Encyclopedia of Environmental Science and Engineering, Volume I and II

HYDROLOGY 473

MULTIPLE REGRESSION AND STREAMFLOW

The similarity of the fi nite difference unit hydrograph approach

to multiple regression analysis is immediately apparent. The

fl ow in terms of precipitation can be written as

QaPaPaP ap

QaPaPaP

Rtt t ntn

Rt t t

  


 

 

 

12132 1

111223

Λ

3 ΛΛ








ap

QaP

ntn

Rt 212 etc.

(9)

Again we can write

{ Q } 
 [ P ]{ a }. (10)

The similarity with Eq. (6) is obvious and may be complete

if we have selected the correct precipitation data to correlate

precipitation at 6 a.m. with downstream fl ow at 9 a.m. when

we know that there is a 3-hour lag in the system. Therefore,

using multiple regression as most hydrologists do, the method

can become identical with the unit hydrograph approach.

LAKE, RESERVOIR AND RIVER ROUTING

The run-off calculations of the previous sections enable esti-

mates to be made of the fl ow in the headwaters of the river

system tributaries. The river system consists of reaches of

channels, lakes, and perhaps reservoirs. The water travels

downstream in the various reaches and through the lakes

and reservoirs. Tributaries combine their fl ows into the main

stream fl ow and also distributed lateral infl ows contribute to

the total fl ow. This total channel system infl uences the fl ow

in two principal ways, fi rst the fl ow takes time to progress

through the system and secondly, some of the fl ow goes into

temporary storage in the system. Channel storage is usually

only moderate compared with the total river fl ow quantities,

but lake and reservoir storage can have a considerable infl u-

ence on the pattern of fl ow.

Calculation procedures are needed which will allow for

this delay of the water as it fl ows through lakes and chan-

nels and for the modifying infl uence of storage. The problem

is correctly and fully described by two physical equations,

namely a continuity equation and an equation of motion.

Continuity is simply a conversion of mass relationship while

the equation of motion relates the mass accelerations to the

forces controlling the movement of water in the system. Open

channel fl uid mechanics deals with the solution of such equa-

tions, but at present the solutions have had little application

to hydrological work because the solutions demand detailed

data which is not usually available and the computations are

usually very complex, even with a large computer.

Hydrologists resort to an alternative approach which is

empirical; it uses the continuity equation but replaces the equa-

tion of motion with a relationship between the storage and the

fl ow in the system. This assumption is not unreasonable and is

consistent with the assumption of a stage–discharge relation-

ship which is widely utilized in stream gauging.

RESERVOIR ROUTING

The simplest routing procedure is so-called reservoir rout-

ing, which also applies to natural lakes. The continuity equa-

tion is usually written as;

IO

S

dT



d

’

(11)

where

I = Infl ow to reservoir or lake

O = Outfl ow

S = Storage

The second equation relates storage purely to the outfl ow, which

is true for lakes and reservoirs, where the outfl ow depends only

on the lake level. The outfl ow relationship may be of the form:

O 
 KBH 3/2^ (12)

if the outfl ow is controlled by a rectangular weir, or:

O 
 K′H n (13)

where K′ and n depend on the nature of the outfl ow channel.

Such relationships can be turned into outfl ow—storage

relationships because storage is a function of H, the lake level.

The Eqs. (12) and (13) can then be rewritten in the form

O 
 K ′′ S m. (14)

Alternatively, there may be no simple functional relationship,

but a graphical relationship between O and S can be plotted

or stored in the computer. The continuity equation and the

outfl ow storage relationship can then be solved either graph-

ically or numerically, so that, given certain infl ows, the out-

fl ows can be calculated. Notice the assumption that a lake or

reservoir responds very rapidly to an infl ow, and the whole

lake surface rises uniformly.

During the development of the kinematic routing model

described later, a reservoir routing technique was developed

which has proved to be very useful. Because reservoir rout-

ing is such an important and basic requirement in hydrology,

the method will be presented in full.

Reservoir routing can be greatly simplifi ed by recogniz-

ing that complex stage–discharge relationships can be lin-

earised for a limited range of fl ows. It is even more simple

to relate stage levels to storage and then to linearise the

storage–discharge relationship. The approach described below

can then be applied to any lake or reservoir situation, ranging

from natural outfl ow control to the operation of gated spill-

ways and turbine discharge characteristics.

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