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

QUANTILE STREAMFLOW ESTIMATES BASED ON THE NEYMANSCOTT RAINFALL MODEL
Carlo Mondonedo1, Yasuto Tachikawa2, and Kaoru Takara3
1

Graduate Student, Department of Urban and Environmental Engineering, Kyoto University
Nishikyo-ku, Kyoto, 615-8540, Japan, e-mail: mondonedo@flood.dpri.kyoto-u.ac.jp
2
Associate Professor, Department of Urban and Environmental Engineering, Kyoto University
Nishikyo-ku, Kyoto, 615-8540, Japan, e-mail: tachikawa@mbox.kudpc.kyoto-u.ac.jp
3
Professor, Disaster Prevention Research Institute, Kyoto University
Uji-shi, Kyoto, 611-0011, Japan, e-mail: takara@flood.dpri.kyoto-u.ac.jp

ABSTRACT
Flood control decisions in areas that are poorly gauged can be based on synthetic data
if and only if such data can be made consistent to the historical counterpart. However,established methods that do not involve synthetic data are preferred for this problem such as
the procedure adopted by the Japan Ministry of Land, Infrastructure, Transport, and Tourism(MLIT). Here, we develop an alternative methodology based on the Neyman-Scott clustered
Poisson rectangular pulse rainfall model (NSM) suited for rainfall extreme value consistency.
It is shown that simulated streamflow based on the NSM rainfall is a competent estimate toextreme flood magnitudes. We observe that the latter method is more rational due to its
careful emphasis on extreme value rainfall.
Keywords: Synthetic Rainfall, Synthetic Streamflow, Point Processes, Neyman-Scott model1.

INTRODUCTION

Historical streamflow records are the prime basis of effective flood control decisions
such that a safe coexistence can be established between river basin and inhabitants.
However, the lack of recorded data in some river basins complicates the determination ofdesign flooding events. Fortunately, methodologies are available to systematically solve thisproblem. The approach we adopt in this study incorporates techniques that simulate thetransformation of rainfall into streamflow in the form of distributed hydrological modeling(Beven, 2002). More importantly, we couple this tool with an enhanced model for
generating synthetic rainfall from scarce historical rainfall records based on the Neyman-Scottclustered Poisson rectangular pulse rainfall model, or NSM (Rodriguez-Iturbe et al., 1987).
Consistency of NSM synthetic extreme rainfall to those of historical counterpart has
been explored in the past (Cowpertwait, 1998). We have also contributed to this problem bydeveloping the so-called NSM Fano factor exponent, or FFE (see Sec 2). Emphasis on
consistency in rainfall extremes leads to a sufficient synthetic basis that is sound, reliable andeconomical. Only then can one base design flood evaluation on a synthetic technique suchthat the lack of historical data is no longer a hindering factor in the decision-making process.
Our interest here is in showing the advantage of synthetic rainfall generation in
evaluating design streamflow by comparing two associated methodologies. One method
excluding synthetic rainfall generation is based on the Japan Ministry of Land, Infrastructure,
Transport, and Tourism (MLIT). Another method is developed by the authors following the
NSM framework. Both methods involve distributed hydrological modeling based on Kyoto
University’s Object Oriented Hydrologic Modeling System (OHyMOS).

2.

METHODOLOGY

2.1 The NSM and its governing equations
The NSM is a clustered Poisson point process (Sefozo, 1990) in which: a) storms
arrive following a temporal Poisson process (mean recurrence rate λ), b) storms consist of ageometric random number of rain cells, no storm containing zero cells (mean cell number μc),c) each cell arrival relative to the storm arrival follows an exponential distribution (mean lag
time 1/β), d) each cell duration follows an exponential duration (mean duration 1/η), and e)each cell intensity follows a two-parameter gamma distribution (shape parameter α and scaleparameter θ). The superposition of these rain cell pulses in the rainfall intensity-time plane
results in the target synthetic rainfall. Six parameters are therefore required to tune the NSM
for a particular application. The following T-duration aggregated moments were derived forparameter estimation in the NSM (Rodriguez-Iturbe et al., 1987 and Cowpertwait, 1998).

E[Y ] =
Var [Y ] =Cov[YT , YT + k ] =λμ c αθ
T
η

(1)

[

]

4λμ c (αθ ) A1 (T ) 2λμ c αθ 2 (1 + α )(μ c − 1) β3 A1 (T ) − η3 B1 (T )+η3βη 3 β 2 − η2
A1 (T ) = ηT − 1 + exp(- ηT ) ; B1 (T ) = βT − 1 + exp(- βT )
2

(

[

)

(2)

]

4λμ c (αθ ) A 2 (T, k ) 2λμ c αθ 2 (1 + α )(μ c − 1) β3A 2 (T, k ) − η3B2 (T, k )+η3βη3 β 2 − η2
2

(

)

(3)
A 2 (T , k , η) =1
(1 − ηT )2 exp[(− ηT )(k − 1)] ; B 2 (T ,k ,β) = 1 (1 − βT )2 exp[(− βT )(k − 1)]
2
2

Cor[YT , YT + k ] =TCM[Y ] =[ ](Cov[YT , YT + k ]Var[Y ]

[ ]

)

(4)

6λμ c E I3 ηT − 2 + ηTe − ηT + 2e − ηT
3λE[I]E I 2 E{C(C − 1)}f (η, β, T )+2η42η4β β 2 − η2
λ (E[I]) E{C(C − 1)(C − 2 )}g(η, β, T )2η4β η2 − β 2 (η − β )(2β + η)(2η + β)
(5)

[

)]

(

3

+

[

(

]

)

[ ]

[ ]

E[I] = αθ ; E I 2 = α(1 + α )θ 2 ; E I 3 = θ 3

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)2

f (η, β, T ) = −2η3β 2 e − ηT − 2η 3β 2 e −βT + η 2 β 3 e −2 ηT + 2η 4 β e − ηT + 2η 3β 2 e − (η+β )T − 2η 4 β e − (η+β )T− 8η 3β 3 T + 11η 2 β 3 − 2η 4 β + 2η 3β 2 + 4ηβ 5 T + 4βη 5 T − 7β 5 − 4η 5 + 8β 5 e −ηT − β 5 e − 2 ηT
− 2Tη3β 3 e −ηT − 12η 2 β 3 e −ηT + 2Tηβ 5 e −ηT + 4η5 e −βT

g(η, β, T ) = 12η5β e −βT + 9η 4 β 2 + 12ηβ 5 e − ηT + 9η 2 β 4 + 12η 3β 3 e − (η+β )T − η 2 β 4 e −2 ηT− 9η5β − 12η 3β 3 e −βT − 9β 5 η − 3ηβ 5 e − 2 ηT − η 4 β 2 e − 2βT − 12η3β 3 e −ηT − 3βη 5 e − 2βT− 10β 4 η 3 T + 6β 5 η 2 T − 10β 3 η 4 T + 4β 6 ηT − 8β 2 η 4 e −βT + 4Tη 6 β + 12η3β 3 − 8η 2 β 4 e −ηT− 6η 6 − 6β 6 − 2η 6 e − 2βT − 2β 6 e − 2 ηT + 8β 6 e −ηT + 8η 6 e −βT + 6β 2 η5 T
where E[Y] is mean rainfall depth, Var[Y] is the rainfall depth variance, Cov[YT,YT+k] is therainfall depth autocovariance at lag-k, Cov[YT,YT+k] is the rainfall depth autocorrelation at
lag-k, TCM[Y] = third central moment (TCM) of rainfall depth.
2.2

The Fano factor exponent of the NSM

We also include the expression for the Fano factor exponent (FFE) of the NSM(Mondonedo et al., 2008) in the parameter search. The derivation of the Fano Factor
exponent expression starts with the Peaks Over Threshold (POT) rainfall point process,shown in Figure 1. In this case, historical hourly data pooled monthly Ui(T) is determinedusing the threshold zi, taken as the smallest monthly maximum rainfall (Figure 1a). The
POT rainfall Qi(T), the corresponding hourly historical rainfall greater than or equal to thisthreshold (Figure 1b), is given a unit count Bi(T) (Figure 1c). The point process Ni(T*)(Figure 1d)is the sum of the unit counts within adjacent non-overlapping windows of size T*.
Ni(T*) is our basis for the FFE.
(a)

Sample(b)Ui(T)

Sample

Qi(T)

19

25

rainfall (mm)rainfall (mm)
30

30

25

25

20

20

15

15

10

10

zi

5
0 1

7

13

19

25

31

37

43

49

5
0

time (hr)

1

7

13

31

37

43

49

37

43

49

time (hr)

(c)

(d)

Sample Ni(T*=20 hours)

Sample Bi(T)
Ni(T*1)=11

1

7

13

19

25
time (hr)

31

37

43

49

1

7

13

Ni(T*2)=2

19

25

31

time (hr)Figure 1. Determining the Peaks Over Threshold (POT) rainfall point process and counting
process Ni(T*). (a) sample rainfall Ui(T) with previously determined threshold zi, (b) POT
rainfall Ui(T) values greater than or equal to zi (c) unit counts Bi(T) assigned for eachrainfall occurrence, (d) counting process Ni(T*).
The Fano factor (Fano, 1947) is defined as the ratio of the variance of Ni(T*) and its
mean, or:

FFi (T * ) =E [N i (T * )] − E N i (T * )22

E N i (T * )where:
E = operation to obtain expected value.
When the Fano factor is evaluated for the set of windows W = {2, 10, 20, …100 hours} we
can observe the typical plot shown in Figure 2. A power law relationship was proposed forthis scaling behavior by Lowen and Teich (1995, 2005) as well as Telesca et al. (2007),
shown here as Eq. 6.

⎛ T*⎞
⎟⎟
FF(T *) = 1 + ⎜⎜⎝ T0 ⎠

10⎛ T* ⎞⎟⎟
FF(T * ) = 1 + ⎜⎜⎝ T0 ⎠

ξ(6)ξ

FFi(T*)11

10
T* (hr)100

Figure 2 Scaling in the counting process Ni(T*) obtained from Kamishiiba POT series Qi(t) in
June.
where ξi is the Fano factor exponent, or the FFE. Alternatively, we assume that the averageof the point process Ni(T*) has as a power law relationship with respect to T*, while theaverage can be taken as a linear one, then the following approximation for the overall FFE
scaling behavior holds:

Var N i (T *) = A i T *Bi(7)

E N i (T * ) = C i T *(8)Ai
T * Bi −1
Ci(9)

FFi (T *) ≈After curve fitting operations for Eqs. 7 and 8 (i.e.: linear regression of historical Qi(T) for
determining Ai, Bi, and Ci), we may use Eq. 9 to write the approximate Fano Factor of
historical Qi(T), FFHi(TMi), at window T* = mean storm duration TMi as:
FFHi (TMi ) ≈

Ai
B −1
(10)
TMi i
Ci
Similarly, the same Fano factor at mean storm duration FFSi(TMi) can be written
explicitly using Eq. 6 such that:⎛T ⎞FFSi (TMi ) = 1 + ⎜⎜ Mi ⎟⎟⎝ T0 ⎠ξSi(11)

For our purposes, an ideal simulation should yield synthetic rainfall with Qi(T) suchthat historical and synthetic Fano factors at mean storm duration are equal, or based on Eqs.
10 and 11:

⎛T ⎞
Ai
B −1
TMi i = 1 + ⎜⎜ Mi ⎟⎟Ci⎝ T0 ⎠

ξSi(12)

After some algebra, we may isolate the synthetic Fano factor exponent ξSi shown hereas Eq. 13:⎡A⎤( B −1)ln ⎢ i TMi i − 1 ⎥Ci⎦ξSi = ⎣
ln[TMi ] − ln[T0 ](13)The expression for TMi in terms of the NSM parameters is based on Cowpertwait’s (1991):
⎛
⎡
η ⎤⎞
⎟
TMi = 1 ⎜⎜ γ + ln ⎢(μ c − 1)β
η − β ⎥⎦ ⎟⎠
⎣
⎝

(14)

where γ is the Euler constant (0.577…). Substituting this approximation for TMi in Eq. 13,
we have:ξ Si =(Bi −1)⎡⎤⎧⎪ 1 ⎛
⎞⎫⎪
A
⎡
⎤
η
i
⎢
⎜
⎟
ln
−1⎥
⎨ γ + ln ⎢(μ c − 1)⎬
⎢ C i ⎪⎩ β ⎜⎝⎥
η − β ⎥⎦ ⎟⎠⎪⎭
⎣
⎣
⎦

⎡1 ⎛
⎡
η ⎤ ⎞⎤⎟⎥ − ln[T0 ]ln ⎢ ⎜⎜ γ + ln ⎢(μ c − 1)
η − β ⎥⎦ ⎟⎠⎦⎥⎣⎣⎢ β ⎝(15)

Normally, plots similar to Figure 2 taken from the historical data have shown that thevalue of T0 in Eq. 15 can be taken as 1 hour. In this case, our NSM FFE reduces to:ξ Si =(Bi −1)⎡⎤⎧⎪ 1 ⎛
⎞⎫⎪
A
⎡
⎤
η
i
⎥
⎜
⎟
γ
+
μ
−
ln ⎢
−
ln
1
1
(
)
⎨
⎬
⎢ c
⎥
⎢ C i ⎪⎩ β ⎜⎝
η − β ⎥⎦ ⎟⎠⎪⎭
⎣
⎣
⎦

⎡1 ⎛
⎡
η ⎤ ⎞⎤⎟⎥ln ⎢ ⎜⎜ γ + ln ⎢(μ c − 1)
η − β ⎥⎦ ⎟⎠⎦⎥⎣⎣⎢ β ⎝(16)

Moreover, using Eq. 6 to estimate the historical counterpart ξHi, it is now possible toincorporate the FFE in the NSM parameter estimation (shown in the next subsection).

2.3

Parameter estimation of the NSM

The parameter estimation of the NSM is based on the solution of the following
objective function:⎛
NSM i ⎞⎟
O(Ω ) = ⎜⎜1 −HISi ⎟⎠⎝

2(17)

where NSMi pertains to the component expression listed in Eqs. 1 – 5, and 16 while HISi
pertains to the historical counterpart (see Appendix for historical counterpart of eq. (6)).
Two combinations of components are used for parameter estimation for synthetic rainfall
generation from the NSM. Scheme A is a configuration based on the TCM in which eq. (17)adopts the hourly mean, hourly variance, hourly autocorrelation at lag-1, hourly third central
moment, 12-hourly autocorrelation at lag-1, 24-hourly variance, and 24-hourly
autocorrelation at lag-1. Scheme B is essentially Scheme A, only that eq. (17) adopts thehourly FFE instead of the hourly TCM.

2.4

Description of adopted streamflow modeling method

Two distributed hydrologic models (DHM) were developed from OHyMOS (Ichikawa,
2000). One model was developed for the Kamo river basin located in Kyoto Prefecture while
another one was developed for Kamishiiba river basin in Miyazaki Prefectures in Japan.
Both study areas were less than 300 Km2 in size for which the use of a single rain gauge isjustified. Rainfall-runoff conversion is implemented following a kinematic wave model
based on a function similar to a discharge-stage relationship (Tachikawa et al., 2007).
Parameters for these conversion equations are assumed constant for all the calculation
elements in the river basin although internal systems are available for changing theseparameters per sub-basin.
These parameters were estimated in separate studies (Tachikawa
et al., 2007 for Kamo and Lee et al., 2007 for Kamishiiba). The simulations conducted withthese tuned river basin models were set for monthly runs at 5 minute computation time withstreamflow results displayed per hour.

3.

EXPERIMENTAL SETUP

Methods for calculating extreme floods are based on two procedures. One procedure
is based on the Japan Ministry of Land, Infrastructure, Transport and Tourism (MLIT) while
another is based on the NSM synthetic rainfall. For stationarity considerations, bothmethods are applied for historical rainfall pooled monthly during June, assumed to be
representative of the rainy season in all study regions.

3.1

MLIT method

A procedure for estimating q-return period design floods from historical rainfall andstreamflow simulation based on the Japan Ministry of Land, Infrastructure, Transport, andTourism (MLIT, 2008) is designated here as M-I. This procedure starts with assigning adesign duration from which the quantile rainfall for all simulations is based. For river basin
areas such as Kamo and Kamishiiba, each less than 300 Km2 in size, this duration is specified
as 24 hours. The corresponding q-return period quantile rainfall depth is then calculatedbased on 24-hourly aggregated historical rainfall record. A 24-hourly distribution to thistotal magnitude is assigned by searching through the historical records for the corresponding
24-hourly maximum rainfall per month. These maximum storms are then proportionately
modified such that the total rainfall within the 24-hour period is the basis quantile rainfall.
The MLIT then runs each 24-hourly storm in an appropriate model to simulate
streamflow. There are thus as many resulting hydrographs from these simulations as there
are historical rainfall records. Each hydrograph is then searched for its maximum value,
yielding the estimate for the design flood, say Smq. Consequently, M-I gives several q-return
period design flood estimates based on the Smq of each simulated hydrograph.

3.2

Method based on NSM rainfall
A second streamflow generating procedure is designated here as M-II. We generate
100 synthetic records of the target month for our applications here where return periods arewithin the 100 year value. Each monthly record is then run through the appropriate river
basin model for streamflow simulation.
The resulting hydrographs are searched for maximum streamflow, resulting in 100
values for flood frequency analysis. The required quantile event, say Qmq, is then estimatedfrom the cumulative distribution that best fits these empirical maxima (a log-normal
distribution fitted by least squares is adopted although other distributions may be used).
Two variants of this procedure correspond to the two NSM parameter estimation schemes ofSec. 2.3. M-IIA adopts NSM O(Ω) Scheme A while M-II adopts that of Scheme B forparameter generation.

4.

RESULTING STREAMFLOW ESTIMATES

Historical streamflow data was limited in this study. Each historical June rainfall
record is also run through the appropriate DHM and is considered a suitable substitute forhistorical streamflow. Resulting quantile estimates from the synthetic streamflow andpseudo-historical counterparts shown here are limited to the hourly duration. Quantile
streamflow are estimated for the 10-, 20-, 30-, 50-, and 100-year return periods.

4.1

Kamo river basin streamflow

The quantile-quantile (q-q) plot of Kamo pseudo-historical hourly streamflow maxima(Dis) appears in Figures 3a-b. Each maximum streamflow value is given a plotting position
pp proportionate to its rank in the overall record (pp=1/(i+1) in which i is rank). The lognormal distribution quantiles are used in this figure such that the independent variable (related
to the event return period) corresponds to the inverse of the standard normal distribution(Gaussian distribution with zero mean and variance of unity) Φ−1 of pp while the dependent
variable corresponds to the logarithm of Dis. Linear regression gives us the parameters forthe fit of this log-normal model that leads to the 95% confidence bands.
We may then project the best fit line for the pseudo-historical streamflow (Figure 3a)
to extrapolate the trend at return periods 10-, 20-, 30-, 50-, and 100-years. There is atendency of M-I to give a wide range of estimates for these target quantile floods in Figure 3a.
In fact, the variation of M-I estimates becomes wider along with increasing return period.
There is thus a pronounced ambiguity in the quantile estimates of M-I, making it
disadvantageous despite its simple approach of using historical data alone. In other words, itwould be difficult to depend on M-I to quantify the quantile events given that we cannot
justify which among the multiple estimates is the most likely value.

Kamo Maximum Discharge
Log-Normal Quantile-Quantile Plot

Kamo Maximum Discharge
Log-Normal Quantile-Quantile Plot

8.00

8.00
R2 = 0.9906

7.00

7.00

6.00

403.4288 CMS

ln(Dis), ln(Qm)ln(Dis), ln(Sm)

6.00
5.00

54.5982 CMS

4.00
3.00

4.00

54.5982 CMS

3.00
7.3890 CMS

2.00

7.3890 CMS

2.00

403.4288 CMS

5.00

Return Pd.
T = 1.02 years

1.00
1.00

Return Pd.
T = 1.02 years

Return Pd.
T = 1.19 years

-2.00

-1.00

0.00
-3.00

Return Pd.
T = 2.0 years

Return Pd.
T = 6.30 years

Return Pd.
T = 43.96 years

0.00

1.00

2.00

0.00
-2.50

-2.00

Return Pd.
T = 1.19 years

-1.50

-1.00

Return Pd.
T = 2.0 years-0.50

Return Pd.
T = 6.30 years

0.00

3.00

0.50

1.00

Return Pd.
T = 43.96 years

1.50

2.00

2.50

Φ-1(pp)Φ-1(pp)Pseudo-Historical

10Y-MIIA

20Y-MIIA

30Y-MIIA

50Y-MIIA

Pseudo-Historical

10-YEAR

20-YEAR

30-YEAR

100Y-MIIA

10Y-MIIB

20Y-MIIB

30Y-MIIB

50Y-MIIB

50-YEAR

100-YEAR

95% confidence band

Linear (Pseudo-Historical)

100Y-MIIB

95% confidence band

Linear (Pseudo-Historical)

Figure 3. Quantile estimates from Kamo River Basin generated from (a) MLIT procedure M-I
and (b) NSM based procedure M-II.
This ambiguity does not appear in M-II, shown in Figure 3b. Estimates appear to bequite consistent to the projections of pseudo-historical data. Scheme B results appear to be
the more rational estimate since this scheme includes POT rainfall maxima information in the
FFE. The advantage of using M-II, which involves synthetic rainfall generation, is therefore
its clearer and unambiguous estimates of the quantile events.

4.2

Kamishiiba river basin streamflow

Similar q-q plots based on Kamishiiba results appear in Figures 4a-b. Not all
quantile estimates generated from M-I are within reasonable proximity to what can be drawn
from the pseudo-historical counterpart, as shown in the lower return periods (10-year and 20year estimates), indicating poor performance. In fact, quantiles should be evaluated at
higher return periods (i.e.: higher than 100 years) before the M-I method yields estimates withhigh variation that lie along the pseudo-historical 95% region. This however is not the ideal
application since at times, one needs an estimate of lower return periods (i.e.: urban
conditions/low priority flood protection works). The M-I scheme therefore generates poor
estimates of the required quantiles in Kamishiiba River Basin.
Figure 4b shows the same estimates generated from M-IIA and M-IIB. Most
quantile estimates are within the historical 95%region of historical quantiles, indicating better
performance over the M-I estimates (of Figure 4a). In fact, both schemes perform
appreciably well given that both yield almost the same low return period estimates andgradually diverge at higher return periods within the 95% historical region. Therefore, either
adopting M-IIA or M-IIB yields reasonable quantile estimates.
Kamishiiba Maximum Discharge
Log-Normal Quantile-Quantile Plot

Kamishiiba Maximum Discharge
Log-Normal Quantile-Quantile Plot
8.00

10.00
8.00

2980.9580 CMS

7.00
6.00

2980.9580 CMS

403.4288 CMS

6.00

ln(Qm)ln(Sm)

5.00
403.4288 CMS

4.00
3.00

4.00

2.00
2.00

7.3890 CMS

0.00
-2.00

7.3890 CMS

1.00
Return Pd.
T = 2.0 years

Return Pd.
T = 1.19 years

-1.50

-1.00

-0.50

0.00

Return Pd.
T = 6.30 years

0.50

1.00

Return Pd.
T = 43.96 years

1.50

2.00

0.00
-2.00
2.50

Return Pd.
T = 1.19 years

-1.50

-1.00

Return Pd.
T = 6.30 years

Return Pd.
T = 2.0 years

-0.50

3.00

0.00

0.50

1.00

Return Pd.
T = 43.96 years

1.50

2.00

2.50

3.00

Φ-1(pp)Φ-1(pp)Pseudo-Historical

100Y-MI

10Y-MIIA

20Y-MIIA

30Y-MIIA

Pseudo-Historical

10-YEAR

20-YEAR

30-YEAR

50Y-MIIA

100Y-MIIA

10Y-MIIB

20Y-MIIB

30Y-MIIB

100-YEAR

50-YEAR

95% Confidence Band

Linear (Pseudo-Historical)

50Y-MIIB

5% confidence band

Linear (Pseudo-Historical)

Figure 4. Quantile estimates from Kamishiiba River Basin generated from (a) MLIT
procedure M-I and (b) NSM based procedure M-II.

5.

CONCLUSION

A comparison between several methods was conducted that led to the advantage ofusing synthetic rainfall in the estimation of critical streamflow in river basins with limited
historical rainfall
and/or streamflow data. Synthetic rainfall was based on the NeymanScott clustered Poisson rectangular pulse rainfall model (NSM). The streamflow generated
from modeling historical rainfall through a distributed hydrological modeling (DHM) was
assumed as an equivalent to historical streamflow (referred to as pseudo-historicalstreamflow).
Results indicate that an established method from the Japan Ministry of Land,
Infrastructure, Transport, and Tourism (MLIT) for estimating design floods from historicalrainfall have several limitations. Estimates from this method (for the Kamo River Basin in
Kyoto) vary widely for any return period due the use of multiple design rainfall that are eachplausible occurrences of the quantile event. In one application (Kamishiiba River Basin in
Miyazaki), results were in gross error for low return periods. Though this method involved
only historical rainfall, estimates were found to be generally unreliable.
Another method involving NSM synthetic rainfall generation appears to be morerational in both form and delivered results. There were no ambiguous or erroneous estimatesfrom the results of this method. Estimates based on this method are therefore more reliable
and recommendable than those of the former type.

REFERENCES
Beven, K. (2002): Rainfall Runoff Modelling: The Primer, John Wiley & Sons Ltd, West
Sussex.
Cowpertwait, P. S. P. (1998): A Poisson-cluster model of rainfall: high-order moments andextreme values, Proceedings of the Royal Society of London A, Vol. 454, pp. 885-898.
Cowpertwait, P. S. P. (1991): Further developments of the Neyman-Scott clustered point
process for modeling rainfall, Water Resources Research, Vol. 27, No. 7, pp. 1431-1438.
Fano, U. (1947): Ionization yield of radiations II. The fluctuations of the number of ions,Physics Review 72, pp. 26-29.
Ichikawa Y., Tachikawa, Y., Takara, K., & Shiiba, M. (2000): Object-oriented HydrologicalModeling System, Proc. of Hydroinformatics 2000, [CD-ROM].
Lowen, S. B. and Teich, M. C. (2005): Fractal-Based Point Processes, John Wiley & Sons,
Inc., New Jersey.
Lowen, S. B. and Teich, M. C. (1995): Estimation and simulation of fractal stochastic point
processes, Fractals, Vol. 3, No. 1, pp. 183-210.
Lee, G., Tachikawa, Y., and Takara, K. (2007): Identification of model structural stability
through comparison of hydrologic models, Annual Journal of Hydraulic Engineering,
JSCE, Vol. 51, pp. 49-54.
MLIT (2008): 国土交通省 河川砂防技術基準, Japan Ministry of Land, Infrastructure,
Transport, and Tourism, retrieved Feb. 26, 2008, <http://www.mlit.go.jp/river/shishin_guideline/gijutsu/gijutsukijunn/gijutsukijunn.pdf>.
Mondonedo, C., Tachikawa, Y., and Takara K. (2008): POT normalized variance parametersearch of the temporal Neyman-Scott rainfall model, Annual Journal of HydraulicEngineering, JSCE, Vol. 52, pp. 97-102.
Rodriguez-Iturbe, I., Cox, D. R., and Isham, V. (1987): Some models for rainfall based onstochastic point processes, Proceedings of the Royal Society of London A, Vol. 410, pp.
269-288.

Sefozo, R. F. (1990): ‘Point Processes’ in Heyman, D. P. and Sobel, M. J. (eds.) Nemhauser,
G. L. and Rinnooy Kan, A. H. G. (series eds.), Handbooks in Operations Research andManagement Science Vol. 2: Stochastic Models, Elsevier Science Publishers, pp. 1-93.
Telesca, L., Lapenna, V., Scalcione, E., and Summa, D. (2007): Searching for time-scaling
features in rainfall sequences, Chaos, Solitons, and Fractals, No. 32, pp. 35-41.
Tachikawa, Y., Takubo, Y. Sayama, T., and Takara, K. (2007): Large flood prediction inpoorly gauged basins: the 2004 largest-ever flood in Fukui, Japan, Methodology inHydrology (Proceedings of the Second International Symposium on Methodology inHydrology held in Nanjing China October-November 2005), IAHS Publ 311, pp. 18-24.