R . L K ennedy and W.Å . Ê'àé

5. MANOCHI~, Ê . (1958): Woprosy î Ü orografi czeskom ciklogenezie. M et. i Gidr . 6.— MANOCHIN, Ê . (1958): Problems of orographical cyclogenesis. M eteorological and Hydro-

logi cal Review, 6.

6. Ì iñí ëüczsw sx r, J. (1953): M eteorologiczna analiza powodzi z 10 i 11 maj a 1951 ã. Przegl.M et. i Hydr ., 3-4.

— MICHALCZEWSKI, J. (1953): Meteorological analysis of the 10-11 Ì àó 1951 fl ood. M eteoro-logi cal and Hydrologi cal Revi ew, 3-4.

7. Ì iñí ëüñêàæçêò, J. i Ì òñæüçêë, Í . (1963): Meteorologiczne przyczyny powodzi w Polscew lipcu 1960 ã. Prace PI H M , 74.

— Ì iñí ëüñêàú çêi , J. i Ì òñæüçêë, Í . (1963): Meteorological factors of the Polish liood ofJuly 1960. Sci entif ic papers of the Poli sh M eteorologi cal and Hydrological I nsti tute, 74.

8. Ì iüëòë, W . (1955): Synoptika wielkich opadow atmosferycznych w K arpatach. Przegl . M et.1 Hydr ., 3-4.

— Ì iüëòë, W. (1955): Synoptics of the great atmospheric precipitations. M eteorologi cal andHydrological Review, 3-4.

9. Ì òñæüçêë, Í . (1965): Przyklad ilustruj acy zwiazek pomiedzy polozenem frontu àrozniieszczeniem maksymalnych opadow. Wi ad. SL . Hydr . i M et. 3.

— Ì òñæüçêë, Í . (1965): A n il lustration of the front distribution and of the distribution of themaxima precipitations. News of the Hydrological and M eteorological Service, 3.

10. PARCzsw SKI, W. (1965) : Warunki wystepowania naglych wezbran na malych ciekach.Wi ad. Sl . Hydr . i M et. VI I I , 3.

— Ðëàñêçæçêi , W. (1965): Conditions of occurrence of sudden high water in small waterstreams. News of the Hydrologi cal and M eteorologi cal Service VI I I , 3.

11. Êëè í .î ú çêi, S. (1953): Problemy meteorologicznej prognozy powodzi . Przegl. M et. i Hydr .,3-4.

— Êëçëüî æçêi, S. (1953): Problems of the meteorological forecasting of fl oods. M eteorologi caland Hydrologi cal Review, 3-4.

12. Wonzrrts~ , Ì . i inni . (1961): Òóðó cyrkulacj i atmosferycznej w Europe i ich zastosowanieprzy opracowywaniu prognoz 5-cio dniowych dla Polski. Wi ad. Sl. Hydr . i M et. VI I I , 5.

— WODZINSKA, Ì . et al. (1961): Òóðå of atmospheric circulation in Europe and their uti l izationfor the elaboration of fi ve-day forecasts for Poland. # èè of the Hydrological and M eteoro-logi cal Servi ce VI I I , 5.

13. Zv BJAN, Z .D . (1949): Wozniknowienie i razwitie ciklonow i anticiklonow, Leningrad,Gi drometeoi zdat.

— Åï â1ëû, Z. D . (1949): Origin and development of cyclones and anti-cyclones.

The relationship between lag time and the physicalcharacteristics of drainage basins in Southern Ontario

R.J. K ennedy and W.Å. Watt , Queen' s University

K ingston, Canada

SUMMARY : This study is part of the Canadian IH D Programme intended to develop à standardmethod of computing the peak rate of runoff corresponding to à selected probability, for drainagebasins in the area range of ten to one hundred and fi fty square miles.

8áá

L ag ti me and the physi cal characteri sti cs of dràinàge basi ns i n Souther n Ontari o

The physical factors thought to have à maj or åéåñã on the lag t ime, here defi ned as the timeinterval from the midpoint of the excess rain to the centre of gravity of runoff , have been combinedinto an equation. Then the records of à number of isolated storms from each of à dozen drainagebasins have been analysed and the actual lag t imes compared with those indicated by the equation.The fi nal equation, with exponents adj usted, gives lag t ime in terms of area, shape factor, equiv-alent slope, storage factor and population factor.

êéàï ì é : Cette htude fait partie du programme canadien de la Dhcennie Hydrologique Inter-nationale (proj et R — SR- 13, Ontario — 36) dont le but est de dhvelopper une mhthode standardishepour calculer le dhbit maximum qui correspond à la probabilit6 äåÿ ãåå pour les bassins versantsayant une superfi cie entre 10 et 150 milles carrhs.

Les facteurs physiques qui, comme on le suppose, d6terminent le temps de d6caIage, ont åãårhunis dans une hquation (le temps de dhcalage est dhfi ni ici comme Ã|ï 1åã÷à11å entre le pointmoyen des pluies eff ectives et le centre de gravith de Ãåñî è1åò åï 1). Ensuite on à analys6 lesdonnbes pour une shrie d'averses isolbes dans chacun des bassins versants et le temps rhel ded6caIage est ñî ò ðàãá avec celui qui est 66éø | de Ãáöèàã1î ï . L 'bquation fi nale avec les exposants

corrig6s donne le temps de d6calage en fonction de la superfi cie, de Ã|ï é ñå de là forme, de lapente 6quivaIente, du facteur de Ãåï ï ï àäàÿ ï åò åï ã d' eau et de celui qui refl hte Ãinfl uence de1'activith humaine.

This study is part of Canadian IHD Programme R-SR-13, Ontario-36 which is intendedto develop à standard method of computing peak rate of runoff corresponding to àselected probability, for drainage basins in the area range of ten to one hundred and fi ftysquare miles.

Previous work [1, 4] has shown that à peak reduct ion factor which is dependent on lagtime, here defi ned as the t ime interval from the mid-period of excess rain to the centre ofgravity of direct storm runoff , can be used to predict with reasonable accuracy the peakrate of discharge generated by à given quantity of excess ãàø . The object of this study isto develop an equation which can be used to predict from their measurable physicalcharacter istics the lag times for basins in the selected size range.

Comparatively few such drainage basins ø Southern Ontario have been equipped withrecording stream gauges and where this has been done the records are usually of only àfew years duration. Records of several isolated intense storms on each of 12 drainagebasins have been analysed to determine the actual lag time. These basins are distr ibutedover à distance of about 200 miles east to west and less than half that north to south.

They are in the âàò å climatological area and since they are all south of the CanadianShield and show few rock outcrops they may be classifi ed as lying roughly ø onegeographical area. The slopes, shapes, soil types and population densities vary consid-erably.

Since the sample is small from à statistical point of view, it is especially important toestimate the exponents ø the proposed equations from the known relationships of openchannel hydraul ics, and to accept only an equation which has exponents in the properrange and as well fi ts the observed data.

ÒÍ Å D ETER M I N A T I ON OF ÒÍ Å OBSERV ED L A G T I M E

Records of runofi ' from the selected drainage basins were obtained from the Water

Resources Branch of the Department of Energy, M ines and Resources. Correspondingrecords of rainfall were made available by the Meteorological Branch of the Departmentof Transport .

For each basin à typical recession curve was determined by analysis of the recessionsegments of several storm hydrographs. The following procedure was employed todetermine the observed lag time for each storm.

8 6 7

R . J . K e n n ed y a n d % . Å . W a t t

5. Storage, ST (dimensionless)

The surface area of marshes, lakes and ponds on the upper two thirds of the drainagebasin, designated as AÄ was measured. The storage factor, ST, was then defi ned as

[1 + (20À,/A)] .

6. Population Factor, PThe population density ø persons per square mile, was determined for each townshipand then the average population density f or each drainage basin, ð , was est imated byproportions of areas. Where towns occupying very small proportions of the area weresituated on à basin, their populations were not taken into the average.

The population factor P = [1 + (p/5000)] .

DESI GN OF EQUATION S RELATING ÒÎ ÒÍ Å PHYSICALCHARACTERISTICS

An expression relating lag t ime to basin physical characterist ics should be relativelysimple. This requirement, together with the small size of the sample available, l imited thenumber of independent variables to 3 or at most 4. In addition it is essent ial that thisexpression be compatible with the known principles of hydraulics.

Consider an equation of the type

Ò = ;1 A S ß ~ ' 3 ß Ò 4

where À, S4, SF and ST are as previously defi ned.The simplest geometry for à drainage basin would be that of à semicircle. For it the

fl ow distance is proportional to À~ ~ and for similar channels T~ should be proport ionalto À~ ~. However, as À increases the quantity of discharge and size of the channel willincrease, with consequent increase in the hydraul ic radius and probably à decrease in therelative roughness. This reduces the eff ect of À on lag time and it may be expected that à,will have à value of 0.3 to 0.4.

The shape factor is intended to account for diff ering geometry among basins. I t shouldbe à function of the distance from the gauging station to the centroid of the basin if theslopes and channels were constant. Since they are not constant the shape factor may bemore useful if related to some representative length measured to à point farther up thebasin. The writers decided to use the basin length ~, and on this basis it may be reasonedin à general way that à3 should be somewhat less than 1.0.

The velocity in à given channel is well known to be proportional to Ó' 5. However,when slopes are steeper the channels are usually rougher or longer or both with the resultthat a~ probably should be in the — 0.25 to — 0.35 range.

The eff ect of area of storage on time of runoff depends on the geographical location ofthe storage and on the level of the water in the storage areas at the beginning of runoff .Since the information necessary to make even à rough estimate of the proper eff ect waslacking, the writers decided to adj ust the defi nition of storage unt il an exponent ø therange of 0.75 to 1.5 was obtained. The use of this storage factor is justifi ed solely by theimprovement in fi t which it brings about.

For the reasons advanced above it was expected that equat ion (1) should be

Ò = A Î .Ç 0 .4 ß 0 ' 2 5 0 .3 5 ~ Ò Î .7 5 1 5 ó ð 0 . 5 1 00 = È ()

Consideration of the å1Òåñ1 of the factors À and SF in equation (1) leads to the thoughtthat the two might be replaced by à representat ive length This is even more attractivefrom à statistical point of view if the selected L and à slope, S, are combined into one

870

L ag t i m e a nd t he p hy si ca l char a ct er i st i cs of d r a i nag e b asi ns i n So u th er n O n ta r i o

T ABLE 1. Ph y si ca l C h ar act er i st i cs o f t h e B asi n s

 àâ| ï Gauging À SFStation Sq. m i . ~ü ~ ç ~ ~ ~ ë ~ ç ßæ Á~mi . m i.

Par ts per 10,000

1 2 3 4 5 á 7 8 9 10 11 12

C a n a g a g i q u e C r .

C o l d C r .

C o n e s t o g o R .

D o n W e s t R .

D u ffi n C r .

E t o b i c o k e C r .

H i g h l a n d C r .

L a u r e l C r .

O a k v i l l e C r .

P a r k h i l l C r .

T r o u t C r .

W e s t H u m b e r R 2 G A — 2 3 4 2 . 0 2 Í Ñ — 2 3 2 2 . 7 2 G A — 1 7 1 2 5 . 0 2 H C — 5 3 4 . 4 2 H C - 6 1 0 0 . 0 2 H C - 2 6 3 . 8 2 Í Ñ — 1 3 3 4 . 0 2 G A — 2 4 2 3 . 0 2 Í Â — 5 3 3 . 0 2 F F — 3 4 8 . 0 2 G D — 9 5 1 . 0

2 H C - 8 7 9 . 0 1 . 7 7 1 1 . 5 1 . 0 7 5 . 1 0 1 . 3 4 1 5 . 0 2 . 0 3 1 1 . 9 1 . 3 4 1 3 . 4 2 . 3 5 1 8 . 8 1 . 4 5 8 . 5 1 . 5 3 7 . 3 1 . 7 7 1 0 . 2 2 . 4 7 1 7 . 1 1 . 7 2 1 2 . 31 . 8 3 1 6 . 3 1 5 . 9 3 7 . 5 3 0 . 2 4 2 . 9

6 . 1 4 9 . 9 5 7 . 4 7 2 . 5

2 2 . 6 2 0 . 5 2 1 . 7 2 5 . 9

1 5 . 8 4 7 . 0 4 8 . 5 5 8 . 3

1 5 . 7 6 9 . 9 6 7 . 1 8 8 . 9

2 6 . 1 3 9 . 9 4 3 . 5 4 1 . 6

1 3 . 3 3 4 . 3 3 9 . 6 4 2 . 2

1 0 . 6 2 7 . 7 2 6 . 8 3 2 . 1

1 3 . 5 5 6 . 4 6 7 . 7 7 2 . 2

2 6 . 8 1 3 . 3 1 5 . 3 1 4 . 1

1 7 . 0 9 . 3 1 2 . 1 1 6 . 8

2 2 . 8 3 5 . 2 4 1 . 4 4 3 . 5 6 3 . 4 1 . 3 2 1 . 0 1 1 2 0 . 0 1 . ô Î 1 . 0 2 3 2 . 8 1 . 1 1 1 . 0 0 7 2 . 4 1 . 0 2 1 . 3 6 1 1 3 . 0 1 . 2 1 1 . 0 3 5 6 . 4 1 . 1 5 1 . 0 7 6 5 . 9 1 . 0 0 1 . 7 6 4 5 . 1 1 . 8 5 1 . 0 1 1 0 0 . 4 1 . 1 3 1 . 0 2 1 9 . 1 1 . 0 2 1 . 0 0 2 5 . 4 1 . 0 9 1 . 0 1

6 4 . 4 1 . 0 3 1 . 1 1

parameter L /S0.s, à form used by some investigators. Using stream length and averagemain stream slope this equation has the form:

Ò — Ü~ Ü, ÁÒ (2)

The coeffi cient b, should have à value of 0.6 to 0.8 according to the arguments advancedpreviously. Àë equation that is somewhat easier to use is produced if L b and S4 are chosenas the typical length and slope.

Ò — Ñ S T

T ABÄE 2 . C o m p ar i so n o f O b ser v ed an d P r ed i ct ed L a g T i m es

 àâû Obser vedL ag Òèï å,

Üã. Predicted L ag Òèï åâ, Üã.

Equati on 5 Equat iou á Equat ion 7 Equation 8

C a n a g a g i q u e C r ' .

C o l d Ñ ã

C o n e s t o g o R .

D o n W e s t R .

D u ffi n C r .

E t o b i c o k e C r .

H i g h l a n d C r .

L a u r e l C r .

O a k v i l l e C r .

P a r k h i l l C r .

T r o u t C r .

W e s t H u m b e r R . 1 5 . 9 1 0 . 9 2 0 . 0 1 3 . 2 1 4 . 1 2 0 . 7 7 . 0 1 8 . 5 1 1 . 0 3 0 . 8 1 7 . 01 5 . 5 1 7 . 1 8 . 9 1 9 . 4 1 2 . 1 1 3 . 6 2 2 . 6 8 . 7 2 0 . 9 1 0 . 8 2 4 . 7 1 8 . 21 6 . 0 1 6 . 8 9 . 5 2 0 . 3 1 1 . 1 1 1 . 8 1 9 . 9 1 0 . 8 2 1 . 4 1 0 . 5 2 5 . 0 1 9 . 01 5 . 8 1 6 . 6 9 . 1 2 0 . 9 1 1 . 7 1 3 . 4 2 1 . 0 9 . 1 2 0 . 9 1 0 . 5 2 4 . 8 1 9 . 21 5 . 7 1 6 . 9 9 . 7 2 1 . 3 1 1 . 1 1 4 . 2 2 0 . 4 7 . 7 1 9 . 0 1 1 . 8 2 5 . 2 1 9 . 9

1 6 . 0

C o e ffi c i e n t o f M u l t i p l e

D eterm inat ion 0 .88 0 .78 0 .87 ' 0 .92

87 1

R . 1. K ennedy and W.Å . Wat t

Finally, if increasing population density is accompanied by the elimination of swampyareas and the improvement of drains and stream channels, the populat ion factor Pshould be added to give

Ò — d S T ð " . ( 4 )

R EG R ESSI ON OF T0 ON ÒÍ Å BA SI N CH A R A CTER I ST I CS

The values of the exponents were found by à stepwise mult iple l inear regression analysisof log T~ on the logarithms of the independent variables. With the exception of P thereduction in variance due to the addition of an independent var iable was ÿ äø éñàï 1 atthe 95% level or higher .

I t should be noted that all 4 defi nitions of slope were used in equation (1). Âàì ï slope,$4, would appear to be à logical measure to use with basin area and preferable becauseof the ease with which it ò àó be calculated. Since the use of the other 3 defi nitions ofslope did not improve the fi t , only the results using S4 in equation (1) are reported here.

The results of the regression analyses are given by equations (5), (6), (7) and (8).

ó = 5 . 5 2 À 0 . 3 9 ~ - 0 . 3 1 S T 1 .4 0 S F 0 . 9 6

0 .6 6T = 6.71 ' ßÒ '~'

0 ' ~ ~ 0 3

ç ( 5 )

( 6 )

0 .7 3ó 8 80 ' ~ ~ ó 1.300 ' ~ ~ 0 34 ( 7 )

0 .6 31(, . 6 ü SToë 6ð - î 630 ' ~ ~ 0 34. ( 8 )

Observed lag times and the lag t imes predicted Úó these equations together with thecoeffi cients of multiple determination are listed in table 2.

D I SCU SSI ON OF R ESU L T S

À comparison of lag t imes predicted by equat ions (5), (6) and (7) suggests à preferencefor equation (7). This relationship contains two fewer independent var iables than equa-tion (5) and yet yields essentially the âàò å goodness of fi t . The basin length and slope ofequation (7) can be determined more easily than the corresponding stream values ofequation (6), and use of the former results 1ï à better fi t .

With regard to equation (7) it is apparent from table 2 or fi g. 3 that the discrepancybetween predicted and observed values is greatest for the two basins with the longest andshortest observed lag t imes. The latter, Highland Creek, has à population density ofabout 3 800 persons per square mile while an unusually fl at slope exists on the upper ÜàÈof the other basin, Parkhil l Creek . When ò î ãå data become available it should bepossible to derive an equation of the form of (7) useful for predominantly rural basinsand which is more certain of not being unduly infl uenced by one or two odd basins.

The improvement in predict ion resulting from the addition of the population factor, P,is obvious from inspection of table 2 and éä. 3. The coeffi cient of mult iple determinationincreases from 0.87 for equation (7) to 0.92 for equation (8). On the basis of the present

8 7 2

S. N ; ,Gup ta, À .P . Bhat tacharya and S. R . Ji ndal

2. Fr om physical considerat ions, for basins in this size range and type, à storage factorof some k ind appears t o be necessary . A lthough the f orm of the storage factor used hasï î sat isfactory theoret ical basis it does signifi cantly impr ove the predicted lag t imes.

3. The populat ion factor , which has à sound physical basis, causes à moderate impr ove-ment in the predicted lag t imes. Because of the small sample size à defi n ite conclusion asto it s usefulness must await fur ther invest igat ion .

R E F E R E N C E S

Cttow , × .Ò. (1962): Hydrologic determination of waterway areas for the design of drainagestructures in small drainage basins, Uni versi ty of I lli noi s Engi neeri ng Experi ment Stati onBulleti n 462.

NASH, J.Å. (1960): À unit hydrograph study with particular reference to British catchments,Pr oc. I nst. Ci vi l Engrs., vol . 17, ðð. 249-282.

TAYLOR, À . Â. and SCHWARTZ, Í .Å. (1952): Unit hydrograph lag and peak fl ow related to basincharacteristics, Trans. Àò . Geophys. Union, vol . 33, ðð. 235-246.

WAr r , W .Å. (1964): Towards à rational determination of peak fl oods on small drainage basins øSouthern Ontario, Ì . Sc. thesis, Queen' s University.

Statistical correlation of Himalayan and Bundelkhandbasin characteristics with fl ood fl ows

S.N . G upta, À .P. Bhattacharya and S.R . Jindal , I rr igat ion Â.åâåàãñÛ ï â1|1Ø åR oorkee, I ndia

âî ì ì ëêò: The fl ood éî ÷ for à basin is generally evaluated either by: (i) empirical formulae,(é) statist ical or probability methods and (ø ) unit hydrograph method. Each method util izes oneor more elements of basin characteristics but none can be considered of universal applicabil ityfor the prediction of fl ood discharge. A n attempt has been made in this paper to correlate peakfl ood derived by the unit hydrograph method for eight H imalayan and Bundelkhand catchmentsin Uttar Pradesh with four signifi cant elements of basin characteristics, viz. catchment area,stream length, average stream slope and length from gauging station to the centre of the catch-ment . The statist ical derivation can be usefully applied to ungauged catchments for evaluatingfl ood fl ow.

RssvMk : Üå debit de crue pour un bassin est en g6nbral 6valu6 par : (1) des formules empiriques,(2) des ò é Üî éåâ statistiques ou de probabilit6, (3) la mhthode de ÃÜóéãî 8ãàø ø å unitaire.Chaque máthode utilise une ou plusieurs caractáristiques du bassin, mais aucune máthode ne peutetre ñî ï â|ñ1åãåå comme universelle pour la prbdiction des d6bits de crue. Ðàï â ñå rapport on àessay6 ñÃåãàÛ |ã la corrhlation du d6bit maximum, obtenue par la ø áãÜî éå de 1 hydrogrammeunitai re pour huit bassins versants de 1'H imalaya et du Bundelkhand dans Uttar Pradesh, avec

quatre ñàãàñé ã|âãù èåâ importantes du bassin, à savoir la superfi cie du bassin, la longueur dufl euve, la pente moyenne du fl euve, la distance de la station Üóéãî ø áããù èå j usqu'au centre du

bassin versant . Les conclusions statist iques peuvent etre appliqu6es aux bassins sans stat ionsÜóñ1ãî ò é ãù èåâ pour l '6valuation des dhbits de crue.

8 7 4