Journal of Hydrology (2007) 336, 186–198

ava i lab le at www.sc iencedi rec t . com

journal homepage: www.elsevier .com/ locate / jhydro l

Temporal scaling comparison of real hydrologicaldata and model runoff records

V. Livina a,*, Z. Kizner b, P. Braun c, T. Molnar c, A. Bunde d, S. Havlin b

a School of Environmental Sciences, University of East Anglia, Norwich, UKb Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israelc Bayerisches Landesamt fur Wasserwirtschaft, Lazarettstrasse 67, D-80636 Munchen, Germanyd Institut fur Theoretische Physik III, Justus-Liebig-Universitat Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany

Received 16 February 2005; received in revised form 21 December 2006; accepted 2 January 2007

00do

KEYWORDSASGi model;Time series analysis;Detrended fluctuationanalysis;Multifractality;Nonlinear volatility;Statistical evaluationof models

22-1694/$ - see front mattei:10.1016/j.jhydrol.2007.01

* Corresponding author. Tel.E-mail address: v.livina@u

r ª 200.014

: +44 78ea.ac.uk

Summary We show that the scaling properties of river runoff records represent a usefultool for evaluating precipitation-runoff models that are widely used in hydrology forassessment of the water balance in a given river catchment. In this respect, it is importantthat the model maps the processes that control the water balance. The main field of appli-cation is therefore water management in a given area over a long time scale (at least sev-eral years). Here, we compare the temporal scaling properties of the runoff of threeBavarian rivers (Naab, Regnitz, and Vils) with the corresponding ASGi model records. Inthe evaluation, we use: (i) detrended fluctuation analysis (DFA); (ii) multifractal analysis;(iii) periodic volatility analysis; and (iv) long-term volatility analysis. Our study generallyshows close similarity between real and simulated data for the main statistical parameters(e.g., correlation and multifractal exponents). Therefore, the ASGi model output seems toadequately describe real basin processes and might be useful for hydrological purposes,such as a posteriori estimation of water balance in a river catchment.ª 2007 Elsevier B.V. All rights reserved.

Introduction

Over the last few decades, many hydrological models havebeen developed, allowing to simulate records of river fluxfor short- and long-term intervals (Beven and Moore, 1993;Beven, 1998; Dutta et al., 2000; Beven, 2004). The model-ling was spurred in 1980–1990s, when computers enabled

7 Elsevier B.V. All rights reserved

96330375.(V. Livina).

processing a large amount of geophysical data and solvingnumerically complex fluid dynamics equations.

There are different approaches in hydrological model-ling, and their intercomparison has been discussed inten-sively (see Refsgaard and Knudsen, 1996; also Marshallet al., 2005). The linear stochastic modelling, where autore-gressive processes are implemented (e.g., fractional ARIMA,Montanari et al., 2000), by definition, is unable to reproducenonlinear features of river runoff series and will not be con-sidered in this paper. Another approach is deterministic

.

Temporal scaling comparison of real hydrological data and model runoff records 187

modelling. Distributed physically based models are devel-oped on partial differential equations governing hydrologi-cal systems (TOPMODEL, MIKE SHE, WATFLOOD, IISDHM,and others; for review see Beven, 2004). Such models usefield measurements and remote sensing records as inputdata and can be exploited for water management. Third,there are parametric conceptual models, where the systemsof deterministic equations are supplemented by someempirical relations (Jones, 1997).

The distributed physically-based models incorporatestreamflow, precipitations, evaporation, averaged stochas-tic groundwater flow and spatial properties of the catch-ment, e.g., elevation data, topographic indices, glaciergrid, land use, and soil types. The spatial scaling parametersare different for different river subcatchments, and only inexceptional cases is it possible to verify them by means ofexplicit variations of the grid width in use. Another problemwhich is difficult to resolve is the exact initial and boundaryconditions for solving partial differential equations, andthose conditions are, to a great extent, unknown, being astochastic component of the system. This explains whymany models fail in reproducing and predicting the patternsof real time series (Wood et al., 1988; Bloschl, 2001). For in-stance, the models of the atmosphere/ocean system(AOGCM), which are based on the same partial differentialequations of fluid dynamics, demonstrated lack of long-term memory in temperature records, compared to realtime series, as has been recently shown using modern toolsof statistical physics (Govindan et al., 2002; Vyushin et al.,2004).

A contemporary parametric conceptual model tunes adetereministic kernel by implementing some empirical rela-tions, and this is supposed to bring the output records closerto real observations. Although the conceptual models re-quire additional calibration to obtain proper values of theparameters, they allow to solve the problem of the lack ofinitialisation data in physically-based models. Also, the con-ceptual models provide a better adjustment to a particularcatchment (Refsgaard and Knudsen, 1996). A typical exam-ple of a parametric model which combines advances of dis-tributed models with empirical balance, is the conceptualgrid-based ASGi model1 (Braun et al., 1998; Becker andBraun, 1999).

The model data comparison is traditionally based onquantitative criteria of model accuracy (for instance, theNash–Sutcliffe coefficient). However, there are recent at-tempts to apply more advanced techniques, like Bayesianapproach (Marshall et al., 2005). Still, this analysis doesnot take into account nonlinear characteristics of the datawhich are important for understanding of the processdynamics.

Here, we perform novel validation of a hydrological mod-el by applying techniques that have been recently devel-oped in statistical physics, such as Detrended FluctuationAnalysis (the major paper by (Peng et al., 1994), with multi-ple applications in many fields of the modern physics in thelast 10 years – see e.g., Santhanam et al., 2006; Kantel-hardt et al., 2006), nonlinear volatility analysis (Ashkenazyet al., 2001; Livina et al., 2003a), and multifractal analysis

1 Kontinuierlicher Abfluss und Stofftransport-Integrierte Modellie-rung unter Nutzung von Geoinformationssystemen, Germany.

(Muzy et al., 1991; Arneodo et al., 1995; Ivanov et al., 1999;Struzik, 2000; Kantelhardt et al., 2002; Enescu et al., 2006).These techniques allow to quantify essential statistical fea-tures (correlations, scaling and nonlinearity) in time seriesand thus provide understanding of the underlying dynamicalprocesses in the system.

The ASGi model and the rivers studied

We study the daily discharge time series generated by theASGi model for three Bavarian rivers. Implementing empiri-cal hydrological relations, the model, in contrast with‘pure’ PDE-based models, does not map the physical pro-cesses by means of exact conservation laws, but establishesan empirical balance between sinks and sources in the gridelements. It is based on the water balance simulationmodel, CH (WaSiM-ETH, Schulla, 2000; Jasper et al., 2002;Gurtz et al., 2003; Verbunt et al., 2003).

The WaSiM-ETH model has been used in more thantwenty institutions in Switzerland, Germany, Austria, Ice-land and Slovakia. A more specialised ASGi model has beenused in three German institutions. Due to the model layoutwhich is well adapted to basins of European rivers, themodel can be useful for many European users.

The model consists of the following main components(Fig. 1): (i) cell interpolation and adjustment of input mete-orological data; (ii) estimation of evapotranspiration and, ifnecessary, irrigation; (iii) snow-glacier submodel; (iv) inter-ception submodel; (v) infiltration submodel; (vi) soil sub-model; (vii) groundwater submodel; and (viii) dischargerouting. Coupling different parts of the model and takinginto account losses due to evapotranspiration, the mainsources of river discharge (surface runoff, interflow andbaseflow) are integrated. The baseflow is generated forthe entire subbasin as an average value. Interflow is gener-ated for each grid cell separately and then averaged overthe area. The surface runoff is routed to subbasin outlet,using a subdivision of the basin into flow time zones, whichare zones of equal flow times for surface runoff to reach thesubbasin outlet. Because of the local soil type, the ASGimodel uses the first version of WaSiM, with TOPMODELgroundwater modelling, instead of the second version,implementing Richard’s equation for groundwater. Suffi-cient adequacy of the model can be illustrated directly bymeans of visual comparison of real and simulated fluxes inhydrographs (Fig. 2). The calibration of the model was car-ried on by trial and error estimation using Nash–Sutcliffeefficiency criterion, and the flow was found to be reason-ably well predicted (see ‘Discussion’).

The Naab river basin area (about 5500 km2) is locatedmainly in the Oberpfalz region in Bavaria, and partly inthe Czech Republic. The Naab is a left tributary of the Dan-ube river, which it joins near Regensburg, in the Danube’supper basin. The catchment is placed in the FrankischeAlb (Eastern continuation of the Schwabische Alb in Ba-den-Wurttemberg), a mountain chain up to 1000 m high(500 m at average) in Bavaria. At its origin, the Naab is amountain stream, and in the lower part it is a slow plainstream. Therefore, the catchment is highly spatially vari-able and difficult for hydrological modelling and forecast-ing. Geologically, the basin is a compound of sand, karst,and slate grit.

Figure 1 Schematic of the WaSiM-ETH model.

188 V. Livina et al.

The catchment of the Danube’s tributary Vils lies in thenorthern part of Bavaria, the Oberpfalz. This area is charac-terized by complex geological conditions (Jura). The maindifficulty in this context is existence of karst areas, whichshows some unusual water storage properties. In order toconsider this specific behaviour, it was necessary to intro-duce some empirical corrections in the model.

The Regnitz river is a left tributary of the Main river inthe Lower Bavaria. It is formed by the confluence of the

Regnitz and Pegnitz and flows into the Main below Bamberg.The catchment area is about 7000 km2 and varies in heightbetween 240 and 650 m. The geology is comprised of slate,clay, and malm-soil types which are of different permeabil-ity and storage capacity.

The maps of precipitation, evaporation and runoff of thepart of Bavaria where the considered river basin are placedare shown in Figs. 3–5. The hydrological characteristics ofthe basins are summarised in Table 1.

0

50

real data

1986 1987 1988 1989 1990

s [year]

0

model data

Flux

[m

3 /sec

]

Figure 2 Five-year cycles of real and simulated flux records for Vils river; model data demonstrate similar pattern with slightlydifferent magnitudes of the fluctuations.

Figure 3 Yearly averaged precipitation map of the part of Bavaria where the basins are placed.

Temporal scaling comparison of real hydrological data and model runoff records 189

Detrended fluctuation analysis

Methodology

In recent years, the detrended fluctuation analysis (DFA)method has become a widely used tool for the study of sta-tistical scaling properties of nonstationary time series (Peng

et al., 1994). It has been applied successfully, e.g., to DNAsequences (Buldyrev et al., 1995), heart-rate dynamics(Peng et al., 1995; Bunde et al., 2000; Ashkenazy et al.,2001), to econometric time series (Mantegna and Stanley,1999; Matia et al., 2003), climate dynamics (Koscielny-Bunde et al., 1998; Govindan et al., 2002; Kantelhardtet al., 2006), and other fields of the modern physics. The

Figure 4 Yearly averaged evaporation map of the part of Bavaria where the basins are placed.

Figure 5 Yearly averaged runoff map of the part of Bavaria where the basins are placed.

190 V. Livina et al.

essence of the method is studying the properties of the fluc-tuations in the data after removing the seasonal trend andother nonstationarities, and this is achieved by multiple

averaging of the mean-root square variance over windowsof variable length in the integral of the series. The methodprovides robust results on two-point correlations, as com-

Table 1 Hydrological characteristics of the three rivers (1961–1990)

River Station Elevation Area(km2)

Precipitation(mm/year)

Evaporation(mm/year)

Runoff

Minimum(m)

Maximum(m)

(1/(s km2)) Averagehigh (m3/s)

Averagelow (m3/s)

Naab Heitzenhofen 350 1000 5246 700–900 400–700 2.18 280 18.9Vils Schmidmuhlen 350 620 756 550–900 450–570 – 130 –Regniz Pettstadt 240 645 6999 650–1000 450–600 4.8–12.7 287 20.7

Temporal scaling comparison of real hydrological data and model runoff records 191

pared to conventional power-spectrum and auto-correlationfunction analyses. Moreover, in case of highly nonstationarydata (in particular, hydrological), the auto-correlation func-tion is incapable to reveal the correlations properly; in thepower spectrum, the analysis is often deviated by the nois-iness of the power amplitude. The DFA is devoid of theseweakness, and this makes it an efficient tool for studyingthe statistical properties of the time series of complexsystems.

Since we are concerned here with studying flux fluctua-tions, before applying statistical scaling techniques weeliminate the seasonal periodicities in the data by removingthe seasonal average (deseasoning). In the framework of thek-order DFA method (DFAk), we integrate the fluctuationseries and divide the range of definition of the obtained pro-file function into windows of size s. Next, within each win-dow, we calculate the best polynomial fit and evaluate thedifference between the polynomial and profile function. Weaverage the obtained values over all windows, and after-wards we repeat the procedure for different window scaless, deriving F(s):

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real data

F(s)

s [days

α=1.25α=0.82

α

α=0.83

α=1.14

α=1.07

α=0.86

Figure 6 Detrended fluctuation analysis of orders 0 (top curve) acrossovers, where the scaling exponents are changing; for all the thfor Naab river) and demonstrate close correlation exponents.

F2ðsÞ ¼ 1

K

XKm¼1

1

s

Xmsðm�1Þsþ1

½YmðiÞ � pkmðiÞ�

2;

where Yn ¼Pn

i¼1DXi is the profile function of the desea-soned time series DXi, pk

mðiÞ is the best polynomial fit of or-der k, K = 2N/s is the number of non-overlapping windows ofsize s (indexed m), and N is the length of the time series.

When the series satisfies a scaling law, we observe apower-law behaviour of the fluctuation function:

FðsÞ � sa;

where a is the scaling exponent. For uncorrelated records,a = 0.5, while for long-term correlated (persistent) records,a > 0.5, where the auto-correlation function decays asC(s) � s�c with an increasing time lag s, and a is related toc by a ¼ 1� c

2.

Results

We analysed daily runoff of the Naab (26 years), the Regnitz(30 years) and the Vils (26 years) rivers and corresponding

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model data

]

Naab

Regnitz

Vils

α=0.85

α=1.35

α=1.18

α=0.95

α=0.90

=1.19

nd 1–3 (three curves below); arrows denote average points ofree rivers, simulated data reproduce crossovers (slightly shifted

192 V. Livina et al.

records simulated by the ASGi model for the same period oftime. Applying DFA analysis to real and simulated series, weobserved high correlations (a � 1.1) for a period of about30–70 days and a relatively smooth change to a smallerexponent (a � 0.8) for longer periods of time. Fig. 6 showsthe DFA0-3 results for real and simulated data. The modelreproduces both the large correlation exponent in theshort-term range and the lower exponent in the asymptotictime range. The crossover in the simulated Naab data is onlyslightly shifted towards larger time scales, and the short-term correlation exponent is slightly overestimated (1.35,as compared to 1.25 in the real data). For the Regniz andthe Vils rivers, the model reproduces scaling behaviour ina nearly perfect manner.

Because of the DFA0 restriction on the exponent value(DFA0 exponent cannot be higher than one), in the short-term regime, DFA0 underestimates the correlation expo-nent a (as compared to DFA1-3 which provide values higherthan one). In long-term regime, for all time series, DFA0curves become straight and parallel to DFA1-3, with expo-nents close to those of DFA1. This indicates the absenceof a linear trend in both real and simulated data, despitethe generally claimed global warming trend. We suggestthat this is because the data were recorded in the 1960–1990s under almost the same industrial conditions in thearea, and, therefore, the records were not significantlyinfluenced by the global warming trend.

Conventional hydrological statistics and itscomparison with the scaling analysis

Methodology

In this section, three statistical criteria conventionally usedin hydrology are described. We also discuss the advantagesof the scaling analysis and its difference from the conven-tional methods. The criteria are the following (see, for in-stance, Hogue et al., 2006):

0.4

0.8

Nas

h-Su

tclif

fe

0

2

4

6

DR

MS

Naab Re

-10

0

10

%bi

as

Figure 7 Conventional hydrological criteria for three rive

Daily Root Mean Square

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=n

XNi¼1ðqi

m � qioÞ

2

vuut ;

Nash–Sutcliffe efficiency 1�XNi¼1ðqi

m � qioÞ

2

,XNi¼1ðqi

o � �qÞ2 !

;

Percent biasð%biasÞXNi¼1ðqi

m � qioÞ,XN

i¼1ðqi

oÞ" #

� 100;

where qo and qm are the observed and modelled fluxes,respectively, i = 1, . . .,N is time index (daily flux), �q is themean value of the series q. It is easy to see that for twoidentical time series DRMS = 0, NS = 1, and %bias = 0.

Results

In Fig. 7, we plot the values of these traditional hydrologicalcharacteristics for the rivers under consideration. As wasshown in hydrograph of the Vils river (Fig. 2), the patternsof observed and modelled data are similar, but the valuesof the traditional criteria do not show this. Since the criteriaprovide point-wise comparison of the series, their values arevery fragile towards minimal changes in temporal organiza-tion of the data array.

This can be explained by a simple experiment. Consider,for example, the flux of the Naab river. By shift of the datathree time steps back, we obtain new series with just thesame pattern. However, the three criteria applied for com-parison of the new series with the initial one provide esti-mates that differ dramatically from what could beexpected for identical series: NS = 0.65 (instead of 1),DRMS = 4.52 (instead of 0), %bias = 0.03 (instead of 0). Ifwe look at the values of the DFA correlation exponent a,it does not change at all, because the DFA algorithm takesinto account the temporal organization of the data and, inthis example, recognizes that the two series have the samescaling properties. In fact, here we compared a series withitself, but the conventional criteria failed. In another exper-

gnitz Vils

rs: Nash–Sutcliffe, daily root mean square, and %bias.

Temporal scaling comparison of real hydrological data and model runoff records 193

iment, introduction of just one spike in the series (whichcan be caused by a measurement error), changes NS signif-icantly, and two practically identical series are qualified asincomparable. Thus, in practice, quite a successful modelrun might be accepted as an inappropriate when testedusing conventional criteria, whereas it would require justa minimal adjustment (if any) when its temporal organiza-tion is analyzed.

The scaling exponent describes a very important prop-erty of a series, the temporal memory. Unlike NS and otherconventional measures, it does not monitor the coincidenceof values at each time step, but rather assesses the typicalbehaviour of the flow. Scaling properties of a river flux canbe regarded as a ‘fingerprint’ of the latter, with specificshape DFA curve, values of a at different time scales, andposition of crossover reflecting the change in the memoryat particular time scale. Although the traditional criteriaserve for comparison of observed and simulated flowspoint-by-point, they can actually be the same for differentrivers, while telling nothing about the dynamics of a partic-ular river system.

Therefore, the traditional analysis would flourish by theaddition of the scaling exponent a. The point-wise criteriacan be very useful at the final stage of the fine tuning ofthe model, when the scaling properties are reproduced welland are detected by the scaling exponent a.

Multifractal analysis

Methodology

Multifractality is a unique property of a nonlinear dynamicsystem. It characterizes a multiplicative mixture of contrib-uting subprocesses and provides essential information aboutstable states of the system. Studying multifractal propertiesis crucial for revealing complexity of the dynamical processunder consideration (Tessier et al., 1996; Pandey et al.,1998; Kantelhardt et al., 2002, 2003). Here, our tool for thisstudy is the multifractal detrended fluctuation analysis(MFDFA) method (Koscielny-Bunde et al., 1998; Kantelhardtet al., 2002), which is a generalised DFA method. It has beenrecently shown (Kantelhardt et al., 2003) that this methoddemonstrates results equivalent to those obtained bythe well-established wavelet transform modulus maxima(WTMM) technique (Arneodo et al., 1995; Muzy et al.,1991).

In the MFDFA procedure, the moments Fq(s) are calcu-lated by (i) integrating the initial series, (ii) splitting theseries into non-overlapping segments of length s, (iii) calcu-lating the mean-square deviations F2(m,s) from polynomialfits in each segment, (iv) averaging [F2(m,s)]q/2 over all seg-ments, and (v) taking the qth root:

FqðsÞ ¼1

2Ns

X2Ns

m¼1½F2ðm; sÞ�q=2

( )1=q

: ð1Þ

Afterwards, we determine the scaling behaviour of the fluc-tuation function by analysing the log–log plots of Fq(s) ver-sus s for each value of q. The series is said to possess apower-law scaling if

FqðsÞ � shðqÞ:

For stationary time series, h(2) is identical to the well-known Hurst exponent H (Feder, 1988), and h(q) is termeda generalized Hurst exponent. Only if small and large fluctu-ations scale differently, will there be a significant depen-dence of h(q) on q. For positive values of q, h(q) describesthe scaling behaviour of the segments with large fluctua-tions. For negative values of q, h(q) describes the scalingbehaviour of the segments with small fluctuations.

We use third-order polynomials in the fitting procedureof step (iii) (MFDFA3), thus eliminating quadratic trends inthe data. We consider both positive and negative momentsFq(s) and determine them for time scales s between s = 5and s = N/4.

In the standard multifractal formalism, the scaling expo-nent s(q) is determined via the partition function

ZqðsÞ ¼1

Ns

XNs

m¼1jYðmsÞ � Yððm� 1ÞsÞ�jq � ssðqÞ;

where Y is the profile of the series and q is a real parameteras in MFDFA. The exponent h(q) can be related to the expo-nent s(q) as follows:

sðqÞ ¼ qhðqÞ � 1:

Another way to characterize a multifractal time series is tocalculate the singularity spectrum f(a), which is related tos(q) via the first-order Legendre transform (Feder, 1988):

a ¼ s0ðqÞ; fðaÞ ¼ qa� sðqÞ;

where a is the Holder exponent. Therefore, we can obtainthe following relations:

a ¼ hðqÞ þ qh0ðqÞ; fðaÞ ¼ q½a� hðqÞ� þ 1:

The width of the spectrum f(a) characterizes the strength ofmultifractal effects in the data. For monofractal data, thespectrum f(a) collapses into a single point, and both func-tions s(q) and h(q) are linear.

Results

To reduce the effect of nonstationarities in the data, we ap-ply the MFDFA to the deseasoned time series DXi in theasymptotic scale (above 90 days). In Fig. 8 one can seethe partition function Fq(s) (Eq. (1)), for q = �8, �6, �4,�2, �1, 1, 2, 4, 6, 8. The simulated series demonstratebehaviour similar to that of the real data, though the corre-sponding curves for negative moments have slightly differ-ent exponents. For the Naab river at negative moments,the curves for simulated data have sharper crossover, ascompared to real data, and, in consequence, different scal-ing exponents.

In Fig. 9, curves of the function h(q) for real and simu-lated data (circles) are plotted. Dashed lines denote theconfidence intervals of h(q), which were calculated basedon the ensemble of subsets of each record in the followingmanner. Using a sliding window of 6000 days in length, weobtain N�6000+1 subsamples, and then for each of themcalculate h(q). Next, we average the results over theensemble of subsamples and arrive at the estimates ofmathematical expectation and a confidence interval ofh(q). This is, in essence, one of the resampling method(Efron, 1982). It does not necessarily provide a fully reliable

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10310

0

104

108

100

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real data

Fq(s

)

model data

s [days]

Naab

Regnitz

Vils

Figure 8 Multifractal fluctuation function Fq(s) versus time scale s obtained with MFDFA3. The curves correspond to momentvalues q = �8, �6, �4, �2, �1, 1, 2, 4, 6, 8, from top to bottom (solid symbols for q = 2, conventional DFA3).

0.5

1

0.5

1

-5 0 5

0.5

1

-50 5

real data

h(q)

model data

q [moments]

Naab

Regnitz

Vils

Figure 9 Generalized Hurst exponent measured in the scale above 90 days. Dashed lines represent confidence interval calculatedfor data subsets of length 6000 days.

194 V. Livina et al.

estimate of the confidence interval, but can be used in ourcase for comparison of the results of the multifractal anal-ysis for the real and simulated data.

In Fig. 9, the confidence intervals for the Regnitz, forboth real and simulated data, are largest among the con-sidered time series. This can be caused by nonstationarit-

ies in the data (subsets of length 6000 days). Forsimulated Naab data at negative q, the generalized Hurstexponent is underestimated, and for the Vils – overesti-mated. For Regnitz, the simulated data have overesti-mated exponents for positive q. In Fig. 10, again for thesimulated Regnitz data, the discrepancy is highest when

0

0.5

1

0

0.5

1

0.5 10

0.5

1

0.5 1

real data

f(α)

model data

α

Naab

Regnitz

Vils

Figure 10 Multifractal spectra measured in time scale above 90 days. Filled circles correspond to data under consideration,dashed lines represent confidence interval calculated for data subsets of length 6000 days.

Temporal scaling comparison of real hydrological data and model runoff records 195

one compares the multifractal spectra calculated for thewhole set and the confidence interval based on the ensem-ble of subsets. Note that the width of the spectra (orstrength of the multifractal effect) is reproduced properlyin the simulated data, but the exponent a is overestimatedfor all three rivers.

Nonlinear volatility analysis

Methodology

Recently, we have identified nonlinear long-term and peri-odic volatilities in river flow records (Livina et al., 2003a).We define nonlinearity with respect to the Fourier phasesin the following way: if the statistical properties of a timeseries depend only on the power spectrum and the proba-bility distribution, regardless of the Fourier phases, theseries is considered to be linear. Otherwise, the series isdefined as nonlinear (for more details see Livina et al.,2003a,b; Ashkenazy et al., 2001, 2003). We consider theabsolute values of river flow increments (volatility), andin order to study nonlinearity of river flow data, we use asurrogate data test (Schreiber and Schmitz, 2000). The sur-rogate data have the same probability distribution and al-most the same power spectrum as the original series, butwith random Fourier phases. When the data are nonlinearin the above sense, the power spectrum of the volatilityseries has a seasonal peak (periodic volatility), which disap-pears after the phase randomization procedure (Livinaet al., 2003a). After elimination of the nonlinearities byrandomizing Fourier phases of the river flow increment ser-ies, the seasonal periodicity in the volatility is significantlyweakened. Moreover, the volatility series shows long-term

correlations, which are also eliminated after randomizingthe Fourier phases of the river flow increment series. Weuse periodic and long-term volatilities as measures of non-linearity, which should be reproduced in the simulatedseries.

Results

To apply the long-term volatility analysis, we consider theabsolute values of the river flux increments (deseasoned),the volatility series eXk ¼ jXkþ1 � Xkj; k ¼ 1; . . . ;N � 1. Here,to study correlations in the volatility data, DFA3 is used.In the time range 1 week – 1 year, we obtain a correlationexponent a � 0.85, and for window scales larger than 1year, a � 0.59 (see Fig. 11). After applying the surrogatedata test for nonlinearity, the exponent decreases toa � 0.53 for window scales larger than one week. The re-sults of the surrogate data test, i.e., the decrease of thevolatility exponent from a large value to uncorrelated va-lue, indicates the nonlinearity of the river flow. The sameeffect is observed in the simulated data. For each data sam-ple, we generated ten surrogate series and calculated meanvalue and standard deviation of their correlation exponents.The summary of the results for real and simulated data is gi-ven in Tables 2 and 3.

In Fig. 12, one can see the power spectra of the volatil-ities for the real and simulated data, before (thick curves)and after (thin curves) applying the surrogate data test.The simulated data demonstrate spectra similar to thoseof the real data. After randomizing the phases in both thereal and simulated data, the periodic peaks vanish. There-fore, the ASGi model reproduces the nonlinear periodic vol-atility property in the river flow data.

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real data

F(s)

model data

s [days]

Naab

Regnitz

Vils

α=0.62 α=0.65

α=0.91

α=0.62

α=0.87α=0.83

α=0.85

α=0.57

α=0.58

α=0.86 α=0.89

α=0.61

α=0.55 α=0.53

α=0.53 α=0.53

α=0.55α=0.55

Figure 11 Long-term volatility effect (open symbols for initial data, filled symbols for phase randomized data); correlations arewell reproduced by the model, both in short- and long-term ranges. After phase randomization, real and model data becomeuncorrelated, indicating the nonlinearity in the initial data.

Table 2 DFA3 exponents in long-term volatilities of realdata

River Data Short-range Long-range

Naab Real 0.85 0.62Surrogate 0.54 ± 0.02

Regnitz Real 0.83 0.57Surrogate 0.51 ± 0.03

Vils Real 0.86 0.58Surrogate 0.53 ± 0.02

Table 3 DFA3 exponents in long-term volatilities of modeldata

River Data Short-range Long-range

Naab Model 0.91 0.65Surrogate 0.53 ± 0.02

Regnitz Model 0.87 0.62Surrogate 0.52 ± 0.02

Vils Model 0.89 0.61Surrogate 0.53 ± 0.02

196 V. Livina et al.

Conclusion

We have applied the DFA method to compare the scalingproperties of real and ASGi simulated river flux series. Wefound that flux records are highly short-term correlatedand less correlated in the asymptotic range. We suggest that

this type of memory in the river data is caused by the inertiaof river basin aquifers (groundwater storage), which are themain sources for the river stream: the high water level ismost likely followed by a still high level, and vice-versa.We applied multifractal analysis and found small differencesbetween real and simulated flux records. To study correla-tion properties of absolute values of river flow increments(i.e., the volatility), DFA3 was used. The volatility seriesare correlated for time scales smaller than one year, andthese correlations become much less pronounced for timescales larger than one year. Through the use of a surrogatedata test which randomizes the Fourier phases of the series,we show that volatility correlations is an indication of non-linearity, and that the nonlinearity decreases for timescales larger than one year. These effects have been ob-served in both real and simulated data, which confirms goodperformance of the ASGi model.

Our main conclusion is that the ASGi model does repro-duce the main statistical properties of the river flux fluctu-ations: (i) values of scaling exponents and crossover in DFAcurves; (ii) general multifractal behaviour of the data,though not precisely, since there is a difference in multi-fractal exponents between real and simulated data; (iii)scaling exponents of the volatility series for small and largetime scales; and finally, (iv) the periodic volatility of thevolatility series.

Our methods provide advanced statistical measures(scaling exponents, width of multifractal spectrum, peakheight of normalised power spectrum, and correlation expo-nent of volatilities) for studying hydrological records. For in-stance, as we showed in ‘‘Conventional hydrologicalstatistics and its comparison with the scaling analysis’’ sec-tion in a simple experiment, the scaling exponent properlydetects similarity of series in cases when the conventional

10-4

10-2

100

10-4

10-2

100

1 210-4

10-2

100

1 2 3

real data

S(f)

model data

f [1/year]

Naab

Regnitz

Vils

* *

**

* *

Figure 12 Periodic volatility effect: periodicity peak in power spectrum of real and model volatility series (thick lines) and resultsof phase randomization test (thin lines). Periodicity peaks denoted by stars become much less pronounced after randomization ofthe phases, both in real and simulated data, indicating existence of nonlinearity of the volatility series.

Temporal scaling comparison of real hydrological data and model runoff records 197

criteria fail. Moreover, conventional criteria tell nothingabout nonlinearity and its strength, which is an importantfeature of the observed river data that should be repro-duced by hydrological models. Different model sets can pro-duce the same values of the Nash–Sutcliffe coefficient, andone of the series might be linear while the other might benot. Therefore, the modern techniques of nonlinear statisti-cal analysis are to be employed in hydrological modelling toprovide advanced comparison of real and model data anddeeper evaluation of model performance.

To summarize, the ASGi model data generally demon-strate close similarity to real river flux. By means of ourmethods, some statistical characteristics have been esti-mated. We suggest that they can be used for evaluatinghydrological models. The ensemble confidence intervalcan be used to quantify the reliability of multifractal esti-mates. We hypothesize that the difference in the width ofmultifractal spectra might be caused by noise artefacts inthe simulated data (like improper seasonal pattern), andthis should be taken into account in further developmentof the ASGi model. We suggest that closer similarity be-tween the multifractal properties of real and simulatedtime series would improve the model’s performance.

Acknowledgements

We thank Dr. Y. Ashkenazy for useful discussions.

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