Annual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, February

A SIMPLE SCALING CHARATERISTICS OF RAINFALL
IN TIME AND SPACE TO DERIVE INTENSITYDURATION FREQUENCY RELATIONSHIPSLe Minh NHAT 1, Yasuto TACHIKAWA2, Takahiro SAYAMA3 and Kaoru TAKARA4
1Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 606-8501, Japan) E-mail: nhat@flood.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
3Member of JSCE, M. Eng., Assistant Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
4Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

The properties of time and space scale invariance of rainfall are investigated and applied toIntensity-Duration-Frequency (IDF) relationships. The hypothesis of the simple scaling is examined intime and space in Yodo River basin and the simple scaling properties in time are confirmed, which is used
to derive IDF relationships for short-duration rainfall of several hours from the statistical characteristics
of daily data only. The derived IDF matches well with the one derived from historical observed data. Also,
the simple scaling properties in space are examined. It is found that two ranges less than 1000 km2 andmore than 1000 km2 show different scale properties.
Key Words: rainfall intensity, scaling invariance, IDF curves, 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 structuresin urban and urban areas. Local IDF equations areoften 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 of recording rainfall gauging stations
operating for a time period sufficiently long to yielda reliable estimation of the IDF relationships; inmany other regions, especially in developing
countries, however, these stations are either
non-existent or their sample sizes are too small.
Because daily precipitation data is the mostaccessible 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.
Over the last two decades, concepts of scale
invariance have come to the fore in both modellingand data analysis in hydrological precipitation
research. Gupta and Waymire1) studied rainfall
spatial variability by introducing the concepts ofsimple and multiple scaling to characterise theprobabilistic structure of the precipitation processes.
Kuzuha et al.2) showed the scaling framework andregional flood frequency analysis. Burlando andRosso3) showed that both the simple scaling andmultiscaling lognormal models can be used toderive Depth Duration Frequency (DDF) curves ofpoint precipitation. Menabde et al.4) developed asimple scaling methodology to use daily rainfall
statistics to infer the IDF curves for rainfall duration
less than 1 day. The scaling hypothesis was verified
by fitting the model to two different sets of data(from Australia and South Africa). Yu et al.5) 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.
Until now, time scaling characteristics have beenstudied by many researchers, while space scaling
properties that will link to derivation of
Area-Intensity-Duration-Frequency (AIDF), are notwell studied. Thus, in this paper, the properties oftime and space scale invariance of rainfall are
examined in the Yodo River catchment. Then the
IDF relationships for short-duration rainfall fromdaily data are derived according to Menabde3) andcompared with the obtained from the traditional
method.
The paper is organized as follows: The next

- 73 -

section presents the theoretical background as
related to the time and space scale invarianceproperties of short-duration rainfall. In section 3,
these properties are verified for data observed at
rainfall gauging stations located in the Yodo River
catchment in Japan. The IDF relationship, derived
from daily rainfall data, then compared to the
traditional estimated curve in section 4 andconclusions are given in the last section.

2. SIMPLE SCALING CHARATERISTICS
OF RAINFALL IN TIME AND SPACEIn 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 Waymire1) andKuzuha et al.2)
Let X(t) and X( λ t) denote measurements at two
distinct time or spatial scales t and λt , respectively.
Definition of scaling of the probability of the X(t) is
dist

X (t ) = λ− H X (λt )

(1)

dist

The equality “ = ” refers to identical probability
distributions in both side of the equations, λdenotes a scale factor and H is a scaling exponent.
Gupta and Waymire1) introduced the notions ofstrict and wide sense simple scaling. The strict senseq
simple scaling in Eq.(1) implies that X (t ) and(λ − H X (λ t )) q have the same probability distribution.
The wide sense simple scaling is expressed as they
have the same moments, i.e.

E[X (t )q ] = λ− HqE[X (λt )q ](2)

The scaling exponent, Hq can be estimated from theslope of linear regression relationship between thelog transformed values of moment, log E [X (λt ) q ]and scale parameters log λ for various order ofmoment q. This is definition of a “wide sense”
simple scaling1). A “wide sense” simple scaling with
t=1 and λt = λ , is given by

E[X (1) q ] = λ− Hq E[X (λ) q ](3)

If the scaling exponent H is not constant andchanges probabilistically, Eq.(3) are described as

E [ X (1)q ] = λ − K ( q) E [ X (λ)q ](4)

where K(q) is a function of the moment order. The
procedure to test the suitability of scale invariant
model to describe the rainfall process is briefly
described in Fig.1. The moments E X (λt ) q areplotted on the logarithmic chart versus the scale λfor different moments’ order q. The slope K(q) is
plotted on the linear chart versus the moment orderq. If the resulting graph is a straight line, the field issimple scaling, while in other cases, themulti-scaling approach has to be considered1).

[

]

q5
q4
q3
q2
q1

Log [moment]

Log [scale]

K (q)

K (q)
q
Simple scalingq
Mutil-scaling

Fig.1 Simple and multiscaling in term of statistical moments.
First step, moments of different orders q are plotted as functionof scale in a log-log plot. From the 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 multiscaling.

(1) Time scaling of rainfall intensity
Let the random variable I(d) the maximum
annual value of local rainfall intensity over aduration d. It is defined as:⎡ 1 t +d / 2⎤
I (d ) = max ⎢X (ξ )dξ ⎥∫0≤t ≤1 year ⎣ d⎦
t −d / 2(5)where X (ξ ) is a time continuous stochastic processrepresenting rainfall intensity and d is duration. It is
supposed that I(d) represents the Annual Maximum
Rainfall Intensity (AMRI) of duration d, defined by
the maximum value of moving average of width dof the continuous rainfall process. Here, someconcepts are introduced about scaling of theprobability distribution of random functions. A
generic random function I(d) is denoted by simple
scaling properties if it obeys the following:
dist DI (d ) = ⎛⎜ ⎞⎟
⎝d⎠

− Hd

I ( D)(6)

D is a aggregated time duration, i.e.: 2, 3, ..., 24 hours.
D
Defining the scaling ratio is λd =ddist

I (d ) = λd − H d I (λd d )- 74 -(7)

The Eq.(7) is rewritten in terms of the moments oforder q about the origin, denoted by E [I (d ) q ] . The
resulting expression is:

E [ I (d )q ] = λd − Hd q E [ I (λd d )q ](8)

If one assumes the wide sense simple scaling exists,
the distribution of IDF for short-duration of rainfall
intensity can be derived from daily rainfall.
(2) Space scaling of rainfall intensity
Let consider a continuous precipitation process
I(d,a) which represents the maximum annual valueof average rainfall intensity over a duration d, andan area a. It is defined as

1 t +d / 2(9)I (d , a) = max ⎡⎢ ∫t −d / 2 ∫a X (ξ , ω)dωdξ ⎤⎥0≤t ≤1 year⎣ da⎦where X (ξ , ω ) is a time-space continuous stochastic
process representing rainfall intensity.
Since the probability of extreme events is
usually examined, the maximum annual rainfall
intensity can be defined as the maximum value of amoving average (with area span) in a given year.
The concepts are introduced about scaling of theprobability distribution of random functions. A
generic random function I(d,a),with fixed durationd, is denoted by the simple scaling properties ofspace if it obeys the following relationship:

1.E+09− Ha(10)I (d , A)A
Defining the space scaling ratio is λa =adist

I (d , a ) = λa − H a I (d , λa )(11)This also implies that the moments of any order are
scale-invariance. The resulting expression is

E [ I (d , a) ] = λaq

HIKONE Station

1.E+08Sample moment of order qA
I (d , a ) = ⎛⎜ ⎞⎟
⎝a⎠
dist

plots of the moments against their duration and area.
Fig.2 and Fig.3 illustrate the relationships between
moments versus duration (time) and area (space)and a plot of the scaling exponents versus the orderof moments for the Yodo River catchment.
In order to examine the time scale invariance ofrainfall in the Yodo River catchment, the four
stations Hikone, Hirakata, Ieno and Kamo wereselected from Yodo River catchment. 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 λd =1, 2...24. Fig.2a) shows therelationships between the log-transformed values ofmoment of various orders against values duration at
the Hikone station, the slopes of linear regression
define the scaling exponents Hd . Fig.2b) shows thescaling exponent decreases with the order ofmoment and a linear relationship exists between
scaling exponents and orders of moment, which
implies that the property of wide sense simple
scaling of rainfall intensity exists in this station.
In Table 1, the results of the scaling exponent
factor Hd of the four 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 timeseries.

− Ha qE [ I (d , λa a) ]q

1.E+07
1.E+06
1.E+05
1.E+04q6=5

1.E+03

q5=4

a)

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

1.E+02
1.E+01
1.E+00
1

(12)

10

100

Duration (hour)

where q denotes the order of the moment. If one
assumes the wide sense simple scaling exists, theAIDF curves can be derived by developing a generalframework based on the estimation of a common
scaling exponent for any frequency level. Thisapproach is used to derive the distribution of the
AIDF where data for the spatial scale of interest
does not exist.

3. SCALE INVARIANCE PROPERTIESOF RAINFALL IN TIME AND SPACE IN
YODO RIVER CATCHMENT, JAPANThe scaling properties of rainfall data wasinvestigated by computing the moment for each
duration and area and then by examining the log-log

0
Hikone Station

-0.5

K(q)

-1
-1.5

b)

-2
-2.5

y = -0.6058x - 0.0103
R2 = 0.9999

-3
-3.5
0

1

2

3

4

5

6

q

Fig.2 The time scale invariance at the Hikone station, the Yodocatchment, Japan a) Relationship between moment of order q
and duration (hour) b) Relationship between K(q) and thesample moment order q.

- 75 -

1.00E+08

01.00E+07

y = -0.0474x-0.1

1.00E+06

R2 = 0.99

M o m e n t E [I(6 ,A )]

-0.2
K(q)

1.00E+05
1.00E+04
1.00E+03

A=1000 to 5150.25-0.3

y = -0.1278x
R = 0.99-0.4

A=2.25 to 1000 Km2-0.5

1.00E+02
1.00E+01

2

2

-0.6

a)

b)

-0.7

1.00E+00
1

10

100

1000

10000

Area (km 2)

0

1

2

q

3

4

5

Fig.3 The space scale invariance at the Yodo River catchment for 6 hours. a) Relationship between
moment of order q and area A (km2). b) Relationship between K(q) and the sample moment order q.
Table 1. The scaling exponent factor of the four stations in the
Yodo River catchment

No of
years
recordScaling
exponent
Hd

R2

Hikone21-0.6050.99

Hirakata

16-0.678

1.00

Ieno

14

-0.681

0.99

Kamo

21

-0.693

1.00

Name ofthe station

To examine the space scaling invariance, at firsthourly data from 1982 to 2002 with 1.5 x 1.5 km
spatial resolution, which is generated by AMeDAS
rain gauge data and covers the Yodo River basin arearranged. To obtain spatial mean rainfall intensity
the Otorii station is put at the center of the area, andthe area is expanded in the form of circle withincreased radius by 0.5 km.
The data used for examined spatial scaling factor
are annual maximum rainfall series for the area from
2.25 km2 to 5150.25 km2, with λa =1,5...2289 and
fixed duration d. In Fig.3 a difference of the degreeof steepness in the slopes for the area (2.25 km2 to1000 km2) and larger area (1000 km2 to 5000 km2)
area is found, which indicates that two different
scaling space regimes exist for rainfall spacescaling. A steeper slope is found in the smaller area
compared to the lager area. The plots indicate thatthe relationships between moments and areas arelinear having two different slopes with a breaking
point at 1000 km2. This property suggests anexistence of the two different regimes with atransition in storm dynamics from high variability
convective storms with less than 1000 km2 to frontal
storms in an area larger than 1000 km2.

4. DERIVATION OF IDF FOR SHORTDURATION
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 factor6). There are several commonly usedfunctions found in the literature of hydrology
applications7),8),9),10). Koutsoyiannis et al.11) havemodified the IDF relationship for a given return
period as particular cases, using the following
general empirical formula
w
(13)
i=
(d + θ )η
where i denotes the rainfall intensity for duration dand w, θ and η represent non-negative coefficients.
In fact, these arguments justify the formulation of
the following general model for the IDF
relationships:i=a (T )b( d )(14)

In Equation (14), b(d) = (d + θ)η with θ>0 and
0<η<1, whereas a(T) is completely defined by theprobability distribution function of the maximum
rainfall intensity. The form of Equation (14) is
consistent with most IDF empirical equations
estimated for many locations12): For example Nhat
et al.13) established the IDF curves for precipitationin the monsoon area of Vietnam.
The random variable rainfall intensity I(d) for
duration d, has a cumulative probability distribution
CDF, which is given byPr( I ( d ) ≤ i ) = Fd (i ) = 1 −

1
T (i )(15)

According to the scaling theory by Menabde et al. 4),
the scaling property in a strict sense can be written
explicitly using the CDF:

- 76 -

(16)For many parametric forms, left hand side ofEquation (16) may be expressed in terms of standard
variant, as in⎡ i − μd ⎤Fd (i) = F ⎢⎥⎣ σd ⎦(17)

where F (.) is a function independent of d. Under
this form, it can be deduced from Equation (16) that(18)μd = λd − H d μλd d

σ d = λd − H d σ λd dfor the Hikone station can be reconstructed byscaling methods with Equation (26). Figure 4 shows
well matching of the IDF relationships calculated by
Equation (26) and plots obtained by the historical
data as the return periods from 5 to 100 years. The
scale factor Hd along with the parameters σ
and μ , may be interpreted as regional climaticcharacteristics.
80
Rainfall intensity (mm/h)

Fd (i) = Fλd d [λd − Hd i](19)

Substituting Equations (17), (18) and (19) into
Equation (15) and investing with respect to i, one
obtains:
id ,T =

μλd d (λd d ) − H d + σ λd d (λd d ) − H d F −1 (1 − 1/ T )− Hd

Rainfall intensity (mm/h)

(24)

30
20

a)

10
5

10

15
Duration(hour)

20

25

30

50 Years return period

70
60

Scaling method

50

Rainfall Intensity

40
30

b)

20
10
0

μ + σF (1 − 1 / T )

0

(25)

dη
d

31.56 − 17.81ln(− ln(1 − 1 / T ))(26)
d 0.605510

15
Duration(hour)

20

25

30

60

are

constants. It is worthwhile to note that the simple
scaling hypothesis leads to the equality between thescale factor and the exponent in the expression
relating rainfall intensity and duration.
Application of the simple scaling model for
rainfall intensities at the Hikone station, the scale
factor can be estimated H d = −0.6058 . The IDF
relationship for a short duration rainfall can be
deduced from daily data by Equation (25) with theestimates of μ D and σ D with D=24h. From
24-hour data collected at the Hikone recording
station, the sample of 21 years of 24 hours annual
maximum rainfall intensity yields, which can be
estimated μ D = 24 = 4 .615 and σ D =24 = 2.604 .
Back with these estimates and the Gumbel inverse
function, the deduced IDF relationship for thelocation of Hikone may be written as Equation (25)with μ = λη μ 24 = 31.56 and σ = λησ 24 = 17.81
then

i=

Rainfall Intensity

40

80

−1

where μ = μλ d (λd d ) − H d and σ = σ λ d (λd d ) − H ddScaling method

500

Rainfall intensity (mm/h)

i d ,T =

60

0

(20)

d
By equaling Equation (20) to the general modelfor IDF relationship, given by Equation (13), it iseasy to verify that(21)
η = −Hd
(22)
θ =0
η
(23)
b( d ) = da(T ) = μλd d (λd d )− Hd + σ λd d (λd d )− Hd F −1 (1 −1/ T )100 Years return period

70

10 Years return period

50

Scaling method

40

Rainfall Intensity

30
20
10
0
0

5

10

15
Duration(hour)

20

25

30

Rainfall intensity (mm/h)

60
5 Years return period

50

Scaling method

40

Rainfall Intensity
(
/ )

30

c)

20
10
0
0

5

10

15
Duration(hour)

20

25

30

Fig.4 The Rainfall Intensity Frequency Curves at the Hikone
station, the Yodo Catchments, Japan by the scaling method.

The rainfall Intensity Duration Frequency curves- 77 -a) Area=110.25km 2

Rainfall intensity(mm/hr)

Rainfall intensity(mm/hr)

70
65
60
55
50
45
40
35
30
25
20

Year ret urn period

0

5

200

100

50

25

10

5

10
15
20
Duration(hour)

25

70
65
60
55
50
45
40
35
30
25
20

b) Area=992.25km 2Year return period

0

30

5

200

100

50

25

10

5

10
15
20
Duration(hour)

2530Fig. 5 The Area Rainfall Intensity Frequency Curves at the Yodo River catchment, Japan by the scaling method.
a)Area=110.25Km2 and b)Area=992.25.25Km2.

The AIDF curves can be derived by the simple
scaling method. For example, with a fixed area
equal to 110.25 km2 and a scaling factor
Hd=-0.51; μ = λη μ 24 = 25.08; σ = λη σ 24 = 8.22
are found, the AIDF curves can be derived by:i=

25.08 − 8.22 ln(− ln(1 − 1 / T ))d 0.514(27)

For an area of 992.25 km2, the IDF curve can beconstructed with a scaling factor Hd= -0.46:

21.09 − 7.35 ln(− ln(1 − 1 / T ))i=d 0.46(28)

The Figure 5 shows a result of the AIDF based on
the scaling method in the Yodo River catchment by
the equations (27) and (28).

5. CONCLUSION
The major findings of the present study can besummarized as follows: The properties of the timeand space scale invariance of rainfall quantiles are
examined in the Yodo River basin: For time scaling,rainfall properties follows a simple scaling; For
space scaling, a simple scaling process with the twodifferent regimes of 2.25 km2 to 1000 km2 and 1000
km2 to 5000 km2 are found.
Then, according to Menabde et al.4) the rainfall
IDF curves for a short duration (hourly) were
derived from 24-hour data. The simple scaling
properties are verified by local data; the IDF
relationships are deduced from daily rainfall, whichshows good results in comparison with the IDF
curves obtained from at-site short-duration rainfall
data.
Results of this study are of significant practicalimportance because statistical rainfall inferences canbe made with the use of a simple scaling in time andspace. Furthermore, daily data are more widely
available from standard rain gauge measurements,
but data for short durations are often not available
for the required site. Further works will focus on a
combination between the time scaling and the spacescaling and theoretical description of the AIDF
applicable for a less gauged basin.
ACKNOWLEDGMENT:This study was supported by Grant-in-Aid for
Scientific Research (C) 18560497 (PI: Assoc. Prof. Y.
Tachikawa, Kyoto Univ.).
REFERENCES
1) Gupta, V. K. and Waymire, E.: Multiscaling properties ofspatial rainfall and river flow distributions. Journal ofGeophysical Research, 95(D3): 1999-2009, 1990.
2) Kuzuha, Y., Komatsu, Y., Tomosugi, K., Kishii, T.:
Regional Flood Frequency Analysis, Scaling and PUB,
Journal Japan Soc. Hydrol. and Water Resources
Vol.18,No.4, pp. 441-458, 2005.
3) Burlando, P. and Rosso, R.: Scaling and multiscaling
models of depth-duration-frequency curves for storm
precipitation. Journal of Hydrology, 187, pp. 45-65, 1996.
4) Menabde, M., Seed, A., Pegram, G.: A simple scaling
model for extreme rainfall. Water Resources Research, Vol.
35, No.1, pp. 335-339, 1999.
5) Yu, P. S., Yang, T.C. and Lin, C. S.: Regional rainfall
intensity formulas based on scaling property of rainfall
Journal of Hydrology, 295 (1-4), pp.108-123, 2004.
6) Chow, V.T., Maidment, D.R. & Mays, L.W: Applied
Hydrology, McGraw-Hill, 1988.
7) Chen, C. L.: Rainfall intensity duration frequency formulas,
Journal of Hydraulic Engineering, ASCE, 109(12),pp.1603-1621, 1983.
8) Hershfied, D. M.: Estimating the Probable Maximum
Precipitation, Journal of the Hydraulic Division, Proceedingof the ASCE, HY5, pp. 99-116, 1961.
9) Van Nguyen, V. T., Nguyen, T.-D., Ashkar, F.: Regionalfrequency analysis of extreme rainfalls, Water Sci. Technol.
45(2), pp.75–81, 2002.
10) Bell, F. C.: Generalized rainfall duration frequency
relationships. Journal of Hydraulic Div., ASCE, 95(1),pp.311-327, 1969.
11) Koutsoyiannis, D., Manetas, A.: A mathematical frameworkfor studying rainfall intensity duration frequency
relationships, Journal of Hydrology, 206, pp.118–135, 1998.
12) Kothyari, U. C. and Grade, R.J.: Rainfall intensity duration
frequency formula for India, J. Hydr. Eng., ASCE, 118(2),pp.323-336, 1992.
13) Nhat, L. M., Tachikawa, Y., Takara, K.: Establishment ofIntensity-Duration-Frequency curves for precipitation in the
monsoon area of Vietnam, Annuals of Disas. Prev. Res.
Inst., Kyoto Univ., No. 49B, 2006.
(Received September 30, 2006)- 78 -