Use of Analog Models 

in the Analysis of 

Flood Runoff 

G E 0 L 0 G I CAL SURVEY P R 0 FE S S I 0 N A L PAPER 50 6- A 

Use of Analog Models 

in the Analysis of 

Flood Runoff 

By JOHN SHEN 

S Y NTH E SIS IN H Y DR 0 L O.G Y 

GEOLOGICAL SURVEY PROFESS ION AL PAPER 5 06-A 

An analog-model study of rainfall excess 

versus flood runoff with special emphasis 

on the determination of flood frequency 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1965 
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary 

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 25 cents {paper cover) 
CONTENTS 

Symbols------------------------------------------- Abstract __________________________________________ _ 

Introduction ______________________________________ _ 

Hydrologic models for rainfall-excess versus flood-runoff 

system __________________________________________ _ 

Development of a quasi-linear analog modeL __________ _ 

Some annotated results derived from the analog-model 

studY--------------------------------------------

Illustrative examples ___________________________ _ 

Page 

iv 

A1 

1 

2 

4 

8 

8 

Some annotated results derived from the analog-model 

study-Continued 

Effect of nonlinear storage in a hydrologic system __ 

Description of a hydrograph in statistical parameters_ 

Effect of storage on peak flow ___________________ _ 

Comparison with Mitchell's formula ______________ _ 

Analog-model study of flood-frequency distribution _____ _ Summary and outlook ______________________________ _ 

References---------------------------------~-------

ILLUSTRATIONS 

Page 

A10 

14 

15 

16 

18 

22 

23 

Page 

FIGURE 1. Schematic representation of an equivalent drainage channeL______________________________________________ A3 

2. A simplified routing scheme _______________________________________ +__________________________________ 3 

3. Linearized admittance function_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 

4. A segment of transmission line_________________________________________________________________________ 5 

5. A variable transmission line___________________________________________________________________________ 5 

6. Model of a linearized drainage system__________________________________________________________________ 5 

7. A linear-reservoir model______________________________________________________________________________ 6 

8. A phase-shifting circuit_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 

9. A nonlinear-reservoir model___________________________________________________________________________ 6 

10. A nonlinear voltage amplifier__________________________________________________________________________ 7 

11. Distribution of translation and storage effects of a basin__________________________________________________ 7 

12. Alag-and-routeprocedure----------------------------------------------------------------------------- 7 

13. A modified time-area diagram_________________________________________________________________________ 8 

14. Outflow hydrographs through a chain of equal reservoirs__________________________________________________ 9 

15. Time lags related to number of reservoirs_______________________________________________________________ 10 

16. Reduction in peak size due to a chain of reservoirs_______________________________________________________ 11 

17. A typical outflow hydrograph from a linear reservoir, K/1'=0.3-------------------------------------------- 11 

18. Effect of basin shape on flood runoff-------------------------------------------------------------------- 12 

19. Hydrographs from a basin having its width increased geometrically in downstream direction___________________ 13 

20. Effect of storm movement_____________________________________________________________________________ 14 

21. Composite hydrographs from 4 tributaries _____________________________________________________________ -- 14 

22. Comparison of outflow hydrographs, linear and nonlinear reservoirs (S=KQ2) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 

23. Comparison of outflow hydrographs, linear and nonlinear reservoirs (S K Qtl2) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 

24. Definition sketch of a hydrograph in terms of statistical parameters _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 

25. Peak-reduction ratio, Qpflv, as a function of £ 1/K-------------------------------------------------------- 16 

26. Peak-reduction ratio, Qvflv, as a function of ulK-------------------------------------------------------- 17 

27. Comparison of Mitchell's formula with figure 25--------------------------------------------------------- 17 

28. Schematic diagram of the frequency and duration analyzer________________________________________________ 18 

29. Random input and output through a linear reservoir_____________________________________________________ 19 

30. Probability distributions of inflow and outflow, linear reservoir_____________________________________________ 19 

31. Random input and output through a nonlinear reservoir, 8 KQ2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ __ _ _ __ _ _ _ _ _ 20 

32. Probability distributions of inflow and outflow, nonlinear reservoir_________________________________________ 20 

33. Probability distributions of inflow and outflow, linear reservoir, log-normal input_____________________________ 21 

34. Duration distributions of inflow and outflow, linear reservoir __________________________________________ ---- 22 

m 
A 

g 

I 

lv 

i 

K 

k 

L 

Lv 

L, 

L2 

Mo 

P(f) 

Q 

SYMBOLS 

Cross-sectional area of a stream channel. Also, an 

admittance function. 

Area of a subbasin. 

Width of the water surface in a stream channel. 

Capacitance. 

Chezy's coefficient. 

Duration of rainfall. 

A constant voltage gradient. 

Input and output voltage. 

Voltage. 

A factor accounting for the increase in peak due to the 

increase in impervious area. 

Acceleration of gravity. 

Rate of inflow. 

Peak of inflow hydrograph. 

Current. 

A storage constant. 

A storage constant describing minor storage. 

Inductance. 

Time to peak of inflow hydrograph. 

First moment of a hydrograph. 

Second moment of a hydrograph. 

Mode. 

Power spectrum. 

Rate of flow. Also, rate of outflow. 

IV 

q 

Q'P 

R 

8 

Bo 

s, 

Bx 

T 

W(f) 

X 

y 

X 

T 

Rate of lateral inflow per unit length of a channel. 

Peak of outflow hydrograph. 

Resistance. 

Quantity of storage. 

Bed slope of a stream channel. 

Friction slope of the flow. 

Pearson's coefficient of skewness. 

A translation lag. Also, a characteristic time for a 

drainage basin. 

Time. 

Time to centroid of outflow hydrograph. 

Time of travel through a reservoir. 

Mean velocity in a stream channel. 

A transfer admittance. 

Distance along a stream channel in the direction of flow. 

Also, an exponent. 

Depth of water in a stream channel. 

Lag between centroids of inflow and outflow hydrographs. 

Lag between centroid of inflow and peak of outflow. 

A ratio. 

Standard deviation. 

A total lag time, between rainfall excess and flood 

hydrograph. 
SYNTHESIS IN HYDROLOGY 

USE OF ANALOG MODELS IN THE ANALYSIS OF FLOOD RUNOFF 

By JOHN SHEN 

ABSTRACT 

The analog technique is applied to the analysis of flood runoff. 

A quasi-linear analog model has been developed for simulating 

the runoff-producing characteristics of a drainage system. 

Where storage is linear a unique relationship correlating the inflow and outflow peaks is derived. A technique for synthesizing 

flood-frequency distribution is also proposed, whereby the effects 

of a linear- or a nonlinear-basin system upon its inflow probability 

distributions are examined. 

INTRODUCTION 

With the growing interest in the development of the 

Nation's water resources, there is an increasing need 

for better understanding of the complex behavior of 

various hydrologic systems. Current practices used in 

many engineering designs are largely empirical. In 

general, these approaches do not render consideration 

of the many hydrologic variables involved in a given 

problem, but rather attempt to treat these variables as 

a lump, usually by means of a coefficient. Some judgment and experience are therefore necessary for the 

successful application of the empirical method. 

Inevitably, all empirical approaches are based on 

historical hydrologic records, which are subject to certain limitations. In many instances, the records do 

not covar a sufficiently long period to permit a complete analysis. Moreover, such information cannot 

clearly reflect the effects of man-made changes that are 

rapidly taking place. Consequently, one must rely on 

the means of synthesis, through modeling technique, to 

reproduce the behavior of a hydrologic system, to study 

the interrelations of different variables for the purposes 

of future planning and management. 

For example, in order to predict the effect of urbanization on the flood potential of a drainage system, one 

must synthesize its hydrologic characteristics in every 

important aspect on the basis of available information. 

And, by properly modifying these characteristics, one 

will be able to forecast their possible consequences. 

Likewise, to achieve the optimum design of a multipurpose water-resources system, one must synthesize 

various combinations of system units, levels of output, 

and allocation of reservoir capacities, in accordance 

with the requirements of the design. 

Presently, there are two general types of scheme used 

in system modeling-the "statistical" and the "hydrological'' models. 

Statistical model.-This is the technique used largely 

by the Harvard group, represented by the work of 

Thomas and Fiering (1962). The behavior of a waterresources system is simulated by means of a synthetic 

sequence which is derived from the historical hydrologic 

events. The historical records need to be sufficiently 

long to include representative samples of dry, wet, and 

normal periods whereby certain statistical parameters 

characterizing the data are estimated. These parameters, together with some basic assumptions about the 

temporal and spatial distributions of the historical data, 

enable one to construct models for generating extended 

synthetic sequences. An excellent summary and discussion of the statistical techniques was presented by 

N. C. Matalas (1962). 

Hydrological model.-This approach differs 'from the 

previous one in that it attempts to employ for the 

modeling such physical characteristics of a hydrologic 

system as the surface and channel roughness, the land 

and channel slopes, and the overland and subsurface 

storage. The problem essentially consists of two parts: 

(a) determining the runoff hydrograph for a specific 

rainfall event: and, (b) determining the probability 

distribution of a classified discharge resulting from a 

given distribution of rainfall. Thus, by the use of a 

simulated drainage system, one may assess the relative 

importance of the many variables involved. 

It is known that various investigations involving the 

use of hydrologic models have existed for many years. 

However, these studies in the past were generally of a 

descriptive nature. It is the intent of this study to 

explore quantitatively the fundamental relations among 

the hydrologic variables pertaining to the events of 

rainfall-excess versus flood-runoff. Special emphasis 

Is placed upon the determination of flood frequency. 

Al 
A2 SYNTHESIS IN HYDROLOGY 

One's first thought regarding the hydrological model 

might be to think of it as a physical laboratory catchment. An example of this type of approach is represented by the work of J. Amorocho and G. T. Orlob 

(1961). In a pioneering manner, Amorocho and Orlob 

succeeded in illustrating some of the fundamental 

behavior of a simple experimental basin. The possibility of such an approach was, in fact, seriously considered during the initial planning of this study. 

However, in view of the large number of variable 

parameters that would be involved-such as the distribution of rainfall pattern, the variations of land slope 

and surface roughness, and the behavior of infiltration 

and evapotranspiration losses-the required model 

would be very complex. Indeed, one could not hope 

to accomplish an extensive physical-model study 

without an enormous amount of expense. Consequently this type of approach was not adopted at this 

time. 

A mathematical model describing the physical behavior of a drainage system affords another possible 

means of hydrological modeling. In this approach 

the hydrologic process would be expressed purely in 

terms of mathematical functions. Dooge's work 

(Dooge, 1959) is an outstanding example of this class. 

The use of mathematical models, however, is often 

handicapped by one's inability to obtain their solutions, 

especially with the more complex, nonlinear cases. 

Nevertheless, the mathematical tool is an essential 

part for all analytical studies and is thus considered as 

a complementary effort to the analog-model approach 

described subsequently. 

An ann:log model is necessarily based upon the many 

similarities which exist between the behavior of fluid 

and electricity flows. Thus, it offers a convenient 

means for hydrological modeling. The advantage of 

such a model lies in its simplicity and flexibility. It 

bypasses many complex mathematical processes and 

allows the causes and effects to be observed readily. 

Hence, it is a very useful tool for r.esearch purposes. 

In the sections that follow, the development of various 

hydrologic-modeling techniques are discussed. In particular, the technique of analog simulation is largely 

applied. 

The author wishes to express his gratitude to Mr. 

R. W. Carter for his constant encouragement in the 

pursuance of this project. Acknowledgments are due 

Messrs. M.A. Benson, N.C. Matalas, D. R. Dawdy, 

and W. D. Mitchell for their invaluable suggestions in 

the planning of this study, and for their constructive 

criticism in the review of this paper. The author is 

indebted to Mr. H. E. Skibitzke, whose initial ideas and 

helpful advice in the analog scheme greatly facilitated 

the progress of this work. Special thanks are also due 

Mr. L. P. Arnold for his aid in the analysis of the data. 

Professor A. Amorocho, University of California, 

critically reviewed the preliminary draft of this paper. 

His thoughtful comments are greatly appreciated. 

HYDROLOGIC MODELS FOR RAINFALL-EXCESS VERSUS 

FLOOD-RUNOFF SYSTEM 

The process of converting rainfall excess into surface 

runoff is one of the oldest problems in hydrology. 

It is also one of the most difficult endeavors. Various 

methods of treating such a system are known-some 

are based on rational analysis, others are based on 

empirical approaches. These methods are often controversial. Undoubtedly, the disputes ~.are due largely 

to the subjective points of view of th~ investigators. 

A more perspective outlook is indeed es~ential. 

In a broad sense, the techniques for the analyses of 

hydrologic systems may be classified into two schools: 

the "linear system" and the "nonlinear system." 

Briefly, a linear system relates the dependent variable 

to a weighted sum of the independent variables, 

whereas, a nonlinear system takes into account the 

interactions of the independent variables. The linear 

system is represented by the work of Paynter (1952), 

Rockwood (1956), Dooge (1959), and Kalinin (l960); 

the nonlinear system is represented by the work of 

Ishihara (1956), Liggett (1959), Harder (1962), and 

Amorocho (1961). Although these individual efforts 

are of the same general nature, the emphases are nevertheless placed upon different applications. Whereas 

all real phenomena in nature are nonlinear in some 

degree, experience from various sources seems to indicate that the use of linear tools with appropriate 

caution and modification can lead to many valuable 

approximations. 

Thus, before abandoning too hastily the linear 

concept, one should go back and examine some of the 

fundamental principles governing the mechanics of 

surface runoff. Liggett (1959) pointed out that there 

are two distinctly different types of problems associated 

with the determination of streamflow in hydrology. 

The first classification is that of the "upstream problem," which consists, in its elements, of a long channel 

into which there is a continuous inflow along the sides 

with little or no inflow at the upper end. There are 

no points with large concentrations of inflow, although 

the lateral inflow may vary with distance. The second 

classification is that of the "downstream problem," 

which consists of a large channel with a very small 

amount of lateral inflow, although there may be large 

concentrations of lateral inflow at the junctions with 

its tributaries. The main source consists of flow into 

the upper end. 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF A3 

\\\ \Jv 

y 

u 

dx 

Y+ _!_L dx 

IJ X 

FIGURE I.-Schematic representation of an equivalent drainage channel. 

The downstream problem is commonly referred to 

as the routing problem which is most satisfactorily 

solved by either the conventional flood-routing procedure, such as the Muskingum method, or the more 

sophisticated method (Stoker, 1957) involving the 

unsteady-flow equations. 

On the other hand, the treatment of the upstream 

problem has thus far been only cursory. Investigators 

rely largely on empirical procedures which were often 

found to be inadequate because of the greater range 

of variables involved. It is in this light that Liggett's 

analytical study represents a significant contribution. 

The upstream problem· is perhaps more critical in 

hydrology today, especially with small drainage basins. 

Assuming that one has an equivalent drainage channel, 

like that suggested by Liggett, he may derive a system 

of :Bow equations as follows: 

oQ oy ox+ B "8t=q, (1) 

(2) 

in which Q is the rate of :Bow, u is the mean velocity 

in the channel, B is the width of the water surface in 

the channel, y is the depth of water in the channel, 

So is the slope of the channel, S1 is the friction slope of 

the flow, x is the distance along the channel, g is the 

acceleration of gravity, t is the time, and q is the rate 

of lateral inflow per unit length of the channel. (See 

under Symbols.) A schematic representation of the 

equivalent channel is shown in figure 1. 

With the use of Chezy's relationship, 

QIQI 

8,= C.2A2r' 

and assuming that the wave height is small in comparison with the water depth, equation 2 may be 

reduced to 

(3) 

where A is the cross-sectional area of the channel, 

and r is the hydraulic radius. 

Experience has shown that for long-period flood 

waves, the acceleration term g~ ~~is usually of small 

magnitude with respect to other terms and may thus 

be neglected under ordinary circumstances. Hence, 

equation 3 may be further simplified into 

(4) 

It may be readily recognized that the product of 

0 2Ar; in equation 4 is a form of "conveyance function." 

Ordinarily it varies nonlinearly with the stage. In 

combination with the continuity equation {eq. 1), 

it represents a nonlinear "admittance function." 

Thus, the procedure of converting rainfall excess into 

surface runoff may be simply stated as a process of 

transformation via the admittance function as illustrated in figure 2. 

Paynter's experience (Paynter, 1952) indicates that 

in many cases, the admittance function of river reaches 

can be represented with surprising accuracy by a 

simple function of delay and linear storage as shown 

in figure 3. Consequently, this function can be 

described by two constants-that is, a "translation 

lag," T, and a "storage constant," K. The translation 

lag is associated with the time of wave propagation 

down the river reach, while the storage constant Is 

related to the overland and the valley storage. 

Inflow r---- Admittance 1--- Outflow function 

FIGURE 2.-A simplified routing scheme. 
A4 SYNTHESIS IN HYDROLOGY 

A 

r---+- K-1 ---~---r-------- 1 I 

I I 

I I I h 

I 

Time 

FIGURE 3.-Linearized admittance function. 

A further hypothesis was put forth by Dooge {1959) 

that the process of converting rainfall excess into 

surface runoff is a mixture of translation and reservoir 

action. Hence, it may be represented by a series of 

"linear channels" and "linear reservoirs." On such 

a basis, Dooge derived a general equation for the 

instantaneous unit hydrograph. 

Thus, the linear system may be considered as a 

special case of the more general nonlinear systems. 

Its advantage lies, of course, in the simplicity of manipulation-that is, the principles of superposition and 

proportionality may be readily applied. 

The selection of a linear or a nonlinear model depends 

upon the degree of refinement required of a given 

problem as well as one's knowledge of a particular 

basin. For example, in the prediction of a flood crest 

as it travels down a major river channel, a certain 

degree of ~ccuracy would undoubtedly be required, 

~s human lives and large amounts of property may be 

Involved. Moreover, in many instances the hydrologic parameters and the channel geom~try are well 

defined. The choice of a more complex nonlinear 

model is therefore justified. On the other hand, in 

the planning of drainage structures for small basins, 

the hydrologic information is generally very crude and 

may even be totally absent; in this case the use of 

any elaborate model may be unfeasible. A linear 

mod.el is likely to be selected as a first-order approximation for the preliminary design purposes. 

In simulating a dr ... ainage system, either linear or 

nonlinear, one is inevitably concerned with the mathematical complexities that are involved. For instance 

in studying the interrelations of the fundamentai 

variables, one must consider a vast number of possible 

combinations within the specified ranges. Consequently, the simulation of such a system must be 

facilitated by the use of high-speed electronic computers-digital or analog. 

DEVELOPMENT OF A QUASI-UNEAR ANALOG MODEL 

Examples of various applications of the computer 

technique in a hydrologic study are known. Harder 

(1962) used a nonlinear analog model to simulate the 

flood-control system; Crawford and Linsley (1962) 

used a digital computer to synthesize the streamflows 

for three small watersheds; Baltzer and Shen (1961) 

utilized a power-series technique to solve the unsteadyflow equations {eq. 1 and 2) on a digital computer. 

In this study, the analog technique is adopted for 

several reasons: 

1. The laws governing the behavior of fluid flow and 

electrical current are, in many instances, identical. 

Thus, an analog model may be developed on the 

basis of direct simulation instead of on the exact 

mathematical expressions which would be required 

by a digital computer. 

2. The input and output of an analog model are expressed in continuous time-varying graphs. They 

are easily recognizable by an investigator. Moreover, the parameters can be readily varied. 

Hence, an analog model is a more flexible 

and convenient tool for research purposes. 

3. In a hypothetical cause-and-effect study, the data 

is primarily derived by means of synthesis which 

is often limited in scope. Hence, the operation 

of an analog computer is generally more economical 

than that of a digital computer which is ordinarily 

a data-processing device. 

If one examines a segment of an electrical ''transmission line" (fig. 4), one may derive the following 

system of equations: 

oe +L oi +R '=Eo 

ox ot ~ ' 

(5) 

(6) 

where e is the voltage, i is the current, C is the unit 

capacitance, L is the unit inductance, R is the unit 

resistance, x is the distance in the direction of current 

flow, and t is the time. 

Comparing equations 5 and 6 with equations 1 and 

3, it may be seen that the following variables are 

equvialent: 

Electric variables Hydraulic variables 

Voltage ___________ (e) ---+ Water depth _________ (y) 

Current ___________ (i) ---+ Discharge ____________ (Q) 

Inductance ________ (L) ---+ Inertia coefficient ___ (g~) 

Capacitance _______ (C) ---+ Surface width ________ (B) 
ANALOG MODELS, lN ANALYSIS OF FLOOD RUNOFF A5 

0»------r-'1 0 0 0 ~ E~ 

CI 

t-1 .... -----Unit length 

FIGURE 4.-A segment of transmission line. 

Variable 

voltage 

fl (y) 

Variable 

resistance 

f2 (y,Q) 

FIGURE 5.-A variable transmission line. 

Furthermore, the constant voltage gradient (E0) 

represents the fixed channel slope (80), and, the current 

gradient (i') represents the lateral inflow (q). 

Thus, the two systems of equations are analogous. 

The only difference lies in the friction terms. In the 

electrical system the resistance term is linear, whereas 

in the hydraulic system the resistance term is commonly nonlinear. Thus, in order to accomplish the 

complete analogy, one must replace R by a variable 

resistance which induces a voltage drop proportional 

to the square of the current (Einstein and Harder, 

1959). Moreover, for natural basins the storage effect 

may be nonlinear-that is, B is a function of stage. 

In this case a variable capacitance would also have to 

be used in the transmission-line model. The method 

for simulating such nonlinear reservoirs will be discussed in a later section. Figure 5 shows a variable 

transmission-line scheme. 

Simple 

RC-network 

Outflow 

FIGURE 6.-Model of a linearized drainage system. 

In view of the previous discussion that the acceleration term g~ ~~is generally of small magnitude for flood 

waves, the inductance element can thus be omitted 

from the transmission-line scheme. Accordingly, the 

model becomes a variable RO-type of network or an 

admittance function. Further simplification can be 

made if the system is linearized in accordance with 

Dooge's and Paynter's hypotheses that the admittance 

function is made up of two fundamental elementsthe linear channel and the linear reservoir. 

Electrically, the two elements may be simulated by 

a delay line and a simple RO-network. A schematic 

representation is shown in figure 6. 

A linear reservoir is one in which the storage is 

linearly proportional to the outflow: 

S=KQ, (7) 

in which Sis the storage, Q is the rate of outflow, and 

K is a storage constant. Thus, in combination with 

the continuity equation, 

dS 1-Q=dt' (8) 

one obtains the inflow-outflow relationship for a linear 

reservoir, 

dQ 1-Q=Kdt, 

where I is the rate of inflow. 

(9) 

Equation 9 is a diffusion type of equation. It is 

equivalent to a simple RC circuit shown in figure 7, 

for which one may derive the relationship 

(10) 

Thus, comparing equations 9 and 10, Et is equivalent 

to I; E2 to Q·; and, RC to K. Unlike the transmission-
A6 SYNTHESIS IN HYDROLOGY 

FIGURE 7 .-A linear-reservoir model. 

line model, the inflow and outflow are simulated by 

voltages rather than electric currents. The operational 

amplifier shown in the circuit is a voltage-transferring 

device which provides a means of interconnecting a 

series of reservoirs. 

A linear channel is defined by Dooge (1959) as a 

reach in which the rating curve at every point has a 

linear relationship between discharge and cross-sectional area. This implies that at any point the velocity 

is constant for all discharges, but may vary from point 

to point along the reach. Thus, 

A=T'.Q, (11) 

where A is the channel cross-sectional area and 

T' C>T • h fir d . . f · ()x Is t e st envat1ve o a translatiOn lag, T. 

Combining with the continuity equation, 

FIGURE 8.-A phase-shifting circuit. (Mter G. A. Philbrick 

Researches, Inc.) 

This system may then be used as a time-delaying 

trigger mechanism to initiate the input signals at 

various time lags. 

Whereas linear storage is found to be applicable to 

many natural basins, Mitchell's experience (Mitchell, 

1962) indicates that the nonlinear storage is a condition 

which occurs with sufficient frequency to warrant 

careful consideration. 

For a nonlinear reservoir the relation between storage 

and outflow may be expressed by 

(15) 

in which x is an exponent. Accordingly, the routing 

equation becomes 

1-Q=KxQz-l dQ dt . (16) 

(12) To simulate such a nonlinear reservoir, it is necessary 

one obtains 

(13) 

which has the solution 

Q(t- T) =Constant. (14) 

This solution corresponds to the case of a pure translation. It indicates that a linear channel will translate 

any inflow hydrograph without a change of its shape. 

The linear channel may be simulated electrically by 

a delay line. One of these devices is the phase shifter 

as shown in figure 8. By interconnecting a cascade of 

such delays, a total delay equal to the sum of the 

individual delays and a. rise time equal to the root mean 

square of the individual rise time can be accomplished. 

to have a variable capacitor so that its capacitance is 

a nonlinear function of Q. Figure 9 shows a nonlinearFIGURE 9.-A nonlinear-reservoir model. 
ANALOG MODELS, DN ANALYSIS OF FLOOD RUNOFF A7 

reservoir model. It may be shown (Shen, 1962) that 

this model bears the relationship 

Et-lh=RCJJ~z-t d/h Lll2 dt ' (17) 

if 

(18) 

Equation 17 is seen to be analogous to equation 16 

with the condition that 

RC=Kx. (19) 

To produce the required nonlinear voltage, E0, a 

variable voltage amplifier must be used. Figure 10 

shows such a circuit which consists of an operational 

amplifier and a group of diodes, each having a series 

resistance that conducts at a specific voltage level. 

By interconnecting this circuit to the circuit shown 

in figure 9, a nonlinear reservoir is accomplished. A 

model such as this is entirely flexible. It may be 

arranged to represent any nonlinear-storage behavior 

of a drainage system. 

In this manner, a complete drainage system may be 

simulated. Figure 11 illustrates such a scheme, in 

which a basin is subdivided into a number of subareas 

separated by isochrones, that is, contour lines joining 

all points in the basin having equal translation time to 

the outlet. A time-area diagram may then be constructed in accordance with the subareas enclosed by 

these isochrones. If uniform rainfall excess occurs 

within the entire basin, the time-area diagram would 

represent the distribution of the volume of runoff. 

Otherwise, it must be readjusted to account for the 

uneven distribution of rainfall. 

Furthermore, the storage effect of each of the subareas 

may be unequal. It is thus necessary to assign different 

FIGURE 10.-A nonlinear voltage amplifier. 

0 1 2 3 4 5 

Time 

FIGURE 1L-Distribution of translation and storage effects 

of a basin. Dashed line represents an isochrone. 

FIGURE 12.-A lag-and-route procedure. 

admittance functions, At, A2, A3, - - -, to these subareas. 

Accordingly, by routing the individual contributions 

through their respective admittance functions, the resulting outflow hydrograph at the outlet is determined. 

This lag-and-route procedure is illustrated in figure 12. 

Because of the manner in which the time, area diagram is constructed, such a hydrograph would be the 

result of an instantaneous rainfall (duration=O hr.). 

Hence, it is called the instantaneous hydrograph. To 

derive a hydrograph due to longer duration of rainfall, 

it is necessary to convert the instantaneous time-area 

diagram into a modified time-area diagram. This procedure first requires the subdivision of the drainage 

basin into a system of isochrones each having a time 

increment equal to the duration of the rain. An instantaneous time-area diagram is then constructed. 

Next, the rectangular inflows, 111 /2, / 3, - - -, are modified into a series of isosceles triangles (fig. 13) to account 

for the effect of duration. In like manner, by routing· 

these triangular inflows through the basin system, an 

outflow hydrograph due to a finite-duration rainfall may 

be obtained. 

A remarkable simplification can be made if the admittance functions are all linear, since the conversion 

may be simply achieved by the principle of superposition. Letting u(O, t) be the ordinate of an inst_antaneous outflow hydrograph and u(D, t) be the ordinate 
AS SYNTHESIS IN HYDROLOGY 

Time 

FIGURE 13.-A modified time-area diagram. 

of the corresponding outflow hydrograph having finite 

duration, D, due to the same amount of rainfall excess, 

then 

u(D, t) =-D li' u(O, t)dt· t-D 

(20) 

In practice, an instantaneous hydrograph is first integrated over a period of time. Then the same integral 

is delayed by D time. The difference of the two integrals divided by D is then the converted hydrograph 

u(D, t). This procedure, commonly referred to as the 

S-curve method, allows the transformation of an instantaneous hydrograph to any finite-duration hydrograph. 

The analog system proposed in the foregoing discussions is necessarily a quasi-linear model, since the admittance functions may be linear as well as nonlinear. 

It can be carried out to varied degrees of refinement in 

accordance with the climatic and the physiographic 

features of a drainage basin. Certain simplified approaches are known to have been made. Crawford and 

Linsley (1962) used a linear reservoir to approximate 

the overall admittance effect of a small basin. Mitchell 

(1962) found that two linear characteristic storage functions generally suffice for the descriptions of many small 

basins in Illinois. In any event, one should not be overconcerned with the complexities that may be involved 

since the endeavor is largely facilitated through the use 

of electronic equipment. 

SOME ANNOTATED RESULTS DERIVED FROM THE 

ANALOG-MODEL STUDY 

It has been demonstrated that the analog model 

constitutes a potentially valuable tool in hydrologic 

investigations. Of even greater importance, perhaps, 

is the fact that it facilitates observation of the fundamental behavoir of a hydrologic system. It would be 

gratifying also if one could gain certain insight into 

some of the significant parameters that could be correlated universally with the physical characteristics 

of a drainage basin. 

ILLUSTRATIVE EXAMPLES 

To illustrate some of the elementary models, figure 

14 shows the routed hydrographs through a chain of 

1-10 equal linear reservoirs from a rectangular inflow 

diagram. 

From the results of figure 14 the ratio of the storage 

lag between the centroids of these hydrographs or travel 

time, "X, -to the storage constant, K, is plotted against 

the number of reservoirs as shown in figure 15. It is 

interesting to note that the lag of centroids is equal to 

the storage constant, K, of the reservoirs. Also shown 

in the same figure is the ratio of the lag of peaks, "Xp, 

to K for different numbers of reservoirs in the series. 

Accordingly, figure 16 illustrates the reduction in 

peak magnitude as the flow is being routed through 

the chain of reservoirs. It may be observed that the 

efficiency of peak reduction declines rapidly as the 

number of reservoirs is increased. Also of interest is 

the fact that the skewness of these outflow hydrographs 

tends to decrease in such a systematic fashion that at 

the end of the tenth reservoir the hydrograph closely 

approximates a normal curve.1 

The accuracy derived from the analog models is 

generally within the tolerable limits for hydrologic 

studies if high-quality electronic components are used. 

For example, figure 17 illustrates an outflow hydrograph from a linear reservoir for a triangular inflow. 

In this case the ratio of the storage constant of the 

reservoir, K, and the time base of the triangular inflow 

diagram, T, is 0.3. For comparison, the hand-conlputed result is shown by the dashed line. The difference between the two curves is less than 2 percent 

everywhere. 

To illustrate the effect of basin shape on flood runoff, 

three drainage basins of equal size and slope but of 

different shape are synthesized: one is rectangular in 

shape, one is triangular in shape with its apex facing 

upstream, and the third is also triangular in shape but 

with its apex facing downstream. Each basin is divided 

into 4 subareas separated by isochrones. Assuming 

that uniform instantaneous rainfall occurs over the 

three basins, the total volume of runoff would be equal 

for each basin. Furthermore, if the storage constant, 

K, of each of the subareas is proportional to its size, 

a, the ratio of Kfa would be constant for all cases. 

Figure 18 A, B and o· depict the resulting hydrographs. Basin 18B appears to have manifested a higher 

peak when compared with the other two basins. However, the differences are not significant enough to be 

of any important consequence. 

t It can be shown (Nash 1958) that the output from a chain or cascading linear reservoirs closely approximates a gamma distribution which converges to a normal 

distribution as the number of reservoirs become large. 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF A9 

In ut 20r---_.~+-------~------+-------~------~------~------~------+-------+-------1 

LIJ 

<!' 

0:: 

<( 

:::c 

~ 

2S 

0:: 

0 

LIJ 

<!' 

1 Reservoir 

2 Reservoirs 

3 Reservoirs 

~10~~--~~+-~-*~------~------~-------+-------4--------~------+-------~------~ 

~ 

1 5 7 9 10 

TIME, IN MILLISECONDS 

FIGURE 14.-0utflow hydrographs through a chain of equal reservoirs. 

This result is, in fact, not too surprising if one compares notes with Richards (1955). In a hypothetical 

study, Richards assumed two extreme cases of linearly 

varying inflow hydrographs: one with a maximum 

concentration at the beginning and zero at the end, and 

one with zero concentration at the beginning and a 

maximum at the end. His results indicate that for 

ordinary small basins-time of concentration less than 

6 hours-the variation in peak discharge due to these 

two extreme cases is from + 13 percent to -20 percent 

approximately, as compared with the case of a uniformly 

distributed inflow hydrograph. 

A more realistic model ,perhaps is that shown In 

figure 19, for which the b asin width is assumed to 

753-202 0---<65---2 

increase geometrically in the downstream direction. 

Here the basin is divided into 4 subareas that are 

equal in size. Moreover, the slope is assumed to 

decrease geometrically downstream such that KfT= 

~=constant for each subarea. The resulting hydrograph is shown as figure 19A. In order to illustrate 

the travel of the flood wave, the corresponding hydrographs at various upstream points are also shown in 

figure 19 B and 0. 

The result shown in figure 19 is notable in that it 

exhibits the case of a flashy mountain stream discharging into a flood plain where large impoundage takes 

place abruptly. 
AlO SYNTHESIS IN HYDROLOGY 

~ 

........ 

0.. .-< 

,_ 

0 

~ .-< 

u.. 

0 

0 

f= 

< 

a::: 

10 

9 

8 

7 

6 

5 

4 

3 

2 

- 1 

0 1/ 

0 

1/ It 

v 

/ v 

/ v 

XfK / / 7 / Xp/K 

/ / 

/ 7 

/ / 

~ / 

v ~ 

1 2 3 4 5 6 7 8 9 10 11 12 13 

NUMBER OF RESERVOIRS 

FIGURE 15.-Time lags related to number of reservoirs. 

Storm movement also constitutes an important 

element in flood runoff. If a rectangular basin system 

such as the one shown in figure 18A is subjected to a 

moving storm in either the upstream or the downstream 

direction, the resulting patterns of runoff would be 

greatly different. Figure 20A illustrates a case in 

which the storm is n10ving upstream at twice the speed 

of travel of the flow. In like manner, figure 20B 

depicts the effect of the same storm when it is moving 

downstream at an identical speed. It is interesting 

to note that the downstream movement contributed 

a much higher rate of flood runoff. 

Timing of the tributary inflows is another determinative factor of the magnitude of flood runoff. Assuming 

that a drainage system consists of four tributaries which 

are. equal in size but that their storage constants are 

in the ratios of 1, 2, 3 and 4, the resulting peak flow 

would be largely dependent upon the translation time of each of the tributaries. Figure 21A depicts an 

extreme case for which the tributaries are assumed 

to have equal translation time. Conversely, figure 

21B shows another case when the tributaries have 

unequal translation time such that K/T=~=constant. 

It may be seen that the peak flow in case A nearly 

doubles that in case B due to the same amount of 

rainfall. 

EFFECT OF NONLINEAR STORAGE IN A HYDROLOGIC 

SYSTEM 

The foregoing examples illustrate a few simple 

models which are linear in assumption. If nonlinearity 

exists, the degree of nonlinearity of a basin would 

play an important role in the magnitude of the flood 

runoff. As an illustration, if an isosceles triangle 

having a peak inflow of 4 units is routed through a 

linear and a nonlinear reservoir (S=KQ2

), the outflow 

hydrographs would show nearly equal flood peaks, as 

shown in figure 22A. However, if the inflow is ampli-
1.0 

0.8 

.::::. a..0.6 

0 

LL 

0 

0 

i= <C 0.4 0::: 

0 

0.2 

0 

0 

4.0 

3.0 

2.0 

1.0 

I 

I 

I 

I) 

0 

ANALOG MODELS, IN ANALYSlS OF FLOOD RUNOFF All 

~ 

"' ~ ~ ~ ~ ~ 

-------- ~ 

1 2 3 4 5 6 7 8 9 10 

NUMBER OF RESERVOIRS 

FIGURE 16.-Reduction in peak size due to a chain of reservoirs. 

I 1\ 

I \ 

I ~ KIT=0.3 

I V\ 1\ I ~ \~ 

I \ '\ 

v \ \ 

\ ' ' 1'--------~-- - 0.5 1.0 1.5 2.0 2.5 3.0 

tiT 

FIGURE 17.-A typical outflow hydrograph from a linear reservoir K/T=0.3. Dashed line represents values computed by 

hand. Solid line represents values obtained by analog. 
:( 

(~ 

\ 

\ 

!'-...._ 

........ 

J 

I ~-r-1 /I II I I 1_1-1 I 

v II 

v 

I 

~ <.;: E--~ 

r-

--- e I ---- @) ---- @) 

6 Q) 

---- E 

--- @ +=' 

---- @) 1 --- ® e I-

~--1 ! 

·--· I 

T 

~ 

) 

~ 
.-----

.----

---- I 

CD I® ® ® 

r-T1+T+T1 +~ KIT= 1

/2 =CONSTANT 

~ 

N 

I I 

,--, I 

\ , __ , 

\ ......_ -

A B c 

FIGURE 19.-A, B, C, Hydrographs from a basin having its width increased geometrically in downstream direction. A, At end of subarea 1; B, At end of 

subarea 2; C, At end of subarea 3. 

~ 

8 

0 

~ 

g 

tr.l 

t'4 

Sf 

~ 

t'4 

~ ~ 

fT.J 

1-4 

UJ 

0 

":rj 

~ 0 

0 

1::1 

=:tl 

~ ~ 

> 

~ "'""' 
A14 SYNTHESIS IN HYDROLOGY 

-

-+-

-

... 

1,/. ~ 

t I I I I I I I I I I I I I I I I I I 

1 !\ ~'~ I~ ~ 

7 'V~ ~ ~ 

"'~ 

r '~ 1----

A 

__; 

__; 

__; 

-

-

~ -

1- -""" I \ ~ 

ITT Till I I II T r I I 

'=~ I I T TIT I I T TT IT 

I 

I -

- \ - 1\. 

j ~ "' ~ 

~ l'---_ ~ 

B 

FIGURE 20.-Effect of storm movement. A, Storm moving 

upstream; B, Storm moving downstream. 

fied 10 times, the resulting outflow from the nonlinear 

reservoir would be greatly decreased in relative magnitude, whereas the inflow-outflow ratio would remain 

unchanged for the linear reservoir (fig. 22B). In 

either case, the storage constants are identical. This 

example shows that the relation of proportionality 

cannot be applied to nonlinear elements. A similar 

example is also shown in figure 23, in which case the 

reservoir has a nonlinear storage-outflow relationship 

of S=KQ 1

'

2

, and a peak inflow of 40 units. The 

upper outflow curve is for the nonlinear case while the 

lower outflow curve is for the linear one. 

The effect of scale on nonlinear reservoirs may be 

realized if one examines the routing equation (eq. 16) 

dQ I -Q=KxQx-1 ([[" 

Assuming that the scales of the inflow and the outflow 

are altered such that I' = p[ and Q' = pQ, then the 

corresponding routing equation becomes 

dQ p(l -Q) =pxK'xQx-1 di' (21) 

-

-

-

/'\. 

II \ 

" ~ 

J \ I I I -1 I I I I I I I I I I 

l TT ~ -

I ~ '--

~ "'r---r--_ - 1- -

A 

-

+ 

-

-

I I I I I rT I TTT T I I TTT T I TT T-f I TT T 

('"'...... r-- J -..... 

7 ~ 

~ 

I} ... 

~ 

~ 

--~ t---- ... 

B 

FIGURE 21.-Composite hydrographs from 4 tributaries. 

A, Tributaries having equal translation time; B, Tributaries having unequal translation time. 

in which K' is a modified storage constant. Dividing 

equation 21 by equation 16, one obtains 

or, 

K' 

p= (px) K' 

K K'=-· Px-1 (22) 

Thus the storage constant must be modified accordingly 

if th~ same proportionality is to be maintained. The 

relation shown in equation 22 is very useful in the 

design of a nonlinear model. It pr?vid~s a. necessary 

clue in detecting the degree of nonlinearity In natural 

basins. A practicable method of determining the 

values of K and x from the recession hydrograph has 

also been suggested in another paper (Shen, 1962). 

DESCRIPTION OF A HYDROGRAPH IN STATISTICAL 

PARAMETERS 

A general description of the storage effect on the 

inflow-outflow relation is complicated by the fact that 

the inflow diagram to a drainage system may have 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF A15 

f--

f--

f-

,/ 1\ I I I I I I I v '\ f-. I I I I I I I I I I I I I I I 

f--

fI ~ f-

~ ~~ \ r---;;;;; 

- A 

I ~ t-

+-

! .. ,\ 1-

f--

f--

_l I I 1- .I I _l I I I I I I I _l '/ ~- f-- I I 

f--

t-

~ ""- t- ; __ vv \ F--- t- r--- t-

- r- tt-

+-

-f.-

B 

FIGURE 22.-Comparison of outflow hydrographs, linear 

and nonlinear reservoirs (8 = KQ2). A, Peak flow is 4 

units; B, Peak flow is 40 units. 

unlimited range of variation in time distribution. 

Thus, to derive any universally applicable rule, this 

effect of time distribution must be adequately accounted 

for. From an engineer's standpoint, however, it is 

not often of particular importance to know with great 

accuracy the exact shape of the out-flow hydrograph 

as long as its peak can be determined fairly closely. 

Amorocho and Orlob (1961) used a two-parameter 

gamma distribution to approximate the shape of a 

unit hydrograph. Edson (1951) and Nash (1958) also 

proposed similar types of schemes. Mitchell (1962) 

employed two dimensionless ratios, kfT and trfto, to 

account for the time variation of inflow and outflow, 

in which, T is a characteristic time for a drainage 

basin; k is a storage constant describing the effect of 

minor storage upstream from the principal reservoir; 

tr is the time of travel through the reservoir; and to is 

the time to the centroid of the outflow hydrograph. 

Intuitively, one should be able to describe a hydrograph by means of certain statistical parameters, that 

is, its mode (M0 ), its first moment (L1), its second 

FIGURE 23.-Comparison of outflow hydrographs, linear and 

nonlinear reservoirs (S = KQ 1!2). 

or Qp 

I 

I • 

I I 

I I 

Time 

FIGURE 24.-Definition sketch of a hydrograph in terms of 

statistical parameters 

moment (k), and its coefficient of skewness (Sk). 

Thus the mode is equivalent to L111 and the first 

mom~nt is equivalent to the time of travel (centroid) 

The second moment, in conjunction with the first 

moment determines the standard deviation (u), which 

is a me~sure of dispersion of the hydrograph. Additionally, the coefficient ·of skewness is a measure of t~e 

skew of the hydrograph. Figure 24 shows the definition sketch of a hydrograph in terms of these parameters. By definition, the standard deviation is 

(23) 

and the coefficient of skewness (Pearson's) is 

(24) 

EFFECT OF STORAGE ON PEAK FLOW 

In the following analysis an attempt is made to 

generalize the effect of linear storage on the peak inflow-
.A16 SYNTHESIS IN HYDROLOGY 

1.0 

0.8 

0.6 

0.5 

0.4 _a.. 

........ 

O"o.3 

0.2 

0.1 

0.1 

~ 

~ v. 1- ~ 

~~ v 

• 

/ v 

0.2 0.3 0.4 0.6 

L,_. 

.......... ~ ~ . • ~I" 

./ ~· . 

7.• 

• v ~-

~ 

2.0 3.0 4.0 6.0 8.0 10.0 

FIGURE 25.-Peak-reduction ratio, Q 11 /1 'P' as a function of L1/ K. 

outflow relation. Accordingly, a large number of 

routings were made on the analog computer to cover 

a practical range of variation for each of these statistical 

parameters. The inflows used in these runs consisted of 

various single-peaked hydrographs that were derived 

from both rectangular and triangular time-area diagrams. The corresponding results are presented in 

dimensionless form. 

Figure 25 shows a plotting of the peak-reduction 

ratio, expressed by QP/lp, against the ratio of LtfK, 

where K is the storage constant of the respective reservoir. It may be clearly observed that the plotted 

points prescribe a uniquely defined trend. In fact, 

a single curve may be fitted through these points. 

Whereas all plotted points are within ± 15 percent of 

this curve, 86 percent of the data are within ± 10 

percent. 

A second trial is made in a plotting of QP/IP versus 

u/K (figure 26). The plotting also shows a fairly welldefined relationship. However, the correlation is less 

pronounced. 

In a similar manner, correlations with the other 

ratios of L2/K and Lp/K have been tried. However, 

these plottings indicate a considerable amount of 

scatter and are thus not presented. Further attempts 

have also been made to improve the relation between 

QP/IP and Lt!K by correlating its scatter with the skewness, S/k, as well as the dispersion, u/K, of the inflow 

hydrographs. The ranges of S/k and u/K covered in 

these trials are O-Q.78 and 0.065-3.50 respectively. 

The lack of correlation in either case indicates that no 

further improvement can be made at this time. 

COMPARISON WITH MITCHELL'S FORMULA 

On the basis of 128 synthetic outflow hydrographs, 

Mitchell derived a generalized expression (Mitchell, 

1962, p. 22) for the ratio of peak inflow and outflow, 

IP/QP, as 

I P/QP= 1 + [7.0- 2.5(k/ T) ](tr/to)2, (25) 

in which, k, T, tr and to are the basin parameters defined 

previously. The above relationship was obtained by 

routing consecutively an isosceles dimensionless hydrograph through a preliminary linear storage, k, and a 

principal linear storage, K, with consideration given 

to different durations of rainfall excess. Equation 25 

may also be expressed as a family of curves correlating 

lp/QP and tr/to for different values of k/T. 

In comparing Mitchell's formula with the general 

curve of QP/ I P versus L1/K (fig. 25), a number of routings 

have also been made on the analog models. These runs 

were arranged in accordance with the parameters established by Mitchell. For each run, an isosceles inflow 

diagram was routed first through a preliminary storage 

and then a principal storage. Hence, the ratios of 

k/T, trfto, and Lt!K were determined. Accordingly, 

the value of lp/Qp, and its reciprocal, QP/lp, were 

computed from equation 25. 
1.0 

0.8 

0.6 

0 .. 4 

a. 

.:::::. 0.3 a. 

0 

Q. 

0.2 

0.1 

0.1 

1.0 

0.8 

0.6 

0.4 

~0.3 

0 

0.2 

0.1 

0.1 

/ ~ • 

./ 

0.2 

ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF 

v 

/ VA ... 

v • 

/. 

0.3 0.4 0.6 

- .. ~ ~ 

~ • 

v v •• / 

v • 

• 

0.8 1.0 

ufK 

. 

2.0 

·- • 

3.0 

FIGURE 26.-Peak-reduction ratio, QP/1 PI as a function of u/K. 

• 

4.0 

---- ~ • 

--

/ ~ . /• 

v v • 

_,/ 

v. / 

• / 

/v v 

v 

/ 

0.2 0.3 0.4 0.6 2.0 3.0 4.0 

A17 

6.0 8.0 10.0 

. 

6.0 8.0 10.0 

FIGURE 27.-Comparison of Mitchell's formula with figure 25. Solid circles show values from Mitchell's formula. 

Figure 27 shows a comparison of the computed values 

of Q11 /111 with the curve shown in figure 25. In general 

the plotted points appear to have shown good agreement 

with this curve. 

Thus, the general relationship of Q11/l11 versus Lz/K 

shown in figure 25 is believed to be valid within the 

range of parameters tested. It provides a simple and 

yet useful tool for estimating the outflow peaks from 
A18 SYNTHESIS IN HYDROLOGY 

the inflow hydrographs if the reservoir is linear. 

Moreover, it is applicable irrespective of the shape of 

the inflow hydrographs. It should be borne in mind, 

however, that this relationship can be used only for 

simple hydrographs, that is, single-peaked hydrographs. 

For multiple-peaked hydrographs, it would be necessary 

to undergo a separation procedure before this relationship can be applied. 

From a practical point of view, the independent 

parameter, L., would be related to the physical as well 

as the climatical characteristics of a basin. Mitchell 

(1962) suggested that L., or tt in his notation, is a dependent function of the rainfall duration, D, and the 

preliminary storage constant, k. Furthermore, he 

found that the variation in the ratio of kfTis generally 

insignificant, thus a simplified relation in dimensionless 

form may be expressed as 

Lz (D) 7'=1+0.7 T. (26) 

ANALOG-MODEL STUDY OF FLOOD-FREQUENCY 

DISTRIBUTION 

One of the fundamental problems in hydrology is 

defining the probability distribution of runoff. The 

traditional approach to this problem has been based 

on a procedure involving the fitting of various probability distributions to the observed data of runoff. 

Frequently, this procedure met with only limited success 

owing to the lack of sufficient data needed by a 

statistical analysis. Nevertheless, these studies have 

set forth a basic understanding that the probability 

distribution of runoff is a function of the probability 

distribution of rainfall and the runoff-producing charRandom Nonlinear Basin signal r--- amplifier f-.- model ,..._ generator 

~ 

--

acteristics of a river basin. A few outstanding examples 

are cited as follows: 

By means of multiple correlation, Benson (1962) has 

derived formulas for New England correlating the 

flood magnitude at different return periods with the 

drainage-area size, a main-channel slope index, a 

precipitation intensity-frequency factor, a wintertemperature index, and an orographic factor. 

On the basis of the rational method, Snyder (1958) 

has developed a procedure for computing the probability 

of flood discharge from the given rainfall-duration 

frequency distributions of specific drainage basins. 

Empirical coefficients were derived for the Washington, 

D.C., area to account for the runoff-producing characteristics of area, length, slope, friction, and shape 

that are associated with the overland-flow areas as well 

as the sewered areas. The effects of rainfall duration 

and basin storage were also incorporated as a correction 

factor. 

Paulhus and Miller (1957) employed a system of 

index precipitation networks in synthesizing the floodfrequency characteristics of a group of basins from the 

rainfall data. The procedure involves the use of a 

typical unit hydrograph for each basin and an empirically 

derived adjustment curve to account for the variations 

in flood magnitude and base flow. 

Analyses of the influence of the basin characteristics 

on the rainfall-runoff probability distributions are 

handicapped by various sampling problems. It is 

seldom possible to select a sufficient number of identical 

drainage basins in the field. Moreover, the rainfall 

pattern is generally different within each basin. Thus, 

the principal problem is that of overcoming the sampling 

Level Frequency Output 

r---- frequency detector counter distribution 

Level Output 

discriminator Integrator f-- duration 

distribution 

FIGURE 28.-Schematic diagram of the frequency and duration analyzer. 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF Al9 

difficulties. An approach to this problem is offered by 

the use of synthetic models. 

Assuming that there is a stationary, random-signal 

generator which simulates a known frequency distribution of the inflow or rainfall excess, then by feeding 

the signal into a given basin model, one can examine 

the output from the system and analyze its frequency 

distribution. In order to reproduce the complete 

characteristics of an input distribution, however, one 

must be able to simulate its duration distribution 

(flow volume) as well as its peak distribution. Thus, 

a complete scheme would be one such as that shown 

in figure 28. 

Accordingly, three basic electronic components have 

been constructed consisting of a frequency synthesizer 

which produces random signals having various types 

of peak distribution; a frequency analyzer which 

determines the frequency of occurrence of the peaks 

above a certain level of magnitude; and, a duration 

analyzer which determines the percentage in time during 

which the signal equals or exceeds a certain level. 

LLJ 

<-' 

~ 

~ 

30 

20 

9 10 

0 

:I: 

en 

LLJ 

0:: 

:I: 

1-

0 

-10 

0.01 

~ 

' "-r--. )( 

'x 

"" ~ ~ 

"" ~ 

0.1 0.5 

~ 

~ ~ 

"' ~ '~ X 

~ ~ .......... 

2 5 10 20 

FIGURE 29.-Random input and output through a linear 

reservoir. 

As a simple illustration, figure 29 shows the appearance of the input and output signals through a linear 

reservoir, for which the input has a normal distribution. 

Accordingly, figure 30 shows the probability distributions of the simulated peak inflow and outflow as 

~ ~ " ~ 

""' k 0 ~ 

""' ......... "'· ~ ~ ~ 

40 60 80 90 95 98 99 99.9 

PROBABILITY, IN PERCENT 

FIGURE 30.-Probability distributions of inflow and outflow, linear reservoir, S=KQ. x, Inflow, normal; e, outflow. 
A20 SYNTHESIS IN HYDROLOGY 

FIGURE 31.-Random input and output through a nonlinear 

reservoir, S= KQ2. 

plotted on a piece of normal-probability paper. It is 

interesting to note that both distributions show the 

appearance of straight lines which indicate that the 

output is again normally distributed. 

On the other hand, if the same input signal is fed into 

a nonlinear reservoir (fig. 31), for which S=KQ2, one 

LLJ 

(.!) 

<( 

~ 0 

> 

30 

20 

0 10 ....J 

0 

J: 

(/) 

LLJ 

0::: 

J: 

1-

0 

-10 

0.01 

' ""' X 

"x_ 

~ 

-r--!"-!-r--

0.1 0.5 

" ~ 

~ 

--... r--._ 

~ ~ . ---1----. !'-. 

! 

2 5 10 20 

would observe that the frequency distribution of the 

peak outflow becomes skewed (fig. 32). 

In a similar manner, figure 33 depicts the distribution 

of an input signal that is log-normally distributed. 

Again, the output from a linear system shows a likely 

log-normal distribution. 2 

The analysis of flow volume or duration distribution 

may be treated by a similar procedure. To illustrate, 

figure 34 depicts a case in which the simulated inflow to 

a linear reservoir has a log-normal duration distribution. 

For the curve shown, the inflow at different levels is 

plotted against the percent in time during which the 

flow is less than that indicated. It is seen that the 

outflow exhibits another log-normal distribution in this 

case. Furthermore, the two distributions have equal 

mean values of discharge at 20.4. The standard 

deviations are 6.38 and 2.95 for the inflow and outflow, 

respectively. Thus, a linear-basin system acts to 

reduce the variance and the skewness of its inflow 

2 Although the results appear to be consistent within the scope of the present 

investigation, no general validity is presumed. 

""' K 

r--..... ~ ~ ~ 

~ ~~ 

" I• """' 

~ "' "' ~ 

40 60 80 90 95 98 99 99.9 

PROBABILITY, IN PERCENT 

FIGURE 32.-Probability distributions of inflow and outflow, nonlinear reservoir, S=KQ2• x, Inflow, normal; e, outflow. 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF A21 

100 

50 

... 

~ ......... 

...... 

~ 

• " """ ~ r-.... 

~ ~ 

G "' I ~ !'-... ~ ~ 

-............. " 

20 LLI 

(!) 

0::: 

<( 

:::c 

(.) 

en 

iS 

0::: 10 0 

LLI ['....... ...... 

'"' "'"-... (!) 

<( 

1-

"" ~ ...J 

0 

> 

5 "" ~ 

I ~ 

""' "" 

~'....... "-., ~ "'-.. ........ 

~~ I ~ • ............... ~ 

~ ~ 

2 

1 

0.01 0.05 0.5 1 2 5 10 20 30 50 70 80 90 95 98 99.5 

PROBABILITY, IN PERCENT 

FIGURE 33.-Probability distributions of inflow and outflow, linear reservoir, log-normal input. X, Inflow, log-normal 

distribution; e, outflow, chain of two linear reservoirs. 

probability distribution. It may be shown (Matalas, 

1963) that the coefficient of skewness decreases systematically with the linear-storage or carryover effect, 

such that it approaches zero as the carryover tends to 

infinity. 

The foregoing examples illustrate, in principle, that 

the relation of inflow and outflow can be described by 

an analysis of duration distribution and an analysis 

of peak distribution. Thus, for a particular problem, it 

would be necessary to simulate these two properties 

individually such that the inflow characteristics may 

be completely represented. The techniques described 

may, of course, be applied to more complex drainage 

systems as well. 

Mathematical treatment, in particular the powerspectrum analysis, of the linear transformation on a 

stationary random process is well known in the field 

of communication, where the power spectrum of the 

output, P 0 (j), min be simply related to the power 

spectrum of the input, P.dj), by (Freeman, 1958) 

(27a) 

where W(j) is the transfer admittance of the linear 

system. Accordingly, the variance of the output may 

be determined as 

(27b) 

For nonlinear systems, however, the process becomes 

rather involved-for example, in the work of Wiener 

(1942) and George (1959). A different treatment of 

this type of problem was offered by Langbein (1958). 
A22 SYNTHESIS IN HYDROLOGY 

100 

50 

v ~ ~ 

./ 

~ ~ ..--....--

?- ___... ~ ~ 

--=M" 

--~ / 

___,_.. ~ ~ / v 

.--- ~ ~ ..--- ~ 

~ v ol 

_.,... 

LLJ 

(.!) 

a:: 

<( 

:J: 20 

() 

en 

0 

a:: 

0 

LLJ 

(.!) 

10 <( 

I-

.....I 

0 ~ 

..,..,... v > 

..,..,... ~ 

X 

5 

2 0.05 0.5 1 2 5 10 20 40 60 80 90 95 99 99.9 

PROBABILITY THAT FLOW IS LESS THAN INDICATED 

FIGURE 34.-Duration distributions of inflow and outflow, linear reservoir. X, Inflow; e, outflow. 

Using the queuing theory, Langbein demonstrated a 

technique of probability routing pertinent to reservoirstorage analyses. The method is nonparametric in 

that it is unaffected by the kind of frequency distribution back of the probabilities. It is valid for nonlinear 

as well as linear reservoirs. 

Thus the principal advantage of using the analog 

technique in flood-frequency analysis lies in its simplicity in dealing with the more complex drainage 

systems when the mathematical process becomes too 

complicated. Once a system model is constructed, 

the inflow-outflow frequency relationship can be 

readily determined. The variation of the system 

model is virtually unlimited. 

SUMMARY AND OUTLOOK 

One of the fundamental questions that often arises 

in hydrology today is: "What are the effects of manmade changes on the frequency distribution of flood 

flows?" This is the kind of question that has not been 

satisfactorily answered by field investigations. It is 

also in this area that the analog technique probably 

can play a very important role. 

The primary objective of this study is to bridge some 

of the missing links encountered in field studies. One 

specific example is that of determining the effect of 

urbanization on the frequency and magnitude of peak 

flow. Carter (1961), in a study on the magnitude and 

frequency of floods in suburban areas at Washington, 

D.C., found that the lag time, r, between rainfall 

excess and the flood hydrograph is decreased in the 

ratio 1.20/3.10 because of storm sewers and improvements to the principal stream channels. Formulas 

were developed to express r in terms of the total length 

of the channel and a weighted slope for the developed 

and undeveloped areas. Furthermore, using 18 streams 

in the area, Carter derived a regression equation correlating the annual flood, Q, with r and the drainage 

area, A, expressed in square miles. It is 

(28) 

where F is a factor accounting for the increase in peak 

due to the percentage increase in the impervious area. 

Accordingly, if the effect of imperviousness is first 

accounted for, the variation in peak magnitude would 
ANALOG MODELS, IN ANALYSIS OF FLOOD RUNOFF A23 

be inversely proportional to the 0.45th power of the 

change in T due to suburban development: 

Q' (T')-0.46 =-ex: - Q T (29) 

Carter's conclusion is extremely interesting in that it 

appears to concur with some of the results derived from 

this study. Assuming that uniform rainfall of a short 

duration occurs over a drainage basin before and after 

the change, then the inflow time-area diagrams could 

be approximately represented by isosceles triangles 

(Mitchell, 1962; Snyder, 1958). Accordingly, for an 

equal volume of rainfall excess, the peak inflows, lp, 

would be inversely proportional to the first moments, 

Lt, of the triangular diagrams. Hence, 

(30) 

Furthermore, from the general curve in figure 25, an 

average slope of 0.56 may be obtained for the common 

range, of LI/K=0.5-2.0; that is 

(31) 

Assuming that the storage effect, K, of the basin was 

not altered by the suburban development, then by 

combining equations 30 and 31, one obtains the relation: 

Q,' ex: (L1')-o.44 

QQ L1 (32) 

The resemblance between equation 32 and Carter's 

equation 29 is indeed remarkable. In essence, T is 

equivalent to Lt+K-D/2 if the storage is linear, where 

Dis the duration of rainfall. Thus for rainfalls having 

durations approximately equal to 2K, T~L • Consequently, equations 29 and 32 are nearly identical. 

A more rigorous approach to this problem can be 

made by using the analog technique of frequency analysis described previously. If it is possible to estimate 

the change in T or L1 due to urbanization-such as the 

formulas suggested by Carter and Snyder-models may 

then be built to account for these effects. Thus, by 

knowing the frequency distribution of runoff before 

the change and the frequency distribution of rainfall, 

the expected frequency distribution of runoff subsequent to the change may be synthesized. The advantage of this type of analysis lies in the fact that it 

not only accounts for the di~tribution of peak flows 

but also renders consideration to the effects of duration 

distribution and sequential correlation. Work along 

this line is in progress. 

The foregoing discussion thus far involves only surface runoff. It begins from rainfall excess with the assumption that the losses due to such factors as evaporation and infiltration are already taken into account. 

Admittedly, these losses can be of a very significant 

nature especially in the arid lands. The most complete 

efforts known to the author are those due to Crawford 

and Linsley (1962) and Chow (1962). However, these 

suggested methods are still largely empirical at this 

time. It appears that further fundamental researches 

in the physical processes are necessary. 

During a private discussion Dr. Jacob Rubin indicated that he is currently undertaking a project on 

the mechanics of infiltration in the Menlo Park office 

of the U.S. Geological Survey. His approach, which 

incorporates the dynamic behavior of soil-moisture 

profile, appears to show considerable promise. An 

analog-model study of this infiltration aspect probably 

would be mutually beneficial. 

The technique of frequency synthesis described previously also carries certain statistical inferences. For 

example, it enables one to analyze the expected errors 

due to short-term samples. By synthesizing such varying types of probability distributions as normal, lognormal, Gumbel, and Pearson, it is also possible to 

determine the most suitable sampling scheme to be 

used for each type of distribution. 

Other proposed endeavors consist of applying the 

suggested modeling techniques to actual field basins. 

As an initial effort three existing experimental basins 

might be selected: one in the humid region, one in the 

arid region, and one in the urbanized region. Accordingly, linear and nonlinear models would be constructed 

for each basin so as to evaluate the degree of nonlinearity that might be encountered in these different environments. 

It is apparent that the high-speed computers, in 

particular the analog simulator, offer a convenient 

means for analyzing complex hydrologic systems. 

The analysis may be accomplished either by the direct 

simulation or by- the indirect solution of the mathematical relations describing such systems. With the 

benefit of the rapid advancement of modern electronic 

techniques, a hydrologist may greatly broaden his realm 

of hydrologic know-how. 

REFERENCES 

Amorocho, Jaime, and Orlob, G. T., 1961, Nonlinear analysis 

of hydrologic systems: Water Resources Center, Contr. 

40, Univ. California, Berkeley. 

Baltzer, R. A., and Shen, John, 1961, Computation of flows in 

tidal reaches by finite-difference technique: National 

Coastal and Shallow Water Research Conf. 1st, Los Angeles, 

Calif., 1961, Proc., p. 258-264. 

Benson, M. A., 1962, Factors influencing the occurrence of 

floods in a humid region of diverse terrain: U.S. Geol. 

Survey Water-Supply paper 158Q-B. 
A24 SYNTHESIS IN HYDROLOGY 

Carter, R. W., 1961, Magnitude and frequency of floods in 

suburban areas: U."S. Geol. Survey Prof. Paper 424-B, 

p. 9-11. 

Chow, V. T., 1962, Hydrologic determination of waterway 

areas for the design of drainage structures in small drainage 

basins: Univ. Illinois, Eng. Expt. Sta. Bull. 462. 

Crawford, N. H., and Linsley, R. K., 1962, Synthesis of continuous streamflow hydrographs on a digital computer: 

Stanford Univ., Dept. Civil Eng. Tech. Rept. 12. 

Dooge, J. C. I., 1959, A general theory of the unit hydrograph: 

Jour. Geophys. Research, v. 64, No. 2, Feb., p. 241-256. 

Edson, C. G., 1951, Parameters for relating unit hydrographs 

to watershed characteristics: Am. Geophys. Union Trans., 

v. 32, p. 951-956. 

Einstein, H. A., and Harder, J. A., 1959, An electric analog 

model of a tidal estuary: Jour. Waterways and Harbors 

Div., Am. Soc. Civil Engineers Proc., v. 85, No. WW3, 

Paper 2173, Sept., p. 153-165. 

Freeman, J. J., 1958, Principles of Noise: New York, John 

Wiley & Sons, p. 104-106. 

George, D. A., 1959, Continuous nonlinear systems: Research 

Lab. Electronics, Massachusetts Inst. Technology, Tech. 

Rept. 355. 

Harder, J. A., 1962, Analog models for flood control systems: 

Jour. Hydraulics Div., Am. Soc. Civil Engineers Proc., 

v. 88, No. HY 2, Paper 3074, March, p. 63-74. 

Ishihara, Tojiro, and others, 1956, Electronic analog computer 

for flood flows: Proc. Regional Tech. Conf. on Water 

Resources Development in Asia and the Far East, U.N. 

Economic Comm., Flood Control Series, No. 9, Bangkok, 

p. 17D-174. 

Kalinin, G. P., and Levin, A. G., 1960, Use of an electronic 

analog computer in the forecasting of rain floods: Meterologiya i Gidrologiya, No. 12, Moscow, Dec., p. 14-18. 

Translation by U.S. Joint Publications Research Service, 

Washington, D.C. 

Langbein, W. B., 1958, Queuing theory and water storage: 

Am. Soc. Civil Engineering Proc. Paper 1811, v. 84, No. 

HY 3, 24 p. 

Liggett, J. A., 1959, Unsteady open channel flow with lateral 

inflow: Tech. Rept. 2, July, Dept. Civil Eng., Stanford 

Univ. 

0 

Matalas, N. C., 1962, Some comments on synthetic hydrology: 

unpublished paper presented at the Symposium on Synthetic Hydrology and Simulation Techniques, July, Phoenix, 

Ariz. 

---1963, Statistics of a runoff-precipitation relation: U.S. 

Geol. Survey Prof. Paper 434-D. 

Mitchell, W. D., 1962, Effect of reservoir storage on peak flow: 

U.S. Geol. Survey Water-Supply Paper 158Q-C. 

Nash, J. E., 1958, The form of the instantaneous unit hydrograph: General Assembly Toronto Proc., Internat. Assoc. 

Sci. Hydrology, Toronto, 1957, v. 3, p. 114-121. 

Paulhus, Joseph L. H., and Miller, J. F., 1957, Flood frequencies 

derived from rainfall data: Am. Soc. Civil Engineers 

Proc. Paper 1451, v. 83, No. HY 6, 18 p. 

Paynter, H. M., 1952, Flood routing by admittance methods: 

Boston Soc. Civil Engineers Jour., v. 39, No. 2, April. 

Reprinted by G. A. Philbrick Researches, A Palimpsest on 

the Electronic Analog Art, 1960, Boston, Mass., p. 24Q-245. 

Richards, B. D., 1955, Flood estimating and control: London, 

Chapman & Hall. 

Rockwood, D. M., and Hildebrand, C. E., 1956, An electronic 

analog for multiple-state reservoir-type storage routing: 

Corps of Engineers, U.S. Army, Tech. Bull. 18, March, 

Portland, Oreg. 

Shen, John, 1962a, Simulation techniques for hydrologic models: 

unpublished paper presented at the Symposium on Synthetic Hydrology and Simulation Techniques, July, Phoenix, 

Ariz. 

---1962b, A method for determining the storage-outflow 

characteristics for nonlinear reservoirs: U.S. Geol. Survey 

Prof. Paper 45Q-E, p. 167. 

Snyder, Franklin F., 1958, Synthetic flood frequency: Am. 

Soc. Civil Engineers Proc. Paper 1808, v. 84, No. HY 5, 22 p. 

Stoker, J. J., 1957, Water waves: New York, Interscience 

Publishers, p. 291-308, 451-505. 

Thomas, H. A., and Fiering, M. B., 1962, Mathematical synthesis 

of streamflow sequences for the analysis of river basins 

by simulation: Design of Water-Resources Systems, Harvard Univ. Press, Cambridge, Mass., p. 459-493. 

Wiener, N., 1942, Response of a nonlinear device to noise: 

Radiation Lab., Massachusetts Inst. Technology, Rept. 129.