Hydrodynamic Derivation of a Variable Parameter Muskingum Method

7/28/2019 Hydrodynamic Derivation of a Variable Parameter Muskingum Method

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Hydrological Sciences -Journal- des Sciences Hydrologiques,39,5, October 1994 4 4 3

Hydrodynamic derivation of a variableparameter Muskingum method:2. VerificationMUTHIAH PERUMALDepartment of Continuing Education, University ofRoorkee, Roorkee 247667,IndiaAbstract The hydrodynamic derivation of a variable parameterMuskingum method and its solution procedure for estimating a routedhydrograph were presented in Part I of this series (Perumal, 1994a). Inthis paper, the limitations of the method, the criterion for its applicabilityand its accuracy are discussed based on the assumptions used. Themethod is verified by routing a given hypothetical inflow hydrographthrough uniform rectangular cross-section channels and comparing theresults with the corresponding numerical solutions of the St. Venantequations. The stage hydrographs as computed by the method are alsocompared with the corresponding St. Venant solutions. It is demonstratedthat the method closely reproduces the St. Venant solutions for thedischarge and stage hydrographs subject to the compliance of the assumptions of the method by the routing process.Dmonstration hydrodynamique d'une mthode de Muskingum paramtres variables: 2. ValidationRsum La dmonstration hydrodynamique d'une mthode deMuskingum paramtres variables et la description de la procdurepermettant d'estimer un hydrogramme ont fait l'objet de la premirepartie de ce papier (Perumal, 1994a). Dans cette seconde partie ondiscutera les limites de la mthode ainsi que les conditions sous lesquelleselle est applicable et prcise en s'appuyant sur les hypothses formules.La mthode est valide en tudiant le routage d'un hypothtiquehydrogramme d'entre travers des biefs de sections uniformmentrectangulaires et en comparant les rsultats la solution numrique desquations de Saint Venant. Les limnigrammes calculs par notre mthodesont galement compars aux solutions de Saint Venant correspondantes.Il apparat que la mthode reproduit fidlement les solutions desquations de Saint Venant pour les dbits et les niveaux pourvu que leshypothses de la mthode soient satisfaites au cours du processus deroutage.

I N T R O D U C T I O NAn approach for the derivation of a variable parameter Muskingum methodfrom the St. Venant equations for routing floods in channels having any shapeof prismatic cross-section and flow following either Manning's or Chezy'sfriction law was presented in Part I of this series (Perumal, 1994a) along withthe solution procedure for estimating the routed hydrograph using this method.Open for discussion until I April 1995


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444 Muthiah PerumalThe definition sketch of the Muskingum routing reach of length Ax is shownin Fig. 1. According to one of the assumptions used in the development of themethod, during unsteady flow the stage at the middle of the reach correspondsto the normal depth of that discharge which passes at the same instant of timeat a distance / downstream from the middle of the reach. Let this discharge bedenoted as Q 3 and the section where it occurs be denoted as section (3). Theinflow and outflow sections are represented as sections (1) and (2) respectively.This paper brings out the limitations, the applicability criterion and theaccuracy of the method based on the assumptions of the method.

Fig. 1 Definition sketch of the Muskingum reach.EVALUATION OF THE APPLICABILITY OF THE METHODIt was noted while deriving this methodology (Perumal, 1994a) that thehydrograph to be routed should satisfy the criterion | l/S0(dy/dx) | < 1 at anytime for a successful application, with S0 and dy/dx denoting the channel bedslope and the water surface slope respectively. Fortunately, there is no need toseparately evaluate the applicability criterion, as the method has the inherentability to bring out its inapplicability in the form of computational instabilitywhile routing a given inflow hydrograph for a small single reach length. Thesmall single reach length is such that routing results arrived at the samelocation by using further sub-divisions of this small single reach would notproduce significantly different results from that of the single reach solution.However, if further sub-division of the small reach length leads to acomputational problem then the inapplicability of the method for routing agiven flood hydrograph is evident. The method may be forced to work byidentifying a minimum reach, by a trial and error approach, for which there isno computational problem while routing. However, when a longer single reachlength is used to avoid this computational problem, the inaccuracy in the resultswill increase as the assumption of linear variations of water surface anddischarge for such a long reach may not hold good. Nevertheless, one may usejudgement in accepting such results depending on their utility.


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Muskingum method derivation: verification 445ACCURACY OF THE METH ODThe accuracy of the solution using the proposed method depends on thesatisfaction of the assumptions involved in the development of the method. Asa consequence of these assumptions, there are two major sources ofapproximations involved:(a) app roxim ation with reference to the use of the assum ption of a linearvariation of water surface along the routing reach; and(b) approxim ation due to a Binomial series expansion of the energy slope inestimating the distance / between the mid-section and the normal flowsection (section 3) of the reach.

The error introduced due to the former approximation can be minimizedby reducing the length of the routing reach. However, if the truncation errorintroduced by retaining only the first two terms of the Binomial seriesexpansion, as adopted in the present method, is significant, then routing usingan increased number of sub-reaches would compound the inaccuracy in thesolution due to the recursive nature of the solution equation, especially whenthe magnitude of the term | HSJJdyldx) | is near to unity.

The accuracy of the computed stage hydrograph depends on the accuracyof the computed discharge hy drograph which in turn depends on the satisfactionof the assumptions of the method. The implications of this statement will beexplained later by routing a given flow hydrograph through rectangular cross-section channels using this method.A P P L I C A T I O NThe flow hydrograph defined by a four parameter Pearson Type-Ill distributionexpressed by the following equation was used in all the test runs:

tTp

i( 7 - 1 ) exp (i-f/gy. ^~ l) .where Ib is the initial steady flow (100 m 3 s~'), Ip is the peak flow (1000 m 3 s 1) ,t is the time to peak (10 h), and y is the skewness factor (1.15).This form of inflow hydrograph has been used for testing flood routingmethodologies by various investigators (NE RC , 1975; W einmann & Laurenso n,1979; Ponce & Theurer, 1982; Garbrecht & Brunner, 1991).A rectangular channel with a uniform bed width of 50 m was used forall the test runs and the hydrograph was routed following the solutionproce dure d escribed by Perum al (1994a) for a distance of 40 km from the

inflow section by subdividing the reach into a single reach and various num bersof sub-reaches. The method was tested on three different channels as given inTable 1. Ten test runs with details as indicated in Table 2 were made in orderto have a better understanding of the proposed methodology. A routing intervalof 15 min was used on all runs.


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4 4 6 Muthiah PerumalTable 1 Channel configurationsCh ann el type Bed slope (S)

0.00020.00020.002

Manning'sroughness (n )0.040.020.04

Table 2 Test runs detailsTest run no. Chan nel type Solutionrequiredat (km)

Total reachlength(km)No. ofsubreaches Length ofsubreach(km)1

2345678910

1112223332

4040404040404040405

4040404040404040405

123184018401

402013.33405140515

THE PERFORMANCE CRITERIAThe performance of the method was evaluated by comparing its solution withthe exact solution obtained using the St. Venant equations based on thefollowing criteria as adopted by Weinmann (1977):(a) accuracy of the simulated peak outflow;(b) accuracy of the simulated time to peak outflow;(c) accuracy of the conservation of mass;(d) accuracy of the simulated peak stage; and(e) accuracy of the simulated time to peak stage.The accuracy of the reproduction of the hydrograph shape and size wasevaluated by the Nash-Sutcliffe criterion (Nash & Sutcliffe, 1970).

RESULTS AND DISCUSSIONHydrograph reproductionTable 3 shows the summary results of all the test runs with reference toreproducing some pertinent characteristics of the discharge hydrographs undervarious unsteady flow conditions. Table 4 shows the summary results of thecorresponding test runs with reference to the reproduction of peak stagecharacteristics.

Figure 2 shows the results of discharge hydrograph routing in channeltype 1 corresponding to test runs 1-3. It shows the inflow hydrograph, theexact solution and the corresponding routed hydrographs arrived at using thepresent method for three cases, referring to the sub-division of the 40 km reach


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Muskingum method derivation: verification 447Table 3 Summary of results showing reproduction of pertinent characteristicsof discharge hydrographsTest run Peak flow and time to peak

St. Venant'Q (m 3 s"')

1 7652 7653 7654 8855 8856 8857 9938 9939 99310 985

s solutiontW14.2514.2514.2513.0013.0013.0012.2512.2512.2510.25

2(m371 167 766 986885 685 698 399 399397 9

Proposed"') t (h)

14.2514.2514.2513.0013.2513.2512.2512.2512.2510.25

methode(%)- 7 . 0 6- 1 1 . 5 0- 1 2 . 5 5- 1 . 9 2- 3 . 2 8- 3 . 2 8- 1 . 0 00.000.00- 0 . 6 1

tP (h)0.000.000.000.000.250.250.000.000.000.00

Varianceexplained97.5097.8897.4399.2799.7399.7299.1399.9799.9899.99

E V O L ( % )- 0 . 2 90.110.500.120.120.120.320.410.420.03

Table 4 Summary of results showing reproduction of pertinentcharacteristics of stage frydrographsTest run

i2345678910

Maximum stage and time to maximumSt. Venant'syP (m)10.2110.2110.217.437.437.436.146.146.147.93

solution' , ( h )16.0016.0016.0013.7513.7513.7512.2512.2512.2511.25

yP (m )11.4710.3010.037.827.467.436.116.146.148.13

Proposed method' , 0017.0016.7517.0014.0014.2514.2512.2512.2512.2511.50

yper ( m )1.260.09- 0 . 1 80.390.030.00- 0 . 0 30.000.000.20

' , (h)1.000.751.000.250.500.500.000.000.000.25

1000

3 0T i m e ( h r )Fig. 2 Routing results for channel type 1.


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448 Muthiah Perumalinto a singe reach and two and three equal sub-reaches for routing the giveninflow hydrograph. Routing using more than three equal sub-reaches, in thecase of channel type 1, resulted in the abortion of solution due to thedevelopment of high negative values of the Muskingum weighting parameter6 in the first reach.Figure 3 refers to the results of the estimated stage hydrographscorresponding to the routed discharge hydrographs of test runs 1-3 arrived atby using the stage-discharge relationship given by Perumal (1994a).

-

-

--

- \ yi

ifififi f

iiii li i

' . Single reach so lutio n" \ N \ Two sub-reaches solut ion\ - \ \ Three sub- reache s so lu t ion

\ \

\ VS*.N V" ^ V .

1 I i , 1 , ! i10 20 40 50

Fig. 330

Tim e ( hr )Com puted stage hydrographs for channel type 1.60

3 s u b - r e a c h e s- 2 s u b - r e a c h e s- s i n g l e r e a c h

. 3 s u b - r e a c h e s2 s u b - r e a c h e ss i n g l e r e a c h40 50 600T i m e ( h r )Fig. 4 Variation of 1/SJdy/dx) comp uted at the inlet of the reach

(channel type 1).


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Muskingum method derivation: verification 449Figures 5 and 7 refer to the results of discharge hydrograph routing inchannel types 2 and 3 corresponding to test runs 4-6 and 7-9 respectively. Theyshow the inflow hydrograph, the exact solutions and the corresponding routed

hydrographs arrived at by using the present method for three cases, referringto the subdivision of the 40 km channel reach into a single reach, and 8 and 40equal sub-reaches for routing the given inflow hydrograph. Figures 6 and 8refer to the results of estimated stage hydrographs corresponding to the routeddischarge hydrographs shown in Figs 5 and 7 respectively.1000

8 0 0 -

e 600

4 0 0

2 0 0

1 v >1 ipi If ii M *

; f; jr1 II; if

- < 1i firi i

\ \

\\

~i

. x x

\ V\ \\ \

. j i i

I n f l o wS i n g l e r e a c h s o l u t i o nM u l t i p l e r e a c h s o l u t i o n s( S a n d 4 0 s u b - r e a c h e s )

. i a _ i i i10 20 30T i m e ( h r )

4 0 5 0 6 0

Fig. 5 Routing results for channel type 2.

3 0T i m e ( h r )Fig. 6 Comp uted stage hydrographs for channel type 2.


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450 Muthiah PerumalIt is seen from Fig. 2 and Table 3 that the reproduction of the exactsolution characteristics including that of peak flow is achieved better by singlereach rou ting, except at the beginning where the infamous "dip" characteristic

of the Muskingum method dominates. It is also seen from Fig. 2 that routingwith two and three sub-reaches reduces the size of the dip.

1000

Ti me (h r )Fig. 7 Routing results for channel type 3.

a

5 -

3 -

0

E x a c t s o l u t i o n S i n g l e r e a c h s o l u t i o n

M u l ti p le r e a c h s o l u t i o n s( 8 a n d 4 0 s u b - r e a c h e s )

10 20 40 5 00T i m e ( h r )Fig. 8 Computed stage hydrographs for channel type 3.60


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Muskingum method derivation: verification 451It was shown by Perumal (1992) that the assumption of linear variationof discharge along with the condition that section (3) lies between the midsection and the outflow section of the reach is responsible for the formation of

negative or reduced outflow at the beginning of a routing solution. Therefore,the higher the magnitude of | l/S0(dy/dx) | , and hence the higher the dischargevariation in space, dQ/dx, and the length of the reach, the higher w ould be thedip or reduced outflow at the beginning of routing. The factors which influencethe formation of higher \l/S0(dy/dx)\ are the steepness of the hydrograph,small bed slope and higher roughness coefficient.There is almost no difference between the routing results obtained usingtwo and three sub-reaches. Although the size of the dip reduces considerablyin these cases, the peak flow estimated is overattenuated when compared withthat of single reach routing. However, it is not appropriate to discuss theresults of discharge hydrograph routing in isolation from the results of thecorresponding estimated stage hydrograph. It is seen from Fig. 3 that theestimated stage hydrograph corresponding to single reach routing is over-predicted for the entire exact solution in general, and very much so near thepeak region. Also, it is interesting to note that the stage hydrographs estimatedcorrespond ing to two and three sub-reaches routing of the discharge hyd rographreproduce the exact solution of the stage hydrograph closely. Therefore, theinability of the method to reproduce both the stage and discharge hydrographsin a consistently better manner may be attributed to the non-compliance of theassumptions used in the development of the method by the routing process inchannel type 1.Figu re 4 show s the variation of the nondimen sionalized slope l/S0(dy/dx)of the exact solution vs . time at the inlet of the reach corresponding to thegiven inflow hydrograph. Also shown are the corresponding variations ofl/S0(dy/dx) estimated from a single reach, and two and three sub reaches routingusing the relationship:

1 ?1 = - _L ?. (2)So dx Be2 dtwhere c is the w ave ce lerity, B is the top width of the water surface and dQ/dtdenotes the rate of change of discharge with reference to time.

It is seen from Fig. 4 that the estimated magnitudes of l/S0(dy/dx) for allthe cases of routings based on a single reach and two and three sub-reaches ismuch greater than the corresponding actual l/S0(dy/dx) even though it is lessthan unity for the entire inflow hydrograph, thus satisfying the applicabilitycriterion of the method. It may be noted that when the absolute maximum ofl/S0(dy/dx) during the rising part of the hydrograph is less than, but nearer tounity, the truncation error introduced by retaining only the first two terms ofthe Binomial series expansion of S f causes the solution of the method to bedifferent from the exact solution. Due to the recursive nature of the solutionequation, the truncation error also propagates resulting in the deviation fromthe actual solution. This aspect can be visualized from Fig. 5, especially, when


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452 Muthiah Perumall/So(dy/dx)>0. The truncation error increases with the increase in the subdivisions of the reach. It may be noted that while theoretically the maximumvalue of \IS0(dyldx) has to be nearer to unity, the estimated + l/S0(dy/dx)> 1due to truncation errors.It is observed that when the number of sub-reaches is increased beyondthree in channel type 1, the routing leads to high negative values of 6 duringthe first sub-reach routing while both the inflow and outflow are receding. Asseen from Fig. 4, routing based on an increasing number of sub-reaches greatlyincreases the truncation error in the estimated + l/S0(dy/dx), resulting in thelocation of section (3) (where the normal discharge corresponding to the stageat the middle of the reach occurs) far downstream of the outflow section(section (2)), thus estimating a higher magnitude of negative 6. Due to therecursive form of the solution equation, section (3) is subsequently located stillfarther away resulting in unbound negative values of 6 leading to the abortionof the solution algorithm. Despite the introduction of large truncation errors,the overall reproduction of exact discharge and stage hydrographs are notseverely affected by the routing solutions obtained using two and threesubreaches of the 40 km reach. Besides, the conservation of volume criterionis not violated with the parameter E VO L < 0 .5 % . One may accept or rejectsuch results depending on their practical utility.Figures 5 and 7, corresponding to test runs 4-6 and 7-9 respectively,show the close reproduction of the respective exact solutions, but with thedevelopment of the dip at the beginning of the single reach routing solution.These cases demonstrate that the approximations involved in arriving at thepresent method are not too severe to affect the close reproduction of the exactsolutions. A close reproduction of the stage hydrographs of the exact solutionsby this method can be seen from Figs 6 and 8. In the cases of routings inchannel types 2 and 3, it can be seen from Figs 5 and 7 that there is no diff-erence between the results based on the routings using 8 and 40 equal subdivisions of the 40 km reach. This leads to the inference that there exists a limiton the use of the number of subdivisions of a reach for yielding improvedresults, beyond which no improvement can be obtained with increased computational effort.Wave attenuation characteristicsAs the same inflow hydrograph was routed in all these studies for the samedistance, it may be inferred that the attenuation characteristics of the routedhyd rograp h can be studied effectively in terms of the characteristics of th erating curves observed during the passage of the inflow hydrograph at the inletof the reach corresponding to each channel type. The variations in the ratingcurve characteristics are the result of differing roughness and slope characteristics of the channels. Figure 9 shows the rating curves obtained at the inletof the reach for all three channel types. The narrowing of the loop for channeltype 2 compared with that of channel type 1 is due only to the reduction in the


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Muskingum method derivation: verification 453Man ning roug hness value from 0.04 for channel type 1 to 0.02 for channe ltype 2. The rating curve obtained at the inlet of channel type 3 is notdistinguishable from the steady flow rating curve. It is seen that the higher thebed slope and the lower the Manning roughness coefficient the narrower willbe the loop of the rating curve. Under such conditions, the routing results usingthe described method closely reproduce the exact solutions, and w hen the ratingcurve almost corresponds to that of a steady state relationship, the flood waveis governed by kinematic wave characteristics as shown in Fig. 7. Also thepeak stage values of the computed and exact solutions shown in Fig. 8 coincidewith the corresponding peak discharge value of the routed hydrograph shownin Fig. 7.

I n f l o w d i s c h a r g e ( m s )Fig. 9 Loop rating curves at inlet of the reaches for all chann eltypes.When the rating curves are characterized by a wider loop width, theinflow hydrograph attenuates as shown in Figs 2 and 5, with less attenuationin channel type 2 than in type 1. It is to be noted that when the inflow hydro-graph is characterized by a w ide loop rating curve , as in the case of rou ting inchannel type 1, the overall reproduction of the exact solution is not satisfactorywhen compared with the reproduction of the exact solution in channel type 2.The formation of a wide loop rating curve implies the development of signif-icantly higher water surface slope dy/dx when compared with the bed slope S0.

Variation of travel timeFigu re 10 shows the variation of the Muskingum travel time parame ter K, withreference to inflow discharge during a single reach routing in all three channel


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454 Muthiah Perumaltypes. The inflow discharge hydrograph remained the same for all the routingstudies made herein. Since the wave celerity was alw ays positive, the inferencearrived at from Fig. 10 for a single reach routing can be generalized formu ltiple reach routings also. Th e relationship of the present method e xhibits theformation of a loop with larger K for a given discharge in a rising limb of thehydrograph and a smaller K for the same discharge observed in the falling limbof the inflow hydrograph due to the accounting of water surface slope. It is alsoseen that the range of variation of K and its magnitude decreases with increasein the bed slope and reduction in channel roug hne ss. It can be inferred fromFigs 9 and 10 that a wide loop rating curve results in a longer travel time offlood discharge with a large variation range.

100 300 500 700 90 0 1000i n f l o w ( m 3 s 1 )Fig. 10 Variation of K for single reach routing.

Variation of weighting parameterFigure 11 shows the variation of the other Muskingum parameter, theweighting parameter 6, with the inflow discharge corresponding to a singlereach routing. Before discussing these results, it is necessary to look brieflyinto the aspects of the variation of 6 from a physical consideration. An insightinto the variation of 6 can be gained by studying the expression for 6 given bythe equation (Perumal, 1994a):

6 = 1 - . 1 ( 3)2 AxWhen section (3) lies between the midsection and the outflow section ofthe routing reach, 0 < 6 < 0.5. When it coincides with section (2), 6 = 0 asin the Kalinin-Milyukov method. When it is located downstream of the midsection of the reach as well as downstream of section (2), 6 < 0.


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Muskingum method derivation: verification 455

0 1 ! , I 1 1 I I 1100 300 50 0 700 90 0 1000i n f l o w ( m s )Fig. 11 Variation of 6 for single reach routing.

It is seen from Fig. 11 that a wide variation in the magnitude of 6 isobserved with reference to inflow for channel types 1 and 2. The relationshipexhibits a wide loop with a higher magnitude of 6 for a given discharge in therising limb of the inflow hydrog raph and a smaller m agnitude of 6 for the samedischarge in the falling limb of the hydrograph. The wider the loop of therating curve, as noted in Fig. 9, the wider also the loop of this relationshipwhich characterizes the attenuation of the inflow hydrograph while routing forthe same length of single reach. However, very insignificant variations of 6with the magnitude being closer to 0.5 are observed corresponding to inflowin channel type 3. This is due to the insignificant role of the term l/S0(dy/dx)in characterizing the inflow hydrograph in channel type 3. The absence of aloop with regard to the relationship of 8 and its presence with regard to therelationship of K suggest that the flood wave propagation in channel type 3 isgoverned by the celerity of the flood wave with no effect of channel storage.This behaviour is characteristic of kinematic wave propagation.

When / > Ax/2, then 6 < 0. Under such a condition, the outflowdischarge would be greater than the normal discharge Q 3 passing at section (3).This situation was realized in test run 10 in which the 6 values correspondingto each time level of routing were negative and thus the outflow discharge wasgreater than normal discharge Q3 at all the routing time levels. Figu re 12 show sthe routing results of test run 10 for a single reach solution, along with theexact solution. The estimated normal discharge corresponding to this singlereach routing is also shown. Confirming the interpretation of equation (3), theoutflow discharge hyd rograph is observed ahead of the normal discharge hy dro -graph. Figure 13 shows the variation of 6 with inflow estimated for this single


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456 Muthiah Perumal1000

8 0 0

B6 0 0

o 4 0 0

2 0 0 h10 0

--.-

-

J ^ p

VVf i

fi i1.TlV

j lj if

\ \

V\ \VVV -VV

V^X

1 1 ^ * - J

Inf lowExact so lu t ionD i r ec t so lu t i onEs t ima ted no r ma l d i scha r gehyd rogra ph a t sec t ion (3 )

_ , l , i ,10 20 3 0T i m e ( h r )

4 0 50 60

Fig. 12 Hydrogra ph at 5 km due to direct routing when outflowdischarge section is upstream of normal discharge section (channeltype 2).

aa

5 _

10 0 300 500 700I n f low ( m 3 s 1 ) 90 0 1 0 0 0

Fig. 13 Variation of 6 for single reach routing at 5 km (channeltype 2).


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Muskingum method derivation: verification 457reach routing. Negative values of 6 were estimated for the entire routingprocess with the limits -2.21 < 6 < - 0 . 0 8 . Although the possibility of 6


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4 5 8 Muthiah PerumalGarbrecht, J. & Brunner, G. (1991) Hydrologie channel-flow routing for compound sections. J.Hydraul. Engng 117(5), 629-642.Nash, J. E. & Sutcliffe, J. V. (1970) River flow forecasting through conceptual models part I - Adiscussion of principles, /. Hydrol. 10, 282-290.Natura l Environm ent Research Council (1975) Flood studies report, volume III - Flood routing studies.London, UK.Perum al, M . (1992) The cause of negative initial outflow with the M uskingum m ethod. Eydrol. Sci.J. 37(4), 391-401.Perum al, M. (1994a) Hydrodynamic derivation of a variable parameter Muskingum method: 1. Theoryand solution procedure. Hydrol. Sci. J. 39(5) (this issue)Ponce, V. M. & Theurer, F. D. (1982) Accuracy criteria in diffusion routing. /. Hydraul. Div. ASCE108(6), 747-757.Strupczewski, W. G. & Kundzewicz, Z. W. (1980) Muskingum method revisited. /. Hydrol. 48, 327-342.Weinmann, P. E. (1977) Comparison of flood routing methods for natural rivers. Report no. 2/1977,Dept. of Civ. Engng, M onash University, Victoria, A ustralia.Weinmann, P. E. & Laurenson, E. M. (1979) Approximate flood routing methods: a review. J.Hydraul. Div. ASCE 105(12), 1521-1536.Wong, T. H. F. (1984) Improved parameters and procedures for flood routing in rivers. Ph.D.dissertation, Monash Univ., Victoria, Australia.Received 8 February 1993; accepted 7 April 1994

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Hydrodynamic Derivation of a Variable Parameter Muskingum Method