京 都 大 学 防 災 研 究 所 年 報 第 50 号 B 平 成 19 年 4 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 50 B, 2007

Regional Rainfall Intensity-Duration-Frequency Relationships
For Ungauged Catchments Based on Scaling Properties

Le MINH NHAT*, Yasuto TACHIKAWA*, Takahiro SAYAMA and Kaoru TAKARA*Graduate school of Urban and Environment Engineering, Kyoto University
Synopsis
This study developed regional Intensity–Duration–Frequency (IDF) relationship for
ungauged locations based on the scaling theory. The scaling properties of extreme rainfall
are examined to establish scaling relationship behavior of statistical moments over different
durations. The results show that a rainfall property in time does follow a simple scaling
process. A scale invariance concept is explored for disaggregation (or downscaling) ofrainfall intensity from low to high resolution and is applied to the derivation of scaling IDF
curves. These curves are developed for gauged sites based on scaling of the Extreme Value
type 1 (EV1) or Gumbel probability distributions. The IDF relationships are deduced fromdaily rainfall with the scaling approach, which shows good results in comparison with the
IDF curves obtained from traditional techniques. The spatial distribution maps of threeparameters: the scaling exponent and two statistical parameters are constructed. These maps
are used to derive the rainfall intensity duration frequency at ungauged sites.
Keywords: Rainfall Intensity-Duration-Frequency relationship (IDF), Scaling invariance,
Regional frequency analysis, Design rainfall.

1. Introduction
The Intensity Duration Frequency (IDF)
relationship of heavy storms is one of the mostimportant hydrologic tools utilized by engineers fordesigning flood alleviation and drainage structures inurban and rural areas. Local IDF equations are often
estimated on the basis of records of intensities
abstracted from rainfall depths of different durations,observed at a given recording rainfall gauging station.
In some regions, there may exist a number ofrecording rainfall gauging stations operating for atime period sufficiently long to yield a reliable
estimation of the IDF relationships; in many otherregions, especially in developing countries, however,these stations are either non-existent or their sample
sizes are too small. Because daily precipitation data is
the most accessible and abundant source of rainfall

information, it seems natural, at least for the regions
where data at higher time resolution are scarce, todevelop and apply methods to derive the IDF
characteristics of short-duration events from daily
rainfall statistics.
There have been considerable attentions andresearches on the IDF relationship: Hershfield (1961)
developed various rainfall contour maps to provide
the design rain depths for various return periods anddurations. Bell (1969) proposed a generalized IDF
formula using the one hour, 10 years rainfall depths;
P110, as an index. Chen (1983) further developed ageneralized IDF formula for any location in the
United States using three base rainfall depths: P110,
P2410, P1100, which describe the geographical variationof rainfall. Kouthyari and Garde (1992) presented a
relationship between rainfall intensity and P242 forIndia. Koutsoyiannis et al. (1998) cited that the IDF－ 33 －

relationship is a mathematical relationship between
the rainfall intensity I the duration D and the return
period T. Sivapalan and Bloschl (1998) proposed an
approach for constricting catchments IDF curves
based on the spatial correlation structure of rainfall.
Nhat et al. (2006a and 2006b) estimated IDF curves,
constructed the parameter contour maps of IDF
functions and generalized IDF formulas for monsoon
area of Vietnam, which allow incorporating data fromrecording stations for constructing the IDF curves atungauged sites.
Over the last two decades, concepts of scale
invariance have come to the fore in both modelingand data analysis in hydrological precipitation
research. Gupta and Waymire (1990) studied rainfall
spatial variability by introducing the concepts ofsimple and multiple scaling to characterize theprobabilistic structure of the precipitation processes.
Burlando and Rosso (1996) showed that both thesimple scaling and multiscaling lognormal models
can be used to derive Depth Duration Frequency(DDF) curves of point precipitation. Van-Nguyen, T.
V. et al. (1998) proposed a Generalized Extreme
Value distribution model for regional estimation ofshort duration rainfall extremes base on the scaling
theory. Menabde et al. (1999) developed a simple
scaling methodology to use daily rainfall statistics toinfer the IDF curves for rainfall duration less than 1
day. The scaling hypothesis was verified by fitting themodel to two different sets of data (from Australia
and South Africa). Pao-shan, et al. (2004) is an
example of methodology in which the theories ofscaling properties and employed to infer the IDF
characteristics of short-duration rainfall from daily
data. Kuzuha et al. (2005) showed the scaling
framework and regional flood frequency analysis.
Nhat et al. (2006c and 2007) showed the existence ofthe simple scaling in time and space.
The purpose of this study is to investigate ‘scale
invariance’ or ‘scaling’ properties of rainfall forderivation of IDF relationships. The scaling
properties of extreme rainfall are examined toestablish scaling behavior of statistical moments overdifferent durations. The results show that a rainfall
property in time does follow a simple scaling process.
A scale invariance concept is explored for
disaggregation (or downscaling) of rainfall intensityfrom low to high resolution and is applied to thederivation of scaling IDF curves. These curves aredeveloped for gauged sites based on scaling of theExtreme Value type 1 (EV1) or Gumbel probability
distributions. The IDF relationships are deduced fromdaily rainfall with the scaling approach, which shows
good results in comparison with the IDF curves
obtained from traditional techniques. The spatial
distribution maps of three parameters: the scaling
exponent and two statistical parameters areconstructed. These maps are used to derive therainfall intensity duration frequency at ungauged sites.
2. Definitions
In this section, a general theoretical framework
for the simple scaling is introduced. The scaling or
scale-invariant models enable us to transform datafrom one temporal or spatial model to another one,and thus, help to overcome the difficulty of
inadequate data. A natural process fulfills the simple
scaling property if the underlying probability
distribution of some physical measurements at one
scale is identical to the distribution at another scale,multiplied by a factor that is a power function of theratio of the two scales. The basic theoreticaldevelopment of scaling has been investigated bymany authors including Gupta and Waymire (1990)
and Kuzuha et al. (2005).
The random field rainfall intensity with duration
D, I(D) exhibits a simple scale invariance behavior ifd
I (O D ) O H I ( D )

(1)
dist

holds. The equality “” refers to identicalprobability distributions in both sides of theequations; Ȝ denotes a scale factor and H is a scaling
exponent. Referring for simplicity to a one
dimensional random field I(D). The Equation (1) isalso defined as simple scaling property. From (1)comes that: Moment of the statistical distribution
scale with a simple scaling law.
d qHqq
E[ I ( O D ] O
E[ I ( D )](2)

Where the E[] is expected value operator, q is
the moment order and Ȝ is a scale factor. The random
field I(D) exhibits a simple scale invariance in a wide

－ 34 －

sense if Equation (2) holds. If H is a non-linear
function of q, the I(D) is a general case of multiscaling (see Fig.1).

The moments E[] are plotted on the logarithmic
chart versus the scale Ȝ for different moments’ order q.
The slope function of order moment K(q)= f(q) is
plotted on the linear chart versus the moment order q.
If the resulting graph is a straight line, the field issimple scaling, while in other cases, the multi-scaling
approach has to be considered (Gupta, V. K. andWaymire, E., 1990).
3. Study area and rainfall data

Fig.1 Simple and multiscaling in term of statistical
moments. First step, moments of different orders q
are plotted as function of scale in a log-log plot. Fromthe slope, values of the function K(q) are obtained. If
K(q) is linear, the process is simple scaling. If K(q) is
non-linear, the process is multi-scaling.

The Yodo River Catchments (approximately
7281 km2) was selected as study area. The rainfall
data for analysis herein were collected from 12 hourlyrain gauges of which locations are indicated in Fig. 2.
The name and record length for recording rain gauge
are listed in Table 1. The Annual Maximum Rainfall
Intensity (AMRI) series for various durations,
including 1, 2, 3...24 h; were taken from AutomatedMetrological Data Acquisition System (AMeDAS).

Fig. 2 Study area and rain gauge locations for analysis.

－ 35 －

1

Ieno

1982-1996

2

Itiba

1982-2002

3

Oogawara

1982-2002

4

Ootorii

1982-2002

follows simple scaling or multi-scaling, the scaling
exponent, Hq, was plotted versus the moment, q, as
illustrated in Fig.4 shows the scaling exponent
decreases with the order of moment and a linear
relationship exists between scaling exponents andorders of moment, which implies that the property ofwide sense simple scaling of rainfall intensity exists
in this station. Once again the data were linearly related,

5

Katura1982-2002

with an R2 value of 0.9999 for the Hikone station.

6

Kamo

1982-2001

7

Kouga

1982-2002

1.E+098

Syuuzann

1982-2002

1.E+089Tarao

1982-1997

1.E+07

10

Hikone1982-2002

11

Hirakata

1982-2002

12

Makino1982-2002

Table 1 The stations and length of data.
Name of stations

Length of data

S am ple m om ent of order q

No

HIKONE Station

1.E+06
1.E+05
1.E+04q6=5

1.E+03

q5=4
q4=3
q3=2
q2=1
q1=0.5

1.E+02
1.E+01

4. Scale invariance of short duration data in the
Yodo River Cathment

1.E+001

10100Duration (hour)

The data in the Yodo River area was analyzed toensure that they did indeed conform to the simple
scaling. Annual maximum rainfall intensity series(AMRI) data for Hikone station, with complete
records from 1h to 24h durations, were analyzed.
From Equation (2), the simple scaling in the wide
sense can be expressed as Menabde et al. (1999):
f (q) D

 Hqapplicable at Hikone station in The Yodo River
catchment, Japan.

0

-1

(3)

where f(q) some function of q (the qth moment);
E[Iq(D)] is qth moment of the intensity data (mm/h);
D is the duration (hour) and H is the scale exponent.
The scaling properties of rainfall data wasinvestigated by computing the moment for each
durations and then by examining the log-log plots ofthe moments against their duration. The record
lengths are 21 years and ranged from 1982 to 2002.
The analysis was performed on annual maximum
rainfall series for storm durations from 1 hour to 24
hours, with Ȝ=1, 2...24. For illustration, in Fig. 3 theqth moment of the intensity (mm/h) is plotted against
the duration for the Hikone station located in the
Yodo River Catchment.
It is clear that the data is linearly related, andthere is not a break between the short and long
duration data. In order to determine if the data

Hikone Station

-0.5

K (q)

q
E[ I ( D)]Fig.3 Linear relationships indicate scaling may be

-1.5
-2
-2.5

y = -0.6058x - 0.0103
R2 = 0.9999

-3
-3.5
0

1

2

3

4

5

6

q

Fig.4 Relationship between K(q) and the sample
moment order q.
Therefore, the simple scaling can be assumed in thissituation.

(Nonlinear

data

would

have

indicated

multiscaling.) The slope of this line is the scale exponent
parameter, which has a value of 0.605 for the Hikone
station (Nhat, et al. 2007).

－ 36 －

5. Derivation of IDF for Short Duration
All forms of the generalized IDF relationships
assume that rainfall depth or intensity is inversely
related to the duration of a storm raised to a power, or
scale factor (Chow, V.T., 1988). There are several
commonly used functions found in the literature ofhydrology applications. Koutsoyiannis et al. (1998)
have modified the IDF relationship for a given return
period as particular cases, using the following generalempirical formulaIw(DT )KI PºF (I ) F ª«d
¬ V »¼a(T )b( D )(5)

In Equation (5), b(D) = (D + ș) Ș with ș>0 and
0<Ș<1, whereas a(T) is completely defined by theprobability distribution function of the maximum
rainfall intensity. The form of Equation (5) is
consistent with most IDF empirical equations
estimated for many locations (Kothyari, U. C. andGrade, R.J, 1992): For example (Nhat et al. 2006a)
established the IDF curves for precipitation in the
monsoon area of Vietnam.

(8)Where F(.) is a function independent of D. Under thisform, it can be deduced from Equation (7) that

(4)

where i denotes the rainfall intensity for duration dand w, ș and Ș represent non-negative coefficients. In
fact, these arguments justify the formulation of thefollowing general model for the IDF relationships:i

For many parametric forms, left hand side ofEquation (7) may be expressed in terms of standard
variant, as inPDO  H PO D

(9)

VD

O H V OD(10)

Substituting Equations (9), (10) and (7) into Equation(8) and investing with respect to I, one obtains:

I D,T

PO D (O ) H V O D (O ) H F 1(11/ T )
D H

(11)

By equaling Equation (11) to the general model for
IDF relationship, given by Equation (4), it is easy toverify that

K

H

(12)

T

0

(13)

b( D )

KD(14)

a(T ) POD (O) H  V OD (O) H F 1 (1 1/ T ) (15)The random variable rainfall intensity I(D) for
duration D, has a cumulative probability distribution
CDF, which is given byPr( I ( D ) d I )FD ( I ) 1 

1
T (I )(6)

According to the scaling theory by Menabde et al.,(1999) the scaling property in a strict sense can be
written explicitly using the CDF:

FD (I ) FOD[O H I ]­P  V F 1 (1  1/ T )° I D,TK
D°°
H® P PO D ( O )
°
H
°V V
O D (O )°¯(16)The ȝ and ı are constants. It is worthwhile to notethat the simple scaling hypothesis leads to theequality between the scale factor and the exponent in
the expression relating rainfall intensity and duration.

(7)－ 37 －

80

100 Years return period

70
60

Rainfall intensity (m m /h)Rainfall intensity (m m /h)80

Scaling method

50

Rainfall Intensity

40
30
20
10
0

50 Years return period

70
60

Scaling method

50

Rainfall Intensity

40
30
20
10
0

0

5

10

15
20
Duration(hour)

25

30

0

60

5

10

25

30

60
5 Years return period

10 Years return period
50

Rainfall intensity (m m /h)Rainfall intensity (m m /h)

15
20
Duration(hour)

Scaling method

40

Rainfall Intensity

30
20
10
0

50
Scaling method

40
Rainfall Intensity

30
20
10
0

0

5

10

15
Duration(hour)

20

25

30

0

5

10

15
Duration(hour)

20

25

30

Fig. 5 The Rainfall Intensity Frequency Curves at the Hikone station by the scaling method.
With application of the simple scaling model for
rainfall intensities at the Hikone station, the scale
factor can be estimated H=-0.6058. The IDF
relationship for a short duration rainfall can be
deduced from daily data by Equation (16) with theestimates of ȝD and ıD with D=24h. From 24-hourdata collected at the Hikone recording station, theparameters of statistical can be estimated for the
sample of 21 years of 24 hours annual maximum
rainfall intensity yields :- The location parameter ȝD=3.8724
ıD=1.288- The scale parameter
Back with these estimates and the Gumbel inverse
function, the deduced IDF relationship for thelocation of Hikone may be written as Equation (16)with ȝ=26.52 and ı=8.82,the rainfall IDF can bederived as:

26.52  8.82 ln( ln(1  1/ T ))(17)
D 0.605
The rainfall Intensity Duration Frequency curves
for the Hikone station can be reconstructed by scaling
methods with Equation (17). Figure 5 shows wellmatching between the IDF relationships calculated by
Equation (17) and plots obtained by the historical
data as the return periods from 5 to 100 years. The
I DT

scale factor H along with the statistical parameters ȝand ı may be interpreted as regional climaticcharacteristics.
6.

Regional the scale invariance of rainfall intime and derivation of IDF formulas.

The formula of IDF curves (Equation 16) can bederived from scaling invariance of rainfall. It is afunction of scaling exponent H, and the 2 maps ofstatistical parameters: Location parameter ȝ of 24
hour rainfall and scale parameter ı of 24 hour rainfall.
I DT

f ( H , P,V )(18)

Where T is return periods and D is duration of rainfall
intensity.
The regional IDF formula can be developed
based on the three parameters above. First, the scaling
exponent H needs to be examined for all stations in
the Yodo River catchment. Base on the testing results,
it is likely to conclude where the simple scaling if
existing for areas or regions.
Second, the two parameters of statisticalanalysis: ȝ and ı can be deriving from distribution of
24h ARMI data series.
Using the values of three parameters of all
stations, three maps of spatial distribution are－ 38 －

constructed with the GIS interpolations technique.
The IDF relationship for any point (ungauged) can bederived base on these maps.
To this end, after verification of scaling of IDF
estimates at Hikone stations in the Yodo River
catchment, the assumption of scaling was examinedat a representative sample of stations throughout the
Yodo River catchment. The available data at thesestations are one-hour continuous one which recording
by AMeDAS. An AMRI data set was derived fromthe continuous data, and the first five moments were
calculated for durations of 1hour up to 24 hours. As
with the Hikone data, these moments were plottedagainst duration on a log–log scale to determine
whether scaling could be assumed.

Fig. 6 Relationship between moment of order q andduration (hour) at Ootorii station.

Fig. 7 Relationship between K(q) and the sample
moment order q at Ootorii station.

Fig. 6 shows the scaling of the moments for the
Ootorii station in central of The Yodo River
catchment. The linearity of the moments seen here issimilar to the other 12 stations that were examined,
indicating that for durations greater than 1h, scaling
appears to be applicable throughout The Yodo River
catchment. For each station, the scaling exponent, Hq,

was plotted versus the moment, q, to determine if thesimple scaling is applicable and to estimate the scale
exponent for stations throughout the catchment.
As shown in Fig. 7, the scaling exponent
coefficient for the Ootorii station is found to be 0.55,
with an R2 value of 0.9989. This indicates that thesimple scaling could be assumed at this station
instead of the more complex multi-scaling.
Data from other gauges showed a similar scaling
relationship, indicating that scaling may be applicablein the Yodo River area for the durations considered.
In the Table 2, the results of the scaling exponent
factor H of the 12 stations in the Yodo River
catchment are shown with the high coefficients ofdetermination for each station ranging from 0.98 to 1.
The result indicates a strong validity of the simple
scaling property of the extreme rainfall in time series.
The scaling coefficient for the Ootorii station of
0.55 was typical for the 12 stations examined, with amean value of 0.62 and a standard deviation of 0.07.
However, a few stations had significantly higher
scalar exponents, with the highest being 0.77 (Itiba
station). These stations were generally located on thenorthern part of the Yodo River Catchment, locatedon the mountain, as seen in Fig. 8, indicating that theeffect of the mountain or topography should bestudied further. It can also be seen in Fig. 9 that thescaling coefficients were generally lower in theeastern part of the Yodo River catchments.
Construction of the three maps can beinterpolated by using Geographical Information
System (GIS) ACR-Map with Inverse-Distance
Weighted interpolation method. The Fig.8 to Fig.10
displayed maps of spatial distributions of the scaling
exponent parameters generated by twelve stations thesimple scaling properties in time of rainfall maximum.
The location parameter ȝ and the scaleparameter ı of the EV1 (Gumbel) distribution of 24
hour of AMRI can be derived by statistical analysis.
With the same technical, 2 maps of statisticalparameters can be constructed.
It is expected that those three maps can beapplicable for any ungauged location. Further study isrequired to verify these maps for regional IDF
relationships.

－ 39 －

Table 2 The values scale exponent of 12 stations at Yodo River Catchment.

No
1
2
3
4
5
6
7
8
9
10
11
12

Name of stations
Ieno
Itiba
Oogawara
Ootorii
KaturaKamo
Kouga
Syuuzann
Tarao
HikoneHirakata
MakinoThe scaleexponent(H)
0.668600.770300.550100.55400
0.66130
0.68900
0.551900.52600
0.532100.605800.67751
0.60280

Fig. 8 Spatial distribution maps of scale exponent H.

－ 40 －

Fig. 9 Spatial distribution maps of the statistical parameters: the location parameter (ȝ) of 24h rainfall AMRI.

Fig. 10 Spatial distribution maps of the statistical parameters: the scale parameter (ı) of 24h rainfall AMRI.

－ 41 －

7.

Conclusion

The major findings of the present study can besummarized as follows: The properties of the timescale invariance of rainfall quantiles are examined in
the Yodo River basin: for time scaling, rainfall
properties follows the simple scaling; then, according
to Menabde et al.(1999) the rainfall IDF curves for ashort duration (hourly) were derived from 24-hourdata. The simple scaling properties are verified by
local data; the IDF relationships are deduced fromdaily rainfall, which shows good results incomparison with the IDF curves obtained from at-site
short-duration rainfall data. Results of this study areof significant practical importance because statisticalrainfall inferences can be made with the use of asimple scaling in time. Furthermore, daily data are
more widely available from standard rain gaugemeasurements, but data for short durations are often
not available for the required site.
The hypothesis of piecewise simple scaling
combined with Gumbel distribution was used todevelop the IDF scaling formulas depend on threeparameters: the scaling exponent, the two statisticalparameters: location and scale parameters of 24 hour
rainfall. The spatial distribution maps of theseparameters are used to derive the rainfall intensity
duration frequency at ungauged points which wereinterested for designing analysis.
This study established the regional rainfall
intensity duration frequency relationship for
ungauged catchments based on three spatial
distribution maps from scale invariance of rainfall intime. These analyzed results required validation of
the scaling from daily to hourly data, the simulationsand verifications should be studied further.
Acknowledgments

This study was supported by Grant-in-Aid for
Scientific Research (C) 18560497 (PI: Assoc. Prof. Y.
Tachikawa, Kyoto Univ.).
References

Bell, F.C. (1969). Generalized rainfall duration
frequency relationships, Journal of HydraulicDiv., ASCE, Vol. 95(1), pp.311-327.

Burlando, P., Rosso, R., (1996). Scaling andmultiscaling models of depth–duration–
frequency curves for storm precipitation. J.
Hydrol. 187, pp. 45–64.
Chen, C.L. (1983). Rainfall intensity-duration frequency formulas, Journal of HydraulicEngineering, ASCE, Vol. 109(12), pp.1603-1621.
Chow, V. T. (1964). Handbook of Applied Hydrology,
McGraw-Hill, New York, pp.1-1450.
Chow, V.T., Maidment, D.R. & Mays, L.W. (1988).
Applied Hydrology, McGraw-Hill.
Gupta, V.K., Waymire, E., (1990): Multiscaling
properties of spatial and river flow distributions.
J. Geophys. Res. Vol. 95(D3), pp.1999–2009.
Hershfied David M. (1961). Estimating the Probable
Maximum Precipitation, Journal of theHydraulic Division, Proceeding of the ASCE,
Vol.HY5, pp. 99-116.
Kothyari, U.C. and Grade, R.J. (1992). Rainfall
intensity duration frequency formula for India, J.
Hydr. Engrg., ASCE, 118(2), 323-336.
Koutsoyiannis, D., Manetas, A. (1998). A
mathematical framework for studying rainfall
intensity-duration-frequency
relationships,
Journal of Hydrology, Vol.206, pp.118–135.
Koutsoyiannis, D., Foufoula-Georgiu, E.,(1993): A
scaling model of storm hyetograph. Water
Resour. Res. Vol.29(7), pp.2345–2361.
Kuzuha, Y., Komatsu, Y., Tomosugi, K., Kishii,
T.(2005): Regional Flood Frequency Analysis,
Scaling and PUB, Journal Japan Soc. Hydrol.
and Water Resources Vol.18,No.4, pp. 441-458.
Menabde, M., Seed, A., Pegram, G. (1999): A simple
scaling model for extreme rainfall. Water
Resources Research, Vol. 35, No.1, pp. 335-339.
Nhat, L. M., Tachikawa, Y. and Takaka, K. (2006a):
Establishment of IDF relationships for monsoon
areas, Annual of Disas. Prev. Res. Inst., Kyoto
University, No. 49B, pp.93-103.
Nhat, L. M., Tachikawa, Y., Sayama, T., and Takaka,
K. (2006b): Generalized IDF for Precipitation in
the Monsoon Areas, Proc. of the 7th Inter.
Workshop on Precipitation in Urban Areas, St.
Moritz, Switzeland, 7-10 Dec. 2006, pp.180-184.
Nhat, L. M., Tachikawa, Y., Sayama, T., and Takaka,
K. (2006c): Derivation of rainfall IDF
relationship for short-duration rainfall fromdaily data, Proc. of International Symposium on－ 42 －

Managing Water Supply for Growing Demand,
Bangkok, 16-20, October, 2006, IHP TechnicalDocuments in Hydrology, no. 6, pp. 89-96.
Nhat, L. M., Tachikawa, Y., Sayama, T., and Takaka,
K. (2007): A simple scaling characteristic of
rainfall in time and space to derive intensity
duration frequency relationships, Annual Journalof Hydaulic Engineering, JSCE, Vol.51, pp.7378.
Pao-shan, Y., T.C. and Lin, C. S. (2004): Regional
rainfall intensity formulas based on scaling
property of rainfall, Journal of Hydrology,

Vol.295 (1-4), pp.108-123.
Sivapalan, M., Bloschl, G. (1998): Transformation ofpoint rainfall to areal rainfall: intensity–
duration–frequency curves. J. Hydrol, Vol
204(1/2), pp.150–167.
Van-Nguyen, V. T., Nguyen, T.-D., Ashkar.,
F.(2002): Regional frequency analysis ofextreme rainfalls, Water Sci. Technol. Vol.45(2),pp.75–81.

᷹ⷰߩਇචಽߥᵹၞߩߚ߼ߩࠬࠤ࡯࡞․ᕈߦၮߠߊ㒠᳓ᒝᐲ㧙ᦼ㑆㧙㗫ᐲ㑐ଥ

.GOKPJ0*#6 ࡮┙Ꮉᐽੱ ࡮૒ጊᢘᵗ࡮ቲ 㚌੩ㇺᄢቇᄢቇ㒮Ꮏቇ⎇ⓥ⑼ㇺᏒⅣႺᎿቇኾ᡹ⷐ ᣦ᷹ⷰߩਇචಽߥᵹၞߦ߅ߌࠆ᳓Ꮏ⸳⸘ߩߚ߼ߦ㧘ࠬࠤ࡯࡞․ᕈࠍၮߦߒߚ+PVGPUKV[̄&WTCVKQP̄(TGSWGPE[+&( 㑐ଥࠍ᭴▽ߒߚ‫ޕ‬㒠᳓ߩᭂ୯᳓ᢥ㊂ߩࠬࠤ࡯࡞․ᕈࠍಽᨆߒ㧘⇣ߥࠆ㒠᳓ᦼ㑆ߩᭂ୯᳓ᢥ㊂ߩⓍ₸ߩ㑐ଥࠍ⏕⹺ߒߚ‫ߚ߹ޕ‬ૐᤨ㑆ಽ⸃⢻࠺࡯࠲ࠍ㜞ᤨ㑆ಽ⸃⢻࠺࡯࠲ߦ࠳࠙ࡦࠬࠤ࡯࡝ࡦࠣߔࠆߚ߼ߦࠬࠤ࡯࡞ਇᄌᕈ
߇᭎ᔨ㧔5KORNG5ECNKPI㧕ࠍ↪޿㧘⇣ߥࠆ㒠᳓ᦼ㑆ߦኻߔࠆ+&(ᦛ✢ࠍᓧߚ‫ޕ‬+&(ᦛ✢ߪᭂ୯ಽᏓ㧵ဳ޽ࠆ޿ߪࠟ
ࡦࡌ࡞ಽᏓࠍ߽ߣߦዉ಴ߐࠇࠆ‫ޕ‬ಽᨆߩ⚿ᨐ㧘㒠㔎ߩᭂ୯᳓ᢥ㊂ߪࠪࡦࡊ࡞ࠬࠤ࡯࡝ࡦࠣࠍ↪޿ߡᤨ㑆ࠬࠤ࡯࡞․ᕈࠍ⴫⃻ߔࠆߎߣ߇น⢻ߢ޽ࠆߎߣ߇␜ߐࠇ㧘ࠪࡦࡊ࡞ࠬࠤ࡯࡝ࡦࠣࠍ↪޿ߡផቯߐࠇߚᭂ୯᳓ᢥ㊂ߪ㧘⋥ធ࠺࡯࠲߆ࠄផቯߒߚᭂ୯᳓ᢥ㊂ߣࠃߊኻᔕߔࠆߎߣ߇ࠊ߆ߞߚ‫ޕ‬7PICWIGFߥ࿾ὐߦ߅ߌࠆ㒠㔎ᒝᐲߩ㧵㧰㧲ᦛ✢ࠍផቯߔࠆߚ߼ߩ㧟ߟߩࡄ࡜ࡔ࡯࠲ߩⓨ㑆ಽᏓ࿑ࠍᶰᎹᵹၞࠍኻ⽎ߣߒߡ૞ᚑߒߚ‫ޕ‬ࠠ࡯ࡢ࡯࠼ 㒠㔎+&(㑐ଥ㧘ࠬࠤ࡯࡞ਇᄌᕈ㧘࿾ၞ㗫ᐲ⸃ᨆ㧘⸘↹㒠㔎

－ 43 －