Encyclopedia of Environmental Science and Engineering, Volume I and II

HYDROLOGY 479

STATISTICAL TECHNIQUES

Many simulations, particularly run-off forecasts, have been

based on multiple regression analysis. Flow has been treated

as the dependent variable and many factors such as precipi-

tation, antecedent precipitation, sunshine hours, etc. have

been used as the independent variables.^13 Such methods

have had at least moderate success. The diffi culties with the

method have been that many of the so-called independent

variables are in reality somewhat related, for example, rain-

fall is inversely correlated with sunshine hours. Also, the

lack of true functional relationship usually means that coef-

fi cients for one year’s data vary for other years or for periods

of higher or lower fl ow, and therefore forecast precision in

inherently limited and evaluation of changes to the system,

such as introduction of reservoirs is not feasible.

The defi ciencies of multiple regression analysis have

been somewhat overcome, according to some investigators,

by the use of multivariate analysis. The principle component

techniques select the most signifi cant relationship even if

the data is interrelated. However, this type of analysis is still

blind to the physics of the system. Examples of this technique

and the theory are given in Kendall, Snyder and Wallis 10,14,15

and the reader can consult these for more details.

Monte Carlo methods, of which the stream generation is

a simple example, have been proposed and used. Such meth-

ods have had staggering success in some fi elds of physics,

such as nuclear studies. The classic example was the original

study of critical mass for the atomic bomb. Such success led

people to hope for more general applications. But even the

keenest exponents of such methods agree that they are no

substitute for an understanding of the physical behavior of

the system. The writer considers that such statistical tech-

niques are a reasonable approach if there is no physically

based alternative. Experience with constructing physically

based simulation models shows immediately how valuable

it is to incorporate a functional relationship which bears

some resemblance to the physical system. An example of

this statement is the unit hydrograph approach which can be

considered as a simple simulation model. The actual physi-

cal behavior of a catchment seems to be well described by a

unit hydrograph. As soon as such a unit hydrograph is intro-

duced, it becomes a relatively simple matter to relate cause

and effect, which in that case is precipitation and run-off.

A similar result can be achieved if the correct data is used in

a multiple regression analysis, but such data selection pre-

supposes a knowledge of the system behavior.

PHYSICAL COMPUTER SIMULATION MODELS

Qualitatively, we can describe the behavior of a hydro-

logic system. The catchment soil layers have storage which

decreases and delays run-off and permits evapo-transpiration

to occur. The lake and channel system further delays fl ows and

modifi es the shape of the outfl ow hydrograph. Groundwater

supplies a highly damped outfl ow which is signifi cant during

dry spells. Quantitatively we may know the precipitation

input and the run-off output, although there may be data error,

especially in the precipitation. There may also be data from

which estimates of potential evapo-transpiration can be made.

Presumably data will also be available of such matters as lake

areas, stage-discharge relationships, catchment areas, and ele-

vations, and the details of the streamfl ow network.

It is important to realize what can be discovered about a

system simply from studying the input to the system and the

output from the system, as has been very well demonstrated

by Nash.^23 He considers a simple electrical system made up

of a capacitor and a resistance as shown in Figure 8. Then

if E ( t ) is considered to be the input and e ( t ) the output it can

be shown that

et

RCD

()
 Et()



1

1

(46)

where D is the differential operator. Hence we can solve for

e ( t ) if E ( t ) is given. Alternatively if e ( t ) and E ( t ) are given we

can solve for RC as a lumped term, but we can never fi nd the

separate values for R and C. Therefore when we construct

a simulation model, we may be able to correctly model the

output from the input, although the parameters used in the

model may be lumped terms describing various factors in

the real system.

Hydrological systems are considerably more complex

than the above example and it is important to realize that we

cannot start to “fi t” the model to the data until we have simu-

lated the total system behavior. Also, in general, the system

will not be linear like the simple electrical network, so that

response becomes a function of fl ow.

In constructing a simulation model we must fi rst of all

decide the factors or processes which should be included to

correctly and adequately describe the system. Each factor

must then be approximately fi tted using any data or knowl-

edge which we may have. It is a basic rule that these factors,

such as evaporation, unit hydrograph, soil moisture storage,

etc. should be modeled with as few parameters as possible,

as long as adequate description of a process is not jeopar-

dized. This minimization of parameters has been well named

E(t) = e(t) + Ri(t)

i(t) = Cde

dt

e(t) =^1

1+CRD

... E(t)

e(t)

C

i

E(t)

R

FIGURE 8 Simple system demonstrating

non-separability of R and C (after Nash).

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