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Application of Intensity-Duration-Frequency Curvefor Korean Rainfall Data

using Cumulative Distribution Function

Kewtae Kim

The Graduate School

Yonsei University

Department of Civil Engineering



Application of Intensity-Duration-Frequency Curvefor Korean Rainfall Data


A Masters Thesis

Submitted to the Department of Civil Engineering

and the Graduate School of Yonsei University

in partial fulfillment of the

requirements for the degree of

Master of Science

Kewtae Kim

July 2008



This certifies that the masters thesis

of Kewtae Kim is approved.

___________________________

Thesis Supervisor: Prof. Jun-Haeng Heo, Ph.D

___________________________

Prof. Woncheol Cho, Ph.D

___________________________

Prof. Sung-Uk Choi, Ph.D

The Graduate School

Yonsei University

July 2008



ACKNOWLEDGEMENT

I am very grateful to my advisor, Professor Jun-Haeng Heo, whose guidance, patience,

encouragement, great help, and invaluable support during the course of study possible. He

taught me hydrological statistics and his lectures were very useful. Without his guidance, I

would not have been able to finish this study. I would also like to thank Professor Woncheol

Cho, Sung-Uk Choi on my thesis committee.

Also, I am thanks professor Won-Hwan Lee who showed extraordinary appreciation

toward water. Im also thank professors Hak-Joo Whang, Bock-Mo Yeu, Keun Joo Byun, Soo

Il Kim, Youn-Kyoo Choung, Moon Kyum Kim, Kwang Baik Ko, Sanh-Hyo Kim, Ha-Won

Song, Sangseom Jeong, Sang-Ho Lee, Yun Mook Lim, Seung Heon Han, Hong-Gyoo Sohn,

Jun-Hwan Lee, Joonhong Park, Joon Heo, Hyoungkwan Kim, Tong Seok Han, Jang-Ho Kim,

and Ho Jeong Kang for my knowledge and personal development at Yonsei University.

Thank all members of Hydro-Laboratory, Yonsei University. Especially, Kyung-Duk Kim,

Kwonsu Kang, Chang-Sam Jeong, Dong-Jin Lee, Youn-Woo Kho, Kyung Hoi Heo, Chang-

Yong Cha, Eunseok Lee, Sang-Bok Lee, Young-Seok Lee, Ji-Hoon Kim, Jiyoun Sung, Mun-

Hyoung Park, Boae Kwon, Jung-Woo Lee In-Chan Park, Haennim Park, Hyeongsik Kang,

Jung-Hwan Ahn, Taebum Kim, Donggyun Lim, Daeryoung Park, Shanghyun Ahn, Myoung Jin

Um, Wonjoon Yang, Younghoon Jung, Seungkyu Han, Sanghwa Cheong, Seonmee Jeon, Han-

Seong Jo, Jungwon Kim, Hee-Sun Jung, Changhun Cho, and Woonghyeon Jeon showed me

the basic attitude to the study, and many fellows including Taesoon Kim, Hongjoon Shin, Woo

Sung Nam, Sooyoung Kim, Jeong-Eun Lee, Ji Hye Kwon, and Jun Hak Lee gave me various

kinds of assistances. And I wish my graduates in the same class, Seong Teak Lim, Hae-Eun

Lee achieve their goal in life. I believe that Younghun Jung, Hye-Seon Yun, Ju-Young Shin,

Heon Cheol Jeong, Young-Il Kim, Gyeong Ho Yun, Sonkyu Yun, Jung Pyo Seo, Gian Choi, In



Young Jang, Byung Woong Choi, Won Geun Lee, Kyoungjoo Lee, Kwanghee Han, Min Hye

Yun, Young Joo Kim, Jaekook Shin, Kyungmin Sung, and Dong Seok Kim can produce good

research results.

And I am thankful to Professor Byong Ho Jun, Professor Mon mo Kim, Professor Chang

Eon Park, Je Hyeong Kim, Jong Yong Kim, member of WATERES, Suk Jin Jang, Ju Il Song,

Jae Kwon Kim, and members of K.G. civil59. Especially, I would like to thank Soomin Kim,

Sunchan Lee, Kilsoo Jung, Daham Yang, my friends, members of C.U.G. the Universe

Conquest, Jae Hwi Lee, Jae Moo Lee, Gihae Kim, Wha Jung Lee, Seong Ho Choi, Eun Sang

Lee, Ki Joo Kim, Do Won Park, Ji Hyoun Woo, So Yeon Park, Yoon Mi Lee, Jae Won Park,

and members of Jang Wi Joongang Presbyterian Church for their special affection to my study.

Finally, my sincere gratitude and appreciation go to my parents, and glory & honor to God.



i

TABLE OF CONTENTS

TABLE OF CONTENTS.................................................. .......................................................... i

LIST OF FIQURES...................................................................................................................iii

LIST OF TABLES ................................................... ........................................................... ...... iv

ABSTRACT........... ........................................................... ........................................................ vi

Chapter 1. Introduction ..................................................... ......................................................... 1

1.1 General Description........................ ........................................................... ....... 1

1.2 Research Objectives .................................................... ..................................... 2

1.3 Literature Review ........................................................ ..................................... 2

Chapter 2. IDF Curve using Cumulative Distribution Function................................................. 6

2.1 Definition of Variables, Notation and Clarification.......................................... 6

2.2 The IDF Relationship for a Specified Return Period............. ........................... 9

2.3 The IDF Curve using Cumulative Distribution Function................................ 12

2.4 Alternative Distribution Functions .................................................. ............... 14

Chapter 3. Parameter Estimation Methods ...................................................... ......................... 17

3.1 Typical Procedure.............................................. ............................................. 17

3.2 Robust Estimation............................................................................ ............... 19

3.3 One-Step Least Squares Method..................................................................... 21

3.4 Parameter Estimation Methods using Genetic Algorithms ............................. 22

Chapter 4. Application ...................................................... ....................................................... 28

4.1 Description of Sites and Data .......................................................... ............... 28

4.2 Application of IDF Curve using Cumulative Distribution Function............... 31

4.3. Parameter Estimation of Conventional IDF Curves....................................... 32



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4.4. Parameter Estimation using Separation of Short- and Long-Rainfall Duration

Method..................................................... ....................................................... 34

Chapter 5. Comparison...................................................... ....................................................... 36

5.1 Total Rainfall Duration................................................ ................................... 36

5.2 Separation of Rainfall Duration....................................................... ............... 39

Chapter 6. Conclusions........................................................................................................ ..... 41

REFERENCES..................................... ........................................................... ......................... 43

APPENDIX A .......................................................... ........................................................... ..... 50

........................................................ ........................................................... ............... 73



iii

LIST OF FIQURES

Figure 3.1 Pareto-optimal solution indicated as "RMSE" that makes an objective function

about RMSE the smallest value and the other objective function about RRMSE

worst, and vice versa .................................................. ............................................. 25

Figure 3.2 Comparison of absolute bias of rainfall quantiles estimated by IDF curve whose

parameters are estimated by two extreme Pareto-optimal solutions in Figure 3.1... 26

Figure 4. 1 Location of 76 rainfall recording sites overall KMA ............................................. 28

Figure 5. 1 IDF curve in case of Modified Sherman at site code number 130 (Uljin).............. 38

Figure 5. 2 IDF curve represented Figure 5. 1 in detail at site code number 130 (Uljin)......... 38



iv

LIST OF TABLES

Table 3. 1 Separation of short- and long-rainfall durations method ......................................... 27

Table 4. 1 Information of 76 sites controlled by Korea KMA ................................................. 29

Table 5. 1 Number of the smallest RMSE or RRMSE of total sites using 12 different IDF

curves in case of Gumbel and GEV...................................... ................................... 37

Table 5. 2 Number of the smallest RMSE or RRMSE of total sites using 8 different IDF curves

in case of Gumbel and GEV.................... ........................................................... ..... 39

Table 5. 3 Number of the smallest RMSE or RRMSE of total sites using 3 different IDF curves

in case of Gumbel and GEV.................... ........................................................... ..... 40

Table A. 1 Average RMSE and RRMSE of each site using 12 different IDF curve in case of

Gumbel.................................................... ........................................................... ..... 51

Table A. 2 Average RMSE and RRMSE of each site using 12 different IDF curve in case of

GEV......................................................... ........................................................... ..... 54

Table A. 3 Average RMSE and RRMSE excluded regression analysis IDF curves of each site

in case of Gumbel.......................... ........................................................... ............... 57

Table A. 4 Average RMSE and RRMSE excluded regression analysis IDF curves of each site

in case of GEV ........................................................... ............................................. 60

Table A. 5 Average RMSE and RRMSE of IDF curves separated short- and long-rainfall

durations for each site in case of Gumbel....................................... ......................... 63

Table A. 6 Average RMSE and RRMSE of IDF curves separated short- and long-rainfall

durations for each site in case of GEV ..................................................... ............... 65

Table A. 7 Parameters of IDF curve using CDF estimated using SEP_DUR method in case of

Gumbel................................................... ............................................................ ..... 67


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v

Table A. 8 Parameters of IDF curve using CDF estimated using SEP_DUR method in case of

GEV......................................................... ........................................................... ..... 70



vi

ABSTRACT

Application of Intensity-Duration-Frequency Curve

for Korean Rainfall Data


Kewtae, Kim

Dept. of Civil Engineering

The Graduate School

Yonsei University

The conventional intensity-duration-frequency (IDF) curve has generally come from the

empirical intuition of researcher, and linear or non-linear regression analysis also has been

usually applied to estimate the parameters of IDF curves. Hence, one can say that the

conventional IDF curves hardly have statistical characteristics of rainfall data at the site of

interest.

In this study, IDF curve using cumulative distribution function (CDF) was applied to

Korean rainfall data, and the parameters of IDF curve using CDF were estimated using single-

objective genetic algorithms and multi-objective genetic algorithms (MOGA) to improve the

accuracy. And then, the parameters were estimated using separation of short- and long-rainfall

durations method (SEP_DUR) to derive more precise formulas. The parameters of

conventional IDF curves were newly estimated to compare the results each other.

As the results, root-mean-square-error (RMSE) and relative RMSE (RRMSE) of IDF curve

using CDF have the smallest values for both cases of total rainfall duration and separation of



vii

short- and long-rainfall durations. The MOGA shows the smallest RMSE and RRMSE among

the applied parameter estimation methods for separation of short- and long-rainfall durations.

It is found that the IDF curve using CDF for SEP_DUR with MOGA is the more accurate than

any other methods.

Key words : Intensity-Duration-Frequency Curve, Multi-objective Genetic Algorithms,

Separation of Short-and Long-Rainfall Durations, Cumulative Distribution Function



1

Chapter 1. Introduction

1.1 General Description

Generally, at-site frequency analysis and regional frequency analysis are performed to

estimate rainfall quantile considering various durations and frequency (return period) at the

site of interest. In order to perform these frequency analysis methods, it should be essential to

compile rainfall data recorded enough length about demanded duration and to estimate

parameters of various probability distributions and to perform the goodness-of-fit tests

corresponding for analysis site.

Although there is an advantage that rainfall quantile can be estimated by at-site frequency

analysis and regional frequency analysis, there are defects; rainfall data should be compiled

more than 30 year and it is necessary to estimate the parameters of a given probability

distribution and to perform the goodness-of-fit test.

IDF curve has widely been used, when hydraulic structure is designed in Korea. Talbot,

Sherman, Japanese and Semi-Log type are the conventional intensity-duration-frequency

(IDF) curves that have widely been used in Korea(Lee (1997), Yoon (1998)). And Talbot,

Sherman, Japanese and Semi-Log also are modified to improve the accuracy of IDF curves.

And other IDF curves have been developed by Lee (1980), Lee (1993), Heo (1999) and

Ministry of Construction and Transportation (2000). However, the conventional IDF curves

formulation has generally come from the empirical intuition of researcher, and regression

analysis also has been usually applied to estimate the parameters of IDF curves. Hence, one

can say that the conventional IDF curves hardly have statistical characteristics of rainfall data

having rationale at the site of interest (Koutsoyiannis et al., 1998).

In this study, in order to overcome such a problem, theoretically derived IDF curve using

cumulative distribution function (CDF) is applied to Korean rainfall data. The parameters of



2

IDF curve using CDF were estimated using single objective genetic algorithms (SGA) and

multi-objective genetic algorithms (MOGA) to improve the accuracy

1.2 Research Objectives

The major objective of this study is to apply the IDF curve using CDF reflecting the

statistical characteristics of rainfall data at the site of interest. The specific objectives are:

(1) to collect annual maximum rainfall data corresponding to each duration for site overall

Korea Meteorological Administration (KMA).

(2) to adopt IDF curve using CDF for Korean rainfall data.

(3) to estimate the parameters of adopting IDF curve using genetic algorithms to improve

the accuracy.

(4) to newly estimate the parameters of conventional IDF curves to compare the results

each other.

(5) to compare the accuracy of total IDF curves.

1.3 Literature Review

The rainfall intensity-duration-frequency relationship is one of the most generally used

methods in urban drainage design and floodplain management. The establishment of such

relationships goes back to as early as 1928 (Meyer, 1928). After a few late Sherman (1931)

derived applicable general intensity duration formula to other localities, and Bernard (1932)

was to make available for any locality within the limits of the study, rainfall intensity formulas

for frequencies of 5, 10, 15, 25, 50 and 100 year, applicable to rainfall duration of 120 to 6000

min. Bell (1969) developed IDF relationship using formula computed the depth-duration ratio

for the U.S.S.R. Chen (1983) developed a simple method to derive a generalized rainfall



3

intensity-duration-frequency formula for any location in the United States using three iso-

pluvial maps of the U.S Weather Bureau Technical Paper No.40.

In the 1990s, some other approaches mathematically more consistent had been proposed,

Burlando and Rosso (1996) proposed the mathematical framework to model extreme storm

probabilities from the scaling properties of observed data of station precipitation, and the

simple scaling and the multiple scaling conjectures was thus introduced to describe the

temporal structure of extreme storm rainfall. Also, Koutsoyiannis (1994; 1996; 1998) proposed

a new approach to the formulation and construction of the intensity-duration-frequency curves

using data from both recording and non-recording stations. More specially, it discussed a

general rigorous formula for the Intensity-Duration-Frequency relationship whose specific

forms had been explicitly derived from the underlying probability distribution function of

maximum intensities. And it also proposed two methods for a reliable parameter estimation of

the IDF relationship. Finally, it discussed a framework for the regionalization of IDF

relationships by also incorporating data from non-recording stations.

More recently, Garcia-Bartual and Schneider(2001) used statistical distribution and found

the Gumbel Extreme Value (GEV) distribution fitted to data well. Yu et al.(2004) developed

regional rainfall intensity-duration-frequency relations for non-recording sites based on scaling

theory, which uses the hypothesis of piecewise simple scaling combines with the Gumbel

distribution. Mohymont et al.(2004) assessed IDF-curves for precipitation for three stations in

Central Africa and proposed more physically based models for the IDF-curves. Di Baldassarre

et al.(2006) analyzed to test the capability of seven different depth-duration-frequency curves

characterized by two or three parameters to provide an estimate of the design rainfall for storm

durations shorter than 1 hour, when their parameterization is performed by using data referred

to longer storms. Karahan et al.(2007) estimated parameters of a mathematical framework for

IDF relationship presented by Koutsoyiannis et al. (1998) using genetic algorithm approach.



4

Singh and Zhang (2007) derived intensity-duration-frequency (IDF) curves from bivariate

rainfall frequency analysis using the copula method.

In Korea, Lee (1980) derived the rainfall intensity probability formulas by the least square

method and constructed the rainfall intensity-duration-frequency curves. And Lee et al. (1992;

1993) derived a typical probable rainfall intensity formula by analyzing pre-issued probable

rainfall intensity formulas over principal rainfall observation stations, and to obtain the

regional characteristics based on the rainfall patterns by evaluating probable rainfall amount.

Yoo (1995) analyzed generalization of probability IDF curve.

Han et al. (1996) and Heo et al. (1999) derived the rainfall intensity-duration-frequency

equation based on the Linearizing Method and the appropriate probability distribution.

Especially, Heo (1999) performed frequency analysis of annual maximum rainfall data for 22

rainfall gauging stations in Korea and estimated parameter using the method of moments

(MOM), maximum likelihood (ML), and probability weighted moments (PWM). And the GEV

distribution was selected as an appropriate model for annual maximum rainfall data based in

parameter validity condition, graphical analysis, separation effect, and goodness of fit tests.

For the selected GEV model, spatial analysis was performed and rainfall intensity-duration-

frequency formula was derived by using linearization technique.

Seoh et al. (1999) analyzed return period due to rainfall duration in 98 Pusan rainfall

datum. Lee and Lee (1999) derived rainfall intensity formula based on the representative

probability distribution in Korea and divided whole region into five zones for 12-rainfall

durations. Choi et al. (2000) derived the probable rainfall depths and the probable rainfall

intensity formulas for Inchon. Park et al. (2000) analyzed the variation in peak discharge

according to the type of probable rainfall intensity formula for Wi stream basin. Song and

Seoh(2000) estimated probable rainfall intensity formula and applied the optimization

technique using Simplex method and Powell method. Lee et al. (2001) derived the rainfall

intensity formula by the regional frequency analysis of individual zone based in representative



5

probability distribution in Korea. Seoh et al. (2001) constructed data base of minutely rainfall

intensity. Yoo et al. (2001) proposed a theoretical methodology for deriving a rainfall

intensity-duration-frequency curve using a simple rectangular pulsed Poisson process model.

Heo and Chae (2002) improved reading method of rainfall recording paper to estimate

accurate probable rainfall intensity. Kim et al. (2002) estimated probable rainfall intensity

corresponding to separate of rainfall duration. Lee and Seong (2003) analyzed the smoothing

of rainfall intensity-duration-frequency relationship curve by the Box-Cox transformation. Yoo

and Rho (2004) computed probable rainfall intensity and flood discharge for culvert design.

Choi (2006) estimated real time rainfall intensity using rainfall radar and rain gauges. Han et al.

(2006) suggested probable rainfall intensity formula considering the pattern change of

maximum rainfall at Incheon city.

Recently, Kim et al. (2007a) estimated IDF curve considering climate change using GCM

model. Yoo et al. (2007) proposed and evaluated a methodology for deriving the rainfall

intensity-duration-frequency relationship for durations less than 10 minutes used for designing

drainage systems in small urban catchments and roads. Kim et al. (2007b) estimated parameter

of intensity-duration-frequency curve using genetic algorithm and compared with study of

existing estimation method. Also, Shin et al. (2007) suggested the separation of short and long

durations for estimation the parameters of IDF curve using Multi-objective Genetic Algorithm

(MOGA).



6

E quati onChapter2Sect i on1Chapter 2. IDF Curve using Cumulative Distribution

Function

It is self-evident that the IDF relationship is a mathematical relationship among the rainfall

intensity i the duration d, and the return period T (or, equivalently, the annual frequency

of excess, typically referred to as frequency only). However, these terms may have different

meanings in different contexts of engineering hydrology and this may lead to confusion or

ambiguity (Koutsoyiannis et al., 1998). For the sake of a comprehensive presentation and

unambiguousness in the material that follows we include in sections 2.1 and 2.2 the definitions,

clarifications, and description of the general properties of the idf relationships. The reader

familiar with these issues may proceed directly to section 2.3.

2.1 Definition of Variables, Notation and Clarification

Let ( )t denoted the instantaneous rainfall intensity process, where t denotes time. Let

d be a selected (arbitrary) time duration (typically from a few minutes to several hours or few

days), which serves as the length of a time window over which we integrate the instantaneous

rainfall intensity process ( )t . Moving this time window along time we form the moving

average process, given by

1( ) ( )

t

dt d

t s dsd

= (2.1)



7

In reality, because we do not know the instantaneous intensity in continuous time, but

rather have measurements of the average intensity ( )t for a given resolution (typically 5-10

min to 1 h), Eq. (2.1) becomes

1

0

( ) ( )N

d

i

t t id

=

= (2.2)

where it is assumed that the duration d is an integer multiple of the resolution , i.e.

d N= . Given the stochastic process ( )d t we can form the series of the maximum average

intensities (or simply maximum intensities) ( )li d ( 1,..., )l n= , which consists of n values,

where n is the number of (hydrological) years through which we have available

measurements of rainfall intensities. This can be done in two ways. According to the first way,

we form the series of annual maxima (or annual maximum series) by

( ) max { ( )}l d

l t l

i d t +

< > + = (2.4)

where the three conditions of the right-hand part must hold all together, otherwise the point lt

and the respective intensity ( )d lt are not selected for the series { ( )}li d .



8

In practice, the construction of the series of maximum intensities is performed

simultaneously for a number kof durations , 1,...,jd j k= , starting from a minimum duration

equal to the time resolution of observations (e.g. from 5-10 min to 1 h depending on the

measuring device) and ending with a maximum duration of interest in engineering problems

(typically 24 or 48 h). Normally, all k series must have the same length n but, owing to

missing values, it is possible to have different lengths jn for different durations jd .

The above description of the construction of the series of maximum intensity allows us to

observe that the duration d is not a random variable but, rather, a parameter for the intensity.

It is not related to the actual duration of rainfall events, but is simply the length of the time

window for averaging the process of intensity. On the contrary, the series of maximum

intensities ( )li d is considered as a random sample of a random variable ( )I d (Koutsoyiannis

et al., 1998).

The return period T for a given duration d and maximum intensity ( )i d is the average

time interval between excess of the value ( )i d . It is well known (e.g. see Kottegoda (1980))

that for the annual series, under the assumption that consecutive values are independent, the

return period of an event is the reciprocal of the probability of excess of that event, i.e.

1

1T

F=

(2.5)

where F denotes the probability distribution function of ( )I d which, of course, is

evaluated at the particular magnitude of interest. It is also known (e.g. see Raudkivi (1979))

that the return period 'T for the series above threshold is related to that of the series of

annual maxima by

1

1 exp( 1/ ')T

T=

(2.6)



10

( )i

d

=

+

(2.8)

where , , , and are non-negative coefficients with 1 . The latter inequality is

easily derived from the demand that the rainfall depth h id= is an increasing function of d.

Equation (2.8) is not obtained by any theoretical reasoning, but is an empirical formula,

encapsulating the experience from several idf studies. In the bibliography, they found

simplified versions of Eq. (2.8), which are derived by adopting one or two of the restrictions

1, 1, = = and 0 = .

It should be noted that considering 1 and 1 results in overparameterization of

Eq. (2.8). Indeed, the quantity 1/( )d + can be adequately approximated by *1/( ')d +

where ' and * are coefficients depending on and , which can be determined

numerically in terms of minimization of the root-mean-square error. Consequently,

1/( )d + is approximated by'1/( ')d + , where ' * = . A numerical investigation

was done to show how adequate the approximation of 1/( )d + by 1/( ')d + is. The

duration d was restricted between the values min 1/12d h= (i.e. 5 min) and max 120d h= , an

interval much wider than the one typically used. The parameter varied between 0 and

max min12d = (i.e. 1 h), and the parameter between 0 and 1. The root-mean-square

standardized error (rmsse) of the approximation took a maximum value of 2.3% for 0.55 =

and max = ; the corresponding maximum absolute standardized error (mase) was 4.3%. For

the most frequent case that mind , the corresponding errors are 0.7% (rmsse) and 1.3%

(rmse). These errors are much less than the typical estimation errors and the uncertainty due to

the limited sizes of the typical samples available. In conclusion, the parameter in the

denominator of Eq. (2.8) can be neglected and the remaining two parameters suffice. Hence,

hereafter we will assume that 1 = .



11

Initially, the coefficients , , and can be considered as dependent on the return

period. However, their functional dependence cannot be arbitrary, because the relationships for

any two return periods 1T and 2 1T T< must not intersect. If 1 1 1{ , , } and 2 2 2{ , , }

are the two parameter sets for 1T and 2T respectively, then it can be shown that there exist at

least two sets of constraints leading to feasible (i.e. not intersecting) idf curves. These are

1

2

1 11 1 1

1 2

2 2 2 2 2

0, 0, 1, ,

> > > (2.9)

and

1 2

1 1 2

1 2 1 2

2 1 2

0, 0, , 1,

= = > (2.10)

To both these sets, the following obvious inequalities are additional constraints

1 2 1 20, 0, 0 1, 0 1 > > < < < < (2.11)

The essential difference between the sets of constraints in Eq. (2.9) and Eq. (2.10) is that

the former does not allow to take zero value, whereas the latter does allow this special

value. Furthermore, it can be shown that, if is allowed to take zero value, then the

exponent in Eq. (2.8) must be constant and independent of the return period. Because the

case 0 = must not be excluded, it is reasonable to adopt the set of constraints of Eq. (2.10)

for the subsequent analysis. For convenience, it is reasonable to consider independent of

the return period as well, thus leading to the following final set of restrictions

1 2 1 2 1 20, 0 1, 0 = = < = = < > > (2.12)

In this final set of restrictions, the only parameter that is considered as an (increasing)

function of the return period T is . This leads, indeed, in a strong simplification of the

problem of construction of idf curves. This theoretical discussion is empirically verified, as



12

numerous studies have shown that real world families of idf curves can be well described with

constant parameters and .

2.3 The IDF Curve using Cumulative Distribution Function

After the above discussion they could formulate a mathematical framework for IDF

relationship in the form (Koutsoyiannis et al., 1998)

( )

( )

a Ti

b d= (2.13)

which has the advantage of a separable functional dependence of i on T and d. The

function ( )b d is

( ) ( )b d d = + (2.14)

where and are parameters to be estimated ( 0 > , 0 1< < ). The function ( )a T

(which coincides with of section 2.2) is given in the bibliography (e.g. Raudkivi (1979);

Shaw (1983); Subramanya (1984); Chow (1988); Wanielista (1990); Singh (1992)) by the

following alternative relations

( )a T T= (2.15)

( ) lna T c T = + (2.16)

The first is the oldest (Bernard, 1932) yet the most common until recently (e.g. see

Kouthyari (1992) and Pagliara (1993)). These relations are rather empirical and their use has

been dictated by their simplicity and computational convenience rather than their theoretical

consistency with the probability distribution functions which are appropriate for the maximum

rainfall intensity. Chen (1983) applied a more theoretical analysis to obtain similar



14

(2.16), and in some cases1 ( )

YF cannot be expressed with an explicit analytical equation.

However, as shown below, approximate analytical expressions and always be gotten

adequately simple and more accurate than the empirical functions Eq. (2.15) and Eq. (2.16).

In section 2.5, it is shown to examine the most typical distribution functions of maximum

intensities and to obtain for each distribution function the corresponding function ( )a T .

Notably, it is shown that the empirical functions Eq. (2.15) and Eq. (2.16) can be obtained

by our general methodology, but they correspond to distribution functions that may not be

appropriate for maximum rainfall intensities.

2.4 Alternative Distribution Functions

To better serve our purpose, the mathematical expressions of the alternative distribution

functions ( )YF y given below may have been written intentionally in a slightly different form

from that typically used in the literature. In all distributions, and denote dimensionless

parameters whereas and c denote parameters having the same dimensions as the random

variable y (or lny in the case of logarithmic transformation of the variable)(Koutsoyiannis

et al., 1998).

2.4.1 Gumbel Distribution FunctionThe type I distribution of maxima, also termed the Gumbel distribution function (Gumbel,

1958), is the most widely used distribution for IDF analysis owing to its suitability for

modelling maxima. Given that the rainfall intensity ( )I d has a Gumbel distribution for any

duration d, so will Y, and thus



15

/( ) exp( )yY

F y e +

= (2.20)

where and are the scale and location parameters respectively of the distribution

function. Combining Eqs. (2.19) and (2.20) we directly get

1( ) ln ln 1

Ty a T

T

=

(2.21)

which is an exact yet simple expression of ( )a T (Koutsoyiannis et al., 1998).

2.4.2 Generalized Extreme Value Distribution

This general distribution, which incorporates type I, II, and III extreme value distributions

of maxima can be written in the form

1/

( ) exp 1 ( 1/ )Y

yF y y

= +

(2.22)

where 0 > , 0 > , and are shape, scale, and location parameters respectively. For

0 = the generalized extreme value (GEV) distribution turns into the Gumbel distribution;

the case where 0 < is not considered here because it implies an upper bound of the variable,

which is not the case in maximum rainfall intensity. We directly obtain from Eq. (2.22) that

1ln 1 1

( )

1' ' ln 1

T

Ty a T

T

= +

= +

(2.23)



16

where for simplification we have set ' / = and ' 1 = . Again we have an exact

expression of ( )a T for the GEV distribution that remains relatively simple (Koutsoyiannis et

al., 1998).



17

E quati onChapter3Sect i on1Chapter 3. Parameter Estimation Methods

The parameters of the IDF curve using CDF fall into two categories: those of the function

( )a T (i.e. , , , etc., depending on the distribution function adopted) and those of the

function ( )b d (i.e. and ) (Koutsoyiannis et al., 1998). In all procedures Koutsoyiannis

at al (1998) assume that they was given groups each holding the historical intensities of a

particular duration jd , 1,...,j k= . They denoted by jn the length of the group j , and by

jli the intensity values of this group (samples of the random variables ( )j jI I d= ) with

1,...,j

l n= denoting the rank of the value jli in the group j arranged in descending order.

3.1 Typical Procedure

The typical parameter estimation procedure for idf curves (Raudkivi (1979); Chow (1988);

Wanielista (1990); Singh (1992)) consists of three steps. The first step consists of fitting a

probability distribution function to each group comprised of the data values for a specific

duration jd . In the second step the rainfall intensities for each jd and a set of selected return

periods (e.g. 5, 10, 20, 50, 100 years, etc.) are calculated. This is done by using the probability

distribution functions of the first step. In the third step the final idf curves are obtained in two

different ways: either (a) for each selected return period the intensities of the second step are

treated and a relationship of i as a function of d (i.e. ( )Ti i d= )is established by (bivariate)

least squares, or (b) the intensities of the second step for all selected return periods are treated

simultaneously and a relationship of i as a function of both d and T (i.e. ( , )i i T d = )is

established by (three-variate) least squares. In case (a) different values of the parameters ,



18

and are obtained for each T. In case (b) unique values of the parameters and are

obtained, whereas is determined as a function ( )a T = The form of this function

(typically Eq. (2.15) or Eq. (2.16)) is selected a priori. In the case that ( )a T is given by the

power relationship in Eq. (2.15), the estimation procedure is simplified, because Eq. (2.13)

becomes linear by taking logarithms of both sides.

The main advantage of this parameter estimation procedure is its computational simplicity,

which in fact imposes the separation of the calculations in three steps, so that the calculations

of each step are as simple as possible. However, the procedure has some flaws, which are not

unavoidable. First, it bears the weakness of using an empirically established function ( )a T

(step 3) instead of the one consistent with the probability distribution function (step 1). This

has been already discussed in section 2. Second, it is subjective, in the sense that the final

parameters depend on the selected return periods in step 2. This dependence may be essential

if the selected empirical function ( )a T departs significantly from that implied by the

probability distribution function (Koutsoyiannis, 1996). Third, it treats the three involved

variables (i , d , T ) as having the same nature, in spite of the fact that they are

fundamentally different in nature, i.e. i represents a random variable, d is a (non-random)

parameter of this random variable, and T is a transformation of the probability distribution

function of the random variable.

In sections 3.2 and 3.3 Koutsoyiannis at al (1998) proposed two different parameter

estimation methods that are free of the flaws of the above-described typical procedure and

harmonize with the general formulation of IDF curves given in section 2. These procedures

need more complicated calculations than the typical procedure, yet remain computationally

simple. Both can be applied sing a typical spreadsheet package and do not require the

development of specialized computer programs.



19

3.2 Robust Estimation

The first proposed method estimates the parameters in two steps, the first concerning the

parameters of function ( )b d and the second those of ( )a T (Koutsoyiannis et al., 1998).

This method is based on the identity of the distribution functions of the variables ( )j j jY I b d =

of all k groups, regardless of the duration jd of each separate group. This identity leads

them to the Kruskal-Wallis statistic, which is used to test whether several sample groups

belong to the same population. Let them assume that the parameters and of ( )b d are

known. Then all values ( )jl jl jy i b d= can be found. The overall number of data values is

1

k

j

j

m n=

= (3.1)

Ranks jlr were assigned to all of the m data values jly (using average ranks in the

event of ties). For each group we compute the average rank jr of the jn values of that

group. If all groups have identical distribution then each jr must be very close to ( 1) / 2m + .

This leads to the following statistic (Kruskal-Wallis) which combines the results of all groups

2

1

12 1

( 1) 2

k

KW j j

j

mk n r

m m =

+ =

+ (3.2)

The smaller the value of KWk , the greater the evidence that all groups of y values belong

to the same population. Obviously, the ranks jlr (and hence KWk ) depend on the parameters

and that were assumed as known. Consequently, the estimation problem is reduced to

an optimization problem defined as Eq.(3.2)

minimize 1( , )KWk f = (3.3)



20

Apparently, it is not possible to establish an analytical optimization method for our case. A

numerical search technique for optimization that makes no use of derivatives (see Pierre

(1986) and Press (1992)) is appropriate. However, it may be simpler to use a trial-and-error

method based on a common spreadsheet computer program. The advantages of the Kruskal-

Wallis statistic are its non-parametric character and its robustness, i.e. its ability not to be

affected by the presence of extreme values in the samples. They clarify, however, that the

minimum value of KWk determined by the minimization process cannot be used further to

perform the typical Kruskal-Wallis statistical test (actually, the testing is not really needed).

The reason is that this test assumes that all k groups are mutually independent. In our case, the

intensities jI of the different groups are stochastically dependent variables, as is evident from

their construction (see section 2.2). Thus, we do not know the distribution function of the

statistic KWk to perform any statistical test. Nevertheless, the minimization of its value is

achievable because the distribution function does not need to be known.

For the sake of improving the fitting of( )b d

in the region of higher intensities (and also

to simplify the calculations) it may be preferable to use in this first step of calculations a part

of the data values of each group instead of the complete series. For example, we can use the

highest 1/2 or 1/3 of intensity values for each duration.

Given the values of and , they proceed to the second step of calculations, which is

very easy. Assuming that, with these values, all groups have identical distribution, they append

all k groups of values jly thus forming a unique (compound) sample. For this sample they

choose an appropriate distribution function, and estimate its parameters using the appropriate

estimators for that distribution (e.g. those obtained by the methods of maximum likelihood,

moments, L-moments, etc.; for a concise presentation of such estimators see Stedinger (1993)).

This defines completely the form and the parameters of ( )a T .



21

3.3 One-Step Least Squares Method

The second method estimates all parameters of both functions ( )a T and ( )b d in one

step, minimizing the total square error of the fitted idf relationship to the data (Koutsoyiannis

et al., 1998). To this aim, to each data value jli Koutsoyiannis et al.(1998) assign an

empirical return period using, e.g. the Gringorten formula

0.12

0.44

j

jl

nT

l

+=

(3.4)

So, for each data value we have a triplet of numbers ( lji , ljT , jd ). On the other hand,

given a specific form of ( )a T , chosen among those of section 2.5 from preliminary

investigations of the type of the distribution function of intensity, they obtain the modeled

intensity

( )

( )

jljl

j

a Ti

b d=

(3.5)

and the corresponding error

ln ln ln( / )jl jljl jl jle i i i i= = (3.6)

where they have applied the logarithmic transformation to keep balance among the errors of

the intensities of greater durations (which are lower) and those of lower ones. The overall

mean square error is

2 2

1 1

1 1 jnk

jl

j lj

e ek n= =

= (3.7)

Again the estimation problem is reduced into an optimization problem, defined as

minimize 2 ( , , , , , ... )e f = (3.8)



22

A numerical search technique for optimization that makes no use of derivatives, such as the

Powell method (see Pierre(1986) and Press et al.(1992)), is appropriate for this problem.

However, it may be simpler to perform the optimization using the embedded solver tools of

common spreadsheet packages.

They note that the least squares method in fitting a theoretical to an empirical distribution

function is not a novelty of the proposed method. Rather, the innovative element of the

proposed method is the simultaneous estimation of the parameters of both the distribution

function and the duration function ( )b d .

3.4 Parameter Estimation Methods using Genetic Algorithms

Genetic Algorithms (GAs) are based on the mechanics of natural selection and natural

genetics (Goldberg, 1989). They were developed in the 1960s and refined throughout the

1970s by John Holland and his coworkers at the University of Michigan to explain and model

the adaptability of natural systems. Hollands monograph (Holland, 1975), Adaptation in

Natural and Artificial Systems, is the seminal book in this field, and Goldbergs book, Genetic

Algorithms in Search, Optimization and Machine Learning (Goldberg, 1989), deals with the

most practical optimization problems in evolutionary computation (Kim, 2005).

GAs are a very effective optimization method for solving multi-objective problems

(MOPs) because they exploit not only a single solution but also a set of many possible

solutions, namely a population, for searching the global optimum. Hence, they can easily

determine which solution is superior to the others, even though only one run of the

optimization model is performed; this is a major advantage of GAs when Pareto-optimality,

which was originally introduced by (Edgeworth, 1881) and later generalized by (Pareto, 1896),

is applied to achieve global or near-global optimum solutions for MOPs. Readers who are



23

interested in GAs and Pareto-optimality can refer to the textbooks written by (Mitchell, 1996)

and (Deb, 2001), and a number of technical papers, e.g. (Van Veldhuizen, 2000), are good

references (Kim et al., 2008a).

Nondominated sorting genetic algorithms-II (NSGA-II) (Deb et al., 2002) is widely used to

solve MOPs in the field of engineering (Deb(2003a); Deb(2003b); ISI(2004)). The key

features of NSGA-II that are different from those of the former NSGA (Srinivas, 1994) are a

reduction in time complexity, a parameterless sharing procedure that uses crowding distance

for ensuring diversity in a population, and elitism that can speed up the performance of GAs

and can also help in preventing the loss of good solutions once they are found (Deb et al.,

2002).

Genetic algorithm would be effective tool for parameter estimation of equations(Giustolisi

et al., 2006), especially for non linear relationship like IDF curves. In addition, the

compromised solutions, namely Pareto-optimal solutions, would be achieved using NSGA-II

satisfying with multi-objectives of this study (Kim et al., 2008b).

3.4.1 Parameter Estimation Method using Single-objective Genetic

Algorithms

Single-objective Genetic Algorithms (SGA) is used as the parameter estimation method. In

order to minimize deviation between objective-value and estimated-value, root-mean-squared-

error (RMSE, Eq.(3.9), Objective-function-1, OF-1) and relative RMSE (RRMSE, Eq.(3.10),

Objective-function-2, OF-2) were selected as to single-objective functions of SGA (Kim et

al., 2007b).

( )2 1 2

1 1

1 n n

ijij

j i

RMSE Q Qn = =

= (3.9)



24

2 12

1 1

1

n n

ijij

j i ij

Q QRRMSE

n Q= =

=

(3.10)

where n is the number of total data, 1n and 2n are the number of rainfall duration and

return period, ij

Q is the estimated rainfall quantile by IDF curve with i -th duration and j -

th return period, and ijQ is the computed rainfall quantile by at-site frequency analysis

software (FARD2006) corresponding to i -th duration and j -th return period.

At-site frequency analysis result was selected as to objective-value (more details in the

section 4.1). The SGA source code using C language can be freely downloaded from

http://www.iitk.ac.in/kangal/soft.htm.

3.4.2 Parameter Estimation Method using NSGA-II

In this method, NSGA-II, which is multi-objective genetic algorithms (MOGA), was used

to derive more precise formulas of IDF curve. In order to achieve Pareto-optimal solutions,

two multi-objective functions (Eqs. (3.9) and (3.10)) are used in this study.

RMSE represents a king of absolute squared error as compared with RRMSE. Therefore, if

the computed quantiles ( ijQ ) which is true value become larger, IDF curve with a parameter

set with smaller RMSE is more accurate than that with smaller RRMSE that is a relative

squared error measure. On the contrary, if the quantile ijQ

become smaller, RRMSE is an

effective tool to measure the accuracy of IDF curve.

Figure 3. 1 shows Pareto-front calculated by NSGA-II with Eqs. (3.9) and (3.10) as

multi-objective functions, and the grey-colored circles are Pareto-optimal solutions. It can be

seen from the figure that a Pareto-optimal solution defined as RMSE has the smallest RMSE



25

among Pareto-front; the other extreme Pareto-optimal solution identified as RRMSE has the

smallest RRMSE.

2.60 2.64 2.68 2.72 2.76 2.80

RMSE

0.0180

0.0182

0.0184

0.0186

0.0188

0.0190

RRMSE

RMSE

RRMSE

Figure 3. 1 Pareto-optimal solution indicated as "RMSE" that makes an objective functionabout RMSE the smallest value and the other objective function about RRMSEworst, and vice versa

Figure 3. 2 depicts more clearly the characteristics of RMSE and RRMSE of Figure 3.

1. The absolute bias means the difference between the estimated rainfall quantiles by IDF

curve that is derived using RMSE Pareto-optimal solution of Figure 3. 1. On the other hand,

RRMSE means the absolute bias of the estimated quantiles using RRMSE of Figure 3. 1.

It is interesting to note that the absolute biases for RMSE and RRMSE corresponding to

the respective short- and long-rainfall duration show the different features. For example, the

absolute biases of RRMSE are smaller than those of RMSE in the short rainfall durations like

1, 2, and 3 hours. Whereas the absolutes biases of RRMSE is smaller than those of RMSE for

the long rainfall durations such as from 6 hours to 48 hours. As a result, the IDF curve derived



26

using the Pareto-optimal solution indicated RRMSE in Figure 3. 1 is more effective to

compute the rainfall quantiles for short duration. On the other hand, the IDF curve using

RMSE is more useful to calculate quantiles for long duration.

0

5

10

15

20

25

30

35

40

45

50

1 2 3 6 9 12 15 18 24 48

Duration

AbsoluteB

ias

RMSE

RRMSE

Figure 3. 2 Comparison of absolute bias of rainfall quantiles estimated by IDF curve whoseparameters are estimated by two extreme Pareto-optimal solutions in Figure 3. 1

Table 3. 1 shows a method of discriminating short- and long-rainfall durations as an

example. RRMSE in the table means that the absolute bias of the rainfall quantiles using

RRMSE is smaller than that using RMSE in the Figure 3. 1, and RMSE shows that the

absolute bias using RMSE is smaller than that of RRMSE. According to the comparison

result, short-duration is by 2 hours and from 3 hours it can be said to be long-duration

(Shin(2007), Kim(2008b)). This separation of short- and long-rainfall durations method is

called SEP_DUR in this study.



27

Table 3. 1 Separation of short- and long-rainfall durations method

Duration(hr)Return Period(year) 1 2 3 6 9

235

10203050

7080100

200300500

RRMSERRMSERRMSE

RRMSERRMSERRMSERRMSE

RRMSERRMSERRMSE

RRMSERRMSERRMSE

RRMSERRMSERRMSE

RRMSERRMSERRMSERMSE

RMSERMSERMSE

RRMSERRMSERRMSE

RRMSERRMSERRMSE

RMSERMSERMSERMSE

RMSERMSERMSE

RRMSERRMSERRMSE

RMSERMSERMSE

RMSERMSERMSERMSE

RMSERMSERMSE

RRMSERMSERMSE

RRMSERRMSERRMSE

RMSERMSERMSERMSE

RMSERMSERMSE

RMSERMSERMSE

13 9 6 1 3

0 4 7 12 10Total

RRMSE RRMSE EQUAL RMSE RMSE



28

E quati onChapter4Sect i on1Chapter 4. Application

4.1 Description of Sites and Data

In this study, rainfall data of 76 rainfall recording sites overall the Korea Meteorological

Administration (KMA) were used, which the 76 sites have been controlled by KMA. 59 sites

have rainfall records at least 35 years until 2007 and the longest record is 47 years. The rainfall

data have maintained recording rain gauges well by KMA and there are almost no incidents of

incorrect measurement. Figure 4. 1 and Table 4. 1 show the location and sites information.

Figure 4. 1 Location of 76 rainfall recording sites overall KMA



29

Table 4. 1 Information of 76 sites controlled by Korea KMA

No. Site CodeNumber Site name First year Last year Recordlength Tm-X Tm-Y

1 90 Sokcho 1968 2007 40 336901.3380 528692.8062

2 95 Cheorwon 1988 2007 20 226596.0977 516148.6695

3 98 Dongducheon 1998 2007 10 205268.5961 488792.3465

4 99 Munsan 2001 2007 7 179392.4277 487055.2332

5 100 Daegwallyeong 1981 2007 27 355053.8601 466708.3982

6 101 Chuncheon 1966 2007 42 264632.8290 489139.6378

7 102 Baengnyeongdo 2000 2007 8 -8264.1643 498889.6621

8 105 Gangneung 1961 2007 47 366590.9666 473801.8625

9 106 Donghae 1992 2007 16 387762.7503 447101.8778

10 108 Seoul 1961 2007 47 196910.0557 452124.694011 112 Incheon 1961 2007 47 166726.1348 441767.7569

12 114 Wonju 1986 2007 22 283809.5290 426585.6786

13 115 Ulleung 1961 2007 47 544819.2612 449544.5517

14 119 Suwon 1964 2007 44 198647.5245 418989.7164

15 121 Yeongwol 1995 2007 13 329356.9309 409814.2162

16 127 Chungju 1973 2007 35 284762.5150 385843.0390

17 129 Seosan 1968 2007 40 154750.0717 364036.9158

18 130 Uljin 1972 2007 36 414744.1560 390517.2335

19 131 Cheongju 1967 2007 41 239338.2020 348761.2530

20 133 Daejeon 1969 2007 39 233323.4500 319077.6045

21 135 Chupongnyeong 1961 2007 47 289344.0003 302935.0398

22 136 Andong 1983 2007 25 352745.6260 342680.1573

23 137 Sangji 2002 2007 6 303745.5927 323671.6535

24 138 Pohang 1961 2007 47 414413.4139 283980.4690

25 140 Gunsan 1968 2007 40 178416.1763 278352.2514

26 143 Daegu 1961 2007 47 346121.0024 266199.7322

27 146 Jeonju 1961 2007 47 213929.7374 257938.2599

28 152 Ulsan 1961 2007 47 410303.2726 231404.6510

29 155 Masan 1985 2007 23 343218.9864 186796.4925

30 156 Gwangju 1961 2007 47 190059.5180 185983.9736

31 159 Busan 1961 2007 47 385202.6520 180285.821132 162 Tongyeong 1968 2007 40 331231.1314 150578.2598

33 165 Mokpo 1961 2007 47 143317.2550 146646.2432

34 168 Yoesu 1961 2007 47 267755.5205 138111.6945

35 169 Heuksando 1997 2007 11 57999.9404 133181.7697

36 170 Wando 1973 2007 35 172513.4340 99807.1125

37 175 Jindo 2002 2007 6 137807.2058 108427.6668

38 184 Cheju 1962 2007 44 156231.9161 2073.4148

TM : Transverse Mercator System



30

Table 4. 1 Information of 76 sites controlled by KMA ( Continued )

No. Site CodeNumber

Site name First year Last year Recordlength

Tm-X Tm-Y

39 185 Gosan 1988 2007 20 121953.0496 -22148.3464

40 187 Sungsan 1997 2007 11 188798.8911 -11788.9053

41 189 Seogwipo 1961 2007 47 159421.4034 -27665.8060

42 192 Jinju 1969 2007 39 294680.3729 185451.3788

43 201 Ganghwa 1973 2007 35 151141.4454 467677.8921

44 202 Yangpyung 1973 2007 35 243666.7505 443049.8459

45 203 Icheon 1973 2007 35 242882.7111 418114.8414

46 211 Inje 1973 2007 35 302345.2968 506990.5046

47 212 Hongcheon 1973 2007 35 277582.3317 464963.1361

48 216 Taeback 1985 2007 23 376611.5448 409468.408449 221 Jecheon 1973 2007 35 291050.0770 404473.6873

50 226 Boeun 1973 2007 35 265710.8250 332092.2665

51 232 Cheonan 1973 2007 35 210566.2477 364577.1789

52 235 Boryeong 1973 2007 35 160192.1333 314136.2998

53 236 Buyeo 1973 2007 35 192812.3510 307961.1641

54 238 Geumsan 1973 2007 35 243307.4232 289563.1791

55 243 Buan 1973 2007 35 174284.8680 247758.8935

56 244 Imsil 1973 2007 35 225800.7405 234746.7007

57 245 Jeongeup 1973 2007 35 187789.2871 229270.5808

58 247 namwon 1973 2007 35 230187.7254 211806.9945

59 248 Jangsu 1988 2007 20 247040.2209 239794.9323

60 256 suncheon 1973 2007 35 221740.5489 175126.6100

61 260 Jangheung 1973 2007 35 192554.5186 132252.6687

62 261 Haenam 1973 2007 35 160368.2896 117347.7116

63 262 Goheung 1973 2007 35 225217.3329 124464.0061

64 271 Bonghwa 1988 2007 20 370473.8296 384158.2743

65 272 Yeongju 1973 2007 35 335179.0246 375556.2048

66 273 Mungyeong 1973 2007 35 302681.8413 347955.6113

67 277 Yeongdeok 1973 2007 35 415702.0564 339940.2625

68 278 Uiseong 1973 2007 35 351495.3460 318565.7020

69 279 Gumi 1973 2007 35 318803.1841 293030.012770 281 Yeongcheon 1973 2007 35 375934.1279 276987.8510

71 284 Geochang 1973 2007 35 282410.6441 241633.2775

72 285 Hapcheon 1973 2007 35 305984.5469 230093.0786

73 288 Miryang 1973 2007 35 358196.6952 222709.9471

74 289 Sancheong 1973 2007 35 279776.1745 212955.9483

75 294 Geoje 1973 2007 35 346607.9993 155551.8548

76 295 Namhae 1973 2007 35 284687.9352 146830.8019

TM : Transverse Mercator System



31

At-site frequency analysis result was selected as to objective-value of SGA and NSGA-II.

Then, as a first step, at-site frequency analysis was performed at 72 sites recorded by KMA. It

is for this reason why analysis site should have record length more than 10 year for performing

at-site frequency analysis. Quantiles were computed about return periods (2, 3, 5, 10, 20, 30,

50, 70, 80, 100, 200, 300, 500 year) and duration (60, 120, 180, 360, 540, 720, 900, 1080,

1440, 2880 min). Gumbel distribution and GEV distribution was selected as the proper

probability distribution for the annual maximum rainfall by the goodness of fit test such as

Kolmogorov-Smirnov test,

2

-test, Cramer von Mises test, and Probability Plot Correlation

Coefficient(PPCC) test. And method of probability weighted moments (PWM) was selected as

to parameter estimation method.

In order to at-site frequency analysis, Frequency Analysis of Rainfall Data 2006

(FARD2006) program has been used (NIDP, 2007). FARD2006 was developed by National

Institute for Disaster Prevention (NIDP) and can be freely downloaded from

http://www.nidp.go.kr.

4.2 Application of IDF Curve using Cumulative Distribution

Function

In this section, IDF curve using CDF is applied for Korean rainfall data to assess

applicable suitability. Equation form of IDF curve using CDF is Eq. (2.13) which was

proposed by Koutsoyiannis et al. (1998) (see chapter 2.4). Mostly selected probability

distributions are GEV and Gumbel distribution in Korea. Where probability distribution is

GEV and Gumbel, IDF curve using CDF is



32

1

ln 1 1

( ): GEV

( ) ( )

T

a Ti

b d d

+

= =+

(4.1)

1ln ln 1

( ): Gumbel

( ) ( )

Ta Ti

b d d

= =

+

(4.2)

where, , , , , are the parameters of IDF curve using CDF for the each probability

distribution function. Robust estimation (ROBUST) and One-step least squares method

(ONESTEP) are performed to estimate the parameters (see sections 3.2 and 3.3); in order to

efficiently minimize KWk and e , especially genetic algorithms was used instead of using

trial-and-error method based on a common spreadsheet computer program. Also, the

parameters of IDF curve using CDF were estimated by SGA to improve accuracy; in this study,

at-site frequency analysis result until 2007 was selected as objective-value and RMSE and

RRMSE were seleted as objective-functions. The parameters of 72 rainfall recording sites of

KMA were estimated using SGA and IDF curve using CDF. In this study, the case of using

RMSE as objective-function called IDF_CDF(OF_I) and the case of using RRMSE as

objective-function called IDF_CDF(OF_II).

4.3. Parameter Estimation of Conventional IDF Curves

Rainfall quantiles corresponding to any specific return period and rainfall duration are

essential for constructing hydraulic structures. In Korea, FARD2006 for computing rainfall

quantiles using at-site frequency analysis is already developed, but IDF curve is still very



33

useful to calculate rainfall intensity or quantile for arbitrary rainfall duration in which rainfall

data has not been recorded (Kim et al., 2008b).

The following six IDF curves are widely used in Korea. These IDF curves have been

essentially used to calculate rainfall quantiles for designing hydraulic structures in Korea.

Equations. (4.3), (4.4), (4.5) and (4.6) are Modified Talbot, Modified Sherman, Modified

Japanese and Modified Semi-Log type IDF curves; Equations (4.7) and (4.8) are developed

by Lee et al. (1993), Heo et al. (1999) and Ministry of Construction and Transportation (2000)

( ) :aI t c Modified Talbott b

= ++

(4.3)

( ) :( )b

aI t Modified Sherman

t c=

+ (4.4)

( ) :a

I t c Modified Japaneset b

= +

+ (4.5)

( ) log( ) :I t a b t c Modified Semi Log= + + (4.6)

log( , ) :n

a b TI T t LEEt c

+=+

(4.7)

ln

( , ) :

ln

n

Ta b

tI T t HEOT

c d tt

+

=

+ +

(4.8)

where ( )I t and ( , )I t T are rainfall intensity (mm/hr), t is duration, T (min) is return

period (year), , , , ,a b c d n are the parameters for the each recording site. The latter to IDF

curves make up for defect of the former four IDF curves that dont have return period ( T) in

the IDF formula and show the reverse intensity as rainfall duration increases And then, the

parameters of the former four IDF curves should be estimated by regression analysis

corresponding to each return period. On the contrary, the parameters of the latter two IDF



34

curves can be estimated as one set. Therefore, the latter two IDF curves are more general than

the former four IDF curve.

In this study, in order to compare accuracy and assess applicable suitability of IDF curve

using CDF, the parameters of the six IDF curves at 72 sites of KMA were newly estimated. In

case of Modified Talbot, Modified Sherman, Modified Japanese and Modified Semi-Log, the

parameters were estimated by linear regression analysis using at-site frequency analysis result

until 2007. Moreover, in case of Lee and HEO, the parameters were estimated by SGA

(section 3.4.1); at-site frequency analysis result until 2007 was selected as objective-value and

RMSE and RRMSE as objective-functions. The case of using RMSE as objective-functions

called LEE(OF_I), HEO(OF_I) and the case of using RRMSE as objective-functions called

LEE(OF_II),HEO(OF_II).

4.4. Parameter Estimation using Separation of Short- and Long-

Rainfall Durations Method

IDF curve using CDF was applied for 72 sites of KMA and the parameters were estimated

using SEP_DUR (section 3.4.2) to derive more precise formulas improve accuracy. SEP_DUR

uses two parameter sets for short- and long-rainfall durations, and can be employed

automatically. Object-value was selected at-site frequency analysis result and the parameters of

IDF curve using CDF were estimated using SEP_DUR (IDF_CDF(SEP_DUR)) at the 72

sites of KMA.

In order to compare accuracy, the parameters of the HEO , which is the more accurate IDF

curve among the conventional IDF curves, at 72 sites of KMA were newly estimated using the

multiple non-linear regression as conventional method separating rainfall duration



35

(HEO(REG)) (Ministry of Construction and Transportation, 2000). And the parameters of

the HEO were estimated using the SEP_DUR (HEO(SEP_DUR)).



36

Chapter 5. Comparison

5.1 Total Rainfall Duration

In this study, IDF curve using CDF was applied to Korean rainfall data and the parameters

IDF curve using CDF and conventional IDF curves were estimated using linear regression,

ROBUST, ONESTEP, and SGA. Modified Talbot, Modified Sherman, Modified Japanese and

Modified Semi-Log were estimated using linear regression analysis using at-site frequency

analysis result until 2007, and then LEE and HEO were estimated using SGA; in this study, at-

site frequency analysis result until 2007 was selected as objective-value and RMSE(OF-1),

RRMSE(OF-2) as objective-functions. Lastly, the parameters of IDF curve using CDF were

estimated using ROBUST, ONESTEP, and SGA.

Intensity of each IDF curve was calculated, and at-site frequency analysis result was

transformed into intensity to compare the accuracy. Then RMSE and RRMSE between

estimated intensity ( jli ) and transformed intensity ( jli ) were calculated. Where iji is the

estimated rainfall intensity by IDF curve with i -th duration and j -th return period, and iji is

the computed rainfall intensity by at-site frequency analysis software (FARD2006)

corresponding to i -th duration and j -th return period.

Table 5. 1 shows result of accuracy comparison (details in Table A. 1 and Table A. 2 of

Appendix A). It was performed to search method having smallest RMSE or RRMSE

corresponding to each site and to count number of smallest RMSE or RRMSE corresponding

to each method. And then it decided that method numerous number of smallest RMSE or

RRMSE is accurate.

The bold italic dark gray figures mean the 1staccurate method, the bold italic light gray

figures mean the 2ndaccurate one and the bold italic figures mean the 3rdaccurate one. Table 5.



37

1 shows that Modified Sherman is the 1staccurate IDF curve for both the Gumbel and GEV

models. And, IDF_CDF(OF_II) is the 2nd and HEO (OF_II) is the 3rd accurate one for the

Gumbel model while HEO (OF_II) is the 2ndand IDF_CDF(OF_II) is the 3rdaccurate one for

the GEV model.

Table 5. 1 Number of the smallest RMSE or RRMSE of total sites using 12 different IDFcurves in case of Gumbel and GEV

IDF curves and Parameter estimation methods

LEE HEO IDF_CDFProbability

distributionAccuracy

Modified

TALBOTModifiedSHERMAN

ModifiedJAPANESE

Modified

SEMILOG OF_I OF_II OF_I OF_II

ONE

STEP ROBUSTOF_I OF_II

Total 0 106 0 2 0 0 0 9 0 0 1 24

RMSE 0 44 0 1 0 0 0 8 0 0 1 17Gumbel

RRMSE 0 62 0 1 0 0 0 1 0 0 0 7

Total 0 119 0 1 0 0 0 15 0 0 0 7

RMSE 0 53 0 1 0 0 0 12 0 0 0 4GEV

RRMSE 0 66 0 0 0 0 0 3 0 0 0 3

Modified Talbot, Modified Sherman, Modified Japanese and Modified Semi-Log dont

have return period ( T) in the IDF formula. Therefore, these IDF curves should be estimated

by regression analysis corresponding to each return period. Then, these four IDF curves are not

general and useful. And alse, the four IDF curves show the reverse intensity as rainfall duration

increases. Figure 5. 1 and Figure 5. 2 well shows the one. Figure 5. 1 is a figure which is IDF

curve in case of Modified Sherman at site code number 130 (Uljin). And Figure 5. 2 represents

Figure 5. 1 in detail. Figure 5. 2 shows that a dotted line corresponding to the return period 70

year reverses intensity a bold solid line corresponding to the return period 80 year as rainfall

duration increases between duration 150 min and duration 500 min.



38

10 100 1000 10000Duration (min)

1

10

100

RainfallIntensity

(mm)

1HR

2HR

3HR

6HR

9HR

12HR15HR18HR

24HR

48HR

Return Period

500 300 200 100 80 70 50 30 10 5 3 2

Sherman

000130

Figure 5. 1 IDF curve in case of Modified Sherman at site code number 130 (Uljin)

100 1000

Duration (min)

50

40

30

20

RainfallIntensity

(mm)

2HR

3HR

6HR

9HR

12HR

15HR

Return Period 500 300 200 100 80 70 50 30 10 5

3 2

Sherman

000130

Figure 5. 2 IDF curve represented Figure 5. 1in detail at site code number 130 (Uljin)



39

These results lead us to the conclusion the four IDF curves should not be compared with

other IDF curves, therefore the four IDF curves should be excluded out of accuracy

comparison. Table 5. 2 shows to compare the accuracy of the rest one (details in Table A. 3

and Table A. 4 of Appendix A).

Table 5. 2 Number of the smallest RMSE or RRMSE of total sites using 8 different IDF curvesin case of Gumbel and GEV

IDF curves and Parameter estimation methods

LEE HEO IDF_CDFProbability


OF_I OF_II OF_I OF_II

ONE

STEP ROBUST OF_I OF_II

Total 1 9 0 22 0 0 3 109

RMSE 1 5 0 13 0 0 2 51Gumbel

RRMSE 0 4 0 9 0 0 1 58

Total 0 16 1 57 0 0 1 69

RMSE 0 14 1 31 0 0 1 25GEV

RRMSE 0 2 0 26 0 0 0 44

Table 5. 2 shows that the bold italic dark gray figures mean the 1

st

accurate method, The

bold italic light gray figures mean the 2ndaccurate one and the bold italic figures mean the 3 rd

accurate one. IDF_CDF(OF_II) is a most accurate IDF curve for both the Gumbel and GEV

models. And, HEO (OF_II) is the 2ndaccurate one and LEE (OF_II) is the 3rdaccurate one.

5.2 Separation of Rainfall Duration

In order to derive more precise formulas, the rainfall durations were separated into two

such as short- and long-rainfall durations and the parameters of IDF curves were estimated for

separation of rainfall durations. The parameters of HEO were estimated using multiple non-

linear regression analysis and SEP_DUR. And, the parameters of IDF curve using CDF were

estimated by SEP_DUR.



40

Therefore, accuracy of each method was compared using RMSE and RRMSE between

estimated results and results of at-site frequency analysis. Table 5. 3 shows accuracy

comparison of the three methods (details in Table A. 5 and Table A. 6 of Appendix A. Table A.

7 and Table A. 8 of Appendix A show the parameters of IDF_CDF(SEP_DUR)).

Table 5. 3 Number of the smallest RMSE or RRMSE of total sites using 3 different IDF curvesin case of Gumbel and GEV

IDF curves and Parameter estimation methodsProbability


HEO(REG) HEO(SEP_DUR) IDF_CDF(SEP_DUR)

Total 5 18 121

RMSE 5 9 58Gumbel

RRMSE 0 9 63

Total 16 27 101

RMSE 14 15 43GEV

RRMSE 2 12 58

Table 5. 3 shows comparison of RMSE and RRMSE. The bold italic dark gray figures

mean the 1staccurate method, the bold italic light gray figures mean the 2

ndaccurate one and

the bold italic figures mean the 3rd

accurate one. Table 5. 3 shows that IDF_CDF(SEP_DUR)

is a most accurate IDF curve for both the Gumbel and GEV models and HEO (SEP_DUR) is

the 2nd

accurate one and HEO (REG) is the 3rd

accurate one.



41

Chapter 6. Conclusions

In this study, IDF curve using CDF was applied to Korean rainfall data. And the

parameters of conventional IDF curves and IDF curve using CDF were estimated using linear

regression, non-linear regression, ROBUST, ONESTEP, SGA, and MOGA to improve

accuracy. And then, the accuracy of each method was compared each other. The obtained

conclusions are as follows.

Accuracy of each IDF curve was compared using RMSE and RRMSE between estimated

results and results of at-site frequency analysis. As the results, it was found that Modified

Sherman was the 1st accurate IDF curve for both the Gumbel and GEV models. And,

IDF_CDF (OF_II) was the 2nd and HEO (OF_II) was the 3rd accurate one for the Gumbel

model while HEO (OF_II) was the 2ndand IDF_CDF (OF_II) was the 3 rdaccurate one for the

GEV model.

Modified Talbot, Modified Sherman, Modified Japanese and Modified Semi-Log should

be estimated by regression analysis corresponding to each return period. And these four IDF

curves show the reverse intensity as rainfall duration increases. Therefore, accuracy of the rest

of the IDF curves excluding these IDF curves was compared. As the results, IDF_CDF (OF_II)

was the most accurate IDF curve for both the Gumbel and GEV models, HEO (OF_II) was the

2ndone and LEE (OF_II) was the 3rdone.

In order to derive more precise formulas, the rainfall durations were separated into two

such as short- and long-rainfall durations and the parameters

(1)applicationofintensity duration...

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