Flood Routing in Long Channels

8/6/2019 Flood Routing in Long Channels

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Journal of the Chinese Institute of Engineers, Vol. 28, No. 7, pp. 23-35 (2005) 23

FLOOD ROUTING IN LONG CHANNELS:

ALLEVIATION OF INCONSISTENCY AND DISCHARGE DIP IN

MUSKINGUM-BASED MODELS

Weihao Chung* and Yi-Lung Kang

ABSTRACT

The Muskingum parameters are often expressed as a function of discharge and

channel properties (Cunge, 1969; Chow et al., 1988) by referring to the coefficients

of the advection-diffusion (A-D) equation or the equivalent parabolicized Saint Venantequation. As extensively investigated in this paper by Taylor series expansion, the

Muskingum model and its variants are found tangibly inconsistent to the A-D equation.

In addition, these Muskingum-based models experience the effect of negative outflows,

i.e., the well-known dip phenomenon. To avoid these problems, which are both present

in the conventional routing procedure, this paper introduces an extra term to the tradi-

tional Muskingum storage function, that is then linked to Gills concept of initial

storage. Through this technique, not only the dip phenomenon but also the model

inconsistency can be alleviated, and a fairly satisfactory outflow prediction can thereby

be achieved. A proper time to employ the extra term is sought by a convolution inte-

gral which is a result of the Laplace transformation applied to the Muskingum model.

It also represents the analytical expression of outflow discharge. The convolution

integral enables us to trace the origin of discharge dip and quantify the shape varia-

tion of outflow hydrographs. With the convolution integral, compact models for com-

puting the maximum flow dip, dip, and its occurrence time, tc, are also offered in

this study. As a concluding example, routings by the traditional Muskingum model,

Gills procedure, and the newly developed algorithms having the extra term are per-

formed in a long channel reach of 90 kilometers to test the robustness of each model.

Key Words: model inconsistency, dip phenomenon, Muskingum model, flood routing,

storage function.

*Corresponding author. (Tel : 886-7-7456290; Email :

[email protected])

W. H. Chung and Y. L. Kang are with the Department of Civil

Engineering of the Chinese Military Academy, Fengshan, Taiwan830, R.O.C.

I. INTRODUCTION

Due to limited field data and marked topogra-phy changes of river channels, it is usually impos-

sible and unnecessary to perform high-dimensional

river flow analyses using complex mathematical

models. In this situation, flood wave transport and

dissipation are often simplified as one-dimensional

problems in a semi-infinite flow domain, provided that

detailed flow velocity distribution and backwater

effects induced by downstream perturbations are

trivial. The Muskingum model is suitable in this

situation and plays a fairly important role due to itssimplicity and practicability. Generally, one may per-

form flood routings and risk analyses for a flood event,

once the Muskingum parameters, Kand X, that ac-

count for wave traveling time and dispersion,

respectively, are calibrated using flow data measured

in the past. Of course strong material scouring or

deposition along with flood flows must not change

the parameter values significantly.

Stability analysis of the Muskingum model or

its variant Koussiss formulation (Koussis, 1978)can be carried out through the use of von Neumanns

method (Smith, 1985) which represents the modelsas a series of complex exponentials. By dampening


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24 Journal of the Chinese Institute of Engineers, Vol. 28, No. 7 (2005)

out the round-off error, the Muskingum model can

be shown to be unconditionally stable, while Koussiss

formulation has the restriction that the Courant num-

ber must be less than 1 (Chung, 1994). Chow and

Kulandaiswamy (1971) developed a general lumped

system model (not concerning the initial storage) by

Taylors series expansion to simulate various hydro-

logic phenomena. The input and output of a water-

shed system were linked by a transfer function which

could be used to formulate the instantaneous unit

hydrograph. A five-term model, consisting of inflow

and outflow discharges and their derivatives, was rec-

ommended for practical application but the article

lacked a discussion of discharge dip and model in-

consistency (see the following paragraph for

definition). The impact of calibrated parameters on

varying outflow hydrograph shape was also not

mentioned. To produce a stable general hydrologicmodel, they outlined the restriction condition but it

seemed easily violated since the transfer function led

to three roots that resulted in more unexpected com-

binations than usual.

The flaw of the Muskingum model falls in the

outcome of equation/model inconsistency (Smith,

1985), that is, a model differs from its original dif-

ferential equation as it is inverted from a discrete form

to a differential one. Despite the fact that Muskingum

model can be regarded as the discrete form of the A-

D equation or the equivalent parabolicized Saint

Venant equation, it is shown in section II that the

former can not be inverted to the latter exactly (i.e.,

inconsistent with A-D equation). The worst thing

about model inconsistency is that the inconsistency

may converge our numerical solutions to unexpected

ones and thereby mislead our judgment in spite of

the fact that all parameters are carefully calibrated.

As shown in this paper, the inconsistency represents

the very factor leading to the underestimation of peak

discharge and possibly the time to peak. Satisfac-

tory routing results in a long channel are seldom ob-

tained by using the conventional Muskingum model

or its variants due to the inconsistency problem.

A n o t h e r p r o b l e m o f t h e c o n v e n t i o n a lMuskingum method is negative discharges or the so-

called dip phenomenon. The dip is often interpreted

as a result of choosing either inadequate time inter-

vals (Hjelmfelt, 1985) or grid sizes (Ponce and

Theurer, 1982), or the nature of the storage function

(Nash, 1959) defined by Kand X. Boneh and Golan

(1979) considered it possible to have a negative in-

stantaneous unit hydrograph in nature and took it as

the cause of negative discharges. Gill (1980) viewed

the same problem as a consequence of introducing

inadequate initial conditions, but this view was re-

jected by Singh and McCann (1980). Gill (1980) alsoproposed a time lag period within which outflow

hydrographs do not change due to the arrival of flood

at the inlet section of the flood routing reach. This is

consistent with reality but seems useless in improv-

ing present-day models. To get a proper fit of out-

flow hydrographs, Gill (1977) proposed the concept

of relative storage in which the Muskingum param-

eters Kand Xare re-calibrated. In the light of Gills

concept, Aldama (1990) proposed the formula for

determining K, X, and the initial storage that links a

relative storage to an absolute one. In his technical

report he shows the calibration formula with relative

storage instead of outflow rates. That leads to more

accurate predictions (Heggen, 1984, and ODonnell,

1985, for comparisons ) of the time to peak discharge

and results in smaller root-mean-square errors, with

a trade off of more marked discharge dips.

If lengthy complicated mathematical deduction

is implicated, the Kalman filtering method (Wood andSzollosi-Nagy, 1978; Ngan and Rusell, 1986; Kan,

1995; and Van Geer et al., 1991) can be employed to

eliminate the dip phenomenon. The greatest advan-

tage one may take from the filtering method is to re-

lax the restriction of fixed Muskingum parameters for

a given flood event. By substituting optimized pa-

rameters within each time interval, Kalmans filter-

ing offers an excellent alternative to route outflow

hydrographs with almost perfect performance (Kan,

1995), even if inflow hydrographs with tributary flows

vary with time in a complex way. This seems to be

good for the conventional Muskingum method but still

does not address the real problem causing the dip

phenomenon. Besides, due to the nature of temporal

variations, Kalmans filtering parameters can not be

interchangeably employed through all flood events.

That is, the parameters must be recalibrated from one

flood event to another. Thus, before clarifying the

cause of dip phenomena, it would still be valuable to

re-study either the Muskingum model or its variants.

In order to solve the problems mentioned above

(model inconsistency and discharge dip), a time con-

stant R is introduced in the conventional Muskingum

storage function. By substituting the so obtained stor-

age function into the continuity mass equation it willhopefully yield an extra term to counterbalance the

factor causing model inconsistency, and consequently

reduce the dip phenomenon substantially. Following

this idea, this paper presents two models essentially

good for flood routings in long channels. The first is

called the modified Muskingum algorithm (MMA)

being a variant of the Muskingum model withR. The

second is called the modified Gills algorithm (MGA)

containingR and Gills relative storage. Thus, MMA

includes three basic parameters while MGA has four.

Robustness of the new algorithms will be tested and

confirmed through a series of comparisons with theconventional Muskingum model (CMM) and Gills


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W. Chung and Y. L. Kang: Flood Routing in Long Channels: Alleviation of Inconsistency and Discharge Dip 25

procedure with Aldamas calibration formula (GILL)

in a super long channel of 90 kilometers. Test re-

sults can be used further for confirming the

postulation, as addressed by Laurenson (1959), that

a single-valued storage function is not adequate for

storage routing in a fairly long channel.

In brief, this paper reveals first the model in-

consistency embedded in CMM by Taylors series

expansions, then under various conditions obtains the

MMAs analytical solutions of outflow rates in a form

of convolution integral. With these solutions, the ori-

gin of discharge dip can be found along with the shape

variation of outflow hydrographs. The connection be-

tween model inconsistency and discharge dips can

thereby be explored. By short time approximation,

the forcing and response functions in the convolu-

tion integral are simplified to formulate a series of

compact models for computing tc and dip. Resultsare verified by a sensitivity analysis example. About

R, its application range is found through the study of

the response function that may be unbounded under

certain flow conditions. The best timing for adopt-

ing R is also proposed through the analysis of the

modified equation and by the model prediction ofdip .

Finally, explicit algorithms are developed for param-

eter estimations of MMA and MGA. Model compari-

sons are then made to test their applicability and

robustness in a fairly long channel.

II. MODEL INCONSISTENCY

Since the mass balance equation has no need

to be modified, attaching one or more terms to the

Muskingum storage function, S, may be the only al-

ternative for avoiding the inherent inconsistency and

flow dips. As a rule of thumb, S must be changed in

such a way that the modified equation remains in the

same form as the A-D equation (Qt + UQx =DQxx, Q

= discharge, U= advection coefficient, D = disper-

sion coefficient) that basically governs diffusion

waves in open channel flows. It is found, after sev-

eral trials, that the most compact way is to introduce

in S an extra term RdO/dtwhere R has a unit of timesquare. This transforms S into the following

expression:

S = KO + KX(I O) + RdOdt

(1)

where t denotes t ime and I and O represent,

respectively, the discharges flowing in and out of a

finite channel reach. In addition to the storage shape

described by the second and third terms of Eq. (1),

the newly added term RdOdt

accounts for the additional

volume induced by the rate of rise of outflowhydrographs which, aside from nonlinear effects, may

contribute to a looped rating curve, hence making the

storage function physically and mathematically more

complete.

Substituting Eq. (1) into the mass balance equation:

dSdt

= I O (2)

yields the modified CMM of discharge as below:

Rd2O

dt2+ K(1 X)dO

dt+ O = KXdI

dt+ I (3)

which can be reorganized and discretized into the

following difference form:

XIn + 1 In

t+ (1 X)

On + 1 On

t+

On + 1 In + 1

2K

+ On

In

2K+ R

K(O

n + 1

2On

+ On 1

t2) = 0 (4)

where n denotes the time level.

To derive the modified form of Eq. (4), the dis-

charges associated with different time levels and lo-

cations in Eq. (4) are first expanded about j and n as

a Taylors series. For example:

In + 1 = Qjn + 1 = Qj

n + Qtjnt+ t

2

2Qttj

n + t3

6Qtttj

n +

On = Qj + 1n = Qj

n + Qxjnx + x

2

2Qxxj

n + x3

6Qxxxj

n +

wherej represents the spatial location along a channel,

etc. The discharge On + 1 (= Qj + 1n + 1 ) can be similarly

expressed as a function ofOjn (= Qj + 1

n ) by Taylors

series expansion for a two variable function (Kaplan,

1981). Substituting all the results into Eq. (4) with K

= x/Uyields an equation in Xand U. Applying thedifferential operator /t=D2/x2U/x to the re-sultant equation and denoting Oj

n as Q yields the fol-

lowing modified equation after some lengthy algebra:

Qt + 1Qx = 2Qxx + 3Qxxx + 4Qxxxx + ... (5)

where 1 = x/Kand

2 = (12

X)Ux RU3

x

3 = (13

Cr4

Cr

2

12 1

Pg(1

Cr2

))Ux2

+ ( 1Pg

12

+Cr2

)XUx2 (1 2Pg

)RU3

4 = (1

Pg(1

2 Cr)X

Cr

2Pg2

1Pg

(12

34

Cr))Ux3

(1 +

Cr2

12 +1

Pg2

2Pg )RU

3

x


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In the above equation, x = grid size (channel reachlength), t= time interval, Cr = Ut/x = the Cou-rant number and Pg = Ux/D = the grid Peclet number.

Now, letting the numerical dispersion equal the

physical one, that is, 2 = D, produces the formulafor computing X:

X = 12

1Pg

RU2

x2(6)

To make Eq. (5) consistent with the A-D equation,

we have 3 = 0 which with Eq. (6) yields the analyti-cal expression ofR as followings:

R = x2

U2(

1 Cr2 12Pg

2

6 + 6Cr 12Pg1

) (7)

Eq. (7) can be also re-expressed as an explicit func-tion ofK and X by substituting Eq. (6) into Eq.(7)

with U= x/K, which gives:

RK2

= 18

(2 6X tK

)

+ 12

X2

4+ X

2 5

48(t

K)2 +

(3X 1)4

tK

112

(8)

Aside from the above analytical expressions, the

following limitations also result for very small x andt:

R DU3

x , X 12

2Pg

, and 4 D3

U2

This gives the limit truncation error of Eq. (5):

ET = D3

U2Qxxxx (9)

Therefore, introducingR in CMM does relax the in-

consistency embedded in Qxxx terms (see Appendix

for comparisons), but evokes another unexpected non-zero term 4Qxxxx. Though the limit value of4 maynot be zero, (i.e., the inconsistency problem still

exists), the discrepancy is alleviated from a third-or-

der to a fourth-order derivative. Eq.(9) can be further

inverted to a function ofKand Xthrough the use of

Eq.(6) and K= x/U:

ET = x4

K(1

2 X R

K2)3Qxxxx (10)

Examining Eq. (9) or (10) reveals that model incon-

sistency becomes worse for a decreasing U (i.e., in-

creasing K) or an increasing D and x. That is, it iseasy to produce phase errors and underestimate the

outflow peak discharge of a flood moving slowly but

undergoing strong dispersion in a long channel. Indeed,

channel length has dominated impacts on ET due to

the fourth power ofx in Eq. (10). It is understoodfrom Eq. (5) that ET behaves like a forcing function

constantly acting on the discharge Q for a period of

x/U, hence leading to the unavoidable modelinconsistency. This is the fate of CMM and its vari-

ants with constant coefficients. The inconsistency can

not be easily removed by simply adjusting time intervals.

III. ANALYTICAL EXPRESSIONS

1. The Analytical Solution ofO

By deducting the base flow rate Qb from O and

letting I = I Qb and O = O Qb, Eq. (3) can be

solved for O by Laplace transformation with O = 0and dO/dt = 0 at t= 0. By so doing, we obtain the

Laplace-transformed O which is divided into three

branches as below:

(i) for R = 0: O = 1K(1 X)s + 1

f;

(ii) for R < 0: O = 1 ( 1s 1

s )f;

(iii) forR > 0:

O = 2(

2R)2

(s + K(1 X)2R

)2 + (2R

)2f, if < 0

same as (ii), if 0

where

= e stdt0

,

= K2(1 X)2 4R,

= ,

=1

2R [ K(1 X) + ] ,

= 12R

[ K(1 X) ] ,

and the forcing function of Eq. (3) is given by:

f = f(t) = KXd Idt

+ I (11)

Generally,f(t) has a great chance to be negative for a

short tsince in this case d Idt I/t and tseldom ex-

ceeds KX.

The resultant analytical solution of cases (i) (ii)

and (iii) is then generally expressed by a convolutionintegral of the form:


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O = f(t u)g(u)du0

t

(12)

in which the response function g(t) reads (Doetsch,

1947):

g(t) = 1K(1 X)

exp( tK(1 X)

) (13)

for R = 0;

g(t) = 2 sin(2R

t)exp( K(1 X)

2Rt) (14)

forand R > 0 and < 0;

g(t) = 1 [exp(t) exp(t)] (15)

otherwise. Eqs. (14) and (15) are the expression of

the MMA approach, which becomes MGA if Gillsrelative storage is considered. The storage will lead

to KandXvalues different from those of MMA; there-

fore producing different g(t) values and outflow

hydrographs.

2. Model Restrictions Stability Analysis

Due to the quadratic differential form of Eq. (3),

the required mathematical work for von Neumanns

stability analysis will become rather discouraging. In

this case, it will be more straightforward to achieve

the goal by looking into the response function of Eq.

(12). Apparently, is always positive ifR < 0. Thisfact makes g(t) and eventually O unbounded as tgoes

to infinity, meaning routing procedures with R < 0

must be excluded. It is also noteworthy that, forR0, and are both negative and || < || since K(1

X) is always positive. Thus, g(t) will remain finite

positive and will be that given by Eq. (13). The same

conclusion applies to Eq. (14) considering that the

dip phenomenon usually occurs at the initial stage of

a flood event (i.e., a short range of time) which makes

sin(t) positive. This assures the convergence of cal-

culated O .

To keepR positive, the numerator and denomi-nator in the parenthesis of Eq. (7) must have the same

sign, i.e.,

1 Cr2 12Pg

2 > 0 and 6 + 6Cr 12Pg 1 > 0 ,

or

1 Cr2 12Pg

2 < 0 and 6 + 6Cr 12Pg 1 < 0 .

The above criteria can be further transformed into:

Pg 2 < (1 Cr

2)/12 and Pg 2 < (1 + Cr)

2/4 ,

or

Pg 2 > (1 Cr

2)/12 and Pg 2 > (1 + Cr)

2/4

which yields, respectively, the following restrictions

recognizing that Pg > 0:

Pg >12

1 Cr2

, if Cr < 1 (16)

or

Pg >2

1 + Cr, for all Cr (17)

A series of numerical tests shows that Eq. (4) is con-

ditionally stable for R > 0 as has been reasoned by

Eq. (12) for case (ii). When channel flow is subjected

to a small perturbation, Eqs. (16) and (17) based on

the linear theory are useful and become the equiva-

lent criteria to assure numerical stability.

3. The Impact ofg(t) on O

Likef(t) being connected to inflow hydrographs,

g(t) has a strong impact on the shape of predicted

outflow hydrographs. By carefully examining Eqs.

(13) - (15), it is found that the g(t) of each equation

differ completely from each other in regard to varia-

tion trends. For example, g(t) of Eq. (14) is an expo-

nentially decayed sine function. It becomes however,

a monotonically decreased exponential function for

Eq. (13), and a combination of Eqs. (13)(14) for Eq.

(15). The g(t) of Eq. (15) rises from zero at t = 0,

then increases monotonically with time all the way

to its maximum value, then falls down monotonically

again to approach the time abscissa as t goes to

infinity. Accordingly, the MMA corresponding to Eq.

(14) will evolve the outflow hydrgraphs with up-

and-down oscillatory tails, depending totally on the

relative importance ofK2(1 X)2 and 4R. It is im-possible however, for the CMM associated with Eq.

(13) to generate such outflow hydrographs. By in-

corporating with Gills relative storage, MGA has a

greater chance to meet the criterion for Eq. (15), hence

it is also unlikely to produce oscillatory tails.Since g(t) affects outflow hydrograph shape, it

may also change the corresponding phase error of

time to peak discharge. For example, ifg(t) decays

exponentially as described by Eq. (13), the negative

value off(tu) will be amplified but concentrated ina small u through the convolution integral of Eq. (12).

That is, the dip phenomenon will occur at the initial

stage of a flood and it quickly decays out as t

increases, accompanying the underestimation of the

time to peak discharge. The situation becomes less

serious for g(t) of Eq. (14) that increases then de-

creases and takes a longer time to become zero, mak-ing an aggregate of the negative discharges difficult.


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Result in dip is postponed and the time to peak dis-

charge is over-predicted.

IV. FLOW DIPS

1. The Cause

It has been shown that all g(t)s are positive as

long as tis adequately small. The function value of

f(tu) based on Eq. (12) must therefore be negativeat a certain time or within some period if negative

outflow rates appear. Accordingly, factors causing

the dip phenomenon are explained as the nature of

f(t). Since f(t) does not include R, flow dips fully

depend on KX values and the shape of inflow

hydrographs . The smal ler KX or the milder

hydrograph shape is (generates possibly a kinematic

wave), the more vague the dip phenomenon appears.

2. The Occurrence Time

In fact, the dip phenomenon seems to be

unavoidable, otherwise f(t) must remain positive all

the way. Taking a Gaussian inflow hydrograph:

I= Qb + Qp exp{[(ttp)/]2} (18)

for example, the forcing function can be expressed

as:

f(t) = (2KXt tp

2+ 1)Qpexp[ (

t tp

2)2] (19)

Its function values are all negative as tis smaller than

the following critical time:

tc = tp 2

2KX(20)

In above, Qp = peak inflow rate, tp = the time to peak

discharge, = constant. Eventually, the function val-ues off(tu) are also negative for all us less than t,given t< tc. It is obvious then the area enclosed by

f(tu) but below the time abscissa reaches its maxi-

mum as t= tc which thus can be roughly identified asthe moment yielding dip . We use the word roughly

because tc still depends more or less on g(t).

3. The Magnitude

To evaluate the magnitude ofdip, attention is

particularly paid to the extreme case t 0. In thiscase, Eq. (13) and Eqs. (14) and (15) are approached

by g(t) = [K(1 X)]1 and g(t) = t/R, respectively. Asfor Eq. (19), it can be approximated by f(t) =

(2KXt tp

2+ 1)Qp in which represents a sufficiently

small positive number introduced for replacing the

exponential function of Eq. (19). With all the results

substituted into Eq. (12), dip can be computed by:

dip = Xtc

2

(1 X)2Qp (21)

for R = 0, and

dip = KXtc

3

R2Qp (22)

forR > 0 since dip usually occurs at the initial stage

of a flood.

Although (21) and (22) are simplified models

based on the approximations within a short time range,

their validity can be extended for a long time to ex-

plore the impacts of Muskingum parameters on dip

without knowing the precise value. For example,

increasing Kor Xwill increase tc (see Eq. (20)) andeventually the dip; a rapidly rising inflow hydrograph

(i.e., smaller ) with high peak discharge may induceserious flow dips as well; a smallerR value may cause

the same result. By comparing Eqs. (9) and (10) with

(21) and (22), one realizes that model inconsistency

seems to worsen while the dip decays by just varying

X, but both become more serious as Kor x increases.It is deduced then thatR or the newly developed stor-

age model (1) contributes the most to the flood routings

in a long slightly rough channel carrying rapidly in-

creased inflow discharges. Generally, conventional

Muskingum algorithms failed in this flow condition

due to the prevalent inconsistency and discharge dips.

V. VERIFICATION

In what follows, a sensitivity analysis serving

as a supplemental example, is performed to help

verify Eqs. (20)-(22). To achieve the verification, it

requires, aside from the linear theory developed in

section 3, a complete model to compute tc and dip for

comparison. We differentiate Eq. (12) with respect

to tand let the result equal zero. This gives the fol-

lowing formula by using the Newton-Leibnitz inte-

gration rule with f(0) = 0:

J(u; tc)g(u)du0

tc= 0 (23)

where J(u; tc) =f(t u)

t t= tc. Sincef(t) involving the

shape of inflow hydrographs may become too com-

plicated to integrate Eq. (23), it is quite difficult to

express explicitly Eq. (23) as a formula for tc. So we

are using the numerical integration method to seek

the equivalent Nvalues:

J(u; tc)g(u) = 0j = 1

N (24)


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where u = (j 12

)u and tc = Nu. The worst dis-charge dip is thus calculated according to:

dip = f(tc u)g(u)du0

tc(25)

which can be also estimated by numerical integration

with an adequately small time interval u.

1. Determination of Influential Parameters

The inflow hydrograph is again assumed as a

Gaussian distribution, as given by Eq. (18). To sat-isfy the initial condition that I(0) = 0, an equivalent

approximation ofI/Qp = 0.001 (i.e., 0.1% error) is

imposed on Eq. (18). This allows us to obtain a value for a given tp by the formula 0.001 = exp[(tp/)2], hence inflow hydrograph shape on tp and Qponly. Accordingly, the factors affecting tc and dip/

Qp (see Eqs. (24)-(25)) are just tp, Kand X, since tcand dip/Qp are independent ofQp and R is a function

ofKandXonly. xs impact can not be ignored sinceit involves the action time off(t). To meet the

purpose, we choose a fairly long channel reach, say

90 km, to route outflow hydrographs. Model robust-ness can thereby be tested.

2. Results

Figures 1~3 are the calculated results based on

Eqs.(24)(25) forR = 0 and R 0. The adopted peakdischarge Qp and the time to peak tp of Eq. (18) are

assumed as 12.5 m3 /s and 6.69 hr, respectively. Any

nonzero R value is theoretically determined from Eq.

(8). It can be found from Figs. 1~3 that the critical

time tc increases monotonically at a decaying rate with

the increasing K, Xand tp, whether R is included or

not. The increment oftc becomes even more obvious

when R is included. This coincides with the afore-

mentioned deduction that g(t) of Eq. (14) does putoff the time to peak discharge and increase tc as a

result. It is also found from the figures that tc seems

to be bounded by an upper-limit, tp, forR = 0, imply-

ing that the worst discharge dip induced by CMM

appears before the peak time of an inflow hydrograph.

As shown in Figs. 1~3, dip exhibits a more com-

plicated variation trend compared with that oftc. In-

stead of monotonically increasing, dip increases then

decreases with the increasing tp. Dip also increases

with the increasing Kand Xwhether R is included or

not. Fig. 1 reveals that the incorporation ofR will

weaken the dip for Kgreater than about 10 hours. Thegreater the K is, the stronger the impact ofR on

K(hr)

tc

(R = 0)

tc

dip (R = 0)

dip

tc

(R = 0)

tc

dip (R = 0)

dip

K(hr)

10

8

6

4

2

0

-2

-4

-6

-8

tc

(hr)anddi

p

(m3/s)

10

8

6

4

2

0

-2

-4

-6

-8

tc

(hr)anddip

(m3/s)

5 10 15 20 250

5 10 15 20 250

X

X

tc

(R = 0)

tc

dip (R = 0)

dip

tc

(R = 0)

tc

dip (R = 0)

dip

7

6

5

4

3

2

1

0

-1

-2

-3

tc

(hr)andd

ip

(m3/s)

10

8

6

4

2

0

-2

-4

-6

-8

tc

(hr)anddip

(m3/s)

0.60.50.40.30.20.10

0.60.50.40.30.20.10

Fig. 1 Influence of Kon the variation of the minimum negative

discharge (dip ) and the corresponding critical time ( tc)

computed with or without the incorporation ofR. (Qp =

12.5 m3/s, tp = 6.69 hr, top: X= 0.2, bottom: X= 0.4)

Fig. 2 Influence of Xon the variation of the minimum negative

discharge (dip ) and the corresponding critical time (tc)

computed with or without the incorporation ofR. (Qp =

12.5 m3/s, tp = 6.69 hr, top: K= 5 hr, bottom: K= 15 hr)


8/13


reducing the dip becomes. Increasing Xfrom 0.2 to

0.4 also strengthens the effects ofKbut worsens the

dip value at the same time. This indicates that for a

long channel or a slowly moving flood flow, one

should try to incorporate R as wave dissipation gets

weaker. In brief, R contributes the most to the

routings of outflow hydrographs with nearly un-dis-

sipated flood peaks in a long channel.

Figure 2 makes clear the influence ofXon flow

dips. As mentioned previously, the increase ofXdoesincrease the absolute dip value of Eq. (25) and the

increment grows rapidly with the increasing X. In

addition, R has no impact on the reduction of dis-

charge dips, as Kis less than about 5 hours. So, there

is no need to introduceR for routing high-speed flood

flows in short channels. At this point, CMM would

be sufficient.

Figure 3 shows that it is more necessary for a

flood event with quick increasing inflow rates (i.e.,

small tp) than a slow one (i.e., big tp) to adopt R for

reducing dip values. Based on the point where two

dip curves intersect, it can be deduced that the effec-tive range ofR on dip increases with the increasing

K. Nevertheless, R becomes less important as tp ex-

ceeds a certain value, indicating there is no necessity

to include R for mild inflow hydrographs. All the

results are consistent with the conclusions of models

proposed in previous sections.

VI. OPTIMIZATION PROCEDURES FOR NEW

MODELS WITH CONSTANT R

1. Modified Muskingum Algorithm (MMA)

In this section, MMA is developed with the cali-

bration ofK, Xand R by the least square optimiza-

tion rather than the linear theory (i.e., by K= x/U,Eq. (6), and Eq. (8)), recognizing that most hydro-

logical engineering problems are nonlinear. To

achieve this and also for symbol saving, Eq. (4) is

reorganized into a more compact form of second-or-der accuracy as follows:

On + 1 = (4+ 6 1)On 2On 1

+ (1 2+ 2 2)In

+ (1 2 2 2)In + 1 (26)

where

= 1K(1 X)t

= 1KXt

= 1R

with the denominator = 2K(1 X)t+ 2R + t2.The optimal , , and values for Eq. (26) will

be determined by minimizing a target function de-

fined below using the least square optimization:

Tg = (On + 1 Qn + 1)2

n = 1

(27)

where On + 1 and Qn + 1 denote, respectively, the pre-

dicted and measured outflow discharges at the time

level n + 1. Differentiation of the target function withrespect to , , and and by letting the results equalzero leads to three simultaneous algebraic equations

in the matrix form of:

A2 BA CAAB B2 CBAC BC C2

=

F1F2F3

(28)

whereA = 4On 2In 2In + 1,B = 2In 2In + 1, C= 6On

2On 1 2In 2In + 1,D = OnInIn + 1, F1 =A(Qn + 1

+D), F2 =B(Qn + 1

+D), F3 = C(Qn + 1

+D). Symbol denotes the summation from n = 1 toN. Ndenotes the

tp (hr)

tp (hr)

tc (R = 0)

tc

dip (R = 0)

dip

tc (R = 0)

tc

dip (R = 0)

dip

8

6

4

2

0

-2

-4

-6

tc

(hr)anddip(m

3/s)

10

8

6

4

2

0

-2

-4

-6

tc

(hr)anddip(m3/s)

1210864

20

12

108

6420

Fig. 3 Influence of tp on the variation of the minimum negative

discharge (dip ) and the corresponding critical time ( tc)

computed with or without the incorporation ofR. (Qp =12.5 m3/s, X= 0.3, top: K= 5 hr, bottom: K= 10 hr)


9/13


total number of time steps. K, X, and R are readily

obtained once , , and are given. Flood routingthen proceeds by Eq. (26).

2. Modified Gills Algorithm (MGA)

In what follows, MGA will be developed by in-

corporating the CMM with R and Gills relative

storage. The storage S in Eq. (2) can be identified as

either the absolute or the relative storage of a river

reach, without affecting mass conservation. To make

it consistent, let the S in Eq. (1) denote the absolute

storage but in Eq. (2) it is regarded as relative storage.

Since absolute storage is rarely obtained, Gill

(1977) proposed the modified expression S = Sr where Sr represents the relative storage and theinitial storage. With the initial storage, the total num-

ber of involved calibrated parameters is increasedwithout yielding the unwanted roots of the general

sys tem mode l recommended by Chow and

Kulandaiswamy (1971). Following Gills modifica-

tion and the least square technique proposed by

Aldama (1990) the sum of square errors between pre-

dicted and measured relative storages becomes:

Tg = (+ 1In + 2O

n + 3On

Srn)2

j = 1

N

(29)

where O= dO/dt, 1 = KX, 2 = K(1 X), 3 =R andsr

n = the measured relative storage at time level n. Dif-

ferentiation of Eq. (29) with respect to and lettingthe result equal zero yield the explicit expression for

. Substituting this expression into the minimized

Tg for j, j = 1 ~ 3, gives the following matrix equa-tion in tensor notation:

Mijj = fi (30)

where:

M11 =InIn 1NImIn

M12 =InOn 1NImOn

M13 =InO

n 1

NImO

n

M22 =OnOn 1NOmOn

M23 =O nOn

1NO

mOn

M33 =OnO

n 1

NOmO

n

and M21 = M12, M31 = M13, M32 = M23. The forcingfunctionsfi read:

f1 =InSrn 1

NImSr

n

f2 =OnSrn 1NOmSr

n

f3 =On

Srn

1NOm

Srn

In the above, Sr at the time level n is computed ac-

cording to the discrete form of Eq. (2) with Sr= 0 for

n = 1. Once variables , j,j = 1 ~ 3, are computed byEq. (30) using Cramers rule, K, X, and R can be de-

termined and Eq. (26) is again employed to compute

the outflow rate at each time level. One of the most

attractive things about MMA or MGA is that they are

formulated without breaking the beauty/simplicity of

Muskingum-type models and all parameters are cali-

brated out explicitly.

VII. MODEL COMPARISONS

This section compares the traditional (i.e., CMM

& GILL) and modified (MMA & MGA) routing mod-

els via numerical experiments. The equation govern-

ing the experiment is the parabolicized Saint Venant

equation that neglects inertia terms but implicitly in-

cludes pressure forces. For wide rectangular

channels, the parabolicized equation has a form of

advection-diffusion properties and reads:

Q

t+ 5

3

Q

Bh

Q

x=

Q

2BSf

2Q

x2(31)

Its solution will be sought by solving simultaneously

with the continuity equation below:

ht

= 1B

Q

x(32)

where Q = discharge, B = channel width, h = water

depth, and Sf= the energy gradient based on Mannings

formula. The observed outflow hydrograph required

for parameter estimations is borrowed from the nu-

merical solution of Eqs. (31)-(32) at x = 0.8L so as to

reduce the influence of the downstream boundary

condition and satisfy the basic assumption that routingsare performed for a semi-infinite river reach. The

numerical solution serves as a benchmark solution for

model comparisons aside from parameter estimations.

Solving Eq. (31) requires the prescription of up- and

down-stream boundary conditions of which the former

is introduced as a Gaussian distributed inflow

hydrograph and the latter is stated as a fixed value of

normal water depth corresponding to a given base flow

rate.

Figure 4 shows the Gaussian inflow hydrograph

and the four predicted outflow hydrographs routed

by CMM, GILL, MMA, and MGA. Concerning thetest of model forecasting ability and robustness, the


10/13


numerical experiment is performed in a 10 m wide,

90 km long rectangular channel with a bottom slope

of 0.0004 and a Mannings roughness coefficient of

0.029. The base flow rate is assumed to be 17.5 m3/s

with which normal velocity and water depth are

computed. Since the channel length for this particu-

lar flow case is quite long, the numerically solved

outflow peak discharge will be much lower than that

of the inflow, in spite of the fact that the channel bed

is only a little rougher. This condition satisfies fairly

well the best criteria for employing R.

As one can see clearly from Fig. 4, CMM com-

pletely fails due to its significant model inconsistency,

regardless of the nonobvious discharge dip. The large

under-estimated outflow peak discharge and time to

peak (phase error) reveal that CMM is not adequate

for flood routing in long channels. When Gills con-

cept of relative storage is adopted, the outflowhydrograph labeled GILL is predicted with better

precision, though the dip value becomes greater. The

predicted outflow peak discharge is now quite close

to the benchmark solution in spite of the evident phase

error. For the case with R but without using Gills

relative storage (i.e., MMA), the dip in the routed

outflow hydrograph is much reduced as expected.

Also, the predicted outflow peak discharge is just

slightly higher than the benchmark solution. Though

there exists the problem of anti-phase-error (i.e., the

time to peak is over-estimated), what really discour-

aged us was the oscillatory tail attached to the pre-dicted outflow hydrograph since its amplitude was

too large to be ignored. The tail, though damping

out rapidly, resulted in a large root-mean-square er-

ror defined as Erms =1

N(Oj Qj)

2j = 1

N

, hence re-

ducing the applicability of MMA. The reason why

MMA evolves such an apparent oscillatory tail is be-

cause it strongly meets the criteria for adopting Eq.(14) the value is as low as 61.4 hr2 being thehighest of the four models in Table 1. In contrast,

GILL is good for the prediction of recession curves

while MMA is for the rising limb. Consequently,

MGA has been adopted for predicting the outflow

hydrograph. It is found that the prediction is so ac-

curate that the hydrograph almost has coincided with

the target one. Indeed, the phase error has become

trivial, thus the outflow peak discharge is deemed

adequate, and the dip value shows lower than that of

the GILL curve. By realizing the least square opti-

mization minimizes the overall instead of the localerror of a predicted outflow hydrograph, MGA may

not remedy its dip to the minimum. Aside from the

smallest root-mean-square error, the most pleasant

result about MGA is that it does not evolve the oscil-

latory tail. This is because its absolute value justexceeds 11 hr2 and itsR value is 2.1 times lower than

that of MMA; this fact serves to fasten the exponen-

tially decaying rate of the sinusoidal wave and mak-

ing the oscillatory tail invisible. Indeed, the newly

introducedR, after linked to the relative storage, re-

duces the dip of GILL, alleviates model inconsistency,

and shrinks the range of phase errors to the minimum

among the four models. As a result, MGA offers thebest predictability for the flood routing in long

channels. The same conclusion can be drawn from

Table 1 by comparing the corresponding parameter

values.

Table 1 lists the root-mean-square errors of all

models and the parameter values used for routing the

outflow hydrographs in Fig. 4. In the last two

columns, tc and dip values are obtained from Eqs. (20)

~(22). The associative values are 0.08 and 0.02 forEqs. (21) and (22), respectively. It can be seen eas-

ily that the K and X values of CMM differ greatly

from those of other models. The main reason is surelydue to the super long channel reach and the adopted

t(hr)

Inflor

Outflow

CMM

GILL

MMA

MGA

220

170

120

70

20

-30

-80

Q(

m3/s)

10 20 30 40 50 600

Fig. 4 Comparisons of the routed outflow hydrographs by CMM,

GILL, MMA, and MGA for a 90 km long channel with n =

0.029 and S0 = 0.0004

Table 1 The calibrated parameters and root-mean-

square errors of all models in Fig. 4

K X R E rms tc dip

(hr) (hr2) (m3/s) (hr) (m3/s)

CMM 127 0.085 26.6 6.38 -9.01GILL 7.43 0.392 22.0 5.57 -47.6

MMA 3.87 0.472 16.4 21.2 4.91 -7.86

MGA 7.23 0.379 7.79 9.71 5.51 -34.9


11/13


storage function. Nevertheless, the least square op-

timization is the key procedure leading to such a large

Kvalue for CMM, since in this case K is no longer

determined by x/U, but is deduced from the statisti-cal point of view. Owing to the excessively large K

value (about an order of magnitude greater than the

others) of CMM,R becomes quite effective and plays

a vital role in the routing procedures of MMA and

MGA. Furthermore, the KXvalue of CMM, given in

Table 1, is 3.7 times higher than those of other models,

implying the largest tc outcome as a result. The

smallest Xvalue in Table 1 also implies that CMM

shows the most significant inconsistency with the

correct A-D expression as indicated by Eqs. (9) and

(10). Since the predicted tc and dip values are so close

to those in Fig. 4 (as can be clearly seen), models

(20)(21)(22) are successfully developed regardless of

their simplicity. Finally, it will not help too much toadopt R or relative storage alone since GILL and

MMA produce roughly the same root-mean-square

errors. Only by using both can MGA obtain the small-

est error. For a shorter channel of 30 km, Fig. 5(a)-

(c) shows about the same results: MGA is superior to

all other variants of CMM. At this point, the postu-

lation by Laurenson (1959) may be further addressed:

flood routings in fairly long channels can be improved

greatly by modifying the conventional storage func-

tion to include RdO/dt. As for the threshold length

used to classify a channel as long or short, it is

suggested to be 30 km in regard to the present analy-

sis and most of the routing length successfully

adopted for CMM in the past. The threshold value is

chosen without rigorous theoretical background but

based on our best educated judgment.

VIII. CONCLUSIONS

In order to solve the inconsistency problem, this

paper introduces a time constantR to the conventional

Muskingum storage function. The newly added term

improves the inaccuracy from Qxxx to Qxxxx, and hence,

reduces the phase error and refines outflow peak

discharge. The inaccuracy is found to be proportionalto x4 or to D3/U2 by the linear theory of Taylor se-ries expansion. As a rule of thumb,R must be dropped

out from the Muskingum storage function if it is cali-

brated as negative.

Based on a small time approximation, the oc-

currence time ofdip (i.e., tc) and its magnitude can

be easily determined according to Eqs. (20)-(22).

Accordingly, dip occurs before an inflow hydrograph

reaches its peak. The higher the peak value is, the

larger the dip becomes. Other parameters such as K,

X, and tp also have positive effects on discharge dips.

Increasing R or decreasing the growth rate of aninflow hydrograph, however, decreases the dip

A careful study of the convolution integral re-

veals that flow dips result from the negativef(t) and

predicted outflow hydrograph shapes relate closely

to g(t) which governs as well the phase error. Also,

parameters affecting the normalized dip (dip/Qp) are

only tp, K, and X, of which Xvaries in an opposite

way with model inconsistency as Kdoes in a positive

way. Thus, the best timing for usingR is to route the

outflow hydrographs in a long but a slightly rough

channel that carries rapidly rising inflow dischargeswith high peak values.

Fig. 5 Performance of traditional and modified routing models

for a 30 km long channel

t(hr)

(a) n = 0.029, S0 = 0.004

Inflor

Outflow

CMM

GILLMMA

MGA

220

170

120

70

20

-30

Q(

m3/

s)

5 10 15 20 25 300

t(hr)

(b) n = 0.08, S0 = 0.0004

Inflor

Outflow

CMMGILL

MMA

MGA

220

170

120

70

20

-30

Q(

m3/s)

5 10 15 20 25 300

t(hr)

(c) n = 0.08, S0 = 0.0001

Inflor

OutflowCMM

GILL

MMA

MGA

220

170

120

70

20

-30

Q(

m3/s)

5 10 15 20 25 300


12/13


Adopting Gills concept of relative storage will

increase the dip value by trading off a more precise

portrayal of outflow peak discharge. CMM and GILL

tend to underestimate the time to peak discharge but

MMA overestimates it, primarily due to the nature of

the response function g(t). As


13/13


Koussis, A. D., 1983 Unified Theory for Flood and

Pollution Routing, Journal of Hydraulic

Engineering, ASCE, Vol. 109, No. 12, pp. 1652-

1663.

Laurenson, E. M., 1959, Storage Analysis and Flood

Routing in Long River Reaches,Journal of Geo-

physical Research, Vol. 64, pp. 2423-2431.

Nash, J. E., 1959, A Note on the Muskingum Flood-

Routing Method,Journal of Geophysical Research,

Vol. 64, pp. 1053-1056.

Ngan, P., and Rusell, S. O., 1986, Example of Flow

Forecasting with Kalman Filter, Journal of Hy-

draulic Engineering, ASCE, Vol. 112, No. 9, pp.

818-832.

ODonnell, T., 1985, A Direct Three-Parameter

Muskingum Procedure Incorporating Lateral

Inflow,Hydrological Science Journal, Vol. 30,

No. 4, pp. 479-496.Ponce, V. M., and Theurer, F. D., 1982, Accuracy

Criteria in Diffusion Routing,Journal of Hydraulic

Division, ASCE, Vol. 108, HY6, pp. 747-757.

Singh, V. P., and McCann, R. C., 1980, Some Note

on Muskingum Method on Flood Routing,Jour-

nal of Hydrology, Vol. 48, pp. 343-361.

Smith, G. D., 1985,Numerical Solution of Partial Dif-

ferential Equations: Finite Difference Methods,

3rd edition, Oxford University Press, New York,

USA.

Van Geer, F. C., Te Stroet, C. B. M., and Zhou, Y.,

1991, Using Kalman Filtering to Improve and

Quantify the Uncertainty of Numerical Ground-

water Simulations 1: the Role of System Noise

and Its Calibration, Water Resources Research,

Vol. 27, No. 8, pp. 1987-1994.

Weinmann, P. E., and Laurenson, E. M., 1979, Ap-

proximate Flood Routing Methods: A Review,

Journal of Hydraulic Division , Vol. 105, HY12,

pp. 1521-1536.

Wood, E. F., and Szollosi-Nagy, A., 1978, An Adaptive

Algorithm for Analyzing Storm-Term Structural

and Parameter Changes in Hydraulic Prediction

Models, Water Resources Research, Vol. 14, No.

4, pp. 577-581.

Manuscript Received: May 11, 2004

Revision Received: Apr. 10, 2005

and Accepted: May 24, 2005

APPENDIX

Of all the approximate methods, as addressed

by Weinmann and Laurenson (1979), the one

typified by the Koussis model (Koussis, 1978;

Koussis, 1983; Koussis and Snenz, 1983) is the most

general. Koussis model assumes basically the lin-

ear variation of discharge within any time interval.

The Laplace transformation yields this model of the

form:

Qj + 1n + 1 = Qj

n + 1(1 + 1

Cr) + Qj

n( +1

Cr)

+ Qj + 1n () (A1)

where

=2 + Pg PgCr2 + Pg + PgCr

(A2)

To derive the modified equation of Eq. (A1), the

same procedure as described in section II is used toyield:

Qt + UQx = Qxx [(1 + 1

)U2t

2 Ux

2]

numerican dispersion

+ ET (A3)

where the truncation errorET reads:

ET

= Qxxx[(2 + )U3t2

6(1 )+ Ux

2

6 U

2xt2(1 )

+DUt

1 ]

Qxxxx[DUxt2(1 )

(1 + )DU2t2

2(1 )+ D

2t2

]

Qxxxxx[UD 2t2

2(1 )] Qxxxxxx(

D3t2

6)

+ O(x3, x2t, xt2, t3) (A4)

Equating the numerical dispersion term in the bracket

of Eq. (A3) to the physical dispersion D yields the

expression for as given by Eq. (A2). Further analy-sis of the extreme case that x 0 and t 0 leads

the truncation error given by Eq. (A4) to the follow-ing nonzero result

ET = D2

UQxxx (A5)

Since has a limit value of 1 and t/(1 ) a limitvalue ofD/U2. It is apparent then that the Koussiss

model of differential form is not identical to the ex-

pected A-D equation, that is, model inconsistency

occurs.

Flood Routing in Long Channels

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