Applying multi-resolution analysis to differential hydrological grey models with dual series

Journal of Hydrology (2007) 332, 174–186

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

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

Applying multi-resolution analysis to differentialhydrological grey models with dual series

Chien-ming Chou *

Department of Computer Science and Information Engineering, MingDao University, 369 Wen-Hua Road, Peetow,Changhau 52345, Taiwan, ROC

Received 20 August 2005; received in revised form 29 June 2006; accepted 29 June 2006

Summary This investigation proposes a new approach for modeling rainfall–runoff processesby wavelet-based multi-resolution analysis. Firstly, a redundant wavelet transform is used todecompose the observed effective rainfall and direct runoff time series to obtain wavelet coef-ficients at each resolution level. Then, these wavelet coefficients are applied to model rain-fall–runoff processes using differential hydrological grey models with dual series at eachresolution level. The average of the estimated grey parameters at each resolution level not onlyprovides information for validating the proposed approach, but also represents the average sys-tem characteristics at each resolution level. The summation of the forecast results at variousresolution levels yields the overall forecast and reveals that this procedure is suitable for mod-eling the rainfall–runoff process.ª 2006 Elsevier B.V. All rights reserved.

KEYWORDSWavelet transform;Multi-resolution analysis;Differential hydrologicalgrey models withdual series;Rainfall–runoff process

0d

Introduction

As Taiwan is located in the major typhoon track in the wes-tern Pacific Ocean, typhoons are an influential weather phe-nomenon in Taiwan, where short, steep upstream channelscharacterize all watersheds. The associated heavy rainfalland flooding are one of the disasters which cause the great-est loss of property and life in the area. It is thereforeessential to study the relationship of the rainfall and runoffprocesses and develop a flood forecasting system to provideprotection and warning systems.

022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reservedoi:10.1016/j.jhydrol.2006.06.031

* Tel./fax: +886 4 8780447.E-mail address: [email protected].

In grey systems theory, all systems can be divided intothree parts – white, grey and black. The white part hascompletely certain and clear messages in a system, andthe black part has entirely unknown characteristics. Themessage released in the grey part can be easily describedas an interval or a grey number. Therefore, the grey uncer-tainty includes both known and unknown messages. Greysystem theory, first proposed by Deng in 1982, avoids theinherent defects of conventional, statistical methods andrequires only a limited number of data to estimate thebehavior of an uncertain system.

Grey system theory is an effective mathematical meansof solving problems that involve uncertainty and indeter-minism. Grey system theory involves grey relational analy-sis, modeling and prediction, as well as decision-making

.

mailto:[email protected]

Applying multi-resolution analysis to differential hydrological grey models with dual series 175

and control. During the last two decades, grey system the-ory has been successfully applied to industrial systems, so-cial systems, ecological systems, the economy, geography,traffic, management, education and environmental sci-ences. It has been successfully adapted to analyze uncertainsystems that have multi-data inputs, discrete data or insuf-ficient data.

The accumulated generating operation (AGO) and the in-verse accumulated generating operation (IAGO) are themain practicable approaches to treating disorganized evi-dence. The series generated can be utilized to construct agrey forecasting model. A grey system model is a dynamicmodel that is described by a differential equation. Such amodel reduces the number of random effects because theaccumulated data series, rather than the data series them-selves, are utilized to derive the differential equation. Greysystem theory is a relatively new method in the field ofhydrology. Grey system theory provides a means of elucidat-ing the relationship of an input–output process with unclearinner relationships, uncertain mechanisms and insufficientinformation.

Based on the grey system theory developed by Deng(1982), rainfall–runoff is viewed as a grey process that in-cludes a set of uncertain parameters. The differentialhydrological grey model with dual series, developed by Xia(1989a,b) and abbreviated as DHGM(2,2), is introduced tosimulate the causal relationship between rainfall and run-off. Lee and Wang (1998) proposed the estimation of param-eters given colored noise effect associated with DHGM(2,2).Model parameters are estimated by the least squares meth-od with the colored noise effect, to improve the determina-tion of parameters. Xia (2000) posited that DHGM(2,2),which is a real-time grey forecasting method, could be ap-plied to storms or floods with a large range. Trivedi andSingh (2005) attempted to model the rainfall–runoff pro-cess using grey system theory. They considered typhoonevents to develop and predict differential hydrological greymodels. Lower values of various error indices and higher val-ues of correlation indices verify the capacity of the model topredict storm runoff with reasonable accuracy in the area ofstudy.

Although widely available for analyzing and simulatingthe rainfall–runoff process, most methods and models in-volve hydrological time series with the original data alone.In practice, studying hydrological time series is difficult be-cause they are controlled and influenced by complex fac-tors. From a time–frequency perspective, eachhydrological time series includes several frequency compo-nents, which satisfy various rules and constraints. Using thecomponent at only one resolution level to model hydrologi-cal processes makes the internal mechanism difficult tounderstand. The application of wavelet-based multi-resolu-tion analysis (MRA) can provide tools for modeling hydrolog-ical processes at various resolution levels.

The distinct feature of a wavelet is its multi-resolutioncharacteristic that is becoming an increasingly importanttool to process image and signal. Since a wavelet is a local-ized function both in time and frequency domain, it can beused to represent an abrupt variation or a local functionvanishing outside a short time interval adaptively. In thefield of hydrology, wavelets have been essentially used inthe analysis of rainfall rates and runoffs. Szilagyi et al.

(1999) applied the wavelets to identify coherent convectivestorm structures and characterize their temporal variabil-ity. Bayazit and Aksoy (2001) treated wavelets as a non-parametric data generation tool and applied them to annualand monthly streamflow series from Turkey and the USA.Smith et al. (2004) defined an index of the variability ofstreamflow discharge that involves filtering the rainfall sig-nal associated with an event.

Wavelets analysis also has been applied in the investiga-tion of the rainfall–runoff relationship. Nakken (1999) ap-plied continuous wavelet transforms to identify thetemporal variability of rainfall and runoff data and the rela-tionship between them. Labat et al. (2000) used wavelets toexplain the non-stationarity of karstic watersheds. In addi-tion, wavelets have been applied to the development ofrunoff prediction model. Coulibaly et al. (2000) appliedrecurrent neural networks and wavelet analysis to discussthe climate index of low-frequency, so as to predict the an-nual runoff. Chou and Wang (2002) applied the wavelet-based linearly constrained least mean squares (WLCLMS)algorithm estimate on-line the wavelet coefficients of theUH. Chou and Wang (2004) proposed a multi-model methodusing a wavelet-based Kalman filter (WKF) bank to simulta-neously estimate decomposed state variables and unknownparameters for real-time flood forecasting.

The central idea of this study is that the original hydro-logical time series, including effective rainfall and directrunoff, can be decomposed into detailed signals and anapproximation by a redundant wavelet transform. Thehydrosystem is simulated using DHGM(2,2) at each resolu-tion level. The rest of this article is organized as follows.First, the redundant wavelet transform is defined. Then,the structure of the DHGM(2,2) is described. A case studyof a small watershed in Taiwan is provided to show theeffectiveness of the presented method. Finally, the resultsare discussed and conclusions drawn.

Wavelet transform

Introduction of wavelet transform

The continuous wavelet transform of a continuous functionoutputs a continuum of scales. However, the input data aregenerally discretely sampled, and may be in the form ofhydrological time series. The discrete wavelet transform(DWT) of a vector is the outcome of a linear transformationthat yields a new vector whose dimensions are equal tothose of the primeval vector. This transformation is alsocalled decomposition, and can be performed efficientlyusing Mallat’s MRA algorithm (Mallat, 1989).

However, the orthonormal DWT requires that the inputdata have a number of values which is an integer power oftwo. The number of resolution levels is naturally limitedby log2 of the number of values in the input. This limitationis inappropriate for a hydrological time series, especiallyduring the short duration of typhoon events that occur inTaiwan.

Aussem and Murtagh (2001) introduced the ‘‘a trous’’wavelet decomposition. The fundamental idea that under-lies MRA or multi-scale analysis is the application of a wave-let transform to decompose signals into different resolution

176 C.-m. Chou

levels or scales. The signal, which is decomposed intocoarse resolution level, is either an approximation signalor a smooth trend. The signal is decomposed into fine reso-lution level, and is called a detailed signal. The wavelettransform can be considered to be a bridge among signalsat various resolution levels. In contrast, the input data ofthe a trous wavelet transform may take any values, suchthat the number of resolution levels is unlimited. Not onlyis this a trous wavelet transform parsimonious but the filteroutputs can also be meaningfully interpreted (Aussem andMurtagh, 2001). The calculation of the wavelets is triviallyperformed in a cascaded scheme, and involves an appealingreconstruction formula that will be exploited.

Other advantages of the a trous wavelet transform are asfollows (Chibani and Houacine, 2003). (1) The evolution ofthe wavelet decomposition can be followed from level to le-vel. (2) The algorithm generates a single wavelet coeffi-cients plan at each level of the decomposition. (3) Thewavelet coefficients are computed for each location, facil-itating the detection of a dominant feature. (4) The algo-rithm is easily implemented. Additionally, the recursivecomputation of this algorithm is very effective and can beachieved using filter banks (Lee et al., 1999).

Redundant wavelet transform

The a trous algorithm is a redundant wavelet transform(Albert and Frederic, 1995; Murtagh, 1998; Lee et al.,1999; Chibani and Houacine, 2003; Renaud et al., 2005).The wavelet transform decomposes the input hydrologicaltime series into detailed signals, and an approximation (ora residual), so the original hydrological time series is ex-pressed as an additive combination of wavelet coefficients,at various resolution levels. The procedure for decomposingthe discrete hydrological time series s(k) is firstly to performsuccessive convolutions using a discrete low-pass filter c(Aussem and Murtagh, 2001):

siþ1ðkÞ ¼X1l¼�1

cðlÞsiðkþ 2ilÞ ð1Þ

where si denotes the approximation signal at revolution le-vel i. The finest scale is used to specify the original hydro-logical time series x(k), i.e., s0(k) = x(k). The increase indistances between the sampled points (2il) explains theapplication of the name ‘‘a trous’’ to this method (Aussemand Murtagh, 2001). A B3 spline, defined as (1/16, 1/4, 3/8,1/4, 1/16), is generally used in a low-pass filter c (Lee et al.,1999; Aussem and Murtagh, 2001), to fulfill the compactsupport condition (required for a wavelet transform), andto be point-symmetric. Wavelet coefficients wi are obtainedfrom determining the differences among successivesmoothed versions of the signal, as shown below (Aussemand Murtagh, 2001);

wiðkÞ ¼ si�1ðkÞ � siðkÞ ð2Þ

Wavelet coefficients provide the ‘‘detailed’’ signal, whichin practice can be used to capture the tiny but meaningfulcharacteristics in the data. Such a characterization will notlose any information only if the original data vector can bereconstructed from the wavelet components. Moreover, the‘‘residual’’ terms sp, representing the ‘‘background’’ data,

are added to the wavelet coefficients. The setR ¼ fw1;w2; . . . ;wp; spg represents the wavelet transformof the data, i.e., wavelet coefficients, up to a resolution levelof p.

Aussem and Murtagh (2001) presented the waveletexpansion of hydrological time series, in terms of waveletcoefficients

xðkÞ ¼ spðkÞ þXpi¼1

wiðkÞ ð3Þ

DHGM(2,2)

The structure and behavior of DHGM(2,2) used in this inves-tigation were taken from Lee and Wang (1998), and are pre-sented again for convenience. The many greycharacteristics of hydrosystems are such that grey systemtheory can be applied by treating the random process in ahydrosystem as a grey process. The AGO technique is usedto transform disorderly raw data into a regular series, to ex-ploit grey modeling, and reduce the noise in the raw data. Aset of differential equations are then established as the gov-erning equations of the given grey information.

DHGM(2,2) is a grey differential equation, which hasbeen applied successfully in hydrological research (Deng,1989; Xia, 1989b; Lee and Wang, 1998). In this study, theDHGM(2,2) is used to model the rainfall–runoff process ateach resolution level, facilitating the simulation of hydro-logical subsystems. The behavior of the significantly causalrelation in the rainfall–runoff process is governed byDHGM(2,2) and given by Xia (1989a,b),

dQ ð1ÞðtÞdt

þ a1ð�ÞQ ð1ÞðtÞ ¼ b0ð�ÞdIð1ÞðtÞdt

þ b1ð�ÞIð1ÞðtÞ ð4Þ

where a1(�), b0(�) and b1(�) are grey parameters; Q(1)(t) isthe first-order accumulated generating operation (1-AGO)series of Q(0)(t); I(1)(t) is the 1-AGO series of I(0)(t); Q(1)(t)is the raw direct runoff and I(1)(t) is the raw effective areamean rainfall.

The discrete forms of I(0)(t) and Q(1)(t) are definedrespectively as,

Ið1ÞðkÞ ¼Xki¼1

IðiÞ

Q ð1ÞðkÞ ¼Xki¼1

QðiÞ

where k is the time index.Applying the Laplace transformation and convolution

integral yields (Xia, 1989a),

Q ð1ÞðtÞ ¼ cð�Þe�a1ð�Þt þ b0ð�ÞIð1ÞðtÞ

þ dð�ÞZ t

0

e�a1ð�ÞsIð1Þðt� sÞds ð5Þ

in which, c(�) = Q(0)(1) � b0(�)I(0)(1) d(�) = b1(�) �a1(�)b0(�).

The theoretical DHGM(2,2), given by Eq. (5), includesterms that each have a particular physical interpretation.The first term specifies the initial flood status; the secondis the function of the rate of change of rainfall, and the


third is the response relation of a grey system (Xia, 1989a).Grey derivatives and grey parameters are whitened accord-ing to Eq. (4). Based on grey system theory, the grey deriv-atives in discrete data are whitened according to Xia(1989a,b),

dQ ð1ÞðtÞdt

��t¼tk

¼ að1Þ½Q ð1ÞðkÞ� ¼ Q ð0ÞðkÞ ð6Þ

Q ð1ÞðtÞjt¼tk ¼1

2½Q ð1ÞðkÞ þ Q ð1Þðk� 1Þ� ð7Þ

dIð1ÞðtÞdt

��t¼tk

¼ að1Þ½Ið1ÞðkÞ� ¼ Ið0ÞðkÞ ð8Þ

Ið1ÞðtÞjt¼tk ¼ Ið1ÞðkÞ ð9Þ

where a(1) is the first-order inverse accumulated generatingoperator (1-IAGO).

Substituting Eqs. (6)–(9) into Eq. (4) yields the matrixequations (Xia, 1989a),

Q ð0Þð2ÞQ ð0Þð3Þ

..

.

Q ð0ÞðNÞ

2666664

3777775ðN�1Þ�1

¼

� 12½Q ð1Þð1Þ þ Q ð1Þð2Þ� Ið1Þ Ið1Þð1Þ

� 12½Q ð1Þð2Þ þ Q ð1Þð3Þ� Ið2Þ Ið1Þð2Þ

..

. ... ..

.

� 12½Q ð1ÞðN � 1Þ þ Q ð1ÞðNÞ� IðNÞ Ið1ÞðNÞ

2666664

3777775ðN�1Þ�3

�a1ð�Þb0ð�Þb1ð�Þ

264

375

3�1

þ

eð2Þeð3Þ...

eðNÞ

266664

377775ðN�1Þ�1

ð10Þ

which are simplified to,

Q ðN�1Þ�1 ¼ XðN�1Þ�3 � h3�1 þ e ð11Þ

where e is the model residual. Q and X are the given whiten-ing matrices, so the whitening value of the grey parameter

Figure 1 The map of Wu–Tu watershed sho

vector h is obtained by the least squares method. It is givenby Lee and Wang (1998),

h3�1 ¼a1ð�Þb0ð�Þb1ð�Þ

264

375 ¼ ðXTXÞ�1XTQ ð12Þ

where a1ð�Þ, b0ð�Þ and b1ð�Þ are the identified whiteningvalues of a1(�), b0(�) and b1(�).

After the values of the grey parameters were whitened,the whitened discrete form of Eq. (5) is given by Xia(1989a,b),

bQ ð1Þðkþ 1Þ ¼ cð�Þe�a1ð�Þk þ b0ð�ÞIð1ÞðkÞ

þ dð�ÞXki¼1

e�a1ð�ÞiIð1Þðk� iþ 1Þð13Þ

where, cð�Þ ¼ Q ð0Þð1Þ � b0ð�ÞIð0Þð1Þ and dð�Þ ¼ b1ð�Þ�a1ð�Þb0ð�Þ. Therefore, applying the 1-IAGO approach yields

the simulated discharge bQ ð0Þ,bQ ð0Þðkþ 1Þ ¼ bQ ð1Þðkþ 1Þ � bQ ð1ÞðkÞ ð14Þ

where bQ ð1Þ is the 1-AGO series of simulated discharge.

wing the study area near Taipei, Taiwan.

178 C.-m. Chou

Application and analysis

Study basin

This work demonstrates feasibility of applying the wavelet-based method to runoff prediction by selecting the Wu–Tuwatershed located in northern Taiwan as the study area.The area of the upstream watershed of Wu–Tu is 203 km2,as displayed in Fig. 1. The mean annual precipitation in thesewatersheds is 2500 mm. Due to the topography of thesewatersheds, the runoff pathlines are short and steep, andthe rainfall is non-uniform in both time and space. Largefloods arrive quickly in the middle-to-downstream reachesof this watershed, leading to severe damage. Thirty typhoon

Table 1(a) The detailed information of all calibratedtyphoon events

Events Year Duration(h)

Shapeof peak

Peakdischarge(m3/s)

ELAINE 1968 132 Single 1038IRIS 1976 79 Single 148ANDY 1982 45 Single 347GERALD 1984 168 Multiple 586ALEX 1987 48 Single 520LYNN 1987 144 Multiple 1872KIT 1988 178 Multiple 482STORM(19880929) 1988 96 Multiple 670SARAh 1989 72 Multiple 401OFFLIA 1990 63 Single 500YANCY 1990 72 Single 825STORM(19900901) 1990 48 Single 301RUTH 1991 96 Multiple 556POLLY 1992 57 Multiple 279FRED 1994 85 Single 243SETH 1994 86 Single 451HERB 1996 85 Single 1083WINNIE 1997 84 Single 1035

Table 1(b) The detailed information of all validatedtyphoon events

Events Year Duration(h)

Shapeof peak

Peakdischarge(m3/s)

NADINE 1968 50 Single 220ELSIE 1969 60 Single 663BESS 1971 66 Single 994STORM(19771115) 1977 96 Single 538FREDA 1984 59 Single 502STORM(19850208) 1985 72 Multiple 236ABBY 1986 118 Multiple 579STORM(19910922) 1991 53 Single 890DOUG 1994 67 Multiple 320GLADYS 1994 96 Single 434ZANE 1996 112 Multiple 666AMBER 1997 71 Single 954

or storm events over the Wu–Tu watershed were collectedfor a case study. Eighteen of these events were used to cali-brate model, respectively, while the other 12 were used toverify the performance of the proposed method. The detailedinformation of calibrated and validated typhoon or stormevents was provided in Tables 1(a) and 1(b), respectively.

Comparison of model performances

In this paper, DHGM(2,2) and WDHGM(2,2) represents theresults of the adoption of DHGM(2,2) method without andwith the redundant wavelet transform, respectively. Toquantitatively compare the DHGM(2,2) with WDHGM(2,2),the 1-h-ahead predicted results were evaluated based onfour different kinds of criteria, as illustrated below. Thisinvestigation focuses on the fitness of the estimated resultsas determined by comparison with observed data. The mostimportant criterion is CE.

(I) Coefficient of efficiency, CE, is defined as:

CE ¼ 1�PN

i¼1½qðiÞ � qðiÞ�2PNi¼1½qðiÞ � �q�2

ð15Þ

where qðiÞ denotes the discharge of the simulatedhydrograph for time period i (m3/s), q(i) is the dis-charge of the observed hydrograph for time period i(m3/s), �q represents the average discharge of the ob-served hydrograph for time period i (m3/s) and N isnumber of data. The CE quantifies the goodness offit between the estimated hydrograph and the ob-served hydrograph. A better fit is indicated by a CEthat is closer to unity.

(II) Error of total volume (EV)

EV ¼PN

i¼1½qðiÞ � qðiÞ�PNi¼1qðiÞ

ð16Þ

where qðiÞ denotes the discharge of the simulatedhydrograph for time period i (m3/s), q(i) is the dis-charge of the observed hydrograph for time period i(m3/s). The EV specifies the mean error betweenthe estimated hydrograph and the observed hydro-graph. When the value of EV is positive, the meanestimated discharge exceeds the observed discharge,and vice versa. A better fit is represented by a smallerabsolute value of EV.

(III) The error of peak discharge, EQP (%), is defined as:

EQPð%Þ ¼qP � qP

qp

� 100% ð17Þ

where qP denotes the peak discharge of the simulatedhydrograph (m3/s) and qP is the peak discharge of theobserved hydrograph (m3/s). When the EQp is posi-tive, the estimated peak discharge exceeds the ob-served peak discharge. When EQp is negative, theestimated peak discharge is smaller than the observedpeak discharge. A better is indicated by a smallerabsolute value of EQp.

(IV) The error of the time for peak to arrive, ETp, isdefined as:

ETP ¼ bT P � TP ð18Þ


where bT P denotes the time for the simulated hydrographpeak to arrive (hours) and TP represents the timerequired for the observed hydrograph peak to arrive(hours). When ETp is negative, the estimated peak dis-charge precedes the observed peak discharge. WhenETp is positive, the estimated peak discharge followsthe observed peak discharge. A better fit is representedby a smaller absolute value of ETp.

Results and discussion

This investigation applies a redundant wavelet transformand DHGM(2,2) to model a multi-resolution rainfall–runoffprocess. Firstly, the observed effective rainfall and directrunoff are decomposed using a redundant wavelet trans-form. Second, the rainfall–runoff process at each resolutionlevel is modeled using wavelet coefficients and DHGM(2,2).

Table 2 presents the calibrated results for 18 typhoon orstorm events obtained using DHGM(2,2) without and with atrous redundant wavelet transform. The calibrated resultsdemonstrate that the CE of the WDHGM(2,2) (with an aver-age value of 0.866) is better than that for DHGM(2,2) (withan average value of 0.820). Based on the ETP criteria, theWDHGM(2,2) (with an average value of 0.61) slightly outper-forms the DHGM(2,2) (with an average value of 1.56). Basedon the EV and EQP criteria, the DHGM(2,2) slightly outper-forms the WDHGM(2,2).

Figs. 2 and 3 display two representative calibrated re-sults; S3 is an approximation at resolution level 3, and

Table 2 The calibrated results for 18 typhoon or storm eventstransform

Events Year EV (%) CE

DHGM (2,2) WDHGM (2,2) DHGM (2,2) W

ELAINE 1968 �5.98 �7.01 0.945 0.IRIS 1976 �7.26 �8.35 0.470 0.ANDY 1982 �9.62 �13.42 0.867 0.GERALD 1984 �8.15 �11.53 0.763 0.ALEX 1987 �9.11 �11.19 0.843 0.LYNN 1987 �3.38 �5.05 0.815 0.KIT 1988 �2.64 �6.43 0.702 0.STORM(19880929)

1988 �9.07 �12.49 0.944 0.

SARAH 1989 �6.72 �6.95 0.701 0.OFFLIA 1990 �7.18 �5.75 0.875 0.YANCY 1990 �8.40 �10.21 0.829 0.STORM(19900901)

1990 �7.05 �4.83 0.741 0.

RUTH 1991 �4.59 0.81 0.868 0.POLLY 1992 �7.20 �5.74 0.816 0.FRED 1994 �5.44 �5.46 0.845 0.SETH 1994 �7.17 �9.27 0.952 0.HERB 1996 �8.21 �9.90 0.900 0.WINNIE 1997 �11.46 �18.54 0.881 0.

Average 7.15 8.50 0.820 0.

Note: The columns for EV, EQp and ETp all contain negative values, an

W1, W2 and W3 are detailed signals at resolution levels 1,2 and 3, respectively. Figs. 2 and 3 indicate that the resultssimulated using DHGM(2,2) can be used to produce easilyshocked hydrographs, whereas WDHGM(2,2) (i.e., the com-bination by summing W1, W2, W3 and S3) yields smoothinghydrographs, because in wavelet decomposition, thesmoothing of detailed signals and the approximation im-prove with the number of resolution levels. Furthermore,detailed signals, such as W1, W2 and W3, are dominatedby noise and contribute little to the overall results. Anapproximation, such as S3, represents the smooth trend ofthe overall results, yielding a smoother hydrograph than isobtained by DHGM(2,2).

Table 3 presents the estimated grey parameters at eachresolution level, obtained from the 18 calibrated events.The grey parameters a1(�), b0(�) and b1(�) vary with thecharacteristics of the study basin, the rainfall pattern andthe climate conditions, respectively.

The grey parameter a1(�) responds to the internal char-acteristics of runoff in grey rainfall–runoff processes,including hydraulic storage, infiltration capacity, soil frontcondition and geographical characteristics. This is associ-ated with the transformation of the grey series responsefunction e�a1ð�Þt, so a1(�) must be positive to satisfy therules that govern the rainfall–runoff process. Table 3 indi-cates that, for a resolution level of S0, which representsthe original data, a1(�) for each typhoon or storm event ispositive, and so satisfies the above rules that govern rain-fall–runoff processes. Table 3 also demonstrates that, foran approximation at resolution level S3, a1(�) for each ty-phoon or storm event is positive, also satisfying these rules.

using DHGM(2,2) without and with a trous redundant wavelet

EQp (%) ETp (h)

DHGM (2,2) DHGM (2,2) WDHGM (2,2) DHGM(2,2)

WDHGM(2,2)

962 �2.64 �12.12 �1 �1702 �13.20 �21.25 �4 �2895 �9.75 �18.55 �1 0794 �39.66 �41.84 �1 0879 �19.97 �27.15 �2 0817 �21.38 �25.73 �2 �3643 �14.57 �17.41 0 1961 0.33 �5.49 1 1

732 5.03 �19.89 �1 �1902 �13.41 �26.15 �2 0897 �17.47 �20.02 0 0930 �20.37 �19.04 �3 0

865 �14.42 �25.42 0 0904 7.72 �19.97 �3 0940 �4.75 �18.92 �3 0954 �6.00 �14.34 �2 2905 �12.75 �22.80 �2 0905 �6.88 �15.59 0 0

866 12.80 20.65 1.56 0.61

d these columns present the averages of the absolute values.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Time (hours)

-100

0

100

200

300

400

500

600D

isch

arge

(cm

s)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

0

10

20

30

40Rai

nfal

l(m

m/h

r)

KIT1988.09.16

Obs. discharge

Est. discharge (DHGM)

Est. discharge (Combination)

Est. discharge (W1)

Est. discharge (W2)

Est. discharge (W3)

Est. discharge (S3)

Figure 2 The calibrated results using DHGM(2,2) without and with a trous redundant wavelet transform, for Typhoon KIT.

180 C.-m. Chou

However, at resolution levels W1, W2 and W3, the detailedsignals do not purely represent the actual physical compo-nents. Therefore, parameters a1(�) appear negative forall events.

The grey parameter b1(�) specifies the internal charac-teristics of weather for the rainfall as it affects the rain-fall–runoff process, and is positive. Like a1(�), the greyparameter b1(�) in both S0 and S3 is positive. At resolutionlevels W1, W2 and W3, the parameter b1(�) is negative forall events.

In relation to runoff associated with the grey rainfall–runoff process, parameter b0(�) is an external factor, andis the action of rainfall rate on rainfall. Hence, it is positiveor negative, depending on the rainfall characteristics andthe pattern of rainfall in the study basin. In Table 3, all val-ues of b0(�) are negative at resolution level S0. The resultsobtained at a resolution level of S0 are the overall results,so cannot clearly reflect the rainfall characteristics or pat-tern. However, b0(�) can be both positive and negative atresolution levels W1 and S3. This parameter reflects thecharacteristics and pattern of the rainfall by separatingthe approximation from the detailed signals of rainfall andrunoff, respectively, using a wavelet transform. Waveletdecomposition can be used to investigate the rainfall–run-off relationship at each resolution level, and so is appropri-ate for modeling the rainfall–runoff process at eachresolution level.

Table 4 displays the average estimated grey parametersat each resolution level, obtained from the 18 calibrated

events in Table 3. Trivedi and Singh (2005) utilized the aver-age value of the model parameters at the original resolutionlevel without decomposition, S0 in this investigation, toconfirm the performance of their developed model and pre-dict the storm runoff with high accuracy. The above idea isapplied herein and extended to various resolution levels W1,W2, W3 and S3. The average of the estimated grey parame-ters at each resolution level not only provides informationfor validating the proposed approach, but also representsthe average system characteristics at each resolution level.

Table 5 presents the validation results for 12 typhoonevents using the representative average value of the esti-mated grey parameters at each resolution level in Table4, which also develops a quantitative comparison of DHGMand WDHGM, based on the four selected model efficiencycriteria. The validation results indicate that the EV of theDHGM (with an average value of 7.18) is better than thatfor the WDHGM (with an average value of 13.26). Based onthe CE criterion, the average value for DHGM is 0.817. Thedeveloped WDHGM offers a real improvement (with an aver-age value of 0.860) over CE, which is the most important cri-terion herein. According to the EQp criterion, the DHGMslightly outperforms the WDHGM. The validation resultsdemonstrate the significant superiority of the ETp associ-ated with WDHGM (with an average value of 1.00 hour) overthat of DHGM (with an average value of 2.25 hours). Figs. 4and 5 plot two representative validation results. Overall,the estimated hydrograph obtained from DHGM is uneven.In contrast, the estimated hydrograph obtained from

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Figure 3 The calibrated results using DHGM(2,2) without and with a trous redundant wavelet transform, for STORM(19880929).


WDHGM is smooth. As expected, wavelet decompositionsmoothes the hydrographs, which may help in building amore meaningful (robust) model.

The number of resolution levels in applying a wavelettransform to model rainfall–runoff processes is deter-mined. The smoothing of the detailed signals and theapproximation improve as the number of resolution levelsincreases. However, the computed error arises during thedecomposition. The error increases with the number of res-olution levels. Therefore, the number of resolution levelsused in the wavelet decomposition may not be excessiveor too few. Xu et al. (2001) recommended that the numberof resolution levels be between three and five, when thetime series that to be estimated must not be too long.Lee et al. (1999) also applied the redundant wavelet trans-form to three resolution levels to decompose the dailystreamflow. The determination of the optimal number ofresolution levels was quantitatively justified by comparingthe validation results by applying the CE criterion for variousnumbers of resolution levels, in a manner similar to that inXu et al. (2001), and presented in Table 6. The table indi-cates that the average CE is the best for decomposition withthree resolution levels. Hence, in this investigation, thenumber of resolution levels is chosen as three in the appli-cation of the a trous redundant wavelet transform.

The best number of resolution levels is independent ofthe wavelet used in the a trous wavelet transform. Theorthonormal DWT requires that the input data have a num-ber of values which is an integer power of two. The number

of resolution levels is naturally limited by log2 of the num-ber of values in the input. For the a trous wavelet trans-form, the best number of resolution levels depends on theerror in the modeling and is given by the minimum error(Xu et al., 2001).

Wavelet functions and wavelet transforms are of manyforms. The choices of wavelet function and wavelet trans-form are critical, and depend on the object of the trans-form, meaning the information to be captured. The basicidea herein is to analyze rainfall–runoff processes at eachresolution level using the wavelet transform of hydrologicaltime series. Accordingly, the a trous algorithm of the wave-let transform was selected because it is redundant and ableto provide the detail signal by wavelet coefficients, and socaptures small ‘‘features’’ of interpretational value in thedata (Murtagh, 1998).

Conclusions

In this study, a redundant wavelet transform was used todecompose the observed effective rainfall and direct run-off, to yield the wavelet coefficients at each resolution le-vel. The wavelet transform sensibly decomposes the datato make the underlying temporal structures of the originaltime series more tractable. Then, the wavelet coefficientswere adopted to estimate the grey parameters inDHGM(2,2). They represent the rainfall–runoff relationship,at each resolution level. Calibrated results indicate that themethod that is based on MRA outperforms the traditional

Table 3 The estimated grey parameters at each resolution level, obtained from the 18 calibrated events

Events Year Resolution level a1(�) b0(�) b1(�)ELAINE 1968 S0 0.1059 �7.8754 5.9752

W1 0.0266 1.4571 �2.7128W2 �0.0075 �3.6168 �0.8373W3 0.0676 �13.0099 4.7691S3 0.1176 �7.8371 6.6228

IRIS 1976 S0 0.0587 �2.4426 3.6898W1 �0.0097 �0.3162 0.1227W2 0.0040 0.2371 �0.4064W3 �0.0176 �6.6322 0.4996S3 0.0704 4.5616 4.4372

ANDY 1982 S0 0.0921 �19.7940 5.5762W1 �0.0372 �2.6597 2.1626W2 �0.0069 �10.0360 1.3406W3 0.0387 �21.9175 0.7663S3 0.0939 �17.5644 6.2890

GERALD 1984 S0 0.0825 �8.1102 4.8321W1 �0.0028 �0.4338 0.0870W2 �0.0142 �4.5662 1.4836W3 0.0382 �12.7269 4.1151S3 0.0880 �8.5103 5.1693

ALEX 1987 S0 0.0949 �10.7631 5.6370W1 0.0584 �0.3581 �1.7610W2 �0.0116 �7.3457 1.4658W3 0.1221 �18.8837 1.8710S3 0.1149 �12.5630 7.2804

LYNN 1987 S0 0.0734 �5.8373 4.1702W1 0.0167 1.2000 �1.8870W2 �0.0046 �3.8055 0.0319W3 0.0081 �3.5885 1.7660S3 0.0792 �9.5039 4.5229

KIT 1988 S0 0.0493 �5.9748 2.8235W1 �0.0158 �0.7559 0.7861W2 0.0456 �4.5739 0.3869W3 0.1350 �14.1349 1.0224S3 0.0424 3.7342 2.5539

STORM (I9880929) 1988 S0 0.1287 �8.0544 7.3789W1 0.0091 �0.0615 �1.1620W2 0.0179 �5.5983 2.2439W3 0.0307 �6.4402 4.1568S3 0.1471 �15.2005 8.4611

SARAH 1989 S0 0.1180 �7.4873 6.7089W1 �0.0060 �0.5096 �0.1132W2 0.0289 �6.6694 0.5442W3 0.0693 �13.9064 2.3405S3 0.0736 12.6710 4.6438

OFFLIA 1990 S0 0.1066 �10.3957 6.1433W1 �0.0156 �1.0186 �0.6173W2 �0.0055 �4.5069 �0.5232W3 0.1083 �18.5122 4.1127S3 0.1189 �11.7679 7.1285

182 C.-m. Chou

Table 3 (continued)

Events Year Resolution level a1(�) b0(�) b1(�)YANCY 1990 S0 0.1252 �16.2688 7.1296

W1 �0.0443 �3.5564 2.2501W2 0.0314 �15.6673 4.2339W3 0.1904 �30.4356 3.4811S3 0.1311 �13.7442 7.8229

STORM (19900901) 1990 S0 0.0640 �7.0010 3.9669S1 0.0026 �0.2077 �0.2372W2 0.0045 �3.7689 0.0380W3 0.0877 �11.8447 0.8796S3 0.0854 �6.6464 8.2017

RUTH 1991 S0 0.0835 �10.8340 4.7612W1 �0.0159 �1.5331 1.2998W2 �0.0066 �8.3674 3.9409W3 �0.0220 �10.2183 5.5532S3 0.0619 �1.3970 3.6981

POLLY 1992 S0 0.0904 �4.6857 5.4661W1 �0.0101 0.2914 �0.4555W2 �0.0215 �1.1219 �2.4376W3 �0.0676 �9.8033 0.1524S3 0.0866 �5.7282 5.5484

FRED 1994 S0 0.0760 �7.5960 4.3882W1 0.0097 �0.1705 �0.4979W2 �0.0078 �3.0825 0.6351W3 0.0404 �12.3315 4.0215S3 0.0897 �14.0184 5.0884

SETH 1994 S0 0.1176 �8.6508 6.6567W1 �0.0032 0.1229 �0.6181W2 �0.0115 �0.9744 �0.8626W3 0.0379 �3.2277 2.0862S3 0.1419 �18.5846 8.0872

HERB 1996 S0 0.1368 �20.9733 7.6276W1 0.0002 �0.6707 �0.0709W2 0.0248 �8.6056 2.6393W3 0.2112 �45.2475 2.2810S3 0.1316 �13.7233 7.8143

WINNIE 1997 S0 0.1496 �31.9636 8.4772W1 �0.0120 �1.8921 1.7546W2 0.0200 �19.7780 5.3133W3 0.1393 �38.4404 5.0435S3 0.1585 �30.9140 9.2676

Table 4 The average estimated grey parameters at eachresolution level, obtained from the 18 calibrated events

Resolution level a1(�) b0(�) b1(�)S0 0.0974 �10.8171 5.6338WI �0.0027 �0.6151 �0.0928W2 0.0044 �6.2138 1.0684W3 0.0677 �16.1834 2.7177S3 0.1018 �9.2631 6.2576


method that operates at only one resolution level. The aver-age estimated grey parameters were calculated at each res-olution level, from the calibrated results. Then, thosevalues were applied to verify the performance of the devel-oped model. Validation results reveal that the average val-ues of the calibrated grey parameters at each resolutionlevel can thus represent the average system characteristicsof each respective resolution level.

In practice, studying hydrological time series is difficultbecause such series are governed by complex factors. Theresults obtained using the conventional hydrological modelare overall results and cannot easily be used to explore the

Table 5 The validation results using the representative average value of the estimated grey parameters at each resolution levelin Table 4

Events Year EV (%) CE EQp (%) ETp (h)

DHGM (2,2) WDHGM (2,2) DHGM (2,2) WDHGM (2,2) DHGM (2,2) WDHGM (2,2) DHGM(2,2)

WDHGM(2,2)

NADINE 1968 �4.62 �11.69 0.824 0.913 �10.8 �16.72 �1 0ELSIE 1969 �7.10 �12.76 0.835 0.834 �7.89 �16.89 �1 0BESS 1971 �7.67 �14.64 0.862 0.806 �26.40 �27.87 2 3STORM(19771115)

1977 �8.56 �15.04 0.917 0.893 �6.58 �6.44 �3 1

FREDA 1984 �5.43 �12.64 0.744 0.855 �21.56 �31.92 �2 0STORM(19850208)

1985 �6.04 �11.93 0.832 0.886 11.57 �9.17 �3 0

ABBY 1986 �8.51 �15.46 0.914 0.911 10.76 8.71 1 2STORM(19910922)

1991 �6.14 �13.20 0.565 0.847 41.3 2.59 �7 �4

DOUG 1994 �7.50 �12.88 0.780 0.839 �19.19 �24.72 �3 0GLADYS 1994 �8.28 �13.81 0.778 0.852 �6.57 �19.71 �2 0ZANE 1996 �8.20 �14.89 0.884 0.857 �3.21 �15.76 �1 0AMBER 1997 �8.07 �14.50 0.870 0.823 �30.13 �35.08 �1 2

Average 7.18 13.26 0.817 0.860 16.33 17.97 2.25 1.00

Note: The columns for EV, EQp and ETp all contain negative values, and these columns present the averages of the absolute values.

184 C.-m. Chou

internal dynamic mechanism of the hydrological model.The original hydrological time series includes componentsat various resolution levels, which are determined bywavelet decomposition. There are many kinds of wavelet

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transform. The a trous wavelet transform, applied in thispaper, is redundant and able to provide the detail signalby wavelet coefficients, and so captures small ‘‘features’’of interpretational value in the data, so as to be suitable

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Figure 5 The validated results using DHGM(2,2) without and with a trous redundant wavelet transform, for Typhoon ABBY.

Table 6 The comparison of the validation results by applying the CE criterion for various numbers of resolution levels

The number of resolution levels 0 1 2 3 4 5 6 7

Average value of CE 0.817 0.836 0.849 0.860 0.858 0.853 0.852 0.850


for the modeling of the rainfall–runoff process at eachresolution level.

The wavelet coefficients at each resolution level can bedetermined by applying a redundant wavelet transform toeffective rainfall and direct runoff time series. An AGO isapplied to the wavelet coefficients to simplify the analysisof rainfall–runoff processes. A simulation that usesDHGM(2,2) at various resolution levels has the advantagesof a simple structure and rapid computation. The character-istics of wavelet decomposition are exploited and the runoffhydrograph estimated by the combination is smoother thanthat obtained using a single resolution level. Accordingly, ahydrosystem could be exactly simulated. Based on the fourcriteria, the validation results reveal that the presentedmethod, which applies wavelet decomposition, outperformsthe conventional method, which uses only the original data,because of the MRA property of the wavelet transform.

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Documents

Applying multi-resolution analysis to differential hydrological grey models with dual series