LUO BOKUN Bureau Office, Wuhan, People's Republic of China ...hydrologie.org/hsj/320/hysj_32_04_0485.pdfABSTRACT The Muskingum flood routing method is widely used by hydrologists and

Hydrological Sciences - Journal- des Sciences Hydrologiques, 32, 4,12/1987

Some problems with the Muskingum method

LUO BOKUN Bureau of Hydrology, Yangtze Valley Planning Office, Wuhan, People's Republic of China QIAN XUEWEI Heilongjiang Provincial Hydrological Service, People's Republic of China

ABSTRACT The Muskingum flood routing method is widely used by hydrologists and good results are frequently achieved. However, there is still some dispute about the Muskingum method. In this paper it is shown that the Muskingum method is an approximate solution of the instantaneous unit hydrograph (IUH) of the lag and route flood routing method. The integration solution of the Muskingum method for multiple river reaches is also derived. The negative response issue is discussed in relation to the basis of the linear theory of hydrological systems.

Certains problèmes rencontrés dans l'application de la méthode de Muskingum RESUME La méthode de Muskingum pour l'étude de la propagation de la crue vers l'aval est largement utilisée par les hydrologues et on aboutit souvent à de bons résultats. Cependant il y a encore certaines contestations à propos de cette méthode de Muskingum. Dans cet article, on montre que la méthode de Muskingum est une solution approchée du temps de réponse et de la propagation de l'onde de crue de 1'hydrogramme unitaire instantané (HUI). On en déduit aussi la solution d'intégration de la méthode de Muskingum pour plusieurs biefs de la rivière. Le résultat correspondant a une réponse négative est discuté en relation avec les bases de la théorie linéaire des systèmes hydrologiques.

INTRODUCTION

The Muskingum flood routing method is based on the assumption of a linear relationship between the inflow to and the outflow from a river reach and the reach storage. This method uses two equations, namely the storage equation which is written as:

S(t) = K[xl(t) + (1 - x)Q(t)] (1)

and the continuity equation:

dS(t)/dt = I(t) - Q(t) (2)

Open for discussion until 1 June 1988. 485

486 Luo Bokum & Qian Xuewei

where S(t) is the reach storage between the upstream and the downstream routing sections, I(t) and Q(t) are the rates of inflow and outflow, respectively, x is a weighting factor and K is called the storage coefficient and has the dimension of time.

Equations (1) and (2) can be combined into:

K(l - x)f + Q = I - Kxf (3)

If the inflow I(t) is the Dirac delta function, <5(t), equation (3) can be integrated to yield (Venetis, 1969):

u ( t ) = K(i - x)» e x p ^ ¥ a ^ > " T^l ô ( t ) ( 4 )

Equation (4) is the expression for the IUH of the Muskingum method for a single river reach.

When equation (3) is solved by a finite difference solution, the following equations are easily obtained:

QmAt = VmAt + ClI(m-l)At + C2Q(m-l)At (5)

where :

C = -(Kx - 0.5At)/(K - Kx + 0.5At)

C = (Kx + 0.5At)/(K - Kx + 0.5At) (6)

C2 = (K - Kx - 0.5At)/(K - Kx + 0.5At)

where At is the finite difference sample interval and m is an integer which indicates the number of intervals from the time origin.

The lag and route flood routing method considers that the propagation of a flood wave is subjected to the effects of translation and attenuation; the impulse response is that of a single linear reservoir reach delayed by the lag, x (Nash, 1960; Dooge, 1973). This model has the system response:

u(t)=iexp(-^ (7)

where x and K^ are parameters for the single reach, If the routing reach of this model is divided into n subreaches

each delayed by the lag, x, and having the storage constant, KL, the system response for the overall routing n times is (Luo et al., 1978; Wang, 1982):

, s ! ft - nx-i f t - nTi u ( t ) = i ^ a H ^ e*p(—"É -̂J < 8 )

In this paper we will derive the integration solution of the Muskingum method for multiple river reaches. The relationship between the Muskingum method and the lag and route method for multiple reaches will be discussed on the basis of the linear theory of hydrological systems. The issue of initial negative outflows with the Muskingum method will also be explored.

Some problems with the Muskingum method 487

THE INTEGRATION SOLUTION OF THE MUSKINGUM METHOD FOR MULTIPLE RIVER REACHES

Taking the Laplace transform of equation (3) and the initial conditions of Q(0) = 0 and Q'(0) = 0, we obtain:

L[Q] = K(l - x) s

S + K(l - x)

L(I)

In equation (9), if the inflow, I(t), is a Birac delta function, ô(t), then the outflow, Q(t), is the IUH. Taking L[6(t)] = 1, equation (9) becomes:

L[u(t)] = K(l - x) 2 1 - x

K(l - x)

(9)

(10)

When equation (10) is applied to multiple river reaches, i.e. the total routing reach is divided into n identical subreaches, then we have a system composed of n subsystems in series with the identical values, K and x, for each subreach. Thus,, by using the outflow of a preceding subreach for the inflow to the succeeding one and performing outflow routings one after another from the first subreach down to the n-th subreach, equation (10) becomes:

L[u(t)] = K(l - x) 2

S + K(l - x)

Expanding equation (11) by means of the binomial theorem, and evaluating the inverse image of each term, we can obtain:

u(t) Li=0

1 II

(-1) (1 ) K(l - x)n_i+1r(n)

exp K(l - x)

K(l - x)

+ (-D

n-i-1

<5(t)

(ID

(12)

Equation (12) is the IUH of the Muskingum method for multiple identical river reaches, i.e. the integration solution of the Muskingum method by successive routing.

It will be proved that the following necessary condition holds for equation (12):

J ^ u ( t ) d t = 1 (13)

S i n c e :

„ n - l ( - l ) 1 ( i n ) x 1

I u ( t ) d t = I. „ • —— L J0 i=0 ( 1 _ x ) n r ( n _ ± ) Jo K( l - x )

n - i - 1

exp K( l - x ) K( l - x ) + ( - 1 )

^1 - xJ jTô(t)dt

488 Luo Bokun & Qian Xuewei

and:

/™<5(t)dt = 1

and:

CO

Jo n- i -1

f •'o

K(l - x )

n - i - i -m

exp

dm

K(l - x )

r ( n - i )

K(l - x )

where m = t / [ K ( l - x ) ] , t h e r e f o r e :

r u(t)dt = in~l ( - 1 ) 1 ( i n )n

x i + ^ ^

J0 1=0 ., , n , , n (1 - x )

i=0

1

(1 - x )

„n

(1 - x )

= 1

n i=0 (-DW

( 1 4 )

The first three moments of the integration solution of the multiple reach Muskingum method can be shown to be:

M nK (1) u

N ( 2 ) = nK2(l - 2x) u

N ( 3 ) = 2nK3(3x2 - 3x + 1) u

(15)

(16)

(17)

Where M^D is the first moment about the time origin, and N<r> is the rth (central) moment about the mean. Since the first three moments of an inflow and corresponding outflow can be computed from the given inflow and outflow, the values of n, K and x defining the IUH can be found from equations (15), (16) and (17).

THE IUH OF THE MUSKINGUM METHOD AS AN APPROXIMATE SOLUTION OF THE LAG AND ROUTE METHOD

Derivation of formulae for the IUH of the Muskingum method and the lag and route method by system analysis

The relationship between the inflow and the outflow of the generalized linear hydrological system can be expressed by (Chow, 1975):

Q(t) =

«m+l „m _ a D + a ,D +....+ a„D m ïii-1 O . _n+l n b D + b nD + n n-1

Kt) (18) + bQD + 1

Q<t> =|§}i(t> (19)


where D = d/dt is the differential operator and aj, bj are coefficients. If both the a^ and the bj are constants or independent of I(t) and Q(t), equation (18) or (19) governs the behaviour of a linear system. The IUH of the Muskingum method and that of the lag and route method can be derived from these formulae (Qian & Luo, 1981).

For a single river reach Let M(D) = e~aoD and N(D) = b0D + 1. Equation (19) can then be reduced to:

Q(t) = - * e~aoD I(t) (20) bQD + 1

The function I(t - a0) can be expanded in a Taylor's series and written as:

I(t - aQ) = e_ a ° D I(t) (21)

When the inflow is a Dirac delta function, 6(t), the outflow becomes the IUH, i.e.:

u ( t ) = 4-^W e ~ a ° D ô ( t )

and so:

u ( t ) = ^ - p + (\ / b o ) 6(t-a Q) (22)

Using the Laplace transform of equation (22):

1 * ~ ao u(t) = — exp(—— ) a0 i t bo °o (23)

u(t) = 0 a Q > t

can be obtained. When a 0 = T, b 0 = KL, equation (23) becomes:

u(t) = ~- exp(- :^^ L) (24a) K L &L

which is the expression of the IUH for a single river reach with both the translation effect and the attenuation effect of flow (Qian & Luo, 1981) as shown in equation (7).

When a 0 = Kx, b Q = K(l - x ) , equation (23) becomes:

u ( t) _ 2 exp[- • t-~—~1 (24b) v ' K(l - x) vt K(l - x ) J K '

This is a version of the expression for the IUH of a single river reach. It also embodies these two effects of translation and attenuation except that the translation effect is reflected in the product of the two coefficients K and x.

If the expression for the expanded function of e ° is truncated to the second order, that is, the inflow is such that its second and higher order time derivatives are small enough in magnitude to be negligible, then:

e~ a° D « 1 - aQD

490 Luo Bokum & Qian Xuewei

If aQ = Kx, b 0

equation (22). K(l - x), then equation (4) can be derived from

For multiple river reaches When the length of a river reach between the upstream section and the downstream section is quite long, the reach should be divided into n identical subreaches and the flood routing carried out reach by reach. From equation (22), the following equation holds:

-na„D u(t) =

b n VD + <l/b0)J

6(t) (25)

Using the Laplace transform, equation (25) becomes:

1 A ~ na0. u ( t ) =^7(nT H-g^H

n-l

exp f- (26)

If aQ = T, b Q = KL, equation (26) can be reduced to the formula for the IUH of the lag and route method for multiple reach reaches (viz. equation (8)):

u(t) ft - nti

n-l

KLr(n) ^ KL exp I

t - nr-i

KT -1 (27a)

If a Q = Kx, b Q = K(l - x ) , equation (26) can be reduced to

another form of the lag and route method for multiple river reaches:

u(t) = K(l - x)T(n)

t - nKx

K(l - x)

n-l

exp t - nKx

K(l - x) (27b)

Similarly, the Laplace transformation may be applied to equation (25)

where e -anD 1 - a0D is taken.

If a 0 = Kx, b 0 = K(l - x ) , the integration solution for the

Muskingum method for multiple river reaches (equation (12)) can

be obtained.

Relationship between the storage equations for the Muskingum method and the lag and route method

Changing equation (2) into (bQD + l)Q(t) = e-aouI(t), and expanding

e-a° in a Taylor's series, a simultaneous solution with the continuity equation can be performed. If a0 = Kx, b 0 = K(l - x) , we obtain:

a a2

S(t) = bQQ(t) + ~ I(t) - ̂ y DI(t) + ...

+ (-1) — D mi

I(t)

= K(l - x)Q(t) + Kxl(t) - ^-jf- DI(t) +

+ (.!)»-! HELL- D»-1 I(t) + . m!

(28)


This is the storage equation corresponding to the lag and route method. Comparing equation (28) with equation (1), obviously the latter is an approximate expression for the former when its second and higher order time derivatives are neglected.

Two important properties of the lag and route method

The lag and route method becomes the model of linear reservoirs in series in which pure attenuation is considered when either T = 0 in equations (24a) and (27a) or x = 0 in equations (24b) and (27b).

x) = 0. If x = 1, then a0 = Kx = K, and b 0 = K(l a0 and b 0 into equation (20), whence Q(t) = e "°"I(t). simultaneously with equation (21), finally we get:

Substitute Solving

Q(t) = I(t - K) (29)

For T = Kx = K:

Q(t) = I(t - T)

and its IUH can be expressed as

u(t) = <5(t - T )

(30)

(31)

which indicates the concept of a single linear channel in which there is pure translation effect but no attenuation effect exists.

The parameter, x, is used for modifying the shape of a flood hydrograph and is representative of the translation effect

The channel characteristics can be visualized according to the range of values of x, as shown in Table 1, so that a choice between different methods of flood routing can be made.

Table 1

x range Characteristics Common routing method

x = 0

o < x < y 2

x = %

y2 < x < 1

x = 1

Pure attenuation effect

Attenuation effect is dominant

Attenuation effect and translation effect are the same

Translation effect is dominant

Pure translation effect

Nash model

Muskingum method

Lag and route method



In general, if the inflow to a reach is gentle and the duration of the rising limb of the inflow approaches the travel time between


the inflow section and the outflow section, the Muskingum method is customarily used; conversely, if the inflow rises or recedes steeply and the duration of the rising limb of the inflow is much shorter than the travel time between the inflow section and the outflow section, it is not appropriate to use the Muskingum method. In this case, however, the Muskingum method of successive routing through subreaches can be used. Of course, it is more appropriate to use the lag and route method in this case since the method cannot only avoid any negative outflow, but also guarantees certain computational accuracy.

PROBLEM OF THE NEGATIVE RESPONSE

Negative response of the integration solution of the Muskingum method

The integration solution of the Muskingum method is as stated above. Now consider the following properties of the IUH of the Muskingum method.

Roots of the IUH of the Muskingum method Since u(t) in equation (12) is not equal to zero at t = 0 , i.e. u(0) £ 0, then t = 0 is not a root of u(t) = 0. The roots of u(t) = 0 must be found from values of t other than t = 0. In addition, based on the properties of the impulse function (<5(t) = °° at t = 0, and ô(t) = 0 at t ? 0), the term 6(t) in equation (12) must be zero in determining the roots of u(t) = 0. Therefore the roots of u(t) are obtained provided that the following equation of the (n - l)th order is solved:

u(t) = r. n-l (-DV1 1) i=0 T(n - i) K(l - x)

n-i-1 x1 = 0 (32)

The number of roots indicates the number of times that the response function changes sign.

Properties of the IUH of the Muskingum method at t = 0 As indicated above, u(0) ̂ 0. However, when t approaches zero from the positive direction, that is t ~> 0+, the terms including t in equation (12) can be considered as zero; the term 6(t) in equation (12) is equal to zero based on the same properties of the impulse function, ô(t), as described in the preceding subparagraph. The following equation can be derived from equation (12):

u(0+) =

n-l n (-1) (n 1) x

n-l

, n+1 (33)

K(l - x)

The impulse of u(t) at t = 0 is (-l)n [x/(l - x)]n6(t). When n is an odd number, a negative impulse emerges at t = 0 , which indicates that positive and negative impulses of u(t) at t = 0 emerge alternatively.

The value of u(t) at t = 0 can be expressed by:


/ " i \ n - 1 , n -i \ n - 1 n

u ( 0 ) = Azl> < n - Dx + ( _ 1 > n rj^ 6 ( t ) ( 3 4 )

K(l - x) X l X

t As the result of exp[- . _ ] -»- 0 for t -> <*>, so u(t) -> 0 for

t -*• °°.

The above properties are shown in Table 2. Obviously, the IUH of the Muskingum method is an oscillatory curve along the t-axis. The curve intersects the t-axis at the beginning of the curve. The number of intersection points on the t-axis increases with n and is equal to (n - 1).

Negative outflow of the finite difference solution of the Muskingum method

The formulae for this method are equations (5) and (6). The formulae for the finite difference solution of successive routing subreaches are given in the form (East China College of Hydraulic Engineering, 1977; Yangtze Valley Planning Office, 1979):

in

P on

P mn

which

cn

o En i=

A =

=lBi

= C1

for

cn-: o

+ C o

m =

l c . .

°2-

0

-i A:

and

for m > 0 and (m - i) > 0 (35)

_ n! (m - 1)! i _ i! (i - 1)! (n - i) ! (m - i)!

where n = the number of subreaches; m = an integer which indicates the number of routing steps.

When x > 0 and At < 2Kx, the formula for the Muskingum method, equation (5), may yield a negative outflow, since C0 is negative. Further, when n is an odd number, the formula for the Muskingum method, equation (35), Pon = C° < 0. Thus, this method may also yield a negative outflow.

When x > 0 and At > 2K(1 - x), the tail of the outflow hydrograph defined by equation (5) may produce negative outflow or pulsative phenomena. Similarly, the tail of the outflow hydrograph described by equation (35) may produce negative outflow. Therefore, in order to avoid negative outflow, the following restrictions should be considered in selecting the routing period

2K(1 - x) S At S 2Kx (36)

The reason that a negative response is produced under some circumstances is that the storage equation of the Muskingum method cannot actually reflect the fact that the river flow does not vary linearly with time or along the river. As shown in Fig.l, both at t = tx and t = t2 the inflow and outflow are unvarying. Although at these times the water surface profiles and hence the storages in the river channel are different, the


o i z>

CM

"

7 Q

:*-

?

CO

II c

(

j /

r-u

o 3 II o " O -rj

CC -S

o II .

O

CM 09

si TO

H

§ > ï= C to 0) 3 -C

LU . 2 .

c

X -M CM

Il II

eE

c~ co CM r-> « - • * '

X X

°2 "t o" co

X

< * CM

O

II X

CM

1 >

CM

O II

X to

+ X

> to 1 ^

CO

X

> to co

+ X Ci > CM

1 >-•*

o

o

o CM

+ X

> O CM


Fig, 1 Storage varying with time in a river reach.

storage equation at these times remains the same. In summary, an unreliable storage equation is used in the

Muskingum method; in addition this method is an approximate solution of a generalized linear hydrological system model. Therefore, both the integration solution and the finite difference solution of the Muskingum method may have a negative response. In order to avoid a negative response without lowering the routing accuracy the routing period should be restricted.

To eliminate the negative response of the Muskingum method, other methods may be adopted. For example, the Nash model (which can be regarded as the best among the versions for the IUH of the Muskingum method) can eliminate the negative response by setting x to zero.

The lag and route method is another method to eliminate the negative response by means of the translation effect. In essence, these methods can be used to eliminate the negative response by modifying their storage equations.

CONCLUSION

Both the Muskingum method and the lag and route method are derived from the equation for a linear hydrological system, and it is shown from their storage equations that the Muskingum method is an approximate solution for the lag and route method. The integration solution for the Muskingum method of successive routing through subreaches is derived.

The range of x in equation (27b) is from 0 to 1, and its value can be used as an index in choosing among the different flood routing methods.

Because the storage equation of the Muskingum method cannot accurately describe the change of storage in the river reach, the negative response of the Muskingum method is unavoidable from the mathematical derivation, but it is physically unrealistic. Nevertheless, the negative response can be avoided by limiting the finite difference interval, At.


ACKNOWLEDGEMENTS Special thanks are given to Yang Ganghe for his suggestions in English and a critical review of this paper.

REFERENCES

Chow, V.T. (1975) Hydrologie modeling. Selected Works in Water Resources, IWRA, March.

Dooge, J.C.I. (1973) Linear theory of hydrological systems. USDA Rgric. Res. Service Tech. Bull. 1468. US Government Printing Office, Washington, DC, USA.

East China College of Hydraulic Engineering (ECCHE) (1977) Flood Forecasting Method for Humid Regions of China. ECCHE, Nanking, China.

Luo, B.K., Yang, G.H. et al. (1978) An application of Nash's IUH to flood routing in channels (in Chinese). Selected Works in Techniques and Experiences of Hydrologie Forecasting. China Water Resources and Electric Power Press.

Nash, J.E. (1960) A unit hydrograph study, with particular reference to British catchments. Proc. Instn Civ. Engrs 17, 249-282.

Qian, X.W. & Luo, B.K. (1981) Relationship and review between the IUH of lag and route method and the IUH of Muskingum method (in Chinese). Yangtze River, No.2.

Venetis, C. (1969) The IUH of the Muskingum channel reach. J. Hydrol. 4, 185-200.

Wang, Qinliang (1982) The model of lag-instantaneous flow concentration. Hydrology, Beijing, China, No.l, 13-19.

Yangtze Valley Planning Office (1979) Hydrologie Forecast Methods in China (in Chinese). China Water Resources and Electric Power

Received 18 April 1986; accepted 21 April 1987.

LUO BOKUN Bureau Office, Wuhan, People's Republic of China ...hydrologie.org/hsj/320/hysj_32_04_0485.pdfABSTRACT The Muskingum flood routing method is widely used by hydrologists and

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