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

EVALUATION OF THE QUANTILES OF THE
NEYMAN-SCOTT RAINFALL MODEL
Carlo Arturo MONDONEDO 1, Yasuto TACHIKAWA2, and Kaoru TAKARA3
1Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Katsura, Kyoto 615-8530, Japan) E-mail:mondonedo@flood.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
3Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

The serial application of a stochastic rainfall time series model and a rainfall-runoff model is aneffective option in the event when historical discharge records are inadequate for determining quantile
flood flows crucial in flood control applications. In this study, the Neyman-Scott clustered Poisson
rectangular pulse rainfall model was applied to generate synthetic hourly and daily rainfall time series.
This model was evaluated for quantiles for the different areas of Kamishiiba (Kyushu), Naha (Okinawa),
and Sapporo (Hokkaido) in Japan. Sets of parameters were determined based on combinations of
moments of the historical records. The quantiles of each synthetic time series were compared to thehistorical counterparts, yielding mixed results. In general, the model proved effective in matching thehistorical quantile except in periods in which a combination of rainfall sources such as convective,
typhoon and/or frontal rainfall was dominant. An ideal combination of historical moments forgenerating model parameters for the objective was determined for the mentioned study areas, although it
is recommended to employ a similar methodical test of moment combinations for regions elsewhere.
Key Words: stochastic rainfall modeling, Neyman-Scott rainfall model, design flood, quantile rainfall

Rodriguez-Iturbe, et al.3), NSM here for brevity.
Based on a clustered point process1, 2) approach,this model is theoretically capable of generatingtime series that are consistent at several levels of
aggregation. The NSM was widely studied tomodel the moment characteristics of historicalrainfall records. The matching of the synthetic
extreme values to the historical ones however, hasbeen examined less. Unlike previous studies
therefore, the emphasis here is to investigate theability of the NSM to preserve the historicalquantile rainfall depths of 1-hour and 24-hour
duration, which may be basic informationnecessary for risk analysis in flood control
applications. In so doing, longer NSM rainfall
records, although synthetic, can be available as
reliable bases of quantile flood events.
The historical data used in this study wereobtained from three locations in Japan toincorporate the effect on the rainfall by local
conditions such as susceptibility to fronts and/or
typhoons. Sixteen yearly records (1988 to 2003)
of hourly rainfall were collected from Kamishiiba(Kyushu Island), while 27 years (1976-2002) eachof hourly rainfall were collected from Naha

1. INTRODUCTION
Flood control decision variables connected todesign and operation depend on an appropriate
availability of discharge records. Ideally, suchdecision variables should be determined based ona quantile of flood discharge. Unfortunately,
discharge records may be short (less than 30
years) and/or nonstationary due to land usechange and/or river intrusive construction,
restricting effective extreme value analysis.
Generating this data, via input of synthetic rainfall
time series into a rainfall-runoff model provides avaluable option in this case. As such, stochastic
techniques based on the point process1, 2) approach
can be employed to generate synthetic rainfall.
However, to be applicable in hydraulic design
applications, the quantiles of the historical rainfall
record should be well represented in the synthetic
time series as well. This study emphasizes on
the quality of the synthetic quantiles generated by
one stochastic rainfall model of interest.
The
Neyman–Scottclustered
Poissonrectangular pulse rainfall model applied in thisstudy follows the basic formulation of- 79 -

stationarity in the mean and variance. In applying
the model therefore, it would be beneficial to haveas long a historical rainfall record as possible(about 20 years if possible). The historicalrainfall record can be expressed in terms of themodel parameters as shown in Eqs. (1) – (3)(Rodriguez-Iturbe, et al.3)).

(Okinawa) and Sapporo (Hokkaido) as the main
historical records of this study.
Section 2 of this paper presents a review of the
NSM in its governing equations and parameter
estimation requirements. The details of themodel are limited here, although past studies(Rodriguez-Iturbe, et al.3), Burlando and Rosso4),
Cowpertwait5)) present a more thoroughdiscussion. The applications of the model to theabove-mentioned study areas appear in section 3,
followed by a discussion in section 4. Section 5
presents the conclusions of this study.

E Yi (h ) = λμ c h / (ημ x )()λ ⋅ μ c − 1 [β 3 A1 (h ) − η 3 B1 (h )]βμ x 2η 3 (β 2 − η 2 )4λμ c A1 (h )+μ x 2η 3

var Yi (h ) =

2

(1)

(2)

) 4λμ A (h, k )(cov Yi (h ) , Yi +(hk) =

(

)

2

c

μx η 32

λ ⋅ μ c 2 − 1 [β 3 A2 (h, k ) − η 3 B2 (h, k )]+βμ x 2η 3 (β 2 − η 2 )(3)

The previous equations were derived from themethod of moments3) in which:

A1 (h ) = ηh − 1 + e −ηh
B1 (h ) = βh − 1 + e − βh()
B (h, k ) = 0.5(1 − e ) e ( )A2 (h, k ) = 0.5 1 − e −ηh e −ηh (k −1)2− βh 2− βh k −1

2

Fig. 1. Schematic drawing of the Neyman-Scott Rainfall Model.

where:i= time interval counterh= integer specifying time step interval ofdata (1 for 1 hour, 24 for 1 day, etc.)hYi= rainfall depth in the i-th time of interval h

2. NEYMAN-SCOTT RAINFALL MODEL(1) Governing equations
This version of the NSM consists of fiveprobability
distributions,
basedonRodriguez-Iturbe’s original formulation3) (see Fig.
1). Clusters of rain cells are linked integrally to astorm origin with mean occurrence rate λ, regarded
as a Poisson process. Each cluster can have arandom number of rain cells described by ageometric distribution (with all clusters containing
at least one cell) with mean cell number μc.
Relative to the cluster origin, the random arrival ofeach cell is based on an exponential distribution
with mean displacement 1/β. Each cell has acorresponding independent identically distributed(iid) random intensity and duration, also based on
the exponential distribution with mean intensity μx
and mean cell life span 1/η. The total rainfall
intensity is then the superposition of the effects ofthese random cell intensities.
Inherent in the model is the assumption of

E Yi (h )= mean rainfall depth record at
h-hours(h )var Yi= variance of rainfall record at
h-hours
cov Yi h , Yi +h k = covariance of rainfall record at
h-hours at lag k()(2) Parameter Estimation
Parameter estimation following the method of
moments approach was adopted here althoughmaximum likelihood estimators may be used aswell4). Nontrivial combinations of Eqs. (1) – (3)are required to form the objective function for theestimation of parameters λ, η, μc, μx, and β in
NSM. Normally, the system formed by these
combinations is solved for the required
parameters by minimization.
At least fiveequations are therefore required in one- 80 -

Table 2.
Range of NSM parameters used for
optimization.
Parameter
Min
Max
0.001
0.050
λ (1/h)0.01
0.50
β (1/h)0.30
15.0
μx (mm/h)
2.00
100.00μc
0.10
5.0η(1/h)combination. For the estimation, the objectivefunction was adopted from previous studies6) toensure that large numerical values do not
dominate the fitting procedure. This objectivefunction appears here as Eq. (4).

⎛ f (Y ) ⎞F = ∑ ⎜ j i − 1⎟⎜⎟
j =1 ⎝ W j
⎠
M

2

(4)

Table 3. Sample NSM parameters for July.
Naha
SapporoNSM Param. Kamishiiba
λ (1/h)0.006360.001850.00828β (1/h)0.071070.01000
0.21714μx (mm/h)
4.4948110.51604
2.99253μc
44.33524
13.93186
10.46855η (1/h)2.17691
1.34515
2.65969

where:f j (Yi ) = j-th NS moment equation of rainfall

Wj
M

time series Yi (from Eqs. (1) – (3)).
= historical moment value from rainfall
record.
= number of equations to be adopted in theestimation.

generated synthetic data should be sufficiently
close to those of the historical maxima in thisstudy (see Section 4). Since Eqs. (1), (2), and(3) does not directly employ information from the
maxima in the estimation of the parameters, itwould be appropriate to assume that Test Set I
may not cover this requirement for all possible
regional conditions of the selected study areas.
To cover such possible dependencies of the target
maxima on the short-term and long-term momentsof the historical data, five more combination test
sets were adopted, as shown in Table 1.
The minimization techniques adopted here
employ the constraints shown in Table 2.
Calenda and Napolitano7) adopted these ranges intheir study on unbiased parameter estimates for
the NSM. The application of these ranges in thisstudy include the use of the Nelder-Meadsimplex9)andtheLevenberg-Marquardt9)
minimization techniques in tandem for initial andrefined estimation, respectively.
A sample of the estimated July parameters Test
Set I for the three study areas appear in Table 3.
In general, the characteristics of the study areas
are manifest through this table of sample
parameters (see (1)). For instance, the higher λand the moderate β in the Kamishiiba region
indicates more frequent clusters with moderate
gaps between storm birth and rainfall initiation,
respectively. When a cluster does arrive in thismountainous region, it is on the average densely
packed (high μc), with moderate intensity (μx) andshort duration (η), possibly characterizing theorographic effect. In contrast, clusters are lessdense in Naha (moderate μc), where theorographic effect is less prevalent. Althoughclusters are moderately frequent, rain cells occur
long after the storm origin arrival, with higher
intensity and longer duration, possibly reflecting
the relatively high vapor storage capacity in thisCombinations of Eqs. (1) – (3) are required in(4), depending on the target use of the final NSM.
Previous combinations include those employed byRodriguez-Iturbe et al.3), Burlando and Rosso3),
Cowpertwait, et al.6), Calenda and Napolitano7),
and Favre et al.8), to cite a few. For instance, thedetermination of the five parameters can be basedon the basic combination Test Set I (see Table 1).
The frequency distributions of the hourly anddaily maximum in one month per record of theTable 1. Proposed test sets of historical moment
combinations for NSM parameter estimation problem.
Test Set
1
6
12
24
48
Mean
○
Var
I
○
○
Cov1*
○
○
Mean
○
Var
II
○
○
○
Cov1
○
○
○
Mean
○
Var
III
○
○
○
Cov1
○
○
○
Mean
○
Var
IV
○
○
○
Cov1
○
○
○
Mean
○
Var
V
○
○
○
○
Cov1
○
○
○
○
Mean
○
Var
VI
○
○
○
○
Cov1
○
○
○
○
*Covariance at lag 1- 81 -

most likely from the same population. The
governing equations of the KS test applied here
appear in Eqs. (5) – (8) (see Press, et al.9)).

Table 4. Sample moments generated from July NSM.
Kamishiiba

Naha

Sapporo

His.

Syn.

His.

Syn.

His.

Syn.

HM

0.583

0.578

0.202

0.186

0.097

0.097

HSD

2.531

2.487

1.746

1.737

0.698

0.681

HC1

0.672

0.672

0.489

0.500

0.517

0.511

DM

13.982

13.870

4.846

4.471

2.339

2.328

DSD

37.60036.76215.92816.031

7.116

6.856

∞

(

)

QKS (γ ) = 2∑ 4 j 2γ 2 − 1 e −2 j γ(5)

QKS (0) = 1 ; QKS (∞ ) = 0(6)

PKS = QKS ([ N e + 0.155 + 0.24 / N e ]D)

(7)

2 2

j =1

DC1
0.348
0.352
0.376
0.369
0.096
0.093
HM = Hourly Mean, HSD, Hourly Stand. Dev., HC1 = Hourly
Covariance at lag 1
DM = Daily Mean, DSD, Daily Stand. Dev., DC1 = Daily
Covariance at lag 1

Ne =

Nh Ns
Nh + Ns(8)

where:PKS = Kolmogorov-Smirnov probability
D = maximum deviation between cumulative
frequency distributions of historical andsynthetic rainfall maxima.
Nh = length of historical record.
Ns = length of synthetic record.

tropical maritime region. Sapporo historicallyexperiences less rainfall, with relatively low raincell count, low intensity and short duration.
Also, despite the moderate storm frequency, thegap from origin to rain initiation is quite small.
Table 4 shows sample moments that weregenerated for Kamishiiba from the NSM modelusing TS I parameters for July (Table 3). In
general, the moments of the historical data were
well-maintained in the generated synthetic data.

This test was performed for each synthetic timeseries and its corresponding historical counterpart.

4. APPLICATION RESULTS

3. MODEL APPLICATION

The three regions in which the NSM wasappliedhavevarying
regional
rainfallcharacteristics. Isolated from the effects of most
typhoons, the Sapporo area of Hokkaido in
Northern Japan has a smaller average rainfall
during the rainy season (with a 2-14 mm/day
average rainfall during the rainy months of Juneto October). In contrast, the southern tropicalmaritime region of Naha in Okinawa is more
frequented by typhoons (with a 5-30 mm/day
average rainfall during the rainy months).
Finally, located closer to Okinawa, themountainous Kamishiiba region is the area thatreceives the most rainfall among the three study
areas (with a 4-30 mm/day average rainfall during
the rainy months). Based on section 2 (2), these
conditions directly affected the simulation results.
The mean and variances of the synthetic data
match those of the historical records, based onStudent’s t-tests for significantly different means9)and F-tests for significantly different variances9).
The results of these tests were omitted here as thiswas the expected result in the conception of the
NSM. The previously described differing regional
conditions appear to have no effect on theproximity of the synthetic moments to thehistorical moments. This indicates in generalthat the clustered rectangular rainfall pulses usedin NSM can account for the varying rainfall
properties inherent in the study areas.
However, not all the maxima of the synthetic

It was necessary to divide the historical records
on a monthly basis from June to October, theaccepted “rainy season” of Japan. This was done
to affect a stationarity of historical data requiredin the NSM3).
Parameter sets were thenestimated for each combination for each monthly
historical record of each study area (see Section
4).
The uniform deviate generator used here wasdeveloped from the Park-Miller “minimal
standard” generator based on the simple
multiplicative congruential algorithm9). The
inverse CDF method10) (ICDFM) was used forgenerating continuous and discrete randomvariables (through a look-up table implementationof the ICDFM for the latter).
One hundred month-hours of rainfall were thengenerated for each month (i.e.: 72,000 hours ofsynthetic records for June, 74,400 for July, etc.).
Each synthetic hourly time series was thenaggregated for every 24 hours to generate its
corresponding synthetic daily time series.
The maxima of each month was searched fromeach synthetic record and compared to its
historical counterpart. A Kolmogorov-Smirnov(KS) test was conducted to check for significanceof match. The KS probability PKS of 95% wasdesignated to be the ideal result at which thesynthetic maxima and historical maxima were

- 82 -

D a ily M a x im u m R a in fa ll

Cummulative Frequency

Cummulative Frequency

H o u rly M a x im u m R a in fa ll
1 .0 00 .9 00 .8 00 .7 00 .6 00 .5 00 .4 00 .3 00 .2 00 .1 00 .0 0
0

10

20

30

40

50

60

70

1 .0 00 .9 00 .8 00 .7 00 .6 00 .5 00 .4 00 .3 00 .2 00 .1 00 .0 0
0

80

100

200

S y n th e tic R e c o rd

S y n th e tic R e c o rd

H is to ric a l R e c o rd

Jul Test I (PKS = 0.99536)

10

15

20

25

30

35

40

H is to ric a l R e c o rd

1 .0 00 .9 00 .8 00 .7 00 .6 00 .5 00 .4 00 .3 00 .2 00 .1 00 .0 00

R a in fa ll (m m )S y n th e tic R e c o rd500

D a ily M a x im u m R a in fa ll
Cummulative Frequency

Cummulative Frequency

H o u rly M a x im u m R a in fa ll

5400

Jul Test I (PKS = 0.999)

1 .0 00 .9 00 .8 00 .7 00 .6 00 .5 00 .4 00 .3 00 .2 00 .1 00 .0 00300

R a in fa ll (m m )R a in fa ll (m m )

50

100

150

200

250

R a in fa ll (m m )H is to ric a l R e c o rd

S y n th e tic R e c o rd

Oct Test II (PKS = 0.88421)

H is to ric a l R e c o rd

Oct Test II (PKS = 0.50603)

Fig. 2. Sample KS-Test results from NSM (taken from Kamishiiba application).

convective rainfall appears to occur in October.
However, although typhoons can occur in therainy month of June over Naha, it is possible that
this region also experiences a frontal rainfall
source.
Incorporating covariances of higher order may
rectify the limitations due to the mix of sources,
although an adequate representation of the mixed
temporal structure appears to be a more thoroughapproach. In his formulation, Cowpertwait5)
proposed a cluster composition consisting of
so-called “light cells” of long expected duration
and “heavy cells” of short duration.
This
formulation appears to be more appropriate for themixed-source conditions.
Test IV and VI could not be applied
successfully to estimate the NSM parameters fromthe Sapporo historical data. It appears that therainfall is poorly correlated at the 48 hour period.
Thus, no synthetic records were generated forthese combinations. The June synthetic timeseries of this region were also ignored since therainfall during this period appeared to be very
little and poorly correlated as well.
Although the proposed methodology led tosynthetic time series that closely matched themoments of the historical records, a few
combinations appeared to yield synthetic maxima
that were closely matched to the historicalmaxima consistently.
However, these ideal
combinations did not appear to be consistent for
all the regions involved. The reason for thisappears to be the varied correlation structure ofeach regional historical record.
Within a region however, certain combinations
seemed to model the maxima better than othercombinations. It is thus possible that a dominantcorrelation structure of the historical record exists

records appear to match the maxima of thehistorical records. Figure 2 shows a sample ofKamishiiba cumulative distributions of historicaland synthetic monthly maximum rainfall intensity
in which 16 historical monthly maximum werecompared to 100 synthetic monthly maximum forthe hourly and daily records. KS test results
gathered were classified into 3 types. Type I
results reflect a 95% KS probability for the hourly
and daily maxima, indicating significant similarity
between historical and synthetic records. Type II
results reflect KS probabilities below 95% forboth the hourly and daily maxima.
Type III
results reflect at least a 95% KS probability for
either the hourly or daily synthetic maxima.
Table 5 summarizes the results of KS tests forhourly and daily data at the three study regions.
Although the majority of the KS results wereType I, most Type II and III results wereconcentrated in the August-September period of
Kamishiiba, and during June and October inNaha. However, during these periods, the arrivaland duration of rain cells of a cluster of thepresent formulation may not be adequate. In
particular, during August and September in
Kamishiiba, it is possible for the region toexperience the usual share of convective rainfall
as well as the irregular passing of a typhoon.
Under such conditions, the rain cell compositionof the historical records is a mix from the two
rainfall sources. This affected the maxima of thesynthetic record although the synthetic moments
were still closely matched to those of thehistorical records.
Type II and III results were likewise
concentrated on the months of June and Octoberin Naha. A similar pattern in which the rainfall
seems to be mixed in sources as typhoon and- 83 -

Table 5. Kolmogorov-Smirnoff Tests for synthetic and historical maxima.
KAMISHIIBA
JUNE
JULY
AUGUSTSEPTEMBEROCTOBERTESTHOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY
0.998
0.998
0.995
0.999
0.624
0.864
0.999
0.669
0.997
0.982
I
0.994
0.996
0.999
0.999
0.981
0.945
0.884
0.990
0.884
0.506
II
0.884
0.996
0.999
0.997
0.449
0.903
0.994
0.550
0.995
0.999
III
0.449
0.991
0.999
0.864
0.793
0.934
0.976
0.911
0.726
0.998
IV
0.994
0.998
0.998
0.988
0.740
0.972
0.972
0.874
0.999
0.999
V
0.968
0.999
0.999
0.994
0.683
0.754
0.976
0.286
0.999
0.948
VI
NAHA
TEST
I
II
III
IV
V
VI

JUNE
JULY
AUGUSTSEPTEMBEROCTOBERHOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY
0.844
0.126
0.743
0.601
0.999
0.976
0.999
0.999
0.536
0.965
0.993
0.241
0.957
0.980
0.999
0.999
0.999
0.928
0.630
0.993
0.949
0.481
0.994
0.979
0.999
0.998
0.999
0.972
0.354
0.966
0.982
0.196
0.957
0.999
0.991
0.856
0.996
0.959
0.920
0.579
0.936
0.697
0.992
0.816
0.999
0.994
0.999
0.959
0.927
0.723
0.982
0.077
0.999
0.996
0.772
0.798
0.995
0.971
0.920
0.848

SAPPORO
TEST
I
II
III
IV
V
VI

JUNE
JULY
AUGUSTSEPTEMBEROCTOBERHOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY HOURLY DAILY
0.997
0.805
0.995
0.966
0.998
0.953
0.993
0.909
0.603
1.000
0.891
0.971
0.998
0.999
0.998
0.406
0.964
0.957
0.999
0.999
0.947
0.998
0.973
0.999
0.999
0.953
0.999
0.971
0.999
0.947
-not applied-not applied-not applied-not applied-not applied0.999
0.944
0.999
0.999
0.999
0.544
0.999
0.997
0.999
0.999
-not applied-not applied-not applied-not applied-not applied-for each synthetic and historical maximum set isthus recommended for regions elsewhere.

for all the months within a region. The ideal
combinations appear to be Tests I, II, and V for
Kamishiiba, Tests II and III for Naha, and Test III
for Sapporo. However, the ideal combination ofhistorical moments to parameterize a NSM thatsufficiently and consistently model historicalrainfall maxima is not straightforward. The
systematic check of combinations conducted here
is recommended in future applications.

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

5. CONCLUSION
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pp. 269-288, 1987.
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The Neyman-Scott Poisson rectangular pulse
rainfall model was applied to generate synthetic
rainfall time series for the regions of Kamishiiba,
Naha, and Sapporo in Japan. A majority of theresulting synthetic records appear to match thehistorical moments and maxima. However, themaxima in several months in which the rainfall
was due to a mix of sources such as convective
rainfall with typhoons and/or fronts were
inadequately matched. A more detailed NSM
formulation, incorporating two rain cell types withshort and long expected durations, is expected torectify the limitations of the current study.
The concept of an ideal test set for estimating
NSM parameters that lead to well-matching
synthetic maxima seems ill-advised as of thiswriting. Although all combinations appear toyield synthetic data with moments closely
matching the historical moments, the same is notnecessarily true for the case of the maxima.
Since a historical data set will have a correlation
structure that is more dominant in several
aggregation levels, it is also possible to have morethan one ideal combination. A systematic checkof possible combinations, followed by a KS-test(Received September 30, 2006)- 84 -