Encyclopedia of Environmental Science and Engineering, Volume I and II

HYDROLOGY 471

for two hours. This effective precipitation will appear some

time later in the stream system, but will now be spread out

over a much longer time period and will vary from zero fl ow,

rising gradually to a maximum fl ow and then slowly decreas-

ing back to zero. Figure 3 shows the block of uniform precip-

itation and the corresponding outfl ow in the stream system.

The outfl ow diagram can be reduced to the unit hydrograph

for the two hour storm by dividing the ordinates by three.

The outfl ow diagram will then contain the volume of run-off

equivalent to one inch of precipitation over the given catch-

ment area. For instance, one inch of precipitation over one

hundred square miles will give an area under the unit hydro-

graph of 2690 c.f.s. days.

When a rainstorm has occurred the hydrologist must fi rst

calculate how much will become effective rainfall and will

contribute to run-off. This can best be done in the framework

of a total hydrological run-off model as will be discussed later.

The effective rainfall hydrograph must then be broken down

into blocks of rainfall corresponding to the time interval for

the unit hydrograph. Each block of rain may contain P inches

of water and the corresponding outfl ow hydrograph will have

ordinates P times as large as the unit hydrograph ordinates.

Also, several of these scaled outfl ow hydrographs will have to

be added together. This process is known as convolution and

is illustrated in Figure 4 and 5.

The underlying assumption of unit hydrograph theory is

that the run-off process is linear, not in the trivial straight line

sense, but in the deeper mathematical sense that each incre-

mental run-off event is independent of any other run-off. In

the early development, Sherman^16 proposed a unit hydro-

graph arising from a certain storm duration. Later workers

such as Nash 17,23 showed that Laplace transform theory, as

already highly developed for electric circuit theory, could be

used. This led to the instantaneous unit hydrograph and gave

rise to a number of fascinating studies by such workers as

Dooge,^18 Singh,^19 and many others. They introduced expo-

nential models which are interpretable in terms of instanta-

neous unit hydrograph theory. Basically, however, there is

no difference in concept and the convolution integral, Eq. (1)

can be arrived at by either the unit hydrograph or the instan-

taneous unit hydrograph approach. The convolution integral

can be written as:

Qt u P

tt

()
∫ ( ) ( )



1

0

0

tttd (1)

Figure 4 shows the defi nition diagram for the formulation is

only useful if both P, the precipitation rate, and u, the instan-

taneous unit hydrograph ordinate are expressible as continu-

ous functions of time. In real hydrograph applications it is

more useful to proceed to a fi nite difference from of Eq. (1)

in which the integral is replaced by a summation, Eq. (2),

and Figure 5.

QumPntR

M


∑ ( ) ( )

l

(2)

where M is the number of unit hydrograph time increments,

and m, n and R are specifi ed in Figure 2. It should be noted

that from Figure 5,

m + n 
 R + 1. (3)

EFFECTIVE RAIN STREAM RUN-OFF

Rain (.ins./hr.)

Time (Hours) Time (Hours)

(^02) t 0612 18 t
0.5
1.0
1.5
3 ins. of Rain
1000
2000
3000
CFS.
Ordinates
divided by 3
(inches of Rain)
Actual Run-off
Q
P
Unit
Hydrograph
Area
equals 1 inch
Rain
FIGURE 3 Hydrograph and unit hydrograph of run-off from effective rain.
C008_003_r03.indd 471C008_003_r03.indd 471 11/18/2005 10:29:26 AM11/18/2005 10:29:26 AM