A Newly Proposed Methodology andParadigm Shift Regarding to
Flood Control as Disaster Reduction
under the Uncertainty of Global Warming

Yamada Tadashi
Dept. of Civil and Environmental engineering, Facultyof Science and Engineering, Chuo universityChapter 1

Recent Heavy Rainfall Disaster
In JapanIntroduction

Serious flood disasters in JapanThe 2015 Kanto-Tohoku heavy rain disaster in Japan3

Introduction

Recent serious flood disasters in Japan

July 2018 “Nishi Nihon Heavy Rain”· Recordable heavy rain occurred in various parts of western Japan due to typhoon andbaiu front.
· Floods of rivers and sediment disasters occurred in many areas, mainly in western Japan.

July 2017 “Northern Kyushu Heavy Rain”· By influence of typhoon and Baiu front, flooding of rivers and large-scale landslides
occurred.
· Damage caused by driftwood flowing into rivers was remarkable.

August 2016 “Hokkaido Heavy Rain”· Recordable heavy rain over Hokkaido due to the landing and approach of the four
typhoons.
· Unprecedented wide area damage (flooding, outflow of pier, agricultural damage).

September 2015 “Kanto-Tohoku Heavy Rain”· Recordable heavy rainfall occurred in various places in the Tohoku region from the Kita
Kanto region.
· Rainfall precipitation concentrated in the Kinugawa river system, resulting breaking oflevee.

4

Introduction

Recent serious flood disasters in Japan

July 2018 “Nishi Nihon Heavy Rain”
・The number of dead：220
・The number of flooded houses：More than 34,200

July 2017 “Northern Kyushu Heavy Rain”

・The number of dead：37
・The number of flooded houses：More than 2,100

August 2016 “Hokkaido Heavy Rain”

・Damaged area：40,258 ha (3.5% of arable land area in Hokkaido)
・Total damage amount：3 billion USD (the highest amount ever recordedin Hokkaido)※1 USD = 100 yen

September 2015 “Kanto-Tohoku Heavy Rain”
・The number of dead：8
・Flooded house :More than 12,000

5July 2018 “Nishi-nihon Heavy Rain”

Seno River, Hiroshima Prefecture(Source) 道路構造物ジャーナル NET

Hiroshima Prefecture(Source) ふるさとチョイス

Hiroshima Prefecture(Source)ふるさとチョイス

Hiroshima Prefecture(Source)ふるさとチョイス

6

July 2017 “Northern Kyushu Heavy Rain” (As of 2018/09/19 12:00)(Source)毎日新聞,https://mainichi.jp/articles/20170920/k00/00m/050/171000c(Source)毎日新聞,https://mainichi.jp/articles/20170810/ddl/k44/040/298000c

Photographed on 21st August 2017
A large amount of driftwood is scattered upstream of Myoukengawa
river.

Photographed date unknown
A driftwood group approaching the private house in Turukawachi
district along with the muddy stream(Source)NET IB News,https://www.data-max.co.jp/article/18367
Photographed date unknown
Immediately after a disaster. You can see a vehicle drifted by driftwood.

(Source)NET IB News,https://www.data-max.co.jp/article/18367
Photographed date unknown
In Asakura city, near the Yamada intersection

7

July 2017 “Northern Kyushu Heavy Rain” (As of 2018/09/19 12:00)
流向(Source)NHK HP, http://blog.hitachinet.jp/archives/51662516.html(Source)時事ドットコム,
https://www.jiji.com/jc/d4?p=tyh017jpp024418545&d=d4_aum(Source)時事ドットコム,
https://www.jiji.com/jc/d4?p=tyh017jpp024418481&d=d4_aum

July 6 afternoon, Asakura city, Fukuoka prefectureRiver filled with a large amount of driftwood.

(Source)朝日デジタル,
http://www.asahicom.jp/articles/images/AS201707060
04556_comm.jpg

July 6 afternoon, Asakura city, Fukuoka prefecture Massive driftwood and private houses due to heavy rain.

8August 2016 “Hokkaido Heavy Rain”

Omoto River, Hokkaido Prefecture(Source)毎日新聞

Omoto River, Hokkaido Prefecture(Source)毎日新聞
Sorachi River, Hokkaido Prefecture

Sorachi River, Hokkaido Prefecture(Source)国土交通省「平成２８年台風第１０号による出水状況について」 (Source)国土交通省「平成２８年台風第１０号による出水状況について」9September 2015 “Kanto-Tohoku Heavy Rain”(Source)Signal(Source)ピースボート災害ボランティアセンター(Source) 国土交通省関東地方整備局(Source) ほっとメール＠ひたち

10Chapter 2

Relation between evacuation informationand situation of inundation at flooding

How to deal with serious flood disaster?
Heavy rain in the Kanto and Tohoku regions, September 2015

Tonegawa
River Basin

Kinugawa
River Basinoverflow
Junction point

Kinugawa River Basin area： 1,761[km2]
Tonegawa River Basin area：16,840[km2]Observed rainfall by “X Band MP Rader”(2015/9/9 9:00~)

12

How to deal with serious flood disaster?
Heavy rain in the Kanto and Tohoku regions, September 2015

線状降水帯

線状降水帯

2015/9/9 12:00

18:00

21:00

24:00
At Kinugawa river
Basin (Especially
upstream area), heavy
rainfall continued from
12a.m 9 Sep. to 10a.m
10 Sep. because ofBand-Shaped
Precipitation
System.(Precipitation:50~100[mm/h])

2015/9/10 3:00

6:00

9:00

13

Heavy rain in the Kanto and Tohoku regions, Sep.2015
Outline of Kinugawa River Basin

Until early days of Edo Period, Kokai riverjointed to Kinugawa river.
And Kinugawa river jointed Hitachi
river(Tonegawa river).
In 1629, Kinugawa river and Kokai river are
separated.

Quoted from「第1回鬼怒川・小貝川有識者会議」
Kanto Regional Development Bureau, Ministry of Land, Infrastructure, Transportation and Tourism

14

Heavy rain in the Kanto and Tohoku regions, Sep.2015
Outline of Kinugawa River Basin

鬼怒川流域

River is narrow and precipitous at 35～40km
section from the junction of Tonegawa river.

利根川流域
縦断距離

80km
Basin of Kinugawa river：1760km2
Length of main river：177km
Population inKinugawa river basin：About 550,000

Vertical distribution of river width

Quoted from『第1回鬼怒川堤防調査委員会資料』Ministry of Land, Infrastructure, Transportation and Tourism

Characteristic of basin：Over 60% is mountains, levelground is about 30%
15

Heavy rain in the Kanto and Tohoku regions, Sep.2015
Essentials of the Kinugawa River Basin

Eiju Yatsu(1966)：Rock Control in Geomorphology

elevation[m]The riverbed profile of the
Kinugawa River has two
exponential curves.

4
5

Longitudinal distance[km](from merging sections of Tone River)『Rock Control in Geomorphology(1966)』 by Prof. YaTsu

It pointed out that there are rivers
with two exponential curves forthe first time in the world (themost of the river longitudinal
profile rivers is an exponential
curve), the river bed sedimentparticle size at the place where
the river bed longitudinal form
folds I revealed that it ischanging by Prof. Yatsu．

16

Heavy rain in the Kanto and Tohoku regions, Sep.2015
Topographic characteristics of inundation area and flood condition of urban area
After 6:00
Start flooding

100m

12: 50Levee
breakdown

小
貝
川

鬼
怒
川

The natural embankment was excavatedby installing a solar panel※(Captured image on February 2, 2015)
The sandbag was piled up to the original
height when the flood happened※ Ministry of Land, Infrastructure andTransport Kanto Region Development
Bureau "About flood damage andrestoration situation related to the KantoTohoku heavy rain disaster in September
2015"＜ Inundation situation of urban area by inhabitant hearing survey ＞
Approximately 2 hours after levee Approximately 3 hours after
breakdown

Approximately 8 hours after

市街地

1km

Elevation map around the flood area Created byGeographical Information Authority of Japan (10m
DEM)

① Inland water flooded
in the tributary② the depth ofimmersion stopped③ Flood water fromthe main stream
reached→ The habitants mistake the Immersions from the main
steam for flood water and then too late to escape .

17

Heavy rain in the Kanto and Tohoku regions, Sep.2015
Reproduction of inundation situation in Joso city by flood inundation analysis
Basic equations (shallow water equations

八間堀川境界条件(計算流
2 量)
5
2
0
1
5
1
0
5

𝜕ℎ 𝜕𝑀 𝜕𝑁++=0
𝜕𝑡 𝜕𝑥 𝜕𝑦𝜕𝑀 𝜕 𝑀2
𝜕 𝑀𝑁
𝜕𝐻 𝑔𝑛2 𝑢 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑥
ℎ1/3
𝜕𝑁 𝜕 𝑀𝑁
𝜕 𝑁2
𝜕𝐻 𝑔𝑛2 𝑣 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑦
ℎ1/3

After 6:00
Start flooding

12: 50Levee
breakdown0M, N：x, y Flow flux
t： Time coordinatesx, y： Plane coordinates
h： Depth 、 g：Gravity
n：Roughness Coefficient
H： Water levelu, v：x, y flow velocity

-10

計算水深
3
(m)

6

標高(m)
10

30

Around 14:00
Inundation occurredin urban areas

Differentiated equations by Leap-frog method

Δx=Δy=10m、 Δt =0.2s
The roughness coefficient of a river
channel and a flood plain was
equally set to 0.03 [m-1/3 s]

18

Research on the behavior of evacuation
Evacuation situation by district at the time of disaster
Kinugawa river
overflowing place（Wakamiyado
left bank
25Km）

Kinugawa
鬼怒川riverOutburst
place
決壊箇所（Misakacho(三坂町 左岸21km)
Left bank 21Km）

避難し な
かっ た

(58.7% )N=516→There were 59% of the entire survey households
evacuated to shelters, and another 41% were at home
without evacuation.

70.0

Hachikenbori river
Outburst place

避難し た( 4 0 .5 % )A district(N=93) B district(N=82) C district(N=44) D district(N=94)
68.8
68.2
62.2

61.7

60.0
50.0
38.3

40.0

37.8
31.2

31.8

30.0
20.0
10.0

2.5km
District division map of survey

0.0
I evacuated

I did not evacuate

Most residents in district D did not evacuate.

19

Research on the behavior of evacuation

District A：Around the overflow area of the embankment of the Kinugawa River
Kinugawa river
overflowing place（Wakamiyado
left bank
25Km）

Kinugawa
鬼怒川riverOutburst
place
決壊箇所（Misakacho
Left bank 21Km）(三坂町 左岸21km)

Many residents started to evacuate at①1 hourinafter
most
residents
obtainedthe
Residents
District
A caneasily recognize
disaster information(at3am 10 Sep)
risk of flooding,Residents
evacuated
immediately② Overflowing
time (at
6am 10 Sep)after getting
evacuation information

Hachikenbori river
Outburst place

2.5km
District division map of survey

20Research on the behavior of evacuation

District B：Around the broken part of the embankment of the Kinugawa River
Kinugawa river
overflowing place（Wakamiyado
left bank
25Km）・Many residents started to evacuate
right after the embankment of the
Kinugawa river collapsed. (at 1pm 10 Sep)
Kinugawa
鬼怒川riverItimagine
was 11 where
hours theafter
most residents
It is hard toembankment
break
Outburst
place
決壊箇所
obtained disaster（Misakacho
down information.
(三坂町 左岸21km)
Left bank 21Km）

Hachikenbori river
Outburst place

Even though they got a evacuation
information, Most residents didn't evacuate
immediately until the embankment broke andwater started to overflow

2.5km
District division map of survey

21Research on the behavior of evacuation

District C：Between the broken part of the embankment of the Kinugawa River and a city area
Kinugawa river
overflowing place（Wakamiyado
left bank
25Km）

Kinugawa
鬼怒川riverOutburst
place
決壊箇所（Misakacho
Left bank 21Km）(三坂町 左岸21km)

・Most residents started to evacuate
is hardafter
to imagine
where the evacuation
embankment
5It hours
they obtained
breaks down and6pm
It is10away
from area where
information(atSep.)
embankment broke
Even getting evacuation information, Most
residents did not evacuate
before when overflow arrive

Hachikenbori river
Outburst place

2.5km
District division map of survey

22Research on the behavior of evacuation
District D: A city area of Mitsukaido
Kinugawa river
overflowing place（Wakamiyado
left bank
25Km）

Kinugawa
鬼怒川riverOutburst
place
決壊箇所（Misakacho
Left bank 21Km）(三坂町 左岸21km)

Time most residents started to
Itevacuate;
is away from area where embankment broke
and Inundation inside the levee of the① 2 hours after
the embankmentof theHachikenbori
river
Kinugawa
2:00pm with
10 Sep)Therewereriverfewcollapsed.(atresidents comparedother
districts.
There
were
evacuated②The nextday(atresidents
12:00 11who
Sep)
when inundation occurred and the others
evacuated the next day because of power outage.

Hachikenbori river
Outburst place

2.5km
District division map of survey

23Research on the behavior of evacuation

Acquisition of the disaster information and evacuation situation (All the areas that surveyed)
Kinugawa river
overflowing place（Wakamiyado
left bank 25Km）

Kinugawa river
Outburst place（Misakacho
Left bank 21Km）

Hachikenbori river
Outburst place

A District：Residents recognized a risk of the inundation
B, C District：It is uncertain and which area seems to be flooded
and it is hard to recognize where the rip of dike occurs
D District： The possibility that the inland waters flooding
caused by the flooding of affluent had an influence on
to a refuge action from the rip spot of the Kinugawa
dike if a long time ago

2.5km
Figure of division of the hearing point

I discovered that there was a difference about time whenresidents evacuate after getting evacuation inform.

24

Evacuation Triggers (Multiple Answers)
Evacuation Triggers (Multiple Answers)

Fact-Information occurred and observed （ex. Information of rainfall and water level）
Probability-Information：The outbreak probability of some kind of phenomena being high(ex.
Emergency warning、Evacuation order)
0.0

10.0

20.0

35.3

避難指示を見聞きしたから
避難勧告を見聞きしたから

18.4

大雨洪水警報を見聞きしたから

17.9

避難準備情報を見聞きしたから

9.0

特別警報を見聞きしたから

8.5

23.4

川の様子を直接自分で見たから
13.9

堤防が決壊したことを見聞きしたから

12.4

河川の水位情報を見聞きしたから

Sum ofFact-Information

11.4

若宮戸地区で越水したことを見聞きしたから
7.0

台風情報を見聞きしたから

(74%)

6.0

記録的短時間大雨情報を見聞きしたから

34.3

いた場所が浸水する危険を感じたから
13.4

いた場所が浸水したから雨が強くなり怖くなったから

3.0

停電したから

2.5

Sum ofCircumstance
(55%)

1.5

断水したから

過去の洪水経験から

Sum of
Probability-Information

(93%)

4.0

土砂災害警戒情報を見聞きしたから携帯電話がつながらなくなったから

（％）
40.0

30.0

0.0
6.0

N=201

Evacuation triggers had most totals of the probability information and was 93% followed by fact information, asurrounding.

25

Evacuation Triggers (Multiple Answers)
A District（Around Overtopping point）Evacuation Triggers
0
.
0

1
0
.
0

20
.
0

30
.
0

40
.
0

避難指⽰を⾒聞きしたから32.
6

⼤⾬洪⽔警報を⾒聞きしたから

9.
3

避難準備情報を⾒聞きしたから

20.
9特別警報を⾒聞きしたから

9
.
3
2.
3

川の様⼦を直接⾃分で⾒たから堤防が決壊したことを⾒聞きしたから

2.
3
1
6.
3

若宮⼾地区で越⽔したことを⾒聞きしたから
2.
3
4.
7

いた場所が浸⽔する危険を感じたから

27.
9

いた場所が浸⽔したから

7
.
0
0
.
0

停電したから

2.
3

断⽔したから2.
3

携帯電話がつながらなくなったから
過去の洪⽔経験から

Sum ofFact-Information(about 54%)1
8.
6

記録的短時間⼤⾬情報を⾒聞きしたから

⾬が強くなり怖くなったから

Sum of
Probability-Information(about 121%)9.
3河川の⽔位情報を⾒聞きしたから

台⾵情報を⾒聞きしたから

60
.
0
（（ （

46.
5

避難勧告を⾒聞きしたから

⼟砂災害警戒情報を⾒聞きしたから

50.
0Sum ofCircumstance(about 40%)

0
.
0
7
.
0

N=43

・Evacuation Triggers of the A district has the most probability information.
・Residents evacuated through probability information in the A district
that could usually recognize a risk of the inundation easily.

26

Evacuation Triggers (Multiple Answers)
B District（Around Overtopping point）Evacuation Triggers
0
.
0

1
0
.
0

20
.
0

30
.
0

避難指⽰を⾒聞きしたから

60
.
0
（（ （

20
.
5

⼤⾬洪⽔警報を⾒聞きしたから

1
2.
8

避難準備情報を⾒聞きしたから

5.
1

特別警報を⾒聞きしたから

1
0
.
3

⼟砂災害警戒情報を⾒聞きしたから

2.
6川の様⼦を直接⾃分で⾒たから

Sum of
Probability-Information(about 87%)28.
2

堤防が決壊したことを⾒聞きしたから

1
0
.
3

河川の⽔位情報を⾒聞きしたから

7.
7

若宮⼾地区で越⽔したことを⾒聞きしたから

Sum ofFact-Information(about 62%)1
2.
8

台⾵情報を⾒聞きしたから

2.
6
0
.
0

いた場所が浸⽔する危険を感じたから

30.
8いた場所が浸⽔したから

2.
6

⾬が強くなり怖くなったから

2.
6

停電したから

2.
6

断⽔したから0.
0

携帯電話がつながらなくなったから

0.
0

過去の洪⽔経験から

50
.
0

35.
9

避難勧告を⾒聞きしたから

記録的短時間⼤⾬情報を⾒聞きしたから

40.
0

Sum ofCircumstance(about 39%)
7
.
7

N=39

・ Evacuation Triggers of the B district has most probability information
・ Evacuation order(13:08) is just after the rip of the dike(12:50), and thepeak of the evacuation is after a rip. Evacuation order that received a rip
might lead to the evacuation.

27

Evacuation Triggers (Multiple Answers)
C District（Around Overtopping point）Evacuation Triggers
0
.
0

1
0
.
0

20
.
0

30
.
0

40
.
0

避難指⽰を⾒聞きしたから

60
.
0（
（（
52.
2

避難勧告を⾒聞きしたから26.
1

⼤⾬洪⽔警報を⾒聞きしたから

1
7
.
4

避難準備情報を⾒聞きしたから

確率情報の合計(約135%)

1
3.
0

特別警報を⾒聞きしたから

1
7
.
4

⼟砂災害警戒情報を⾒聞きしたから

8.
7

川の様⼦を直接⾃分で⾒たから

34.
8

堤防が決壊したことを⾒聞きしたから

34.
8河川の⽔位情報を⾒聞きしたから

1
3.
0若宮⼾地区で越⽔したことを⾒聞きしたから

26.
1

台⾵情報を⾒聞きしたから

8.
7

記録的短時間⼤⾬情報を⾒聞きしたから

1
3.
0

いた場所が浸⽔する危険を感じたから

1
3.
0

いた場所が浸⽔したから

事実情報の合計(約130%)

21.
7

⾬が強くなり怖くなったから

0.
0

停電したから0.
0

断⽔したから0.
0

携帯電話がつながらなくなったから

0.
0

過去の洪⽔経験から

50.
0

周囲の状況の合計(約35%)
1
3.
0

N=23

・factual information and probability information(5:5)
・It took time for residents to evacuate after they got the information.
Focusing on total amount for factual information and probability
information, residents based to evacuate on these kinds ofinformation.

28

Evacuation Triggers (Multiple Answers)
D District（Around Overtopping point）Evacuation Triggers
0
.
0

1
0
.
0

20
.
0

避難指⽰を⾒聞きしたから

確率情報の合計(約60%)

3.
8

特別警報を⾒聞きしたから

5.
8
1
.
9

川の様⼦を直接⾃分で⾒たから

1
7
.
3

堤防が決壊したことを⾒聞きしたから

21.
2

河川の⽔位情報を⾒聞きしたから

9.
6

事実情報の合計(約62%)

0.
0

台⾵情報を⾒聞きしたから

9.
6

記録的短時間⼤⾬情報を⾒聞きしたから

3.
8

いた場所が浸⽔する危険を感じたから

40.
4

いた場所が浸⽔したから

15.
4

⾬が強くなり怖くなったから

5.
8

停電したから

5.
8

断⽔したから

過去の洪⽔経験から

60
.
0（
（（

7
.
7

避難準備情報を⾒聞きしたから携帯電話がつながらなくなったから

50
.
0

9
.
6

⼤⾬洪⽔警報を⾒聞きしたから若宮⼾地区で越⽔したことを⾒聞きしたから

40
.
0

30
.
8

避難勧告を⾒聞きしたから

⼟砂災害警戒情報を⾒聞きしたから

30
.
0

3.
8
0
.
0

周囲の状況の合計(約71%)5.
8・Trigger of evacuation in area D have the most circumstances
・Trigger of evacuation in district D is changing circumstance rather
than probability-information and factual-information.
N=52

29

Evacuation Triggers (Multiple Answers)
A地区(N=43)

B地区(N=39)

134.8

140.0

C地区(N=23)

D地区(N=52)

130.4

120.9
120.0
100.0

87.2
71.2

80.0
59.6
60.0

61.5

61.5

53.5
39.5 38.5

40.0

34.8

20.0
0.0
確率情報

事実情報

危機感や周囲の状況

District A：District A has the most probability information. It seems that probability
information is helpful for evacuation , because district could recognize easily
flood risk.
District B：District B has the most probability information.The trigger is evacuation orderthat ordered right after a river bank breach.
District C：Probability information and factual information are almost the same rate.
Residents who live in district C evacuated from judging with plurality of
information.
District D：Trigger of evacuation in district D is changing circumstance rather thanprobability-information and factual-information.

30

information that was effective for evacuation
U： whole event＜Conditional Probability＞
Possibility of event Y is happened by Event X
condition.Then, it call that possibility of Event Y‘scondition what is based on Event X

P YXP(X∩Y)
=
①
P(X)

Y

X

X∩Y

＜Multiplicative theorem＞

P YX =
From①and②

P(X∩Y)P(X)①

P XY =

P(X∩Y)P(Y)②P X ∩ Y = P Y X P X = P X Y P(Y) ③

31

Information of evacuation effect𝑝 x ∩ y = 𝑝 x y p(y) = y x 𝑝 x③

In ③ equation, plus x of all possible X,
from the definition of probability, σ𝑥 𝑝 𝑥 𝑦 = 1
Therefore ③ equation become𝑝 y = σ𝑥 𝑝 𝑦 𝑥 𝑝(𝑥)④Bayes' theorem
And ③ equation divided by ④ equation,
Posterior probability

Prior probability

𝑝 𝑦 𝑥 𝑝(𝑥)𝑝 𝑥𝑦 =
σx 𝑝 𝑦 𝑥 𝑝(𝑥)32Information of evacuation effect𝑝 x ∩ y = 𝑝 x y p(y) = y x 𝑝 x③

In ③ equation, plus x of all possible X,
from the definition of probability, σ𝑥 𝑝 𝑥 𝑦 = 1
Therefore ③ equation become𝑝 y = σ𝑥 𝑝 𝑦 𝑥 𝑝(𝑥)④x：The residents evacuated、 y：The residents heard the information
Bayes'
theoremベイズの定理p(y|x)： The ratio of the information that the residents
evacuatedheard (possibility)

And ③ equation divided by ④ equation,
Posterior probability

Prior probability

𝑝 𝑦 𝑥 𝑝(𝑥)𝑝 𝑥𝑦 =
σx 𝑝 𝑦 𝑥 𝑝(𝑥)p(x|y)：The ratio of the residents that heard the information
33

33

Posterior probability of the case that Prior probability of the residents evacuated is 59.2％(actual ratio of the residents evacuated in all survey)
WE DON’T KNOW!!
％

The ratio
The residents
evacuated
59.2%

事後確率(%)The ratio
The residents
Not evacuated40.8%

We gotInformation

34

A whole Joso-shi The relationship between Posterior probability of the residents
evacuated that got information and prior probability of the residents evacuatedFactual Information ：The information that occured and observed information by the time
Probability Information：Enhancing event of possibility after that event
59.2%(actual ratio of the residents
evacuated in all survey)

Sediment-related disaster warning information

Evacuation
order
Evacuation preparation information%

Evacuation order
Emergency warning

Information that heavy rain is falling in a brief time

Information on the waterlevel of the river
Heavy rainfall flood warning
Disaster information in each area
Typhoon information→In a whole Joso-shi, the probability information is more effective
than the factual information

35

35

District A(Around the overflow area of ​the embankment )Information that heavy rain is fallingin a brief time
Information on theriver water level
Evacuation advisory
Disaster informationin each area

Effect of information [%](事後確率)-(事前確率)=(情報の効果)%

Sediment-related disasterwarning information
Evacuation
order

Typhoon information
Heavy rainfall
flood warning

68.8%(Percentage of residents
actually evacuated in the district A
0％

10%

20%

30%

40%

50%

60%

Evacuation preparation information
Emergency warning

70%

Prior probability (%)

80%

90%

100％

・Effective Information for evacuation has bothprobability information and fact information.
Particularly, advance information about rainfall and riverwater level is effective for evacuation.

36

District B(Around the broken part of the embankment )Effect of information [%](事後確率)-(事前確率)=(情報の効果)%

Emergency warning
Information that heavy
rain is falling in a brieftime
Heavy rainfall
flood warning

Typhooninformation

Evacuation
preparation
information
Evacuation
order
Information on theriver water level

Evacuation
advisory

Disasterinformation ineach area

62.2%(Percentageof residents
actually evacuatedin the district B

Sediment-related disasterwarning information

0%

10%

20%

30%

40%

50%

60%

70%

80%

90% 100%

Prior probability (%)

Many residents evacuated
immediately after the collapse→Evacuate immediately after abreakdown occurs

Information on evacuation(recommendation, instruction ·evacuation effect of preparation
information) is small effect→There was no time delay to utilize
since the evacuation direction was
issued immediately after the collapse
37

District C(Between the broken part of the embankment
and a city area)%

Information on theriver water level

Effect of information [%](事後確率)-(事前確率)=(情報の効果)

Evacuation order

Emergency warning

Evacuation advisory
Disaster informationin each area
Information that heavy rain
is falling in a brief time
Heavy rainfall flood warning

68.2%(Percentage of residents
actually evacuated in the district C
0％

10%

20%

30%

40%

50%

Evacuation preparation information

60%

70%

Prior probability (%)

80%

90% 100％

・There are few information effective for evacuation,
information on evacuation instructions and river water level.

38

District D(around the city area)
Evacuation preparation information
Evacuation advisory
Information that heavy rain
is falling in a brief time
Sediment-related disasterwarning information

Effect of information [%](事後確率)-(事前確率)=(情報の効果)%

Evacuation order
Information on the waterlevel of the river
Disaster informationin each area

Heavy rainfall
flood warning

38.3%(Percentage ofresidents actually evacuatedin the district D
0％

10%

20%

30%

40%

50%

Emergency warning
Typhoon information

60%

70%

80%

Prior probability (%)

90%

100%

・Effective information for evacuation has both probability
information and fact information→ In particular, it was information on evacuation such as
evacuation advisory and evacuation preparation information.

39

Summary of the disaster information and evacuation situation by
district
Information effective forDistrictfeaturesevacuation(Using Bayes Theorem)

District A(around theoverfloware)

Residents recognize
the risk of flooding
from daily andevacuate immediately
after obtaining
evacuation
information.
→Disaster prevention
consciousness is highIn particular, advance information to beissued before the occurrence of thedisaster of rainfall amount and river waterlevel.

District B(Around thebroken part
area)

Difficult ofembankment
breakdown occurred.
Many people
evacuated
immediately after thecollapse.

Information on evacuation such as
evacuation instructions has less effect on
evacuation. Because the evacuation
direction was the issuance immediately
after the collapse, I could not afford atthat time to make use of it.

District C(Between
the brokenpart area)

Evacuation start islate.

Two less effective information.
The effective is evacuation instructions andthe water level of the river.

District D(A cityarea)

There are few people
who evacuated.

In particular, information on evacuationsuch as evacuation recommendations andevacuation preparation information.

40

40

Awareness of Hazard Maps

Question：Have you seen your local hazard map?

Awareness of Hazard Maps

2015年11⽉調査(N=516)
0.0

10.0
7.0

家族でハザードマップの内容を確認しているハザードマップを⾒て、⾃分の家が2017年11⽉調査(N=372)
20.0

ハザードマップを⾒なくても、⾃分の家が

6.4

70.0

61.0

28.7
18.4
16.9

どこにしまってあるか分からない
ハザードマップをしまってある場所

3.9
5.6

は分かっているが、内容は⾒ていない“Residents who do not know the hazard map at all”
decreased from 61% to about 29%(About 30%
reduction).

0.8
0.8
0.0

16.0

内容を知らない“Residents who would only read the hazard map” is16%.

Question : Have you seen the hazard map after the disaster?
0
.
0

1
0
.
0

20
.
0

Question : Reason for becoming to see hazard map after a disaster
30
.
0

40
.
0

災害後に⾒た未回答（（ （(2.7%)

50.
0

その他（⾃由記述）
は精査中

39
.
2

災害前から⾒ていた

その他(28.8％)

25.
5

災害前も災害後も⾒たことがない
未回答

60.0

2.3
4.5

ハザードマップを⾒たことはあるが、ハザードマップを⾒たことあるが、50.0“Residents who have reviewed the contents of
the hazard map” increased from about 13% toabout39%(About 25% increase).

21.9

ハザードマップを知らない、⾒たことがない

⼤⾬時や緊急時に⾒るから良い

40.0

16.9

どの程度浸⽔する可能性があるか分かっている

どの程度浸⽔する恐れがあるか分かっている

30.0

⾒聞きして

3.
5
N=37
2

About 40% of the 65%
people (Residents who saw ahazard map) answered “theysaw a hazard map after
disaster”.

災害を経験したか

他の災害を

31
.
7

ら (67.8%)(1.4％)

市などの防災訓練に参加したから(3.4%)

There were 68% who answered 「Residents who experienced adisaster」 was the largest. In Joso City, disaster drills such as“Because residents who participated in disaster drills such as
municipalities” was as low as about 4%”.

41

Evacuation location decision

Question：Do you decide where to evacuate with your family?
避難場所を決めている

Question：Did you change the evacuation site
after the disaster?

避難場所を決めていない

100.0
90.0
80.0
70.0
60.0

Residents who did notchange the evacuation site
（８１％）

The propotion of those who decide
the evacuationsite almost
69.4
73.6
unchanged.

50.0
40.0

Residents whochanged theevacuation site
（１９％）

30.0
20.0
10.0

26.0

30.1

2015年11⽉調査

2017年11⽉調査(N=516)(N=372)0.0

Question：Why did you change the evacuation site after the disaster?
0.0

10.0

20.0

19.0

災害時に、避難が困難だったから
0.0
4.8

鬼怒川を渡らず避難したが、渡って避難した方が良いと思った
防災訓練やマイタイムラインなどの活動を通して、前の避難場所が危険だと思ったから

0.0

There were 38% people answered that
“Because residents thought that theformer evacuation site was dangerous
through experiencing the disaster “ it was
the largest.

Other (free description) is under scrutiny
42.9

その他
未回答

（％）
50.0

40.0
38.1

災害を経験して、前の避難場所が危険だと思ったから鬼怒川を渡って避難したが、渡らない方が良いと思った

30.0

0.0
N=21

42

Reproduction of inundation situation in Joso City
by flood inundation analysis
Analysis of river and flooplain
integrated
Basic equation(shallow
water equations)

八間堀川境界条件(計算流量)

20
15
10
5

𝜕ℎ 𝜕𝑀 𝜕𝑁++=0
𝜕𝑡 𝜕𝑥 𝜕𝑦𝜕𝑀 𝜕 𝑀2
𝜕 𝑀𝑁
𝜕𝐻 𝑔𝑛2 𝑢 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑥
ℎ1/3
𝜕𝑁 𝜕 𝑀𝑁
𝜕 𝑁2
𝜕𝐻 𝑔𝑛2 𝑣 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑦
ℎ1/3

M, N：Discharge flux in x and y direction
t：Time coordinates、 x, y：plane
coordinates
h：Water depth、 g：gravitational
acceleration
n：roughness length、 H：water levelu, v： flow verocity in x and y directions

25

Past 6:00
Overflowing

12:50
Outburst

0

計算水深(m)
3

6

-10

標高(m)
10

30

Around 14:00
Inundation ina city area

Differentiated by a Leap-frog algorithm

Δx=Δy=10m、 Δt =0.2sLong calculation time！
（It takes one day to reproduce
the data for one day）

43

Inundation flow analysis by Topography Fitting
Grid Model

Basic equation(shallow water equations)𝜕ℎ 𝜕𝑀 𝜕𝑁++=0
𝜕𝑡 𝜕𝑥 𝜕𝑦Ignore the advection term𝜕𝑀 𝜕 𝑀2
𝜕 𝑀𝑁
𝜕𝐻 𝑔𝑛2 𝑢 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑥
ℎ1/3
𝜕𝑁 𝜕 𝑀𝑁
𝜕 𝑁2
𝜕𝐻 𝑔𝑛2 𝑣 𝑢2 + 𝑣 2
+
+
= −𝑔ℎ
−
𝜕𝑡 𝜕𝑥 ℎ
𝜕𝑦 ℎ
𝜕𝑦
ℎ1/3

M, N：discharge flux in x and y direction
t：time coordinates、 x, y：plane
coordinates
h：water depth、 g：gravitational
acceleration
n：roughness length、 H：water levelu, v： flow verocity in x and y directions

Q4

l4
Q3

l1
Extend to linear flooding model that can becalculated using topography-fitting
grid(Yasuda・Yamada*)
𝑁

𝜕𝜂 1
=
෍ 𝑄𝑖
𝜕𝑡 𝐴
𝑖=1

𝜕𝑄
𝜕𝜂
𝑔𝑛2 |𝑄|𝑄
+ 𝑔ℎ𝑙
=−
𝜕𝑡
𝜕𝑠
ℎ7/3 𝑙η：water level of flood、 h： water
depth 、
t： time coordinates 、
s： plane coordinates(distance of center of figure between
adjacent grids)、
A：grid area
g： gravitational acceleration
n： roughness length 、
Qi：inflow from adjacent grid、
N：total number of edge of grid i 、l：length of edge of grid

Q1

l3
l2

Q2

Variable definition ofequation of continuity

li+1

si
li

si+1
s= si+1+si

Variable definition of equation of motion

44

Inundation flow analysis by Topography Fitting
Q4
η：water level of flood、 h： water
Grid Modell4depth 、
𝑁

𝜕𝜂 1
=
෍ 𝑄𝑖
𝜕𝑡 𝐴
𝑖=1

𝜕𝑄
𝜕𝜂
𝑔𝑛2 |𝑄|𝑄
+ 𝑔ℎ𝑙
=−
𝜕𝑡
𝜕𝑠
ℎ7/3 𝑙t： time coordinates 、
s： plane coordinates(distance of center of figure between
adjacent grids)、
A：grid area
g： gravitational acceleration
n： roughness length 、
Qi：inflow from adjacent grid、
N：total number of edge of grid i 、l：length of edge of grid

Q3

l1
Q1

l3

A
l2

Q2

Variable definition ofequation of continuity

li+1

si

z

Altitude of linear boundary
Average altitude of theground in the grid

di
ηi

si+1

li

About linear boundary

di+1

ηi+1
Datum level

Linear boundary

𝑑𝑖+1 > 0
𝑑𝑖 + 𝑑𝑖+1
𝑑𝑖 > 0 ℎ𝑖+1/2 =2
𝑑𝑖+1
𝑑𝑖 ≤ 0ℎ𝑖+1/2 =2

s= si+1+si
Variable definition of equation of motion

𝑑𝑖+1 ≤ 0ℎ𝑖+1/2 =

𝑑𝑖
2

𝑄=0

45

Inundation flow analysis by Topography Fitting
Grid Model
It is shown by Fukuoka and others (1994,1998) andgrid division

Inoue, Toda and others that it is necessary to divide aroad and the ridge into a case to ① lane ② obstacle tothe spread of the flooding water by a pitch differencewith width and neighboring ground height in flooding
analysis.

Overflowpoint

Rip point① The example that a roadplays a role as the lane

Estimatedinundation
range(Geographical
SurveyInstitute)② The example that a roadobstacles to the spread of
the flooding water

A road and theridge assume ita linear borderin defiance of width

We divided an analysis domain into alattice by road centerline shown in
OpenStreetMap (free database)46

Inundation flow analysis by Topography Fitting
Topography Fitting Grid Model
Grid Model
Rectangle GridTerm
9/10 4:00~20:00

6:00
overflow

Rectangle Grid
Number of grid：
1232065
Grid size：10m×10m
Calculation time：about 24 hour
（about 1440 minutes）

12:50
The levee
collapsed

grid adapting terrain
Number of grid ：3337
Calculation time:
About 10 minutes

Though Canal（Hachikenbori
river）,We canunderstand thatimmersion is
occurred at theearly time in thecity

144 times faster!
Real time forecast of flood+ Apply the simulation of
the evacuation behavior

0

計算水深(m)
3

6

-10

標高(m)
10

30

47

Chapter 3

Rainfall-runoff analysis considered theuncertainty of rainfall based on Ito’s
stochastic differential equation theory

Introduction

Serious flood disasters in Japan

Modelized the basin, think the rainfall as input,
and then we can get the time evolution of thewater level.
According to the result, government
can give warnings to the citizens.
H.W.L.
dangerous
water level

leveeHigh water level(H.W.L.) ：The most important index in flood control which considered as the design external force of levee. This
index is calculated by the theory of extreme value statistic using historical hydrology data.
Flood monitoring and forecasting：After H.W.L. had been designed, The levee will be designed strong enough to resist the H.W.L.,so, it is very important monitor and forecast the water level in a flood event. By compare the water level to H.W.L.(or otherevaluation index like dangerous water level), we can know how

49

Deterministic rainfall-runoff models
A brief description about rainfall-runoff problemthyetographinput(t)Rainfall process[mm/h]

input(t)Rainfall-runoff model

Rainfall runoff models

output(t)output(t)hydrograph

Quantity[m3/s]
Water level[m]Runoff height[mm/h]tModeling of rainfall-runoff system

50Deterministic rainfall-runoff models
The basic equation of rainfall-runoff process for simple slope¶h ¶q+ = re (t)¶t ¶x𝑣 = 𝛼ℎ𝑚 ,𝑞 = 𝑣ℎ = 𝛼ℎ𝑚+1

Using the continuous equation andthe momentum equation we can get:¶q(x, t)¶q(x, t)+ (m +1)a m+1q(x, t)b= (m +1)a m+1 q(x, t)b re ( t )¶t¶xLumped1

1Concept of the simple
slope model

Assuming that the direct outflow will only take placenear the river channel, so the outflow will be in inproportion to the length of slope

𝑞(𝑥, 𝑡) ≅ 𝑥𝑞∗ (𝑡)The outflow take place at x=Ldq*b= a0 q* ( re (t) - q* )
dt

𝛼
𝑘𝑠 𝑖
= 𝛾−1 𝛾
𝐷 𝑤

𝑎0 =

𝛽=

𝑚
𝑚+1

𝛽
𝛼 1−𝛽
1−𝛽 𝐿

v:Average velocity of cross section[mm/h],h:Submerged depth [mm]q* ：Flow rate[mm/h] a,m:Parameters51

Deterministic rainfall-runoff models
Expand the model to multi-layers modeln層
Vn-1(t)n：Layer index
m：runoff index foreach layer
According to Yoshimi, Yamada’s research, thebasic equation for simple slope can be expand tomulti layers. By doing so, the model can deal withbasins with multi layer soil structure and consider
the vertical flow between this layers.

rnm(t)=anm(sn(t)-hnm)
hnm

sn(t)
Vn(t) =bnsn(t)Basic equation for each layer:
斜面計算：

dqnm (t)= a nm qnm (t)bnm (rnm - qnm (t))
dtExpand the rainfall-runoff
model to multi-layers

52

Deterministic rainfall-runoff models

Practical use of the basic equation for simple slope(Case study in Kusaki dam basin)

Observation
Calculation resultof 1-layer modelParameters
q0 [mm/h]
D [mm]

L [mm]
ks [mm/h]w
mi

Kusaki dam basin
CaptionInitial condition of therunoff heightThickness of thesurface soil layer
Length of modelized
Slope
Permeation
coefficient of soil
Effective void ratio
Non dimensionalparameter represents
the resistance of soil
Gradient of slope

Simulation result of 1983-08-14 rainfall
event in Kusaki dam basin
Values
0.1
200

30000
360
0.42
0.667

〇 Using 1-layer model, the generalshape of the runoff series is matching
the observation series.
〇 However, the rising part and peakof the runoff series is not quite
matching the observation series.

0.174

53

Deterministic rainfall-runoff models

Practical use of the 2-tanks-3-layers model(Case study in Kusaki dam basin)

Observation

2-tanks-3layers model

Simulation result of 1983-08-14 rainfall event in Kusaki dam basin
By compare the results of 1-layer model and 2-tanks-3-layers model, we can tell thatthe result of 2-tanks-3-layers matches the rising part and peak of the runoff series
better.

54

Deterministic rainfall-runoff models
Compare the 2-tanks-3-layers model to 1-layer model

Simulation result of 1982-07-31 rainfall event in Kusaki dam basin

Simulation result of 1989-08-24 rainfall event in Kusaki dam basin

55

Uncertainty of rainfall intensity(Temporal & spatial distribution)

mm
60min
[mm/60min]
rain gauge(Tokyo)
Radar XRAIN雨量:関東

100

XRAIN

Temporal Resolution:

Rain
Gauge

1 minute

80

Rain
Gauge

Spatial Resolution:
250m×250m

60

Rad arRain Gau g e

Rain
Gauge

Rain
Gauge

Radar(XRAIN) and Ground rain gauge

40

20

0
0

20

40

60

80

Rain
gauge
[mm/60min]
地上雨量
mm
60min

100
difference between radar rain gauge and rain gauge

There is always a difference between the measurement of the rain
gauge and the radar rain gauge system and there is no way to tellwhich one is the " true" rainfall.

56

Uncertainty of rainfall intensity(Temporal & spatial distribution)

Change the pattern of rainfall

The peak discharge ofYattajima station is
22,000m3/s

This is a reproduce calculation of
the typhoon Kathleen 1947 flood
event in Tonegawa catchment area.
Changing the pattern of rainfall
between sub catchments can causea difference of ±𝟕% in peakdischarge.

57

Uncertainty of rainfall intensity(Temporal & spatial distribution)

One hour
1時間雨量
averagerainfallData of laser rainfall(raindrop) rain gauge system developed by Yamada(1994)

It implies that one way to look at the rainfall intensity timeseries is to consider the average part as the deterministic
part and the rest as stochastic part.
58

Deterministic rainfall-runoff models

Deterministic models cannot consider the uncertainty of rainfall-runoff processthyetographinput(t)Rainfall process[mm/h]

input(t)𝒅𝒒= 𝒂𝒒𝒃 𝒓 𝒕 − 𝒒𝒅𝒕Rainfall-runoff model

output(t)output(t)hydrograph

Quantity[m3/s]
Water level[m]Runoff height[mm/h]tModeling of rainfall-runoff system

59

How to consider the uncertainty of rainfall
Using stochastic differential equationt𝒓 𝒕 = 𝒓ത 𝒕 + 𝒓′

input(t)output(t)uncertainty of rainfall

uncertainty of outflow

𝒅𝒒= 𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒 + 𝒂𝒒𝒃 𝒓′
𝒅𝒕t

We want to know uncertainty of runoff
caused by uncertainty of rainfall.
60

Physical systems with random external force
The relation between Ito stochastic differential eq. and Fokker-Planck eq.

Ito Stochastic differential equationdx(t) = y(x(t), t)dt + z(x(t), t)dw(t)one sample pathp(x(t),t)t

Fokker-Planck equation¶p(x(t), t)¶y(x(t), t)p(x(t), t) 1 ¶2 z 2 (x(t), t)p(x(t), t)
=+
¶t
¶x
2
¶x 2
61

Background

Runoff analysis introducing stochastic process theory
From Ito’s stochastic differential equation to Fokker-Planck equation𝒅𝑿 𝒕 = 𝒚 𝑿 𝒕 , 𝒕 𝒅𝒕 + 𝝈 𝑿 𝒕 , 𝒕 𝒅𝒘(𝒅𝑿)𝟐 = 𝒚 𝑿 𝒕 , 𝒕 𝟐 (𝒅𝒕)𝟐 +𝟐𝒚 𝑿 𝒕 , 𝒕 𝝈 𝑿 𝒕 , 𝒕 𝒅𝒕𝒅𝒘 + 𝝈 𝑿 𝒕 , 𝒕 𝟐 (𝒅𝒘)𝟐(𝒅𝑿)𝟐 = 𝒚 𝑿 𝒕 , 𝒕 𝟐 (𝒅𝒕)𝟐 +𝟐𝒚 𝑿 𝒕 , 𝒕 𝝈 𝑿 𝒕 , 𝒕 𝒅𝒕𝒅𝒘 + 𝝈 𝑿 𝒕 , 𝒕 𝟐 𝒅𝒕(𝒅𝒘)𝟐 = 𝒅𝒕

𝒅
𝑬 𝒉 𝑿 𝒕
𝒅𝒕

=𝑬

𝒅
𝒉 𝑿 𝒕
𝒅𝒕

𝒅
𝑬 𝒉 𝑿 𝒕𝒅𝒕𝒅𝝏𝑷 𝒙, 𝒕=න 𝒉 𝒙 𝑷 𝒙, 𝒕 𝒅𝒙 = න 𝒉 𝒙
𝒅𝒙
𝒅𝒕
𝝏𝒕

𝒅
𝑬
𝒉 𝑿 𝒕
𝒅𝒕

𝒅𝒉
𝟏 𝒅𝟐 𝒉
= 𝑬((
𝒅𝑿 +
𝒅𝑿
𝟐 𝒅𝑿𝟐

∞

∞

−∞

−∞

Using theproperty ofWinnier process(𝒅𝑿)𝟐 𝒉𝒂𝒔 𝒂𝒏 𝒊𝒕𝒆𝒎 𝒐𝒇𝒅𝒕 ′𝒔 𝒐𝒓𝒅𝒆𝒓𝒅𝑿 𝟐 )/𝒅𝒕)

In the case where there is no uncertainty，(𝒅𝑿) becomes order of(𝒅𝒕) and𝟐

𝟐

𝟏 𝒅𝟐 𝒉𝟐 𝒅𝑿𝟐

𝒅𝑿 𝟐 𝐠𝐨𝐞𝐬 𝐭𝐨０, it becomes a general chain law

62

Background

Runoff analysis introducing stochastic process theory
From Ito’s stochastic differential equation to Fokker-Planck equation

𝒅
𝑬
𝒉 𝑿 𝒕
𝒅𝒕

𝒅𝒉
𝟏 𝒅𝟐 𝒉
=𝑬 (
𝒅𝑿 +
𝒅𝑿
𝟐 𝒅𝑿𝟐

𝒅𝑿 𝟐 )/𝒅𝒕

𝒅𝒉=𝑬(𝒚 𝑿, 𝒕 𝒅𝒕 + 𝝈 𝑿, 𝒕 𝒅𝒘)/𝒅𝒕
𝒅𝑿

property ofWinnier process
𝑬 𝒅𝒘 = 𝟎

𝟏 𝒅𝟐 𝒉𝟐 𝒅𝒕 𝟐 + 𝟐𝒚 𝑿 𝒕 , 𝒕 𝝈 𝑿 𝒕 , 𝒕 𝒅𝒕𝒅𝒘 + 𝝈 𝑿 𝒕 , 𝒕 𝟐 𝒅𝒕)/𝒅𝒕
+𝑬
(𝒚
𝑿
𝒕
,
𝒕
𝟐 𝒅𝑿𝟐
Ignore (𝒅𝒕)𝟐 order or more𝒅𝒉=𝑬𝒚 𝑿, 𝒕𝒅𝑿

𝟏 𝒅𝟐 𝒉+𝑬
𝝈 𝑿 𝒕 ,𝒕 𝟐
𝟐
𝟐 𝒅𝑿

∞

= න 𝒉′ 𝒙 𝒚 𝒙, 𝒕 𝑷 𝒙, 𝒕 𝒅𝒙 +−∞∞𝟏
න 𝒉′′ 𝒙 𝝈(𝒙, 𝒕)𝟐 𝑷 𝒙, 𝒕 𝒅𝒙
𝟐
−∞

63

Background

Runoff analysis introducing stochastic process theory
From Ito’s stochastic differential equation to Fokker-Planck equation𝒅
𝑬 𝒉 𝑿 𝒕
𝒅𝒕
∞

න𝒉 𝒙
−∞

=𝑬

𝒅
𝒉 𝑿 𝒕
𝒅𝒕
∞

∞

𝝏𝑷 𝒙, 𝒕𝟏
𝒅𝒙 = න 𝒉′ 𝒙 𝒚 𝒙, 𝒕 𝑷 𝒙, 𝒕 𝒅𝒙 + න 𝒉′′ 𝒙 𝝈(𝒙, 𝒕)𝟐 𝑷 𝒙, 𝒕 𝒅𝒙
𝝏𝒕
𝟐
−∞
−∞
PartialPartialintegral
integral
X２
∞

∞

−∞

−∞

𝝏𝒚 𝒙, 𝒕 𝑷 𝒙, 𝒕
𝟏
𝝏𝟐 𝝈(𝒙, 𝒕)𝟐 𝑷 𝒙, 𝒕=− න𝒉 𝒙𝒅𝒙 + න 𝒉 𝒙
𝒅𝒙
𝝏𝒙
𝟐
𝝏𝒙𝟐
Establishment
against any 𝒉 𝒙 ：𝝏𝑷 𝒙, 𝒕𝝏𝒚 𝒙, 𝒕 𝑷 𝒙, 𝒕
𝟏 𝝏𝟐 𝝈(𝒙, 𝒕)𝟐 𝑷 𝒙, 𝒕
=−
+
𝝏𝒕
𝝏𝒙
𝟐
𝝏𝒙𝟐
Fokker-Planck equation

64

How to consider the uncertainty of rainfall
Using stochastic differential equation

Langevin equation𝒅𝒙= 𝒚(𝒙) + 𝜻′(𝒙, 𝒕)
𝒅𝒕

Step1:Devide the input
into a random part andan average part𝒅𝒒
= 𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒 + 𝒂𝒒𝒃 𝒓′
𝒅𝒕

Step2:Write the equationsin the Ito stochastic
differential equation form

𝒅𝒒 = 𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒 𝒅𝒕
+𝒂𝒒𝒃 𝝈 𝑻𝑳 𝒅𝒘

Step3:Drive the governing
equations of the probability
density function

𝝏𝑷(𝒒) 𝝏𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒 𝑷(𝒒)+𝝏𝒕𝝏𝒒𝟏 𝝏𝟐 (𝒂𝒒𝒃 𝝈 𝑻𝑳 )𝟐 𝑷(𝒒)=𝟐𝝏𝒒𝟐

Itô stochastic
differential equation
𝒅𝒙 𝒕 = 𝒚 𝒙 𝒕 , 𝒕 𝒅𝒕
+𝒛 𝒙 𝒕 , 𝒕 𝒅𝒘
Fokker-Planck equation
𝝏𝑷(𝒙, 𝒕)𝝏𝒚 𝒙 𝑷 𝒙, 𝒕=−𝝏𝒕𝝏𝒙
𝟏 𝝏𝟐 𝒛𝟐 𝑷(𝒙, 𝒕)
+
𝟐
𝝏𝒙𝟐

65

How to consider the uncertainty of rainfall
Using stochastic differential equation

Consider the uncertainty ofrainfall intensity𝑟ҧ = 40𝑚𝑚/ℎ

discharge [mm/h]

Deterministic analysis

Time [Hour]

Basic equation of
the deterministic
model
Fokker-Planck eq.

𝜎𝑟𝑎𝑖𝑛 = 4𝑚𝑚/ℎ𝑟ҧ = 40𝑚𝑚/ℎ

Time [Hour]𝒅𝒒= 𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒𝒅𝒕𝝏𝑷(𝒒, 𝒕)𝝏𝒂𝒒𝒃 𝒓ത 𝒕 − 𝒒 𝑷 𝒒, 𝒕
=−
𝝏𝒕
𝝏𝒒
𝟐

𝟏 𝝏𝟐 𝒂𝒒𝒃 𝝈 𝑻𝑳 𝑷(𝒒, 𝒕)
+
𝟐
𝝏𝒒𝟐

66

The basic of filter theory(Prediction and Innovation)

Basic Concept of filtering

67The basic of filter theory(Prediction and Innovation)
Step2

Step1
Update the mean value

Calculate the Kalman gain

Update the mean value

Update the variance

Update the variance

Do not havephysical meanings
Input the initialcondition

Algorithm ofKalman filter

68Rainfall-runoff analysis consider theuncertainty of rainfall intensity

Simulation result of the 1983-08-14 rainfall event using
stochastic differential equation method69

Result of the new filter(1983-08-14 rainfall event)

1987-07-31 rainfall event

1990-09-19 rainfall event
Some other results of the new filter

70One dimensional open channel
simulation
LegendConception grahp of one dimensional open channel

Governing equations
𝜕ℎ 𝜕𝑞
+
=0
𝜕𝑡 𝜕𝑥

𝑳(km)

Length of the open
channel𝒉(m)Water depth𝒒(m2/s)Flow rate𝒗(m/s)

Cross-section
average velocity

𝑩(m)Width of the open
channel𝒊𝟎

Slope of the open
channel𝒊𝒇

Slope of theenergy loss𝜕𝑞 𝜕 𝑞𝑣𝜕ℎ++ 𝑔ℎ− 𝑔ℎ(𝑖0 − 𝑖𝑓 ) = 0
𝜕𝑡
𝜕𝑥
𝜕𝑥

71

One dimensional open channel
simulation

Upper boundary condition

𝑳(km)

50

T(hour)48∆𝒙(km)

0.1

∆𝒕(s)

72

𝑩(m)

200

𝒊𝟎

1/2000𝒊𝒇

Use Manning
Law, roughcoefficient=0.05

Left animation shows the result of a one
dimensional open channel simulation.
The conditions are listed above.

72

One dimensional open channel
consider a random external force
Governing equations
𝜕ℎ 𝜕𝑞
+
=0
𝜕𝑡 𝜕𝑥

𝜕𝑞 𝜕 𝑞𝑣𝜕ℎ++ 𝑔ℎ− 𝑔ℎ(𝑖0 − 𝑖𝑓 ) = 𝑓′𝜕𝑡𝜕𝑥
𝜕𝑥
The random external force
represents the uncertainty of theinformation of the open channelsuch as:1,The uncertainty of energy loss.
2,The uncertainty of crosssection area.
3,The error caused by modellingthe channel in one dimension.

The left animation showed therandom simulation of the above
equations.
Random simulation of one dimensional open channel
under random external force

73One dimensional open channel
consider a random external force
Solve the equations numerically

𝜕ℎ 𝜕𝑞
+
=0
𝜕𝑡 𝜕𝑥

q(x,t)

𝜕𝑞 𝜕 𝑞𝑣𝜕ℎ++ 𝑔ℎ− 𝑔ℎ(𝑖0 − 𝑖𝑓 ) = 𝑓′
𝜕𝑡
𝜕𝑥
𝜕𝑥

Same solutiondh(x,t) =𝑑𝑞(𝑥, 𝑡) =ℎ(𝑥, 𝑡)¶h(x,t)¶h(x,t)
dx +
dt
¶x
¶t

𝜕𝑞(𝑥, 𝑡)𝜕𝑞(𝑥, 𝑡)
𝑑𝑥 +
𝑑𝑡
𝜕𝑥
𝜕𝑡

Add the random external force
𝑑ℎ = 𝑔ℎ (ℎ(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑥 + 𝑓ℎ (ℎ(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑡

𝑓 ′ 𝑑𝑡 = 𝜎𝑑𝑤

𝑑𝑞 = 𝑔𝑞 (𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑥 + 𝑓𝑞 (𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑡 + 𝜎(𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑤(𝑥)74One dimensional open channel
consider a random external force
𝑑ℎ = 𝑔ℎ (ℎ(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑥 + 𝑓ℎ (ℎ(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑡

𝑑𝑞 = 𝑔𝑞 (𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑥 + 𝑓𝑞 (𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑡 + 𝜎(𝑞(𝑥, 𝑡), 𝑥, 𝑡)𝑑𝑤(𝑥)Ito calculus

The governing equations of one dimensional open
channel under random external force

75

The solution of the suggested equation
𝜎 = 0.01𝑚2 /𝑠Random simulation of one dimensional open channel
under random external force

The PDF of h76

The solution of the suggested equation

Random Test
Solution of theEquationssuggested bypresent study

This graph shows the comparison of therandom simulation and the solution of
the equation suggested by the present
study at T=18hour, Section x=25km.

77The solution of the suggested equation

Random Test
Solution of theEquationssuggested bypresent study

This graph shows the comparison of therandom simulation and the solution of
the equation suggested by the present
study at T=18hour, Section x=10km.

78Introduction
Flood forecasting

Modelized the basin, think the rainfall as
input, and then we can get the time
evolution of the water level.

According to the result, government
can give warnings to the citizens.
H.W.L.
dangerous
water level

levee

79

Important applications
Risk management

The most important topic of risk
management is to evaluate the probabilityof the occurrence of disasters

We have to consider theuncertainty of the system
H.W.L.
dangerous
water level

levee

80

Chapter 4

A new theoretical method of flood
forecasting and reliability evaluation oflevee based on uncertainty rainfall by thestochastic process theory• With the global climate change,
the frequency of natural disasteris also change.

2015/09 Kinugawa River (鬼怒川破堤災害)• 随着全球气候的变化，自然灾
害的发生频率也在变化• Most of the past studies on theanalysis of floods are
determinism. It means theanalysis are only two results,
stable and unstable.
• 过去关于洪水的研究分析都是基于确定论的进行的。这也就
是说分析的结果只有安定和不安定两种。

国土交通省 関東地方整備局

Photo from: Ministry of Land, Infrastructure, Transport and Tourism.
Kanto Regional Development Bureau.

82

Study Results❖ The stability analysis of levee with considering theuncertainty of soil parameters❖ The reliability analysis of levee83❖The Stability Analysis of Levee (堤防的安定性分析)• Circular slip method（圆弧滑动面法）• The uncertainty of soil parameters(土质参数的不确定性)• Because of construction method, sites, age of levee and etc.
由于筑堤的方式，选址以及堤坝的建筑年龄。
• However it would be not consider for the safety evaluation in
generally
但是一般来说这当进行风险评估时并不会考虑这些因素• The deviation of soil parameters are referred from :…
….
The cross section of levee
Fs ： the safety factor of slope stabilityc’ ： cohesion (kN/m2 (tf/m2))
φ’ ： friction angle of soil (o)l ： the length of the slice (m)W ： the weight of the slice(kN/m2 (tf/m2))
u ： pore water pressure(kN/m2 (tf/m2))
b ： the width of slides(m)α ： the inclination of the slip surface within the slice tothe horizontal plane [o]土质参数的偏差值参考：

Kok-Kwang Phoon and Fred H. Kulhawy：Characterization of geotechnical variability,
Canadian Geotechnical Journal 36(4), pp.612-624, 1999.

84

❖The Stability Analysis of Levee (堤防的安定性分析)• The calculation conditions(计算条件)➢ Levee(堤坝)✓ Height（高程） 7.5m
✓ Grade（坡度） 1:2(26.4°)• Soil parameters（土质参数）➢ The unit weight of soil is 20kN/m2
Cohesion（内聚力） Friction angle（摩c´擦角）φ´Mean value（均值）

10 kN/m2

34 °Coefficient ofvariation (%)

30

10• The wetting plane inner levee is assumed that thesame to the water level(假定堤坝内的浸润面高
等于河川的水位高)85• The relationship among the cohesion, the friction angle and safety
factor with considering the uncertainty of soil parameters(考虑土质参数的不确定性时内聚力，摩擦角与安全系数的关系)
The times of calculations ：10,000
times（计算次数10,000次）

c’

F.S.

c’
φ´

φ
The correlation of´ cohesion and friction
angel (Correlation coefficient=0.8)

86• The probability of levee broken for thecertain water level(在某个确定水位决
堤的概率)

failureThe calculation method of the levee broken
is as following（用以下方法计算决堤概率）

n
FR =
N
FR： the probability of levee broken
n：n is the case number of levee broken
N：N is the number of all calculation case

87

PDF

The uncertainty of water level based on
the stochastic process theory(基于随机过程理论的河川水位不确定性评价)
A=100 km2𝒓ത =45.5 mm/hm=4
𝝈 =4mm/h
B=50m
n=0,025
i=1/1600

Water level[m]The probability of levee broken❖The Reliability Analysis of Levee (堤防的可靠性分析)The probability of levee broken
for a certain water level(在某个特定水位决堤的概率)

Water level[m]The reliability analysis of levee堤坝的可靠性分析

88

溢流概率
（红色面积
部分）

堤坝滑坡概率基于可靠性分析的堤坝滑坡
概率

用集合论分析的溢流和堤坝滑坡发生的概率89吉見 和紘, 山田 正，山田 朋人：確率微分方程式の導入による降雨流出過程における降雨の不確実性の評価，土木学会論文集B1(水工学)，59，pp.259-264，2015.

❖The Reliability Analysis of Levee(堤防的可靠性分析)The uncertainty of water level based on the stochastic process theory（Yoshimi et. al ,2015）
基于随机过程理论的河川水位不确定性研究（Yoshimi）• It based on the relation between the runoff heights of stochastic differential equation andthe mathematic equation of Fokker-Planck to obtain the uncertainty of rainfall and runoff.
这个研究基于一个关于降雨和径流深的随机微分方程。通过解等价与这个随机微分
方程的Fokker-Planck方程来得到径流深的不确定性。
dq*
dq* = a0 q*b (r - q* )dt + a0 q*bs TL dw= a0 q*b (r(t) - q* )dt¶p(q*, t)¶a0 q*b (r - q* )p(q*, t)=¶t¶q*Fokker-Planck2b2
1 ¶ (a0 q* s TL ) p(q*, t)
+
2
¶q*2

A=100 km2𝒓ത =45.5 mm/hm=4
𝝈 =4mm/h
B=50m
n=0,025
i=1/1600

90

❖The Reliability Analysis of Levee• Here according to the certain water level (like H.W.L.)the failure probability would be estimated from 0 to ∞ :根据在每个特定水位的决堤概率，对水位
从0到无穷积分，可以得到总的决堤概率

s：external force
外力载荷
fS：PDF of external force
外力载荷的概率密度函数
r：resistance force
抵抗强度
fR：PDF of resistance force
抵抗强度的概率密度函数𝑠

𝑃 𝑅 ≤ 𝑠] = න 𝑓𝑅 𝑟 𝑑𝑟 = 𝐹𝑅 𝑠
0

• As the range of S is 𝑠~𝑠 + 𝑑𝑠 and because the failure probability is
independent for R and S like 由于S的范围是S~S+ds，又因为R，S是独立的，所以：𝑃 𝑅 ≤ 𝑠 ∩ 𝑠 < 𝑆 ≤ 𝑠 + 𝑑𝑠] = 𝑓𝑆 𝑠 𝑑𝑠 ⋅ 𝐹𝑅 𝑠 = 𝑓𝑆 𝑠 𝑓𝑅 𝑠 𝑑𝑠

• If the external force s is form -∞ (or 0) to ∞, the failure probability of
the levee may be shown 若外力载荷s是从0到无穷的，那么决堤概率可以表示为：∞∞𝑝𝑓 = න 𝑓𝑆 (𝑠)𝐹𝑅 𝑠 𝑑𝑠∞

0𝑠

= න 𝑓𝑆 (𝑠)𝑑𝑠 න 𝑓𝑅 (𝑟)𝑑𝑟00𝑠

𝑃𝑓 = න න 𝑓𝑆 𝑠 ⋅ 𝑓𝑅 𝑟 𝑑𝑟𝑑𝑠
𝑠

0

∞

0

= න න 𝑓𝑆 𝑠 ⋅ 𝑓𝑅 𝑟 𝑑𝑠𝑑𝑟
0

0

91

❖The Reliability Analysis of Levee• when R is between 𝑟~𝑟 + 𝑑𝑟, the probability 𝑓𝑅 𝑟 𝑑𝑟
is the failure probability of resistance between 0 ~ ∞.

s：external force
外力载荷
fS：PDF of external force
外力载荷的概率密度函数
r：resistance force
抵抗强度
fR：PDF of resistance force
抵抗强度的概率密度函数

当r在r~r+dr之间，概率fR(r)dr的意思是抵抗强度在0~无穷的区间里的决堤概率∞∞

𝑝𝑓 = න 𝑓𝑆 (𝑠)𝐹𝑅 (𝑠)𝑑𝑠 = න 𝑓𝑅 (𝑟) 1 − 𝐹𝑆 (𝑟) 𝑑𝑟00fs(s)FR(s) is the mean value of failure probability when 𝑅 is 𝑟 < 𝑠fR(r)[1-FS(r)] is the mean value of failure probability when s is 𝑆 < 𝑟• The probability of levee broken from the water level 0~a certain waterlevel 水位从0到某个特定水位的条件下的决堤概率为：∞𝑃𝑓 (ℎ𝑆) = න 𝑓𝑆 (ℎ𝑆 , 𝜎𝑆 ; ℎ) 𝐹𝑅 (ℎ𝑅 , 𝜎𝑅 ; ℎ)𝑑ℎ0

fs (hs,σs ; h)： the PDF of external force h with mean hs and standard deviation 𝜎𝑆fR (hR,σR ; h)： the PDF of resistance force h with mean hR and standard deviation 𝜎𝑅

92

❖The Reliability Analysis of Levee• The summation of failure
probability from the water level 0~ H is 𝑃𝑓 (𝐻) and 𝜎𝑆 is assumed
and transferred to ℎ𝑆 . In
numerical methods
可以得出堤防在水位在0~H时决堤的总概率
𝑃𝑓 (𝐻)并将之用数值方法换算成ℎ𝑆 。
𝑃𝑓 (𝐻)𝐻∞= න 𝑑ℎ𝑆 න 𝑓𝑆 (ℎ𝑆 , 𝜎𝑆 ; ℎ) 𝑓𝑅 (ℎ𝑅 , 𝜎𝑅 ; ℎ)𝑑ℎ0∞

s：external force
外力载荷
fS：PDF of external force
外力载荷的概率密度函数
r：resistance force
抵抗强度
fR：PDF of resistance force
抵抗强度的概率密度函数

The probabilityof water level
The probabilityof levee broken0= න 𝑓𝑅 (ℎ𝑅 , 𝜎𝑅 ; ℎ) 1 − 𝐹𝑆 (𝐻, 𝜎𝑆 ; ℎ) 𝑑ℎ
093The failure probability◆ The results of the reliability analysis (可靠性分析的结果)： the probability of levee broken
决堤概率： the probability of overflow
溢流概率： the probability of both(levee broken and overflow)
两者发生的概率

Mean water level [m]①the probability of levee
broken决堤概率②the probability of overflow
溢流概率③the probability of levee
broken and overflow
两者发生的概率

堤防天端
top of levee
堤頂94

Conclusions• The safety factor is estimated then based on the uncertainty rainfall and water level, thereliability analytical solutions of the external force and the resistance force can becalculated.

可以根据降雨和水位的不确定性计算安全因子，可以对外力荷载与抵抗强度作可靠性分析，并得到解析解。
• Because of considering the inhomogeneous soil properties, the safety factor in the same
conditions of water level can be different to about 2.0.
由于考虑了土壤性质的不均一性，在同一水位下安全因子的值相差可以达到2.0.
• In considering the inhomogeneous soil properties, uncertainty rainfall and water level, thereliability evolution can be known. From the 0 m to h of water level, the damage ratio canbe estimated.
通过考虑土壤的不均一性以及降雨与水位的不确定性，可以进行可靠性评价。若对水位从0到h积分，可以得到堤防的破坏概率

95Chapter 5

Uncertainty evaluation inhydrological frequency analysis introducing
confidence interval and prediction interval

Annual maximum rainfall [mm/day]

Difficulty of conventional hydrological frequency analysis

Return period

Non-exceedance probability [%]Observed data
(heavy rainfall disaster in KantoTohoku region in September 2015)

Kanto-Tohoku Heavy Rainfall
in September 2015

Year
Fig. Annual maximum daily rainfall time series
at Ikari observatory

Difficulty of frequency analysis
caused by limited data
Gumbel distribution
fitting with data
Observed data（annual maximum daily rainfall）① Many observed data of heavy rainfall
often deviate from the adopted probability
distribution.
② Estimation accuracy of long-term return
period decreases.

etc.

Annual maximum daily precipitation [mm/day]97Confidence interval of extreme value statistics
Relationship between reliability of estimation and sample size

In mathematical statistics, more than several thousand data is needed to estimate parameter
stably. For example, several thousand trials are needed for us to recognize probability of “1steyes” appearing in a dice is “1/6”.
Relative error[%]＝relative error
error [%]
Relative40

20

𝑀/𝑁 − 1/6
× 100 [%]
1/6

0

M：A roll of the dice，
N：Number of trials

20

40
0

2000

4000

6000

8000

10000

N：Numbercounts of trials

The result of this simulation suggests that extreme hydrological data for several thousand
years are required to estimate the parameters of the frequency analysis model stably.

98Confidence interval of extreme value statistics
An outline of the confidence interval of probability distribution model【Definition】 The range where the probability distribution derived from N ensemble sample extracted
from the same population
For example, the 95% confidence interval means that about 95% of the N probability distribution models are included.
for this reason, the 2.5 percentile value of the probability hydrological distribution is on the 95% lower confidence limit
line and the 97.5 percentile value is on the 95% upper confidence limit line.

Formulation of confidence interval for
probability distribution model𝑃(𝐿 < 𝑌(𝑋) < 𝑈) ≥ 1 − 𝛽
We denote the CDF fitted with the samples {X1, X2, …,
Xn} as Y(X). At this time, the interval [L,U] is defined as100(1-β)% confidence interval of Y(X).
U：upper confidence limit value，
L： lower confidence limit value，β：significance level，
1- β ：confidence coefficient

Formulation of coverage probability＝𝑃 𝐿<𝑌 𝑋 <𝑈
【Definition】 The rate at which probability
distribution models obtained from each ensemble
sample fall within the confidence interval
Fig. Observed data of annual maximum precipitation at Yattajima Observatory andGumbel distribution fitted these observed data, 95% confidence interval of theGumbel distribution

99Confidence interval of extreme value statistics
Relationship between confidence interval and sample size

As the number of data increases, the confidence interval narrows,
and the reliability of estimation improves.

Annual maximum daily precipitation [mm/day]Return period

Non-exceedance probability [%]Sample size=500

Return period

Non-exceedance probability [%]Sample size=50

Annual maximum daily precipitation [mm/day]Fig. Relationship between confidence interval and sample size
Analytical data (red dots) on both probability papers are random numbers according to the Gumbel distribution fitted with theannual maximum daily precipitation for 54 years at the Yattajima Observatory of the Tonegawa River system. Also, 95%
confidence intervals were written in both probability papers.

100Confidence interval of extreme value statistics
Relationship between confidence interval and probability distribution models

Adoption of Gumbel distribution

Adoption of GEV distribution

Gumbel distribution (2 Parameters) : It shows good fit to the maximum value of normal year
and the corresponding confidence interval is narrow.
Generalized Extreme Value Distribution (3 Parameters) : It shows good fit for the whole data
but the corresponding confidence interval is wide.
Fig. Observed data of annual maximum 2days precipitation at Nakanojou Observatory and Gumbel (/GEV) distribution fitted these observed data, 10, 20,
30, 40, 50, 60, 70, 80, 90, 95,99 % confidence interval of the Gumbel (/GEV) distribution

101

Introduction of confidence interval
By Introducing confidence interval, it is possible to intake heavy
rainfall which is considered “unexpected” in flood management.
【Risk evaluation based on C.I.】【Conventional risk evaluation】

Rainfall amount [mm]Return period

Non-exceedance probability [%]Targeted
return period

Return period

Non-exceedance probability [%]Targeted
return period

Width of rainfall which canoccur in targeted return period【=95% Confidence interval】

Rainfall amount [mm]【参考文献】北海道地方における気候変動予測（水分野）技術検討委員会(https://www.hkd.mlit.go.jp/ky/kn/kawa kei/splaat000000vdyw.html)

102

Evaluation of heavy rainfall using confidence interval
95%C.I.(n=41)

Exceedance probability of confidence limit isexpressed by the product of “targeted return
period” and “exceedance probability of C.I.”

95% upper
confidence interval

Exceedance probability of 95% upper
confidence limit of 100-year rainfall

Return period

Non-exceedance probability [%]Targeted
return period1
× 0.025
100
Return period

Exceedance prob. (95% C.I.)= 2.5 × 10−4 (1/4000)
Gumbel distribution fitted withobserved data
95% Confidence interval of theGumbel distribution
Observed data

Annual maximum total rainfall [mm]This probability paper shows 41 observed data of annual maximum total rainfall in Kusaki Dam basin,
Gumbel distribution fitting with these data and 95% confidence interval based on probability limit
method test. n shows total number of observed data.

By considering the confidence intervals, it ispossible to calculate the risk of occurrence of
unprecedented heavy rain.

Relative evaluation of risk realized[ref：the rate of deaths]
traffic accident：1/(2×104) [/year]
air plane accident：1/(50×104)) [/year]
drug accident：1/(200万×104) [/year] 103

Sample size n = 50
Gumbel distribution adopted

Probability density

10％C.I.

99％C.I.

10% ConfidenceInterval
100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=50) on above probability paper are random numbers according to the Gumbel
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, Gumbel distribution fitted with analytical data and5000 Gumbel distribution fitted with ensemble data (n=50), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
Gumbel distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I.[197.8, 238.0] ＝64.7%
Coverage probability of 95% C.I.[179.7, 266.7] ＝95.1%
Coverage probability of 99% C.I.[174.7, 276.8] ＝97.4%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient104

Sample size n = 100
Gumbel distribution adopted

Probability density

10％C.I.

99％C.I.

10% ConfidenceInterval
100-Year annual maximum daily precipitation [mm/day]Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [194.5, 222.3] ＝ 68.0%
Coverage probability of 95% C.I. [181.9, 240.1] ＝ 95.4%
Coverage probability of 99% C.I. [178.2, 246.2] ＝ 97.6%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Annual maximum daily precipitation [mm/day]
Analytical data (n=100) on above probability paper are random numbers according to the Gumbel
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, Gumbel distribution fitted with analytical data and5000 Gumbel distribution fitted with ensemble data (n=100), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
Gumbel distribution fitted with analytical data

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient105

Sample size n = 500
Gumbel distribution adopted

Probability density

10％ C.I.

99％ C.I.

10% ConfidenceInterval
100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=500) on above probability paper are random numbers according to the Gumbel
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, Gumbel distribution fitted with analytical data and5000 Gumbel distribution fitted with ensemble data (n=500), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
Gumbel distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [199.4, 213.5] ＝ 72.8%
Coverage probability of 95% C.I. [193.2, 220.8] ＝ 96.8%
Coverage probability of 99% C.I. [191.3, 223.2] ＝ 98.5%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％ C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient106Sample size n = 1000
Gumbel distribution adopted

Probability density

10％C.I.

10% ConfidenceInterval

99％C.I.

100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=1000) on above probability paper are random numbers according to the Gumbel
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, Gumbel distribution fitted with analytical data and5000 Gumbel distribution fitted with ensemble data (n=1000), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95,
99% were written in this probability paper. Ensemble data is composed of random numbers according to
the Gumbel distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [208.9, 220.1] ＝ 79.0%
Coverage probability of 95% C.I. [204.4, 225.2] ＝ 97.8%
Coverage probability of 99% C.I. [203.0, 226.9] ＝ 99.3%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient107Sample size n = 5000
Gumbel distribution adopted

Probability density

10％ C.I.

99％ C.I.

10% ConfidenceInterval
100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=5000) on above probability paper are random numbers according to the Gumbel
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, Gumbel distribution fitted with analytical data and5000 Gumbel distribution fitted with ensemble data (n=5000), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95,
99% were written in this probability paper. Ensemble data is composed of random numbers according to
the Gumbel distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [211.5, 216.7] ＝ 81.1%
Coverage probability of 95% C.I. [209.4, 218.9] ＝ 98.1%
Coverage probability of 99% C.I. [208.7, 219.7] ＝ 99.1%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％ C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient108

Sample size n = 50
G.E.V. distribution adopted

Probability density

10％ C.I.

Annual maximum daily precipitation [mm/day]
Analytical data (n=50) on above probability paper are random numbers according to the GEV
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, GEV distribution fitted with analytical data and5000 GEV distribution fitted with ensemble data (n=50), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
GEV distribution fitted with analytical data

100-Year annual maximum daily precipitation [mm/day]Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [212.8, 405.8] ＝ 66.9%
Coverage probability of 95% C.I. [161.5, 673.0] ＝ 96.2%
Coverage probability of 99% C.I. [150.1 803.5] ＝ 98.2%

Coverage probability [%]99% ConfidenceInterval

Return period

Non-exceedance probability [%]10% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient109

Probability density

10％ C.I.

99% ConfidenceInterval

95％C.I.
99％C.I.

100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=100) on above probability paper are random numbers according to the GEV
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, GEV distribution fitted with analytical data and5000 GEV distribution fitted with ensemble data (n=100), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
GEV distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [195.8, 299.5] ＝ 70.2%
Coverage probability of 95% C.I. [163.5, 395.1] ＝ 96.9%
Coverage probability of 99% C.I. [155.3, 434.6] ＝ 98.9%

Coverage probability [%]10% ConfidenceInterval

Return period

Non-exceedance probability [%]Sample size n = 100
G.E.V. distribution adopted

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient110Sample size n = 500
G.E.V. distribution adopted

Probability density

10％ C.I.

10% ConfidenceInterval

99％C.I.

100-Year annual maximum daily precipitation [mm/day]Fig. 100-Year quantile distribution and confidence interval

Annual maximum daily precipitation [mm/day]
Analytical data (n=500) on above probability paper are random numbers according to the GEV
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, GEV distribution fitted with analytical data and5000 GEV distribution fitted with ensemble data (n=500), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
GEV distribution fitted with analytical data

Coverage probability of 10% C.I. [245.9, 314.1] ＝ 77.3%
Coverage probability of 95% C.I. [221.0, 358.7] ＝ 98.4%
Coverage probability of 99% C.I. [213.9, 374.8] ＝ 99.4%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient111Sample size n = 1000
G.E.V. distribution adopted

Probability density

10％ C.I.

10% ConfidenceInterval

99％C.I.

100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=1000) on above probability paper are random numbers according to the GEV
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, GEV distribution fitted with analytical data and5000 GEV distribution fitted with ensemble data (n=1000), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
GEV distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [229.2, 267.1] ＝ 77.4%
Coverage probability of 95% C.I. [215.1, 287.4] ＝ 97.4%
Coverage probability of 99% C.I. [210.9, 294.5] ＝ 99.1%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient112

Sample size n = 5000
G.E.V. distribution adopted

Probability density

10％ C.I.

10% ConfidenceInterval

99％C.I.

100-Year annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]
Analytical data (n=5000) on above probability paper are random numbers according to the GEV
distribution fitted with the annual maximum daily precipitation for 54 years at the Yattajima
Observatory of the Tonegawa River system. Also, GEV distribution fitted with analytical data and5000 GEV distribution fitted with ensemble data (n=5000), 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99%
were written in this probability paper. Ensemble data is composed of random numbers according to the
GEV distribution fitted with analytical data

Fig. 100-Year quantile distribution and confidence interval
Coverage probability of 10% C.I. [263.6, 288.7] ＝ 84.6%
Coverage probability of 95% C.I. [254.7, 299.6] ＝ 99.0%
Coverage probability of 99% C.I. [251.9, 303.3] ＝ 99.7%

Coverage probability [%]Return period

Non-exceedance probability [%]99% ConfidenceInterval

95％C.I.

Confidence coefficient [%]Fig. Relationship between coverage probability andconfidence coefficient
113

n=50

Return period

Non-exceedance probability [%]n=500
Annual maximum daily precipitation [mm/day]Return period

Non-exceedance probability [%]Return period

Non-exceedance probability [%]Annual maximum daily precipitation [mm/day]n=100
Annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]

n=1000Return period

Non-exceedance probability [%]Return period

Non-exceedance probability [%]Relationship between sample size and confidence interval (Gumbel)

As the number of data
(sampling number) increases,
the confidence interval
narrows, and the reliability of
estimation improves.

n=5000
Annual maximum daily precipitation [mm/day]Fig. Relationship between sampling number and confidence interval in the case of adopting Gumbel distribution
Analytical data (red dot) in each probability paper is a random number according to the Gumbel distribution fitted to the observed data of the annual maximum daily precipitation for 54 years at the Yattajima Observatory of theTone River system. In each probability paper, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99% confidence intervals were written. Here, n represents the sampling number (total number of analysis data).

114

Relationship between sample size and confidence interval (GEV)
11

n=50

Return period

Non-exceedance probability [%]Return period

Non-exceedance probability [%]Annual maximum daily precipitation [mm/day]

Annual maximum daily precipitation [mm/day]Return period

Non-exceedance probability [%]Annual maximum daily precipitation [mm/day]

n=500

n=100

Annual maximum daily precipitation [mm/day]

n=1000Return period

Non-exceedance probability [%]Return period

Non-exceedance probability [%]5

As the number of data
(sampling number) increases,
the confidence interval
narrows, and the reliability of
estimation improves.

n=5000
Annual maximum daily precipitation [mm/day]Fig. Relationship between sampling number and confidence interval in the case of adopting GEV distribution
Analytical data (red dot) in each probability paper is a random number according to the GEV distribution fitted to the observed data of the annual maximum daily precipitation for 54 years at the Yattajima Observatory of the Tone
River system. In each probability paper, 10, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99% confidence intervals were written. Here, n represents the sampling number (total number of analysis data).

115

Occurrence characteristic ofextreme rainfall in Japan

Periodicity of extreme rain
fall in mountainous area

Annual maximum 2-days rainfall
Annual maximum 2-days
rainfall [mm/2days]

Observed
data※：Nakanojou observatory ofAgatumagawa river in Tonegawa
river system （Elevation：351m）

Year

It is seen that 10 year cycleof annual maximum rainfall
exists in mountainous area.

Spectrum 𝑋 𝑓

2

Energy Spectrum※observed data of annual maximum 2-days rainfall at
Nakanojou observatory from 1942 to 2003. Missing value isinterpolated by average value of data of before and after year.
Data of 1962 and 1963 are missing.

10 years 4 months
12 years 5 months
5 years 7 months
Energy Spectrum

Periodicity [Year]117

Periodicity of extreme rainfall in
Kanto area

Total number of data：44（1960～2003 [year]）Spectrumof annualmaximum 3-days rainfall
年最大
3日累積降雨量のスペクトル1.0

Spectrum
Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

Periodicity
周期 年 [Year]
年最大3日累積降雨量のスペクトル

Manba observatory of
Kannagawa river in Tonegawa
river system (Elevation:320m)

1.0：Periodicity exists：No periodicity
Spectrum
Spectrum・In mountainous area of Kanto area, there
is about 10 years periodicity of annual
maximum 3-days rainfall.
・In plain area of Kanto area, there is no
periodicity of annual maximum 3-days
rainfall.

0.8

0.6

0.4

0.2

0.0
0

5

10

15

Periodicity
周期 年[Year]

20

25

118

Periodicity of extreme rain
fall in plain area

Annual maximum 2-days rainfall
Annual maximum 2-days
rainfall [mm/2days]

Observed
data：Ootemachi observatory in Tokyo（Elevation：6m）

Year

There is no 10 year cycle ofannual maximum rainfall inplain area.

Energy spectrum

Spectrum 𝑋 𝑓

2

Energy Spectrum

Periodicity [Year]

119

Periodicity of extreme rainfall in
Hokkaido area

Total number of data：44（1960～2003 [year]）Spectrum of annual maximum 3-days rainfall
年最大3日累積降雨量のスペクトル1.0

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

周期 年 [Year]
Periodicity

年最大3日累積降雨量のスペクトル

Shirogane observatory ofBieigawa river in Ishikarigawa
river system (Elevation:688m)

1.0：Periodicity exists：No periodicity

・In mountainous area of Hokkaido area,
there is about 10 years periodicity ofannual maximum 3-days rainfall.

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

Periodicity
周期 年 [Year]

20

25

120

Periodicity of extreme rainfall in
Tohoku area

Total number of data：44（1960～2003 [year]）Spectrum of annual maximum 3-days rainfall
年最大3日累積降雨量のスペクトル1.0

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

Periodicity
周期 年 [Year]
年最大3日累積降雨量のスペクトル1.0

Ooshio observatory of Ooshio
river in Aganogawa river
system (Elevation:580m)：Periodicity exists：No periodicity

Spectrum

0.8

0.6

0.4

0.2

・In mountainous area of Tohoku area, there
is about 10 years periodicity of annual
0.00maximum 3-days rainfall.

5

10

15

Periodicity
周期 年 [Year]

20

25

121

Periodicity of extreme rainfall in
Chubu area

Total number of data：44（1960～2003 [year]）Spectrum年最大of annualmaximum 3-days rainfall
3日累積降雨量のスペクトル1.0

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

周期 年 [Year]
Periodicity

年最大3日累積降雨量のスペクトル

Ooma observatory of
Sumatagawa river in Ooigawa
river system (Elevation:538m)

1.0：Periodicity exists：No periodicity

・In mountainous area of Chubu area, there
is about 10 years periodicity of annual
maximum 3-days rainfall.

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

周期 年 [Year]
Periodicity

20

25

122

Periodicity of extreme rainfall in
Kinki area

Total number of data：44（1960～2003 [year]）Spectrumof3annualmaximum 3-days rainfall
年最大日累積降雨量のスペクトル
1.0

0.8

Spectrum

0.6

0.4

0.2

0.0
0

5

10

15

20

25

Periodicity
周期 年 [Year]
年最大3日累積降雨量のスペクトル
1.0

Takihata observatory ofIshikawa river in Yamatogawa
river system (Elevation:360m)：Periodicity exists：No periodicity

Spectrum

0.8

・In mountainous area of Kinki area, there isabout 10 years periodicity of annual
maximum 3-days rainfall.

0.6

0.4

0.2

0.0
0

5

10

15

Periodicity
周期 年 [Year]

20

25

123

Periodicity of extreme rainfall in
Chugoku area

Total number of data：44（1960～2003 [year]）・In mountainous area of
Chugoku area, there isabout 10 years
periodicity of annual
maximum 3-days rainfall.

：Periodicity exists：No periodicity
Spectrumof annualmaximum 3-days rainfall
年最大
3日累積降雨量のスペクトル

年最大3日累積降雨量のスペクトル
Torikami observatory ofHiigawa river in Hiigawa river
system (Elevation:430m)

1.0

1.0

0.8

0.6

Spectrum

Spectrum

0.8

0.6

0.4

0.4

0.2

0.2

0.0
0

5

10

15

Periodicity
周期 年 [Year]

20

25

0.0
0

5

10

15

Periodicity
周期 年[Year]

20

25

124

Periodicity of extreme rainfall in
Shikoku area

Total number of data：44（1960～2003 [year]）Spectrum of annual maximum 3-days rainfall
年最大3日累積降雨量のスペクトル1.0

Spectrum

0.8

0.6

0.4

0.2

：Periodicity exists：No periodicity

0.0
0

5

10

15

20

25

Periodicity
周期 年 [Year]
年最大3日累積降雨量のスペクトル

Oonomi observatory of
Shimantogawa river in
Watarigawa river system(Elevation:420m)

1.0・In mountainous area of Shikoku area,
there is about 10 years periodicity ofannual maximum 3-days rainfall.

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

周期 年 [Year]
Periodicity

20

25

125

Periodicity of extreme rainfall in
Kyushu area

Total number of data：44（1960～2003 [year]）Spectrum of annual maximum 3-days rainfall
年最大3日累積降雨量のスペクトル1.0

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

周期 年 [Year]
Periodicity

年最大3日累積降雨量のスペクトル

Hikosan observatory of
Hikosangawa river in
Ongagawa river system(Elevation:420m)

1.0：Periodicity exists：No periodicity

・In mountainous area of Kyushu area, there
is about 10 years periodicity of annual
maximum 3-days rainfall.

Spectrum

0.8

0.6

0.4

0.2

0.0
0

5

10

15

周期 年 [Year]
Periodicity

20

25

126

Periodicity of extreme rainfall in Japan
There is around 10 years
periodicity of annual maximum
3-days rainfall at 115 points outof 138 point of observatory inJapan’s mountainous area.

：Periodicity exists：No periodicity

127

Frequency analysis ofextreme hydrological quantity
by using prediction intervalIs it possible to predict unprecedented heavy rain ？

Annual maximum daily rainfall [mm]By using observed data of annual maximum daily rainfall at Nagoya observatory from1901 to 1999, we consider whether Tokai heavy rain can be predicted statistically.

Tokai heavy rain(2000)
428 mm/day

Year

129

Frequency analysis introducing prediction interval

99% Prediction
Interval(n=99)

Tokai heavy rain
428 mm/day

Return period

Non-exceedance probability [%]99% ConfidenceInterval(n=99)

By introducing prediction
interval, it can be possible toestimate occurrence risk of
unprecedented heavy rain.

Annual maximum daily rainfall [mm/day]Fig. Observed data of annual maximum daily rainfall at Nagoya
observatory from 1901 to 1999, Gumbel distribution fitting withthese data, 99%confidence interval and 99% prediction interval
based on “Probability limit method test”.

130Evaluation of heavy rainfall using prediction interval
Exceedance probability of prediction
limit value is obtained by product of“targeted return period” and “exceedance
probability of prediction interval”.

Occurrence probability of “Tokai
heavy rain”

Tokai heavy rain
428 mm/day

99%Prediction
Interval(n=99)

Return period

Non-exceedance probability [%]Targeted
return period1
× 0.005
200
Targeted return period

Exceedance prob. (99% P.I.)= 2.5 × 10−5 (1/40000)
By introducing prediction interval, it canbe possible to estimate occurrence risk of
unprecedented heavy rain.

Relative evaluation of risk realized
Annual maximum daily rainfall [mm/day]Fig. Observed data of annual maximum daily rainfall at Nagoya observatory from 1901 to
1999, Gumbel distribution fitting with these data, and 99% prediction interval based on“Probability limit method test”.

[ref：the rate of deaths]
traffic accident：1/(2×104) [/year]
air plane accident：1/(50×104)) [/year]
drug accident：1/(200×104) [/year]131