02626660109492855

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Nonparametric streamflow simulationby wavelet or Fourier analysisM. BAYAZIT a , B. NZ a & H. AKSOY aa Division of Hydraulics, Civil Engineering Faculty , IstanbulTechnical University , Istanbul, 80626, Turkey E-mail:Published online: 29 Dec 2009.

To cite this article: M. BAYAZIT , B. NZ & H. AKSOY (2001) Nonparametric streamflowsimulation by wavelet or Fourier analysis, Hydrological Sciences Journal, 46:4, 623-634, DOI:10.1080/02626660109492855

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HydrologicalSciences-Journal~~des Sciences Hydrologiques, 46(4) August 200i 623

Nonparametric streamflow simulation by wavelet or Fourier analysis

M. BAYAZIT, B. ONOZ & H. AKSOY Division of Hydraulics, Civil Engineering Faculty, Istanbul Technical University, Istanbul 80626, Turkey e-mail: [email protected]. tr

Abstract Wavelet or Fourier analysis is proposed as an alternative nonparametric method to simulate streamflows. An observed series is decomposed into its components at various resolutions and then recombined randomly to generate synthetic series. The mean and standard deviation are perfectly reproduced and coefficient of skewness tends to zero as the number of simulations increases. Normalizing transforms can be used for skewed series. Autocorrelation coefficients and the dependence structure are better preserved when Fourier analysis is used, but the mean and variance remain constant when the simulated and observed series have the same length. Monthly as well as annual flows can be simulated by this technique as illustrated on some examples. Wavelet analysis should be preferred as it generates flow series that exhibit a wider range of required reservoir capacities. Key words streamflow simulation; wavelet analysis; nonparametric modelling; synthetic series; Fourier analysis; harmonics; data generation Simulation non paramtrique de dbits par analyses en ondelettes ou de Fourier Rsum Les analyses en ondelettes et de Fourier sont proposes comme mthodes alternatives non paramtriques pour simuler les dbits. Les sries observes sont dcomposes pour des rsolutions diverses et recomposes de manire alatoire pour former des sries synthtiques. La moyenne et l'cart type sont parfaitement reproduits et le coefficient d'asymtrie tend vers zro avec l'augmentation du nombre de simulations. Les transformations de normalisation peuvent tre employes pour les sries asymtriques. Les coefficients d'autocorrlation et la structure de dpendance sont mieux conservs avec l'analyse de Fourier, mais la moyenne et la variance restent constantes quand les sries simules et observes ont la mme longueur. Les dbits mensuels et annuels peuvent tre simuls avec cette technique, ce que nous illustrons avec quelques exemples. L'analyse en ondelettes doit tre prfre puisqu'elle produit des sries de dbit qui prsentent une plus grande gamme de volumes de rservoir. Mots clefs simulation de dbit; analyse en ondelettes; simulation non paramtrique; sries synthtiques; analyse de Fourier; harmoniques; production de donnes

INTRODUCTION

In water resources engineering planning, design and operation studies it is desired to consider not only the historical inflows but also other inflow series that are statistically similar to the observed series so that various patterns of time series can be taken into account. This requires modelling the streamflows to generate so-called synthetic series that reproduce the historical statistical characteristics. Parametric models, mostly of ARMA(p, q) (autoregressive moving average) type, where p and q are orders of the autoregressive and moving average components, respectively, have been widely used for this purpose (Salas et al, 1980; Loucks et al, 1981).

Open for discussion until I February 2002

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624 M. Bayazit et al.

Certain problems arise in this approach. It is not a straightforward task to choose the appropriate model. Salas et al. (1980) discuss how the orders p and q can be determined. Once the model is identified, its parameters are estimated from the available data, where sampling errors may be of considerable magnitude because of short records. The fitted model usually does not reproduce the whole dependence structure of the flows as only the autocorrelation coefficients of lag one and maybe lag two are considered. State-dependent correlation statistics, persistence and threshold crossings cannot be preserved (Sharma et al, 1997). Streamflows are mostly non-normally distributed. Because ARMA models can be fitted to normal data only, it is necessary to apply an appropriate transformation in the case of non-normal data. Srinivas & Srinivasan (2000) discuss the drawbacks of parametric models.

Attempts have been made to use nonparametric methods for modelling streamflows without having to make assumptions about the probability distribution and autocorrelation structure. Bootstrap resampling techniques have been applied (Vogel & Shallcross, 1996; Lall & Sharma, 1996; Sharma et al, 1997; Tasker & Dunne, 1997). Difficulties are encountered in reproducing the autocorrelation structure of the original time series. Model-based resampling, moving block bootstrap, nearest neighbour bootstrap and post-blackening approaches are developed to model dependent data (Srinivas & Srinivasan, 2000). Sharma et al (1997) used kernel estimates of the joint and conditional probability density functions to generate synthetic streamfiow sequences.

A quite different nonparametric approach is based on the generation of synthetic series by wavelet or Fourier analysis. In this approach, the observed time series is first decomposed into its components at various frequencies, and then reconstructed randomly to generate new data. Feng (1998) and Bayazit & Aksoy (2001) used wavelets to generate synthetic streamflows. In this paper, simulation by wavelet transform is briefly reviewed. Then, the use of the Fourier analysis to generate synthetic streamfiow series is described. Advantages and drawbacks of the method are discussed by means of some examples.

SIMULATION BY WAVELET ANALYSIS

Wavelets are mathematical functions that break up a signal into different frequency components so that they can be studied at different resolutions or scales. They have advantages over the Fourier analysis in analysing the signals with discontinuities and sharp spikes. They were developed independently in various fields of mathematics, physical sciences and engineering. The concept of wavelets in its present form was first proposed by Morlet (1981) and Grossmann & Morlet (1984). Wavelets were employed for signal transmission applications in electronic engineering (Daubechies, 1988; Mallat, 1989), and used in geophysical applications (Foufoula-Georgiou & Kumar, 1994). Smith et al (1998) attempted to apply the wavelet transformation to daily river discharge records to quantify streamfiow variability. The wavelet analysis, analogous to Fourier analysis, is used to decompose a signal by linear filtering into components at various frequencies and then to reconstruct it at various resolutions. Rao & Bopardikar (1998) describe the decomposition of a signal using a Haar wavelet, a very simple wavelet, defined as:

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Nonparametric streamflow simulation by wavelet or Fourier analysis 625

V(0 =

1 0 < / < ^ 2

-1 -


f=- k = 0,l,-,K (7)

where / is the integral part of the number in parentheses; fo is the original series, and fht are all zeroes. The detail functions g^ k = 1, ..., K, are then computed as:

2000 1000

-1000 -2000

1500 500

-500 -1500

1500 500

-500 -1500

1000 500

-500 -1000 1000 -500

-500 -1000 J

1000 -500 -

0 --500 -

-1000 -

500 -

-500

1000

1000

JL . 1

II " 1

D 1

1

n 2

(l 2

(l 2

2

(I n 3 4

I I 3 4

n n 3 4

3 4

Su 9i2

1

1

J 1

1

2

2

n 2

2

n 3 4

3 4

n n 3 4

3 4

...0 . 5 6

n n 5 6

n n 5 6

1 1 5 6

II 5 1

921 922

5 6

I) 7

n 7

n 7

i 7

i 7

7

931

5 6

n n 5 6

y 7

n 7

8

n 8

n 8

[I 8

y

8

P32

u 8

II 8

Observed series fn

1 10 11 12 T3 14 ft W

fm

1 1 "I "2 S 1 { | f2

9 10 11 12

fan

nuint 9m

E D D D

9 10 H 12 ft 14 15 16

92n

Jl .il 9 IB 11 12 13 14 15 16

93n

. 1 1 n 9 10 11 12 | | f |

941 942

/ / 9" IttUfftlltt

1500

500

-500

-1500

/

II f i h

I *J L

5 6 U

J 8 < 3 1 0 11 1

Simulated series

2 13 1 S )

Fig. 1 Simulation by wavelet analysis. Arrows show the detail function elements chosen randomly to generate the first two elements, fi and f2, of a synthetic series.

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gfa=f*-.-ffa k = \,...,K (8) For each n, one now has K detail function values corresponding to different resolutions. Choosing randomly from among TV elements for each g and then summing them up by equation (6) a simulated value for f is obtained. The generation is continued following the steps described above. Figure 1 shows an example with N= 16 (K = 4). The functions gn, g2i, g3i and g are the randomly selected detail function values of the first element of the simulated series at different scales k = 1 to 4, which are summed up to get ft; gn, g22, g32 and g42 are the corresponding values for the second element f^ .

This scheme fails to preserve the correlation structure of the process. Bayazit & Aksoy (2001) proposed to modify the generation algorithm as follows. After obtaining the first element of the synthetic series as described above, the second element is gene-rated by choosing for each k the detail function values gt coming just after the values selected in the previous step. In Fig. 1, gu, g22, g32 and g42 are selected in this way.

Data generation is continued like this, using for the generation of each element, the detail function values right next to those of the previous step for each resolution level k. It was found that the modified scheme was successful in reproducing the autocorre-lation structure of the observed data, as well as its mean and standard deviation. The coefficient of skewness, however, could not be preserved, the generated data having an average skewness tending to zero as the number of simulations increases.

A restriction of the data generation by wavelet analysis is that the number of resolution levels K is related to the size of the available sample as N=2K. For example, for N=32, a sample size that is rarely exceeded for annual flows, this number is K = 5. The number of non-identical synthetic series, each N = 32 years long,

K that can be generated is equal to J~[ N12 -32 768. For N= 16, this is reduced to

k=0 1024.

SIMULATION BY FOURIER ANALYSIS

A sample record fi, fi, ..., fv of a stationary time series {i) sampled at N equally spaced points can be represented by the finite version of a Fourier series:

f- = 1 4 cos^r+S* s m ^ r = 5 X = * > 2> > N q=\ J* q=\ N

q=l

where it is assumed that f has zero mean. The Fourier coefficients Aq and Bq are given by:

9 N 2 ^ , . 2nqn cos-q

Ntt " N 2 v-,. . 2nqn Bn- > fsm

q Nj~t " N

1 N

iv =i

1 0 N 1 q = l,2,..., -l 2

(io)

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Synthetic series generation by Fourier analysis is similar to the generation by wavelet analysis. Harmonics fq of equation (9) correspond to detail functions. Having calculated the Fourier coefficients by equation (10) from the observed values f, n = 1, 2, ..., N, a value of is randomly selected and harmonics corresponding to q = 1 are calculated for that value of. Similarly, for each q a random value of is used to compute the harmonics for that frequency. Summation of harmonics by equation (9) gives an element of the synthetic series to be generated. To reproduce the correlation structure, next element is generated increasing the value of by one for each harmonic over that of the previous element. When n = N+ 1, generation is continued taking n = 1, with a small loss of the autocorrelation. Thus, a synthetic series of iV elements is obtained. Figure 2 shows an example with N = 16. The notation is similar to Fig. 1: fn to fgi are the randomly selected harmonics at scales k = 1 to 4, which are summed up to get the first element fi of the simulated series; fi2 to f82 are harmonics next to fn to f8i, respectively.

When the length of the simulated series is equal to that of the historical series (as in the examples below), the mean of the simulated series will be exactly the same as the historical mean, because each element of the detail functions in the case of the wavelet analysis (and each element of the harmonics in the case of the Fourier ana-lysis) will have been used exactly once in generating the simulated series. This is also true when the length of the simulated series is a multiple of the length of the historical series. For series of other lengths, the mean will be somewhat different from the historical mean. When the Fourier analysis is used, the standard deviation also remains constant when the simulated series is of the same length as the observed series. The simulations will reflect various orders in which the generated values occur even though their means (and, in the case of the Fourier analysis, their standard deviations) do not change. Stedinger & Taylor (1982) argue that the persistence of droughts and other important characteristics of flow sequences are emphasized when the confounding effects of variations in the mean and variance of flow sequences are eliminated.

The number of non-identical series that can be generated is larger in this case, equal to ht112. ForN= 16, one can generate 16 synthetic series, each 16 years long.

APPLICATIONS

Annual streamflows Simulation by Fourier analysis is applied to the annual flow series at Homa station on Manavgat River in Turkey. The record length is 40 years. Flows have a very small coefficient of skewness (Csx = -0.043) and rather high autocorrelation coefficients {rx = 0.509, T2 = 0.476, r3 = 0.366). Thirty synthetic series, each 40 years long, are generated by Fourier analysis. Averages and ranges of the statistics of the synthetic series are shown in Table 1, compared with those of the original series.

Mean and standard deviation are perfectly reproduced. The average skewness coefficient is almost zero, very close to the historical value. Autocorrelation coef-ficients of lag one, two and three are also preserved. Their averages are slightly smaller (3-6%) than the historical values. Although the average skewness coefficient is almost zero, skewness of individual series varies in a rather wide range (from -0.66 to 0.66). Ranges of the autocorrelation coefficients of individual series are not so wide.

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1000 ] 0

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Observed series

2 3 4 5 6 7 8 I 11 12 If 14 IE 16

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-1000

1 2 3 4 5 6 7 6

f21 ^22

m i p a L

2 3 4 5 6 8 I 10 11 12 13 14 TS IB

1000 500

f* , f 31 ^32

M D D y y n "-4r I D -500 1 2 3 H 5 6 7 8 9 11 12 13 14 16 16 1000 -

II 1 2 n 3 4 ii 5 6

n

/ 8 II 9 10 (1 11

500

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-500

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0 0 a -500-I 1 2 4 5 6 ' 7 8 9

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*41 f42 . K

JL

1^ 1 JA9

10 11 12 13 14 15 16

" T " T J " = T T -IO 11 12 13 14 15 TC

0

-500

R1, ffi-

1 2 B 6 7 i 9 10 fl 12 tt 14 15 13 aJL

1000 -500

o-n--500 - 1

-1000 J

f/i f/:

X i i I

ffii 'a;

9 10 "H 12 -rS 14 15 16

fa, 200 100 ^ o\T^n y " u n u n u "THhHhr

-100- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -200 -2000

1000

0

-1000

-2000

f, f2

H Simulated series

3 4 5 6 7 B 10 ' I > T3 14 15 1

Fig. 2 Simulation by Fourier analysis. Arrows show the harmonic elements chosen randomly to generate the first two elements, f! and f2, of a synthetic series.

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Table 1 Statistics of the historical and synthetic annual flow series generated by the Fourier analysis, Manavgat River at Homa.

Statistic

x (106 m3) ^(106m3) C-n r2 ri

Historical series

4692.5 1302.5 -0.043 0.509 0.476 0.366

Synthetic series: Ave. 4692.5 1302.5 -0.030 0.494 0.488 0.345

Min. 4692.5 1302.5 -0.655 0.460 0.397 0.298

Max. 4692.5 1302.5 0.655 0.534 0.506 0.485

Table 2 Statistics of the historical and synthetic annual flow series generated by the wavelet analysis, Manavgat River at Homa.

Statistic

x(106m3) s*(106m3) r

n r2 ?3

Historical series

4757.2 1316.1 -0.100 0.455 0.419 0.295

Synthetic series: Ave. 4757.2 1320.9 0.007 0.356 0.402 0.203

Min. 4757.2 968.9 -0.721 -0.079 0.067 -0.261

Max. 4757.2 1678.3 0.751 0.563 0.622 0.522

A 32-year (32 = 2 ) portion of the annual flow series at Homa is used to generate synthetic series by wavelet analysis. The results of 30 simulations, each 32 years long, are given in Table 2. The mean is perfectly preserved; average standard deviation is slightly higher than the observed value, with a rather wide range of variation. Again, average coefficient of skewness is zero. Averages of the first three autocorrelation coefficients are smaller than the observed values, with differences as high as 22% for the lag-one and 30% for the lag-three coefficients. Moreover, they vary in very wide ranges.

Autoregressive moving average (ARMA) models replicate the first few moments of the dependence structure, whereas nonparametric models may be successful in reproducing the state-dependent (nonlinear) correlation statistics. Sharma et al. (1997) defined forward (backward), above median and forward (backward), below median correlation coefficients to quantify nonlinear dependence in data. For a linear Gaussian process these should be the same. The hypothesis that there is no difference between sample forward (backward) above median and below median correlation coefficients of the Manavgat River data was not rejected by the tests described by Sharma et al. (1997) at the significance level of 0.10. Average values of forward (backward) above median and below median correlation coefficients of 30 simulations by Fourier analysis were practically the same (0.181, 0.204, 0.191 and 0.208, respectively).

As another example, a 128-year long record of annual flows at St Louis on Mississippi River, USA is considered. Flows are positively skewed, with a coefficient of skewness equal to 0.74. They are transformed into a series of almost no skew

is (Csy = -0.027) by the Box-Cox transformation y =x ' . The transformed series simulated by wavelet or Fourier analysis, which is then transformed back to the streamflows. Statistics of the original record and 30 synthetic series, each 128 years long, generated by wavelet or Fourier analysis are given in Table 3.

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Table 3 Statistics of the historical and synthetic annual flow series, Mississippi River at St Louis.

Statistic

,(106m3) *x(106m3) Q!X n r2 r-i

Historical series

5220 1649 0.74 0.34 0.076 0.016

Synthetic series: Wavelet analysis: Ave. 5220 1653 0.68 0.26 0.010 0.087

Min. 5191 1480 0.13 0.10 -0.18 -0.13

Max. 5260 1840 1.19 0.39 0.18 0.25

Fourier Ave. 5220 1649 0.63 0.35 0.066 0.034

analysis: Min. 5216 1564 0.11 0.31 0.022 -0.027

Max. 5224 1758 1.38 0.39 0.103 0.071

Mean and standard deviation of the historical series are very well preserved by both wavelet and Fourier analysis. In fact, the mean remains constant when the length of the generated series is the same as that of the historical series. Statistics of the series generated by Fourier analysis have much narrower ranges. In this case, the standard deviation does not change when the generated series has the same length as the historical series. Coefficient of skewness of synthetic series is on the average somewhat smaller than that of the historical series, difference is a little higher for the series generated by the Fourier analysis. Skew coefficients vary in wide ranges in simulations both by wavelet and Fourier analysis. Fourier analysis is more successful in preserving the autocorrelation coefficients, giving averages almost equal to those of the historical series. Ranges are also narrower when Fourier analysis is used.

A statistic that can be used in the validation of the streamflow generation models is the reservoir capacity required to provide a specified draft with the given flow sequence because synthetic series are frequently used in reservoir design and operation studies (Stedinger & Taylor, 1982). For each of the simulated series, the reservoir storage capacity Csjm required for release of a target draft equal to 2/3 of the mean annual flow is calculated with the sequent peak algorithm, and divided by C0bs, the capacity calculated with the historical flow record. Cumulative distribution functions of the reservoir capacities x = Csim/C0bs are given in Fig. 3. In the same figure, results obtained with the series generated by the AR(2) model are also shown.

It is seen that the reservoir capacities corresponding to series generated by the AR(2) model have the largest range, implying that much drier and wetter periods have been simulated. Wavelet-simulated series have a smaller range, and Fourier-simulated series have the smallest range. As explained before, all the Fourier-simulated series have the same mean and standard deviation, and all the wavelet-simulated series have the same mean (but different variances) when the length of the simulated series is the same as the length of the observed series as in these examples. Therefore, simulations reflect only the different flow sequences in the case of the Fourier analysis, and different variance and different flow sequences in the case of the wavelet analysis.

Monthly streamflows

Periodic (seasonal or monthly) flow series can be generated by Fourier or wavelet analysis using a similar procedure. A 20-year portion of the Manavgat River monthly flows at Homa was simulated by Fourier analysis. Table 4 shows the simulation results

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F(x)

Fig. 3 Cumulative distributions of the required reservoir capacities of the series simulated by wavelets, Fourier analysis and AR(2) model.

Table 4 Statistics of the historical and synthetic annual flow series, Manavgat River at Homa.

Month x(106m3) T(106m3) C^ 7t Historical Synthetic Historical Synthetic Historical Synthetic Historical Synthetic

October November December January February March April May June July August September Annual

177 233 514 726 631 649 605 538 394 302 232 182

5183

(Ave.) 177 234 512 726 632 649 606 539 395 303 232 182

5186

52 116 312 251 217 157 147 154 121 95 74 57

1330

(Ave.) 52

116 311 251 215 157 146 153 120 94 74 56

1327

-0.14 1.04 0.93

-0.70 -0.28 -0.02 -0.58 0.06 0.13

-0.01 0.05 0.04

-0.26

(Ave.) -0.03 0.28 0.11 0.20 0.05 0.06

-0.37 -0.44 -0.43 -0.34 -0.18 -0.08 -0.08

0.560 0.653 0.702 0.520 0.459 0.729 0.950 0.980 0.988 0.987 0.995 0.861 0.482

(Ave.) 0.556 0.663 0.707 0.519 0.460 0.730 0.950 0.980 0.988 0.987 0.995 0.860 0.380

compared with the statistics of the original data. Averages of 10 simulations, each 20 years long, are given for each month and for annual flows.

It is seen that means, standard deviations and lag-one correlation coefficients of the monthly flows are successfully replicated. Averages of the simulated values of these statistics are practically equal to the corresponding historical values. Skew coefficients of monthly flows, however, are not preserved. In this case no trans-formation was applied to monthly data because a different transform would be needed for each month.

Average statistics of the annual flows generated as the sum of monthly flows are close to those of the observed annual flows. Mean and standard deviation are almost the same. The lag-one autocorrelation coefficient is somewhat smaller than the

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historical value. The skew coefficient is again not preserved, having a value close to zero.

CONCLUSIONS

In this study a nonparametric streamflow simulation model is presented that is based on wavelet or Fourier analysis. An observed series is decomposed into components at various frequencies and then randomly reconstructed to obtain a synthetic series. The following conclusions are drawn about the reproduction of the statistics. 1. The mean of the synthetic series is the same as that of the observed series when

they have the same length. Standard deviation also remains constant in the case of Fourier analysis. However, standard deviations of individual simulations vary in a rather wide range when wavelet analysis is used.

2. Coefficients of skew of non-normal series cannot be reproduced. Simulated series have skew coefficients that vary in wide ranges, tending to zero on the average as the number of simulations increases, which may be explained by the central limit theorem. Non-normal series can be normalized by a suitable trans-formation, simulated by wavelet and Fourier analysis, and then transformed back. This procedure approximately reproduces the coefficient of skew of the original series.

3. Fourier analysis reproduces the first few autocorrelation coefficients quite well. Average values are slightly smaller than the observed ones, and the variation among the simulations is reasonably small. Discrepancies are somewhat higher when wavelets are used in the simulation.

4. The state-dependent sample correlation coefficients of a linear Gaussian example are replicated in simulation. The case of nonlinear processes is not investigated.

5. Fourier analysis is not successful in generating streamflow series that require a wide range of required reservoir capacity, an important statistic in water resources studies. A wide range of capacities can be reproduced by wavelet analysis, although still not quite as wide as those of the autoregressive models.

6. Monthly streamflows can also be simulated similarly. Average values of monthly and annual means and standard deviations are very well reproduced. Lag-one correlations of monthly flows are also preserved, although that of annual flows is smaller than the historical value. Coefficients of skew are generally close to zero.

7. The number of non-identical series that can be generated is much larger when Fourier analysis is used.

8. Wavelet analysis seems to be more suitable than Fourier analysis for simulating streamflow series. This may be because wavelets can better analyse signals with sharp spikes as in the case of streamflow series.

Acknowledgements The authors would like to thank two reviewers for their constructive comments, which have helped in improving the quality of the paper. H. Aksoy appreciates supports by Istanbul Technical University (ITU) and the Scientific and Technical Research Council of Turkey (TUBITAK), enabling him to do his postdoctoral researches at the University of California at Davis, USA.

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.Res. 32(6), 1875-1882. Received 8 December 2000; accepted 4 April 2001

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