JC)lqU,AL OF !\PPLIEIl HYIlROLOC;YVol XXI No I 8: 2. 21111X. IW XX - 100

DEVELOPMENT OF INTENSITY DURATION FREQUENCYCURVES USING L-MOMENT AND GIS TECHNIQUER. Venkata Ramana', B. Chakravorty', N.R.SamaIl, N.G. Pandey' and P. Mani'

, CFMS, NIH, Phulwarishrif, Patna - 801 5051 NIT, Patna-800 004

The rainfall Intensity Duration Frequency (lDF) relationship is one of the most commonlyused tools in water resources engineering for planning, designing or operation of waterresources projects. The establishment of such relationship was done in 1932. Since then,many sets of relationships have been developed in different parts of the globe. In the presentpaper rainfall dada of 14 raingauge stations of Punpun basin located in Bihar was taken forregional rainfall frequency analysis based on L-moment approach facilitated to find the robustdistribution for these daily raingauge stations having data availability of 9-17 years. Therobust distribution was used to find the IDF relationship and curves for short duration rainfallfor Punpun basin. from the IDF curves parameters of empirical equations for the gaugedlocations were determined and contour maps were generated. IDF curves for ungaugedlocations were developed from the generated contour maps using GIS and finally a generalizedIDF curve incorporating return period and the duration of rainfall for particular station wasdeveloped.

The intensity duration frequency (IDF) relationship of heavy storms is one of the most important hydrologictools utilized by water resource engineers for planning, designing and operation of water resources projects.Local IDF equations are estimated on the basis of rainfall intensities abstracted from the rainfall depths ofdifferent durations observed at rainfall gauging station. In some regions, there may exist a number ofraingauges operating sufficiently for long time to yield a reliable estimation of IDF relationships. But inmost of the regions, SRRG data are either non-existent or their sample sizes are too small. Daily precipitationdata is the most accessible and available source of rainfall information. Thus for regions where data atshort time interval are not available, it is necessary to derive IDF characteristics of short duration eventsfrom the daily rainfall statistics. The establishment of such relationships was done as early as in 1932(Bernard, 1932). Since then, many sets of relationships have been developed for several parts of the globe.But, such maps with rainfall intensity contours were not developed for many developing countries.

Hershfield (1961) developed various rainfall contours maps to provide the design rain depths for variousreturn periods and durations. Bell (1969) proposed a generalized IDF formula using the one hour, 10 yearsrainfall depths (P,IO) as an index. Chen (1983) further generalized the formulae for any location in theUnited States using three base indices ofraindepthsP,IO'P2~1O,p,lOowhich describe the geographical variationof rainfall. Koutsoyiannis et al. (1998) developed a mathematical relationship between the rainfall intensityi, the duration d, and the return period T for IDF curves.

This paper proposes the approach of formulation and development of IDF curves using rainfall records ofPunpun basin using empirical equations best suited for the basin. Normally rainfall intensity durationfrequency relationship is derived from the network of daily rainfall records of the Punpun basin. Also theparameters of the regional IDF formulas are generated for ungauged areas to estimate the rainfall intensity

lor various return period and duration using L-moment approach. The method proposed in this study hasheen applied to ungauged rainfall locations and verified on dummy station (arbitrarily proposed!. Also aneffort has been made to develop generalized IDF formula with daily rainfall depths and return period.

The following three steps have been developed (i) Identification of the best robust distribution for the['unpun basin using with L-moment approach, (ii) Development ofIDF curves at 14 stations using empiricalfunctions, and (iii) Develop the generalized IDF equation for a particular location.

L-moments are defined as liner combinations of Probability Weighted Moments (PWMs). They are robustto outliers and virtually unbiased for small samples, making them suitable for rainfall frequency analysis,including identi fication of distribution and parameter estimation. Greenwood et al., (1979) defined PWMsof a random variable X with cumulati ve distribution function F(X) by equation (1).

M " = E lX p V (X Y }{ 1 - F ( X s )} Jp,I ..1

(1)

weighted moments ar = Ml.o.rand fJr Ml.r.O .Ior aParticularly useful special cases are the probabilitydistrlhution that has a quantile function X(ll),forp=/ and.l = 0, Eq.(l) gives

Ii, 0= dX {F (X Y }] = f X (II )11 ' dllo

The first four L-moments, expressed as liner combination of PWMs, are:

AI = fJoA:. = 2fJI - fJoA, = 6fJ:. - 6fJ, + 6fJo,,1.4 = 20/3, - 30fJ:. + 12fJ1 - fJoWhere \ (L-mean) is a measure of central tendency, 1..2 (L-standard deviation) is a measure of dispersion.Their ratio ).)1..1 is termed as L-CV (L-coefficient of variation), whereas the ratio, ').,-/1..2 is referred to as T,(L-skewness), and the ratio 1../\ is referred to as '4 (L-kurtosis).Once frequency is known, the maximum rainfall intensity is determined using the best robust distributionfunction (e.g. Generalized Extreme Value (GEV), Generalized Logistic (GLO), Generalized Normal (GNO),Pearson type-II (PE3 by using the L-moments approach of the basin. The rainfall intensities for eachdurations and a set of selected return periods (e.g. 2, 3, 5, 10, 20, 50, 100 years etc.) are calculated. Theempirical formulas (Section) are used to construct the rainfall IDF curves. The least-square method isapplied to determine the parameters of the empirical IDF equation that is present intensity-duration- frequencyrelationships.

The IDr formulas are the empirical equations representing a relationship among maximum rainfall intensity(as dependent variable) and other parameters of interest such'as rainfall duration and frequency (asindependent variables). There are several commonly used functions found in the literature of hydrology

applications (Chow et aI., (988), four basic forms of equations used to describe the rainfall intensity durationrelationship are summarized as follows:

. a1=---

d +b

. a1=-

dl'

a1 =

dl' +b

. a1=

(d+h)"

where I is the rainfall intensity (mm/hr); d is the duration (minutes); a, b, and e are the constant paramctcrsrelated to the metrological conditions.

These empirical equations show rainfall duration for a given return period. All these functions arc widelyused in hydrological applications. The least-square method is applied to determine the parameters ojempirical lDF equations that are used in this study. The values of parameters were choosed on the basis ofminimum Root Mean Square Error (RMSE) between the IDF relationships produced by the frequencyanalysis and

Regionalization of the Para~eter of Rainfall Intensity Duration Frequency Equation

The rainfall lDF curves are derived from the point rain gauges. A set of IDF curves were established. Weneed the IDF curves of Punpun basin for which SRRG data is required. But the network of ORG stationsfor daily records is available in higher density than SRRG records in the basin. Thus the regional lDFformula parameters are generated for ungauged areas to estimate rainfall intensity for various return periodand duration. The method Iproposed in this study had reasonable and good agreement to ungauged rainfalllocations, which was verified. After determining the parameters of IDF formula such a, b, and e for thesame return period using Arc View/GIS interpolation and were generated contour maps of each parameter.The generated map of the parameter was then used for ungauged rainfall station. Now it is possible toestimate the parameter set of any point in the basin for estimation of IDF relations.

Generalized Rainfall Intensity Duration Frequency Formula

A set of intensity duration frequency (lDF) curves constitute a relation between the intensity of precipitation(mm/hr), the duration of the rainfall (min) and the return period (how frequent this event occ'urs) of anevent and is defined by the inverse of the annual exceedance probability which is expressed as:

where i is the rainfall intensity (mm/hr), d is the duration of the rainfall (min) and the return period Tisreturn period (years).

According to Koutsoyannis et al. (1998) the IDF curve is a mathematical relationship between the rainfall

intensity I, the duration d, and return period T(eljui\alent to the annu~tllll'ljUency ot'cxcl:l:dancl: or simplyas frequl:ncy), fhe typical IOF relationship for a specific return period is a special case of the generalizedformula given by I(outsoyannis ('I (//. (Equ:12l

. (/I=---~

(d' + bywhere a. h. e and v are non negative coefficients, Thus, the generalized equation: with v= I and e= I formsTalbot equation; v= I and b=O is Bernard equation; e= I is Kimijima equation and v= I is Sherman equation.This empirical expressioJrn is the outcome of the experiences gathered fi'om several studies. The correspondingerrors associated with the equation (12) assuming v= I has been studied numerically that simplified byequation (13)

. a1=

(d +h)'

Ilknn (1969) proposed a generalized IDF formula using PliO. P21~O,md PII!)!) , Bell developed the following

tw'(\]) generalized [OF relationships for high intensity short duration rainfall which also takes care ofgeographical variation of rainfalL

pi'd (') -4 (II 25 0-0

-, =.) ~ -.) (5

/'(1')= I: = I: =c+Aln(1'). I [' T'II! Idl

Andf~ (eI) is the ratio of I~ and I~, which is the function of rainfall duration

IT ITt,Cd) = _,_I = _,_I = __ a__. - I,~, (;,' (d + b)"After combining equation (16), (17) and (18), the generalized formula of rainfall intensity and frequency

can be written as

I,; = II; (c +A In(T)) aI! (d+br

H

+

The Punpun basin lies between latitude 2411 to 25"OO N and longitude 84"10 to 85"20 E, It is locatedon the right bank of the river Ganga and bounded by the Sone river system on its west and Kiul-Harohar-Falgu river system in the east. On its northern side, it is the river Ganga and on its southern side, boundedby Chotonagpur hills, The drainage map of Punpun river basin with the locations of rain gauge stations is

shown in Fig, 1.

"/1';urthaMal

intensity /, the duration d, and return period T(equivalcnt to the annu~li fll'quellcy of ex ceeda nee or simplyas frequency). fhe typical IDF relationship for a specific return period is a special case of the generalizedformula given by J(outsoyannis 1'1 (II. (Equt 12)

. a1=---'--

(d' +h)'where a, h, l' and v are non negative coefficients. Thus, the generalized equation: with v= I and e= I formsTalbot equation; v= I an:d b=O is Bernard equation; e= I is Kimij ima equation and v= I is Sherman equation.This empirical expressio'" is the outcome of the experiences gathered fi'om several studies. The correspondingerrors associated with the equation (12) assuming v= I has been studied numerically that simplified byequation (13)

. al=---~

(d +h)'llkllll (1969) proposed a generalized IDF formuta using PliO, p,-I~I and PIIIIII . Bell developed the followingUW:'J) generalized IDF relationships for high intensity short duration rainfall which also takes care ofgco:graphical variation of rainfall.

pi'd (') -4/1125 0-0~I = ,J (. - ,) (5

f (T) = I~ = I~ = c + A In(T). I r' r'1/ I ,, d

Andf~ (d) is the ratio of I~' and I~, which is the function of rainfall duration

T ITf (d) = ~ = _d_ = a.2 IT T' (d+b)'"

ii' I d'

After combining equation (16), (17) and (18), the generalized formula of rainfall intensity and frequency

can be written as

I,'~= I'; (c+Aln(T)) ad (d+b)"

The Punpun basi n lies between latitude 24 11 to 2S"OO N and longitude 84" 1O to 85"20 E. It is locatedon the right bank of the river Ganga and bounded by the Sone river system on its west and Kiul-Harohar-Falgu river system in the east. On its northern side, it is the river Ganga and on its southern side, boundedby Chotonagpur hills. The drainage map of Punpun river basin with the locations of rain gauge stations is

shown in Fig. 1.

H

+

40 0 40 80 hilometersI""""~"" __ ""~~~~"" iiii=aiiiii

Fig. 1. Location of the rain gauge stations in the Punpun basin

It is a preliminary screening test of the data set by discordance measures (0). Hosking and Wallis (1997)

defined the discordance measure (D) forN sites of the group by equation (7) where ui = [t2 (i)t3 (i)t4(i) ris a vector containing the sample L-moment ratios t2, t,. and t~values (of each site) for i sites, analogous totheir regional valves termed as t1, t, and t~.T denotes transpose of a vector of matrix.

1 - _I -D, = - N(u; - u)A

1I1(ui - u)3

N

where, ~ = N-1 L ui;=1

The site 'i' is declared to be discordant, if Diis grater than the critical value of the discordance statistics D

i,

given in a tabular form by Hosking and Wallis (1997).

If the variability of the cloud of points on a plot of L-CV versus L-skewness and/or L-skewness versus L-kurtosis is large, the possibi Iity that they do not belong to a single population. This can be tested by meansof the L-moment heterogeneity tests. The L-moment test for heterogeneity fits a four-parameter Kappadistribution to the regional data set, which generates a series of 500 equivalent regions data by numericalsimulation and compares the variability of the L-statistics of the actual region to those of the simulatedseries. Three heterogeneity statistics can be employed to test variability of three different L-statistics: HIfor L-CV, H1 for the combination of L-CV and L-skewness and H, for the combination of L-skewness andL-kurtosis. The HI statistics has much better discrimination power than H1 and H, statistics (Hosking andWallis1997).The general form of H-statistics is given by equation (9).

H = (Vobs - fJ.v)/ av

where Ilv and 0vare mean and standard deviation of the simulated values of V. Vobs is the observed dispersion,calculated from the regional data and is based on a corresponding V-statistics in terms of L-moment ratiot, defined by equations (10 and 11).

N N

tR = LI1,t(i) /Ll1i1=1 i=l

The H-statistics indicate that the region under consideration is homogeneous when H< 1; possiblyhomogeneous when 1$ H< 2; and definitely heterogeneous when H;::: 2. The details L-statistics includingthe value of discordance measure are given in Table 1. It was found that the Di values for 14 sites vary from

O.O') to l.SS. all ()I" \\hlch arc less than the LTitll'al f) \:t1ucs (II"~.'nJ I!loskilli' :llld \\';dlisJl)Ij/]hl,tcro('cneitv measures IH), computed using the data of 14 rain (,augc sitcs (If I'unpun hasin was 1"(1I111l11,thall 1,0. S IIlCC II< I . the regIon may he treatcd as honHlgencous.

..(

\\ Ii\\ ,I

1'01

crl

Tli\ ;\

dl'dl;1'0

\ ;1

l-T:ell

Meanannual

ma ximum Sample DlscoldallName of the rain rainfall L-CY L-CS L-CK S17 mcas 11](','

S\. gauge locations (mm) (t2) (1:\ ) (t.l ) (Years) (/), )I Chatapur 98.79 0.2671 0.0489 0,0902 13 o SO2 Tekari 74,16 0,3381 0.2656 0.2802 13 u:~;

3 Punpun 89.10 0.2423 0.0158 0.1602 12 0.'1'1

4 Aurangahad 114.49 0.1966 0.1735 0.1817 17 () / \.5 raluah 101.36 02183 0.28.5 0.3.546 9 I ',11

() Ibrthargall.l 10424 02~27 0.2995 0.1902 16 () II,

7 Karpi 87.73 0.3166 0.0477 0.0240 13 I 'd I

f) Nahinagar 82.83 0.2707 0.3815 0.2436 13 Of,

9 Inunganj 107.03 0.2990 0.3031 0,2228 12 Ole:

10 KUltha 79.97 03675 0.2842 0.1128 12 I 10

11 Sherghali 117.35 0.2626 0.3499 0,0984 15 I. I!

12 Obra 88.92 0.1991 0.1121 0.2937 13 I. ' ,.13 GJh 100.88 0,2892 0.2284 0.1620 13 (WI

14 Makdampur 113.32 0.2152 0.1496 0,1920 14 () 1(,

Selection of best~flt distribution: Generally goodness of fit measure is used to evaluall' tlJ(' ',lIll.d"llI\ "Idata of a particular site to be consistent with the fitted probability distribution fUIlCliol1 II, ,',11111'1111

where N",,, is the number of simulated regional data sets generated using Kappa distribution in a similarway as for the heterogeneity statistics, the subscript m denotes the mil' simulated region. The distributionfor which Zl)ISI value is very close to zero will be declared the best fit distribution. However, a reasonablecriterion is I Zl)IS I I :s: 1.64.The Zl)lsr statistic for the various distributions is ~iven Table 2. It is observed that the I Zl)lsr I statisticvalue are lower than 1.64 for the four distributions namely GLO. GEY. CiNO, and PE-lII. Further, for GLOdistribution I Zl)IST I value (0.29) is found to be very close to zero. Thus, based on the L-moment ratiodiagram as well as I ZOIST I statistic criteria, the GLO distribution is identified as the robust distributionfor Pun pun basin. The values of the regional parameters for the distributions, which have I ZDIST I statisticvalue less than 1.64 are given in Table 3,

Table 2: ZDlST statistic values for variousdistributions

Distributio illS!

S\.. n statisticGLO 0.29

:2 GEV -0.7, GNO -0.97.'4 PE 111 -1.51

S =0.911S =0.772S =0.902/l =1.000

u. =0.2':+3ex =0.356a =0.4290=0.488

K =-0.210K =-0.061K =-0.434y = 1.269

The GLO distribution was identified as the robust distribution for Punpun basin. The regional quantilefunction of the G LO distribution is expressed as equation (28).

The values of regional parameters of the GLO distribution for Punpun basin were found to be S=0.911,a =0.243, and K=-0.210 and substituting these values in the equation 29.

f(x) = -0.246 + 1.157(_1_)-UlI0. T-I

The GLO probability model is used to calculate the rainfall intensity at different durations and returnperiods to forms the historicallDF curves for each station. Using the GLO distribution function, maximumrain t~tli intensity for considered durations and 2,5, 10,20,50, J 00,200 years return periods, have determined.The results are shown in figure2. The relationship between the maximum rainfall intensities and theduration for ever return periods are determined by fitting empirical functions.

The IDF curves for 14 stations were constructs by using equations (7) to (10): Talbot, Bernard,.Kimijimaand Shennan. Least-square method is applied to determine the parameter for empirical IOF equations usedto present intensity-duration relationships. The value of parameter in the rainfalll OF equations were chosenon the minimum of Root Mean Square Error (RMSE) between the lOr: relationships produced by thefi'equency analysis and the simulated by the lDF equations. The RMSE (mc,1I1SljLWre error) was defined as

III II ( )"2..: 2..: l,~ - I,~* -j~lk --I

where III is the number of various rainfall durations, 11 is the number of various return periods, li~ is tt1l'rainfall intensity derived by GLO distribution fori hour duration, k year return period at the i station, and

l,~* is the rainfall intensity estimated by Equation for j hour duration, k year return period at the i station

I IU".""">.z].5

----._~_._._--~---~---------------;ItlllU I

__ Jl__Fig. 2. Maximum rainfall intensity for different time intervals and return periods obtained from the GLO

distribution at Tekari station

I~~-----14

I

~12

g 10..0..

8..~";j..

6""

Igc 4Ii;:;

2

'1 0

0 50

------1

250 I

JRetul11 PCliod (yca.-s)

~~~~~;;~;=-B;'~;;"d~~

Comparison among the four empirical methods (Equ.7 to 10) for 1DP formula wcrc madc and found thatKimiJima equation (Equ.9) minimum Root Mean Square Error (RMSE) as shown in the figure 3 and fittedwcll. Thus for all the rain gauge stations of Punpun basin, the parameters of the KimiJima equation weredetermined for 100 year return period (table 4) that has Root Mean Square Error (RIvlSE) ranging 5.636 to8.000 mm/hr and its wrrelation coefficient R is approximated 0.98. The results are that the Kimijimaequation is acceptable and fit to the 1DF relationship in Punpun basin. The RMSE with Kimijima equationare less than 8.000 mm/hr.

i )j

Name of the rainSI. gauge locations a b e Coefficient RMSE1 Chatapur 20796.68 20.00 0.989 0.992 6.224

2 Tekmi 15881.21 26.00 0.959 0.981 6.024

3 Punpun 17082.00 20.00 0.949 0.986 6.399

4 Aurangabad 24719.30 20.00 0.989 0.992 6.224

5 Fatuah 19433.00 20.00 0.949 0.988 6.960

6 Haliharganj 19403.00 18.00 0.949 0.986 7.279

7 Karpi 16868,00 20.00 0.950 0.986 6.261

8 Nabinagar 16851.25 24.00 0.950 0.981 6.706

9 Immganj 22401.00 23.00 0.965 0.986 7.674

10 Kurtha 15466.30 20.00 0.952 0.986 5.636

11 Sherghati 26458.59 22.00 0.996 0.991 6.588

12 Obra 17182.00 24.00 0.988 0.977 8.000

13 COh 19809.62 20.00 0.956 0.986 6.930

14 Makdampur 24804.25 22.00 0.994 0.989 6.856

After determining the parameter of a, band e of IDF formula for the same return period interpolationtechniques were applied to generate contour maps using GIS Arc- View.

The parameter contour maps and the IDF relation has been generalized for further use to estimate intensityduration of rainfall in ungauged locations with various return periods. The results has been applied ungaugedlocation and found satisfactory. The results has been verified assuming Rafiganj as an ungauged stationThe paramcters contours map for Kimijima equation as shown in figure 4.

The rainfall intensity duration frequency at Rafiganj (ungauged) can determine. Parameters set: a= 20000.00,b= 21.55, 12=0.963. The 1DF curve at Rafiganj can be follow equation for 100 years return period:

a1=---

d" +b20000.000

dO 963 + 21.550

++~~E", 100.~

.

III 10

L 10iI

I(i)10000 I

Fig. 4 parameter contours of Kimijima equation with 100 years return period and IDF curves atungauged point. i) contour map of a ii) contour map of b iii) contour map of e and iv) Rainfall IDF

curves at Rafiganj (ungauged location) using parameters contour maps

The function~ (T) is the ratio of I,; and I;t is the function of the return period (equation 17). The Tekaristation is used to illustrate hoe to define the generalized IDF formula. For this example: r=lOO year as the

base return period. The ratio is of I,; / I,; '100 for various duration and return periods are given in Table 5.The ratio show little variation with duration, and are a function of period.

The table 5: Average relationship between rainfall intensity and duration (Ratio of /1 /11' 1(1)) same'- d d

duration at Tekari station

Return periodI, (T)

The parameter e is slope value of linear regression relationship between the log-transformed values ofreturn periods (T) and the ratios of rainfall intensity:

T T. Id Id 'J./1 (T) = -1-'/ = -1-'/ = C + Aln(T) = 0.1697 + 0.18111n(T)

I, I // d

The parameter A = 0.181l and c = 0.1697 with correlation coefficient value r = 0.995.

The intensity-duration ratios are calculated for each available data. The calculations are made in order toobtain the average value of the ratios each consideration durations. The ratios 60-minute rainfall intensity

and duration (I; /1;/060 ) for same return period T. the ratio j2 was fitted by Sherman equation:

The parameter a 62.810, b= 16.000 and e=O.956 with con'elation coefficient value r=0.992. Combing equation(32), (33) the generalized Intensity Duration Frequency formula at Tekari station, with rainfall intensity in60 minute and 100 years is 206.981 mm/hr, gives

T 62.810Id = 206.981(0.1697 +0.18111nT) 0956

Cd + 16.00) ..

2206.180 + 2354.3 861nT(d + 16.00)0956

i The rainfall intensity can calculate from (35) equation for any duration (d) and return periods (T) at TekariI station.

The study has been made conducted to the formulation and construction of IDF curves using data fromrecording station by using empirical equations, four empirical functions used to represent intensity-Duration-Frequency relationship for Punpun basin. Using the L-moment to find out the best robust distribution(GLO) for rainfall quantiles at various return periods and 3 parameters functions (Kimijima) has been used

to rainfall intensity quartiles.

The regionalization of the parameters of rainfall intensity duration frequency equation was generated forungauged area to estimate rainfall intensity for various return period and rainfall duration. The parameterscontours maps were made to estimate ungauged rainfall with return periods. More specifically, this researchis to generalize IDF formula using some base rainfall depth and base return period. In fact, IDF curves givethe rainfall intensity at a point. Storm characteristics are important for larger catchments. Intensity-Duration-Area-Frequency curve (IDAF) is studied for the evaluation for design storms using a scaling approach.

Bell, F. C. ,1969. Generalized rainfall duration frequency relationships. 1. o{Hydraulic Dis., ASCE, 95( I),

311-327.Chen, C. L., 1983. Rainfall intensity duration-frequency formulas, 1. o{ Hydraulic Engineering, ASCE,

109(12),1603-1621. WiChow, V. T., 1964. Hand book of Applied Hydrology, McGraw-Hill, New York, 1-1450. I

Chow, V. T., Madiment, D. R. & Mays. L. W., 1988. Applied Hydrology, McGraw-Hill, New York. IGreenwood, 1. A., Landwehr, J. M., Matalas, N. C. and Wallis, J. R., 1979 Probability weighted moments: I~.~.;....'..Definition and relation to parameter of several distributions expressible in inverse form. Water Resource,l .

Research. 15(5), 1049-1054

Hershfield, D.M., 1961. Rainfall Frequencies Atlas of the United States for Durations from 30 Minutes to24 Hours and Return Periods from I to 100 Years. Technical Paper No. 40. Washington, D.C., U.S.Weather

BureauHosking, J. R. M. and J. R. Wallis., 1997 Regional/i'Nlllency analysis-An Approach Based on L-Mo/llents.

New York, Cambridge university Press

Koutsoyiannis, D. and Manets, A., 1998. A mathematical framework for studying rainfall intensity-duration

frequency relationships .. J. of Hydrology, 206, 118-135.