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

NSM Modifications for Improved Generation of Rainfall Maxima

Carlo Arturo MONDONEDO*, Yasuto TACHIKAWA* and Kaoru TAKARA*Graduate School of Urban Environmental Engineering, Kyoto University
Synopsis
A previous application of the Neyman-Scott Rectangular Clustered Poisson Rainfall Model,
NSM here for brevity, focused on the generation of synthetic rainfall records that yielded block
maxima (monthly hourly and daily maximum rainfall values) that matched the historicalcounterpart. Among the findings, it was determined that historical moments used to identify the
NSM model parameters such as the hourly mean, hourly variance, hourly covariance at lag 1,
24-hourly variance and 24-hourly covariance at lag-1 were sufficient for modeling the Kamishiiba
Region in Kyushu. To affect parsimony in the synthetic rainfall generation in this study, thehistorical data from the said region was pooled into the frontal rainfall season (June to July) andtyphoon season (September to October). Results were further improved when the search
information included the hourly third central moment of the NSM. As of this writing, a limited
improvement was observed in the Peaks Over Threshold Maxima as well.
Keywords: stochastic hydrology, rainfall model, design flood, time series generation, simulation

1.

Introduction

Synthetic rainfall records generated via the
Neyman-Scott Rectangular Clustered Poisson Rainfall
Model, NSM here for brevity, must have block maxima(hourly maxima and 24-hourly maxima for instance)that match the historical counterparts to be sufficient.
In a previous study (Mondonedo et al, 2007), it was
shown that a proper combination of moments andcovariances from the historical records is critical inmeeting this objective. From the same study, it was
found that (among other combinations) the hourly
mean, hourly variance, hourly covariance at lag 1,
24-hourly variance and 24-hourly covariance at lag 1
were sufficient for modeling the monthly rainfall block
maxima of hourly and 24-hourly duration in the
Kamishiiba Region in Kyushu. Based on this region,
this study focuses on the improvement of the parametersearch and the synthetic rainfall generation and on theextension of maxima modeling to the Peaks Over
Threshold (POT) definition as well.

2.

NSM Equations

The schematic of the NSM appears in Figure 1.
This model consists of essentially five probability
distributions. A storm is characterized as a cluster ofcells that follows a Poisson process with mean
occurrence rate O. Each storm consists of a random
number of cells with all storms containing at least one
cell. Candidate distributions for this purpose are the
Poisson and geometric distributions. In this study, thegeometric distribution was used extensively with mean
Pc. The arrival of each cell is an exponential variate
with mean E. The duration and intensity of each cellare assumed independent identically distributed (iid)based on the exponential distribution with means K and
1/Px, respectively. The total rainfall intensity is thenthe superposition of the effects of these random cellintensities. However, this study also involved the casewhen the intensity of each cell is rendered as a gamma
variable with shape parameter D and scale parameter T.
From the method of moments, the NSM parameters－ 57 －

can be linked to actual record data moments. Such
expressions include the following (Rodriguez-Iturbe,1987):E Yi hOP c P x h / K

Synthetic
RainfallRainfallintensity

var Yi h>

2OP c E X 2 P c  1 E 3 A1 h  K 3 B1 h
3

2

EK E  K

@

2

2



4OP c P x A1 h

Time

Cell
Intensity

K3

cov Yi h , Yi hk>

2OP c E X 2 P c  1 E 3 A2 h, k  K 3 B2 h, k@

EK 3 E 2  K 2
2



Cell
Duration

Time

Cell
ArrivalTime

4OP c P x A2 h, k

K3

in which:2

Kh  1  e

A1 h

3

0.5 1  e Kh e Kh k 1

B2 h, k0.5 1  e Eh e  Eh k 1

Time

Cluster
Arrival2where:i= time interval counterh= integer specifying time step interval of data
(1-hour, 24-hour, etc.)= rainfall depth in the ith time interval oflength h

E Yi h3

2

A2 h, k

Yi h

Time

Clustersize()

E h  1  e  Eh

B1 h

1

Kh

= mean rainfall depth

Time

Fig. 1 Schematic diagram of the Neyman Scott
Clustered Poisson Rectangular Rainfall Model (NSM).
As of this writing, the formulation of the third
central moment included a cell intensity based solely
on the gamma distribution. A formulation using theoriginal exponential distribution and other distributions
for cell intensity will appear in a future edition of thisstudy. The third central moment of rainfall intensity
appears here as (Cowpertwait, 1998):

var Yi h = variance of rainfall record
cov Yi h , Yi h k = covariance or rainfall record at lag§ T 3* 3  D ·¸¸ *6OP c ¨¨
© *D¹Kh  2  Khe Kh  2e Kh / K 4

E Yi h  Yi hk.

3

 6ODT D 1  D T 2 Pc Pc  1 f K , E , h /The term E X

2>2K E E K @refers to the expectation of thesquare of the cell intensity random variable which canbe based on either an exponential or a gamma
distribution (see Section 3 for details).

4

2

2 2

3

2

 6O DT Pc Pc  1 g K , E , h />2K E K  E K  E 2E  K 2K  E @4－ 58 －

22

2K 3 E 2e Kh  2K 3E 2e  Eh

f K, E , h K 2 E 3e  2Kh  2K 4 Ee Kh  2K 3 E 2e  K  E h
 2K 4 Ee  K  E h  8K 3 E 3h  11K 2 E 3
 2K 4 E  2K 3 E 2  4KE 5h  4EK 5h
 7 E 5  4K 5  8E 5e Kh  E 5e  2Kh
 2hK 3 E 3e Kh  12K 2 E 3e Kh  2hKE 5e Kh

fitting procedure (Favre et al., 2004).

 4K 5e  Eh
g K, E , h

12K 5 Ee  Eh  9K 4 E 2

 12KE 5 e Kh  9K 2 E 4  12K 3 E 3 e  K  E h
 K 2 E 4 e 2Kh  12K 3 E 3e  Eh  9K 5 E
 9 E 5K  3KE 5 e 2Kh  K 4 E 2 e  2 Eh
 12K 3 E 3e Kh  6 E 2K 5 h  10 E 4K 3 h
 6 E 5K 2 h  10 E 3K 4 h  4 E 6Kh
 8E 2K 4 e  Eh  4hK 6 E  12K 3 E 3
 8K 2 E 4 e Kh  6K 6  6 E 6  2K 6 e 2 Eh
 2 E 6 e 2Kh  8E 6 e Kh  8K 6 e  Eh
 3EK 5 e 2 Eh
where:E Yi h  Yi h3= third central moment of rainfall

depth.

Yi h = mean value of rainfall at aggregation level h.
3. NSM Parameter Search
The objective function used in the estimation
follows the form:§ f j Yi
·
 1¸
F ¦¨
¨
¸
j 1 © Wj¹
52

where:

f j Yi= jth NSM moment or equation

covariance equation of rainfall depth Yi.

Wj

Table 1. NSM Parameter search limits.
NSM Parameter Upper limit Lower limit
O, 1/h0.001
0.05
E, 1/h
0.05
0.99
0.5
60
KK
0.3
15
Px, mm/h
1
50
Pc
0.01
20
D
T, mm/hN/A
N/A= actual moment value from rainfall record.

This choice of the objective function was made toensure that large numerical values do not dominate theTwo key parameter searches based on a basic
reconfiguration of the model were used in this study.
The first search, or S1, did not involve the third central
moment with rain cell intensity X rendered as an
exponential variate, requiring parameters O, E, Pc, Px,
and K. In this case, the term E<X2> in the varianceand covariance equations corresponded to 2Px2. Thissearch required the following historical information:
hourly mean rainfall, hourly rainfall variance, hourly
rainfall covariance at lag 1, 24-hourly rainfall variance,and 24-hourly rainfall covariance at lag 1.
The second search, or S2, involved the third central
moment with rain cell intensity X rendered as a gamma
distribution variable, requiring parameters O, E, Pc, D, T,
and K. However, since Px = DT, the parameter T was
rendered in terms of the historical hourly rainfall mean,leaving only 5 parameters in the search. In addition,
the term E<X2> in the variance and covarianceequations corresponded to D(1+D)T2. This searchrequired the following historical information: hourly
rainfall variance, hourly correlation at lag 1, hourly 3rd
central moment, 24-hourly variance and 24-hourly
correlation at lag 1. NSM correlation equations are
the ratio of the covariance equation and the varianceequation at appropriate duration h and lag k.
A 15 year record of hourly rainfall from 1988 to
2002 from a rain gauge based in Kamishiiba served as
the historical data in the study. Rainfall from months
June to July was pooled to form a 15-year long recordof the frontal season rainfall. The same was donewith the rainfall from September to October, covering
the typhoon season. It was assumed that stationarity
was still observed within the pooled data. Each blockof the 15 year historical record then contained 61 days
worth of hourly rainfall. Four searches where thenconducted in total: S1 and S2 for June-July and S1 andS2 for September-October. For brevity, the former
pair was called SF1 and SF2 while the latter pair was－ 59 －

called ST1 and ST2.
Table 1 show the limits used in the constrained
search. A simplex algorithm was used for an initial
estimate and the Levenberg-Marquardt search was used
for refinement (Press, 1986). Table 2 shows theresults of the estimation obtained from all searches.
Table 2. Parameters determined from NSM searches.
NSM Parameter
SF1
SF2
ST1
ST2
O, 1/h
0.0106 0.0106 0.0038 0.0040
E, 1/h
0.0946 0.0946 0.0556 0.0543
2.179
2.179
1.037
0.994
KK
3.784
4.112
Px, mm/h
36.214 40.774 20.124 50.000
Pc
0.794
0.220
D
T, mm/h
4.217
6.965

4. Synthetic Rainfall Generation

applicable to the study data in that for each record, eachmaximum can be assigned to a single independent
variable (Press, 1986). Based on the maximum
deviation of historical and synthetic cumulative
distributions, the KS probability approaches unity when
the null hypothesis that two sets did not derive from thesame population can be rejected (i.e.: statisticaldifference is improbable). An alternative test basedon the fit of the same maxima to extreme valuefunction distributions (Gumbel, Pareto, Pearson, etc.) is
underway as of this writing.
Both block maxima and POT maxima werechecked. The historical block maxima for a period
were used as a basis for determining the POT maxima.
For instance, Table 4 shows the historical block
maxima for the 15 year Kamishiiba record. The
threshold used to determine the POT maxima were the
8th to the 11th largest values from this record.

The synthetic record generated from each search
consisted of 100 realizations of 61-day rainfall. A
summary of the moments and correlations involved in
the searches and the historical counterparts appear in
Table 3.
Table 3. Historical and Synthetic Rainfall Moments
Kamishiiba
rainfall
Jun-Jul
SF1
SF2
Sep-Oct
Mean (1H),
mm
0.665
0.643
0.647
0.305
Variance
6.384
6.280
6.383
3.042
(1H), mm2
Lag-1 Covariance
4.315
4.266
4.346
2.165
(1H), mm2

ST1

ST2

0.306

0.298

3.113

3.120

2.233

2.237

Lag-1 Correlation (1H)
3rd Central
Moment

0.676

0.679

0.681

0.712

0.717

0.717

(1H), mm3

6.614

N/A

6.711

10.653

N/A

10.604

1288.50 1249.05 1314.54

606.99643.84636.03347.46349.98354.48228.65249.25

246.51

0.270

0.280

0.270

0.377

0.387

0.388

Variance(24H), mm2
Lag-1 Covariance(24H), mm2
Lag-1 Correlation(24H)

5. Preliminary Analysis
Rainfall maxima obtained from synthetic andhistorical rainfall records were checked here following
two tests. The standard Kolmogorov-Smirnov or KS
Test in which the unbinned distributions of the two data
sets are compared for maximum deviation served as themain test at this phase of the study. This test was

Table 4. Block Maxima from Kamishiiba rainfall records.
Kamishiba Maximum Rainfall (1988-2002) in mm
Jun-Jul
Sep-Oct
Rank
Hourly24-Hourly Hourly
24-Hourly
1
15
100
9
21
2
20
101
12
28
3
21
120
13
32
4
22
127
15
55
5
23
128
18
55
6
28
131
19
62
7
30
144
22
62
8
31
145
24
81
9
31
150
27
89
10
34
189
30
162
11
36
221
31
163
12
38
225
34
182
13
41
266
36
212
14
48
271
38
232
15
55
380
39
318

Table 5 shows the KS Test results for the 4 searches
conducted in the study for the block maxima and POT
maxima. It appears that both S1 and S2 canadequately model the block maxima for the June-Julyperiod. S2 appears to have a slight advantage over S1for the September-October period in which skewness ismore prominent.
S1 and S2 appear to perform consistently for thefrontal rainfall season. This suggests that the basic
structure of storm and rain cells can be further
improved for the typhoon season.
A possible
correction here is to modify the NSM storm into a－ 60 －

collection of cells that suits the typhoon season better
than the current configuration (Cowpertwait, 1996).
S2 can be modified into a similar search S2` that
includes additional historical information such as lag 2
correlations or the third central moment of daily rainfall
depth for the typhoon season. However, such an
approach may strain the objective functions such thatS2` may perform worse than the original S2.
While S2 suggests an improved search for NSM
parameters, it is not clear how S2 affects the synthetic
rainfall generation. Parameters determined from S2
may have improved from 1) the choice of distribution
for cell intensity, 2) the inclusion of the third momentof rainfall depth, or 3) both.

Table 6. Fitted Gumbel parameters from Block Maxima
Gumbel Distribution: P[X<x]=Exp{Exp[-(x-U)/H]}
SearchJun-Jul SF1

SF2

Sep-Oct

ST1

ST2

U
1H

0.488

0.547

0.558

0.392

0.316

0.35

24H

0.332

0.384

0.336

-0.1720.047-0.052

H
1H

0.329

0.228

0.273

0.448

0.415

0.356

24H

0.356

0.351

0.384

0.716

0.496

0.465

(Takara et al, 1994) taken from historical andsynthetic data. A preliminary summary of this data
appears in Table 6.
6. Further Studies

Table 5. Summary of KS Test Results using generated
synthetic and historical Kamishiiba rainfall
Search
SF1
SF2
ST1
ST2
Block Maxima
1H
0.992
1.000
1.000
0.994
24H
1.000
1.000
0.918
0.966
Hourly POT Maxima
8/15
0.957
0.997
0.992
0.464
9/15
0.957
0.997
0.992
0.991
10/15
0.778
0.944
0.997
0.863
11/15
0.987
0.986
0.985
0.776
24-Hourly POT Maxima
8/15
0.999
0.975
0.961
0.963
9/15
0.933
1.000
0.620
0.699
10/15
0.336
0.984
0.975
1.000
11/15
1.000
0.965
0.726
0.998

As expected from previous NSM performancestudies, moments, covariances, and block maxima of
the historical records were closely matched for all
searches used here. From this information and KS
Test results, it appears that pooling together the frontal
rainfall record of June and July and the typhoon rainfall
record of September and October is justified.
In addition, the POT maxima do not appear to bewell-matched. Neither S1 nor S2 yielded parametersthat led to synthetic rainfall that closely match the POT
maxima of the historical record consistently.
However, this limitation may be rectified by including
more moments or correlations in further investigations.
In a later edition of this study, the historical andsynthetic maxima will be checked through the fit of an
extreme value distribution such as the Gumbel
distribution. Confirmation by this test is based on theproximity of the fitted Gumbel parameters with SLSE

Modeling typhoon rainfall based on the current
NSM configurations that require searches S1 and S2
was limited. Approaches to remedy this limitation
will be conducted.
The KS Tests of the block maxima indicate that theinclusion of the third central moment is beneficial forthe objectives of the study. Parameter searches whererain cell intensity is exponentially distributed, gamma
distributed, or by some other distribution with andwithout the third central moment of rain fall depth will
be assessed.
Neither S1 nor S2 searches convincingly yield POT
maxima that match the historical POT maxima. A
solution to this, whether in the form of a new NSM
equation or the search for ideal combination in theparameter search will be investigated.
Additional block maxima from the synthetic data
will also be checked relative to the historicalcounterparts (6-hour maxima, 12-hour maxima, etc.).
A secondary confirmation, by way of extreme value
distribution parameters, appears to be a necessity toreinforce findings based on the KS tests.
A practical application of this synthetic rainfall data
in rainfall-runoff modeling in the Kamishiiba area will
be conducted to pool synthetic stream flow peakdischarges that correspond to each storm.

References
Cowpertwait, P. S. P., O'Connell, P. E. , Metcalfe, A. V. ,and Mawdsley, J. A., 1996. Stochastic point processmodeling of rainfall. I. Single-site fitting andvalidation. Journal of Hydrology. 175(1-4): 47-65.

－ 61 －

Cowpertwait, 1998, A Poisson-cluster model ofrainfall: high-order moments and extreme values.
J. R. Soc. Lond. A 454, pp.885-898.

Press, W., Teukolsky, S., Vertterling, W., Flannery, B.
1986, Numerical Recipes in FORTRAN,
Cambridge University Press, London.

Favre, A. C., Musy, A., Morgenthaler, S., 2004,
Unbiasedparameter
estimationofthe
Neyman-Scott Model for rainfall simulation withrelated confidence interval, Journal of Hydrology
Vol. 286, pp. 168-178.

Rodriguez-Iturbe, I., Cox, D., Isham, V. 1987, Somemodels for rainfall based on stochastic point
processes, Proc. R. Soc. London A 410, pp.
269-288.
Takara, K., Stedinger, J., 1994, Recent Japanesecontributions to frequency analysis and quantile
lower bound estimators, Stochastic and Statistical
Methods in Hydrology and Environmental
Engineering, Vol 1, Kluwer Academic Publishers,
The Netherlands, pp. 217-234.

Mondonedo, C. 2007, Evaluation of the quantiles of the
Neyman-Scott rainfall model, Annual Journal of
Hydraulic Engineering, JSCE, vol. 51, pp. 79-84.

ᭂ୯᳓ᢥ㊂ߩౣ⃻ᕈะ਄ߩߚ߼ߩNSM ࡁࠗࡑࡦ࡮ࠬࠦ࠶࠻ࡕ࠺࡞ ߩᡷ⦟Carlo Arturo MONDONEDO*࡮┙Ꮉᐽੱ*࡮ቲ 㚌੩ㇺᄢቇᄢቇ㒮Ꮏቇ⎇ⓥ⑼ㇺᏒⅣႺᎿቇኾ᡹ⷐ ᣦࡁࠗࡑࡦ࡮ࠬࠦ࠶࠻ࡕ࠺࡞ߦࠃߞߡᓧࠄࠇࠆ㒠㔎ᤨ♽೉߆ࠄᓧࠄࠇࠆ᦬ᦨᄢᤨ㑆㒠᳓㊂߅ࠃ߮᦬ᦨᄢᣣ㒠᳓㊂߇㧘᷹ⷰ㒠᳓ᤨ♽೉߆ࠄᓧࠄࠇࠆߘࠇࠄߩ୯ߣࠃߊኻᔕߔࠆߎߣࠍ㧘ߎࠇ߹ߢߩ⎇ⓥߦࠃࠅ╩⠪ࠄߪ⏕⹺ߒߚ‫ߚ߹ޕ‬㧘ࡁࠗࡑࡦ࡮ࠬࠦ࠶࠻ࡕ࠺࡞ߩࡄ࡜ࡔ࡯࠲᳿ቯߦ߅޿ߡ㧘ᤨ㑆㒠᳓㊂ߩᐔဋ୯㧘ಽᢔ㧘ᤨ㑆Ꮕᤨ㑆ߩ⥄Ꮖ⋧㑐ଥᢙ㧘ᤨ㑆㔎㊂ߩಽᢔߣᤨ㑆Ꮕᤨ㑆ߩ⥄Ꮖ⋧㑐ଥᢙࠍ↪޿ࠆߎߣߢ㧘਻Ꮊ਄ᬁ⪲࿾ὐߩ㒠᳓㊂ᤨ♽೉ࠍㆡಾߦࡕ࠺࡞ൻߢ
߈ࠆߎߣࠍ⏕⹺ߒߚ‫⎇ᧄޕ‬ⓥߢߪ㧘ߐࠄߦౣ⃻ᕈߩะ਄ࠍ࿑ࠆߚ߼ߦ㧘㒠㔎ᚑ࿃ߩ⇣ߥࠆ᪢㔎ᦼ㧔᦬᦬㧕ߣบ㘑ᦼ㧔᦬᦬㧕ߣߦಽߌߡ㒠᳓ᤨ♽೉࠺࡯࠲ࠍ⊒↢ߐߖߚ‫࠲࡯ࡔ࡜ࡄ࡞࠺ࡕޕ‬ផቯߦᤨ㑆㒠᳓㊂ߩ㧟ᰴࡕ࡯ࡔࡦ࠻ࠍዉ
౉ߔࠆߎߣߦࠃߞߡ㧘ᭂ୯᳓ᢥ㊂ߩㆡวᕈ߇ะ਄ߔࠆߎߣࠍ⏕⹺ߒߚ‫ߚ߹ޕ‬㧘POTᦨᄢ୯ߦߟ޿ߡ߽㧘ߘߩౣ⃻ᕈ߇ะ਄ߔࠆߎߣࠍ⏕⹺ߒߚ‫ޕ‬ࠠ࡯ࡢ࡯࠼⏕₸᳓ᢥቇ㧘㒠㔎ࡕ࠺࡞⸘↹ᵩ᳓㧘ᤨ♽೉⊒↢㧘ࠪࡒࡘ࡟࡯࡚ࠪࡦ
－ 62 －