)12min 089 (0082) 080 (0079) 061 (0086) 050 (0111)24min 081 (0079) 063 (0087) 050 (0112) mdash48min 065 (0085) 052 (0116) mdash mdash

Table 6 Multiscales and multisampling for H of Site 3 (data length = 189967)

Interval 1198951sim5 2sim6 3sim7 4sim8

6min 079 (0047) 097 (0041) 087 (0044) 069 (0048)12min 095 (0046) 089 (0041) 069 (0047) 050 (0059)24min 091 (0044) 071 (0048) 050 (0059) mdash48min 074 (0046) 050 (0060) mdash mdash

that the Hurst parameter 119867 can control all of the scalingproperty The value of 119867 can be received by estimating thelevel The linear multiscale diagram and multiscale diagramare statistically equivalent

To determine the alignment of hydrological data weimpose the synthesized FBM (119867 = 08) known as the ldquolinearformrdquo for comparison The regressions for estimating 120577(119902)for each of the 10 different 119902 are performed for small scales(1198951 1198952) = (2 6) according to the power-law behaviour of

119878(119895 119902) discussed in Section 32 For Site 1ndashSite 10 as well asthe FBM the multiscale diagrams are shown in Figure 4 andthe values of 120577(119902) for all 10 sites are also shown in Table 7 (thesampling interval is 6 minutes)We choose db3 as the waveletof the wavelet transform which is orthogonal to quadratictrends

In Figure 4 the multiscale diagram in (a) shows a lackof alignment for the hydrologic data compared to the clearalignment of FBM for its self-similarity which indicates that

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

8 Discrete Dynamics in Nature and Society

Multiscale diagram (j1 j2) = (2 6)

minus3

minus2

minus1

0

1

2

3

4

5

6

7

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

q

(a)

Linear multiscale diagram

minus04

minus02

0

02

04

06

08

1

12

Hq

4 6 82q

FBMSite 1Site 2Site 3Site 4Site 5

Site 6Site 7Site 8Site 9Site 10

Hq = qq

(b)

Figure 4 Testing for multiscaling (a) Multiscale diagram Compared with the alignment for FBM hydrological data with no alignmentsuggests multiscale behaviour (b) Linear multiscale diagramThe FBM series exhibits horizontal alignment The errors bars are based on theweighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Table 7 120577(119902) for hydrological data of 10 sites 120577(119902) is estimated with a sampling interval of 6min and (1198951 1198952) = (2 6)119902 05 1 15 2 25 3 4 5 6 8Site 1 052 086 113 133 149 161 180 197 215 252Errors bars 0008 0015 0025 0041 0064 0095 0170 0250 0319 0423Site 2 051 097 134 156 168 176 187 197 209 239Errors bars 0010 0023 0046 0083 0123 0160 0233 0308 0383 0526Site 3 052 100 139 168 185 196 207 212 216 226Errors bars 0007 0014 0024 0045 0077 0117 0201 0282 0357 0471Site 4 059 116 161 170 159 147 127 115 107 100Errors bars 0007 0017 0064 0156 0222 0272 0362 0441 0511 0618Site 5 055 094 110 098 076 049 minus006 minus061 minus113 minus210Errors bars 0013 0031 0086 0161 0228 0288 0394 0481 0552 0655Site 6 042 080 112 137 158 173 191 202 210 226Errors bars 0008 0015 0023 0037 0060 0090 0161 0227 0285 0386Site 7 048 077 102 123 142 158 186 208 225 256Errors bars 0016 0031 0050 0075 0110 0157 0282 0420 0537 0684Site 8 058 107 152 177 182 180 174 171 171 176Errors bars 0015 0027 0067 0188 0334 0437 0542 0589 0615 0653Site 9 046 088 125 151 166 176 198 222 246 294Errors bars 0007 0015 0035 0079 0140 0202 0304 0373 0423 0495Site 10 059 112 159 186 195 198 203 211 222 250Errors bars 0012 0024 0057 0135 0220 0286 0383 0455 0513 0599Note The errors bars are based on the weighted linear least square fit which is used to estimate 120577(119902) and reveal 95 confidence interval

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 9

it has multifractal behaviour Thus the multifractal modelmay be relevant to the hydrological dataThe linearmultiscalediagram in (b) confirms this conclusionTheHurst parameter119867 of the synthesized FBM series can be estimated by thehorizontal alignment and its estimator is approximately 08This is not the case for the hydrologic data In summarythe hydrological data have multifractal behaviour and do notcomply with the equation 120577(119902) = 119867119902 The errors bars inFigure 4 are based on the weighted linear least square fitwhich is used to estimate 120577(119902) They are calculated as 196 timesStd 120577(119902) Under the assumption of normality this reveals a95 confidence interval The detailed values of 120577(119902) for othersites can be seen in Table 7

43 Multifractal Analysis and Results In this section weanalyze the daily water level records at the 10 observationsites using MF-DFA analysis for the multifractal behaviourIn order to ensure that there are enough fluctuations in eachsegment so that MF-DFA can be applied the high-frequencycollected daily records are used For sufficient samples weget 100 sets of daily records randomly at different timesof one day The number of records of each sets is aboutsix hundred We also apply wavelet leaders for multifractalanalysis (WLMF) based on the daily water level records for acomparison with MF-DFA

Statistical Tests It is well known that finite-size effects andlack of statistical convergence will introduce biases on theestimate of ℎ(119902) when applying the method of MF-DFAand WLMF Also the robustness of the results obtained bydifferent orders of polynomials in MF-DFA and differentvanishing moments of wavelet inWLMF should be tested Inthis part we perform some statistical tests for these issues

Following the test method reported by Khalil et al [42]we consider the problem of statistical convergence of ℎ(119902)through the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) in thetwo methods Recall that 119865119902119902 (119904) and 119878119871119902(119895) can be transformedinto a continuous integral over 119872

119878119871119902 (119895) = 1119899119895119899119895sum119896=1

119871119883 (119895 119896)119902 = int 119872119902119875119895 (119872) 119889119872

119865119902119902 (119904) = 1119873119904119873119904sumV=1

[1198652 (V 119904)]1199022 = int 119872119902119875119904 (119872) 119889119872(23)

119872 means the modulus of 119871119883(119895 119896) or [1198652(V 119904)]12 119875119895(119872)means the probability density function (pdf) of 119871119883(119895 119896)119875119904(119872) means the probability density function (pdf) of[1198652(V 119904)]12 The well-defined shape of the pdf rsquos plots indi-cates a proper convergence [42 43] Figures 5(b) 5(f) 6(b)and 6(f) show 119875119895(119872) and 119875119904(119872) computed at three differentscales at Site 4 and Site 7 It can be seen that there are notenough statistics to give a well-defined shape of the pdfHowever the statistical convergence of 119865119902119902 (119904) and 119878119871119902(119895) doesnot need so many statistics that can give a well-defined shapeof the pdf Then other statistical tests of convergence areconductedWe need to check the plots of119865119902119902 (119904) or 119878119871119902(119895) versus

the number of datasets used (from 1 to 100 implicating 119899119895 or119873119904) If119865119902119902 (119904) or 119878119871119902(119895) in plots is convergentwith the increase ofnumber of datasets used the statistical convergence of 119865119902119902 (119904)or 119878119871119902(119895) is confirmed Figures 5(b) 5(f) 6(b) and 6(f) alsoshow that the shapes of pdf are close to 0 with the decreaseof scale So we investigate the convergence for 119902 = 10 witha larger scale and 119902 = minus10 with a smaller scale (see Figures5(c) 5(d) 5(g) 5(h) 6(c) 6(d) 6(g) and 6(h) for Site 4 andSite 7) All the convergence plots except for Site 7 show that119865119902119902 (119904) and 119878119871119902(119895) are convergent for the range of 119902 from minus10 to10 the range of 119904 from 25 to 272 and the range of 119895 from 3 to6 From Figures 6(d) and 6(h) it can be seen that 119865119902119902 (119904) and119878119871119902(119895) at Site 7 are not convergent for 119902 lt 0 with some smallscalesThis can be confirmedwith the log-log plots in Figures6(a) and 6(e)

Figures 5(a) 5(e) 6(a) and 6(e) show the log-log plots atSite 4 and Site 7 obtained by three different orders of polyno-mials in MF-DFA and three different vanishing moments ofwavelet inWLMF It can be seen that three different orders ofpolynomials in MF-DFA give the consistent results as threedifferent vanishing moments inWLMF doThe robustness ofthe results obtained by different orders of polynomials inMF-DFA and different vanishing moments of wavelet in WLMFis confirmed

Selection of Parameters Before comparing MF-DFA andWLMF we should select parameters to be used in the twomethods InMF-DFA for detrending following [16] we use athird-order polynomial to calculate the best polynomial fit ofthe profile in each segment which can eliminate the second-order polynomial trend in the original series and is efficientfor the hydrologic data in [16] According to Section 33 thescales 119904 should be lower than 1198734 and the values of 119902 shouldbe both positive and negative For all of the sites the numberof daily water level records is approximately 600 and therange of time scale 119904 is selected from 25 to 272 The upperscale is selected for sufficient statistics The lower scale isselected for avoiding abnormal points caused by small scales(see Figure 7 for details) According to the statistical testsabove and following [14 16] we choose the range minus10 le119902 le 10 In WLMF compared with the results of MF-DFAwe appropriately choose 3 as the number of the vanishingmoment which can eliminate the second-order polynomialtrend in the original series The scale 119895 is chosen to be from 3to 6

Multifractal Model After using MF-DFA and WLMF weapply the least-squaresmethod to fit the generalized binomialmultifractalmodel (see (16)) Some othermultifractalmodelsmay be available such as 119871119878 ansatz model with the functionalform

ℎ (119902) = 1 + 1198671015840 minus 119862 (1199021205721015840minus1 minus 1)(1205721015840 minus 1) 119902 gt 0 (24)

with the three parameters 1198671015840 119862 and 1205721015840 which has been usedsuccessfully to describe the multifractal behaviour of rainfall

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

10 Discrete Dynamics in Nature and Society

(a) MF-DFA log-log plot (e) WLMF log-log plot

q = minus1 q = minus1

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

times104times108

times108

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus10

q = 10

q = minus10

q = 10

j = 3j = 4j = 6

15

35

4

45

5

SL q(j)

0

1

2

3

4

Fq q(s)

0

5

10

SL q(j)

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

6000

7000

8000

9000

Fq q(s)

05 1 15 2 250M

0

1

2

3

ps(M

)

1 2 3 4 5 6 7 80M

0

05

1

15

pj(M

)

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus20

minus15

minus10

minus5

0

5

ＦＩＡ 2

Fq

minus30

minus20

minus10

0

10

20

30

40

50

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

(f) WLMF pj(M) versus M(b) MF-DFA ps(M) versus M

Figure 5 Statistical tests at Site 4 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 11

(a) MF-DFA log-log plot (e) WLMF log-log plot

(d) MF-DFA convergence of Fqq (s) q = minus10 ＦＩＡ2s = 5

(c) MF-DFA convergence of Fqq (s) q = 10 ＦＩＡ2s = 72

(b) MF-DFA ps(M) versus M (f) WLMF pj(M) versus M

times10156 times10117

ＦＩＡ2s = 5ＦＩＡ2s = 6ＦＩＡ2s = 72

q = minus1q = minus10

q = 10q = minus1q = minus10

q = 10

j = 3j = 4j = 6

(h) WLMF convergence of SLq (j) q = minus10 j = 3

(g) WLMF convergence of SLq (j) q = 10 j = 6

minus60

minus50

minus40

minus30

minus20

minus10

0

10

ＦＩＡ 2

Fq

35 4 45 5 55 6 65 73ＦＩＡ2(Ｍ＝Ｆ)

minus100

0

100

200

300

400

500

600

700

ＦＩＡ 2

SL q(j)

2 3 4 5 61j

0

2

4

6

8

ps(M

)

05 1 150M

0

1

2

3

pj(M

)

05 1 15 2 250M

11

12

13

Fq q(s)

20 30 40 50 60 70 80 90 10010Sample size

5000

5500

6000

6500

7000

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

0995

1

1005

101

SL q(j)

20 30 40 50 60 70 80 90 10010Sample size

20 30 40 50 60 70 80 90 10010Sample size

0

2

4

6

Fq q(s)

Figure 6 Statistical tests at Site 7 (a) and (e) show the log-log plots of MF-DFA and WLMF different colors indicate different orders ofpolynomial (or different vanishingmoments of wavelet) (a) 2-order (red) 3-order (blue) and 4-order (green) (e) 2-moment (red) 3-moment(blue) and 4-moment (green) (b) shows the pdf of (1198652(V 119904))12 (c) and (d) show the convergence of 119865119902119902 (119904) The results in (b) (c) and (d)are based on 3-order (f) shows the pdf of 119871119883(119895 119896) (g) and (h) show the convergence of 119878119871119902(119895) The results in (f) (g) and (h) are based on3-moment These results correspond to an averaging over 100 sets of daily records

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

12 Discrete Dynamics in Nature and Society

Site 1 MFDFA Site 2 MFDFA

Site 3 MFDFA Site 4 MFDFA

Site 5 MFDFA Site 6 MFDFA

Site 7 MFDFA Site 8 MFDFA

Site 9 MFDFA Site 10 MFDFA

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

5 55 6 65 745ＦＩＡ2(Ｍ＝Ｆ)

minus6

minus4

minus2

0

2

4

ＦＩＡ 2

Fq

minus5

0

5

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus40

minus20

0

20

ＦＩＡ 2

Fq

minus6

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2

ＦＩＡ 2

Fq

minus4

minus2

0

2ＦＩＡ 2

Fq

Figure 7 The log-scaling plots of the water level dynamics of all of the sites The different symbols and colors indicate different moments119902 = minus10 (blue circles) 119902 = minus1 (red squares) and 119902 = 10 (green triangles) The colored lines indicate the regression lines of the correspondingcolors from scale 25 to scale 272 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained fromthese sets and reveal 95 confidence interval Some abnormal points at small scales are not shown in the figure because of their very smallvalues such as points of scale 244 and scale 248 at Site 8

and runoff records for 119902 gt 0 [16 44 45] However the 119871119878ansatz cannot be used for 119902 lt 0The two-parameter binomialmodel is suitable for both positive and negative values of 119902and has fewer parameters So the two-parameter binomialmodel is relevant

Computational Results By using the results of the fit ofthe generalized binomial multifractal model we calculateΔ120572 via (17) We can also calculate the fluctuation exponent119867 = ℎ(2) with MF-DFA see Table 8 for details 119867 can be

used to determine the long-range dependence because of itsrelationship with the correlation exponent 120574 For a stationarytime series 119867 is the well-known Hurst exponent

Figure 7 shows the log-scaling plots of the water leveldynamics of all of the sites It can be seen that except atSite 7 log2119865119902(119904) increases approximately with log2119904 for 119902 =minus10 minus1 10 and can be fittedwell by a straight line ℎ(119902) can beestimated by the slope of the regression line Moreover MF-DFA fails to estimate ℎ(119902) for 119902 lt 0 at Site 7

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 13

Table 8 The results of MF-DFA and WLMF for 10 sites

MF-DFA WLBMF119867(ℎ(2)) ℎ(2)rsquos errors bars a b Δ120572 a b Δ120572Site 1 088 0069 040 070 081 048 097 100Site 2 110 0043 039 056 052 039 066 076Site 3 090 0040 034 066 095 026 083 166Site 4 089 0028 030 067 117 036 087 127Site 5 092 0014 026 070 144 026 083 166Site 6 095 0029 026 063 127 032 077 126Site 7 088 0039 037 066 082 052 086 073Site 8 099 0018 016 063 199 019 095 228Site 9 120 0027 029 054 088 035 064 087Site 10 137 0017 025 046 085 030 053 081Note Since the MF-DFA and WLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fitting results at Site 7 are based on values of ℎ(119902) when 119902 gt 0

Figure 8 shows the comparison of the shape of ℎ(119902)assessed respectively by MF-DFA and WLMF and the fit ofthe two-parameter binomial model From the comparison ofthe shape of ℎ(119902) although the values of ℎ(119902) estimated by thetwo methods are different the shapes assessed by MF-DFAand WLMF at all sites (except Site 7) are similar and suggestthe two-parameter binomial model Especially at Site 1 Site 4Site 6 and Site 10 the shapes of ℎ(119902) assessed byMF-DFA andWLMF can be consistent by up- and downtranslation Thisindicates that the multifractality of the water level recordsassessed by MF-DFA is credible Since the MF-DFA andWLMF fail to estimate ℎ(119902) for 119902 lt 0 at Site 7 the fittingresults at Site 7 are based on values of ℎ(119902) when 119902 gt 0 FromFigure 8 it can be also seen that there are large errors bars for119902 lt 0 at Site 2 which also indicates poor estimation of ℎ(119902)for 119902 lt 0

To further confirm the results of MF-DFA we applyMF-DFA to the daily records at different times (morningafternoon and night) of a day As seen in Figure 9 the shapesofℎ(119902) at different times are similarThen through themutualverification of the results at different times we can furtherconfirm the results of MF-DFA that are shown in Figure 8

FromFigure 8 we can also find some details of the resultsAs mentioned above the estimation of ℎ(119902) for 119902 lt 0 byMF-DFA and WLMF at Site 2 and Site 7 is poor especiallyat Site 7The adjacent sites are also affected by these two sitesIt can be seen that at Site 6 and Site 8 ℎ(119902) near 0 deviate fromtheir respective fitting curves Excluding these sites ℎ(119902) ofmost other sites can be fitted well with the two parameters(119886 119887) The records of most sites satisfy the two-parameterbinomial model (or ℎ(119902) satisfy (16)) For the interpretationof the special behaviour at Site 2 and Site 7 we note that thefrequent water storage and discharge activities at these twosites lead to a differentwater level propertywhen compared toother sites Moreover apart from Site 2 Site 7 and those thatare seriously affected by these two sites (ie Site 1 Site 6 andSite 8) the two parameters 119886 and 119887 exhibit a relationship suchthat 119886 + 119887 asymp 1 whereas in the standard binomial multifractalmodel [24] the relationship is close to 119886 + 119887 = 1 Howeverfor all of the sites downstream of Site 7 the sum of 119886 and 119887is smaller than 09 The decrease in 119886 + 119887 can also be seen in

the figures for WLMF From Figure 2 we think that the fourtributaries downstream of Site 7 have a significant impact onthe river that is they change the multifractality of this riverrsquoswater level

From Table 8 we can see that all of the Hurst exponentsare larger than 08 This indicates that the water level recordsof the river have the long-range dependence property whichis the same as the result given by the wavelet-based methodThe values ofΔ120572 are larger than 05 which indicates that thereis strong multifractality at all sites

5 Discussion

The long-range dependence property and multifractality wedetected in the water level records are identical to theprevious results of hydrologic data (see eg [14ndash16]) andour results are based on multiple observation sites and a highsampling frequency that we have not yet seen in other studies

In the previous results (see eg [14ndash16]) we would like tonote that the lateral inflows have an effect on the runoffs (orwater level) Our results of the change of 119886+119887 seem to suggestthat the contribution of lateral inflow (tributaries) leads to thechange in the multifractality of the trunk streamrsquos water level

In light of the results obtained by MF-DFA the values of119886 are between 012 and 040 for all of the sites and the valuesof 119887 are between 054 and 070 From the previous results of41 riversrsquo runoffs studied by Koscielny-Bunde et al [14] thevalues of 119886 and 119887 of the Wurm river are in the range of ourvalues We would like to note that the Wurm river is locatedin southern Germany (near Munich) which has a temperatecontinental climate The river is also within a localized areaand its climate and area are similar to those of the analyzedriverThe consistency of the two rivers in parameters validatethe reliability of the obtained multifractal model

As previously mentioned the spurious multifractalitymay be induced by intricate nonstationarities which oftenexist in the hydrologic data for example due to the seasonalcycle or a change in climate We consider that the obtainedmultifractality is reliable First the method we used are basedon wavelet analysis and MF-DFA which can eliminate sometrends and avoid spurious detection of correlations Second

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

14 Discrete Dynamics in Nature and Society

Site 1 Site 2

Site 3Site 4

Site 5 Site 6

Site 7 Site 8

Site 9

MFDFAWLMF

Site 10

MFDFAWLMF

minus5minus10 5 100minus5minus10 5 1000608

112141618

ℎq

0

05

1

15

ℎq

05

1

15

2

ℎq

05

1

15

2

ℎq

0

05

1

ℎq

1

12

14

16

18

2

ℎq

05

1

15

2

25

ℎq

05

1

15

2

ℎq

05

1

15

ℎq

06

08

1

12

14

16

ℎq

Figure 8 Comparison of the shape of ℎ(119902) assessed by MF-DFA and WLMF The curves are obtained by fits of the two-parameter binomialmodel yielding (16) The different colors indicate the corresponding different fitting results The resulting model parameters are reported inTable 8 These results correspond to an averaging over 100 sets of daily records The errors bars are obtained from these sets and reveal 95confidence interval

the multifractality is confirmed consistently via multiplemethods (multiscale diagram MF-DFA and WLMF) TheMF-DFA and WLMF also give the same multifractal modeland this multifractal model is robust in different location anddifferent time Last the results obtained from the analyzedriver are consistent with the previous study of actual example

6 Conclusions

In this paper we proceed in an orderly manner to studythe LRD and multifractal properties of high-frequency water

level records of a river in northern China by using waveletanalysis and MF-DFA

We observe the long-range dependence property andmultiscale behaviour in the time series which indicates themultifractality This multifractality is detected via multiscalediagram Then we focus on the multifractality of recordsusing MF-DFA and WLMF At most sites the generalizedHurst exponent ℎ(119902) can be described with the functionℎ(119902) = 1119902 minus ln(119886119902 + 119887119902)(119902 ln 2) (119886 119887 isin (0 1]) and themultifractal models are robust for different times (morningafternoon and evening) The validity of multifractality is

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 15

Site 1Site 2

Site 3 Site 4

Site 5 Site 6

Site 7Site 8

Site 9

MorningAernoonNight

Site 10

MorningAernoonNight

minus5minus10 5 100minus5minus10 5 100

1

15

2

ℎq

1

2

3ℎq

1

15

2

ℎq

1

15

2

ℎq

06

08

1

12

14

ℎq

05

1

15

2

ℎq

08

1

12

ℎq

1

15

2

ℎq

1

15

2

ℎq

05

1

15ℎq

Figure 9The generalized Hurst exponents ℎ(119902) for ten representative daily water level records at different times of a day The errors bars arebased on the linear least square fit which is used to estimate ℎ(119902) and reveal 95 confidence interval

(partially) supported with the same model obtained from thedifferent methods (MF-DFA and WLMF) and by comparingprevious study of actual example The results also show thatthe poor estimation of ℎ(119902) for 119902 lt 0 at Site 2 and Site 7 andthe decreased change in 119886 + 119887 indicates human activities andtributaries have a great influence on the multifractality of theriver

We obtain and list the values of 119867 119886 119887 and Δ120572 for allavailable records including some abnormal records At allavailable sites MF-DFA yields values of Δ120572 are larger than05 whereas WLMF yields values are larger than 07 whichshows a strong multifractality in the data

As a technical improvement in the estimation of theHurstparameter we show that the sampling intervals and choice

of scales of the discrete wavelet transformation should beconsistent (Tables 4ndash6)

In contrast to the previous studies our studies are basedon a short sampling time which we have not seen beforeThis study contributes a new insight into previous researchapplications and it may also be the extension of the study ofmultifractality Further investigations are needed especiallyfor the massive data cases for example comparing the fittingof ℎ(119902) that is obtained from the records partitioned for eachyear As mentioned above the human activities (frequentwater storage and discharge activities) at Site 2 and Site 7 leadto poor estimation of ℎ(119902) for 119902 lt 0 and the change in 119886 + 119887This is the key point for the future work on detection andprediction A real time multifractal model may be built for

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

16 Discrete Dynamics in Nature and Society

ℎ(119902 119905) 119886(119905) and 119887(119905) The detection and prediction of humanactivities and climate change in real time may be conductedby the change of ℎ(119902 119905) 119886(119905) and 119887(119905) Besides it would bealso interesting to consider the water level dynamics as amultifractional process study 119867(119905) at the different sites withdifferent time lags and provide predictivemodels of thewaterlevel records

Appendix

A Proof of the Consistent Relationshipbetween the Wavelet Scale and theSampling Period in Section 41

As we know that the process 119883(119905) with sampling period 2119897119879 isequal to the process 119883(2119897119905) with sampling period 119879 thereforeprocess 119883(119905) with sampling period series 119879 2119879 4119879 2119897119879is equal to the process 119883(119905) 119883(2119905) 119883(4119905) 119883(2119897119905) withsampling period 119879 For convenience let 119883119897(119905) = 119883(2119897119905)

Assuming 119867(119897 1198951 1198952) is the Hurst exponent of time series119883119897(119905) for scale (1198951 1198952) then119889119883119897+1 (119895 119896) = int

1198772minus1198952120595 (2minus119895119905 minus 119896) 119883 (2119897+1119905) 119889119905 (A1)

Let 119905 = 2119905 we have119889119883119897+1 (119895 119896) = 2minus1 int

1198772minus1198952120595 (2minus(119895+1)119905 minus 119896) 119883 (2119897119905) 119889119905

= 2minus05119889119883119897 (119895 + 1 119896) (A2)

Furthermore

119864119889119883119897+1 (119895 119896)2 = 2minus1119864119889119883119897 (119895 + 1 119896)2 (A3)

According to the estimation method

119867 (119897 + 1 1198951 1198952) = 119867 (119897 1198951 + 1 1198952 + 1) (A4)

Under the discrete condition the degree of the approximationof119883(119905) is different for the different time sequence by the samescale of sampling frequency Then we only have

119864119889119883119897+1 (119895 119896)2 asymp 2minus1119864119889119883119897 (119895 + 1 119896)2 (A5)

More 119867(119897 + 1 1198951 1198952) asymp 119867(119897 1198951 + 1 1198952 + 1)B The Related Materials Download Addresses

The multiscale diagram is implemented in Matlab by Veitchand available from his homepage at httpcrinengutseduausimdarrylindexhtml

The multifractal detrended fluctuation analysis (MF-DFA) is implemented in Matlab by Ihlen and available athttpwwwntnueduinmgerisoftware

The wavelet leaders for multifractal analysis (WLMF) isimplemented in Matlab by Wendt and available from hishomepage at httpswwwiritfrsimHerwigWendtindexhtml

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Basic ResearchProgram of China (973 Program Grant no 2013CB910200)National Science Foundation of China (no 61103136) theopen project of the Hubei Province Key Laboratory of Intel-ligent Robot (no 201007) and the Fundamental ResearchFunds for the Central Universities (no JBK170164 noJBK120509 and no JBK140507)

References

[1] H E Hurst ldquoLong-term storage capacity of reservoirsrdquo Trans-actions of the American Society of Civil Engineers vol 116 pp770ndash808 1951

[2] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[3] J Mielniczuk and P Wojdyłło ldquoEstimation of Hurst exponentrevisitedrdquo Computational Statistics amp Data Analysis vol 51 no9 pp 4510ndash4525 2007

[4] J Barunik and L Kristoufek ldquoOn Hurst exponent estimationunder heavy-tailed distributionsrdquoPhysica A StatisticalMechan-ics and its Applications vol 389 no 18 pp 3844ndash3855 2010

[5] G Poveda ldquoMixed memory (non) Hurst effect and maximumentropy of rainfall in the tropical Andesrdquo Advances in WaterResources vol 34 no 2 pp 243ndash256 2011

[6] M A Sanchez Granero J E Trinidad Segovia and J GarcıaPerez ldquoSome comments on Hurst exponent and the longmemory processes on capital marketsrdquo Physica A StatisticalMechanics and its Applications vol 387 no 22 pp 5543ndash55512008

[7] B B Mandelbrot ldquoIntermittent turbulence in self-similar cas-cades Divergence of high moments and dimension of thecarrierrdquo Journal of Fluid Mechanics vol 62 no 2 pp 331ndash3581974

[8] E Bacry J F Muzy and A Arneodo ldquoSingularity spectrum offractal signals from wavelet analysis exact resultsrdquo Journal ofStatistical Physics vol 70 no 3-4 pp 635ndash674 1993

[9] J F Muzy E Bacry and A Arneodo ldquoThe multifractal formal-ism revisited with waveletsrdquo International Journal of Bifurcationand Chaos vol 4 no 2 pp 245ndash302 1994

[10] A Arneodo E Bacry and J F Muzy ldquoThe thermodynamics offractals revisited with waveletsrdquo Physica A Statistical Mechanicsand its Applications vol 213 no 1-2 pp 232ndash275 1995

[11] U Frisch and G Parisi ldquoOn the singularity structure offully developed turbulencerdquo in Turbulence and Predictabilityin Geophysical Fluid Dynamics and Climate Dynamics M GilR Benzi and G Parisi Eds pp 84ndash88 Elsevier AmsterdamNorth-Holland 1985

[12] J F Muzy E Bacry and A Arneodo ldquoWavelets andmultifractalformalism for singular signals Application to turbulence datardquoPhysical Review Letters vol 67 no 25 pp 3515ndash3518 1991

Discrete Dynamics in Nature and Society 17

[13] D Schertzer and S Lovejoy ldquoPhysical modeling and analysis ofrain and clouds by anisotropic scaling mutiplicative processesrdquoJournal of Geophysical Research Atmospheres vol 92 no 8 pp9693ndash9714 1987

[14] E Koscielny-Bunde J W Kantelhardt P Braun A Bundeand S Havlin ldquoLong-term persistence and multifractality ofriver runoff records Detrended fluctuation studiesrdquo Journal ofHydrology vol 322 no 1-4 pp 120ndash137 2006

[15] B-J Zang and P-J Shang ldquoMultifractal analysis of the YellowRiver flowsrdquo Chinese Physics vol 16 no 3 pp 565ndash569 2007

[16] JWKantelhardtD Rybski and SA Zschiegner ldquoMultifractal-ity of river runoff and precipitation comparison of fluctuationanalysis and wavelet methodsrdquo Physica A Statistical Mechanicsand its Applications vol 330 no 1-2 pp 240ndash245 2003

[17] V Livina Z Kizner P Braun T Molnar A Bunde and SHavlin ldquoTemporal scaling comparison of real hydrological dataand model runoff recordsrdquo Journal of Hydrology vol 336 no1-2 pp 186ndash198 2007

[18] C-M Chou ldquoWavelet-based multi-scale entropy analysis ofcomplex rainfall time seriesrdquo Entropy vol 13 no 1 pp 241ndash2532011

[19] Y-F Sang ldquoA review on the applications of wavelet transform inhydrology time series analysisrdquo Atmospheric Research vol 122pp 8ndash15 2013

[20] Y-F Sang ldquoWavelet entropy-based investigation into the dailyprecipitation variability in the Yangtze River Delta China withrapid urbanizationsrdquo Theoretical and Applied Climatology vol111 no 3-4 pp 361ndash370 2013

[21] V Venugopal S G Roux E Foufoula-Georgiou and AArneodo ldquoScaling behavior of high resolution temporal rainfallNew insights from a wavelet-based cumulant analysisrdquo PhysicsLetters A vol 348 no 3-6 pp 335ndash345 2006

[22] V Venugopal S G Roux E Foufoula-Georgiou and AArneodo ldquoRevisiting multifractality of high-resolution tempo-ral rainfall using a wavelet-based formalismrdquo Water ResourcesResearch vol 42 Article IDW06D14 2006

[23] S G Roux V Venugopal K Fienberg A Arneodo andE Foufoula-Georgiou ldquoEvidence for inherent nonlinearity intemporal rainfallrdquo Advances in Water Resources vol 32 no 1pp 41ndash48 2009

[24] J W Kantelhardt S A Zschiegner E Koscielny-Bunde SHavlin A Bunde and H E Stanley ldquoMultifractal detrendedfluctuation analysis of nonstationary time seriesrdquo Physica AStatistical Mechanics and its Applications vol 316 no 1ndash4 pp87ndash114 2002

[25] D Veitch and P Abry ldquoA wavelet-based joint estimator of theparameters of long-range dependencerdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 45 no 3 pp 878ndash897 1999

[26] P Abry P Flandrin M S Taqqu and D Veitch ldquoWaveletsfor the analysis estimation and synthesis of scaling datardquo Self-Similar Network Traffic and Performance Evaluation pp 39ndash882000

[27] P Abry H Helgason and V Pipiras ldquoWavelet-based analysis ofnon-Gaussian long-range dependent processes and estimationof the Hurst parameterrdquo Lithuanian Mathematical Journal vol51 no 3 pp 287ndash302 2011

[28] L Wu and Y Ding ldquoEstimation of self-similar Gaussian fieldsusing wavelet transformrdquo International Journal of Wavelets

Multiresolution and Information Processing vol 13 no 6 ArticleID 1550044 12 pages 2015

[29] L Wu and Y Ding ldquoWavelet-based estimator for the Hurstparameters of fractional Brownian sheetrdquo Acta MathematicaScientia B vol 37 no 1 pp 205ndash222 2017

[30] R H Riedi M S Crouse V J Ribeiro and R G Baraniuk ldquoAmultifractal wavelet model with application to network trafficrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 45 no 3 pp 992ndash1018 1999

[31] E Serrano and A Figliola ldquoWavelet leaders a new methodto estimate the multifractal singularity spectrardquo Physica AStatistical Mechanics and its Applications vol 388 no 14 pp2793ndash2805 2009

[32] G Xiong S-N Zhang and L Shu ldquoThe legendre multifractalspectrum distribution based on WTMMrdquo in Proceedings ofthe 3rd International Workshop on Chaos-Fractals Theories andApplications IWCFTA 2010 pp 481ndash485 October 2010

[33] E A F Ihlen ldquoIntroduction to multifractal detrended fluctua-tion analysis in Matlabrdquo Frontiers in Physiology vol 3 ArticleID Article 141 2012

[34] B Lashermes S Jaffard and PAbry ldquoWavelet leader basedmul-tifractal analysisrdquo in Proceedings of the 2005 IEEE InternationalConference on Acoustics Speech and Signal Processing ICASSPrsquo05 pp IV161ndashIV164 Philadelphia PA USA March 2005

[35] HWendt P Abry S Jaffard H Ji and Z Shen ldquoWavelet leadermultifractal analysis for texture classificationrdquo in Proceedingsof the 2009 IEEE International Conference on Image ProcessingICIP 2009 pp 3829ndash3832 Cairo Egypt November 2009

[36] S Jaffard B Lashermes and P Abry ldquoWavelet leaders inmultifractal analysisrdquo inWavelet Analysis and Applications pp201ndash246 Birkhauser 2007

[37] HWendt and P Abry ldquoMultifractality tests using bootstrappedwavelet leadersrdquo IEEE Transactions on Signal Processing vol 55no 10 pp 4811ndash4820 2007

[38] B Audit E Bacry J-F Muzy and A Arneodo ldquoWavelet-based estimators of scaling behaviorrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol48 no 11 pp 2938ndash2954 2002

[39] S Jaffard ldquoMultifractal formalism for functions II Self-similarfunctionsrdquo SIAM Journal on Mathematical Analysis vol 28 no4 pp 971ndash998 1997

[40] L Delbeke and W Van Assche ldquoA wavelet based estimatorfor the parameter of self-similarity of fractional Brownianmotionrdquo in 3rd International Conference on Approximationand Optimization in the CARibbean (Puebla 1995) vol 24 ofAportaciones Mat Comun pp 65ndash76 Soc Mat MexicanaMexico 1998

[41] H Wendt P Abry and S Jaffard ldquoBootstrap for empiricalmultifractal analysisrdquo IEEE Signal Processing Magazine vol 24no 4 pp 38ndash48 2007

[42] A Khalil G Joncas F Nekka P Kestener and A ArneodoldquoMorphological analysis of H I features II Wavelet-basedmultifractal formalismrdquo The Astrophysical Journal SupplementSeries vol 165 no 2 pp 512ndash550 2006

[43] A Arneodo N Decoster and S G Roux ldquoA wavelet-basedmethod for multifractal image analysis I Methodology andtest applications on isotropic and anisotropic random roughsurfacesrdquoTheEuropean Physical Journal B vol 15 no 3 pp 567ndash600 2000

18 Discrete Dynamics in Nature and Society

[44] Y Tessier S Lovejoy P Hubert D Schertzer and S PecknoldldquoMultifractal analysis and modeling of rainfall and river flowsand scaling causal transfer functionsrdquo Journal of GeophysicalResearch Atmospheres vol 101 no 21 pp 26427ndash26440 1996

[45] G Pandey S Lovejoy and D Schertzer ldquoMultifractal analysisof daily river flows including extremes for basins of five totwo million square kilometres one day to 75 yearsrdquo Journal ofHydrology vol 208 no 1-2 pp 62ndash81 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

18 Discrete Dynamics in Nature and Society

[44] Y Tessier S Lovejoy P Hubert D Schertzer and S PecknoldldquoMultifractal analysis and modeling of rainfall and river flowsand scaling causal transfer functionsrdquo Journal of GeophysicalResearch Atmospheres vol 101 no 21 pp 26427ndash26440 1996

[45] G Pandey S Lovejoy and D Schertzer ldquoMultifractal analysisof daily river flows including extremes for basins of five totwo million square kilometres one day to 75 yearsrdquo Journal ofHydrology vol 208 no 1-2 pp 62ndash81 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of