Encyclopedia of Environmental Science and Engineering, Volume I and II

HYDROLOGY 475

Channel routing is also based on continuity, Eq. (11).

Before it can be utilized this equation must be rewritten in

fi nite difference form

II OO SS

t

12 1 2 2 1

22









. (29)

Also, some assumption must be made from which storage

can be computed. The Muskingum method linearizes the

problem and assumes that storage in a whole channel reach

is completely expressible in terms of infl ow and outfl ow

from the reach, namely

S 
 K [ xI + (1 − x ) O ]. (30)

Substituting from (16) in (15), following Linsley,^2 the result

obtained is,

OcIcIcO^2021121 
^ (31)

where c 0 , c 1 and c 2 are functions of K, x and t and c 0 + c 1 +

c 2 = 1. The infl uence of x is illustrated in Figure 6. Extending

this equation to N days:

OcI cI ccI ccI

ccI ccI

c

NNN N N

NN

N

=

+

011201122

2

2

02 2

2

13

 











Λ

22

cI 11 c O 2 1

 N^1.

(32)

Usually, because c 0 , c 1 and c 2 are less than one, terms in c 3

and higher are ignored, but whatever the simplifi cation, the

result is of the form

OaI aI aI aI

aI

NNN N N

N

= 1213243

54







 etc.

(33)

Once again the set of equations for O 1 to O R can be written as

{ O } 
 [ I ]{ a }. (34)

(N.B. The use of I for infl ow is unfortunate and leads to con-

fusion with the identity matrix. Also O is confusable with the

null matrix.) It must be admitted that, from the theory, only

a 1 , a 2 and a 3 are independent, but in practice the precision

with which a solution can be obtained for c 0 , c 1 and c 2 does

not justify calculating a 4 , a 5 etc. from the fi rst three. The best

approach at this stage is once more to resort to least squares

fi tting, recasting Eq. (34) in the form of Eq. (8.)

Many hydrologists calculate the routing coeffi cients by

evaluating K and x in the traditional graphical method.

Many assumptions are made in the Muskingum method,

such as the linear relationship between storage and discharge,

and the implied linear variation of water surface along the

reach. In spite of these assumptions the method has proved

its value.

Another diffi culty which occurs with the Muskingum

method usually occurs when the travel time in the reach and

the time increments for the data are approximately equal,

such as when the travel time is about one day and the data

available is mean daily fl ows. From the strict mathematical

viewpoint, the time ∆ t should be a fraction of the travel time

K, otherwise fl ow gradients such as d O /d t are not well rep-

resented on a fi nite difference basis. However, it is still pos-

sible to use time increments ∆ t greater than K if the fl ow is

only changing slowly, but caution is necessary. The reason

for caution is that the C values in Eq. (31) are a function of

∆ t / K and are not constant.

Solving for the C ’s in terms of K, x and ∆ t:

C

xtK

xtK

C

xtK

xtK

C

0

1

2

05

105

05

105





 



 

.

.

.

.

.







( )

( )

( )

( )



 

 

105

105

xtK

xtK

.

.





( )

( )

(35a)

(35b)

(35c)

Hence,

C 0  C 1  C 2 
 1. (35d)

To illustrate the infl uence of ∆ t and K, some synthetic data as

used to construct Figure 7. Values for K and x were chosen

and when values of C 0 , C 1 , and C 2 were calculated for differ-

ent values of ∆ t. In addition, an assumed infl ow was routed

using the K and x values and using a time interval ∆ t which

was small compared with K. These resulting infl ows and

outfl ows were then reanalyzed for C 0 , C 1 , and C 2 using time

increments fi ve times greater than the original ∆ t. Such ∆ t

values exceeded K. The new estimates of C 0 , C 1 , and C 2 are

Time

Discharge

Inflow

x = 0

x = 0.4

x = 0.2

FIGURE 6 Muskingum routing: to illustrate the influence of x on

outflow hydrograph (after Linsey^2 ).

C008_003_r03.indd 475C008_003_r03.indd 475 11/18/2005 10:29:27 AM11/18/2005 10:29:27 AM