Ì . Í . D i s k i n

where:

M q is normal mean-water runoff obtained from stations similar in character and fromprecipitation reduced by evaporation ;

N is drainage area in km (sq. km) andл is the per cent of lakes within the drainage area, adj usted due to the location ø the

drainage area.

À lake situated immediately above the point, where computations are required, is givendouble weight while single weight is given to the other lakes, when computing the P~.

H ighest high-water runoff for à normal 50-year period is generally estimated to 2-3 timesnormal high-water runoff , and the fi gure for every category is obtained from availableobservations within the group.

All the data refer to mean éà11ó values.From experience i t is found that for smaller drainage areas additions up to 100% have

to be added to obtain the instantaneous highest high-water runoff . Lowest low-waterrunoff is obtained in similar way, though ï î computation of the instantaneous lowestlow-water ï Àï î é' exist.

H ighest high-water in Sweden is normally caused by melting snow, but in the smallerrivers, especially in southern Sweden, it is caused by heavy rains.

Examples of M q and HH q (mean daily values) for à normal 50-year period.

N km s . M q 1/s. km s H Hq 1/s. km s

M o u n t ai n o u s a r ea

E ast c o ast

W est c o ast

So u t h er n Sw ed en

N o r t h er n Sw ed en f o r est

N o r t h er n Sw ed en c o m b i n ed f o r est an d

m o u n t a in s 6 3 5 54 5 140 010 6 0 0 22 22 22 6 .3 10 .716 620

50 0

300

2 10

130

1 10

Transfer functions for the analysisof rainfall-runoff relations

Ì .Í . D iskin, Technion-Israel Inst itute of TechnologyÍ àÊà, Israel

SU MMAR Y : R ai n f a l l - ï ø î é r el at i o n s f o r w at er sh ed s ar e u su a l l y st u d i ed b y m ean s o f t h e c o n v o l u t i o n

in t eg r a l :

Q (s) =î

co n si d er in g t h e an al o gy b et w een t h e w at er sh ed an d à l u m p ed in p u t - o u t p u t sy st em , t h e t h r ee

962

Transf er f uncti ons f or the analysi s of r ai nf al l-r unop ' r elati ons

functions that appear in the integral are the output, QÄ>, the input, RÄ>, àï 4 the impulse response,U«>, functions of the system.

The convolution integral is strict ly applicable only È the watershed system is Iø åàã; otherwise,it may be used only in an approximate way by assuming that the response functions varies fromstorm to storm but remains constant during any one storm. N on-linearity of watershed systemsmay be studied by changes in the form of the response functions.

The process of deriving the shape of the response function from rainfall and runoff records istedious and involves t rial and error methods. It is possible however, to study the properties of theresponse function from the properties of the transfer function which is defi ned as the Laplacetransform of the response function. I ï contrast to the complexity of deriving response functions,the transfer function (U ) is obtained directly from transforms g , R, of the output and input functions, as the ratio of these functions

U(p) = (2(~)~~ (, ) •

The paper gives à discussion of the properties of transfer functions for watershed systems,methods obtaining these functions, and their use for the analysis of rainfall-runoff relations.

êâû ì ø : La relation entre les prbcipitations et Ãåñî è1åø åï ã pour les bassins versants est î ãé -nairement etudibe ðàã Ãø âåãø åé àâãå de Ãø Ì ~ãà1å de la convolution :

Q(t) = ~ (â) ~1(â- â) àòî

î è Ãàï à1î à1å entre le bassin et le systbme total d 'åï âãåå et 4å sortie est envi sage . Les t rois fonc-tions dans Ã>ï é äãà1å sont celles 4å la sortie Q«>, de Ãåï âãåå R«, et de 1'impulsion de ãåðî ï çå U«>.

L 'intbgrale de la convolution ne peut etre àðð1ù èåå qu'à la condition du systbme de bassinIinbaire. Dans le cas contraire on n'à que la solution approximative en admettant que la fonctionde la ãåðî ï çå varie d'une averse à Ãàï âãå mais reste constante au cours d'une seule averse. Üàï î ï -linbaritb des çóçâåø åç de bassins peut etre åâèé åå d'àðãåç les changements de la forme des

fonctions de la ãåðî ï çå.Üå processus de derivation de la forme de fonction susdite resultant des observations de

pr6cipitations et 4å Ãåñî è1åø åï â est assez diffi cile et comprend la ø åâÜî áå d'essais et d'erreurs.N>tanmoins Ãåâàðå des ðãî ðã1åãåç de la fonction de ãåðî ï çå est possible (Ãàðãåç les ðãî ðã1åãåç dela fonction de transport qui est 4åéï 1å comme la t r ansf or ms 4å la fonction de ãåðî ï çå de L aplace.1.à derivation de la fonction de ãåðî ï çå est t res compliqu& , tandis que la fonction de t ransfert Uest obtenue directement des transformbes Q, À des fonctions de Ãåï âãåå et de la sortie commele rapport de ces fonctions :

~~(p) = Q(p)>R (p) '

Üå ðãåçåï â rapport envisage les ðãî ðã1åâåç des fonctions de transport pour les systbmes debassins, les máthodes d'obtenir ces fonctions et leur application pour Ãàï àlóçå de la relation

entre les pr6cipitations et Ãåñî ï 1åø åï â.

I N T R O D U C T I O N

The relationship between rainfall and r unoff has many var ied aspects. The methodsadopted for studying th is relat ionship depend on the purpose of the study as well as onother factors such as the availabil ity of data. Thus an approach used when the annualyield of à given r iver is needed wi ll be diff erent f rom that used in the study of fl oodsproduced by the âàò å r iver .

The fl ood producing capacity of à r iver draining à watershed of known character ist icsis usual ly studied by means of the power fu l tool of the unit hydrograph theory. T histheory states that if the runoff hydrograph due to an isolated st orm of à k nown durat ionover the entire watershed is known, then it is possible to der ive à unit hydrograph of àdurat ion equal t o that of the known storm , which is character istic of the watershed

9 6 3

Ì . Í . Di sk i n

investigated. This unit hydrograph, or à mean unit hydrograph derived from à numberof storms of the same duration, ò àó then be used to estimate the runoff hydrograph,and hence the peak fl ood, due to any combination of rainfall expected with à givenfrequency over the watershed. For this purpose the expected rainfall pattern is dividedinto blocks, âî that the duration of each such block is equal to that of the unit hydrograph.The runoff due to rainfall in each block is then computed separately and the total ñèëî é'

is obtained as the sum of the individual runoff curves properly shifted according to thetimes of occurrence of the blocks of rainfall .

The trend in recent years has been to use unit hydrographs of very short durations, orin the limit to use instantaneous unit hydrographs, that are obtained when the durationof the unit hydrograph is allowed to approach zero as à l imiting value. The runoff Q®at any time t, due to à rainfall of known distribution R«> is given, if the instantaneous unithydrograph U«> is known, by the well known convolution integral

The diff icult problem associated with the use of instantaneous unit hydrographs is thederivation of the form of these hydrographs from rainfall and runoff records. Sinceactual extensive storms of very short durations are, as à rule, very rare, the direct methodsèçåé for deriving unit hydrographs cannot be used. Special methods are therefore neededfor the evaluation of instantaneous unit hydrograph. These special methods include thematching of moments of mathematical models to those obtained from the rainfall andrunoff data, à process analogous to long division for obtaining the ordinates of theinstantaneous unit hydrograph, and methods based on the use of transfer functions.

À transfer function for à given system is defi ned as the Laplace transform of its impulseresponse function which, in hydrological terms, is the âàò å as the Laplace transform ofthe instantaneous unit hydrograph.

The use of transfer functions for the analysis of systems is fairly common in the ñàâåof mechanical and electrical systems. I t is common practice in the study of these systemsto deduce the properties of the system investigated and of its impulse response functionfrom the propert ies of the transfer function. The advantage of using transfer functionsis that they are easily obtainable from available data of input and output for the systemsunder consideration. I f the diff erential equations describing the systems are known, itis also possible to der ive the transfer function for the systems considered from the formof the diff erential equations, without actually solving these equations.

I t should be noted that the convolution integral (equation 1), and the concepts of thei nst ant aneous unit hydrograph and of the transfer function are strictly applicable onlyif the watershed system is l inear. À linear system is defi ned here as à system for whichthe principle of superposition is correct . I f the system is not lø åàã and the non-linearityis not excessive it is still possible to derive for each rainfall-runoff event an instantaneousunit hydrograph that represents the transformation of rainfall into runoff for thatparticular event. Eff ects of non-linearity will be expressed in this procedure by changesin the form of the instantaneous unit hydrograph and hence in the form of the transferfunction from one rainfall-runoff event to ano ther .

PR OPER TI ES OF ÒÍ Å I N STA N T A N EOU S U N IT H Y D R OGR A PH

Some of the properties of the instantaneous unit hydrograph (or the impulse-responsefunction) for the watershed system ò àó be deduced from the evaluation of the processof the formation of runoff from rainfall . Áî ò å additional properties of the function areobtained directly from the defi nitions of the rainfall excess and the direct surface runbff '

functions.

9 6 4

Ì . Í . D i sk i n

( 5 )

( 6 )

I f the rainfall excess R«) and the direct surface runoff Q«) are expressed in the sameunits, for example in mm/hr (3.6 mm/hr = 1 m~/sec/km~), and if the time intervals dr

are measured in hours, it follows from equation 1 that the instantaneous unit hydrographis expressed in units of hr. Further, since the total volume of direct surface runoff isby defi nition equal to the total volume of rainfall excess:

f R(t)dt = Q(s)dtî î ( 7 )

it follows that the area under the curve representing the instantaneous unit hydrographmust be unity (dimensionless.)

f è « ) à = I . o o .

î

The requirement expressed by equat ion 8 limits the magnitude of the maximum ordinateof the instantaneous unit hydrograph to be of the order of (1/L). The practical range ofvalues for the maximum ordinate of the U«) curve may be t aken to be from (0.4/L)to (2/L).

ÒÍ Å U SE OF L A PL A CE T R A N SFOR M S FOR R A I N FA L L -R U N OFF A N A L Y SI S

The Laplace transform of à time function Y«), which is à sectionally continuous functionof exponential order in the range t > 0 and which is zero for t ( 0 is defi ned by thefollowing integral (å.g. Fodor, 1965):

( 9 )

The value of the transform function Ó<ð) for any value of its argument p depends on theshape of the function Ó«~ 1Üãî èäÜî è1 its range of defi nition. The correspondence betweenthe transform function Ó<ð) and the î ã® ï à1 time function Y«) is unique âî that it ispossible to study the properties of functions ø the time domain by the behaviour of theirLaplace transforms. The variable p of the transformed function has evidently the inversedimension of the original variable t. I f t is measured in hours, as is common øhydrology, the units of the variable p are hr.

I t may be noted at this stage that the three functions, Q«), R«) and U® related bythe convolution integral are all continuous or sectionally continuous functions ofexponential order âî that their Laplace transforms do exist .

The advantage of using Laplace transforms for the analysis of rainfall-runoff relationsis mostly due to the fact that the Laplace transform of the convolution integral(equation 1) is given simply as à product of the transforms of the two functionsconvoluted. The transform of equation 1 is thus

~ (ð) R (n) ~ (ð) ' ( 10à)

9 6 6

T r a n sf e r f u n c t i o n s f o r t h e a n a l y si s of r ai nf a l l - r u n og " r e l a t i o n s

The result ing equat ion may be directly solved f or the funct ion U (ð)

( 1 0 Ü )

I t follows that if the functions Q«) and R«) are known it is possible to compute theirLaplace transforms Q(» and ß (ð) and then use equation 10 to get the Laplace transformof the instantaneous unit hydrograph U(s), which is by definition the transfer functionof the watershed system.

PR OPERTI ES OF ÒÍ Å TR A N SFER FU N CTI ON

The general shape of the transfer function for watershed systems may be deduced fromits defi nition as the Laplace transform of the instantaneous unit hydrograph

U,(Ä„ , ) —— å ~' Ó(,)dtî

and from the properties of the instantaneous unit hydrograph discussed above.As à fi rst observation it may be noted from equation 11 and 2à that the values of the

function U (» are positive for all values of the variable p. I t follows from equation 8that the value of U (p) for ð = Î is unity. (Values of U(p) are dimensionless.) Further,as p increases the value of U (» decreases continuously tending to zero for large valuesof the variable p:

U ( p ) — — 1 . 0 f o l p = 0 (12à)

0 < ÃÓ(ð) < 1.0 f or ð ) 0 ( 1 2 Ü )

Ó ( ) -+ Î f or ð - + co . ( 12c)

Differentiating equation 11 with respect to the variable ð à function defi ning the slopeof the function U ( ) is obtained:

d U— — å ~ t U dt .(t )

dp î

Equation 13 indicates that the slope of the function throughout its range of def inition(Î < p) is negative and that the absolute value of the slope decreases continuouslytending to zero at large values of the variable p. The slope at the point ð = Î is seen tobe equal to the fi rst moment of the instantaneous unit hydrograph about the origin(compare equation 4)

1Η = Ldp at ð = Î .( 1 4 )

Reference to the second der ivat ive of the transfer funct ion

= + å "' t Ó(,)dtdp î ( 1 5 )

shows that the rate of change of slope is always positive and that it also decreases andtends to zero for large values of p .

On the basis of the above discussion it is ï î è possible to get the general shape of thetransfer function for watershed systems. I t appears that it is à function that has à value of

9 6 7

Ì . Í . D iski n

for comparing special transfer functions, but it cannot be the only criterion ; the behaviourof the special transfer function at other values of the argument pL should also beinvestigated.

The range of values of the maximum ordinates of the special transfer functions for ànumber of watersheds investigated were found to fall in the range of Ã~"' ,„ = 0.04 toÃ~ „ = 0.20 depending on the shape of the instantaneous unit hydrograph. Sharppeaked instantaneous unit hydrographs tend to produce special transfer functions havingà smaller maximum ordinate and fi at instantaneous unit hydrographs produce highvalues of special transfer functions.

Since it is possible to get à given value of maximum ordinate for special transferfunctions by diff erent instantaneous unit hydrographs it is clear that the comparisoncannot be confi ned to the maximum ordinate. The curves of the special transfer functionsshould be compared over à wide range. I t was found, however, that fairly goodcomparisons could be obtained by comparing two ordinates of the special transferfunction other than the maximum ordinate. The two additional ordinates chosen wereat values of pL = 5 and pL = 10.

C O N C L U SI O N S

Considerations of the process of formation of runoff from rainfall and the defi nitionof the rainfall and runoff functions may be used to derive some properties of theinstantaneous unit hydrographs for watershed systems. Corresponding properties of thetransfer functions, defi ned as the Laplace transforms î é ëå instantaneous unit hydrograph,can also be obtained.

Transfer functions are obtained directly from the Laplace transforms of the rainfalland ï ï þ ï ' functions. They form therefore à useful tool for the comparison of the

behaviour of à given watershed during diff erent storm events or for the comparison ofvarious watersheds.

The similarity in the general shape of transfer functions when plotted versus àdimensionless parameter defi ned in the paper led to the development of à special transferfunction which consists of the diff erence between the actual transfer functions and theexponential curve å >L. Properties of the special transfer and their possible use forrainfall-runoff analysis are outlined and Üï åéó discussed.

A C K N O W L E D G EM EN T

The paper is based on material prepared for and included in à thesis (D iskin 1964)written under the direction of Professor × .Ò. Chow, University of I llinois, U .S.À .

N O T A T I O N

å base of natural logarithmsG~ second moment of unit curve about its centroidÍ ~ third moment of unit curve about its centroid

L fi rst moment of unit curve about its origin (hrs)L < distance from origin to centroid of Q«> curve (hrs)LR distance from origin to centroid of R«>' curve (hrs)

Ì maximum ordinate of U«> curveð variable of L aplace transfer functions (hr.)Ä«> direct surface runoff function, (mm/hr.)Q(p) Laplace transform of Q«>, (mm)

9 7 0

Closi ng A ddresses

R«> rainfall excess function (mm/hr .)ß < ) Laplace transform of ß« (mm)t time variable (hrs)U«) instantaneous unit hydrograph function (l /hr.)Ó<ð) Laplace transform of U«) (dimensionless)(Ó,"' > à special Laplace transform, Utl~) = Ó1 )- å ð~

ò à dummy time var iable.

R EFER E N CE S

1. Ðâ êï ÷, Ì . Í . (1964): À Basic Study of the L inearity of the Rainfall-Êèï î é' Process in Water-sheds, Ph. D. Theses, University of I l linois, Urbana, Ø .

2. DISKIN, Ì .Í . (1967): À Laplace Transform Proof of the Theorem of M oments for theInstantaneous Unit Hydrograph, Journal of Water Resources Research, vol . 3, ðð. 385-388.

3. FODOR, G. (1965): Laplace Transforms in Engineering, Àé øå ò!à1 Ki ado, Budapest .4. NASH, J.Å. (1959): Systematic Determination of Unit Hydrograph Parameters, Journal of

Geophysi cal Research, vol . 64, ðð. 111-115.

Closing A ddresses

Prof . À .À . Áî êî Ì.î ÷

ÒÍ Å ESSENCE OF ÒÍ Å PROBLEM AND ÒÍ Å SI GNI FI CANCEOF ÒÍ Å SYMPOSIUM

The problem of fl oods and their computation is one of the main and most complicatedproblems of the science of hydrology, as is evident from the reports and discussionsof the symposium.

The development of the exploitation of water resources depends on fl ood-fl ow control ;the development of railways and highways and the problem of designing à great numberof spillways, the struggle against fl oods and soil erosion, which in many countr ies is àreal national calamity — all this requires increased accuracy reliability of fl ood ñî ò ðè-tation and forecasting.

From à practical standpoint, the essence of the problem under discussion is to basethe design of hydraulic structures such as dams, bridges, levees, etc. on reliable fl ood-fl ow characteristics. On the one hand the security and durabil ity of such structures underfl ood conditions should be assured, but on the other hand there should not be excessivemargins of safety, which raise the cost of the structures and immobilize large capitalinvestments.

From the scientifi c point î Ãì |åì , the task is one of studying the run-off process andworking out the theoretical fl ood discharge so as to achieve à scientifi c regulation ofthe fl ood fl ow from rain and snow melt , and of devising methods of computation incases where hydrometric observations are insuf5cient or absent.

97 1