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Physics and Chemistry of the Earth 31 (2006) 1262–1273

Characteristic scales, temporal variability modes and simulationof monthly Elbe River flow time series at ungauged locations

D. Markovic *, M. Koch

Department of Geohydraulics and Engineering Hydrology, University of Kassel, Kurt-Woltersstr. 3, 34109 Kassel, Germany

Received 19 August 2005; received in revised form 12 April 2006; accepted 14 April 2006Available online 25 October 2006

Abstract

Spatial and temporal patterns of the long-range extreme monthly Elbe River flows across Germany are investigated, using variousstatistical methods, among others, principal component and wavelet analysis. Characteristic time scales are derived for various time ser-ies statistics. The wavelet analysis of the raw river discharge data as well as of the major principal component reveal the main oscillatorycomponents and their temporal behavior, namely low frequency oscillations at interannual (6.9 yr) and interdecadal (13.9 yr) scales. TheEOFs at ungauged stations are estimated from the principal components of the observed time series sampled over a limited time spanwhose length equals the major temporal variability scale (�7 yr). The EOFs (empirical orthogonal functions) obtained in this way aresubsequently used to simulate long-range flows at these locations. A comparison of this method with linear interpolation and ordinarykriging of the EOF shows the superiority of the former in representing the distributional properties of the observed time series. The sim-ulated time series preserve also short and long-memory.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Temporal variability; Principal component analysis; Wavelets; Ungauged locations; Interannual to inderdecadal variability; Elbe River

1. Introduction

Hydrologic engineers are mostly called upon to provideinformation on activities such as designing or operatinghydraulic-conveyance and water-control facilities, prepar-ing for and responding to floods, or regulating floodplainactivities. For example, a flood-damage reduction studymay require an estimate of the extreme flood peaks as wellas estimates of an runoff increase for proposed changes toland use or climate scenarios. To achieve such tasks, char-acteristic flow quantiles, i.e., flows for a certain occurrenceprobability are required. A prerequisite for deriving suchquantiles are long-range time series of flow observationswhich are, almost as a rule, not available at the planed con-struction site.

1474-7065/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pce.2006.04.042

* Corresponding author.E-mail addresses: danijela.markovic@uni-kassel.de (D. Markovic),

kochm@uni-kassel.de (M. Koch).

In spite of significant progress made in recent years inthe understanding of the basic processes that govern therunoff formation through a watershed and propagationof a flood wave through a channel, practical applicationof a physically based hydrological model is often limitedby a paucity of soil, vegetation or channel data. While thismight be less of a problem for an event, lumped, empiricalrainfall-runoff model where mostly, only a short flood dis-charge time series is simulated, classical hydrological mod-els are of a little help when long-term time series areconsidered. The situation is even more worrisome whenthere is lack of calibrating gauge data. In fact, many catch-ments are ungauged, particularly in less developed coun-tries, which in return are often struck by extremehydrological events. Hence, there is an urgent need to haveeasy, consistent, robust and reliable methods for simulatingand predicting streamflow discharges within ungaugedcatchments and this is, thus, one of the most challengingissues in modern hydrology.

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1263

In this study we investigate the performance of the prin-cipal component analysis (PCA) regarding the simulationof the long-term river flows at ungauged locations. ThePCA technique is a pure statistical method frequently alsocalled EOF (empirical orthogonal function) analysis withwidespread application for the multivariate data analysisin oceanography, atmospheric sciences and climatology(see, e.g. Emery and Thomson, 1997). Exemplarily, fromthe numerous PCA studies made up-to-date we note thatof Rodriguez-Puebla et al. (2001) on the spatial and tempo-ral patterns of winter precipitation over the Iberian Penin-sula and their connection to teleconnection indices, andthat of Brunetti et al. (2004), who explored changes in sta-tistical properties of daily precipitation in Italy. An exten-sive PCA-based hydrological study is that of Saco andKumar (2000) on the streamflow wavelet spectra over theUS to identify regions with coherent variability acrossscales, and that of Keyantash and Dracup (2004) whodeveloped an aggregate drought index.

The idea behind the PCA method is that temporal vari-ations in multivariate geophysical time series are often theresult of complex nonlinear interactions between manydegrees of freedom or modes. The PCA reduces the dimen-sionality of the initial system by defining a new coordinatesystem, the so called empirical orthogonal vectors (EOFs),and finds the most important variability modes. However,the physical interpretability of the obtained modes is still amatter of controversy, as these do not necessarily relate tothe physical system modes.

Our study is focussed on the German part of the ElbeRiver Basin, in particular, the Elbe River flow variability.The specific hydrological questions discussed in this paperare as follows:

(a) Are there any characteristic time scales of themonthly Elbe River flows?

(b) What are the main Elbe River flow variabilitymodes?

(c) If there is a statistically or mathematically derivabledependence of the EOFs on the river gauge location,could we use these EOFs for the simulation of flowdata at ungauged locations?

Whereas the answers to questions (b) and (c) comesstraightforward from the PCA analysis, our starting pointin search for an answer to (a) is that the variability ofhydrological variables can be described using notions ofdeterministic and stochastic characteristic scales. Thedeterministic scales relate to the morphometric and hydro-dynamic properties of the catchment area (e.g. area size,aquifer depth, hill slope length, concentration and resi-dence times) while the stochastic variability scales can bederived using signal processing tools (see, e.g. Ghil et al.,2002) or the variogram analysis (see, e.g. Skøien andBloschl, 2003). A signal processing tool that performssimultaneous time-frequency localization is the wavelettool (cf. Foufoula-Georgiou and Kumar, 1994). In addi-

tion to the dominant oscillatory components, their periodsand phases, the wavelet tool also shows how these vary intime (Torrence and Compo, 1998). Mathematically speak-ing, a variogram is an average squared difference betweenmeasured quantities at different locations. Using the vario-gram analysis, Skøien and Bloschl (2003) found an increaseof the characteristic time and space scales with increasingcatchment area, i.e. larger catchments react slower thansmaller ones to precipitation forcing. Koutsoyiannis(2005) emphasized that the significant fluctuations at largetemporal scales indicate the presence of a scaling behaviorand link the latter with the Hurst phenomenon, i.e., non-random walk correlations, through the maximum entropyprinciple. The Hurst phenomenon can be quantifiedthrough the calculation of the Hurst parameter H. The lat-ter is H < 0.5 for series with short-term persistence,H = 0.5 for uncorrelated observations and H > 0.5 for ser-ies with long-term persistence.

As for the present paper, the characteristic scales arederived using the wavelet- and the variogram analysis whilethe joint spatial variability of the monthly Elbe River flowsacross Germany is investigated using principal componentanalysis. The corresponding main variability modes arethen subjected to a wavelet analysis to detect the dominateoscillatory components and their temporal behavior, whilethe EOFs are used to estimate monthly flows at ungaugedtest stations.

2. Theoretical framework

2.1. Wavelet analysis

The wavelet transform is a common tool for perfor-ming a continuous time-frequency localization of a timeseries f(t). The method has been applied in numerous stud-ies in geosciences and climatic research (e.g. Foufoula-Georgiou and Kumar, 1994; Torrence and Compo, 1998;Zoppou et al., 2002; Gray et al., 2003). We performedthe wavelet analysis using the continuous wavelet trans-form (CWT) as proposed by Torrence and Compo(1998). The CWT of a function f(t) can be interpreted asthe inverse Fourier transform (FT) of the product ofFT(f(t)) and the FT of the scaled and translated basic wave-let W(t/s). The basic wavelet used in this study is the Morletwavelet with the non-dimensional frequency set to w0 = 6.The average of the wavelet power over all local waveletspectra along the time axis is called the global wavelet spec-trum (GWS):

W 2ðsÞ ¼ 1

N

XN�1

n¼0

jW nðsÞj2 ð1Þ

It can be shown that the GWS comes closest to a some-what smoothed version of the classical Fourier spectrum.The significance of the periods observed by the GWS istested by the v2 test (Torrence and Compo, 1998) againstthe hypothesis of red noise.

1264 D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273

2.2. Variogram analysis

The temporal variogram (Skøien and Bloschl, 2003)is defined as the average of variograms of all availableElbe River discharge time series over the selected timeinterval:

cðhÞ ¼ 1Pmj¼12njðhÞ

Xm

j¼1

XnjðhÞ

i¼1

ðQðxj; ti þ htÞ � Qðxj; tiÞÞ2 ð2Þ

where h is a time lag, m denotes the number of used timeseries and Q is a discharge at the location xj. The spatialvariagram is analog to the temporal whereas the time lagh is replaced by the spatial lag l. To both variograms, the-oretical variograms of the exponential potential, Weibulland the Sigmoid-type, respectively, are fitted. For theWeibul-type variogram

cðhÞ ¼ ahbð1� expð�ðh=cÞdÞÞ ð3Þ

the slope at short and lags is b + d and b, respectively(Skøien and Bloschl, 2003). The Sigmoid-type variogram

cðhÞ ¼ a=ð1þ bð�ch�dÞÞ ð4Þ

has a typical S-shaped form, where a, �d/c, and b controlthe sill, the slope at small legs, and the range, respectively.

2.3. Principal component analysis (PCA)

PCA compresses time series variability into a set of sta-tistical, orthogonal modes called the principal components(PC) (cf. Emery and Thomson, 1997; Wilks, 1995). The PCare defined as the projection of the [N · M] data matrix X

onto the eigenvectors (EOFs) obtained solving the eigen-value problem X 0X EOF = kEOF where X can be eitherthe original matrix of observations or the matrix of anom-alies or the matrix of standardized anomalies. Since theeigenvectors due to their orthonormality diagonalize theinitial matrix (X 0X) (Wilks, 1995), the sum of the diagonalelements of the (X 0X) matrix (i.e. the trace) equals the sumof the eigenvalues. Consequently, if the X matrix is the ori-ginal data matrix, the sum of the eigenvalues of (X 0X) isequal to the sum of squares of the original time series. Sincethe PCs are simply the projection of the X matrix on theempirical orthogonal vectors (EOFs), it follows that theyare mutually independent and that their total variance isequal to the total variance of the initial time series.

The EOFs define a new coordinate system for the stud-ied data whereas the largest eigenvalues k indicate the dom-inate directions of the time series joint variability.However, the EOFs of the original data matrix or of thevariance–covariance matrix tend to align near the direc-tions of the variables with the highest sum of squares andhighest variance, respectively, wherefore in the PCA ofvariables in unlike physical units the correlation matrix ispreferable (Wilks, 1995). For the time series reconstruction(5), only the k 6M PCs which describe the major portionof the joint variance are usually used:

QðtÞm ¼Xk

j¼1

PCðtÞjEOF mj ; 1 6 m 6 M ; 1 6 k 6 M ð5Þ

where, owing to the orthogonality property, the PCs areobtained by

PCðtÞj ¼XM

m¼1

QðtÞmEOF mj : ð6Þ

Eq. (6) can also be viewed upon as a simple regressionrelationship between the PC’s and the discharge Q, where-fore the EOFs can be computed from the inversion of thenormal equations as:

EOF ¼ ðPC0tjti PCtjtiÞ�1ðPC0tjti Qtj

tiÞ ð7Þ

where ti and tj denote the beginning and the end of the se-lected interval and Q the sample discharge.

Upon determination of the mathematical description ofthe EOFs spatial pattern, the Q at an unknown location x

can be simulated using principal components obtained bythe EOF analysis of the data from the neighbouringgauges. Our basic assumptions for this simulation proce-dure are: (a) At the location x, either a short-range dis-charge time series is available or the location x wasungauged over the entire period of interest; (b) The selecteddischarge statistics Q remains constant (representative) fora selected sampling interval; (c) EOFs at the particularlocation x are time-invariant (which, by definition, theyare); (d) For all separation distances l the increment EOFj

(x + l) � EOFj (x) has a finite variance 2c(l) that dependsonly on l.

Although the best choice for analyzing the joint variabil-ity of the single variable time series like the river dischargeQ would be the variance-covariance matrix, assuming thaton a position x no observations are available favorizes theEOF analysis of the initial data matrix because the maxi-mum number of unknowns is only equal to the length ofthe EOF-vector.

Since our goal is the time series simulation and not thesimulation of their major variability modes, the truncationof the eigenvalues at some prescribed level of the joint var-iability is not necessary. Moreover, if the subject of interestis the simulation of the time series’ large-scale behavior, i.e.of deterministic signals that govern the discharge process,then a better option would be to use the Multichannel Sin-gular Spectrum Analysis (MSSA) of Golyandina andStepanov (2005) or the classical version of this methoddescribed in detail in Ghil et al. (2002). Unlike the classicalMSSA, the method of Golyandina and Stepanov (2005)offers a possibility of detecting deterministic components(slowly varying trend, different periodic components), theirgrouping and forecasting (cf. Markovic and Koch, 2005a).

3. Data and study area

The German part of the Elbe River Basin (Fig. 1)comprises almost two thirds of its total Basin size

Fig. 1. The German part of the Elbe River Basin and the Elbe Rivergauges.

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1265

(148,268 km2), with most of the remaining part belongingto the Czech Republik and about 1% to Poland and Aus-tria. The upper Elbe valley ends just before the SchwarzeElster mouth (199 km), while the lower Elbe valley reachesfrom the Havel river mouth (438 km) to the Elbe estuary atCuxhaven (725 km). The hydrological discharge regime ischaracterised by a pronounced seasonal cycle whose risinglimb is situated between November and April and the fall-ing one between May and October.

The German Federal Agency for Hydrology (BAfG)provided the daily discharge time series for eight gaugesalong the Elbe River whose station name, geographic posi-tion, station altitude, km position and time series span arelisted in Table 1. From the daily data we derived monthlymeans and monthly extremes, defined as the mean monthlydischarges and maximum daily discharges for each month,respectively. Since the results for the two time series do notvary significantly, only the monthly extremes series will bepresented.

Table 1Discharge stations used and record information

Gauge Long.[O]

Lat.[N]

H [m] [km] Startingyear

N

[month]

Dresden 25.74 51.05 102.73 55.6 1852 1788Torgau 22.01 51.55 75.18 154.1 1936 780Wittenberg 21.65 51.86 62.48 214.1 1951 600Aken 21.06 51.86 50.24 274.7 1936 780Barby 20.89 51.99 46.15 294.8 1900 1212Magdeburg 20.65 52.13 39.92 326.6 1931 840Wittenberge 20.76 52.99 16.72 453.9 1900 1212Neu

Darchau19.89 53.23 5.68 536.4 1875 1512

4. Results

4.1. Temporal stationarity scales

Hydrological time series are considered stationary ifthey are free of trends, periodicities and shifts (Salas,1992). Statistically, the strict stationarity condition imposestime invariance of the joint distribution of any possiblesubset of process realizations. When dealing with hydro-logical time series of moderate time spans, instead of astrict stationarity, a covariance stationarity can beassumed, which implies stationarity up to the secondmoment (Hipel and McLeod, 1994). Unlike the monthlytime series which are due to the annual cycle seasonallynonstationary, it is generally assumed that the hydrologicaltime series at annual scale are stationary. Nonstationaritycan be however caused by an anthropogenic action orlarge-scale climatic variations which act at interannual tointerdecadal and even larger scales, wherefore, prior toany kind of time series modelling, the time series temporalstationarity scales have to be investigated.

Classical statistical tests for the equivalence of the sub-sets moments like the Z-, T- or the F-test are based onthe assumption that the data are normally distributedand independent. However, all studied time series are pos-itively skewed and the skewness decreases along the ElbeRiver course (e.g. Dresden 2.47, Neu Darchau 1.52),whereas the short-range memory, i.e. the ‘‘lag-1’’ – correla-tions increase (e.g. Dresden 0.38, Neu Darchau 0.62).Although a logarithmic transformation appears to renderthe data approximately normal, the strong dependencewithin the data rules out the use of the classical statisticaltests.

As a first step in the study of the temporal stationaritywe analyze the influence of the time window width Twin

on the estimates of the time series’ mutual correlations, firstand second moments. As representative gauges we selectedNeu Darchau, Barby and Dresden (see Fig. 1). Fig. 2(a)indicates that above the time scale of about 6 yr (Twin � 70months), the mutual correlations between the analyzedtime series become stationary.

For the estimation of the ‘‘characteristic scale’’ for thetime series’ first two moments, in the second step, theextreme absolute ‘‘error’’ of the moment estimate fromall possible windows (Twin 2 {1, Nmax}) of this length is cal-culated. As the ‘‘true’’ value, the corresponding momentsfrom the whole time series are used. The procedure isrepeated for each of the representative gauges and, finally,the averages of these error estimates are calculated for eachwindow length. Fig. 2(b) shows the results of this proce-dure for two different time spans, namely, 1960–2000 (1)and 1900–2000 (2) i.e. the corresponding maximal windowlength Nmax is set to 10 yr and 25 yr, respectively. Oneobserves from Fig. 2(b) that for the 1900–2000 set (2) theestimate of the standard deviation decays slower than thatof the mean. For all used window widths, the error estimateof the mean is larger for the 1900–2000 set (2) than for the

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

t [month]

r [–

]

0 50 100 150 200 250 3000

20

40

60

80

100

120

t [month]

Max

. Err

. Est

. [%

]

Fig. 2. Influence of the time window width on the estimates of mutual correlations, mean and standard deviation based on extreme monthly time seriesfrom gauges Neu Darchau, Barby and Dresden. (a) Mutual correlations: Neu Darchau–Barby (dashed line), Barby–Dresden (solid line) and NeuDarchau–Dresden (dotted line); (b) Average extreme error of the running mean estimate (solid line) and standard deviation estimate (dashed line) based ontime series from 1960–2000 to 1900–2000.

1266 D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273

1960–2000 set (1). This is not surprising and can beexplained by the entropy principle which says that theuncertainty of the system (e.g. confidence intervals)increases with the number of possible system states, i.e.with the length of the used time series. This simple investi-gation shows that the ‘‘characteristic scales’’, i.e., stationa-rity scales of the mean and the standard deviation are notequal but are approximately in the decadal to interdecadalrange (10–15 yr) which is, however, far longer than the‘‘characteristic scale’’ of the mutual correlations (�6 yr).

For further inspection of the stationarity, the long-rangerecords are split into 3 nonoverlapping sets of equal lengthand the first two moments (mean and variance) are calcu-lated for each of them. At the gauge Dresden, the firstmoment is 10% smaller in the third data subset (1952–2001) than in the first subset (1852–1901). The secondmoment, i.e. the variance, shows a pronounced nonstation-arity. Exemplarily, at the gauge Magdeburg, the variance is15% larger in the third data subset (1967–1999) than in thefirst subset (1931–1953), while at the gauge Dresden thevariance is 35% smaller in the third subset (1952–2001)than in the first subset (1852–1901). Furthermore, a slowdecreasing trend of the mean characterizes the dischargetime series of the gauge Neu Darchau which is closest tothe North Sea estuary of the Elbe River and, therefore,indicates the resulting dynamics of the whole catchment.

We additionally show in Fig. 3 the log-wavelet spectrumof the extreme monthly river discharges from the northestElbe River gauge (Neu Darchau, 1875–2000). One observesthat before 1920 the seasonal cycle (1 yr scale) is the only

statistically significant periodic component whereas afterthat time a broad low frequency spectrum with dominantvariability scales equal to 6.9 yr and 13.9 yr, respectively,is present. Though the time series’ variability at scaleshigher than 1 yr is obviously nonstationary, there is a clearquasiperiodic pattern, also called the Joseph effect (Man-delbrot, 1977; Beran, 1994) characterized by approximately7 yr long and dry and wet periods (see Fig. 4). As this pat-tern has also been found by Markovic and Koch (2005a)applying the singular spectrum analysis of Golyandinaet al. (2001) on the extreme monthly discharge time seriesof the Elbe River gauge Dresden, we have enough confi-dence on the existence of the named periodicity. The mostprobable source of these quasiperiodic cycles is the NorthAtlantic Oscillation (NAO) since similar interannual tointerdecadal oscillations as well as significant correlationswith the NAO are found in the extreme monthly precipita-tions across Germany (Markovic and Koch, 2005b). How-ever, apart from an apparent ‘‘skin effect’’ of thegroundwater aquifer which feeds the baseflow to thestream-sections under study, further causes of the low fre-quency amplification of the river discharge are still a matterof investigation (Markovic, 2006).

4.2. Spatial and temporal variograms

For the variogram analysis, the data should nearly fol-low a normal distribution and be free of periodicities(Skøien and Bloschl, 2003). This can be achieved approxi-mately by logarithmizing the data. Subsequently the wave-

1880 1900 1920 1940 1960 1980 2000

0.25

0.5

1

2

4

8

16

32

64

Per

iod

[yea

r]

t [year]

Fig. 3. Log of the wavelet power spectrum for gauge Neu Darchau. The black solid line marks the cone of influence.

1880 1900 1920 1940 1960 1980 20000

500

1000

1500

2000

2500

3000

3500

4000

4500

t [year]

Q [m

3/s

]

Fig. 4. Elbe River discharge at gauge Neu Darchau (1875–2000): extreme monthly discharge (thin line) and the low pass wavelet filtered signal for scalesP7 yr (thick line).

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1267

let filter (Torrence and Compo, 1998; Markovic and Koch,2005a) is used to remove the periodicities. In Fig. 5 weshow the scatter plots of the temporal (a) and the spatial(b) variability as well as the corresponding theoretical vari-ograms described by Eqs. (3) and (4), respectively.

For the temporal variogram the smallest least squaresfitting error is obtained with the Weibull-type model(RMS = 0.01), whereas the spatial variogram is fittedbetter by the Sigmoid-type function (RMS = 0.02) than

by the Weibull variogram (RMS = 0.024). However,unlike the temporal variogram which converges relativelyfast (�1 yr) to its asymptotic value, the spatial variogramdoes not have any characteristic scale. This propertyalludes to a long-memory in the spatial domain andshort-memory in the time domain (cf. Beran, 1994). Theparameters of the spatial variogram can further be usedfor the geostatisical interpolation (kriging) of the dischargeparameters.

0 10 20 30 40 50 60 700.4

0.6

0.8

1

1.2

1.4

t [month]

γ (t

)

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

γ (l)

l [km]

Fig. 5. Temporal (a) and spatial (b) variogram. The scatter plots are the sample variograms, the line plots are Weibull-type (dashed) and the Sigmoid-type(solid) fitted variograms.

1268 D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273

4.3. Principal component analysis of the Elbe River flow

variability

As mentioned in the theoretical section, for the detectionof the joint variability modes the PCA will be applied to the

0.25 0.5 1 20

5

10x 10

6

Period

σ2 [m

6 /s2 ]

0.25 0.5 1 20

5

10x 10

4

Perio

σ2 [m

6 /s2 ]

0.25 0.5 1 20

2

4x 10

4

Perio

σ2 [m

6 /s2 ]

Fig. 6. Global wavelet spectra of the first (a), second (b) and third (c) principanoise background.

original multivariate data from all available gauges, whichmeans that the data are not transformed prior to theanalysis.

In Fig. 6 we show the global wavelet spectra of the firstthree principal components and the corresponding 95%

4 8 16 [year]

4 8 16d [year]

4 8 16d [year]

l component and the corresponding 95% significance levels assuming a red

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1269

significance levels derived from the v2 test assuming a rednoise background (see Torrence and Compo, 1998). Thepercentage of the joint variability (percentage of the totaltime series sum of squares) explained by the first, secondand the third principal components are 98.7%, 0.86% and0.17%, respectively.

The significance level for the spectra of the first PC indi-cates two statistically significant variability scales, namely,1 yr and 6.9 yr, but, a clear peak at 13.9 yr is also notice-able. If one keeps in mind that the global wavelet spectrais the time-averaged wavelet power it becomes clear thatthe slowly varying oscillation at the scale 13.9 yr is presentin the time series, (but not over the entire investigated timeinterval). Something similar can be observed from thewavelet spectrum of the discharge time series at gaugeNeu Darchau (see Fig. 3) where the low frequency oscilla-tions at scales larger than 10 yr are present only between1920 and 1970. The second PC consists of the periodiccomponent at the scale 5.2 yr, and probably some time-dependent slowly varying components at scale 10–11 yr.The third PC does not have any significant peaks and istherefore considered as a noise. Interestingly, the seasonal(annual) component appears only in the spectra of thedominant PC while the second PC seems to describe theadditional low frequency joint variability. Finally, sincethe 13.9 oscillation is not time-independent, the major jointvariability scale of the German Elbe River must be the6.9 yr period. (�7 yr).

The EOFs derived from the PCA analysis constitute thedirections which explain most of the variability of the sys-tem, i.e. in the present application the discharge time seriesalong the Elbe River course. By construction the EOFs are

50 100 150 200 250 3

–0.4

–0.2

0

0.2

0.4

0.6

[–]

L [k

Fig. 7. The first EOF (circles) and the second EOF (squares) as a function offrom one gauge (here Torgau) are omitted from the analysis because for the s

stationary, i.e. they do not evolve in time. It is only theprincipal component attached to the corresponding EOFthat provides the sign and the overall amplitude of theEOF as a function of time. Moreover, the position of thesevectors is defined for the M-dimensional space in the EOFmatrix itself, but if there is a statistically or mathematicallyderivable dependence of the eigenvectors on the gaugelocation, we could derive the position of these vectors forthe M + 1 dimensional space. Such a dependence is shownin Fig. 7 where the scatter plots of the EOF1 (circles) andthe EOF2 (squares) as a function of the gauge position aredrawn. The third PC is not considered because of the neg-ligible portion of variance explained by this component aswell as to the fact that this component has a white noisespectrum. Obviously, the dependence of EOF1 and EOF2on the gauge location can easily be described by a secondto third order polynomial alluding to a possibility of simu-lating flows at ungauged locations using interpolated ormodeled EOFs.

4.4. Simulation of flow time series at ungauged locations

With an increase of the number of available dischargetime series the uncertainty of the multivariate-baseddescription of the discharge process along the river coursedecreases. However, in practice, often only a small numberof gauges is available, therefore, limiting the application ofmultivariate methods. The easiest possible method for cal-culating values between two neighbouring locations wouldbe linear interpolation (LI). If data from at least threegauges are available it becomes reasonable to apply thePCA while for a higher number ordinary kriging (OK)

00 350 400 450 500 550

m]

the position (km) of the corresponding gauge on the Elbe River. The dataimulation study one gauge will be assumed unknown.

Table 2Statistics of original and simulated data (Torgau)

Statistics Qobs LI PC-LSR PC-LSRc OK OKc

R2 – 0.985 0.985 0.985 0.984 0.984Q 533.9 659.3 512.0 529.2 657.2 509.2SQ 388.1 478.0 378.3 377.3 480.3 473.6Cs 1.57 1.58 1.60 1.61 1.59 1.61r1 0.51 0.53 0.52 0.52 0.53 0.52r2 0.25 0.28 0.27 0.27 0.27 0.27r3 0.16 0.17 0.17 0.17 0.17 0.18H 0.81 0.80 0.80 0.80 0.80 0.81v2 – 138.93 29.97 64.06 117.17 123.65RMS – 171.19 71.24 67.95 172.30 118.59

Abbreviations are: Qobs – sample data; LI – linear interpolation betweentwo closest referent gauges; PC-LSR – statistics of the PCA based simu-lation; PC-LSRc – statistics after corrections of the PC-LRC; OK – geo-statistical interpolation of EOFs; OKc – results after correction of OKsimulations; R2 – coefficient of determination; Q – mean; SQ – standarddeviation; Cs – skew; r1 � r3 are lag-1 to lag-3 autocorrelations; H – DFAHurst parameter estimate; v2

d – test statistic (v20:05 ¼ 38:88); RMS – the

root mean squares error of each model.

1270 D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273

can be used. We will discuss all these possibilities by com-paring their outputs with respect to the time series’ statisti-cal, namely, its distributional properties. Similarly,Haberlandt et al. (2001) compared the performance ofthe regression method, ordinary- and external drift krigingwhen regionalizing the base flow index in the ElbeRiver Basin. The study of Helms et al. (2002) deals withthe simulation of Elbe River flows along the river coursebetween 1985 and 1990 using the ELBA model, which isa diffusion–translation model based on the hourly data.However, the authors argue that the difference is negligibleif instead of each hour of the day the one mean daily valueis used.

For the application of the LI and the PCA method weselected the three gauges Neu Darchau, Barby and Dresden(see Fig. 1) because the gauge Dresden is the most southerngauge, the gauge Neu Darchau is closest to the estuary ofthe Elbe River and the gauge Barby is about ‘‘half way’’between these two gauges. The results show that the firsttwo PCs explain 99.7% of the joint variance and, like inthe PCA of all available time series (Fig. 6), the spectrumof the third PC does not have any statistically significantpeaks. For the calibration of the EOFs, only the first twoPCs and a data subsample with a length of 7 yr (1953–1960) are used. This approach is based upon the followingassumptions: (a) the third PC represents negligible noise,and (b) the calibration sample should be at least as longas the major temporal variability scale.

To estimate the influence of sampling on the EOF esti-mate, we have repeated the subsample selection and theleast squares procedure (LSR) 1000 times and calculatedthe corresponding mean and standard deviations. It turnedout that for the analyzed data sets the standard deviation

1984 1984.2 1984.4 1984.6 1984.8 10

200

400

600

800

1000

1200

1400

1600

t [y

Q [m

3 /s]

Fig. 8. Elbe River discharge time series: Neu Darchau (dashed line), Barby

of the EOF estimates are only up to 5% and 15% of themean EOF1 and EOF2, respectively. Assuming that themeans of the LSR and OK simulations are seasonallybiased, the correction of the models is straightforward,i.e. seasonally dependent (winter and summer) correctionfactors are simple added to the simulated Q values.

A simulation study is performed for one gauge locatedin the upper part of the Elbe River Basin (gauge Torgau)and one located in its lower part (gauge Wittenberge)(Fig. 1). In Fig. 8 we show the corresponding observed timeseries (1984–1986), including those for the selected refer-ence gauges (Neu Darchau, Barby and Dresden). One

985 1985.2 1985.4 1985.6 1985.8 1986

ear]

(triangles), Dresden (circles), Wittenberge (solid line), Torgau (squares).

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1271

notices only few exceptions from the rule that the dischargeincreases with increasing downstream distance (Dresden toBarby to Neu Darchau). In Tables 2 and 3 we list the sta-tistical properties of the sample and simulated data. Thecalibration period for the PCA-based determination ofthe EOFs at the ‘‘ungauged’’ locations is 1953–1960, andthe simulation period is the subsequent time interval(1960–2000).

Interestingly, regarding the portion of varianceexplained by the simulated time series, all methods providealmost equal results (R2 > 0.98) (cf. Tables 2 and 3). How-ever, the simple linear interpolation (LI) between the twoclosest reference gauges significantly overestimates (Tor-gau) or underestimates (Wittenberge) the first two

Table 3Statistics of original and simulated data (Wittenberge)

Statistics Qobs LI PC-LSR PC-LSRc OK OKc

R2 – 0.995 0.995 0.995 0.995 0.995Q 952.5 907.1 934.9 935.7 904.6 934.2SQ 590.6 578.0 588.3 588.4 573.7 597.9Cs 1.31 1.46 1.44 1.44 1.45 1.45r1 0.65 0.64 0.64 0.64 0.64 0.63r2 0.38 0.36 0.36 0.36 0.36 0.35r3 0.22 0.20 0.20 0.21 0.20 0.20H 0.81 0.80 0.80 0.80 0.80 0.80v2 – 47.64 25.89 25.29 51.69 26.92RMS – 75.01 61.50 61.30 77.32 62.76

See Table 2 for explanations.

1960 1965 1970 1975–500

0

500

1000

1500

2000

2500

Q [m

3/s

]

1960 1965 1970 1975–800

–600

–400

–200

0

200

400

Qob

s–Qsi

m [m

3 /s]

a

b

Fig. 9. (a) Measured and simulated discharge time series (Torgau), (b

moments. All simulation methods turn out similar resultsconcerning the lag-1 to lag-3 correlations and the Hurstparameter H. Although the latter are >0.5, this does notmean a priori that a long-term persistence should beassumed, as H- parameter estimations based on the DFAtechnique are usually upwards-biased when the analysedtime series contain periodic signals (cf. Hu et al., 2001;Markovic and Koch, 2005a). The skewness of the simu-lated discharge matches better that of the sample at loca-tion Torgau than at Wittenberge where it is consistentlytoo high. Similar to LI, the OK method either overesti-mates (Torgau) or underestimates (Wittenberge) the firsttwo moments. In any case, the PC-LSR method exhibitsthe best performance regarding the distributional proper-ties of the original sample data and, additionally, resultsalso in the lowest RMS.

In Figs. 9 and 10 we show traces of the sample data, thecorrected PC-LSR-QLSRc and of the OK-simulated timeseries QOKc, as well as of the absolute errors (Qobs � Qsim)obtained with both simulation methods for two Elbe Riverstations. Expectedly, after correction for the mean, theerrors have approximately a zero mean, however, a weak(<5% of the mean) 13.9 yr periodicity was further noticedfrom the inspection of the corresponding global waveletspectra of the error terms, confirming the results of the var-iability scales study of the previous section.

Since the results of the v2 test, as well as the basic sta-tistics listed in Tables 2 and 3 indicate the PC-LSR as thebest method, we conclude that the PCA can indeed be used

1980 1985 1990 1995 2000

t [year]

1980 1985 1990 1995 2000

t [year]

) Qobs � Qsim: Qobs (gray line), QOKc (dotted), QLSRc (solid line).

1960 1965 1970 1975 1980 1985 1990 1995 20000

1000

2000

3000

4000

t [year]

Q [m

3 /s]

1960 1965 1970 1975 1980 1985 1990 1995 2000–400

–200

0

200

400

t [year]

Qob

s–Qsi

m [m

3 /s]

a

b

Fig. 10. (a) Measured and simulated discharge time series (Wittenberge), (b) Qobs � Qsim: Qobs (gray line), QOKc (dotted), QLSRc (solid line).

1272 D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273

for the simulation of discharge time series at ungaugedlocations.

5. Conclusions

Regarding a hydrological process as stationary, equalsbelieve that the process is predetermined, whereas whenassuming a asymptotic nonstationarity, dynamic spontane-ity of the system is allowed (Klemes, 1974). Various statis-tical analyses performed up-to-date confirmed that, insteadof an a priori hypothesis on either stationarity or nonsta-tionarity, basic temporal or spatial variability scales shouldbe investigated.

Applying temporal and spatial variogram analysis andPCA (EOF) analysis, in combination with the wavelet tool,we were able to (a) extract the characteristic time scales ofthe monthly Elbe River flows, (b) to determine the maindeterministic signals which describe the joint variabilityof the Elbe River flow across its Basin within the Germanterritory, and (c) show that the EOF- eigenvectors com-puted from a small subset of the river gauge locationscan be used for the simulation of flow data at ungaugedlocations.

The results indicate that the stationarity scales of thefirst two moments of the analyzed time series are not equalbut are approximately in the decadal to interdecadal range(10–15 yr). Although there is no overall linear trend in thedata, a slow decreasing tendency of the mean characterizesthe discharge time series of the Northern gauge (Neu Dar-

chau), reflecting the aggregate dynamics of the Elbe RiverBasin. Moreover, in addition to the nonstationary charac-ter of the interdecadal oscillations (13.9 yr), there is a clearquasiperiodic pattern distinguished by an approximately7 yr cycle. A wavelet analysis of the first principal compo-nent which describes more than 95% of the joint variabilityalso shows low frequency oscillations at the interannual(6.9 yr) and interdecadal (13.9 yr) scales, with the corre-sponding EOFs depending on the gauge location.

As for the simulations of flow at ungauged locations,three methods are applied and compared, namely linearinterpolation (LI), PC-EOF least squares regression (PC-LSR) and ordinary kriging (OK). With respect to the por-tion of variance explained by the simulated time series, aswell as the lag-1 to lag-3 correlations and the Hurst param-eter H (computed by DFA), all methods appear to fareequally well. However, only the PC-LSR method fits thefirst two moments of the observed time series satisfactorilyand results also in the lowest RMS. We conclude that thePCA can be used as an interesting alternative to determin-istic methods for the long-range simulation of dischargetime series at ungauged locations, provided that a mini-mum calibration period of the order of the major temporalvariability scale is used.

Acknowledgement

We would like to thank the German Federal Agency forHydrology (BAFG) for providing the discharge time series

D. Markovic, M. Koch / Physics and Chemistry of the Earth 31 (2006) 1262–1273 1273

and the Potsdam Institute for Climate Impact Research(PIK) for the GIS data of the Elbe River Basin. We wouldalso like to thank the two anonymous reviewers for helpfulcomments and suggestions.

References

Beran, J., 1994. Statistics for Long-Memory Processes. Chapman andHall.

Brunetti, M., Maugeri, M., Monti, F., Nanni, T., 2004. Changes in dailyprecipitation frequency and distribution in Italy over the last 120 yr. J.Geophys. Res. 109, D05102.

Emery, W.J., Thomson, R.E., 1997. Data Analysis Methods in PhysicalOceanography. Elsevier, Amsterdam.

Foufoula-Georgiou, E., Kumar, P., 1994. Wavelets in Geophysics.Academic Press.

Ghil, M., Allen, R., Dettinger, M.D., Ide, K., Kondrasov, D., Mann,M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., Yiou, P.,2002. Advanced spectral methods for climatic time series. Rev.Geophys. 40 (1), 1–41.

Golyandina, N., Nekrutkin, V., Zhigljavsky, N., 2001. Analysis of TimeSeries Structure: SSA and Related Techniques. Chapman and Hall/CRC, London, New York.

Golyandina, N., Stepanov, D., 2005. SSA-based approaches to analysisand forecast of multidimensional time series. In: Proceedings of the 5thSt.Petersburg Workshop on Simulation, June 26–July 2.

Gray, S.T., Betancourt, J.L., Fastie, C.L., Jackson, S.T., 2003. Patternsand sources of multidecadal oscillations in drought-sensitive tree-ringrecords from the central and southern Rocky Mountains. Geophys.Res. Lett. 30 (6), 49-1–49-4.

Haberlandt, U., Klocking, B., Krysanova, V., Becker, A., 2001. Region-alisation of the base flow index from dynamically simulated flowcomponents-a case study in the Elbe River Basin. J. Hydrology 248,35–53.

Helms, M., Ihringer, J., Nestmann, F., 2002. Analyse und Simulation desAbflussprozesses der Elbe, in Morphodynamik der Elbe by Nestmann,F., Buchele, B. , TH Karlsruhe.

Hipel, K.W., McLeod, A.I., 1994. Time Series Modelling of WaterResources and Enviromental Systems. Elsevier.

Hu, K., Ivanov, P., Chen, Z., Carpena, P., Stanley, H.E., 2001. Effect oftrends on detrended fluctuation analysis. Phys. Rev. E 64, 011114.

Keyantash, J.A., Dracup, J.A., 2004. An aggregate drought index:Assessing drought severity based on fluctuations in the hydrologiccycle and surface water storage. Water Resour. Res. 40, W09304.

Klemes, V., 1974. The Hurst phenomena: a puzzle. Water Resour. Res. 10(4), 675–688.

Koutsoyiannis, D., 2005. Uncertainty, entropy, scaling and hydrologicalstochastics, 2, time dependence of hydrological processes and timescaling. Hydrol. Sci. J 50 (3), 405–426.

Mandelbrot, B.B., 1977. The Fractal Geometry of Nature. W.H. Freemanand Company.

Markovic, D., 2006. Interannual and interdecadal oscillations in hydro-logical variables: Sources and modeling of the persistence in the ElbeRiver Basin, Dissertation, University of Kassel.

Markovic, D., Koch, M., 2005a. Sensitivity of Hurst parameter estimationto periodic signals in time series and filtering approaches. Geophys.Res. Lett. 32, L17401.

Markovic, D., Koch, M., 2005b. Wavelet and scaling analysis of monthlyprecipitation extremes in Germany in the 20th century: Interannual tointerdecadal oscillations and the North Atlantic oscillation influence.Water Resour. Res. 41, W09420.

Rodriguez-Puebla, C., Encinas, A.H., Saenz, J., 2001. Winter precipitationover the Iberian peninsula and its relationship to circulation indices.Hydr. Earth Sys. Sci. 5, 233–244.

Saco, P., Kumar, P., 2000. Coherent modes in multiscale variability ofstreamflow over the United States. Water Resour. Res. 36 (4), 1049–1067.

Salas, J.D., 1992. Analysis and modeling of hydrological time series. In:Maidment, D.R. (Ed.), Handbook of Hydrology. McGraw-Hill, NewYork.

Skøien, J.O., Bloschl, G., 2003. Characteristic space and timescales inhydrology. Water. Resour. Res. 39, 1304.

Torrence, C., Compo, G.P., 1998. A practical guide to wavelet analysis.Bull. Am. Met. Soc. 79 (1), 62–78.

Wilks, D., 1995. Statistical methods in the atmospheric sciences. AcademicPress.

Zoppou, C., Nielsen, O.M., Zhang, L., 2002. Regionalization of dailystream flow in Australia using wavelets and k-means analysis,Available from: <http://www.maths.anu.edu.au/research.reports/mrr/mrr02.003/abs.html>.