1 STATISTICAL ESTIMATION OF WAVE CLIMATES IN A MONSOON REGION Akira Kimura1 and ...

15 Pages·2012·0.93 MB·English

STATISTICAL ESTIMATION OF WAVE CLIMATES IN A MONSOON REGION
Akira Kimura1 and Takao Ota2
This paper deals with a statistics of wave climate. NOWPHAS data from 7 observatories along the Sea of Japan coast
are investigated. Fourier spectrum of one year time series of significant wave height shows inherent
characteristic to the monsoon region. It can be divided into three frequency regions; (A) , (B)
and (C) where . Simple but 𝐻ap𝟑p⁄𝟑ropriate sp𝟑e𝟑ctrum
modelin𝟑g𝟑 for the region𝟑 𝟑is proposed in th𝟑i𝟑s study. It is𝟑 𝟑possible to 𝟑m𝟑ake Monte-Carlo simulatio𝑓ns< of0 w.0a0v6e 𝑑climates
a0p.0p0ly6i𝑑ng th<e 𝑓m<od0el. 2v 𝑑ery easily. All 0st.a2t𝑑istica<l 𝑓pr<ope2r𝑑ties can be c𝑑alcu=late1d⁄ 𝑑fr𝑎o𝑦m the simulated data. No formulation
for the frequency distribution of wave properties such as annual maximum wave height is necessary. Some useful
applications of the present study are shown in the last.

Keywords: wave climate, spectrum of a time series of , simulation of wave statistics, Monte-Carlo method
𝐻𝟑⁄𝟑
INTRODUCTION
Design manual for public structures revised in Japan in 2007. The new manual adopts a concept of
performance based design. Slight failures of a structure are allowed as far as the structure keeps the demanded
performance. In the former manual, a specification based design was used so that damage must not occur during
the lifetime. Design wave conditions were determined applying an extreme value statistics (Goda, 2000). After the
revision, it becomes necessary to estimate all waves which may damage the structure during its lifetime. For better
design and management of structures after the revision, statistics of wave climate become necessary. However the
number of relating researches is very limited. Hirose et al., (1982) showed that a frequency distribution of a daily
averaged is very close to the Weibull distribution. A difficult point in wave climate study is that some data
in the population have correlations. However statistic theory demands the data independency. On the other hand, it
is necessar𝐻y𝟑 ⁄to𝟑 show the wave climates in a simple form for its better application to the life cycle management
(LCM) of the structure. The present study aims to overcome the restriction of the theory and to clarify statistical
characteristics of the wave climates. A spectrum concept is introduced. W data from 7 observatories
(NOWPHAS) along the Sea of Japan coast from 1991 to 2008 (18 years) were investigated and showed one year
time series of have a inherent spectral characteristic to the monsoon regiona vreegardless of the observatories
and years. The spectrum can be divided into three frequency regions, (A) , (B)
and (C𝐻) 𝟑⁄𝟑 where . The spectrum in the region𝟑 (𝟑A) decides the𝟑 s𝟑easonal
chang𝟑e𝟑 of with a𝟑 s𝟑mall numbe𝟑r𝟑 of Fourier 𝟑c𝟑omponents. Very simple m𝑓od<el0in.0g 0f6or 𝑑 the region0s.0 (0B6) 𝑑and <(c)𝑓 ar<e
p0o.2s s𝑑ible. Statistica0l. 2p𝑑rope<rti𝑓es< of2 𝑑wave climat𝑑es a=re 1s⁄im𝑑u𝑎l𝑦ated by Monte-Carlo method using the models. The
present stud𝐻y𝟑 ⁄i𝟑s applicable to study not only wave climates but also extreme wave statistics. Some applications
which may be useful in the LCM are shown in the last.
STATISTICAL PROPERTIES OF THE OBSERVED DATA
The present method bases on the observed data from 7 observatories. Figure 1 shows the locations of the
observatories. Figures 2(a) and 2(b) show examples of one year data (observed every 2 hours). These figures
show the data from the coast in the north and center of the Sea of Japan (Fig.1). The numbers in the horizontal axis
are the days counted from Jan. 1st. Here after the same horizonta𝐻l 𝟑a⁄x𝟑is is used. Data in the both ends of the figures
(winter) are large, and small in the center part (summer). The term “monsoon” in the title of the present study
means that data (wave climate) show seasonal characteristics like Fig.2. Figures 3(a) and 3(b) show spectra
(absolute value) of one year time series of . The spectrum is given by
𝐻𝟑⁄𝟑 exp 𝐹′(𝑓𝟑)
𝟑𝟑𝟑𝟑
in which = , 𝐹′( 𝑓i𝟑s) a= d𝟑iscre𝐻te𝟑⁄ 𝟑f(re𝑡q𝟑)uenc(y− 𝑗w2i𝜋th𝑓 𝟑a𝑡𝟑n) , i n t e r v(a𝑛l =of0 ,1,2,⋯,2190 )w h e r e (1)
𝟑𝟑𝟑
, is a𝟑 t𝟑ime sequence with 2 hours interval, data number of one y𝟑e𝟑ar total is 𝟑43𝟑80
( 𝑗 √). −1 𝑓𝟑 = i𝑛n⁄ F3i6gs5. ∙3𝑑(a) and 3(b) show the shape that looks like very we1ll⁄. 3F6ig5u∙re𝑑 4 shows a p𝑑atter=n
d1i⁄a𝑑g𝑎ra𝑦m 𝑡o𝟑 f( 𝑖th=e 1s,h2a,p⋯e., 4T3h8e0 s)pectrum can be divided into 3 frequency regions. Their frequency region and
c=ha1ra2ct×er3is6ti5c are|𝐹 l′i(s𝑓te𝟑d) |in Table 1. The upper bound of the region in (C) is set to avoid aliasing effect.
In this study, totally 128 years (7 observatories are inve𝟑s𝟑tigated. Due to space limitation,
only limited number of results is shown here after. However there are no 𝑓sig<ni2fi𝑑cant differences between the shown
and other cases. 𝐻𝟑⁄𝟑 ×18 years)

1 NEWJEC, Koyama Kita 6-448-69, Tottori, 680-0941, Japan
2 Tottori Univ., Faculty of Eng., Koyama Minami 4-101, Tottori, 680-8552, Japan
1
2 COASTAL ENGINEERING 2012

Rumoi
Setana
Sea of Japan
ta
Akita
Niigata
Wajima
Tottori
Hamadaa

Figure 1. Locations of observatories.

7
6
5
m) 4 Rumoi 1994
(

1/3 3
H
2
1
0
0 50 100 150 200 250 300 350
days
(a) Rumoi (1994)
7
6
5
4 Niigata 1998
1/3
H 3
2
1
0
0 50 100 150 200 250 300 350
days
(b) Niigata (1998)
Figure 2. Observed one year time series of , (a) Rumoi (1994) and (b) Niigata (1998)

𝐻𝟑⁄𝟑
COMPONENTS FROM 3 FREQUENCY REGIONS
Seasonal characteristics
can be written as sum of components from 3 regions.
𝐻𝟑⁄𝟑(𝑡)
COASTAL ENGINEERING 2012 3

in which , and are wave forms fr𝐻o𝟑m⁄𝟑 e(a𝑡)ch= re𝐻g𝟑i(o𝑡n) A+, 𝐻B𝟑 ′a(n𝑡)d +C 𝐻r𝟑es′(p𝑡e)c t i v e l y . T h e y a r e g i v e n b y (th2e)
inverse Fourier transformations.
𝐻𝟑 𝐻𝟑′ 𝐻𝟑′
exp
𝟑
𝐻𝟑(𝑡𝟑)=𝟑 𝐹′(𝑓𝟑) (𝑗2𝜋𝑓𝟑𝑡𝟑)⎫
𝟑𝟑𝟑 ⎪
𝟑𝟑 exp ⎪
⎪
𝐻𝟑′(𝑡𝟑)=𝟑 𝐹′(𝑓𝟑) (𝑗2𝜋𝑓𝟑𝑡𝟑) (𝑖=1,2,⋯,4380 ) (3)
𝟑𝟑𝟑 exp ⎬
𝟑𝟑𝟑
⎪
⎪
𝐻𝟑′(𝑡𝟑)= 𝟑 𝐹′(𝑓𝟑) (𝑗2𝜋𝑓𝟑𝑡𝟑)⎪
in which is the time (NOWPHAS data𝟑 𝟑e𝟑v𝟑ery 2 hours), , and
⎭
(Table 1). Figures show the examples : , 𝟑𝟑 , 5(c) 𝟑𝟑 and 5(d) 𝟑𝟑
(Tottori, 20𝑡0𝟑0). 𝑓𝟑≈0.006𝑑 𝑓𝟑𝟑=0.2𝑑 𝑓𝟑𝟑𝟑≈2𝑑
5(a)~5(d) 5(a) 𝐻𝟑⁄𝟑(𝑡𝟑) 5(b) 𝐻𝟑(𝑡𝟑) 𝐻𝟑′(𝑡𝟑) 𝐻𝟑′(𝑡𝟑)
1
6 Setana 1991
4
2
) 0.1
(m
6
)| 4 𝟑𝟑
fn 𝑓𝟑
( 2
'
F 0.01
|
6
4
2
0.001
0.001 0.01 0.1 1
fn (d-1)
(a) Setana (1991)
1
6
4 Hamada 2007
2
) 0.1
m
( 6
)| 4
n 𝟑𝟑
(f 2 𝑓𝟑
'
F 0.01
|
6
4
2
0.001
0.001 0.01 0.1 1
fn (d-1)
(b) Hamada (2007)
Figure 3. Spectrum of the observed one year time series of

|𝐹′(𝑓)| 𝐻𝟑⁄𝟑
4 COASTAL ENGINEERING 2012

A B C
1
6
4
2
m) 0.1
( 6
|
) 4 𝟑𝟑
(fn 2 𝑓𝟑
'
F 0.01
|
6
4 Rumoi 2001
2
0.001
0.001 0.01 0.1 1
fn (d-1)
Figure 4. Pattern diagram of the spectrum

Table 1 Frequency regions and characteristics
Region
A 0.0𝟑0𝟑6 quick decrease with
B 0𝑓.𝟑0 0 (6𝑑 0).2 almost constant 𝐹(𝑓𝟑)
C ~0. 2 decrease in proportio𝑓n𝟑 to
~ 𝟑𝟑
7 ~ 2 𝑓𝟑
6
5
(m) 4 Tottori 2000
H 1/33
2
1
0
0 50 100 150 200 250 300 350
days
Figure 5(a) (Tottori, 2000)
2.5
𝐻𝟑⁄𝟑(𝑡𝟑)
2.0
1.5
m) Tottotttoorrii 22000000
(
HA1.0
0.5
0.0
0 50 100 150 200 250 300 350
days
Figure 5(b) (Tottori, 2000)
3
𝐻𝟑(𝑡𝟑)
2
Tottori 2000
) 1
(m
H'B 0
-1
-2
0 50 100 150 200 250 300 350
days
Figure 5 (c) ′ (Tottori, 2000)
𝐻𝟑(𝑡𝟑)
COASTAL ENGINEERING 2012 5

3
2
Tottori 2000
H' (m)C 10
-1
-2
0 50 100 150 200 250 300 350
days
Figure 5 (d) ′ (Tottori, 2000)

𝐻𝟑(𝑡𝟑)
2.0
Tottori 2000
0.6
1.5
HA
HA (m) 1.0 H'Bsm 0.4' (m)sBm
H
0.5 0.2
0.0 0.0
0 50 100 150 200 250 300 350
days
(a) and
2.0 𝟑
𝐻𝟑(𝑡𝟑) 𝐻𝟑𝟑𝟑(𝑡𝟑)
Setana 2003
1.5 0.6
(m) 1.0 HH'ACsm 0.4 (m)sm
HA 'C
H
0.5 0.2
0.0 0.0
0 50 100 150 200 250 300 350
days
(b) and
𝟑
Figure 6. Comparison between (solid line ; 𝐻le𝟑ft( 𝑡a𝟑)xis) and𝐻 m𝟑𝟑o𝟑nt(h𝑡l𝟑y) standard deviation (black circle ; right axis) of
(a) ′ (Tottori, 2000) and (b) ′ (Setana,2003)
𝐻𝟑(𝑡𝟑)
Figures 6(a) and 6(b) show co𝐻m𝟑p(a𝑡r)isons between (l𝐻e𝟑ft( 𝑡a)xis) and the standard deviation of (a)
and (b) (right axis), respectively. and are the monthly standard deviation of
and . is a day𝟑 at ce𝐻nt𝟑e(r𝑡 e𝟑)very𝟑 month. In the both figures, good agreeme𝐻n𝟑ts′ (a𝑡r𝟑e)
observed.𝐻 T𝟑′h(i𝑡s𝟑 )means that seasonal characte𝐻r𝟑is𝟑ti𝟑cs( 𝑡𝟑o)f 𝐻𝟑 𝟑a𝟑nd(𝑡 𝟑) are almost proportional to .
𝐻Th𝟑e′(re𝑡𝟑f)ore the 𝐻se𝟑a′(s𝑡o𝟑n)al 𝑡c𝟑ha(r𝑙a=cte1ri,s⋯tic,1s 2ca)n be removed by dividing and by .
𝐻𝟑′(𝑡) 𝐻𝟑′(𝑡) 𝐻𝟑(𝑡𝟑)
𝐻𝟑′(𝑡) 𝐻𝟑′(𝑡) 𝐻𝟑(𝑡𝟑)
Figures 7(a) and𝐻 7𝟑((b𝑡)𝟑 )sh=ow𝐻 𝟑′(𝑡𝟑)⁄𝐻 𝟑an(𝑡d𝟑 ) 𝐻,𝟑 r(e𝑡s𝟑p)e=cti𝐻ve𝟑l′y(.𝑡 𝟑C)⁄or𝐻r𝟑es(p𝑡𝟑o)n d i n g (𝑖=1, ⋯an,d4 380) a r e s h o (w4n)
in Figs. 5(c) and 5(d). Seasonal characteristics of Figs. 5(c) and 5(d) are not seen in the figures. Since the spectrum
width of and ar𝐻e 𝟑v(e𝑡r𝟑y) narrow𝐻 𝟑a(n𝑡d𝟑 )their monthly standard deviatio𝐻ns𝟑 ′a(r𝑡e𝟑 )almost 𝐻co𝟑n′(s𝑡t𝟑a)nt, it can be
assumed that these wave forms are stationary.
𝐻𝟑(𝑡𝟑) 𝐻𝟑(𝑡𝟑)
6 COASTAL ENGINEERING 2012

3
2 Tottori 2000
1
)
m
(

HB 0
-1
-2
0 50 100 150 200 250 300 350
days
(a)
3
𝐻𝟑(𝑡𝟑)
2
Tottori 2000
1
)
m
(

HC 0
-1
-2
0 50 100 150 200 250 300 350
days
(b)
Figure 7. Seasonal characteristics are removed from data in Figs. 5(c) and 5(d) using eq.(4) (Tottori, 2000).
𝐻𝟑(𝑡𝟑)
0.1
| 8
) 6
n
(f 4
F
|
2

,
|) 0.01 Setana 2006
n 8
(f 6
F' 4
| |F'(fn)|: with seasonal change
2 |F(fn)|: without seasonal change
0.001
2 3 4 5 6 7 8 9 2
0.01 0.1
fn (d-1)
(a) Comparison of the spectrum of and
2
𝐻𝟑′(𝑡𝟑) 𝐻𝟑(𝑡𝟑)
0.1
8
| 6
)
fn 4
(
F
| 2

,
|) 0.01
fn 8 Setana 2006
( 6
'
F 4
| |F'(fn)|: with seasonal change
2 |F(fn)|: without seasonal change
0.001
2 3 4 5 6 7 8 9
-1 1
fn(d )
(b) Comparison of the spectrum of and
Figure 8. Comparison of the spectrum with and without seasonal characteristics (Setana,2006).
𝐻𝟑′(𝑡𝟑) 𝐻𝟑(𝑡𝟑)

New spectrum for and are given as
𝐻𝟑(𝑡𝟑) 𝐻𝟑(𝑡𝟑)
COASTAL ENGINEERING 2012 7

exp
𝟑𝟑𝟑𝟑
𝐹(𝑓𝟑)= 𝟑 𝐻𝟑(𝑡𝟑) (−𝑗2𝜋𝑓𝟑𝑡𝟑), (𝑛=3,⋯,73) ⎫
⎪
𝟑𝟑𝟑
 𝟑𝟑𝟑𝟑 exp (5)
⎬
Figures 8(a) and 8(b) show com𝐹p(a𝑓𝟑ri)so=ns𝟑 betw𝐻e𝟑e(n𝑡 𝟑t)he sp(−ec𝑗t2ra𝜋 f𝑓o𝟑r𝑡 𝟑()a,) (𝑛= a7n4d, ⋯,731) ⎪and (b) and
(Setana, 2006). There are no signific𝟑a𝟑n𝟑t changes in the spectra before and after the rem⎭oval of the seasonal
characteristics. 𝐻𝟑′(𝑡𝟑) 𝐻𝟑(𝑡𝟑) 𝐻𝟑′(𝑡𝟑)
𝐻 𝟑(𝑡𝟑)
Phase of
From eq.(5), phase spectrum of is given as
𝑭(𝒇𝒏)
𝐹(𝑓𝟑)

𝟑𝟑 Im𝟑𝐹(𝑓𝟑)𝟑
𝜀𝟑=tan 𝟑− 𝟑 (𝑛=3,4,⋯,731) (6)
in which and mean the real and imagRinea𝟑r𝐹y( 𝑓p𝟑a)r𝟑ts, respectively. Figures 9(a) and 9(b) show
frequency distributions of every year for (a) (Hamada) and (b) (Wajima). The
distributionRse a(re )almost Iumni(for)ms.
𝐹0.𝑅4(𝜀𝟑) 𝜀𝟑 𝐻𝟑(𝑡𝟑) 𝐻𝟑(𝑡𝟑)
Hamada 1991-2008
0.3
)
(en0.2
R
F
0.1
0.0
1 2 3 4 5 6
en
(a)
0.30
𝜀𝟑 (𝑛=3,⋯,73)
Wajima 1991-2008
0.25
0.20
)
(en0.15
R
F
0.10
0.05
0.00
1 2 3 4 5 6
en
(b)
Figure 9. Frequency distributions of the phase of (a) (Hamada) and (b) (Wajima)
𝜀𝟑 (𝑛=74,⋯,731)
𝐻𝟑 𝐻𝟑
MODELING OF
Equation (2) is rewritten as,
𝑯𝟏⁄𝟑(𝒕)

Spectrum of 𝐻an𝟑⁄d𝟑 (𝑡𝟑)=𝐻 a𝟑r(e𝑡 𝟑a)p{p1r+ox𝐻im𝟑a(t𝑡e𝟑d) +sim 𝐻p𝟑ly( 𝑡a𝟑s) }( F i g . 4 ) (, 𝑖 =1,⋯,4380) (7)
𝐻𝟑(𝑡𝟑) 𝐻𝟑(𝑡𝟑)
|𝐹(𝑓𝟑)|𝟑 =𝛼𝟑
in which and are the abso|lu𝐹t(e𝑓 𝟑va)|lu𝟑e=s o𝛼f 𝟑t𝑓he𝟑 𝟑m o d e l𝟑 s p e c t r u m i n t h e r e g i o n s A a n d B (8)
respectively, and are constants determined by the least square principle.
|𝐹(𝑓𝟑)|𝟑 |𝐹(𝑓𝟑)|𝟑
𝛼𝟑 𝛼𝟑
8 COASTAL ENGINEERING 2012

𝟑𝟑
𝟑 {|𝐹(𝑓𝟑)|−𝛼𝟑}𝟑 → min ⎫
⎪
𝟑𝟑𝟑
𝟑𝟑𝟑 (9)
𝟑𝟑 𝟑 ⎬
𝟑 {|𝐹(𝑓𝟑)|−𝛼𝟑𝑓𝟑 } → min ⎪
Calculated and show alm𝟑𝟑o𝟑s𝟑t constant value with small fl⎭uctuations. Figure 10 shows an
example of the values (Akita, 1991 - 2008).
𝛼𝟑 0.𝛼12𝟑
0.10
0.08

Akita
a 20.06 a1
a1 a
0.04 2
0.02
0.00
1992 1994 1996 1998 2000 2002 2004 2006 2008
year
Figure 10. Yearly change of and

Normal distributions for the fluctuation of and a𝛼r𝟑e assu𝛼m𝟑ed. Their means and standard
deviations , and , are calculated from the observed data.
𝛼𝟑 𝛼𝟑
𝛼𝟑 𝛼𝟑 𝛼𝟑𝟑 𝛼𝟑𝟑
SIMULATION OF
From eq.(7), is approximated as
𝑯𝟏⁄𝟑(𝒕)
𝐻𝟑⁄𝟑(𝑡) cos
𝟑𝟑
𝐻𝟑⁄𝟑(𝑡𝟑)=𝐻𝟑(𝑡𝟑)𝟑1+ 𝟑 (𝛼𝟑+𝛼𝟑𝟑𝑅𝟑𝟑) (2𝜋𝑓𝟑𝟑𝑡𝟑+𝜀𝟑𝟑)
𝟑𝟑𝟑𝟑
cos
𝟑𝟑𝟑
𝟑𝟑
+  𝟑 (𝛼𝟑+𝛼𝟑𝟑𝑅𝟑𝟑)𝑓𝟑𝟑 (2𝜋𝑓𝟑𝟑𝑡𝟑+𝜀𝟑𝟑)𝟑
𝟑𝟑𝟑𝟑𝟑
𝑡𝟑=𝑖𝑑𝑡 (𝑖=0,1,⋯) (10)
where and are independent normal ra𝜀n𝟑d𝟑o=m 2n𝜋um𝑅𝟑b𝟑er𝟑s, w 𝜀it𝟑h𝟑 0= m2e𝜋a𝑅n 𝟑a𝟑n𝟑d 1 . 0 s t a n d a r d d e v i a t i o n , (1an1d)
are also independent uniform random number . If , any value can be used for in
eq.(10).𝑅 H𝟑𝟑owever𝑅, i𝟑t𝟑 is preferable to use the same value as the observed data interval. is approxima𝑅t𝟑ed𝟑𝟑 as
𝑅𝟑𝟑𝟑 (0 ~1) 𝑑𝑡<𝑑⁄4 𝑑𝑡
𝐻𝟑(𝑡)
𝐻𝟑(𝑡𝟑)≈𝐻𝟑(𝑡𝟑)+𝐻𝟑(𝑡𝟑)𝑅𝟑𝟑 (12𝑎)
𝟑
1
𝐻𝟑(𝑡𝟑)= 𝑎𝟑+𝟑 𝟑𝑎𝟑sin(2𝜋𝑓𝟑𝑡𝟑)+𝑏𝟑cos(2𝜋𝑓𝟑𝑡𝟑)𝟑 (13)
2 𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟑⁄𝟑
𝟑
1
𝐻𝟑(𝑡𝟑)=𝟑 𝟑 𝟑𝐻𝟑𝟑(𝑡𝟑)−𝐻𝟑(𝑡𝟑)𝟑𝟑 ( 𝑖=1,2,⋯,4380) (14)
in which , 18𝟑𝟑, 𝟑𝟑𝟑𝟑 and is an average of 18 years
. is an average of 18 years . is the -th year (1991 2008) value of .
𝑎𝟑=Re𝟑𝐹′(𝑓𝟑)𝟑 𝑏𝟑=Im𝟑𝐹′(𝑓𝟑)𝟑 (𝑛=0,1,2) 𝐹′(𝑓𝟑) 𝐹′(𝑓𝟑)
is a standard deviation of 18 years . is a normal random number with 0 mean and 1.0
s{etaqn.d(a1r)d} de𝐻v𝟑ia(t𝑡i𝟑o)n. One year uses the value of s𝐻a𝟑m(e𝑡 𝟑) 𝐻.𝟑 𝟑(𝑡𝟑) 𝑚 ~ 𝐻𝟑(𝑡𝟑)
𝐻𝟑(I𝑡f𝟑 )the observation years are not sufficien𝐻tl𝟑y𝟑 l(o𝑡n𝟑)g, 𝑅fo𝟑l𝟑lowing definition for can be used instead of
eq.(12a). 𝑅𝟑𝟑
𝐻𝟑(𝑡𝟑)

in which is an average of . One year uses also the value of same . Since
itself and its fluctuation are small, there are no𝐻 b𝟑ig(𝑡 d𝟑)if≈fer𝐻en𝟑c(e𝑡s𝟑) b+etw𝐻e𝟑e𝑅n𝟑 e𝟑q s . ( 1 2 a ) a n d ( 1 2 b ) . (12b)
Consta𝐻n𝟑ts for the simulation𝐻 a𝟑t (e𝑡a𝟑c)h,( oib=se1rv,⋯ato,3ry8 a4r0e) listed in the Table 2. 𝑅𝟑𝟑 𝐻𝟑(𝑡𝟑)
COASTAL ENGINEERING 2012 9

Table 2 Constants for the simulation
Rumoi Setana Akita Niigata Wajima Tottori Hamada
1.119 1.188 1.088 1.000 1.197 1.093 1.119
0.777 0.805 0.680 0.726 0.777 0.650 0.597
𝑎𝟑
𝑎𝟑 0.093 0.088 0.144 0.106 0.093 0.062 0.081
0.150 0.057 0.120 0.142 0.150 0.101 0.091
𝑏𝟑
𝑎𝟑 0.067 -0.022 0.020 0.077 0.067 0.078 0.083
0.107 0.120 0.138 0.108 0.129 0.111 0.123
𝑏𝟑
0.0818 0.0817 0.1000 0.0824 0.0741 0.0777 0.0778
𝐻𝟑
0.00718 0.00594 0.00860 0.00634 0.00462 0.00557 0.00722
𝛼𝟑
0.0155 0.0150 0.0184 0.0157 0.0130 0.0134 0.0137
𝛼𝟑𝟑
0.00093 0.00085 0.00150 0.00107 0.00101 0.00105 0.00128
𝛼𝟑
Uinit: : , , : otherwise.
𝛼𝟑𝟑

𝑚⁄𝑑 𝛼𝟑 𝛼𝟑𝟑 𝑚
CORRECTION AND RESULTS
Figures 11(a) and 11(b) show observed and simulated one year (Hamada).
7
𝐻𝟑⁄𝟑(𝑡)
6
5
m) 4 Hamada 2000
(

3
1/ 3
H
2
1
0
0 50 100 150 200 250 300 350
days
(a) (observed)
𝐻𝟑⁄𝟑(𝑡)
6
4
)
m
(

3
/
H1 2
0
0 50 100 150 200 250 300 350
days
(b) (simulated)
Figure 11. Observed (Hamada, 2000) and simulated one year (Hamada)
𝐻𝟑⁄𝟑(𝑡)
Correction 𝐻𝟑⁄𝟑(𝑡)
Simulated in Fig.11(b) shows a similar seasonal characteristic to Fig.11(a). However some data show
negative value. Figure 12 shows frequency distributions of the observed { ; solid line, Wajima
𝐻𝟑⁄𝟑(𝑡)
1991-2008} and simulated { ; dotted line Wajima；10000 years simulation} .
𝑃𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑
𝑃𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑 𝐻𝟑⁄𝟑(𝑡)
10 COASTAL ENGINEERING 2012

0.8
Wajima
0.6
Observed (1991-2008)
Psim Simulated
, 0.4
Pobs
0.2
0.0
-1 0 1 2 3 4 5
H1/3 (m)
Figure 12. Frequency distributions of observed (solid line) and simulated (dotted line)

𝐻𝟑⁄𝟑(𝑡)
There is a difference between and in the area of . About 2% in all
the simulated values are negative. However and show good agreement in the area of
𝑃𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑 𝑃𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑 𝐻𝟑⁄𝟑(𝑡)<1.0 𝑚
. In this study, data are corrected applying frequency distribution conversion method. The
conversion is made using the cumulative distrib𝑃u𝟑t𝟑i𝟑o𝟑n𝐻s 𝟑w⁄𝟑h𝟑ich are 𝑃c𝟑a𝟑l𝟑cu𝟑l𝐻a𝟑te⁄d𝟑 𝟑as,
𝐻𝟑⁄𝟑(𝑡)>1.0 𝑚
𝟑𝟑⁄𝟑
𝐹𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑= 𝟑 𝑃𝟑𝟑𝟑(𝐻)𝑑𝐻 ⎫
⎪
𝟑𝟑𝟑
 𝟑𝟑⁄𝟑 (15)
⎬
𝐹𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑= 𝟑 𝑃𝟑𝟑𝟑(𝐻)𝑑𝐻⎪
in which and are the cumulative distributions𝟑 𝟑o𝟑f𝟑 and ⎭ respectively. Data are converted
using the following relation.
𝐹𝟑𝟑𝟑 𝐹𝟑𝟑𝟑 𝑃𝟑𝟑𝟑 𝑃𝟑𝟑𝟑

∗
in which is a target cumulative distribu𝐹t𝟑i𝟑o𝟑n 𝟑t𝟑o𝐻 c𝟑o⁄𝟑n𝟑v𝟑e𝟑r𝟑te=d 𝐹the𝟑 𝟑d𝐻at𝟑a⁄ 𝟑f𝟑r𝟑o𝟑m𝟑𝟑 t h e s i m u l a t e d d a t a . F o r t h e t a(r1g6e)t
distribution,∗ must be 0 and very close to . However if the observation period is
not sufficie𝐹ntly∗ long, uneven properties appear on the . In that case, modified cumulative distribution of
can be used. 𝐹Figure 13 show𝟑s𝐻 a𝟑 ⁄m𝟑o≤di0fi𝟑cation of the frequen𝐹c𝟑y𝟑𝟑 di𝟑s𝐻tr𝟑ib⁄𝟑ut>ion0.𝟑 Dotted line shows . Solid
line shows the function to modify . 𝐹is𝟑 𝟑g𝟑iven as 𝐹𝟑𝟑𝟑
𝑃𝟑𝟑𝟑−𝑃𝟑𝟑𝟑
𝐸𝟑𝟑𝟑 𝑃𝟑𝟑𝟑 𝐸𝟑𝟑𝟑
−𝑃𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑 A: 𝐻𝟑⁄𝟑≤𝐼𝟑
⎧
⎪ 𝐻𝟑⁄𝟑−𝐼𝟑
𝐸𝟑𝟑𝟑𝟑𝐻𝟑⁄𝟑𝟑= 𝟑2 −1𝟑𝑃𝟑𝟑𝟑(𝐼𝟑) 𝐼𝟑≤𝐻𝟑⁄𝟑≤𝐼𝟑 (17)
⎨ 𝐼𝟑−𝐼𝟑
30x10-3 ⎩⎪ 𝑃𝟑𝟑𝟑𝟑𝐼𝟑+𝐼𝟑−𝐻𝟑⁄𝟑𝟑 B∶ 𝐻𝟑⁄𝟑≥𝐼𝟑
B
Approximation
20 Pobs-Psim
10
Emod 0
C

-10
-20 A

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
H1/3 (m)
Figure 13. Modification of the frequency distribution

is a straight line between the points A and B (Fig.13). is point symmetric for C : .
and are determined so that difference between and becomes minimum .
In F𝐸ig𝟑.1𝟑3𝟑, and . Modified frequency distrib𝐸u𝟑tio𝟑n𝟑 is given by (𝐼𝟑−𝐼𝟑)⁄2,0
𝐼𝟑 𝐼𝟑 𝑃𝟑𝟑𝟑−𝑃𝟑𝟑𝟑 ∗𝐸𝟑𝟑𝟑 𝟑 𝐻𝟑⁄𝟑≥𝐼𝟑𝟑
Substituting𝐼𝟑 =0. 1i𝑚n eq.(15𝐼)𝟑, i=ts 0cu.3m𝑚u∗lative distribution is calculat𝑃e𝟑d𝟑𝟑.
∗ 𝑃𝑠𝑖𝑚𝟑𝐻1⁄3𝟑=𝑃𝑠𝑖𝑚𝟑𝐻∗1⁄3𝟑+𝐸𝑚𝑜𝑑𝟑𝐻1⁄3𝟑 (18)
𝑃𝟑𝟑𝟑 𝐹𝟑𝟑𝟑