A methodology for discharge estimation and rating curve development at ungauged river sites

A methodology for discharge estimation and rating curve

development at ungauged river sites

Muthiah Perumal,1 Tommaso Moramarco,2 Bhabagrahi Sahoo,1 and Silvia Barbetta2

Received 23 September 2005; revised 3 October 2006; accepted 13 October 2006; published 14 February 2007.

[1] A methodology for estimating discharges and development of rating curves atungauged river sites is presented. The methodology employs a routing method, which isan extension of the variable parameter Muskingum stage hydrograph (VPMS) routingmethod developed by M. Perumal and K. G. Ranga Raju, for routing a given upstreamstage hydrograph in a channel reach characterized by trapezoidal compound cross sectionto arrive at the stage hydrograph at the downstream site. Furthermore, the VPMS methodalso enables us to estimate the discharge hydrographs at the upstream and downstreamsites. It is assumed that there is no lateral flow within the routing reach. For establishingthe rating curve at the ungauged river site the following concept employed in thedevelopment of the VPMS method can be conveniently exploited: During unsteady flow,there exists a one-to-one relationship between the stage estimated at the ungauged site andthe corresponding steady discharge, which occurs somewhere downstream of thatlocation, and using the method employed in the VPMS method, one can estimate thatsteady discharge. By linking this discharge with the stage estimated at the ungauged sitethe rating curve can be established. This approach of developing the rating curve isverified for a number of hypothetical data sets, obtained using the MIKE 11 model, whichis used as the benchmark model in this study. Furthermore, the appropriateness of theproposed extended VPMS routing method is verified using two sets of experimental dataon unsteady flows, obtained from a laboratory channel with rectangular compoundflow section. The methodology is also field tested using six sets of concurrent stagehydrograph data obtained at the upstream and downstream sites of a 15 km reach length ofthe Tiber River in central Italy, out of which only one set of data was used forcalibrating the reach-averaged Manning’s roughness coefficient. The close reproductionsof the rating curves and discharge hydrographs recorded at the upstream anddownstream sites demonstrate that the proposed methodology can be confidently used forrating curve development and discharge estimation at ungauged sites, thus avoiding themanual discharge measurement at any river site not subjected to backwater effects.

Citation: Perumal, M., T. Moramarco, B. Sahoo, and S. Barbetta (2007), A methodology for discharge estimation and rating curve

development at ungauged river sites, Water Resour. Res., 43, W02412, doi:10.1029/2005WR004609.

1. Introduction

[2] The discharge measurement at any river gauging siteplays a vital role in comprehensive and coordinated plan-ning for the utilization of water resources of river basins.Barring a few exceptional cases, continuous measurementof stream discharge is difficult to obtain. Further, directmeasurement of discharge is a time consuming and costlyprocedure. Hence a two-step procedure is followed inpractice: First, the discharge in a given stream is relatedto the corresponding elevation of the water surface (gauge)through a series of careful measurements. In the next step,the gauge of the stream is observed routinely, and the

discharge is estimated by using the previously determinedstage-discharge relationship, or alternatively, known asthe rating curve. As development of rating curve at anyriver section requires sufficient number of stage andcorresponding discharge measurements, it becomes all themore difficult to develop a rating curve at an ungauged riversite.[3] Techniques for indirect estimation of high flows are a

critical component of a streamflow information program.Existing indirect flow measurement techniques, such as theslope-area method, are usually employed for the extensionof rating curves in the domain where the concurrent flowand stage records could not be observed. However, meth-odologies for synthesizing the rating curve at a river site,wherein only the recorded stage information is available,have been developed in recent years, by linking the stageinformation with the concurrent discharge informationrecorded at an upstream river site for which the reliablerating curve is also available [Birkhead and James, 1998;Franchini et al., 1999; Moramarco and Singh, 2001]. The

1Department of Hydrology, Indian Institute of Technology Roorkee,Roorkee, India.

2Istituto di Ricerca per la Protezione Idrogeologica, National ResearchCouncil, Perugia, Italy.

Copyright 2007 by the American Geophysical Union.0043-1397/07/2005WR004609

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WATER RESOURCES RESEARCH, VOL. 43, W02412, doi:10.1029/2005WR004609, 2007

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rating curve thus developed can be used for estimatingdischarges at the site where only stage measurements areavailable. All these methods assume insignificant lateralflow in the considered reach. The method proposed byBirkhead and James [1998] relates the upstream flow to theconcurrent downstream stage hydrograph using the classicalMuskingum method and using the assumption that there is aone-to-one relationship between stage and discharge at ariver location during unsteady flow. However Moramarcoand Singh [2001] reported that this method could not servethe intended purpose satisfactorily. Franchini et al. [1999]proposed an alternate method for synthesizing the ratingcurve at an ungauged site, wherein only the recorded stageinformation is available, by linking it with the concurrentdischarge recorded at an upstream site wherein a reliablerating curve is available. The method used a differentformulation of a variable parameter Muskingum-Cungemethod with a specific parameterization scheme for takinginto account of the space-time dynamics of the flood wave.Moramarco et al. [2005] pointed out that this method iscumbersome to use as it requires the estimation of nineparameters. Both the methods advocated by Birkhead andJames [1998] and Franchini et al. [1999] are of limited useas these methods have been developed using one-to-onerelationship between stage and discharge, and involveestimation of many parameters using observed data. There-fore these methods are not suitable for using in streamswherein the unsteady flow is not characterized by a one-to-one relationship between stage and discharge. Further, thesemethods introduce parameter uncertainty due to the require-ment of estimation of many numbers of parameters usinglimited observations. Moramarco and Singh [2001] pro-posed a method for reconstructing the discharge hydrographat a river section, where only the stage is monitored, bylinking the stage to the discharge recorded at an upstreamsection. This approach did not involve any routing proce-dure and was developed using the entropy concept. Later,their method was modified to take into account the presenceof lateral flow in the considered river reach [Moramarco etal., 2005]. However, as far as the authors are aware, nomethod has been developed so far to synthesize the ratingcurves at the upstream and downstream sites of a riverreach, wherein only a concurrent stage hydrographs of aflood event have been recorded. However, the approach ofrouting in trapezoidal cross section channel reach followedin the present study is on the same lines as employed byTang et al. [1999] using the variable parameter Muskingum-Cunge method, even though their routing was for dischargevariable rather than stage variable.[4] It may be noted that the currently available stream-

flow routing packages such as NWS DAMBRK [Fread,1990], HEC-RAS [U.S. Army Corps of Engineers, 2002],and MIKE-11 [Danish Hydrologic Institute (DHI ), 2003]may not be ideally suitable to serve the purpose of ratingcurve extension or its development at a site under the abovesituation, due to their prohibitive requirement of channelcross-section details and roughness information at closerintervals. However, approximate flood routing techniquesmay serve the intended purpose more usefully, as thesetechniques could cope with sparse spatial details. As therating curve sites, where the extension or development ofrating curves are sought, are usually located on main-

streams, the approximate routing technique to be employedshould also be capable of routing flows in compoundchannels consisting of a main channel and an adjoiningfloodplain channel.[5] A methodology for estimating discharges and devel-

opment of rating curves at ungauged river sites is presentedin this study. The term ‘‘ungauged’’ is used herein in a lessrestrictive sense that at least one set of concurrent stagehydrograph measurements are available at the upstream anddownstream ungauged sites of a river reach for the purposeof estimating the reach averaged Manning’s roughnesscoefficient, n, and also, it refers to the situation of non-availability of discharge data at these sites. The methodol-ogy employs a routing method which is an extension of theapproximate stage hydrograph routing method developedby Perumal and Ranga Raju [1998a, 1998b]. It envisagesthe routing of a given upstream stage hydrograph in achannel reach, characterized by a trapezoidal compoundcross section consisting of a main channel section and afloodplain channel section, to the location of a downstreamungauged site using the reach averaged roughness coeffi-cient, which could be calibrated by matching the routedstage hydrograph estimated at the ungauged site with therecorded stage hydrograph. The parameters of the routingmethod, which are the same as that of the Muskingummethod, vary at every routing time interval using therelationship established by Perumal and Ranga Raju[1998a, 1998b]. Using the recorded stage hydrograph andthe corresponding normal discharge hydrograph estimatedby the VPMS method, the rating curve at the ungauged sitemay be developed. This rating curve enables to establish acontinuous discharge record corresponding to a continuousstage measurements recorded automatically and thus avoidsthe conventional measurements of river flows which arecostly, tedious and frequently dangerous during floodevents. The VPMS method also enables to estimate thedischarge hydrographs at the upstream and downstreamsites corresponding to the given input and routed stagehydrographs, respectively. The proposed methodology isverified using a number of hypothetical data sets, simulatedby the MIKE-11 model, which is used as the benchmarkmodel in this study, and two sets of experimental data madeavailable from the laboratory study conducted by Rashidand Chaudhry [1995]. The methodology is also field testedusing six sets of concurrent stage hydrographs data recordedat Pierantonio (the upstream site) and at Ponte Felcino (thedownstream site) of a 15 km reach length of the Tiber Riverin central Italy, out of which only one set of data was usedfor calibrating the reach averaged Manning’s n. The closereproductions of the discharge hydrographs recorded at theupstream and downstream sites of the Tiber River reach bythe proposed method demonstrate that this methodology canbe confidently used for discharge estimation and ratingcurve development and thus avoiding the manual dischargemeasurements at any river site which is not subjected tobackwater effects.[6] A similar approach followed in the proposed meth-

odology was used by Jha [2005] for his Masters’ degreethesis, under the supervision of the first author, wherein theDAMBRK model [Fread, 1990] was used to arrive at thebenchmark solutions required to verify the proposed routingmethod. However, due to limitations of the DAMBRK

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W02412 PERUMAL ET AL.: RATING CURVE ESTIMATION AT UNGAUGED RIVER SITES W02412

model in estimating the upstream stage hydrograph, whichis required as the input hydrograph for routing using theVPMS method, the benchmark solutions obtained had somediscrepancies. In the method presented in this study, thislimitation has been overcome by directly routing the givenstage hydrograph using the MIKE 11 model, to arrive at thebenchmark stage hydrograph at the required downstreamlocation. Besides, a new method of estimating the wavecelerity when the flow spills to the floodplain channelsection has been incorporated in the routing method devel-oped herein.

2. Variable Parameter Muskingum StageHydrograph Routing Method

[7] This method, henceforth referred to as the VPMSmethod, was developed by Perumal and Ranga Raju[1998a, 1998b] directly from the Saint Venant’s equations.The form of the routing equation developed is same as that ofthe Muskingum method, by replacing the discharge variableby the stage variable and hence the reason for adherence ofthe term ‘‘Muskingum.’’ Further, the parameters vary atevery routing time interval and they are related to thechannel and flow characteristics by the same relationshipsas established for the physically based Muskingum method[Apollov et al., 1964; Cunge, 1969; Dooge et al., 1982;Perumal, 1994a, 1994b]. For the sake of better understand-ing of the proposed extension of this method for routing astage hydrograph in a compound section channel reach, abrief description of the method developed by Perumal andRanga Raju [1998a, 1998b] is presented herein.

2.1. Concept

[8] The VPMS method has been developed using thefollowing concept: During steady flow in a river reachhaving any shape of prismatic cross section, the stage,and hence the cross-sectional area of flow at any point ofthe reach is uniquely related to the discharge at the samelocation defining the steady flow rating curve. However,

this situation is altered during unsteady flow, as conceptu-alized in the definition sketch (Figure 1) of the variableparameter Muskingum routing reach of length Dx, in whichthe same unique relationship is maintained between thestage and the corresponding steady discharge at any giveninstant of time, recorded not at the same section, but at adownstream section (section 3 in Figure 1) preceding thecorresponding steady stage section (midsection in Figure 1).

2.2. Theoretical Background

[9] The routing method is derived from the Saint Venantequations, which govern the one-dimensional unsteady flowin channels and rivers without considering lateral flow andare given by

Continuity equation

@Q

@xþ @A

@t¼ 0 ð1Þ

Momentum equation

Sf ¼ So �@y

@x� v

g

@v

@x� 1

g

@v

@tð2Þ

where Q is the discharge; A is the flow area; So is the bedslope; Sf is the energy slope; g is the acceleration due togravity; v is the average velocity over cross section, and y isthe depth of flow. The notations x and t denote the space andtime variables, respectively.[10] The derivation of the method involves some assump-

tions which enable the simplification of the unsteady flowprocess by assuming the channel reach to be prismatic andthe gradients @y/@x, (v/g)(@v/@x) and (1/g)(@v/@t), which,respectively, denote the longitudinal water surface gradient,the convective and the local acceleration gradients, remainapproximately constant at any instant of time in a given

Figure 1. Definition sketch of the stage hydrograph routing method.

W02412 PERUMAL ET AL.: RATING CURVE ESTIMATION AT UNGAUGED RIVER SITES

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reach. The latter assumption implies that the friction slope Sfis constant over the computational reach length at anyinstant of time, and hence the flow depth varies approxi-mately linearly. It has been shown by Perumal and RangaRaju [1998a, 1998b, 1999] that the use of the assumption ofconstant Sf and the Manning’s friction law governing theunsteady flow enable to arrive at the simplified momentumequation expressed as

@Q

@x¼ v

@A

@yþ mP

@R

@y

� �@y

@x

� �ð3Þ

where m is an exponent which depends on the friction lawused (m = 2=3 for Manning’s friction law; m = 1=2 for Chezy’sfriction law); R is the hydraulic radius (A/P); and P is thewetted perimeter.[11] Using equation (3), it may be shown that the celerity

of the flood wave may be expressed as

c ¼ @Q

@A¼ 1þ m

P@R=@y

@A=@y

� �� v ð4Þ

[12] Note that the application of equation (4) for unsteadyflow in rectangular channels yields the same celerity rela-tionship as given in the report of Natural EnvironmentResearch Council [1975].[13] Using equations (1), (2), (3) and the expression of

discharge using Manning’s friction law, Sf can be expressedas

Sf ¼ So 1� 1

So

@y

@x1� mF

P@R=@y

@A=@y

� �� 2( )" #ð5Þ

where F is the Froude number expressed as

F ¼ Q2@A=@y

gA3

� �1=2

ð6Þ

[14] Use of equation (5) in the expression for dischargeQM, at the middle of the computational channel reach, usingthe Manning’s friction law, and its simplification based onbinomial series expansion leads to the simplified expressionfor QM as

QM ¼ Q3 �Q3 1� m2F2

M

P@R=@y

@A=@y

� �2

M

" #

2So@A

@y

3

1þ mP@R=@y

@A=@y

� �� v3

@Q

@x

3

ð7Þ

[15] Equation (7) expresses the discharge QM in terms ofnormal discharge Q3, corresponding to yM, the flow depth atthe middle of the reach. The section where Q3 passescorresponding to yM is located at a distance L downstreamof the midsection and is expressed as

L ¼Q3 1� m2F2

M

P@R=@y

@A=@y

� �2

M

" #

2So@A

@y

3

1þ mP@R=@y

@A=@y

� �� v3

ð8Þ

where the subscripts M and 3 attached with differentvariables denote these variables at the midsection andsection 3, respectively.[16] Use of equations (1), (3) and (4) leads to the

expression [Perumal and Ranga Raju, 1998a,b]

@y

@tþ c

@y

@x¼ 0 ð9Þ

[17] It was pointed out by Perumal and Ranga Raju[1998a, 1998b] that although the form of equation (9) issame as that of the well-known kinematic wave equation[Lighthill and Whitham, 1955], it is capable of approxi-mately modeling a flood wave in the transition rangebetween the zero-inertia wave, governed by the convec-tion-diffusion equation [Hayami, 1951] and the kinematicwave, including the latter. The characteristic of this newwave type governed by equation (9), termed as the approx-imate convection-diffusion (ACD) equation, has beeninvestigated in detail by Perumal and Ranga Raju [1999].[18] Applying equations (4) and (9) at section 3 of

Figure 1 and its simplification leads to [Perumal and RangaRaju, 1998a, 1998b] the governing differential equation ofthe Muskingum type routing, using stage as the operatingvariable in place of discharge, and it is expressed as

yu � yd ¼ Dx

1þ mP@R=@y

@A=@y

� �3

� �v3

� @

@tyd þ

1

2� L

Dx

� �yu � ydð Þ

� �ð10Þ

where yu and yd denote the stages at the upstream anddownstream of the reach. Using the similarity between thegoverning differential equation of the Muskingum methodin discharge formulation and that of equation (10), it isinferred that the traveltime K of the Muskingum type stagerouting method may be expressed as

K ¼ Dx

1þ mP@R=@y

@A=@y

� �3

� �v3

ð11Þ

and the weighting parameter q, after substituting for L fromequation (8), may be obtained as

q ¼ 1

2�

Q3 1� m2F2M

P@R=@y

@A=@y

� �2

M

" #

2So@A

@y

3

1þ mP@R=@y

@A=@y

� �3

� �v3Dx

ð12Þ

[19] The product term (@A/@y)3[1 +m(P@R/@y)/(@A/@y)]3 v3,present in the denominator of equation (12), can be replacedby the simple expression deduced from equation (4) as

@A

@y

3

1þ mP@R=@y

@A=@y

� �3

� �v3 ¼

@Q

@y

3

ð13aÞ

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[20] Expressing in terms of c3, equation (13a) is modifiedas

@Q

@y

3

¼ @A

@y

3

c3 ð13bÞ

[21] Using equation (13a), equation (12) is modified as

q ¼ 1

2�Q3 1� m2F2

M

P@R=@y

@A=@y

� �2

M

" #

2So@Q

@y

3

Dx

ð14aÞ

[22] After neglecting inertial terms of the Saint Venantequations, equation (14a) can be expressed as

q ¼ 1

2� Q3

2So@Q

@y

3

Dx

ð14bÞ

where the subscript 3 attached with these variables denotesthe section 3, wherein the discharge passing is the normaldischarge corresponding to the stage at the middle of theMuskingum reach.[23] Using equations (11) and (14b) in equation (10)

leads to a form similar to that of the Muskingum routingequation, but using stage as the operating variable instead ofdischarge, and it is expressed as

yd; jþ1 ¼ C1yu; jþ1 þ C2yu; j þ C3yd; j ð15Þ

where yu,j+1 and yd,j+1, respectively, denote the upstream anddownstream stages at time (j + 1)Dt; and yu,j and yd,j,respectively, denote the upstream and downstream stages attime jDt, where Dt is the routing time interval, and thecoefficients C1, C2 and C3 are expressed as

C1 ¼�Kqþ 0:5Dt

K 1� qð Þ þ 0:5Dtð16aÞ

C2 ¼Kqþ 0:5Dt

K 1� qð Þ þ 0:5Dtð16bÞ

C3 ¼K 1� qð Þ � 0:5Dt

K 1� qð Þ þ 0:5Dtð16cÞ

[24] The variation of Q with reference to x can be writtenusing equations (3) and (13a) as

@Q

@x

3

¼ @Q

@y

3

@y

@x

3

ð17Þ

[25] Equation (17) can be expressed using forward dif-ference scheme as

Q3 � Qd

Dx=2� L¼ @Q

@y

3

y3 � yd

Dx=2� L

� �ð18Þ

which leads to a generalized relationship between thedischarge Qd, corresponding to an estimated stage yd as

Qd ¼ Q3 þ@Q

@y

3

yd � y3ð Þ ð19Þ

[26] Similarly, using equation (17), the discharge at theupstream gauging site can be expressed using the backwarddifference scheme as

Qu ¼ Q3 þ@Q

@y

3

yu � y3ð Þ ð20Þ

3. Extension of the VPMS Method for Routing ina Two-Stage Compound Cross-Section ChannelReach

3.1. Channel Reach Details

[27] It is assumed that the channel reach is characterizedby a two-stage uniform compound cross section with atrapezoidal main channel flow section and an extendedtrapezoidal floodplain section as shown in Figure 2. It isfurther assumed that the entire channel reach is character-ized by a uniform or representative Manning’s roughnesscoefficient irrespective of main or floodplain channel. Thisassumption may not be strictly valid in practice. However,as the main aim herein is to develop a simplified hydraulicrouting method using stage as the main routing variable,such an assumption helps to reduce complications in thedevelopment of the method.

3.2. Development of Celerity-Stage Relationship

[28] Stage hydrograph routing using the VPMS method,either in a single section (main channel section) or a

Figure 2. Schematic diagram of the cross section of the channel reach adopted in the study (bm =15.0 m, ym = 1.5 m, z1 = z2 = 1, and the maximum flow depth = ymax).


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compound section channel reach, involves the use ofequations (11), (14b), (15), and (16a)–(16c). Estimationof the parameters K and q, given by equations (11) and(14b) at every routing time interval, involves the variablesQ3, c3, and (@Q/@y)3. One of the important parameters inflood routing process is the celerity at which the flood wavetravels along the river reach downstream. The celerity of aflood wave can be estimated as the average traveltime of theflood peaks of the hydrographs recorded at either end of areach [Wong and Laurenson, 1983, 1984]. Celerity,corresponding to any discharge, Q may also be estimatedusing the rating curve at a particular cross section as

c ¼ @Q

@A¼ @Q

@y=@A

@yð21Þ

and also, it can be estimated analytically by equation (4).[29] Note that the wave celerity relationship is not unique

during unsteady flow due to differing relationships estab-lished when the flood is in the rising and falling stages.Following equation (21), it is expected that the celerityversus stage relationship to have a discontinuity at theintersection of the full-bank flood level (corresponding tothe top width of the main channel cross section) and thebottom width of the floodplain cross section, due to suddenincrease of the wetted perimeter.3.2.1. Celerity-Stage Relationship for the MainChannel[30] When the depth of flow in the channel ranges

between zero and ym, the unsteady flow corresponds tothe main channel flow traversing within the simpletrapezoidal section. The celerity of flow at any sectionof the main channel flow reach may be expressed usingequation (4) as

c ¼ 5

3� 2

3

Rmain @Pmain=@yð Þ@Amain=@yð Þ

� �Qmain

Amain

� �ð22Þ

where Qmain is the discharge at the section where celerity iscomputed; and

Rmain ¼Amain

Pmain

ð23aÞ

Amain ¼ bmyþ z1y2 ð23bÞ

Pmain ¼ bm þ 2y

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z21

qð23cÞ

@Amain

@y¼ bm þ 2yz1 ð23dÞ

@Pmain

@y¼ 2


qð23eÞ

[31] The longitudinal gradient of water depth @y/@x maybe estimated using backward difference scheme as

@y

@x¼ yd � yu

Dxð23fÞ

where yu and yd, respectively, correspond to flow depths, atany instant of time, at the upstream and the downstream ofthe conceptual reach length Dx.3.2.2. Celerity-Stage Relationship for the FloodplainChannel[32] When the flow depth exceeds ym, the flow occupies

the floodplain. For the estimation of celerity under thiscondition, the flow is compartmentalized into flow in themain channel and in the two symmetrical compartmentsabove bankfull level as shown in Figure 3. It may be notedthat the main channel flow section above the bankfull levelcorresponds to a rectangular section having the widthcorresponding to the top width of the main trapezoidalchannel section. Accordingly, the compound channel dis-charge encompassing the flow in the floodplain is expressedas

Qcompound ¼ Qmain þ Q1 þ Q2 ð24aÞ

where Qmain is the flow in the main channel, and Q1 and Q2,respectively, are the flow components of the floodplainchannel compartments 1 and 2. Equation (24a) may bewritten in terms of the flow area and velocity of respectivecompartmentalized sections as

Qcompound ¼ Amainvmain þ A1v1 þ A2v2 ð24bÞ

where vmain, v1, and v2 are the velocities in the mainchannel section, and in the floodplain channel compart-ments 1 and 2, respectively.

Figure 3. Compartmentalization of the compound channel section into a main channel section (shaded)and two floodplain channel sections 1 and 2 for computing the celerity, when y > ym.

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[33] Using equation (21), the wave celerity for the com-pound section may be expressed as

ccompound ¼@Qcompound

@Acompound

¼ @Qcompound

@y

� �=

@Acompound

@y

� �ð25Þ

[34] The expression for (@Qcompound)/@y may be estimatedfrom equation (24a) as

@Qcompound

@y¼ @Qmain

@yþ @Q1

@yþ @Q2

@yð26Þ

[35] Each of the derivatives of the right hand side ofequation (26) may be expressed using equation (13a) as

@Qmain

@y¼ 5

3

@Amain

@y� 2

3

Amain

Pmain

@Pmain

@y

� �vmain ð27aÞ

[36] Similarly, the second and the third terms of theequation (26) can be expressed as

@Q1

@y¼ 5

3

@A1

@y� 2

3

A1

P1

@P1

@y

� �v1 ð27bÞ

@Q2

@y¼ 5

3

@A2

@y� 2

3

A2

P2

@P2

@y

� �v2 ð27cÞ

where the velocities of unsteady flow in different compart-ments of the compound channel section can be expressedusing the Manning’s friction law and the momentumequation (2), after neglecting the acceleration terms, as

vmain ¼ vmain;0

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

So

r@y

@xð27dÞ

v1 ¼ v1;0

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

So

r@y

@xð27eÞ

and

v2 ¼ v2;0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

So

@y

@x

sð27f Þ

where

vmain;0 ¼ffiffiffiffiffiSo

p

n

Amain

Pmain

� �2=3

ð27gÞ

v1;0 ¼ffiffiffiffiffiSo

p

n

A1

P1

� �2=3

ð27hÞ

and

v2;0 ¼ffiffiffiffiffiSo

p

n

A2

P2

� �2=3

ð27iÞ

[37] Substituting equations (27a)–(27f) in equation (26)leads to

@Qcompound

@y¼ 5

3

@Amain

@y� 2

3

Amain

Pmain

@Pmain

@y

� �vmain;0


So

@y

@x

s

þ 5

3

@A1

@y� 2

3

A1

P1

@P1

@y

� �v1;0


So

@y

@x

s

þ 5

3

@A2

@y� 2

3

A2

P2

@P2

@y

� �v2;0


So

@y

@x

sð28Þ

[38] Substituting equation (28) in equation (25), thecelerity of the flow when it exceeds the main channelsection can be modified as

ccompound ¼ 5

3

@Amain

@y

� �vmain;0


So

@y

@x

s" #=

@Acompound

@y

� �

þ 5

3

@A1

@y� 2

3

A1

P1

@P1

@y

� �v1;0


So

@y

@x

s" #=

@Acompound

@y

� �

þ 5

3

@A2

@y� 2

3

A2

P2

@P2

@y

� �v2;0


So

@y

@x

s" #=

@Acompound

@y

� �

ð29Þ

where

Acompound ¼ bm þ ymz1ð Þym þ bf þ z2 y� ymð Þ� �

y� ymð Þ ð30aÞ

@Acompound

@y¼ bf þ 2 y� ymð Þz2 ð30bÞ

Amain ¼ bm þ ymz1ð Þym þ bm þ 2ymz1ð Þ y� ymð Þ ð30cÞ

Pmain ¼ bm þ 2ym


qð30dÞ

@Pmain

@y¼ 0 ð30eÞ

A1 ¼ A2 ¼ 0:5 bf � bm � 2ymz1� �

þ z2 y� ymð Þ� �

y� ymð Þ ð30fÞ

P1 ¼ P2 ¼ 0:5 bf � bm � 2ymz1� �

þ y� ymð Þffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z22

qð30gÞ

@A1

@y¼ @A2

@y¼ 0:5 bf � bm � 2ymz1

� �þ z2 y� ymð Þ ð30hÞ

@P1

@y¼ @P2

@y¼


qð30iÞ

3.3. Routing Procedure

[39] The following procedure is adopted while routingusing the VPMS routing method.


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[40] The stage at the outlet of the reach is estimated usingthe recursive equation (15). The parameters K and q vary atevery routing time interval in a two-step process: In the firststep, by estimating the unrefined stage estimate yd,j+1forthe current routing time interval using the values of K and q,estimated at the previous time step, and then, using thisestimate and yu,j+1, the flow depth at the middle of the reachis estimated as

yM ¼ yu; jþ1 þ yd; jþ1

� �=2 ð31Þ

[41] The initial values of K and q are estimated using theinitial steady flow in the reach.[42] Similarly, the depth of flow at section 3 is computed

as

y3 ¼ qyu; jþ1 þ 1� qð Þyd; jþ1 ð32Þ

[43] Using yM, given by equation (31), the discharge atsection 3 is computed in the following manner dependingon whether yM is within the main channel section or in thecompound channel section:

Q3 ¼AM

n

AM

PM

� �2=3

S1=20 ; when yM � ym ð33aÞ

Q3 ¼ Amain vmain½ Mþ A1v1½ Mþ A2v2½ M ; when yM > ym

ð33bÞ

where Amain,M, and A1,M and A2,M are evaluated at themidsection of the reach using equations (30c) and (30f),respectively, and

vmain;M ¼ffiffiffiffiffiSo

p

nR2=3main;M ð34aÞ

v1;M ¼ffiffiffiffiffiSo

p

nR2=31;M ð34bÞ

v2;M ¼ffiffiffiffiffiSo

p

nR2=32;M ð34cÞ

When y3 � ym , the wave celerity c3 is estimated by usingequation (22) corresponding to y3; and when y3 > ym,the wave celerity c3 corresponding to flow depth y3 ofthe compound channel section is estimated by usingequation (29).[44] Corresponding to the estimated values of yM and y3,

obtained using equations (31) and (32), respectively, thevalue of (@Q/@y)3 is estimated using equations (13b),(22) and (23a)–(23f) or (28) and (30a)–(30i) depending onwhether yM is within the main channel section or extendsinto the floodplain channel section, respectively. In thesecond step, using these values of Q3, c3 and (@Q/@y)3, therefined values of K and q are estimated using equations (11)and (14b), respectively, for the current routing time in-

terval, which are, then, used to estimate the refined stage–hydrograph estimate using equations (15) and (16a)–(16c).

3.4. Synthesizing the Rating Curve at the Upstreamand Downstream Sites of the Reach

[45] One of the primary objectives of the study is todevelop the rating curve at an ungauged site. To achieve thisin practice, it is presumed that at least for one flood event,stage hydrograph record is available at the downstream siteof the reach with the concurrent stage record available at theupstream river site, to enable the calibration of the reachaveraged Manning’s roughness coefficient by matching therouted stage hydrograph arrived at by the VPMS methodwith the corresponding stage hydrograph observed at thatdownstream site. Though the VPMS method is capable ofestimating the discharge hydrograph using equation (19) atthe downstream site corresponding to the routed stagehydrograph, the resulting rating curve obtained using therouted stage hydrograph and the estimated discharge hydro-graph is in the form of a loop rating curve which will varyfor every flood event. In the conventional sense, the ratingcurve at a site refers to the steady stage-discharge relation-ship. To arrive at the conventional rating curve at theungauged downstream river site, the concept used in thedevelopment of the VPMS method is convenientlyexploited by linking the stage simulated at the middle ofthe reach, which herein corresponds to the ungauged site,with the discharge estimated at section 3 of the last routingsubreach. The relationship formulated by this stage hydro-graph and the corresponding Q3 values form the requiredconventional stage-discharge relationship at the ungaugeddownstream site.[46] However, in order to synthesize the rating curve at

the ungauged upstream site, the same procedure adopted forthe ungauged downstream site is not applicable fromtheoretical considerations. This is due to the reason thatthe upstream ungauged site cannot be considered as themidsection of a routing subreach. Therefore the rating curvesynthesized for the upstream site by establishing the rela-tionship between Q3, estimated during the first subreachrouting, and the corresponding recorded stage at the upstreamsite forms an approximate rating curve. However, by reduc-ing the routing subreach length Dx, one can reduce thisapproximation considerably to closely reproduce the actualrating curve pertaining to that of the upstream site. Thisstrategy is followed in this study to synthesize the ratingcurve at the upstream ungauged site.

4. Application

4.1. Test Runs Details

[47] The methodology described above was applied forrouting stage hydrograph in the considered compoundchannel section reach characterized by a constantManning’s roughness coefficient. One of the best ways toverify such newly developed methodologies is by simulat-ing the hydrographs obtained using the routing of hypo-thetical input hydrographs by some well-known streamhydrograph routing packages such as the MIKE 11 model[DHI, 2003]. The MIKE 11 model has an option forexclusive stream routing only, by which any stage hydro-graph can be routed in a user defined channel reach for anyreach length. In the context of the present study, a hypo-

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thetical input stage hydrograph of the form of Pearsontype III distribution was used for routing by the VPMSmethod. The routed hydrograph at the outlet of the con-sidered channel reach using MIKE 11 may be consideredas the benchmark stage hydrograph, and it is this hydro-graph, which is required to be simulated by the proposedVPMS method.[48] Different sizes of compound channels defined by

bf /bm ratios of 1.5, 2.0, 2.5, 3.0 3.5, and 4.0, with com-pound section configuration as shown in Figure 2, wereused in this numerical experiments study for testing theproposed VPMS method. Corresponding to each bf /bmratio, twelve channel configurations defined by two differ-ent Manning’s roughness values (n = 0.04 and 0.025), andsix varying channel bed slopes (So = 0.003, 0.002, 0.001,0.0005, 0.0003, and 0.0002) were employed. In all, a totalof 72 channel types as given in Tables 1 and 2 were usedin this study to test the routing potential of the proposedmethod. In all these channel reaches, routing of stagehydrographs was carried out for a reach length of 40 km.For each of the 72 channel types studied, two test runswere made corresponding to routing by the VPMS methodin the 40 km channel reach, considering it as eight andforty subreaches. A uniform routing time interval of30 minutes was used for all the test runs, except in the caseof routing in channel reaches with bed slopes of 0.002 and0.003, wherein a routing time interval of 5 minutes wasused. Use of a larger time interval in such cases resultedin wiggles while simulating the recession of the stagehydrograph at 40 km.[49] To enable the development of rating curve at the

section corresponding to 40 km, the given 40 km uniformrouting reach is extended to 41 km, which is, then, dividedinto 39 subreaches of one kilometer each for the upstreamportion of the reach, and the 40th subreach having a lengthof 2 km with the middle of this subreach coinciding with theungauged site located at 40 km from the upstream site.Routing of the given upstream stage hydrograph in thegiven 40 km channel reach with such reach subdivisionarrangement enables us to arrive at the conventional form ofrating curve at the ungauged site.[50] To enable the development of rating curve at the

upstream section of the 40 km uniform routing reach, thestrategy outlined in section 3.4 is followed, by whichthe Q3, estimated while routing the upstream stage hydro-graph in the first subreach is approximately considered asthe normal discharge corresponding to the recorded stage atthe upstream of the routing reach. It is assumed that thesubreach length of 1 km is small enough that the ratingcurve established by the above strategy closely representsthe actual rating curve at the upstream ungauged site.

4.2. Input Stage Hydrograph Used by the MIKE 11Model

[51] The following is the form of the input stage hydro-graph that was used in the hypothetical routing study:

y tð Þ ¼ yb þ yp � yb� � t

tp

� �1= g�1ð Þexp

1� t=tpg � 1

� �ð35Þ

where yb = flow depth in the channel (m) corresponding toinitial steady flow (considered in this study as 10.0 m3/s for

all test runs); yp = peak stage (4.0 m); tp = time to peak(10 h); and g = shape factor (1.15). A number of differenthypothetical input stage hydrographs have been generatedusing equation (35) corresponding to various channel typereaches used. No lateral flow is considered in this study.[52] The following upstream, downstream and initial

conditions were used in each of the hypothetical channelrouting cases using the MIKE 11 model: The upstream stagehydrograph defined by equation (35) formed the upstreamboundary condition; the normal rating curve estimated foreach of the considered compound section channel reachformed the downstream boundary condition. The down-stream boundary point was considered at a distance of100 km from the upstream boundary point, so that it maynot affect the routing results in the 40 km reach immediatelybelow the upstream boundary point. A steady flow of 10 m3/swas used as the initial flow in the channel reach for all thetest runs.

5. Performance Evaluation Criteria

[53] The following evaluation criteria were adopted forverifying the efficacy of the proposed method in simulatingthe corresponding benchmark solution.

5.1. Variance Explained Criterion

[54] The closeness with which the proposed methodreproduces the benchmark solution, including the closenessof shape and size of the hydrograph, can be measured usingthe Nash-Sutcliffe criterion of variance explained [Nash andSutcliffe, 1970]. The variance explained in percentage forstage and discharge hydrographs reproductions are given,respectively, as

hy ¼ 1�XNi¼1

yoi � ycið Þ2=XNi¼1

yoi � �yoið Þ2" #

� 100 ð36Þ

hq ¼ 1�XNi¼1

Qoi � Qcið Þ2=XNi¼1

Qoi � Qoi

� �2" #� 100 ð37Þ

where yoi is the ith ordinate of the benchmark stagehydrograph at the outlet; �yoi is the mean of the benchmarkstage hydrograph ordinates at the outlet; yci is the ithordinate of the computed stage hydrograph using theproposed VPMS method; Qoi is the ith ordinate of thebenchmark discharge hydrograph at the outlet; Qoi isthe mean of the benchmark discharge hydrograph ordinatesat the outlet; Qci is the ith ordinate of the dischargehydrograph corresponding to yci, computed using theproposed VPMS method; and N is the total number ofstage or discharge hydrograph ordinates to be simulated.

5.2. Magnitude of Peak Consideration

[55] Relative error in peak of the routed stage hydrographis given as

yper ¼ ypc=ypo � 1� �

� 100 ð38Þ

where ypc is the peak of the routed stage hydrograph at theoutlet; and ypo is the peak of the routed benchmark stagehydrograph at the outlet. A positive value of yper indicates


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Table 1. Summary of Performance Criteria Showing Reproduction of Pertinent Characteristics of Routed Stage Hydrographs by the

VPMS Method Considering MIKE 11 as the Benchmark Model (for Eight Subreaches Consideration)

Channel Type bf /bm n So my,a % mq,

a %

Discharge Computation Stage Routing

tpqer,h qper,% EVOL,% hq,% tpyer,h yper,% hy,%

1 1.5 0.04 0.003 0.75 1.53 �0.08 �1.10 �1.86 99.92 0.17 0.30 99.822 1.5 0.04 0.002 1.00 1.96 �0.08 �1.50 �2.26 99.84 0.17 0.12 99.653 1.5 0.04 0.001 2.75 5.20 0.00 �1.89 �1.32 99.78 0.00 �0.04 99.814 1.5 0.04 0.0005 9.00 15.71 0.50 �5.11 �4.33 98.02 1.00 �1.14 99.295 1.5 0.04 0.0003 16.50 29.29 1.50 �9.92 �6.72 95.15 1.00 �3.68 97.476 1.5 0.04 0.0002 22.50 39.83 2.50 �16.57 �8.93 89.27 2.50 �6.73 93.657 1.5 0.025 0.003 0.25 0.47 0.00 �1.62 �1.71 99.95 0.17 0.05 99.968 1.5 0.025 0.002 0.25 0.66 0.00 �1.80 �1.84 99.94 0.17 �0.14 99.949 1.5 0.025 0.001 1.25 2.23 0.50 �1.94 �0.63 99.93 0.50 0.08 99.8610 1.5 0.025 0.0005 4.00 7.45 0.00 �2.66 �2.51 99.48 0.00 �0.47 99.8611 1.5 0.025 0.0003 9.50 16.88 0.50 �5.31 �4.69 98.15 0.50 �1.65 99.1612 1.5 0.025 0.0002 15.00 26.88 1.50 �12.62 �8.00 94.20 1.00 �5.10 96.9413 2.0 0.04 0.003 0.75 1.74 0.00 �1.15 �2.13 99.82 0.00 0.29 99.5714 2.0 0.04 0.002 1.00 2.15 0.00 �1.61 �2.41 99.77 0.08 0.10 99.4615 2.0 0.04 0.001 3.00 5.64 0.00 �2.36 �1.45 99.60 0.50 0.12 99.5516 2.0 0.04 0.0005 9.50 17.55 1.00 �5.74 �4.50 97.60 0.50 �1.24 99.1817 2.0 0.04 0.0003 17.25 31.92 1.50 �12.22 �8.11 92.37 2.00 �4.11 96.1218 2.0 0.04 0.0002 23.50 43.50 2.50 �18.25 �10.69 85.85 2.50 �6.97 91.7519 2.0 0.025 0.003 0.25 0.56 0.08 �1.73 �1.88 99.91 0.17 0.06 99.8720 2.0 0.025 0.002 0.25 0.75 0.00 �1.94 �1.99 99.90 0.08 �0.15 99.8521 2.0 0.025 0.001 1.25 2.52 0.00 �1.79 �0.58 99.90 0.00 0.14 99.8322 2.0 0.025 0.0005 4.50 8.00 0.00 �3.30 �2.83 99.29 0.50 0.04 99.8023 2.0 0.025 0.0003 9.75 18.25 1.00 �7.21 �5.75 96.93 1.00 �2.05 98.7224 2.0 0.025 0.0002 15.75 29.42 1.50 �13.52 �8.20 93.58 1.00 �4.98 97.0325 2.5 0.04 0.003 0.75 1.85 0.00 �0.99 �2.21 99.73 0.00 0.29 99.2826 2.5 0.04 0.002 1.00 2.28 0.00 �1.47 �2.52 99.66 0.17 0.10 99.0927 2.5 0.04 0.001 3.00 6.08 0.50 �2.07 �0.88 99.64 0.00 0.10 99.2828 2.5 0.04 0.0005 9.75 18.81 0.50 �6.13 �4.64 97.07 0.50 �1.35 98.8029 2.5 0.04 0.0003 17.75 34.05 1.50 �12.94 �8.80 90.61 1.50 �4.24 95.0530 2.5 0.04 0.0002 24.25 46.12 2.50 �19.19 �12.28 83.44 3.00 �6.82 90.5731 2.5 0.025 0.003 0.25 0.62 0.08 �1.59 �1.88 99.87 0.17 0.06 99.7732 2.5 0.025 0.002 0.50 0.79 0.00 �1.83 �1.94 99.86 0.25 0.11 99.7233 2.5 0.025 0.001 1.25 2.52 0.00 �1.69 0.09 99.80 0.00 0.18 99.6234 2.5 0.025 0.0005 4.50 8.46 0.50 �3.06 �2.42 99.33 0.00 0.04 99.7635 2.5 0.025 0.0003 10.25 19.51 0.50 �6.92 �5.27 97.04 1.00 �1.57 98.8436 2.5 0.025 0.0002 16.25 31.34 1.00 �14.28 �8.87 92.04 1.50 �4.94 96.3237 3.0 0.04 0.003 0.75 1.90 0.00 �0.81 �2.13 99.65 0.00 0.29 98.9438 3.0 0.04 0.002 1.00 2.39 0.08 �1.28 �2.58 99.54 0.08 0.11 98.6639 3.0 0.04 0.001 3.00 6.56 0.50 �1.58 �0.50 99.55 0.00 0.10 98.7840 3.0 0.04 0.0005 10.25 19.93 1.00 �6.16 �4.41 96.95 0.50 �1.10 98.4441 3.0 0.04 0.0003 18.25 35.78 1.50 �12.80 �8.40 91.14 1.50 �3.84 95.5742 3.0 0.04 0.0002 24.75 48.10 3.00 �19.81 �13.90 81.40 3.00 �6.73 89.5743 3.0 0.025 0.003 0.25 0.65 0.08 �1.42 �1.73 99.82 0.17 0.07 99.6444 3.0 0.025 0.002 0.50 0.83 0.00 �1.67 �2.00 99.79 0.25 0.11 99.4845 3.0 0.025 0.001 1.00 2.53 0.00 �1.54 0.29 99.63 0.00 �0.05 99.3646 3.0 0.025 0.0005 4.50 8.88 0.50 �2.72 �2.14 99.34 0.00 0.02 99.6947 3.0 0.025 0.0003 10.25 20.27 0.50 �7.23 �5.72 96.94 0.50 �1.75 99.0048 3.0 0.025 0.0002 16.50 32.62 1.50 �14.69 �9.69 91.07 1.50 �4.97 95.9649 3.5 0.04 0.003 0.75 1.95 0.00 �0.65 �2.21 99.60 0.00 0.30 98.7250 3.5 0.04 0.002 1.00 2.48 0.00 �1.11 �2.66 99.47 0.00 0.10 98.2851 3.5 0.04 0.001 3.25 6.83 0.00 �1.34 �1.11 99.28 0.00 0.32 98.2552 3.5 0.04 0.0005 10.50 21.01 0.50 �6.19 �5.35 96.39 1.00 �1.28 98.3353 3.5 0.04 0.0003 18.50 37.04 1.50 �13.73 �9.31 89.26 1.50 �4.12 94.1154 3.5 0.04 0.0002 25.00 49.65 3.00 �20.56 �15.58 79.21 2.50 �6.90 88.4955 3.5 0.025 0.003 0.25 0.67 0.00 �1.27 �1.80 99.77 0.17 0.07 99.4556 3.5 0.025 0.002 0.50 0.86 0.00 �1.52 �1.92 99.77 0.17 0.12 99.4157 3.5 0.025 0.001 1.00 2.52 0.00 �1.41 0.81 99.56 0.00 �0.02 99.0358 3.5 0.025 0.0005 4.50 9.26 0.50 �2.36 �1.47 99.37 0.00 0.02 99.4559 3.5 0.025 0.0003 10.50 20.98 1.00 �7.54 �6.18 96.13 1.00 �1.82 98.5660 3.5 0.025 0.0002 16.75 33.79 1.50 �15.16 �9.96 90.30 1.50 �4.92 95.6461 4.0 0.04 0.003 0.75 1.99 0.00 �0.50 �2.18 99.59 0.00 0.30 98.3262 4.0 0.04 0.002 1.00 2.54 0.00 �0.97 �1.99 99.65 0.00 0.10 98.4963 4.0 0.04 0.001 3.25 6.93 0.00 �1.18 0.40 99.44 0.00 0.35 98.1464 4.0 0.04 0.0005 10.50 21.57 0.50 �6.80 �4.97 96.03 0.50 �1.51 98.4565 4.0 0.04 0.0003 18.75 38.13 2.00 �13.89 �9.25 88.62 1.50 �4.12 94.3066 4.0 0.04 0.0002 25.25 50.83 3.00 �21.29 �16.78 77.81 3.00 �6.95 87.6767 4.0 0.025 0.003 0.25 0.68 0.00 �1.14 �1.25 99.85 0.17 0.07 99.5268 4.0 0.025 0.002 0.50 0.88 0.00 �1.38 �1.51 99.84 0.17 0.12 99.4369 4.0 0.025 0.001 1.00 2.51 0.00 �1.30 1.72 99.34 0.00 0.00 98.7670 4.0 0.025 0.0005 4.50 9.56 0.50 �2.13 �1.08 99.38 0.00 �0.02 99.67

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overestimation of the peak from the benchmark value, and anegative value of yper indicates underestimation of the peakfrom the benchmark value.[56] Similarly, the relative error in peak of the computed

discharge hydrograph is given as

qper ¼ qpc=qpo � 1� �

� 100 ð39Þ

where qpc is the peak of the computed discharge hydrographat the outlet; and qpo is the peak of the benchmark dischargehydrograph at the outlet. A positive value of qper indicatesoverestimation of the peak from the benchmark value, and anegative value of qper indicates underestimation of the peakfrom the benchmark value.

5.3. Time of Peak Consideration

[57] Error in time to peak stage (hours) is given as

tpyer ¼ typc � typo ð40Þ

where typc is the time corresponding to routed peak stage atthe outlet; and typo is the time corresponding to peak stage ofthe benchmark hydrograph at the outlet.[58] Similarly, error in time to computed peak discharge

(hours) is given as

tqyer ¼ tqpc � tqpo ð41Þ

where tqpc is the time corresponding to computed peakdischarge at the outlet; and tqpo is the time corresponding topeak discharge of the benchmark hydrograph estimated atthe outlet.

5.4. Conservation of Mass Consideration

[59] The relative error in the flow volume is expressed as

EVOL in %ð Þ ¼XNi¼1

Qci=XNi¼1

Ii � 1

" #� 100 ð42Þ

where Qci is the ith ordinate of the computed dischargehydrograph; and Ii is the ith ordinate of the inflow dischargehydrograph. A negative value of EVOL indicates loss ofmass and positive value of EVOL indicates gain of mass. Avalue close to zero suggests mass conservation ability of themethod.

6. Results and Discussion

[60] Tables 1 and 2 present the results corresponding toall hypothetical routing solutions obtained by considering8 and 40 equal subreaches of the 40 km channel reach,respectively. Tables 1 and 2 show the details of channel

types, routing results related to the reproduction of pertinentcharacteristics of MIKE 11 solutions such as estimates ofvariances explained, hy in percent, and hq in percent,respectively, in reproducing the stage and discharge hydro-graphs; error in volume conservation estimated in reproduc-ing the discharge hydrograph (EVOL in percent), relativeerrors in estimating peak discharge (qper in percent) and itstime to peak (tpqer in h), and the peak stage (yper in percent)and its time to peak (tpyer in h). Figures 4a, 4c, 4e, 4g, 5a,5c, 5e, and 5g show some of the typical routing resultsobtained using the VPMS method at the outlet of the 40 kmreach and their comparison with the corresponding MIKE11 solutions for the channel reach sizes of bf/bm = 2.0 and4.0, respectively. The input stage hydrographs are alsoshown in these figures. Figures 4b, 4d, 4f, and 4h, 5b, 5d,5f, and 5h show the respective computed discharge hydro-graphs at the upstream and downstream of the 40 kmrouting reach corresponding to Figures 4a, 4c, 4e, 4g, 5a,5c, 5e, and 5g, respectively. The discharge hydrographs atthe downstream and upstream of the reach are estimatedusing equations (19) and (20), respectively, with the use ofequation (13b). The MIKE 11 inflow and outflow hydro-graphs are also shown in these figures only for comparisonwith these estimated discharge hydrographs.

6.1. On the Capability of the VPMS Routing Method inReproducing Benchmark Solutions

6.1.1. Stage Hydrographs[61] The stage hydrograph routing results given in Tables 1

and 2 and some of these typical results shown in theFigures 4a, 4c, 4e, 4g, 5a, 5c, 5e, and 5g demonstrate thatthe proposed VPMS routing method is able to reproduce theMIKE 11 solutions closely for channel types having lowerbf/bm values. When the size of floodplain channel sectionincreases from 1.5 to 4.0, the ability of the VPMS method tosimulate stage hydrograph decreases slightly, but not to theextent of rendering the results unacceptable for practicalpurposes. Similarly, it is also found that the varianceexplained estimate decreases slightly when the bed slopedecreases or when the roughness increases or for both. It isexperienced from the test run results corresponding torouting in steep slope channels with the consideration ofeight subreaches that there is a tendency for the formation ofwiggles in the computed hydrograph, whenever the hydro-graph suddenly rises or falls. Some of these typical resultsare shown in Figures 4a, 5a, and 5e of the routed stagehydrographs and also seen in the respective computeddischarge hydrographs as seen from the Figures 4b, 5b,and 5f. This characteristic of the Muskingum routingsolution is well documented in literature by the terms‘‘initial dip’’ or ‘‘negative initial flow’’ [Weinmann andLaurenson, 1979]. The cause for the formation of wiggleshas been brought out by Perumal [1992] and the size of the

Table 1. (continued)

Channel Type bf / bm n So my,a % mq,

a %


tpqer,h qper,% EVOL,% hq,% tpyer,h yper,% hy,%

71 4.0 0.025 0.0003 10.75 21.55 1.00 �7.32 �5.53 96.38 1.00 �1.52 98.9272 4.0 0.025 0.0002 17.00 34.65 1.50 �15.64 �10.44 88.73 1.50 �5.02 95.10

aParameters my and mq are the attenuations in peak stage and discharge by the MIKE 11 model, respectively.


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Table 2. As in Table 1, but for 40 Subreaches Consideration

Channel Type bf/bm n So my,a % mq ,

a %


tpqer,h qper,% EVOL,% hq, % tpyer, h yper, % hy, %

1 1.5 0.04 0.003 0.75 1.53 �0.08 �0.98 �1.37 99.92 0.17 0.36 99.782 1.5 0.04 0.002 1.00 1.96 �0.08 �1.28 �1.41 99.83 0.17 0.21 99.603 1.5 0.04 0.001 2.75 5.20 0.00 �1.07 1.05 99.83 0.00 0.32 99.574 1.5 0.04 0.0005 9.00 15.71 0.00 �1.20 0.30 99.75 0.50 0.70 99.795 1.5 0.04 0.0003 16.50 29.29 1.00 �2.50 �0.07 99.06 0.50 �0.24 99.546 1.5 0.04 0.0002 22.50 39.83 1.50 �7.53 �1.10 96.45 2.00 �2.25 97.917 1.5 0.025 0.003 0.25 0.47 0.00 �1.60 �1.50 99.95 0.17 0.06 99.968 1.5 0.025 0.002 0.25 0.66 0.00 �1.74 �1.47 99.94 0.17 �0.12 99.949 1.5 0.025 0.001 1.25 2.23 0.00 �1.76 0.51 99.85 0.50 0.13 99.7110 1.5 0.025 0.0005 4.00 7.45 0.00 �1.51 0.48 99.90 0.00 0.07 99.7611 1.5 0.025 0.0003 9.50 16.88 0.00 �1.38 0.49 99.78 0.00 0.09 99.7912 1.5 0.025 0.0002 15.00 26.88 1.00 �5.42 �0.61 98.63 0.50 �1.54 99.2513 2.0 0.04 0.003 0.75 1.74 0.00 �0.99 �1.30 99.87 0.08 0.35 99.7414 2.0 0.04 0.002 1.00 2.15 0.00 �1.35 �1.27 99.78 0.08 0.21 99.5615 2.0 0.04 0.001 3.00 5.64 0.00 �1.07 1.92 99.61 0.50 0.47 99.2816 2.0 0.04 0.0005 9.50 17.55 0.50 �0.56 0.87 99.73 0.50 0.95 99.7117 2.0 0.04 0.0003 17.25 31.92 1.00 �3.07 �0.14 98.45 1.00 �0.06 99.1618 2.0 0.04 0.0002 23.50 43.50 1.50 �7.44 �1.64 95.50 2.00 �1.97 97.5219 2.0 0.025 0.003 0.25 0.56 0.08 �1.72 �1.48 99.99 0.17 0.06 99.9020 2.0 0.025 0.002 0.25 0.75 0.00 �1.87 �1.51 99.90 0.08 �0.12 99.8921 2.0 0.025 0.001 1.25 2.52 0.00 �1.65 1.05 99.69 0.00 0.18 99.5622 2.0 0.025 0.0005 4.50 8.00 0.00 �1.56 0.93 99.87 0.50 0.53 99.7323 2.0 0.025 0.0003 9.75 18.25 0.50 �2.13 0.37 99.61 0.50 0.14 99.7124 2.0 0.025 0.0002 15.75 29.42 1.00 �5.33 �0.38 98.26 0.50 �1.27 99.0725 2.5 0.04 0.003 0.75 1.85 0.00 �0.81 �1.05 99.78 0.00 0.36 99.6226 2.5 0.04 0.002 1.00 2.28 0.00 �1.15 �1.13 99.71 0.08 0.22 99.5327 2.5 0.04 0.001 3.00 6.08 0.00 �0.80 2.81 98.98 0.00 0.53 98.6828 2.5 0.04 0.0005 9.75 18.81 0.00 0.24 1.87 99.72 0.00 1.28 99.3929 2.5 0.04 0.0003 17.75 34.05 1.00 �2.19 0.32 98.37 0.50 0.32 98.9630 2.5 0.04 0.0002 24.25 46.12 2.00 �6.68 �2.21 94.96 2.00 �1.49 97.3931 2.5 0.025 0.003 0.25 0.62 0.08 �1.57 �1.42 99.87 0.17 0.06 99.8632 2.5 0.025 0.002 0.50 0.79 0.00 �1.75 �1.33 99.86 0.33 0.13 99.8433 2.5 0.025 0.001 1.25 2.52 0.00 �1.50 1.72 99.26 0.00 0.25 99.2034 2.5 0.025 0.0005 4.50 8.46 0.00 �1.21 1.71 99.72 0.00 0.66 99.5035 2.5 0.025 0.0003 10.25 19.51 0.00 �1.38 0.85 99.62 0.50 0.66 99.6136 2.5 0.025 0.0002 16.25 31.34 0.50 �4.51 0.02 98.06 1.00 �0.82 98.8337 3.0 0.04 0.003 0.75 1.90 0.00 �0.62 �0.51 99.67 0.00 0.37 99.4438 3.0 0.04 0.002 1.00 2.39 0.00 �0.92 �0.74 99.61 0.08 0.23 99.4039 3.0 0.04 0.001 3.00 6.56 0.00 �0.30 3.44 98.23 0.00 0.61 98.1340 3.0 0.04 0.0005 10.25 19.93 0.50 1.01 2.61 99.62 0.00 1.83 98.9841 3.0 0.04 0.0003 18.25 35.78 0.50 �1.08 0.90 98.34 1.00 0.78 98.6042 3.0 0.04 0.0002 24.75 48.10 2.00 �5.89 �2.86 94.60 2.00 �1.08 97.3743 3.0 0.025 0.003 0.25 0.65 0.08 �1.41 �1.39 99.81 0.17 0.06 99.8044 3.0 0.025 0.002 0.50 0.83 0.00 �1.59 �1.13 99.81 0.33 0.14 99.7845 3.0 0.025 0.001 1.00 2.53 0.00 �1.25 2.54 98.71 0.00 0.10 98.7746 3.0 0.025 0.0005 4.50 8.88 0.00 �0.75 2.52 99.45 0.00 0.76 99.1347 3.0 0.025 0.0003 10.25 20.27 0.00 �0.96 1.45 99.64 0.00 0.65 99.4448 3.0 0.025 0.0002 16.50 32.62 1.00 �4.12 0.41 97.95 0.50 �0.65 98.6149 3.5 0.04 0.003 0.75 1.95 0.00 �0.44 0.03 99.47 0.00 0.38 99.0750 3.5 0.04 0.002 1.00 2.48 �0.08 �0.72 �0.41 99.50 0.00 0.25 99.2851 3.5 0.04 0.001 3.25 6.83 0.00 0.14 4.05 96.98 0.00 0.96 97.3252 3.5 0.04 0.0005 10.50 21.01 0.00 2.26 3.51 99.35 0.00 2.12 98.4153 3.5 0.04 0.0003 18.50 37.04 1.00 �0.46 1.39 98.35 0.50 1.01 98.1754 3.5 0.04 0.0002 25.00 49.65 1.50 �5.26 �3.52 94.29 1.50 �0.90 97.3855 3.5 0.025 0.003 0.25 0.67 0.00 �1.26 �1.46 99.74 0.17 0.07 99.7256 3.5 0.025 0.002 0.50 0.86 0.00 �1.43 �0.88 99.74 0.25 0.15 99.7257 3.5 0.025 0.001 1.00 2.52 0.00 �0.99 2.93 98.31 0.00 0.23 98.5158 3.5 0.025 0.0005 4.50 9.26 0.00 �0.27 2.95 98.95 0.00 0.85 98.8759 3.5 0.025 0.0003 10.50 20.98 0.50 �0.38 1.96 99.64 0.50 0.99 99.2860 3.5 0.025 0.0002 16.75 33.79 1.00 �3.32 0.80 97.84 1.00 �0.36 98.2461 4.0 0.04 0.003 0.75 1.99 0.00 �0.28 0.52 99.50 0.00 0.38 99.4462 4.0 0.04 0.002 1.00 2.54 �0.08 �0.53 0.35 99.53 0.00 0.26 99.4863 4.0 0.04 0.001 3.25 6.93 0.00 0.40 5.32 95.44 0.00 1.00 95.8964 4.0 0.04 0.0005 10.50 21.57 0.00 2.85 4.59 99.11 0.00 2.12 98.5565 4.0 0.04 0.0003 18.75 38.13 1.00 0.48 2.14 98.46 0.50 1.23 98.8466 4.0 0.04 0.0002 25.25 50.83 1.50 �4.65 �3.81 93.75 2.00 �0.67 97.1567 4.0 0.025 0.003 0.25 0.68 0.00 �1.12 �1.17 99.67 0.17 0.07 99.5468 4.0 0.025 0.002 0.50 0.88 0.00 �1.29 �0.28 99.74 0.25 0.15 99.7869 4.0 0.025 0.001 1.00 2.51 0.00 �0.67 3.66 97.31 0.00 0.40 97.6770 4.0 0.025 0.0005 4.50 9.56 0.00 0.14 3.88 98.60 0.00 0.92 98.70

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Channel Type bf/bm n So my,a % mq ,

a %


tpqer,h qper,% EVOL,% hq, % tpyer, h yper, % hy, %

71 4.0 0.025 0.0003 10.75 21.55 0.50 0.26 2.91 99.63 0.50 1.36 99.4372 4.0 0.025 0.0002 17.00 34.65 0.50 �2.73 1.51 97.79 1.00 �0.02 98.56

aParameters my and mq are the attenuations in peak stage and discharge by the MIKE 11 model, respectively.

Table 2. (continued)

Figure 4. Typical routed stage and computed discharge hydrographs for the case of bf/bm = 2.0 forchannel types (a and b) 14, (c and d) 18, (e and f) 20, and (g and h) 24. VPMS:8 and VPMS:40 refer tothe outputs of the VPMS model corresponding to 8 and 40 subreaches, respectively.


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wiggle depends directly on the steepness of the bed slope,steepness of the rise or fall of the input stage hydrograph,length of the routing subreach and the magnitude of theroughness coefficient. In essence, when the input stagehydrograph experiences a large variation in the magnitudeof (1/So)(@y/@x), while routing in long subreach steep slopechannels, the tendency for the formation of wiggle is strong.The effect of the wiggle on the hydrograph can be mini-mized by increasing the number of subreaches used forrouting as in the case of routing with 40 equal subreaches.However, it may be noted that the effect of wiggle impactsthe solution results only locally and does not affect the

entire solution, such as the peak and time to peak character-istics. Further, the size of the wiggle, as experienced in thestage hydrograph routing solution, is not reflected in theestimated discharge hydrograph. Therefore the overall per-formance of the VPMS method may be considered satis-factory considering the fact that even when the input stagehydrograph has completely entered into the channel reach,typically as seen from Figures 4c and 5c, with no significantresponse at the outlet of the reach, the VPMS method couldclosely reproduce the MIKE 11 solutions with hy greaterthan 95%. In fact, routing using the VPMS method consid-ering 40 subreaches for each of the channel types resulted in

Figure 5. As in Figure 4, but for the case of bf/bm = 4.0 for channel types (a and b) 62, (c and d) 66,(e and f) 68, and (g and h) 72.

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hy being greater than 95%. Only in a very few cases ofchannel routing considering 8 subreaches, the hy valueswere estimated less than 95%, but greater than 87%. Thisdemonstrates the insufficiency in the number of subreachesused, rather than the inability of the method to route thestage hydrograph accurately. The MIKE 11 solutions stud-ied herein were subjected to attenuation to the extent of 25%and 50%, respectively, for the stage and discharge hydro-graph reproductions, and they could be reproduced verywell by the VPMS method. The errors in reproducing thepeak stages and its corresponding time to peak stages of theMIKE 11 solutions were found to be less than 5% whilerouting in all channel types, except for few casescorresponding to channels characterized by bed slope valuesof 0.0002. On the basis of the past studies of the VPMSrouting method [Perumal and Ranga Raju, 1998b], it wassurmised that the VPMS method may have to be appliedwith caution for routing in channels characterized by smallslopes, of the order of So = 0.0002. Overall, the resultsdemonstrate the capability of the VPMS method for routingin moderate slope channels wherein attenuation of a passingflood wave may be significant.6.1.2. Upstream and Downstream DischargeHydrographs[62] Usually the simplified hydraulic routing methods

available in literature, route the stage hydrograph or dis-charge hydrograph only. If one of these variables is used for

routing, the other is estimated using the rating curvepertaining to that routing point. However, the VPMSrouting method is different from these usually availablesimplified methods in that the discharge hydrographcorresponding to routed stage hydrograph at downstreamof the reach can be estimated using equation (19), and that

Figure 6. Typical normal rating curves at the downstream channel section for different channel typeswith (a) bf /bm = 1.5 and (b) bf/bm = 4.0.

Figure 7. Sectional details of laboratory compoundchannel [Rashid and Chaudhry, 1995].


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at the upstream of the reach corresponding to given inputstage hydrograph can be estimated using equation (20). Theresults of discharge hydrographs estimated at the upstreamof the reach, while routing using the VPMS method in the40 km reach, are compared with the corresponding MIKE 11

solutions only for a qualitative analysis as shown in Figures4 and 5, and therefore the reproduction of pertinent char-acteristics of the MIKE 11 discharge hydrograph solutionsat the upstream of the reach are not presented in Tables 1and 2. Routing results shown in Table 1 corresponding to8 subreaches division of the channel reach reveal that forsteeper bed slope channel cases (So > 0.001) the reproduc-tion of the MIKE 11 discharge hydrographs by the dis-charge estimation equation (20) of the VPMS method maybe considered very close, with the variance explainedestimates being greater than 99%. However, when thedischarge hydrographs are estimated using equation (19),corresponding to routing solution of the VPMS method inthe reaches of the channels with So < 0.001, the varianceexplained estimates in reproducing the MIKE 11 dischargehydrograph decreases with decreasing channel slope. TheNash-Sutcliffe variance explained estimate (hq in percent)varies in the range of 99%, at its best, to 77%, at the lowest.

Table 3. Channel Roughness at Different Flow Depths of the

Laboratory Compound Channela

Depth of Flow, cm Manning’s n

<20.0 0.01320.0 0.01323.0 0.01326.0 0.01629.0 0.01832.0 0.01635.0 0.015

aSee Rashid and Chaudhry [1995].

Figure 8. Comparison plots of routed stage hydrographs by the VPMS method, experimental(observed) data published by Rashid and Chaudhry [1995, p. 83], and simulated results by method 1 ofRashid and Chaudhry [1995]. (a) Station 2, test 1; (b) station 3, test 1; (c) station 4, test 1; (d) station 5,test 1; (e) station 2, test 2; and (f) station 3, test 2.

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One of the factors for the poor reproduction of dischargehydrograph may be attributed to the use of insufficientnumber of subreaches while routing using the VPMSmethod. This decrease in the variance explained estimatesget reversed when routing using the VPMS method is madeby subdividing the given 40 km reach into 40 subreaches.This improvement in the estimate of hq in reproducing thedischarge hydrographs is shown in Table 2 for the casesof routing in smaller bed slope reaches. It is seen thatthe corresponding variations in the estimate of hq in repro-ducing the MIKE 11 discharge hydrographs ranges from99% to 98%, except for routing in two channel types,corresponding to the bed slopes of So = 0.0002, whereinhq was estimated as 94%. This is expected as the longitu-dinal gradient of water depth for small bed slope channelreaches attains such a magnitude close to the limit ofapplicability of the VPMS method which is nearly equalto 0.5 [Perumal and Ranga Raju, 1998b]. It is inferred fromthe solutions of the discharge hydrographs, computed usingequation (19), that the errors in peak discharge estimatesand its timings of the MIKE 11 solutions were found to beless than 5% for the cases of routing considering 40 sub-reaches of the given routing reach, except for the case ofrouting in channel reach with So = 0.0002. However, when

the VPMS routing was made with eight subreaches of thegiven 40 km reach, these errors were estimated greater than5% for smaller slope reaches. A similar inference may bearrived at as far as the estimate of EVOL (in percent) isconcerned, except in the case of the VPMS routingconsidering 40 subreaches of the routing reach, whereinit was found to be less than 5%, even for the case ofrouting in channel type with So = 0.0002. The capability toconserve volume of the benchmark solution by the dis-charge estimation method, even for routing in small slopechannels, demonstrates the robustness of the proposedrouting method.

6.2. On Synthesizing the Rating Curve at anUngauged Site

[63] Figures 6a and 6b show the reproduction of theconventional stage-discharge relationships established atthe end of 40 km reach of all channel types correspondingto channel sizes with bf/bm = 1.5 and 4.0. To arrive at thesestage-discharge relationships, the procedure described insection 3.4 was followed. It is seen that the stage-dischargerelationships developed for all channel types at the end of40 km reach length form a unique relationship and thusestablishing the conventional stage-discharge relationship at

Figure 9. Cross sections of the upper Tiber River at Pierantonio (upstream) and Ponte Felcino(downstream) gauging stations.

Table 4. Summary of Performance Criteria Showing Reproduction of Pertinent Characteristics of the Flood Events of the Upper Tiber

River by the VPMS Method

Events


Lateral Inflow, %tpqer, h qper, % EVOL, % hq, % tpyer,h yper, % hy, %

December 1996 �0.50 �4.13 �4.96 98.45 �0.50 7.27 96.83 0.00April 1997 �1.50 �7.97 �4.63 97.82 �1.50 5.46 98.60 2.0November 1997 �1.00 0.94 �2.36 99.26 �1.00 10.17 96.11 0.00February 1999 0.00 1.85 �3.15 97.26 0.00 11.19 93.74 0.00December 2000 �1.00 7.68 �5.62 96.54 �1.00 12.28 86.04 floodinga

November 2005 �5.50 �6.36 1.10 91.43 �5.50 1.98 91.98 floodinga

aNo discharge measurement was made.


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a river site. However, a couple of rating curve relationshipshave been developed with a very narrow loop, which maybe considered insignificant for all practical purposes. Sim-ilarly, rating curve at the upstream site of the reach may besynthesized by linking the observed input stage with thedischarge, Q3 estimated in the first routing subreach.

6.3. Experimental Verification

[64] Besides verifying the routing capability of this meth-od using field data, the method was also verified using twosets of test data made available from the study of unsteadyflows in laboratory channels, conducted in a rectangularcompound section channel of the laboratory test facility[Rashid and Chaudhry, 1995, p. 83]. The laboratory facilityconsisted of a flume of 21 m length, 0.93 m width, and0.56 m depth. As reported by Rashid and Chaudhry [1995],a free flow boundary condition was provided resulting incritical flow depth downstream boundary condition, whichdid not affect the upstream flows. The cross section ofthe compound channel is shown in Figure 7. The Manning’sn values at different flow depths is presented in Table 3.The bed slope of the channel was kept constant at 0.0021.The location of stations 1, 2, 3, 4, and 5 were 1.22 m, 3.35 m,6.09 m, 7.61 m, and 11.27 m from the inlet point ofthe laboratory flume, respectively. Station 1 was consideredas the input stage hydrograph point (upstream boundary)and stations 2 to 5 were considered as the downstreampoints where the routed stage hydrographs were obtained.The stage hydrographs for test data sets 1 and 2, observed atstation 1, were routed up to stations 2 to 5 by the VPMSmethod, using the sectional Manning’s n values as given in

Table 3. While data set 1 relates to routing results pertainingto unsteady flow occupying the floodplain section predom-inantly, data set 2 relates to routing results pertaining to thecase of unsteady flow taking place in the main section only.The comparison plots of the routed results and the observed(experimental) data are shown in Figures 8a–8f. It isinferred from the results of unsteady flow in floodplainsection, also as shown in Figures 8a–8d, that the ability toclosely reproduce the rising limb of the observed hydro-graph decreases with the increase in the reach length, butconsistently maintaining the close reproduction of therecession limb of the observed hydrograph. Note that theresults obtained by Rashid and Chaudhry [1995] for thisdata set 1 using the full Saint Venant equations, some ofwhich are shown in Figures 8a and 8d, demonstrate that theVPMS solutions are closer to the observed hydrograph thanthe solutions of the Saint Venant equations obtained byRashid and Chaudhry [1995]. The reason for the routedrising limb stage values being higher than the observedvalues may be attributed to the effect of flow interactionformed at the interface between the main channel sectionand floodplain section of the laboratory channel. It is alsoinferred from the results of unsteady flow in main channelsection only, as shown in Figures 8e and 8f, that the VPMSmethod is able to reproduce the laboratory observations inall entity very closely in comparison with the Saint Venantsolutions as given by Rashid and Chaudhry [1995]. On thebasis of the limited experimental results and the consistentlybetter reproduction of these results by the VPMS methodover that of the Saint Venant solutions, one cannot arrive ata firm conclusion that the VPMS method may be moreamenable to handle the practical flood routing problems in abetter way than that handled by the Saint Venant equations.However, one may accept this inference in a qualitativesense pending a detailed investigation in this regard toarrive at a firm conclusion.

6.4. Field Application

[65] The capability of the VPMS method to route a givenstage hydrograph in a river reach and to estimate thecorresponding discharge hydrograph at the routed point isinvestigated using the field data of the upper Tiber River incentral Italy, for the river reach between the gauged sectionsof Pierantonio (upstream) and Ponte Felcino (downstream).Both the stations are equipped with remote ultrasonic waterlevel gauges, while the measurements of velocity are madeusing current meter from cableways. The length of the reachis 15 km with a bed slope of 0.0016. The surveyed crosssections of the two end gauged sections are overlapped anda two-stage trapezoidal compound section geometry (withbm = 30 m, bf = 67 m, z1 = 1.7, and z2 = 2.5), as required forthe prismatic channel reach conceptualization of the VPMSmethod, is sketched between these sections as shown inFigure 9. Six flood events recorded in December 1996,April 1997, November 1997, February 1999, December2000, and November 2005 with minimum lateral flowcontribution, as detailed in Table 4, have been identifiedfor the application of the VPMS method. For two of theseflood events, the flood occupied the floodplain and thevelocity measurement could not be carried out in all rangesof flow. However, stage hydrographs were recorded forthese two events. The pertinent characteristics of these

Figure 10. Comparison plots of (a) routed stage hydro-graphs and (b) corresponding computed discharge hydro-graphs by the VPMS method for the calibrated event ofDecember 1996.

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Figure 11. Comparison plots of routed stage and corresponding computed discharge hydrographs bythe VPMS method for the validated events from (a and b) April 1997, (c and d) November 1997, (e and f)February 1999, (g and h) December 2000, and (i and j) November 2005.


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simulated events are given in Table 4. The event ofDecember 1996 with no lateral flow was used for calibrat-ing the reach averaged Manning’s roughness coefficient.After making many trials, the observed stage hydrograph atthe Ponte Felcino section could be matched closely by therouted hydrograph, obtained by routing the correspondinginput stage hydrograph observed at the Pierantonio sectionusing the VPMS method. The calibrated n for this bestmatching was found to be 0.049. Figure 10 shows thecomparison between the routed stage and computed dis-charge hydrographs with the respective observed hydro-graphs at Ponte Felcino station. The other five eventssimulated at the Ponte Felcino section, by routing the inputstage hydrographs available at the Pierantonio section usingthe VPMS method for the calibrated value of n = 0.049 areshown in Figures 11a–11j. It is seen from Figure 11 andinferred from the measures of the reproduction of pertinentcharacteristics of the events given in Table 4, that thedischarge hydrographs of those events with no or insignif-icant lateral flow contribution could be simulated well, withhy > 94%, in comparison with the discharge hydrographs ofthose events (December 2000 and November 2005) withlateral flow. Further, it is seen from Figure 11 and inferredfrom Table 4 that the upstream flooding significantly dis-torts the matching between the observed stage hydrographand the corresponding routed stage hydrograph. Thereforethe present methodology based on the VPMS routing meth-od should be improved to take into account lateral flow in thereach, in order to match the observed hydrograph at the endsection by the routed hydrograph. However, it is inferredfrom the results shown in Figures 11a–11j that the observeddischarge hydrographs at Ponte Felcino and Pierantonio siteswere reproduced closely by the discharge hydrographsestimated at these sites using equations (19) and (20),respectively. One can infer from the reproduction of dis-charge hydrographs by the proposed methodology that evenfor the cases of flood events occupying the floodplain, theproposed discharge estimation method may be consideredas the potential replacement of the tedious, costly, andfrequently dangerous manual discharge measurements ofthe flood events. Further investigation of this methodology

using more field data is urgently required considering itsdirect relevance to the understanding of hydrology ofungauged basins.[66] With reference to the main objective of this study of

developing rating curves at ungauged sites, the methodologydescribed in section 3.4 of this paper enables the develop-ment of rating curves using the available six sets of concur-rent data at Pierantonio and Ponte Felcino sites, as shown inFigures 12 and 13, respectively. It is inferred from Figure 12that the actual rating curves developed at Pierantonio sitesfor the periods 1996–1997 and 1998–2001 could be veryclosely reproduced by the proposed methodology. However,such a close reproduction could not be realized in reproduc-ing the actual rating curves available at Ponte Felcino sitesfor the periods 1996–1997 and 1998–2001 using the samesets of data used for reproduction of rating curves atPierantonio site. It is seen from Figure 13 showing the

Figure 12. Comparison between the existing rating curves and those developed by the VPMS methodat the Pierantonio section for different flood events.

Figure 13. Comparison between the existing rating curvesand those developed by the VPMS method at the PonteFelcino section for different flood events. Also, the normalrating curve computed by the Manning’s friction law isshown.

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reproduction of actual recorded rating curve at PonteFelcino site that the actual rating curve could be closelyreproduced only in the lower ranges of stage, but deviatingto the extent of estimating discharge by 15 to 20% less thanthe actual in the higher ranges of recorded stages. Thisdifference may be attributed to the presence of local flowwithin the reach in the higher ranges of stage recorded forsome of these six flood events analyzed. However, estima-tion of 15 to 20% less than the actual discharge is acceptableunder many of the field conditions. It may be surmised fromthe above analysis of reproducing the actual rating curvesthat the proposed extended VPMS stage hydrograph routingbased methodology holds a lot of promise for the develop-ment of rating curves at ungauged sites. Further study in thisdirection is on progress.

7. Summary and Conclusions

[67] Establishment of stage-discharge relationship at anungauged site is not only a problem of ungauged basin, butalso of a gauged basin as one cannot install a gaugingstation at all the locations in a river at all the times. Thisstudy carried out with the objective of establishing ratingcurves at ungauged locations resulted in the development ofan extended variable parameter Muskingum stage hydro-graph (VPMS) routing method applicable for routing stagehydrograph in a two-stage trapezoidal compound sectionchannel reach. Considering the term ungauged in a lessrestrictive sense, it is assumed that at least one set of stagehydrograph for a flood event is available at the upstreamand at the ungauged site located downstream of a givenrouting reach for establishing the reach averaged Manning’sroughness coefficient of the given routing reach. The dis-cussions on the results obtained using the VPMS routingmethod demonstrate that the method is capable of reproduc-ing the benchmark stage hydrograph closely for many of thecases. Some of the deviations from this general inferencemay be attributed to the inapplicability of the proposedrouting method for routing in small slope channel reaches orwhen there is significant lateral flow within the routingreach. The extended VPMS method is not only able to routethe flood in the form of stage hydrograph, but it is alsocapable of estimating the corresponding discharge hydro-graph at the point up to which the given input stagehydrograph has been routed or corresponding to the inputstage hydrograph at the upstream site. Using the routedstage hydrograph results arrived at the desired location andthe corresponding estimated normal discharge hydrographestimated at section 3 of the last subreach, one can synthe-size the rating curve at that location bringing out the actualakin to the conventionally estimated rating curves of naturalrivers. The capability of the proposed method for routing incompound section channel reaches is demonstrated usingthe laboratory flume data, made available from the exper-imental study of routing floods in rectangular compoundsection reaches. The capabilities of the VPMS method toroute floods in channel reaches and to estimate the dis-charge hydrographs, corresponding to the routed stagehydrographs, have also been investigated using six sets offield data of the upper Tiber river in central Italy, availablefor the stretch of a 15 km river reach. One set of the eventswas used for calibrating the reach averaged Manning’sroughness coefficient, and the other five events were

simulated using the extended VPMS method and the cali-brated n value. Study of field data using the extendedVPMS method demonstrates that the stage and dischargehydrographs of these events with no significant lateral flowcould be closely simulated by this method. The actual ratingcurves established at the upstream and downstream gaugingsites could be reproduced using the proposed methodologyof rating curve development reported in section 3.4. Thedischarge hydrographs recorded at these sites could beclosely reproduced by equations (19) and (20). It is recog-nized that the assumption of no lateral flow, while routingthe field observed hydrographs, is a real restriction on thepractical utility of this method. The extension of thismethod for rating curve development and discharge estima-tion accounting for lateral flow contribution in the routingreach forms the next phase of the work.

[68] Acknowledgments. The suggestions of reviewers for significantlyimproving the contents of the earlier version of the manuscript are gratefullyacknowledged.

ReferencesApollov, B. A., G. P. Kalinin, and V. D. Komarov (1964), Hydrologicalforecasting, translated from Russian, Isr. Program for Sci. Transl.,Jerusalem.

Birkhead, A. L., and C. S. James (1998), Synthesis of rating curvesfrom local stages and remote discharge monitoring using nonlinearMuskingum routing, J. Hydrol., 205, 52–65.

Cunge, J. A. (1969), On the subject of a flood propagation method (Musk-ingum method), J. Hydraul. Res., 7(2), 205–230.

Danish Hydrologic Institute(DHI) (2003), User’s manual and technicalreferences for MIKE 11 (version 2003b), Hørsholm, Denmark.

Dooge, J. C. I., W. G. Strupezewski, and J. J. Napiorkowski (1982), Hydro-dynamic derivation of storage parameters of the Muskingum model,J. Hydrol., 54, 371–387.

Franchini, M., P. Lamberti, and P. D. Giammarco (1999), Rating curveestimation using local stages, upstream discharge data and a simplifiedhydraulic model, Hydrol. Earth Syst. Sci., 3(4), 541–548.

Fread, D. L. (1990), DAMBRK: The NWS Dam-Break Flood ForecastingModel, report, Off. of Hydrol., Natl. Weather Serv., Silver Spring, Md.

Hayami, S. (1951), On the propagation of flood waves, Bull. Disaster Prev.Res. Inst. Kyoto Univ., 1, 1–16.

Jha, S. K. (2005), Establishment of rating curve at an ungauged river site,M. Tech. thesis, Dep. of Hydrol., Indian Inst. of Technol. Roorkee,Roorkee.

Lighthill, M. J., and G. B. Whitham (1955), On kinematic waves—I, Proc.Roy. Soc., Ser. A., 229, 281–316.

Moramarco, T., and V. P. Singh (2001), Simple method for relating localstage and remote discharge, J. Hydrol. Eng., 6(1), 78–81.

Moramarco, T., S. Barbetta, F. Melone, and V. P. Singh (2005), Relatinglocal stage and remote discharge with significant lateral inflow, J. Hydrol.Eng., 10(1), 58–69.

Nash, J. E., and J. V. Sutcliffe (1970), River flow forecasting throughconceptual models, part 1: A discussion of principles, J. Hydrol., 10,282–290.

Natural Environment Research Council (1975), Flood studies report, Inst.of Hydrol., Wallingford, U.K.

Perumal, M. (1992), The cause of negative initial outflow with theMuskingum method, Hydrol. Sci. J., 37(4), 391–401.

Perumal, M. (1994a), Hydrodynamic derivation of a variable parameterMuskingum method: 1. Theory and solution procedure, Hydrol. Sci. J.,39(5), 431–442.

Perumal, M. (1994b), Hydrodynamic derivation of a variable parameterMuskingum method: 2. Verification, Hydrol. Sci. J., 39(5), 443–458.

Perumal, M., and K. G. Ranga Raju (1998a), Variable-parameter stage-hydrograph routing method. I: Theory, J. Hydrol. Eng., 3(2), 109–114.

Perumal, M., and K. G. Ranga Raju (1998b), Variable-parameter stage-hydrograph routing method. II: Evaluation, J. Hydrol. Eng., 3(2), 115–121.

Perumal, M., and K. G. Ranga Raju (1999), Approximate convection diffu-sion equations, J. Hydrol. Eng., 4(2), 161–164.


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Rashid, R. S. M. M., and M. H. Chaudhry (1995), Flood routing in channelswith floodplains, J. Hydrol., 171, 75–91.

Tang, X., D. W. Knight, and P. G. Samuels (1999), Variable parameterMuskingum-Cunge method for flood routing in a compound channel,J. Hydraul. Res., 37(5), 591–613.

U.S. Army Corps of Engineers (2002), HEC-RAS river analysis systemhydraulic reference manual (version 3.1), Rep. CPD-70, Hydrol. Eng.,Cent., Davis, Calif.

Weinmann, P. E., and E. M. Laurenson (1979), Approximate flood routingmethods: A review, J. Hydraul. Div. Am. Soc. Civ. Eng., 105(12), 1521–1526.

Wong, T. H. F., and E. M. Laurenson (1983), Wave speed-discharge relationin natural channels, Water Resour. Res., 19(3), 701–706.

Wong, T. H. F., and E. M. Laurenson (1984), A model of flood wave speed-discharge characteristics of rivers, Water Resour. Res., 20(12), 1883–1890.

��S. Barbetta and T. Moramarco, IRPI, National Research Council, Via

Madonna Alta 126, I-06128 Perugia, Italy. ([email protected];[email protected])

M. Perumal and B. Sahoo, Department of Hydrology, Indian Institute ofTechnology Roorkee, Roorkee 247 667, India. ([email protected];[email protected])

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A methodology for discharge estimation and rating curve development at ungauged river sites

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