Advances in Hydro-Science and Engineering, vol. VIII
Proc. of the 8th International Conference on Hydro-Science and Engineering, Nagoya, Japan, Sept. 9-12, 2008

ESTIMATION OF INTENSITY-DURATION-AREA-FREQUENCYCURVES USING SCALING PROPERTIES OF HOURLYRAINFALL DATALe Minh NHAT1, Yasuto Tachikawa2, Takahiro Sayama3 and Kaoru Takara4
1

Doctoral Student, Department of Urban and Environmental Engineering, Kyoto University
Nishikyo-ku, Kyoto, 615-8540, Japan, nhat@flood.dpri.kyoto-u.ac.jp
2
Associate Professor, Department of Urban and Environmental Engineering, Kyoto University
Nishikyo-ku, Kyoto, 615-8540, Japan, tachikawa@mbox.kudpc.kyoto-u.ac.jp
3
Assistant Professor, Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji, Kyoto, 611-0011, Japan, sayama@flood.dpri.kyoto-u.ac.jp
4
Professor, Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji, Kyoto, 611-0011, Japan, takara@flood.dpri.kyoto-u.ac.jp

ABSTRACT
Many hydrological and meteorological applications require knowledge about spatial
and temporal variabilities of rainfall over an area. The intensity of point precipitation is only
applicable for relatively small areas. For larger areas, design rainfall needs to be converted toaverage areal depths. Areal Reduction Factors (ARFs) have been commonly used to obtain
this transformation. To estimate ARFs at sparely gauged basins, to derive the IntensityDuration-Area-Frequency (IDAF), it is essential to incorporate the scaling properties ofrainfall in time and space.
The IDAF curves are determined for the evaluation of design rainfall using a scaling
approach. The variabilities of annual maximum rainfall intensity in area and duration are
represented through the scaling properties in time and space. Thus the scaling relationships ofmean rainfall intensity with area and duration are derived using the concepts of scaling
properties. Using Generalized Extreme Value (GEV) the authors obtain a scaling model togive the IDAF curves of extreme rainfall. An application is made to the Yodo River
catchment of Japan.
Keywords: extremes rainfall, scale-invariance, IDF curves, ARF, design rainfall
1.

INTRODUCTION

Many hydrological and meteorological applications require knowledge about spatial
and temporal variabilities of rainfall over an area. The intensity of point precipitation is only
applicable for relatively small areas. For larger areas, design rainfall needs to be converted toaverage areal depths. Areal Reduction Factors (ARFs) have been commonly used to obtain
this transformation. To estimate ARFs at sparely gauged basins, to derive the IntensityDuration-Area-Frequency (IDAF), it is essential to incorporate the scaling properties ofrainfall in time and space.
In the literature, two different types of ARFs are found (Omolayo, 1993; Sivapalan
and Blöschl, 1998): The storm-centered ARFs and the fixed-area ARFs. The storm-centered
ARFs are associated with rainfall intensity within the rainfall isohyets of specific storm
events, they represent the ratio of average storm depths over an area (defined by rainfall
isohyets) and the maximum rainfall depths for the storm (at storm centred). The stormcentered ARFs are used more commonly in PMP (probable maximum flood) estimation. The
fixed-area ARFs relate rainfall estimation at point to the average over catchment which isfixed in space. They are estimated by constructing from all available rainfall data at station,
the time series of catchment average rainfall, performing the same types of extreme value
analyses described above for constructing point IDF curves, and finally relating the catchment
rainfall intensities to the point values, for the same return period and duration. This study isconcerned with estimating fixed area ARFs.
In hydrological risk analysis and design, one is often interested in the rainfall intensity
averaged over a region of area A and duration D, with return period T. Plotting such extreme
rainfall intensity I(A, D, T) against D for given A and T produces so-called Intensity Duration
Area Frequency (IDAF) curves. For A → 0 (precipitation at a point), the IDAF curves reduceto the familiar Intensity Duration Frequency (IDF) curves.
For a given location, the ARFs can be defined as ratio between the mean rainfall
intensity I(T,A) and that of point rainfall intensity.
ARF ( D, A) =I ( D, A) IDAF=
I ( D,0)IDF(1)Some studies have derived the properties of the IDAF curves and the ARFs using nonscaling representations of rainfall. An early attempt in this direction was made by RodriguezIturbe and Mejia, 1974 approach by assuming that the rainfall field is a zero mean stationary
Gaussian process. A different approach to ARF estimation, based on crossing properties ofrandom fields, was proposed by Bacchi and Ranzi (1996). Properties of extremes of randomfunctions were used also by Sivapalan and Blöschl (1998). Finally, Asquith and Famiglietti(2000) derived the ARF as the catchment average of the ratios between the T-year rainfall
depths at distance r from the centroid of the storm and at the centroid itself.
In recent years, the concepts of rainfall scale-invariance have come to the fore in bothmodeling and data analysis in hydrological precipitation research (Gupta and Waymire, 1990;
Nguyen et al., 2002). It opens a new approach to developing a formulation of IDAF. A few
studies have assumed that rainfall intensity has scale invariance and used that analysis toderive scaling properties rainfall (e.g., De Michele et al., 2001; Nhat et al., 2007), with thisapproach, this study is to deal with the question how the rainfall properties at a point scale
linked with areal rainfall in terms of time and space. Scaling properties of extreme rainfall intime and space explored for either disaggregation of rainfall intensity from low to highresolution time scale or aggregation from point to area in spatial scale. Consequently, basedon the scale invariance in time and space of rainfall characteristics, the IDAF curves derived.
The IDAF curves are determined for the evaluation of design rainfall using a scaling
approach. The variability of annual maximum rainfall intensity in area and duration are
represented through the scaling properties in time and space. Thus, the scaling relationships ofmean rainfall intensity with area and duration are derived using the concepts of scaling
properties. Using Extreme Value type 1 (EV1) distribution, the authors obtain a scaling model
to give the IDAF curves of extreme rainfall. An application is made to the Yodo River
catchment of Japan. The approach is expected to be more useful and practical to evaluate
design rainfall for a specified area.

2.
2.1

METHODOLOGY
Scale invariance properties of rainfall in time

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

− Hd

(3)

ID

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

D
d

dist

I d = λd − H d I λd d

(4)

The Equation (4) is rewritten in terms of the moments of order q about the origin,denoted by E ⎡⎣( I d ) q ⎤⎦ . The resulting expression is:
E ⎡⎣( I d ) q ⎤⎦ = λd − H d q E ⎡⎣( I λd d ) q ⎤⎦(5)

If one assumes the wide sense simple scaling exists, the scale-invariant models enable
us to transform data from one temporal to another one, and thus, help to overcome thedifficulty of inadequate. The data distribution of IDF for short-duration of rainfall intensity
can be derived from daily rainfall.
2.2

Scale invariance properties of rainfall in space

Let consider a continuous precipitation process I(d,a) which represents the maximum
annual value of average rainfall intensity over a duration d, and an area a. It is defined as⎡ 1 t +d / 2⎤
I ( d ,a ) = max ⎢ ∫X (ξ , ω )dω dξ ⎥∫0 ≤t ≤1 year da t − d / 2 a
⎣
⎦

(6)

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 a moving average (with area span)in a given year. The concepts are introduced about scaling of the probability distribution of
random functions. A generic random function I(d,a), with fixed duration d, is denoted by thesimple scaling properties of space if it obeys the following relationship:A
I ( d , a ) = ⎛⎜ ⎞⎟
⎝a⎠
dist

− Ha

(7)

I ( d , A)

Defining the space scaling ratio is λa =

A
a

dist

I ( d ,a ) = λa − H a I ( d ,λa a )(8)This also implies that the moments of any order are scale-invariance. The resulting
expression is
E ⎡⎣( I ( d , a ) ) q ⎤⎦ = λa − H a q E ⎡⎣( I ( d ,λa a ) ) q ⎤⎦(9)where q denotes the order of the moment. If one assumes the wide sense simple
scaling exists, the IDAF curves can be derived by developing a general framework based on
the estimation of a common scaling exponent for any frequency level. This approach is used
to derive the distribution of the IDAF where data for the spatial scale of interest does notexist.
2.3

Estimation of Intensity-Duration-Area-Frequency curves

The IDAF curves (with fixed Area) are often fitted to the extreme value type I (EVI)
distribution developed by Gumbel and it is still the most often used distribution by manynational meteorological services in the world to describe rainfall extremes. It will also be usedin this study along with the method of moments. The annual maximum rainfall intensity I(d)has a cumulative probability distribution CDF (Gumbel, 1958), which is given by1= exp[− exp{− [ I (d ) − μ ] / σ }]TI (d ) = μ − σ ln(− ln T )F [ I (d )] = 1 −(10)where the location parameter μ and scale parameter σ to be calculated from data series
based on L moment method .
According to the scaling theory, the IDAF formula can be derived (Nhat et al., 2007)withμ * +σ *[ − ln( − ln(1 − 1/ T )]⎧⎪ I d ,T =−Hd⎨
⎪ μ * = μ( λ d ) .(λ ) − H ; σ = σ ( λ d ) .(λ ) − H⎩(11)

It is worthwhile to note that the simple scaling hypothesis leads to the equality
between the scale factor and the exponent in the expression relating rainfall intensity andduration. The IDAF relationship can be derived from longer duration data series based on
three parameters: scale exponent, the location and scale parameters of EVI distribution.
An application of the scale invariance concept in hydrology is presented for
disaggregation (or downscaling) of rainfall intensity from low resolution (e.g., 1-day or 1hour) down to high resolution (e.g., 1-hour or 10-min).

Figure 1 Derivation of intensity-duration-frequency curves and areal reduction factor.
In the other words, statistical properties of rainfall for short durations can be inferred
form those of rainfall data available for longer duration. In principle, the statistical properties(e.g., moments) of the scaling relationship could be obtained from large scales and then used
to estimate the process properties at smaller scales as will be illustrated in the derivation of
scaling IDAF curves.
The IDAF curves and ARF reflect the variability of rainfall in time and space, thus itis necessary to make joint analysis of scaling properties of the rainfall field in duration andarea (Figure 1)

3.

APPLICATION

For time scaling, the simple scaling theory can be applied to derive IDAF curves
consistent with hourly rainfall series in rain gauges where only daily data are available. These
curves are developed for gauged sites based on scaling of the generalized extreme value(GEV) and Gumbel probability distributions. Statistical analysis was performed on annual
maximum rainfall series for the Yodo River catchment for durations ranging from 1 hour to
24 hours. The results showed that rainfall does follow a simple scaling process in time (Nhatet al., 2007). To investigate the scaling invariance behaviour of the spatial rainfall
representation, analyses of the statistical moment are made, similar as for the time series. At
first hourly 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 catchment are arranged. To obtainspatial mean rainfall intensity, the Otorii station is put at the center of the area, and the area is
expanded in the form of circle with increased radius by 0.5 km (see Figure 2). 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. As previously mentioned, oneof these describes the variation of statistical moments with scale asE ⎡⎣( I ( d ,a ) ) q ⎤⎦ = λa K ( q ) E ⎡⎣( I ( d ,λa a ) ) q ⎤⎦(12)

where E ⎡⎣( I ( d ,a ) ) q ⎤⎦ is the ensemble average qth moment of the field studied at (i.e. averagedover) a scale specified by scale ratio λa.

Figure 2 Spatial mean area rainfall intensity expanded in the form of circle.
1.00E+08
1.00E+07

M oment E[I(6,A)]

1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01a)
1.00E+00
1

10

100

1000

10000

Area (km 2)

Figure 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.
Figure 3a show the qth statistical moment, E ⎡⎣( I ( d ,a ) ) q ⎤⎦ , fixed d=6hour, as function of

λa for one typical example of each investigated rainfall scale invariance. Results are shown fororder of q=1, 2, 3, 4, and 5. The straight-line behavior is well respected between
approximately λa=1 (A =2.25 km2) and λa= 2889 (A~ 5000 km2), with mean R2 of the fitted
regression lines for all q equal to 0.99 (0.96-1.0).
In Figure 3b, the difference of the degree of steepness in the slopes for the area (2.25
km2 to 1000 km2) and larger area (1000 km2 to 5000 km2) are found, which indicates that two
different scaling space regimes exist for rainfall space scaling. A steeper slope is found in thesmaller area compared to the lager area. The plots indicate that the relationships between
moments and areas are linear having two different slopes with a breaking point at 1000 km2.
This property suggests an existence of the two different regimes with a transition in storm
dynamics from high variability convective storms with less than 1000 km2 to frontal storms inan area larger than 1000 km2. The rainfall intensity-duration-area-frequency (IDAF) can bederived from a small area to a larger area based on space scaling present in Figure 4.

Figure 4 The IADF curves constructed by the scaling method and comparison
with traditional method for the A=569 km2
For comparison purposes, empirical IDAF curves for AMRI rainfalls are also
constructed using the traditional method by which the extreme rainfall quantiles were
estimated for each duration independently using the observed rainfall data for that particularduration and area; that is, without considering the scaling properties of rainfall series for otherdurations. Figure 4 show the IDAF curves constructed by the scaling method approach
together with those given by traditional method for the A=569 km2. It can be seen that bothmethods produce IDAF curves that are similar in terms of both the shape and rainfall intensity
estimates.
Hence, it can be concluded that the proposed scaling method could provide IDAF
curves that are comparable to those derived by traditional method. However, the scaling
approach was found to be more efficient since it can provide extreme rainfall estimates for all
rainfall duration simultaneously and using a smaller number of parameters as well, while thetraditional method requires the estimation of rainfall estimates for each duration,
independently. Another interesting feature of the proposed scaling procedure is its ability toderive IDAF curves for rainfall durations that have not been observed or measured. The
traditional method can not be used for cases without data.
We adopt a statistical analysis to obtain the Areal Reduction Factor (ARFs) based on
its scaling properties in space and time. The concepts of the statistical scaling are used to
study the variability of a random process in time and space. The approach is expected to be
more useful and practical to evaluate design rainfall for a specified area. The figure 5 shows
ARF against some of most important empirical ARF-are relationship found in the literature.
- Leclerc - Schaake (1972)
ARF = 1- exp(-1.1d 0.25 ) + exp(-1.1d 0.25 - 0.003863 A)- Horton ' s Re lationship (2003)(13)

ARF = exp(-0.09* A0.23 )It is evident that all of the empirical curves underestimate the ARF for large areas (A)and higher duration (D). Also, it is possible to note that empirical curves overestimate the
ARF for small area (A).

Figure 5 The areal reduction factor derived based on the scale invariance of rainfall

4.

CONCLUSION

The properties of scale invariance of rainfall quantiles are examined in the Yodo River
catchment: For space scaling, a simple scaling process with the two different regimes forareas less than 1000 km2 and more than 1000 km2 are found. For time scaling with fixed A,
rainfall properties follow a simple scaling. The rainfall IDAF curves for short duration(hourly) were derived from daily rainfall data. The IDAF relationships are deduced from daily
rainfall, which show good results in comparison with IDAF curves obtained from at-site
short-duration rainfall data. The Areal Reduction Factor (ARFs) was derived based on scaling
properties of rainfall.
Results of this study are of significant practical importance because statistical rainfall
inferences can be made with the use of a simple scaling in time and space. Furthermore, daily
data are more widely available from standard rain gauge measurements, but data for shorter
durations are often not available for the required site.

ACKNOWLEDGMENTS

The authors would like to thank the members of Innovative Disaster Prevention
Technology and Policy Research Lab., Research Division of Disaster Management for Safeand Secure Society, Disaster Prevention Research Institute, Kyoto University for their helpful
discussions and supports.

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