Derived distribution of floods based on the concept of partial areacoverage with a climatic appeal

Vito IacobellisDipartimento di Ingegneria delle Acque, Politecnico di Bari, Bari, Italy

Mauro FiorentinoDipartimento di Ingegneria e Fisica dell’Ambiente, Universita della Basilicata, Potenza, Italy

Abstract. A new rationale for deriving the probability distribution of floods and help inunderstanding the physical processes underlying the distribution itself is presented. On thebasis of this a model that presents a number of new assumptions is developed. The basicideas are as follows: (1) The peak direct streamflow Q can always be expressed as theproduct of two random variates, namely, the average runoff per unit area ua and the peakcontributing area a; (2) the distribution of ua conditional on a can be related to that ofthe rainfall depth occurring in a duration equal to a characteristic response time ta of thecontributing part of the basin; and (3) ta is assumed to vary with a according to a powerlaw. Consequently, the probability density function of Q can be found as the integral, overthe total basin area A, of that of a times the density function of ua given a. It issuggested that ua can be expressed as a fraction of the excess rainfall and that the annualflood distribution can be related to that of Q by the hypothesis that the flood occurrenceprocess is Poissonian. In the proposed model it is assumed, as an exploratory attempt, thata and ua are gamma and Weibull distributed, respectively. The model was applied to theannual flood series of eight gauged basins in Basilicata (southern Italy) with catchmentareas ranging from 40 to 1600 km2. The results showed strong physical consistence as theparameters tended to assume values in good agreement with well-consolidatedgeomorphologic knowledge and suggested a new key to understanding the climatic controlof the probability distribution of floods.

1. Introduction

The theoretical derivation of flood probability distributionsis a matter of great interest for hydrologists as an attractivetool for flood frequency analysis in regions where sufficient orreliable direct measurements of streamflow are lacking. Fur-thermore, it helps in providing physical support to the classicstatistical analysis and remains a significant way to improveknowledge of the climatic and geomorphologic processes lead-ing to the generation of extreme floods.

Klemes [1987] ascribes the principal differences between ob-served flood series to the mechanisms underlying the genera-tion of floods. In addition, a remarkable influence of the cli-mate on the variability of the statistical behavior of floods wasclearly identified by Farquharson et al. [1992]. They analyzedflood data relative to arid and semiarid areas spread all overthe world (162 sites in Africa, Iran, Jordan, Saudi Arabia,Russia, and the United States), in comparison with other flooddata recorded in the United Kingdom, in humid and temperateclimates. In arid areas a significant homogeneity between theobserved frequency curves was recognized.

All the theoretical models deriving flood distributions owe atribute to Eagleson [1972], who first depicted a completeframework to describe the physical mechanisms underlying theprobability distributions of rainfall-generated floods. In hismodel the kinematic wave theory was used to account for the

surface runoff. The intensity and the duration of rainfall wereexpressed as exponentially distributed and reciprocally inde-pendent, and the infiltration process was schematized by wayof a constant loss rate. Other kinematic wave-based models arethose of Shen et al. [1990] and Cadavid et al. [1991], whoinvestigated, within the same framework, some situations notcompletely developed by Eagleson [1972].

Hebson and Wood [1982] and Wood and Hebson [1986] usedthe geomorphologic instantaneous unit hydrograph theory[Rodriguez-Iturbe and Valdes, 1979; Rodriguez-Iturbe et al.,1982] to model the basin response. Diaz-Granados et al. [1984]introduced the Philip’s equation to model infiltration.Moughamian et al. [1987] tested the Hebson and Wood [1982]and Diaz-Granados et al. [1984] methods and noticed unsatis-fying performances in both cases.

Sivapalan et al. [1990] investigated the case of partial con-tributing areas controlled by different mechanisms such asHortonian infiltration excess and Dunne’s saturation excess[Leopold, 1974]. They assumed that the rainfall intensity wasgamma-distributed and used the geomorphologic unit hydro-graph to account for the basin response. Raines and Valdes[1993] used the Soil Conservation Service curve numbermethod to model infiltration. Kurothe et al. [1997] accountedfor the bivariate probability density function of rainfall inten-sity and duration, which were supposed to be negatively cor-related.

It seems to us that the various quoted authors who havegone through Eagleson’s [1972] model have always limitedtheir action to improve either the rainfall statistic model or the

Copyright 2000 by the American Geophysical Union.

Paper number 1999WR900287.0043-1397/00/1999WR900287$09.00

WATER RESOURCES RESEARCH, VOL. 36, NO. 2, PAGES 469–482, FEBRUARY 2000

469

basin response function. In other words, they only modifiedsome parts of the model, leaving unchanged the general frame-work. The first exception is given by Gottschalk and Weingart-ner [1998] where the peak runoff is synthetically linked to therainfall volume by means of a runoff coefficient which is con-sidered to vary stochastically according to a beta function.

In this paper the basic idea arises from Eagleson’s [1972]paper where he strongly highlighted the important role of thepartial area contributing to the direct runoff. In that paper thisarea was identified as a stochastic variable dependent on stormintensity, duration, and size as well as on antecedent surfaceand subsurface conditions like vegetation and soil moisture.Only because of the difficulty in handling a trivariate jointdistribution, did Eagleson consider the contributing area as abasin feature, being independent of the rainfall variables.

In our approach the partial area a which contributes to theflood peak is considered as a stochastic variable. In addition,the “storm duration,” here intended as the rainfall durationmost significant for flood peak generation, is treated as a vari-able dependent on it. In particular, given a certain a , the lagtime ta, depending on the contributing area a , is supposed tobe the duration of the storm “responsible” for the flood peak.This lag time is here intended as the lag of direct runoffcentroid to effective rainfall centroid.

Such a scheme allows for simplifying the routing model, atthe price of the increased complexity needed to model themarginal distribution of the runoff contributing area. In thispaper it is also suggested that this distribution may be some-how related to the long-term climatic characteristics of thebasin.

The paper begins with a general description of the methodproposed to derive the cumulative distribution function offloods. The method is based on the combination of the prob-ability density functions of the contributing areas with that ofthe peak runoff per unit contributing area. Next, the wholegroup of governing parameters involved in the proposed dis-tribution is surveyed focusing on their physical interpretation,and an expression for the index flood is derived. In section 3,the application of the model to eight annual flood series re-corded in southern Italy suggests that the flood distributionparameters that have a stronger physical sound are sensitive tothe long-term climatic features of the basin. Then, on the basisof a dimensionless formulation of the derived distribution, apreliminary sensitivity analysis also highlights such a climaticcontrol.

2. TheoryIn the presence of a storm of any size, duration, and inten-

sity, whatever process of transformation of effective rainfallinto runoff is considered or whatever model for that transfor-mation is adopted, the peak of direct streamflow is always

Q 5 uaa , (1)

where a is the peak contributing area, that is, the partial areacontributing to the peak flow, and ua is the discharge per unit(contributing) area at the flood hydrograph peak.

The “contributing area” accounts for both the facts that thestorm area is finite and that the direct runoff comes from onlya fraction of the area covered by a storm. Moreover, for short-duration storms the “peak contributing area” is also controlledby the routing process. It is sometimes as low as 5% [Betson,1964].

Considering ua and a as two stochastic dependent variables,it is possible to derive the cumulative distribution function ofpeak streamflow integrating the joint density function g(ua, a)over the region R(q) within which the peak streamflow, de-pending on ua and a as in (1), is smaller then q .

GQ~q! 5 prob@Q , q# 5 EER~q!

g~ua, a! dua da . (2)

Analogously, we get the exceedance probability function ofthe peak streamflow Q , G9Q(q) integrating the same jointdensity function g(ua, a) over the region R9(q), within whichQ is greater then q .

G9Q~q! 5 prob@Q $ q# 5 1 2 GQ~q!

5 EER9~q!

g~ua, a! dua da . (3)

As already stated, ua and a are mutually dependent; thus thejoint density function g(ua, a) may be calculated as

g~ua, a! 5 g~uaua! g~a! 5 g~a uua! g~ua! , (4)

where g(uaua) is the distribution of the average runoff per unitarea given the same contributing area, g(a uua) is the distribu-tion of the peak contributing area conditional on the unitrunoff, while g(a) and g(ua) are the respective marginal dis-tributions.

Substituting (4) into (3), we get

G9Q~q! 5 E0

AEq/a

`

g~uaua! g~a! dua da (5)

or

G9Q~q! 5 E0

`Eq/u

A

g~a uua! g~ua! da dua. (6)

Thus the domain R(q) in (3) becomes explicit as the contrib-uting area a may assume only values lower or equal to the totalarea A of the basin, and then, in (5), for any a , ua must begreater or equal to q/a , while in (6), a must range betweenq/ua and A . In principle, either (5) or (6) could be exploited;in the following, consistent with the assumptions which we willmake in sections 2.1 and 2.2, we will use (5), which can also beexpressed as

G9Q~q! 5 E0

A

G9uauaS qa , aD g~a! dua da , (7)

which may lead to different formulations for the G9Q as differ-ent hypotheses are assumed for G9ua

ua , that is, the exceedanceprobability function of ua given a and g(a).

2.1. Contributing Areas

Let us refer to floods produced by runoff coming from anarea that may represent only a portion of the entire basin. Thishappens for the following arguments. Thinking of the basinarea as composed of some independent subbasins, the stormmay affect only one or a few of them. Furthermore, within anyof these subbasins the conditions needed to produce surfacerunoff, and eventually significant through flow, are not reached

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS470

everywhere during the flood event. Therefore the contributingarea is composed of the sum of the partial areas belonging tothe contributing subbasins. In other words, the flood peak isgiven by the superposition of flows coming from a randomnumber of subcatchments involved by the storm according tothe storm size and movement. This number depends on thebasin size relative to two characteristic space scales, namely,the storm and rain-cell sizes, hereafter indicated as Ss and Sc,respectively. In fact, for very small basins, whose area is com-parable to or less than Sc, there is a high probability that theentire catchment contributes to the peak flood with space-timehomogeneous rainfall. Whereas for basins whose area rangesbetween Sc and Ss, it is much more likely that a is given by thesum of separate subcatchment contributing areas. This is dueto different timing and duration with respect to which separateportions of the basin are involved by the rain cells and todifferent response times of these portions.

We assumed that the probability density function (pdf) ofthe peak contributing area is a gamma distribution for any asmaller then A . Thus letting PA be the finite discrete proba-bility that the whole catchment is contributing to the floodpeak, we get

g~a! 51

aG~b! S aaD

b21

exp S2aaD 1 d~a 2 A! PA, (8)

where, indicating with g( , ) the incomplete gamma func-tion [e.g., Gradshteyn and Ryzhik, 1980, p. 940],

PA 5 prob@a 5 A# 5 g~ A/a , b!

5 EA

` 1aG~b! S a

aDb21

exp S2aaD da , (9)

while the impulsive function d( ) is the generalized functiondefined by

E2`

`

w~ x!d~ x! dx 5 w~0! , (10)

so that

E2`

`

PAd~a 2 A! da 5 PA. (11)

Basically, the choice of the gamma distribution is a workinghypothesis. Nevertheless, it can be justified when it is remindedthat this function arises as the distribution of the sum of bstochastic (independent) variables exponentially distributedwith equal mean value a.

This justification helps in giving a meaning to the distribu-tion parameters. In fact, without any loss of generality, we maythink of any flood peak as being due to the superposition offlows coming from a number of subbasins which can be differ-ently involved by the storm. In addition, it is always possible topostulate that the subbasins that may provide runoff havecomparable sizes, so that it is consequent to suppose, for mod-eling purposes, that their contributing areas have equal mean.They can also be thought of as being independent of eachother, with the warning that in the eventuality of lack of inde-pendence, attention should be paid to the meaning of b.

The number of these subbasins is lower bounded by unity forvery small basins. Instead, for basins whose area ranges be-

tween Sc and Ss, it is possible to think of b as the number Nv

of subbasins of Horton order immediately smaller than that ofthe whole basin. In fact, these subbasins are, on average, ofcomparable sizes and may consequently be modeled by thesame value of a. For larger catchments, b could be evengreater than Nv because of the probable superposition ofdischarges coming from different subbasins covered by morethan one storm. Incidentally, it may be interesting to remem-ber that, according to well-consolidated geomorphologicknowledge, Nv tends to be invariant at any scale and assumesvalues ranging between 3 and 5 in nearly all cases [Horton,1945]. According to Gupta and Waymire [1983], its expectedvalue is close to 4.

However, the estimation of the gamma parameters could beleft to direct observation, and it is pointed out that even otherassumptions for the probability distribution of a could bemade, especially if local information is available to supportthem.

2.2. Peak Runoff per Unit Area and Flood Distribution

Let us make the following basic assumptions: (1) The peakdischarge per unit contributing area ua can be linearly relatedto the total amount of excess rainfall occurring on the contrib-uting part of the basin in a duration equal to a characteristicresponse time ta of the area itself; within this duration theareal and temporal variability of the rainfall intensity can beneglected thanks to the basin storage effect. (2) The durationta depends on the size of the contributing area only and is, onaverage, related to it by a power law. (3) The probabilitydistribution of the areal rainfall depth which may occur in afixed duration presents all the moments of order higher thanone independent of the duration, whereas its mean scales withthe duration according to a power law.

The first assumption is quite often invoked as a first-orderapproximation and leads to a rational type formula. Accord-ingly, the duration ta is assumed as the so-called critical du-ration (different from the flow equilibrium duration), that is,the rainfall duration which leads, for a given return period, tothe maximum peak discharge. To support the assumption, werefer to Fiorentino et al. [1987], who showed that under differ-ent choices of the basin hydrologic response function, the crit-ical duration approaches the lag time. This is particularly truewhen the shape parameter c9 of the intensity-duration function(idf) is in the range (20.5, 20.7). Here the idf is intended suchthat the rainfall intensity is proportional to the rainfall dura-tion to the power c9 . Moreover, with particular regard to thehydrologic response functions more close to the reality, theratio j of the peak direct runoff to the time-averaged excessrainfall intensity in the critical duration appeared very stable.In fact, j, which is a routing factor, was found to vary condi-tionally on c9 in a narrow range (0.6, 0.8) with an average valueclose to 0.7.

The assumption that the time variability of the rainfall in-tensity can be neglected within the critical duration is sup-ported by the above quoted observation that this duration isoften close to the lag time. In fact, according to that and owingto the smoothing effect provided by the routing of excess rain-fall through the basin, intensity fluctuations within the rainfallduration should not be significant. Moreover, since the lagtime tends to scale, as is well known, with the basin area by apower law, the second assumption is also supported.

The third assumption is usually very accurate for rainfallduration ranging from more or less 1 hour to a little more than

471IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

1 day. Thus it can be confidently applied when analyzing catch-ments with area ranging from a few tenths to some thousandssquare kilometers. It is still reasonable out of (but not too farfrom) this range. In the range considered, in fact, the pdfmoments different from the first are controlled by climaticfeatures, for example, the average humidity conditions and theprobability of occurrence of extreme rainfall. These quantitiescan be considered event features and then considered inde-pendent of the within-the-event rainfall characteristics. More-over, in this range the rainfall idf is often expressed as a simplepower law.

Let us make an additional assumption: (4) The excess rain-fall can be derived by finding the difference between the rain-fall depth and the total loss amount.

This is mainly based on the first assumption, according towhich, within the contributing area and during the productionof runoff, the storage and infiltration losses can be clusteredinto a loss factor fa, dependent on the contributing area a . Weexpress this quantity as the ratio of the total water losses to therainfall duration. Thus the peak hydrograph can be predictedwithout discriminating between the initial adsorption and in-filtration. The coefficient fa is strongly dependent, at the eventscale, on the antecedent moisture conditions of the watershed,while it is expected to be much more stable with respect to thebasin permeability and to its degree of vegetation. However,we believe that in the frequency domain, fa can confidently beassumed as mainly controlled by the expected basin status atthe timing of a heavy storm.

All that being stated, we may write:

ua 5 j~ia,t 2 fa! , (12)

where ia ,t is the space-time averaged intensity of the totalrainfall which occurs in a duration ta over the area a contrib-uting to the peak discharge Q . Hence, following the aforemen-tioned assumptions, it is possible to state that the probabilitydistributions of ua and ia ,t are mutually related. In addition,the pdf of ua given a will exploit the above-quoted relation-ships existing between the expectation of ia ,t, ta, and a .

As a distribution of ia ,t, we assume, consistent with theexploratory nature of the paper, a Weibull pdf which is wellsuited to account for the frequently observed skewness of rain-fall intensity. It can be written as

g~ia,t! 5k

ua,tia,t

k21 exp S2ia,t

k

ua,tD , (13)

with

ua,t 5 E@ia,tk # 5 $E@ia,t#/G~1 1 1/k!%k (14)

Accordingly, the cumulative distribution function (cdf) ofia ,t is written as

G~ia,t! 5 1 2 exp S2ia,t

k

ua,tD . (15)

Then, the cdf of ia ,t, given ia ,t greater than fa, is

G~ia,tuia,t . fa! 5 1 2 exp S2ia,t

k 2 f ak

ua,tD . (16)

Following (16), because of the monotonic relation betweenua and ia ,t, (12), the cdf of ua is

G~uauia,t . fa! 5 1 2 exp 32S ua

j1 faD k

2 f ak

ua,t4 . (17)

In addition, let E[ia ,t] be related to a by a simple power law:

E@ia,t# 5 i1a2«, (18)

where i1 is the mean areal rainfall intensity referred to the unitarea and « is a scaling factor that is somehow dependent onboth the exponents of the idf and of the power law relating ta

to a .The local infiltration process may be modeled by means of

the Philip’s equation:

f i 512

Si t21/ 2 1 c , (19)

where Si is the sorptivity and c is the gravitational rate ofinfiltration. Integrating (19) with respect to time, we get thecumulated infiltration volume at time t:

Fi 5 Si t1/ 2 1 ct , (20)

which, divided by t , gives the mean infiltration rate:

f# i 5 Si t21/ 2 1 c . (21)

Inserting the relation between the lag time and the contrib-uting area,

ta 5 t1an, (22)

and neglecting the gravitational term and passing from thelocal scale to that the contributing area, we get

fa 5 f1a2«9, (23)

where f1 is a constant value and «9 5 n/2; «9 tends to 0.25 whenconsidering for the exponent n of (22) the classic value 0.5.However, because of the uncertainty about both the gravita-tional term and the exponent of the Philip’s equation whenused at the basin scale, one should expect that the best esti-mate of «9 could be significantly different from n/2.

The exceedance probability function of ua, given a , is ob-tained by (17), (14), (18), and (23) as

G9~uauia,t . fa! 5 exp H2~ua/j 1 f1a2«9!k 2~ f1a2«9!k

@i1a2«/G~1 11/k!#k J . (24)

By means of (8) and (24), (7) becomes

G9Q~q! 5 E0

A

g~a! exp H2@q/~ja! 1 f1a2«9#k 2 ~ f1a2«9!k

@i1a2«/G~1 11/k!#k J da ,

(25)

where g(a) is given by (8).Since Q is the peak direct streamflow, the flood peak is

found as

Qp 5 Q 1 q0, (26)

with q0 as a base flow.Under the hypothesis of Poissonian occurrence of indepen-

dent flood events the cdf of the annual maximum values of Qp

is

FQp~qp! 5 1 21T 5 exp @2LqG9Qp~qp!# , (27)

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS472

where T is the recurrence interval in years and Lq is theaverage annual number of independent peak streamflowevents.

Finally, after (8), (25), and (26), (27) becomes

FQp~qp! 5 exp H2LqFE0

A

g~a!

z exp F2~~qp 2 q0!/~ja! 1 f1a2«9!k 2~ f1a2«9!k

~i1a2«/G~1 11/k!!k G daG J . (28)

2.3. Governing Parameters

Summarizing, the quantities that have been recognized asresponsible for controlling the probability distribution of an-nual floods can be grouped as (1) those affecting the proba-bility distribution of contributing areas, that is, a and b as in(8); (2) the ones regarding the probability distribution of arealexcess rainfall intensities given a certain contributing area,namely, i1, f1, « , «9 , j , and k as in (24); (3) the base dischargeq0 above which the flood sequence can be schematized as acompound Poisson process; and (4) the annual rate Lq of thisprocess.

Although the meaning of these parameters has been clari-fied in sections 2.1 and 2.2, it is useful to discuss the way theycan be reconnected, when needed, to other measurable andphysically interpretable quantities. Furthermore, somethingmay be said about their typical range and their stability.

With regard to the first group of parameters most have beendiscussed in section 2.1 with respect to b. Regarding a, that is,the ratio of the mean contributing area to b, one can argue thatits ratio to the basin area A should be strongly controlled bythe runoff-generating mechanism and by the climate. In fact,this ratio is expected to be lower in humid basins where asaturation excess mechanism is likely to prevail. This will behereafter confirmed in section 3. In arid zones, where thevegetation is usually scarce or absent, the mean contributingarea may also strongly depend on the watershed geology.

The parameters ascribed to the second group can further bediscriminated into three subcategories. In particular, i1, « , andk depend on the space-time-frequency rainfall pattern, j is arouting coefficient which, as referred to in section 2.2, showshigh stability, and f1 and «9 are related to the infiltration andadsorption mechanisms. With regard to the first subcategorythe necessary information may follow from knowledge of thespace-time rainfall process which, however, can still be consid-ered as an area open for research development. However, away to draw simplified information about the main features ofthis process is provided by empirical formulae relating theareal to the point precipitation. Moreover, when the probabil-ity distribution of the annual maximum rainfall is known, onecan exploit the relationships existing between the annual max-ima and the base process. Incidentally, the use of informationprovided by the annual maxima rainfall process could be seenas more appropriate than investigating the base rainfall mech-anism when inferring extreme flood distribution, as in this case.

Let pt be the annual maximum of the at-site rainfall intensityaveraged on the duration equal to t; pt varies with t accordingto the idf, then its expected value can be written as

E@ pt# 5 p1tn21. (29)

In (29), n 2 1 is equal to the coefficient c9 defined in section2.2.

The mean rainfall intensity scales with the catchment areaand duration in a quite complex manner [e.g., Rodriguez-Iturbeand Mejia, 1974; Ranzi and Bacchi, 1994]. However, accordingto the aim of this paper, it may be sufficient to refer to thesimple and classic method of the areal reduction factor pro-posed by the U.S. Weather Bureau, invoked here as by Eagle-son [1972]. Thus letting pa ,t be the annual maximum of therainfall intensity averaged in time over ta and in space over a ,

E@ pa,t#

E@ pt#5 1 2 exp ~21.1ta

0.25! 1 exp ~21.1ta0.25 2 0.004a! ,

(30)

with ta expressed in hours and a in square kilometers. As it willbe shown below, this method allows for preservation of thesimple scaling relationship of (29) with regard to E[ pa ,t] too.In fact, as a simple exercise, we substituted (22) and (29) into(30) using, in equation (29), p1 5 25 (mm hr21) and n rangingfrom 0.2 to 0.5, values typically observed in Basilicata (south-ern Italy). Also, in (22), let t1 5 0.16 (h km21) and n 5 0.5,values which account for the lag time spatial variability in mostbasins of the same region.

Consequently, the curves shown in Figure 1, where E[ pa ,t]is plotted versus the basin area a , are achieved. Although thesecurves clearly present a scaling break at the value a 5 500km2, it should be pointed out that before and after that pointthe slope does not change dramatically. In addition, the expo-nent of the relative power laws stays in a very narrow rangebetween 20.1 and 20.3.

This result supports the basic assumption leading to (18)because E[ pa ,t] is linearly related to E[ia ,t] in a number ofcompound Poisson processes. In our case we assumed theindependent rainfall to be distributed as Weibull; hence therelative compound Poisson process accounting for the distri-bution of the annual maxima is the so-called power extremevalue distribution [e.g., Villani, 1993]. In such a case the rela-tionship between the mean value E[ia ,t] of the exceedances inthe base process and the mean value E[ pa ,t] of annual max-ima is

E@ pa,t# 5 LE@ia,t#SL, (31)

with

SL 5 Oj50

`~21! jL j

j!~ j 1 1!~1/k11! , (32)

Figure 1. Expectation of the annual maximum value of theareal rainfall intensity in the critical duration ta versus the area a.

473IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

where L is the annual mean number of rainfall occurrencesthat exceed a given intensity threshold. It can be shown thatE[ia ,t

k ] is independent of that threshold, while L depends on it.In particular, assuming the threshold equal to zero, L coincideswith the mean number of independent annual rainfall eventsLp, and the expectation of ia ,t may be calculated as in (31)replacing L with Lp.

As frequently observed, the value of Lp may be considered,at least at an hourly timescale, as independent of t [e.g.,National Environmental Research Council, 1975]. Therefore,following (31), E[ia ,t] may be supposed to scale with a as in(18), where, according to Figure 1, « should range between20.1 and 20.3.

Consistent with (12), the mean annual number of indepen-dent floods Lq must equal the mean annual number of ex-ceedances over a threshold equal to fa. Then, according to thedistribution adopted for ia ,t, the independence of E[ia ,t

k ] fromf a

k allows writing the following relation between Lp and Lq as

Lq 5 Lp exp S2f a

k

E@ia,tk #D , (33)

which also gives

log ~Lp/Lq! 5 f ak/ua,t. (34)

The ratio at the right-hand side of (33) depends on thecontributing area a that, in turn, is related to the critical rain-fall duration. Therefore, since Lq is independent of the rainfallduration, (33) supports the assumption « 5 «9 when Lp isconstant with respect to duration too. A convenient dimen-sionless parameter related to fa is

f * 5 fa/E@ia,t# . (35)

This parameter represents a global loss coefficient that isindependent of the contributing area a and of the rainfallduration as demonstrated by the following relationship derivedby use of (33) and (14)

f * 5@2log ~Lq/Lp!#

1/k

G~1 1 1/k!(36)

2.4. Index Flood

On the basis of the above equations, we can now introducean index flood QI, defined as

QI 5 E@uE@a##E@a#Lq SLa 1 q0. (37)

QI is as closer to the mean annual flood Qm as u and a are lessdependent on each other and as the probability distribution ofu given a is better described by a Weibull law. QI represents aconsistent approximation for Qm, whose exact expression isnot easy to derive because of the complexity of (28). Anyway,the fact that QI provides a good approximation for Qm issupported by a number of numerical experiments presented insection 3.2.

In (37), E[uE[a]] can be obtained once E[ua] is calculatedaccordingly to (17):

E@ua# 5 E@j~ia,t 2 fa! uia,t . fa#

5 jEfa

` kua,t

~ia,t 2 fa!ia,tk21 exp S2

ia,tk 2 f a

k

ua,tD dia,t. (38)

Then, integrating by parts,

E@ua# 5 jF ia,t exp ~ f ak/ua,t!

g~1 1 1/k , f ak/ua,t!

G~1 1 1/k!2 faG . (39)

Thus (37) becomes

QI 5 jH i1

g@1 1 1/k , log ~Lp/Lq!#

G~1 1 1/k!2 f1JLq SLq r12«A12« 1 q0

(40)

where r is the ratio of the mean contributing area to the totalbasin area A , that is,

r 5 E@a#/A . (41)

This parameter is related to basin features such as climateand vegetation as shown in section 3.

3. ValidationAn analysis aimed to test whether the proposed distribution

is able to reproduce real-world data with parameter valuesconsistent with the proper physical interpretation is reportedhere. The proposed distribution was applied to the annualflood series of eight gauged basins in Basilicata (southern It-aly), with at least 15 years of reliable observations, wherefloods were practically rainfall-generated only.

3.1. Estimation Procedure

All the distribution parameters needed for the analysis weredrawn from available studies, except the contributing area ra-tio r as in (41), which we used for calibration purposes. Thisparameter is related to a as in (8) through (41), given thewell-known property of the gamma distribution:

a 5 E@a#/b . (42)

All the available studies were based on annual maximumdata, with regard to both rainfall and floods. The main featuresof the basins are listed in Table 1a, where N , E[Qp], Cn , andCa are length, sample average, coefficient of variation, andskewness of the annual flood series, respectively. In Table 1athe basin area A and the mean annual number of independentfloods Lq, as well as of independent storms Lp, are also listed.In particular, Lq and Lp were taken from Iacobellis et al.[1998], who used the generalized extreme value distribution[Jenkinson, 1955] for fitting floods and the two-component

Table 1a. Observed Basins and Their Main Features WithParticular Regard to the Length, the Moments, andParameters of the Annual Series

SiteA,

km2 NE[Qp],m3 s21 Cv Ca Lq Lp

Bradano—San Giuliano 1657 17 507 0.79 1.03 2.9 21Bradano—Ponte Colonna 462 32 202 0.76 1.21 4.0 21Basento—Menzena 1382 24 401 0.63 1.57 6.6 21Basento—Gallipoli 853 38 353 0.63 2.25 8.5 21Agri—Tarangelo 511 32 189 0.49 0.75 16.8 21Sinni—Valsinni 1140 22 555 0.56 2.42 19.1 21Basento—Pignola 42 28 35 0.43 1.1 19.6 21Sinni—Pizzutello 232 19 255 0.51 0.75 31.0 32

Definitions are as follows: A, basin area; N, length; E[Qp], meanannual flood; Cv, coefficient of variation; Ca, skewness of annualflood series; Lq, mean annual number of independent floods; and Lp,mean annual number of independent storms.

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS474

extreme value distribution [Rossi et al., 1984] for analyzingrainfall annual maxima.

The values of parameters p1 and n , shown in Table 1b, werederived by Claps and Straziuso [1996], who applied a krigingprocedure in order to achieve basin representative values start-ing from the available at-site information.

The values of the basin lag time tA (see Table 1b) weredrawn, with regard to basins 1, 2, 4, 5, and 7, from Rossi [1974],who recognized a local stability of tA for return periods greaterthan 10 years. For basins 3, 6, and 8 we used the empiricalrelationship tA 5 0.16=A (tA in hours and A in squarekilometers), fitted to basins of Bradano, Basento, and Sinni[Dipartimento di Ingegneria e Fisica dell’Ambiente, 1998].

With regard to k , we used annual maximum data of 78gauging stations located within the entire region. In particular,we decided not to deeply detail the regional analysis, and weput k 5 0.8, equal to the overall average of the k at-siteestimates obtained from the annual daily rainfall series.

The base flow q0 (see Table 1b) was given a tentative valueequal to the average monthly flow observed in January andFebruary, which are the months with the highest probability offlood occurrence. Finally, for b as in (8), j as in (12), « and «9as in (18) and (23) respectively, we decided, following theabove discussions, to assume all over the region and as asufficiently good first-order approximation, b 5 4, « 5 «9 50.25, and j 5 0.7. Here f * was estimated by use of (36).

For estimation purposes, in order to overcome the uncer-tainty in dealing with the parameters i1 and f1, we substitutedthe following for (18) and (23):

E@ia,t# 5 iA~a/A!2« (43)

fa 5 fA~a/A!2«9. (44)

Thus iA was calculated by use of (29), (30), (31), and (32) byputting t 5 tA in (29), a 5 A in (30), and L 5 Lp in (31) and(32). The loss coefficient fA was derived by (33) taking thequantities at the right-hand side as referring to the wholewatershed.

3.2. Results

The estimated values of r are shown in Table 4 along withthose of fA and f *; in subsection 3.3 we will focus on theclimate influence on these parameters. In Table 2 the theoret-ical values of QI as from (37), of the mean (E[Qp]c) and of thecoefficient of variation (Cnc), which were calculated by nu-

merical integration based upon the proposed distribution, aredisplayed.

The good agreement between E[Qp]c and Cnc and E[Qp]and Cn , respectively (see Table 1a) stems trivially because ofthe performed calibration. However, it is worth mentioningthat because of the accordance between E[Qp]c and QI, thechoice of the proposed index flood is validated.

Figures 2 and 3 show the found cdfs of the peak flood andthe contributing area pdfs, respectively. In Figure 2 we usedthe plotting position suggested by Cunnane [1978]:

Pi 5 ~i 2 0.4!/~N 1 0.2! .

Figures 2 and 3 clearly display that the theoretically deriveddistribution is able to describe the main statistical features ofthe observed annual flood series. This is not a trivial result ifone notes that, despite the numerous parameters used in thedistribution, only Lq and r were estimated by using the flooddata.

3.3. Climatic Control

As pointed out by T. Dunne (quoted by Leopold [1974]), inhumid and semihumid basins, where it is likely to find highlyvegetated zones, one can observe that, depending on condi-tions, only the part of a basin that is near the channels con-tributes to surface runoff, and the rest of the basin area, fartheraway, makes no contribution. This contributing area expandsand contracts depending on the surface and subsurface condi-tions such as vegetation and soil moisture at the time prior tothe flood event. Accordingly, in such basins the area that con-tributes to overland runoff depends on the antecedent soilconditions and on the storm rainfall depth. The spatial exten-sion of the storm plays a role too. In semiarid zones, accordingto the Horton runoff generation model, in the presence ofintense rainfall all parts of a drainage area contribute to over-land flow. In such a case the antecedent soil conditions arecrucial to allowing activation of overland flow, but, once sur-face runoff has begun, the contributing area is rather con-trolled by the storm extension.

In order to distinguish objectively between areas of differentclimatic characteristics, we used the simple index:

I 5h 2 Ep

Ep, (45)

where h is the mean annual rainfall depth and Ep is the meanannual potential evapotranspiration. Ep was calculated accord-ing to Turc [1961]. This index allows the climate classification

Table 1b. Observed Basins and Their Main Features WithParticular Regard to the Basin Lag, the Intensity-DurationFunction Parameters, and the Base Discharge

SiteA,

km2tA,

hoursp1,

mm hr21 nq0,

m3 s21

Bradano—San Giuliano 1657 7.1 23.52 0.289 10Bradano—Ponte Colonna 462 4.3 22.20 0.283 5Basento—Menzena 1382 6.0 21.48 0.315 25Basento—Gallipoli 853 4.8 20.41 0.315 25Agri—Tarangelo 511 8.9 21.56 0.362 10Sinni—Valsinni 1140 5.6 23.13 0.405 45Basento—Pignola 42 2.9 21.00 0.311 1.5Sinni—Pizzutello 232 2.4 21.56 0.362 15

Definitions are as follows: tA, basin lag time; p1 and n, intensity-duration function parameters; and q0, base flow.

Table 2. Calculated Values of Mean Annual Flood, IndexFlood, and Coefficient of Variation

SiteA,

km2E[Qp]c,m3 s21

QI,m3 s21 Cvc

Bradano—San Giuliano 1657 523 564 0.85Bradano—Ponte Colonna 462 201 213 0.77Basento—Menzena 1382 434 465 0.61Basento—Gallipoli 853 355 370 0.58Agri—Tarangelo 511 199 198 0.50Sinni—Valsinni 1140 556 541 0.48Basento—Pignola 42 37 36 0.49Sinni—Pizzutello 232 239 226 0.44

Abbreviations are as follows: QI, index flood; E[Qp]c, calculatedmean annual flood; and Cvc, calculated coefficient of variation.

475IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

shown in Table 3. The sign of I discriminates between humid(positive) and arid (negative) basins.

The values of I for the observed basins are shown in Table4; they allow for roughly splitting the entire region into twoparts: semiarid and humid. In fact, the climatic index assumesnegative values on the Bradano River basins only and alwaysremains positive elsewhere. It is higher then 0.3 for thosebasins characterized by higher mean altitude and a greaterdegree of vegetation and forested hillslopes. More particularly,one can note that all the parameters in Table 4 show a cleardependence on the climatic index.

First, as already recognized by Iacobellis et al. [1998], the Iindex seems to crucially influence the reduction of Lq with

respect to Lp. This is consistent with the meaning of parametersfA and f * which, according to (36), control the ratio Lq/Lp.

In fact, within the humid areas, where the probability offinding humid antecedent moisture conditions is high, the lossfactor fA should decrease with I . In addition, the ratio Lq/Lp

should tend to grow with I and approach unity in hyperhumidclimate.

Actually, a certain role may also be played by the mean valueof the exceedances in the base rainfall process (as suggested by(35)). However, with regard to this issue, not too much can besaid here since this quantity does not vary sensibly throughoutthe examined basins. The way the ratio Lq/Lp and the param-eters fA and f * depend on I is shown in Figure 4.

Figure 2. Peak flood derived distribution fitted to data. T is in years and Qp in m3/s.

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS476

It is so emphasized that the climate directly interferes withthe hydrological losses, as these are strongly dependent on thecharacteristic moisture conditions. In other words, the basinaridity produces a greater initial adsorption capacity, whichcauses the low yield in the number of flood events compared tothe number of storms. Instead, in humid basins, which arecharacterized by the presence of permanently saturated areasnear the streams, any precipitation event tends to producerunoff, as also pointed out by Leopold [1997, p. 42].

The climate influence can also be noted with regard to theparameter r (Table 4). In fact, the expected fraction of the

basin that contributes to flood peak is quite higher (about 0.5)for the arid Bradano River basin. Instead, humid basins havelower r values in the range 0.2–0.3. This could be related to theway the long-term climate tends to constrain the activation ofa certain runoff generation mechanism. Since the storm sizemay become as large as the entire basin, in arid zones the finiteprobability that the whole watershed contributes to the floodpeak (see (9)) cannot be neglected. Instead, in humid basinsthe presence of highly vegetated areas favors a saturation ex-cess mechanism which allows direct runoff only in those satu-rated parts, usually small, near the streams.

Figure 3. Probability density functions of the contributing area (square kilometers).

477IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

4. Dimensionless Cumulative DistributionFunction and PreliminarySensitivity Analysis

Let us introduce the following dimensionless quantities:

Q*p 5Qp

QI, q*0 5

q0

QI, a* 5

arA , a* 5

a

rA 51b

. (46)

Inserting these quantities into (25), along with the parame-ter f * defined by (36), we get a dimensionless expression forthe exceedance probability function of the peak streamflowQp:

G9Q*p ~q*p! 5 E0

1/r

g~a*!

z expH2G~1 1 1/k!kF S hq*p 2 q*0a* ~12«! 1 f *D k

2 f *kG J da*,

(47)

with

g~a*! 51

a*G~b! S a*a*D

b21

exp S2a*a*D

1 dS a* 21rD gS 1

ra* , bD , (48)

where d( ) and g( ) were introduced in (8) and (9), respec-tively. In (47), h is given by

h 5LqSLq

1 2 q*0HLp

Lq

g@1 1 1/k , log ~Lp/Lq!#

G~1 1 1/k!2 f *J . (49)

Finally, the dimensionless distribution of annual floods isreadily achieved as

FQ*p ~q*p! 5 1 21T 5 exp @2LqG9Q*p ~q*p!# . (50)

This theoretical dimensionless distribution makes it possibleto look at some interesting scaling properties of its momentsand to investigate significant climatic effects on the flood dis-tribution.

The sensitivity analysis of the derived distribution has beencarried out with regard to two climatic conditions. The first isrepresentative of humid regions and is characterized by a lowf * value and consequently by a high Lq/Lp ratio. The secondone, which is more typical of arid zones, presents higher valuesof f * (then, lower Lq/Lp). The following working parametervalues were used: in the first case, f * 5 0.13, Lq 5 16, andLp 5 20 (Lq/Lp 5 0.8) and, in the second, f * 5 1.60, Lq 54, and Lp 5 20 (Lq/Lp 5 0.2). The analysis was carried outwith regard to both configurations, starting from the followingarbitrary set of the remaining parameters: k 5 0.8, b 5 4,« 5 0.3, r 5 0.5, and q*0 5 0. These values were changedone at time within the following ranges: k 5 0.6–1, b 5 1–5,« 5 0.1–0.6, r 5 0.1–0.7, and q*0 5 0–0.2.

Figure 4. (a) The ratio Lq/Lp, (b) the loss factor fa, and (c)the global loss coefficient f * versus the climatic index, withinthe observed region. See Table 3 for climate classification.

Table 3. Climate Classification

Climate I

Hyperhumid 1.0 # IHumid 0.2 # I , 1.0Subhumid 0.0 # I , 0.2Dry subhumid 20.2 # I , 0.0Semiarid 20.4 # I , 20.2Arid 20.6 # I , 20.4

I is climatic index.

Table 4. Parameters Controlled by Climate and ClimaticIndex

SiteA,

km2 r Lq/Lp

fA,mm/hr21 f * I

Bradano—San Giuliano 1657 0.50 0.14 2.170 2.040 20.17Bradano—Ponte Colonna 462 0.50 0.19 2.360 1.645 20.08Basento—Menzena 1382 0.20 0.32 1.210 1.050 0.08Basento—Gallipoli 853 0.28 0.41 0.975 0.776 0.28Agri—Tarangelo 511 0.27 0.80 0.141 0.136 0.47Sinni—Valsinni 1140 0.20 0.91 0.070 0.046 0.57Basento—Pignola 42 0.30 0.93 0.069 0.032 0.70Sinni—Pizzutello 232 0.26 0.97 0.027 0.012 1.26

Parameters are defined in sections 2.2 (fa, a 5 A), 2.3 (Lq, Lp, andf*), and 2.4 (r).

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS478

The parameters Lq and Lp were also varied, keeping theirratio constant; in particular, in the first case we put Lq/Lp 516/ 20, 24/30, and 32/40 (all ratios 5 0.8), and in the othercase, Lq/Lp 5 4/ 20, 6/30, and 8/40 (all ratios 5 0.2). Thiswas not possible while investigating the case of variable k ,since, according to (36), fixing f * and k , does not allow pres-ervation of the ratio Lq/Lp. In this case we chose such valuesof Lp leading to the selected values of f * (see Table 5).

Figures 5 and 6 show the derived dimensionless frequencycurves with reference to the humid and arid areas, respectively.It is possible to observe the different sensitivity of the cdfs tothe parameters in the two climates. In fact, as also frequentlyobserved, flood distributions relative to arid climates aresteeper than others. Furthermore, in comparison it seems thatin humid climate the cdfs are more sensitive to the parametervalues, with particular regard to the Weibull shape parameterk . This could be explained by the observation that in arid areasthe scarcity of flood events makes the process mainly randomthus hiding the influence of physical features.

In Figure 7 the way the mean of the dimensionless distribu-tion varies with Lq within the above-defined range of variabil-ity of k , b , « , r , and q*0, and for f * ranging between 0.1 and2, is displayed. Figure 7 highlights that the calculated values ofthe mean are close to unity, so validating the use of the indexflood QI as an approximation for the mean annual flood.

Thus, on the basis of the definition of the index flood it isalso possible to speculate about the scale features of QI withrespect to the basin area. In fact, dividing QI by the catchmentarea A and neglecting the role of the base flow, (40) suggests

Figure 5. Dimensionless cumulative probability function of Q*p 5 Qp/QI in humid climate: (a) k 50.6–0.8–1; (b) Lq/Lp 5 16/ 20–24/30–32/40; (c) r 5 0.1–0.5–0.7; (d) b 5 1–3–5; (e) « 5 0.1–0.3–0.6;and (f) q*0 5 0–0.1–0.2.

Table 5. Parameters Used in the Analysis of Sensitivity tok in a Humid Climate and an Arid Climate

k Lq Lp Lq/Lp

Humid Climate (f * 5 0.13)0.6 16 23.5 0.680.8 16 20.0 0.801 16 18.3 0.87

Arid Climate (f * 5 1.60)0.6 4 21.8 0.180.8 4 20.0 0.201 4 19.8 0.20

479IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

that this ratio varies with A according to a power law with theexponent equal to 2«. Since the probable values of « range, asseen before, between 0.1 and 0.3, such a result is in fairly goodagreement with real-world observations [e.g., Robinson andSivapalan, 1997].

5. ConclusionsA derived distribution of flood frequency is proposed with

clear reference to the climate influence. It arises from theanalysis of the stochastic features of rainfall and of basin hy-drologic response. The theoretical scheme is founded on theidentification of a probabilistic model of the basin response,which accounts for both the geomorphoclimatic features of thebasin and the spatial distribution of rainfall. The distributionseems to be able to reproduce the observed patterns of floodannual maxima by means of a number of physically basedparameters in a wide range of climatic conditions.

The influence of stochastic and pseudodeterministic factorsinfluencing the process of generation of floods is highlightedtaking into account in a synthetic manner some crucial elementsinvolved in processes like the mechanisms of runoff generation.

This model identifies the combined role played by climaticand physical factors at the basin scale. Conversely, it indicatesthe influence of climate on the water loss processes and shedsmore light on the way to identify the relative importance ofclimate, lithology, and land use. It also highlights that theprobability distribution of floods is controlled by the expectedvalues of some quantities, for example, contributing area andsoil moisture conditions, whose high randomness is significantat the event scale only.

Moreover, the influence of the runoff generation mechanismon the probability distribution of the contributing areas is re-vealed. Thus the need for a deeper knowledge of conditions atwhich a certain mechanism tends to prevail at the basin scaleclearly arises. Further research with regard to the spatial vari-ability of rainfall is also recommended. Therefore field exper-iments aimed at better understanding the physical processesunderlying flood generation processes are also advised, payingparticular attention to the role played by the long-term climateand degree of vegetation of the basin.

In this paper the influences of nonlinear processes of floodgeneration were not taken into account. However, we should

Figure 6. Dimensionless cumulative probability function of Q*p 5 Qp/QI in arid climate: (a) k 5 0.6–0.8–1; (b) Lq/Lp 5 4/ 20–6/30–8/40; (c) r 5 0.1–0.5–0.7; (d) b 5 1–3–5; (e) « 5 0.1–0.3–0.6; and (f)q*0 5 0–0.1–0.2.

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS480

remark that these influences may play a crucial role in ampli-fying the skewness of flood series, mainly in humid basins. Infact, in these watersheds, sensible skew may be induced bypossible activation of the Horton type mechanism in very se-vere storms. In addition, a smoother nonlinear behavior maybe related to the catchment response time ta, which tends todecrease as the rainfall intensity increases. Both these effectsshould be less important in arid basins. In fact, dry soils allowonly high-intensity rainfall to produce floods, and scarce veg-etation tends to make the runoff generation mechanism lessdependent on the rainfall intensity. The developed model,besides suggesting remarkable hints for further scientific re-search and unlike the merely statistical flood frequency meth-ods, succeeds in highlighting the differences, rather than thesimilarities, in the physioclimatic features of the basins.

Acknowledgments. This work was supported by funds granted bythe Consiglio Nazionale delle Ricerche—Gruppo Nazionale per laDifesa dalle Catastrofi Idrogeologiche, Progetto “Valutazione Piene.”Comments and suggestions by two anonymous referees are also ac-knowledged and truly appreciated.

ReferencesBetson, R. P., What is watershed runoff?, J. Geophys. Res., 69(8),

1541–1552, 1964.Cadavid, L., J. T. B. Obeysekara, and H. W. Shen, Flood-frequency

derivation from kinematic wave, J. Hydraul. Eng., 117(4), 489–510,1991.

Claps, P., and E. Straziuso, Analisi regionale delle piogge brevi in

Basilicata (in Italian), internal report, Dip. di Ing. e Fis. dell Ambi-ente, Univ. della Basilicata, Italy, Potenza, 1996.

Cunnane, C., Unbiased plotting positions–A review, J. Hydrol., 37,205–222, 1978.

Diaz-Granados, M. A., J. B. Valdes, and R. L. Bras, A physically basedflood frequency distribution, Water Resour. Res., 20(7), 995–1002,1984.

Dipartimento di Ingegneria e Fisica dell’Ambiente, Valutazione dellePiene in Basilicata (in Italian) report, Univ. della Basilicata, Po-tenza, Italy, 1998.

Eagleson, P. S., Dynamics of flood frequency, Water Resour. Res., 8(4),878–898, 1972.

Farquharson, F. A. K., J. R. Meigh, and J. V. Sutcliffe, Regional floodfrequency analysis in arid and semi-arid areas, J. Hydrol., 138, 487–501, 1992.

Fiorentino, M., F. Rossi, and P. Villani, Effect of the basin geomor-phoclimatic characteristics on the mean annual flood reductioncurve, in Proceedings of the IASTED International Conference, Mod-eling and Simulation, pp. 1777–1784, Univ. of Pittsburgh, Pittsburgh,Pa., 1987.

Gottschalk, L., and R. Weingartner, Distribution of peak flow derivedfrom a distribution of rainfall volume and runoff coefficient, and aunit hydrograph, J. Hydrol., 208, 148–162, 1998.

Gradshteyn, I. S., and I. M. Ryzhik, Table of Integrals, Series, andProducts, Academic, San Diego, Calif., 1980.

Gupta, V. K., and E. Waymire, On the formulation of an analyticalapproach to hydrologic response and similarity at the basin scale, J.Hydrol., 65, 95–123, 1983.

Hebson, C., and E. F. Wood, A derived flood frequency distributionusing Horton order ratio, Water Resour. Res., 18(5), 1509–1518,1982.

Horton, R. E., Erosional development of streams and their drainagebasins: Hydrophysical approach to quantitative geomorphology,Geol. Soc. Am. Bull., 56, 275–370, 1945.

Figure 7. Sensitivity of the expected value of Q*p 5 Qp/QI to parameters: (a) k 5 0.6–0.8–1; (b) b 51–3–5; (c) « 5 0.1–0.3–0.6; (d) q*0 5 0–0.1–0.2; (e) r 5 0.1–0.5–0.7; and (f) f * 5 0.1–1.0–2.0.

481IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS

Iacobellis, V., P. Claps, and M. Fiorentino, Sulla dipendenza dal climadei parametri della distribuzione di probabilita delle piene (in Ital-ian), in Proceedings of the XXVI Convegno di Idraulica e CostruzioniIdrauliche, vol. II, pp. 213–224, Univ. of Catania, Catania, Italy,1998.

Jenkinson, A. F., The frequency distribution of the annual maximum(or minimum) of meteorological elements, Q. J. R. Meteorol. Soc.,81, 158–171, 1955.

Klemes, V., Hydrological and engineering relevance of flood frequencyanalysis, Proceedings of the International Symposium on Flood Fre-quency and Risk Analysis: Application of Frequency and Risk in WaterResources, pp. 1–18, D. Reidel, Norwell, Mass., 1987.

Kurothe, R. S., N. K. Goel, and B. S. Mathur, Derived flood frequencydistribution for negatively correlated rainfall intensity and duration,Water Resour. Res., 33(9), 2103–2107, 1997.

Leopold, L. B., Water: A Primer, pp. 50–54, W. H. Freeman, New York,1974.

Leopold, L. B., Water, Rivers and Creeks, Univ. Sci., Books, Sausalito,Calif., 1997.

Moughamian, M. S., D. B. McLaughlin and R. L. Bras, Estimation offlood frequency: An evaluation of two derived distribution proce-dures, Water Resour. Res., 23(7), 1309–1319, 1987.

Natural Environment Research Council, Flood studies report, inHydrologic Studies, vol. II, p. 15, London, 1975.

Raines, T. H., and J. B. Valdes, Estimation of flood frequencies forungauged catchments, J. Hydraul. Eng., 119(10), 1138–1154, 1993.

Ranzi, R., and B. Bacchi, Analysis and forecasting of rainfall fieldsusing radar, in Advances in Distributed Hydrology, edited by I. Becchiet al., pp. 327–346, Water Resour. Publ., Highlands Ranch, Colo.,1994.

Robinson, J. S., and M. Sivapalan, An investigation into the physicalcauses of scaling and heterogeneity of regional flood frequency,Water Resour. Res., 33(5), 1045–1059, 1997.

Rodriguez-Iturbe, I., and J. M. Mejia, On the transformation frompoint rainfall to areal rainfall, Water Resour. Res., 10(4), 729–735,1974.

Rodriguez-Iturbe, I., and J. B. Valdes, The geomorphologic structureof hydrologic response, Water Resour. Res., 15(6), 1409–1420, 1979.

Rodriguez-Iturbe, I., M. Gonzales-Sanabria, and R. L. Bras., A geo-morphoclimatic theory of the instantaneous unit hydrograph, WaterResour. Res., 18(4), 877–886, 1982.

Rossi, F., Criteri di similitudine idrologica per la stima della portata alcolmo di piena corrispondente ad un assegnato periodo di ritorno(in Italian), in Proceedings of the XIV Convegno di Idraulica e Costru-zioni Idrauliche, vol. II, pp. 235–261, Univ. of Naples, Naples, Italy,1974.

Rossi, F., M. Fiorentino, and P. Versace, Two-component extremevalue distribution for flood frequency analysis, Water Resour. Res.,20(7), 847–856, 1984.

Shen, H. W., G. J. Koch, and J. T. B. Obeysekara, Physically basedflood features and frequencies, J. Hydraul. Eng., 116(4), 495–514,1990.

Sivapalan, M., E. F. Wood, and K. J. Beven, On hydrologic similarity,3, A dimensionless flood frequency model using a generalized geo-morphologic unit hydrograph and partial area runoff generation,Water Resour. Res., 26(1), 43–58, 1990.

Turc, L., Estimation of irrigation water requirements, potential evapo-transpiration: A simple climatic formula evolved up to date, Ann.Agron., 12, 13–14, 1961.

Villani, P., Extreme flood estimation using Power Extreme Value(PEV) distribution, in Proceedings of the IASTED International Con-ference, Modeling and Simulation, edited by M. H. Hamza, pp. 470–476, Univ. of Pittsburgh, Pittsburgh, Pa., 1993.

Wood, E. F., and C. S. Hebson, On hydrologic similarity, 1, Derivationof the dimensionless flood frequency curve, Water Resour. Res.,22(11), 1549–1554, 1986.

M. Fiorentino, Dipartimento di Ingegneria e Fisica dell’Ambiente,Universita della Basilicata, Contrada Macchia Romana, 85100 Po-tenza, Italy. (fiorentino@unibas.it)

V. Iacobellis, Dipartimento di Ingegneria delle Acque, Politecnico diBari, Via E. Orabona 4, I-70125 Bari, Italy. (viacobellis@difa.unibas.it)

(Received January 29, 1999; revised September 13, 1999;accepted September 16, 1999.)

IACOBELLIS AND FIORENTINO: DERIVED DISTRIBUTION OF FLOODS482