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NTPC Sponsored Training Course on August 20 – 25, 2007 Design Flood Analyses for Hydropower Projects

Module-3

REGIONAL RAINFALL INTENSITY - DURATION - FREQUENCY RELATIONSHIPS

Dr. N. K. Goel Professor & Head

Department of Hydrology Indian institute of Technology,

Roorkee-247667

&

Shibayan Sarkar Research Scholar

Department of Hydrology Indian institute of Technology,

Roorkee-247667

INTRODUCTION Rainfall intensity of a particular frequency and duration is required for estimation of average annual flood and sediment yield from the catchments. The design engineers do not have simple and reliable method for estimation of rainfall intensity, particularly for short duration. This necessitates going for the regionalisation of IDF relationships. The limitation of such study is the scarcity of data from self-recording raingauges for sufficiently long time. Again, studies for less than 1-hour data are also scarce. However, an effort has been made to briefly put forth the IDF relationships for various regions (mostly in Indian context) in the forthcoming sections.

REVIEW OF LITERATURE Sherman in 1931 developed an empirical relationship of the form,

I = KTt c

a

b( )+ (1)

Where, t is duration in minutes, T is return period, K, a, b, and c are constants depending on geographical location. This is the most common form of IDF relationship, which is still being used widely.

Bernard (1932) developed an empirical relationship in the form of:

Ia T

ttT o

a

a=1

2 (2)

Where, = rainfall intensity having duration 't' and return period 'T', aI tT

o ,a1 & a2 are constants and depend upon geographical location.

Bilham (1935) published his well - known article on the IDF relationship for U.K. and the frequencies were calculated from the formula as (converted in S.I. Units).

Organized by: Department of Hydrology, IIT Roorkee, Roorkee- 247 667, India

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n = 1.214 * 105 t [ R + 2.54 ]-3.55 (3)

Where, R is depth of rainfall in mm, n is no. of occurrences in 10 years and t is duration of rain in hours.

Bilham's work was modified by Holland (1967), who showed that Bilham's equation over estimates the probabilities of high intensity rainfall (> 35 mm/hr).

Yarnall (1935) presented such data in the form of maps of a region with isohyetal lines indicating total rainfall depth that may be expected in a time 't', at a frequency once in N-years for United States.

In Australia, the Bureau of Meteorology (Hall, 1984) had developed procedures for estimation of IDF values for return period up to 100 years and duration up to 72 hours.

General extreme value (GEV) distribution has been widely used for application to rainfall extremes in Australia, East Africa and U.K. This also forms the essence of rainfall studies in volume - II of U.K. Flood studies report (National Environmental Research Council, 1975). Gumbel distribution (a specific case of GEV distribution) has also been used in several reports of U.S. weather Bureau for durations ranging from 2 minutes to 24 hours.

Bell (1969) proposed the following the following depth - duration - frequency formula:

[ ][ ] 101

0.25Tt R 0.50 - 0.54t 0.52 + T nl 21.0R = (4)

For 2 ≤ T (years) 100 & 5≤ ≤ t (min.) ≤ 100 Where, = T-year and t-hour rainfall depth in inches R t

T

= 10-year and 1-hour rainfall depth in inches R110

Baghirathan and Shaw (1978) made rainfall depth-duration-frequency studies for Sri Lanka. Chen (1983) provided a general relationship for rainfall intensity in U.S.A. Raudkivi of New Zealand presented regional relationship on IDF in 1979. All these authors used Bell's equation in their studies. Ferreri and Ferro (1990) verified the applicability of Bell's equation for Sicily and Sardinia in the Mediterranean.

Neimczynowicz (1982) used Log Pearson Type-III distribution with method of moments for preparing areal IDF curves for short-term rainfall events in Lund, Sweden.

Steel & McGhee (1979) gave the empirical relationship for United States for duration less than 2 hours and for any given frequency as:

I = A

t B+ (unbalanced) (5)

Where, I is intensity in inches/hour, t is duration in min and A & B are constants depending on frequency and climatic condition. Rao et al. (1983) obtained relationship between short duration rainfall and 24 hour rainfall as: I (t) = a + b R24 + C R (6) 24

2

Where, a,b,& c are constants. Gert et al., (1987) obtained the following relationships for Pennsylvania, U.S.A.

I (t) = (1+ 0.42 logt 24) R24 (7)

Where, I(t) is rainfall amount for a duration of 't' hours and R24 is 24 hours rainfall amount.


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Mc Cuen (1989) gave the mathematical representation of IDF curves for computerizing the elements in hydrologic design for Baltimore in Maryland and the equations are:

I =A

t b+for t 2 hours ≤

I = c td for t > 2 hours (8)

Where I is intensity in inches/hour, t = duration in hours and a, b, c & d are coefficients that vary with frequency.

Chen's (1983) represented IDF as:

It t

T = a I log (10 T )

+ b)110 2-x x-1

c( (9)

Where, is Rainfall intensity in inches/hour for T-year and t-min storm duration, x is depth of frequency ratio ( ), a, b & c = storm parameters that are dependent on regional ratio (R

I tT

R Rt100 10/ t

t100/ Rt

10) Chen showed that the 10-year, 1 hr rainfall R1

10 used in Bell's equation alone can not measure the geographical variations of rainfall and equation (8) produces more accurate results.

A comparison of available IDF relationships for short durations, given by Chow (1964), Raudkivi (1979), Gert et al. (1987), and Chen (1983) reveals that the values of exponents of variables 't' & 'T' in Bernard's equation do not vary much from place to place for shorter duration rainfalls. The exponent of 'T' ranges between 0.18 & 0.26. For 't' the exponent varies from 0.7 to 0.85.

Koutsoyiannis et al. (1998) used several appropriate statistical distributions functions ( Gumbel, Gamma, GEV, Log Pearson III Log Normal Exponential, Pareto etc.) for a reliable parameter estimation of IDF relationships and proposed a generalized empirical IDF relationships Equation (10) facilitating the description of the geographical variability with regionalization of IDF curves. Moreover, this formulae allows incorporating data from non-recording stations.

( )i

dν η

ωθ

=+

(10)

Where,ω ,ν ,θ and η are non-negative coefficients with 1≤νη . The inequality is easily derived from the demand that the rainfall depth idh = is an increasing function of . d

Based on the partial duration series (PDS) approach Madsen et al. (1998) proposed regional estimation of extreme precipitation from a high resolution rain gauge network in Denmark. For a preliminary assessment of regional homogeneity and identification of a proper regional distribution L-moment analysis is applied. To analyze the regional variability in more detail, a generalized least squares regression analysis is carried out that relates the PDS model parameters to climatic and physiographic characteristics. A regional parent distribution is identified as the Generalized Pareto distribution.

Sivapalan and Blöschl (1998) presented an methodology for IDF curve based on the spatial correlation structure of rainfall linking scientific theories of space-time rainfall fields with design methods rather using by the use of empirically-derived areal reduction factors (ARFs).


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Yu and Cheng (1998) formulated a generalized regional IDF relationship by pooling annual maximum rainfall series of southern Taiwan.

Naghettini (2000) investigated the properties of time scale invariance of rainfall applied to intensity-duration-frequency relationships for short-duration rainfall for southeastern Brazil from the statistical characteristics of daily data. The hypothesis of simple scaling implies in direct and empirically-verifiable relations among the moments of several orders and also among the probability distributions of rainfall intensities of different durations.

Trefry et al. (2000) applied method of moments or maximum likelihood procedures to fit a suitable probability distribution to annual maximum or partial duration series data for each gage of Michigan to estimates site-specific IDF curve. Further, isopluvial maps have been developed from these using interpolation procedures and judgment.

Davis and Naghettini (2000) aimed to estimate regional intensity-duration-frequency (IDF) curves using partial duration series and for the Brazilian state of Rio de Janeiro, whereas, application of L-Moments have improved parameter and quantile estimates of extreme rainfall intensities for this study.

Trefry and Watkins Jr. (2001) discusse the application of a PDS / Generalized Pareto (GPA) regional index-flood procedure assuming independent peaks and a Poisson-distributed number of threshold exceedances for updating rainfall intensity-duration-frequency (IDF) estimates for the State of Michigan.

Yu et al. (2004) developed regional Intensity–Duration–Frequency (IDF) formulas for non-recording sites of northern Taiwan based on the scaling theory using annual maximum rainfall series for various durations. The hypothesis of piecewise simple scaling combined with Gumbel distribution was used to develop the IDF scaling formulas.

Durrans and Kirby (2004) studied IDF curves and 24-h design storm hyetographs based on US Soil Conservation Service which consist of at-site estimates of generalized extreme value distribution parameters for gauging sites of Alabama.

Maurino (2004) compared the generalized rainfall intensity-duration-frequency relationships proposed by Bell in 1969 with results obtained from data registered in different climatic regions of Argentina.

Amin and Shaaban (2004) used GEV and EV1 distribution with one step least square method for parameter estimation to propose regionalized IDF relationship for Peninsular Malaysia.

Hadadin (2005) constructed IDF curve for Mujib Basin in Jordon which is compared with Gumbel method and Water Authority of Jordon.

Trefry et. Al. (2005) applied a regional frequency analysis approach based on L moments to carry out storm water management plans for the State of Michigan; considering the area as a homogeneous region two regional index flood models were applied: a generalized Pareto distribution fit to partial duration series data (PDS) data, and a generalized extreme value distribution fit to annual maximum series (AMS) data.

Ghahraman and Hosseini (2005) carried out a study to test the performance of commonly used IDF models for three synoptic stations in Iran.

Guo (2006) extended IDF analysis from previous analysis results for Chicago, considering the effect of changing climate conditions of the first and second halves of the last century. Further this IDF is applied on the design and performance of urban drainage systems.


49


Nhat et al. (2006) constructed IDF curves for the monsoon area of Vietnam and to propose a generalized IDF formula using base rainfall depth, and base return period for Red River Delta (RRD) of Vietnam.

Desa et al. (2006) proposed an approach to estimate design rainfall depth at low ARI using the Partial Duration (PD) data series of short duration high quality rainfall data to overcome the drawback of At site frequency analysis for economical and efficient urban drainage design of Sungai. Finally, they showed relatively simple application which is relevant for the at-site and regional IDF relationship.

Bougadis and Adamowski (2006) examined scaling properties of extreme rainfall to establish scaling behaviour of statistical non-central moments over different durations, whereas, a scale invariance concept is explored for disaggregation (or downscaling) of rainfall intensity from low to high resolution and is applied to the derivation of scaling IDF curves based on scaling of the generalized extreme value (GEV) and Gumbel probability distributions for the province of Ontario.

Regalado and Yuste (2006) proposed an “intra-station” regionalization, meaning, a regionalization in the same station for Spain. In addition to this, they incorporate GIS (Geographical Information System) application to improve this relationship called MAXIN.

For small and/or urban catchments of Scotland, Svensson et al. (2007) compared a number of IDF relationship approaches calculated from each artificially fragmented record which is modified considering the existing data gaps of varying degrees of severity.

Raiford et al. (2007) updated the existing intensity-duration-frequency curves for ungauged sites of South Carolina, North Carolina, and Georgia. The L-moment method with X-10 test was used to search for homogeneous regions within the study area. Finally, at-site statistics were calculated to develop frequency relationships. Normal, lognormal, generalized extreme value, Pearson type III, and log Pearson type III probability distribution functions were used to fit the maximum annual precipitation data at each gauging site for each duration. The chi-squared goodness-of-fit test was used to determine the best fit probability distribution.

Salas et al. (2007) carried out IDF studies for Spain incorporating regionalization, and more adequate functions, such as the SQRT-ET max function of distribution.

REGIONAL IDF RELATIONSHIPS FOR INDIAN REGIONS

Parthsarathy and Singh (1961) prepared rainfall intensity duration frequency curves for India for local drainage design.

Based on 15 min tabulations of rainfall for 50 stations and hourly tabulations for 67 self-recording raingauges in India, Ayyar and Tripathi (1973,1974) have prepared generalized charts of 2,5,10,25 & 50 year return period values of 15, 30, 45 min, 3, 6, 9, 12 & 15 hours rainfall.

Ram Babu et al. (1979), after analysing rainfall characteristics for 42 stations, presented IDF equations and nomographs. With the equations and/ or nomographs, the intensity for any desired duration for a given frequency may be determined. Its general form is:

I = K Tt b n

a

( )+ (11)

Where, I is intensity in cm/hr, T is return period in years, t is storm duration in hours. K, a, b & n are constants developed for various stations and zones of India. The values of K, a, b and n are presented in Table 1 for different stations and zones of India.


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Table 1

Intensity duration return period relationship, India

Zone Station K a b n Northern Agra 4.911 0.1667 0.25 0.6293 zone Allahabad 8.57 0.1692 0.5 1.019

Amristar 14.41 0.1304 1.4 1.2963 Dehradun 6 0.22 0.5 0.8 Jaipur 6.219 0.1026 0.5 1.1172 Jodhpur 4.098 0.1677 0.5 1.0359 Lucknow 6.074 0.1813 0.5 1.0331 New Delhi 5.208 0.1574 0.5 1.1072 Srinagar 1.503 0.273 0.25 1.0636 Northern Zone 5.914 0.1623 0.5 1.0127

Central Bagra-tawa 8.5704 0.2214 1.25 0.9331 zone Bhopal 6.9296 0.1892 0.5 0.8767

Indore 6.928 0.1394 0.5 1.0651 Jabalpur 11.379 0.1746 1.25 1.1206 Jagdalpur 4.7065 0.1084 0.25 0.9902 Nagpur 11.45 0.156 1.25 1.0324 Punase 4.7011 0.2608 0.5 0.8653 Raipur 4.683 0.1389 0.15 0.9284 Thikrl 6.088 0.1747 1 0.8587 Central zone 7.4645 0.1712 0.75 0.9599

Western Aurangabad 6.081 0.1459 0.5 1.0923 zone Bhuj 3.823 0.1919 0.25 0.9902

Mahabaleshwar 3.483 0.1267 0 0.4853 Nandurbar 4.254 0.207 0.25 0.7704 Vengurla 6.863 0.167 0.75 0.8683 Veraval 7.787 0.2087 0.5 0.8908 Western Zone 3.974 0.1647 0.15 0.7327

Eastern Agarthala 8.097 0.1177 0.5 0.8191 zone Dumdum 5.94 0.115 0.15 0.9241

Gauhati 7.206 0.1557 0.75 0.9401 Gaya 7.176 0.1483 0.5 0.9459 Imphal 4.939 0.134 0.5 0.9719 Jamshedpur 6.93 0.1307 0.5 9.8737 Jharsuguda 8.598 0.1392 0.75 0.874 North Lakhimpur 14.07 0.1256 1.25 1.073 Sagarisland 16.524 0.1402 1.5 0.9635 Shillong 6.728 0.1502 0.75 0.9575 Eastern Zone 6.933 0.1353 0.5 0.8801

Southern Bangalore 6.275 0.1262 0.5 1.128 zone Hyderabad 5.25 0.1354 0.5 1.0295

Kodaikanal 5.914 0.1711 0.5 1.0088 Madras 6.126 0.1664 0.5 0.8027 Mangalore 6.744 0.1395 0.5 0.9374 Tiruchirapalli 7.135 0.1638 0.5 0.9624 Trivandrum 6.762 0.1536 0.5 0.8158 Visakhapatnam 6.646 0.1692 0.5 0.9963 Southern Zone 6.311 0.1523 0.5 0.9465

Rambabu et al. (1979) also give monograph (Fig 1) to convert one hour rainfall intensity into rainfall intensities of other durations.

For Vasad and Kota, the following relationships have been obtained by Central Soil Water Conservation Research and Training Institute.


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Figure 1 Locations of raingauge stations and zonal boundaries in India Vasad 7.506 T0.1393

I = ------------------------ (12) (t + 0.5)0.3857 Kota 5.79 T0.23

I = ------------------- (13) (t + 0.5)0.85

Kothyari & Garde (1992) developed a general relationship on IDF after analysing 80 raingauge stations (Fig. 2) in India. They have made use of the assumption that general properties of convective cells that are associated with short - period rainfalls are similar in different hydrologic regions Raudkivi (1979). The formula is of the form:

I tT = C 33.02

2471.0

20.0

)R(tT

(14)

Where, I t

T = rainfall intensity in mm/hr for T-year return period & t – hours duration


52


C = constant having value 8.31 for the whole of the considered stations

(Values for different regions are given in Table 2) R = 2-year return period & 24 hr. rainfall in mm . 24

2

Figure 2 Relation to convert one our rainfall to intensities for other durations

Table 2 Values of 'C' for different regions of India

Geographical region Zones in figure 1 Value of C Northern India 1 8.0 Eastern India 4 9.1 Central India 2 7.7 Western India 3 8.3 Southern India 5 7.1

In India, number of studies have been carried out to analyse the data of individual stations e.g. Rama and Krishna (1958) for Delhi, Alipore and Madras Stations, Rama and Bandyopadhyay (1969) for Calcutta.

CONCLUSIONS Gumbel’s extreme value distribution has been most oftenly used for analysis of the short duration rainfall data. The IDF relationships developed for Indian regions are based on limited data of very few stations and there is need to further improve upon the relationships based on larger data base.


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REFERENCES

Amin, M. Z. M., Shaaban, A. J. (2004). The rainfall intensity-Duration-Frequency (IDF) relationship for Ungauged sites in peninsular Malaysia using a Mathematical Formulation. 1st international conference on managing rivers in the 21st century: issue and challenges, Rivers’04

Ayyar, P. S. H & Tripathi, N. (1973) - One day rainfall of India for different return periods, No.5, IMD, N.Delhi.

Ayyar, P. S. H. & Tripathi, N. (1974) - Rainfall frequency maps of India, No.-6, IMD, N. Delhi.

Baghirathan, V. R. & Shaw, E. M. (1978)- Rainfall depth - duration- frequency studies for SriLanka, Journal of Hydrology, 37(3), 223-239.

Bell, F. C. (1969) - Generalized rainfall-duration-frequency relationships, Journal of Hydraulics Engineering, ASAE, 95(1),311-327.

Bernard, M. M. (1932) - Formulas for rainfall intensities of long duration, Trans. ASCE, Vol. 96, pp. 592-624.

Bilham, E. G. (1935) - The classification of heavy falls of rain in short periods, H.M.S.O., London, 1962 (republished).

Bougadis, J. , Adamowski, K. (2006). Scaling model of a rainfall intensity-duration-frequency relationship, Hydrol. Process. 20, 3747–3757

Chen, C. L. (1983) - Rainfall intensity - duration - frequency formula, J. of Hydraulics Engineering, ASCE, 109(12), 1603-1621.

Chow, V. T. (Ed.) (1964) - Handbook of applied hydrology., Mc Graw Hill, New York.

Davis, E. G., Naghettini, M. (2000). Regional Analysis of Intensity-Duration-Frequency of Heavy Storms over the Brazilian State of Rio de Janeiro , Journal of Hydrologic Engineering ,138

Desa, M. N. M., Amin, M. Z. M., Asnor, M. I., Mohamed, F. (2006). Developing a Low Return Period Regional IDF Relationship using Generalized Pareto (GPA) Distribution, National Conference – Water for Sustainable Development Towards a Developed Nation by 2020 , 13 – 14 July, Guoman Resort Port Dickson

Durrans, S. R., Kirby, J. T. (2004). Regionalization of extreme precipitation estimates for the Alabama rainfall atlas, Journal of Hydrology, 295, 101–107

Ferreri, G. B. & Ferro, V. (1990) - Short duration rainfalls in Sicily, Journal of Hydraulics Engineering, ASCE, 116(3), 430-435.

Gert, A., Wall, D. J., White, E.L. & Dunn, C.N. (1987) - Regional rainfall intensity - duration-frequency curves for Pennsylvania, Water Res. Bulletin, 23(3), 479-486.

Ghahraman, B., Hosseini, S. M. (2005). A new investigation on the performance of rainfall IDF models, Iranian Journal of Science & Technology, Transaction B, Engineering, 29 (B3), 333-342

Guo, Y., (2006). Updating Rainfall IDF Relationships to Maintain Urban Drainage Design Standards, Journal of Hydrologic Engineering, 11(5), 506–509


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Hadadin, A. N. (2005). Rainfall Intensity Duration-Frequency Relationship in the Mujib Basin in Jordan. Journal of Applied sciences, 8(10), 1777-1784

Hall, A. J. (1984) - Hydrology in tropical Australia & Papua New Guinea, Hydrological Sciences Bulletin, Vol.-29(4).

Holland, D. J. (1967) - Rainfall intensity - frequency relationships in Britain, British Rainfall 1961, H.M.S.O.

Kothyari, U.C., and Garde, R. J. (1992), - Rainfall intensity - duration-frequency formula for India, Journal of Hydraulics Engineering, ASCE, 118(2).

Koutsoyiannis, D., Kozonis, D., Manetas, A. (1998). A mathematical framework for studying rainfall intensity-duration-frequency relationships .Journal of Hydrology, 206, 118-135

Madsen, H., Mikkelsen, P.S., Rosbjerg, D. and Harremoës, P. (1998). Estimation of regional intensity-duration-frequency curves for extreme precipitation. Water Science and Technology, 37(11), 29-36

Maurino, M. F. (2004). Generalized Rainfall-Duration-Frequency Relationships: Applicability in Different Climatic Regions of Argentina.Journal of Hydrologic Engineering, 9(4), 269–274

Mc. Cuen, R. H. (1989) - Hydrological analysis and design, Englewood Cliffs, New Jersey,: Prentice Hall

Naghettini, M. (2000). A Study of the Properties of Scale Invariance as Applied to Intensity-Duration-Frequency Relationships of Heavy Storms. Journal of Hydrologic Engineering. 139

National Environmental Research Council (1975)- Flood studies report, Vol.-1, Hydrologic studies, NERC, London.

Neimczynowicz, J. (1982) - Areal intensity-duration-frequency curves for short term rainfall events, Nordic Hydrology, Vol.13(4), 193-204.

Nhat, Le M., Tachikawa,Y. , Takara, K. (2006). Establishment of Intensity-Duration-Frequency Curves for Precipitation in the Monsoon Area of Vietnam. Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 49 B, 93-103

Parthasarathy, K. & Singh G., (1961) - Rainfall intensity-duration-frequency for India for local drainage design - Indian Journal of Meteorology & Geophysics, Vol.-12(2), 231-242.

Raiford, J. P., Aziz, N. M., Khanand, A. A., Powell, D. N. (2007). Rainfall Depth-Duration-Frequency Relationships for South Carolina, North Carolina, and Georgia, American Journal of Environmental Sciences, 3 (2), 78-84

Ram Babu, Tejwani, K. K., Agrawal, M. C. & Bhusan, L. S. (1979) - Rainfall intensity-duration-return period equations & nomographs of India, CSWCRTI, ICAR, Dehradun, India.

Raman, P. K. & Krishnan, A. (1958) - Intensity-duration-frequecy relationships of rainfall at selected stations in India, Proceedings Symp. on Meteorological and Hydrological aspects of floods & droughts in India, 21-30.

Raman, V. & Bandyopadhya, M. (1969) - Frequency analysis of rainfall intensities for Calcutta, Journal of Sanitary Engineering Dn., ASCE, 95(6), 1013-1030.


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Rao, D. V. L. N., Goyal, S. C. & Kathuria, S. N. (1983) - Interpolation of short duratin rainfall from 24 hour rainfall in Lower Godavari basin, Mausam Journal, Meteorological Soc., N. Delhi, 34(3): 291-298.

Raudkivi, A. J. (1979) - An advance introduction to Hydrological Processes and Modelling, Oxford Pergamon Press, N. York, N.Y.

Regalado, L. de S., Yuste, J. A. F. (2006). Maximum rainfall intensity analysis using l-moments in Spain, Proceedings of the 7 International Conference on HydroScience and Engineering Philadelphia, USA September 10-13, 2006, http://idea.library.drexel.edu/bitstream/1860/1442/1/2007017070.pdf, Aug, 2007

Salas, L. de, Carrero, L., Fernández, J.A. (2007). MAXIN:Gis application to estimateRainfall Intensity-Duration-Frequency laws in the Spanish peninsular area . MS7/ECAM8 Abstracts, Vol. 4, 7th EMSAnnual Meeting / 8th ECAM ,EMS2007-A-00007

Sivapalan, M., Blöschl, G. (1998). Transformation of point rainfall to areal rainfall: Intensity-duration-frequency curves. Journal of Hydrology, 204, 150-167

Steel, E. W. & McGhee, T. J. (1979) - Water Supply and Sewerage, 5th ed., McGraw-Hill Book Com., New York.

Svensson, C., Clarke, R. T., Jones, D. A. (2007). An experimental comparison of methods for estimating rainfall intensity-duration-frequency relations from fragmentary records. Journal of Hydrology, 341, 79–89

Trefry C. M., Watkins D. W. Jr. (2001). Application of a Partial Duration Series Model for Regional Rainfall Frequency Analysis in Michigan, Journal of Hydrologic Engineering. 41

Trefry, C. M., Watkins, D. Jr., Johnson, D. (2005). Regional Rainfall Frequency Analysis for the State of Michigan, Journal of Hydrologic Engineering, 10(6), 437-449

Trefry, C. M., Watkins, D. W., Johnson, D. L. (2000). Development of Regional Rainfall Intensity-Duration-Frequency Estimates for the State of Michigan, Journal of Hydrologic Engineering, 140

Yarnall, D. L. (1935) - Rainfall intensity frequency data, USDA, Misc. Publ. No. 204, Washington, D. C.

Yu, P. S., Lin, C. S., Yang, T. C. (2004). Regional rainfall intensity formulas based on scaling property of rainfall. Journal of Hydrology, 295, 108–123


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