Annual Journal of Hydraulic Engineering, JSCE, Vol.53, 2009, February

NEYMAN-SCOTT RAINFALL MODEL PARAMETERS
AS INDICATORS OF TEMPORAL CHANGE INHISTORICAL RAINFALL
Carlo Arturo MONDONEDO 1, Yasuto TACHIKAWA2, and Kaoru TAKARA3
1Student Member, Dr. Eng., Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 615-8540, Japan) E-mail:mondonedo@flood.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Associate Professor, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 615-8540, Japan)
3Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

The Neyman-Scott clustered Poisson rectangular pulse rainfall model (NSM) parameters are
estimated to investigate temporal change in rainfall characteristics in a historical record. This was
conducted on partial rainfall series involving monthly pooled rainfall. The historical data used is based on
Kyoto in Japan, where an hourly historical record from 1950 to 2002 was obtained. Each monthly record
is further subdivided into four overlapping 20-year sub-records prior to parameter estimation via
optimization. The method of moments is used to identify the NSM model parameters. Results indicatethat storm rainfall volume in the month of September has increased throughout the 50 year historical
record. Frontal rainfall in June also appears to have increased throughout the record period as well.
Key Words: Point processes, Neyman-Scott rainfall model, climate change, rainfall temporal change

process2) of rain cells with properties such as
intensity, duration, and arrival that can besystematically handled as random processes. The
parameters of the NSM represent the probabilistic
distributions of rain cell intensity, rain cell
duration, rain cell arrival time and so on. We
show that these parameters in turn lead toimportant storm characteristics. The determinationof these characteristics based on the method
proposed here is less arbitrary than similar
methods based on historical data. Through the
NSM model parameters, it is plausible to identify
trends that are not easy to detect by analyzing
historical rainfall time series data alone.
Historical rainfall at Kyoto used here wasobserved by the Automated Meteorological Data
Acquisition System (AMeDAS) from 1976-2002
as well as a paper record from 1950-1976. Eachmonthly record is further subdivided into four
overlapping 20-year sub-records prior to the NSM
parameter
estimation
via
optimization.
Measures based on the parameter values are thenanalyzed. A brief description of the NSM appears
in Section 2 followed by the parameter estimationin Section 3. Trends and other results are
examined in Section 4 followed by concluding
remarks in Section 5.

1. INTRODUCTION
The detection of temporal change of rainfall
properties has been a key issue in topics related to
global climate change. Though spatio-temporal
analysis of historical data offers more detailed
information, the temporal analysis at a single site
is sufficient in cases when regional distributions
are reasonably homogeneous and/or when overall
computation time is to be limited.
Our
motivation here is based on the need to establish
evidence of rainfall temporal change that is basedon a stochastic approach. We propose amethodology that can be applied systematicallythroughout a historical rainfall record that yields
quantitative interval based volumetric measures ofa storm. This in turn can be used as a standard
check on the output of a synthetic rainfall
generating scheme (i.e., rainfall from modelingtechniques) such that the overall performance ofthe scheme can be assessed.
This paper focuses on using the parameters of
the temporal Neyman-Scott clustered Poisson
rectangular pulse rainfall model (NSM) tocontribute such interval based measures.
Originally proposed by Rodriguez-Iturbe et al.1),
the NSM treats rainfall as a temporal point- 109 -

2. MODEL DESCRIPTION(1) Governing equations
Rainfall in the NSM1) is formulated as atemporal cluster of rain cells (storms) which occur
based on a Poisson process with occurrence rate
λ. The cluster size is based on a geometric
distribution with mean μc. The lag from cluster
to rain cell occurrences follows an exponential
distribution with mean 1/β. Rain cell durations
follow an exponential distribution as well withmean 1/η. Rain cell intensities are modeled by agamma distribution with scale parameter θ andshape parameter α. The resulting rainfall is thesuperpositions of these processes as shown in Fig.
1.
The moment and correlation expressions forthe aggregated rainfall point process in terms of
the NSM parameters were derived byRodriguez-Iturbe et al.3) The succeeding
Equations (1) – (3) list these expressions in theaggregated form, specific to the distribution
parameters mentioned: β, λ, μc, η, α, and θ.

E Yn(h ) = λμ c (αθ)h / ηvar Yn(h ) =Rainfallintensity

Time
Cell
Intensity

(

)

3

c

3

2λμ c αθ 2 (1 + α )(μ c − 1)
var Yi(h ) βη 3 β 2 − η 2[β A (h, k ) − η B (h, k )]3

2)

var Yn(h )

×

(h )

(h )

record at h-hours at lag k

A1 (h ) = ηh − 1 + e − ηh ; B1 (h ) = β h − 1 + e −βh

Cowpertwait4) also derived the expression for
the third central moment of rainfall depth in termsof the NSM parameters. The inclusion of this termin the parameter estimation was recommended forhigher consistency between synthetic rainfall
maxima at multiple durations and its historicalcounterpart. The authors5) have previously applieda variant of this recommended estimation
procedure and found that seasonal synthetic
rainfall (i.e., months June and July are designatedas the Frontal Rainfall Season) are indeed moreconsistent due to the dependency of the third
central moment on the rainfall seasonal maxima.
The aggregated form of the NSM expression for
the third central moment, TCM, of rainfall depth
appears here as Eq. (4) where Γ() is the Gamma
distribution function.

()
B (h, k ) = 0.5(1 − e ) e ( )A2 (h, k ) = 0.5 1 − e −ηh e −ηh (k −1)2− βh k −12where:
n = time interval counterh = integer specifying time step interval ofdata (1 for 1 hour, 24 for 1 day, etc.)h
Yn = rainfall depth in the n-th time of intervalh

E Yn(h )= variance of rainfall record at

h-hours
cor Yn , Yn + k = autocorrelation of rainfall(3)

4λμ c (αθ ) A2 (h, k )
η3

−βh 2

Time

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

2

+

Time

Time

(2)

3

(

1
3

1

1

2

Time

Cluster
Arrival) A (h )[β A (h) − η B (h)]+ 4λμ (αθ
η

3

Cell
Arrival(1)2λμ c αθ (1 + α )(μ c − 1)×βη 3 β 2 − η 2

cor Yn(h ) , Yn(+h )k =Time

Clustersize (cells)

2

1

Cell
Duration22

3Synthetic
Rainfall= mean rainfall depth record at
h-hours- 110 -

+(

6λμ c E I 3 ηh − 2 + ηhe − ηh + 2e − ηhη)

3λE I E I 2 E C(C − 1) F (η, β, h )[2η β(β − η ) ]4

Kyoto(1950-2002)

4
5

(4)

4
3

2 2

2

Student's T-ValueTCM Yn(h ) =λ(E I ) E C(C − 1)(C − 2 ) G (η, β, h )3+[2η β(η − β )(η − β)(2β + η)(2η + β)]
4

2

2

2
1
0
-1

Jan

Feb

Mar

Apr

May

Jun

Jul

-3

Sep

Oct

Nov

Dec

5%
Significance
Level

-4

where:

Aug

-2

-5
Month

E I = αθ ; E I 2 = α(1 + α )θ 2 ;E I3 = θ3

Fig. 2 Student’s T values obtained from total hourly
synthetic rainfall vs. historical counterpart.

Γ(3 + α )Γ (α )E C(C − 1) = 2μ c (μ c − 1)
E C(C − 1)(C − 2 ) = 6μ c (μ c − 1)

Figure 2 shows a reproduction of averification resulting from standard Student’s
T-tests on the historical and synthetic rainfall data
based on the NSM. Each T-statistic in Fig. 2 isbased on the average synthetic hourly rainfall, its
historical counterpart, and an appropriate estimateof standard error. The tests show that at the 5%
significance level, statistical and historical rainfall
means are consistent for all months. Tests forvariances and autocorrelations indicate similar
results. Kolomogorov-Smirnov (KS) tests on theother hand show that resulting synthetic maxima
are consistent to historical counterparts at multiple
durations for most months. We therefore adopt
the NSM and the technique to estimate its
parameters as a consistent method for temporal
rainfall generation.

2

F (η, β, h ) = −2η3β 2 e − ηh − 2η3β 2 e −βh + η 2β 3 e −2 ηh+ 2η4 βe −ηh + 2η3β 2 e −(η+β )h − 2η 4βe −(η+β )h − 8η3β 3 h+ 11η 2β 3 − 2η 4 β + 2η3β 2 + 4ηβ 5 h + 4βη 5 h − 7β 5
− 4η5 + 8β 5 e −ηh − β 5 e − 2ηh − 2hη3β 3 e −ηh − 12η 2β 3 e −ηh+ 2hηβ 5 e −ηh + 4η5 e −βh
G (η, β, h ) = 12η5 βe −βh + 9η 4 β 2 + 12ηβ 5 e − ηh + 9η 2 β 4+ 12η3β 3 e −(η+β )h − η 2 β 4 e − 2 ηh − 9η5 β − 12η3β 3 e −βh− 9β 5 η − 3ηβ 5 e − 2 ηh − η 4 β 2 e − 2βh − 12η 3β 3 e −ηh− 3βη 5 e − 2βh − 10β 4 η3 h + 6β 5 η 2 h − 10β 3 η 4 h+ 4β 6 ηh − 8β 2 η 4 e −βh + 4hη 6 β + 12η 3β 3 − 8η 2 β 4 e −ηh

3. EXPERIMENTATION− 6η 6 − 6β 6 − 2η 6 e − 2βh − 2β 6 e − 2 ηh + 8β 6 e −ηh+ 8η 6 e −βh + 6β 2 η 5 h(1) Kyoto historical rainfall data
The rainfall data used here was based on anhourly record obtained at Kyoto from the years
1950 to 2002. Kyoto is located in the central partof the Japanese main island of Honshu where little
rainfall (as well as snowfall) is observed in thewinter months such as from January to March.
Frontal rainfall is observed in May, June and July.
Summer rainfall in August on the other hand isoften convective and torrential. In the autumn
months of September and October, a large amountof rainfall is sometimes observed due to thelandfall of a typhoon. We assume that these trends
are perennial so that we can pool data by themonth for parameter estimation.
Data is grouped into overlapping intervals suchas: a) Y-A (1950 – 1970), b) Y-B (1960 – 1980),c) Y-C (1970-1990), and d) Y-D (1980-2002).
Hourly rainfall is then pooled monthly per group.
Sample moments from each interval is calculated,(2) Model validation
Some validation of the model is necessary
before proposing any intrinsic property based on
its parameters. We assume that if the results
based on the synthetic rainfall generated from theparameters are consistent, then both model andparameters are validated. One of the authors
have shown in a previous study that the NSM iseffective in the temporal modeling of rainfall fromthe Kyoto study region6) that was used in theexperiments of the succeeding section. In thisprevious study, monthly-pooled hourly historicalrainfall was used to estimate the 12 corresponding
NSM parameter sets (i.e., one set per month). A
hundred synthetic data sets were generated per set
and compared statistically to its historicalcounterpart to verify the consistency of the NSM.

- 111 -

including rainfall hourly mean, hourly variance,hourly autocorrelation at lag-1, hourly third
central moment, 12-hourly autocorrelation atlag-1, 24-hourly variance, and 24-hourly
autocorrelation at lag-1. This combination of
moments was found to yield the most consistentsynthetic data generation using the NSM5).

instance, Eq. (7) represents the approximate meanstorm duration9) TS which is written in terms of the
NSM parameters as:(2) Parameter estimation
A basic scheme consisting of the mentioned
NSM moments β, λ, μc, η, α, and θ is proposed
based on previous studies6, 7). The objectivefunction in the parameter search is given by Eq.
(5).

where:
γ = Euler constant, 0.5772.
Ln = natural logarithm.

⎛ NSM m ⎞⎟O = ∑ ⎜⎜1 −HIS m ⎟⎠
m =1 ⎝6⎛⎡η ⎤⎞TS = 1 ⎜⎜ γ + Ln ⎢(μ c − 1)
⎥ ⎟⎟
β
η
−
β
⎣
⎦⎠
⎝

Eq. (7) is a convenient estimate without resorting
to complicated considerations on storm type (i.e.,
only monthly historical moments are required).
In addition, normalizing Eq. (1) by the
duration h, yields the mean intensity of a storm:

2(5)

I S = λμ c (αθ ) / η

where:
m = equation index (i.e., any of equations (1), (2),(3), and (4))
NSMm = NSM expression for rainfall moment
HISm = equivalent historical value of moment.

θ=where E(1)

λμ c α(8)Similar to Eq. (7), the determination of this term
depends simply on historical moments of thehistorical data.
The mean volume VS of rainfall from a storm
event (i.e., regardless if the storm is convective,
frontal, or typhoon-based) is therefore the productof the mean duration (Eq. 7) and mean intensity(Eq. 8), or:

The scale parameter θ is written here in termsof the hourly mean rainfall depth as shown in Eq.
(6) to reduce the computational burden of solving
Eq. (5). The Levenberg-Marquardt scheme8) isused to carry out the optimization of the objectivefunction.

E Yn

(7)

VS =

λμ c (αθ ) ⎛⎡η ⎤⎞⎜ γ + Ln ⎢(μ c − 1)⎟⎜ηβ ⎝
η − β ⎥⎦ ⎟⎠
⎣

(9)

η
The equivalents of these measures based on
historical rainfall cannot be readily determined.
The main problem with deferring to historical datais the identification of a storm’s actually start
and/or end. Conversely, the NSM parameter
identification procedure described previously
shows no bias and arbitrariness in assessing thesestorm features. These are the measures we use
here to determine any apparent temporal trend inhistorical rainfall from our study area.

(6)= operation to obtain expected value.

The NSM parameters were estimated subject tothe constraints of Table 1.
Table 1 Ranges in parameter estimation for NSM.
NSM
Lower limit
Upper limit
Parameter
α (mm/s)
0.0150
β (1/h)
0.05
0.99
0.5
60
η (1/h)λ (1/h)
0.001
0.05
1
50
μc
θ

4. RESULTS(1) Overall trends
The values of the NSM model parameters wereidentified for each month for the four intervals
Y-A (1950-1970), Y-B (1960-1980), Y-C(1970-1990), and Y-D (1980-2002). We limit our
results for brevity, showing measures from several
months of the year. In general, trends are
difficult to detect based on plots of the parametersin time. For instance, Fig. 2 shows the storm
arrival parameter λ obtained from the historical
data. Clearly, this information cannot be easily(3) Quantitative storm measures
Some information is available from observing
the values of the NSM parameters throughout thementioned groupings of historical data.
However, we propose here several measures that
describe the storm as defined in the NSM. For- 112 -

Table 2 NSM parameter estimates and storm
volume from January historical rainfall.
Y-A
Y-B
Y-C
Y-D
1950- 1960- 1970- 1980parameter 1970
1980
1990
2002
α (mm/s) 20.000 20.000 11.542 20.000
β (1/h)
0.316
0.292
0.171
0.369
1.882
0.639
0.741 60.000η (1/h)λ (1/h)
0.007
0.006
0.008
0.011
10.0362.222
2.170 28.284
μc
0.083
0.126
0.228
0.665
θ

NSM Storm Arrival Parameter λBased on Kyoto (1950‐2003)

0.06

Storm Arrival λ (1/hr)

0.05

0.04
Y‐A (1950‐1970)
Y‐B (1960‐1980)
Y‐C (1970‐1990)

0.03

Y‐D (1980‐2003)

0.02

0.01

0

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Fig. 3 Temporal trends in storm arrival parameter λobtained from historical hourly rainfall from Kyoto,
Japan.

Table 3 NSM parameter estimates and storm
volume from June historical rainfall.
Y-A
Y-B
Y-C
Y-D
1950- 1960- 1970- 1980parameter 1970
1980
1990
2002
4.649
1.605
0.123
0.074
α (mm/s)β (1/h)
0.213
0.055
0.157
0.153
59.226
0.873
2.208
1.652
η (1/h)λ (1/h)
0.011
0.016
0.016
0.015
27.6402.924 50.000 50.000μc
13.023
3.853
7.201
9.106
θ

obtained from the historical data at hand. In
contrast, the parameter search used clearly showsthat this parameter is directly obtained throughhistorical moments alone.
However, it isdifficult to determine from this plot if a significant
change occurred in this tendency throughout theentire historical record. In fact, weaker trends
are observed from the other parameters and areomitted here for brevity.

Table 4 NSM parameter estimates and storm
volume from July historical rainfall.
Y-A
Y-B
Y-C
Y-D
1950- 1960- 1970- 1980parameter 1970
1980
1990
2002
α (mm/s)
1.033
1.494
1.911
1.491
β (1/h)
0.177
0.125
0.05
0.05
η (1/h)
1.907
1.681
1.560
1.382
λ (1/h)
0.013
0.011
0.013
0.014
μc
5.852
4.579
3.134
3.00
θ
8.013
6.065
5.784
6.446

(2) Monthly rain cell trends
Aside from the previous trends, other trends
are easier to interpret based on monthly
parameters. Table 2 shows the NSM parameters
obtained from the winter month of January. The
values of λ from Y-A through Y-D indicate that
more storms hit Kyoto in the current 20 year
period than in the more recent 20 year periods.
In addition, months similar to January with low
rainfall show a fluctuating number of cells per
storm (parameter μc) throughout the times
observed.
Table 3 shows the estimated parameters forthe month of June in which frontal rainfall is quite
dominant. Although this month is known toexperience a regular amount of rainfall in theyear, the NSM parameters do not reveal any
temporal trend from Y-A through Y-D. Similar
to the previous findings, rain cell characteristics
such as intensity and arrival do not exhibit atemporal trend throughout the historical record.
Rain cell properties differ slightly in the
succeeding month of July shown in Table 4.
The decreasing values of η indicate that theoverall gap between the birth of a storm and thearrival of the rain cells has increased throughout
the historical record. It also appears that thestorms became more intense based on thedecreasing values of average rain cell number μc
despite the increase in rain cell duration β. The
rain fronts appear to have been more severe at theend of the study period for this month.

August and September storms show an
opposite trend based on Tables 5 and 6. Whileη values show that currently rain cells arrive
sooner after storm development, rain cell number
μc also appears to have increased in the recent
record. These indicate that convective storms
and typhoons (August and September rainfall,respectively) have become more intense in Y-D
than in Y-A.
(3) Storm trends
Figure 4 shows the effect that changes in raincell has on overall storm intensity. There are noapparent trends based on the frontal months ofJune and July as well as the typhoon month of
September. However, average storm rainfall
intensity has steadily decreased from Y-A throughY-D in the summer month of August.

- 113 -

Table 5 NSM parameter estimates and storm
volume from August historical rainfall.
Y-A
Y-B
Y-C
Y-D
1950- 1960- 1970- 1980parameter 1970
1980
1990
2002
α (mm/s)
1.222
1.110
0.808
0.290
β (1/h)
0.053
0.050
0.112
0.104
1.184
1.637
2.875
2.599
η (1/h)λ (1/h)
0.007
0.010
0.014
0.007
3.243
2.893
3.411 13.110μc9.815 10.609 14.820 17.453
θFig. 6 Average storm volume VS in Kyoto, Japan.

Table 6 NSM parameter estimates and storm
volume from September historical rainfall.
Y-A
Y-B
Y-C
Y-D
1950- 1960- 1970- 1980parameter 1970
1980
1990
2002
1.646
1.586
0.554
0.185
α (mm/s)β (1/h)
0.050
0.051
0.132
0.143
1.113
0.979
2.160
2.367
η (1/h)λ (1/h)
0.011
0.008
0.008
0.008
2.884
3.497 19.559 50.000
μc
4.963
5.288
7.311
8.059
θ

5. CONCLUSION
The Neyman-Scott clustered Poisson rectangularpulse rainfall model (NSM) was used to find ameasure of temporal change in historical rainfall
in Kyoto. This validated method was selected to
avoid difficulties in identifying average storm
characteristics such as storm duration, intensity,and volume. A 53-year (1950-2002) hourly
record from Kyoto, Japan reveals trends inaverage overall storm volume. More severe rain
fronts in June and July may have increased therainfall throughout the mentioned period. The
same was observed from the rainfall in September
due to increased typhoon landfall.
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Kyoto University, 2008.
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(Received September 30, 2008)

Fig. 4 Average storm intensity IS in Kyoto, Japan.

Fig. 5 Average storm duration TS in Kyoto, Japan.

Figure 5 on the other hand shows that it is in June
when storms tend to have increased in duration
the most. Finally, the overall volume of rainfrom a storm appears to have increased in Juneand September, as shown in Fig. 6.

- 114 -