Encyclopedia of Environmental Science and Engineering, Volume I and II

472 HYDROLOGY

Expanding Eq. (2) for a particular value of R,

QuPuP uP() ()() ()() 10
 10 1 9 2 .()() (^110) (4)
The whole family of similar equations for Q may be expressed
in matrix form (Snyder^20 )
Q
Q
Q
Q
P
PP
R
1
2
3
1
21
00 0
00
.
.
⎛ ....
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟


PPPP
P
P
P
P
PP
n
n
n
321
1
21
00
0
00
000 0
... ..
.....
.
...
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎜⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
⎟⎟
←→m
m
u
u
u
u
columns
1
2
3
.
.
(5)
Or more briefl y
{ Q }
[ P ]{ u }. (6)
Equation (6) specifi es the river fl ow in terms of the precipita-
tion and the unit hydrograph. In practice Q and P are mea-
sured and u must be determined. Some workers have guessed
a suitable functional form for u with one or two unknown
parameters and have then sought a best fi t with the available
data. For instance, Nash’s series of reservoirs yields 17,23
ut
n
t
K
e
n
n
() tk
()



(^1) 
1
1
!
−
 (7)
in which there are two parameters, K and n. Another approach
is to solve the matrix Eq. (6) as follows (Synder^21 )
{}uPPPQ
⎡⎣ TT⎤⎦ {}
 1

. (8)

It has already been demonstrated that R 
 m + n − 1 so that

there are more equations available than there are unknowns.

The solution expressed by Eq. (8) therefore automatically

yields the least squares values for u. This result will be

referred to after the next section.

t-τ

u(t-τ)

u

O

Q(t)=

τ<4

0

u(t-τ)P(τ)dτ

t

t t

t

Q

O

P

τ dτ

P(τ)

FIGURE 4 Determination of streamflow from precipitation input

using an instantaneous unit hydrograph.

P

P(n)

U(m)

tn

tn

tn

∆t

U

t

t

t

Q

Qn

FIGURE 5 Convolution of precipitation by unit hydrograph on a

finite difference basis.

C008_003_r03.indd 472C008_003_r03.indd 472 11/18/2005 10:29:26 AM11/18/2005 10:29:26 AM