Third Chinese-German Joint Symposium on Coastal and Ocean Engineering
National Cheng Kung University, Tainan
November 8-16, 2006
Joint Probability Analysis of Waves
and Water Level during Typhoons
Sun-Pei Yeh
Coastal Ocean Monitoring Center (COMC), National Cheng Kung University, Tainan
[email protected]
Shan-Hwei Ou
Tajen University, Pingtung
[email protected]
Dong-Jiing Doong* and Chia Chuen Kao**
Coastal Ocean Monitoring Center (COMC), National Cheng Kung University, Tainan
*[email protected] **[email protected]
David W.J. Hsieh
Dapeng Bay National Scenic Area Administration,
Tourism Bureau, Ministry of Transportation and Communications
Abstract
Overtopping and damage to sea defenses tends to be associated with times of
large waves occurring in combination with high water levels. Methods for predicting
extremes of either water level or waves are in common use, but assessment of the
joint probability of high waves and high water level is more important. The
combination of these phenomena leads to extremely high water levels, increasing
the risk of coastal flooding. The objective of this paper is to study the probability of
joint occurrences of high waves and high water levels. The in-situ data during
typhoons are analyzed. Marginal distributions for wave and water level data during
typhoons are examined. Furthermore, the joint probability diagram of waves and
water level is presented. Results of design water level from joint probability method
(JPM) are compared with traditional empirical design approach by frequency
analysis. It is shown that the commonly used empirical method may lead to
overestimation of the design water level.
1. Introduction
The traditional design approach to coastal defenses is essentially deterministic and
empirical. The determination of the required dike height is either based on the
accumulation of maximum historical records from the high water levels due to
spring tides, wind effects and the expected wave run-up, or empirically based on
the joint event of 100-year return period wave height and 10-year return period
water level due to surge and tide. However, the failure of coastal defense due to
flooding is mainly caused by a combination of high water levels and waves. If this
events are considered are as independent variables, the probability of failure can
be calculated easily, but this would be an incorrect assumption. These two
variables are dependent, so the method suitable for multivariate statistics should be
developed, if one does not want to over- or underestimate the probability of failure.
Therefore when one is designing coastal defenses, such as sea walls, groynes,
breakwaters etc., one should look at the likelihood of both conditions occurring
simultaneously. A joint probability study of wave heights and water levels will do
this and is an important research topic. Many studies (Ferreira and Guedes, 2002;
Battjes and Groenendijk, 2000; Memos and Tzanis, 2000; Prevosto et al., 2000 and
Song et al., 2004) focus on the joint distribution of wave heights and periods or sea
surface elevations of two points in the sea. They provide more understanding on
the scientific problem. However, the joint probability problem of wave height and
water level is necessary on the design requirement of coastal defense engineering.
Here in this study the wave height means the significant wave height (SWH), and
the water level (WL) is the summation of astronomical tides and meteorological
residues without the effect of waves or wave setup.
There are two approaches to construct the joint distribution of two variables. One is
the analytical approach; the other one is the illustrative approach. If the
distributions of wave heights and water levels are known and their dependent
correlation is found, the joint distribution model can be determined by statistical
derivation. However, it is difficult to find the correlation coefficient between SWH
and WL since their correlation may not be functioned. The other way to find their
joint distribution is to use the illustrative approach. HR Wallingford (2000)
presented the joint distribution of SWH and WL in UK. Rodríguez et al. (1999)
presented the cases in Spain. The problem in the illustrative analysis of joint
distribution is that the field data is always insufficient. Hawkes et al. (2000) used
Monte Carlo to generate the data. Li and Song (2006) use a third-generation wave
model and a 3D flow model to simulate long term data for analysis of the joint
probability of wave height and storm tide level. This kind of probability-based
design concept has been presented by Plate and Duckstein (1998) on hydraulic
engineering. This concept is also used on coastal engineering (Liu et al., 2000)
however it is still less.
In this paper, the study will focus on the joint probability of significant wave height
and water level for typhoon data since the severe sea state induced by typhoons is
the main impact for coastal defenses. By the traditional method on sea dike design,
extreme wave height and water level are summarized to be the design value. This
is on the assumption of the two events occurring simultaneously. This paper will
study the joint probability of occurrence of these two variables. The joint distribution
of significant wave height and water level for typhoon cases will be presented in
illustrative form. Results of design example of sea dike will be compared by joint
probability method (JPM) with traditional frequency analysis method (FAM).
2. Field Data
Hourly time series of water levels and significant waves are used in this paper.
They are from four in-situ stations, namely Longdong, Hualien, Chiku and Eluanbi
that located at northern, eastern, western and southern of Taiwan, respectively, in
order to study the joint distribution form of SWH and WL by locations. Table 1
shows the information of the in-situ wave stations and the data amount used in this
study. Locations of the sites are shown in Figure 1. The wave data are measured
by pitch-and-roll buoy (Longdong, Hualien and Eluanbi) and pile station (Chiku).
The water level records are obtained from the acoustic sensors in the wave-filter
tube. The buoys locate at the area of water depth between 30 to 45 meters. The
pile station locates at the area of 15m water depth. All data used in this study are
quality checked.
Figure 2 shows an example of the hourly time series of significant wave height and
water level at Hualien during typhoon Dujuan in 2003. From this case, the
maximum significant wave height occurred when the water level was in the low.
When there is the high water level period, the sea-state is not as severe as it at
other time. This case shows clearly that the high water level is not always occurring
together with high waves.
The data used in this study are collected from over 30 typhoons from each station
which moved forward to Taiwan and were alarmed to the public by Central Weather
Bureau from year 2001 to 2005. The sea state is normally strong affected by one
typhoon for one to three days because of the moving speed of typhoon is mainly
from 10 km/hr to 25 km/hr, such as the example shown in Figure 2. Since there are
so many typhoon data, the study on the joint distribution form of SWH and WL by
various typhoon paths is possible. One data set composed of the maximum
significant wave height and its corresponding water level in one typhoon are culled
for analysis. Figure 3 shows the culled wave and water level data which will be
analyzed in this study.
Table 1 Information of in-situ wave stations
Station
Longdong
Hualien
Chiku
Eluanbi
Data
Instruments
wave
Pitch-and roll buoy
water level
acoustic sensor
wave
Pitch-and roll buoy
water level
acoustic sensor
wave
Acoustic wave gauge
water level
acoustic sensor
wave
Pitch-and roll buoy
water level
acoustic sensor
Water
depth
Typhoons
32m
33
30m
31
15m
37
45m
30
Longdong
Hualien
Chiku
Eluanbi
Significant wave height(cm)
Figure 1 Locations of in-situ wave stations
Dujuan typhoon(2003/08/31~2003/09/02)
700
600
500
400
300
200
100
0
423
443
Data series
463
water level(cm)
200
100
0
-100
-200
423
443
Data series
463
Figure 2 Example of simultaneous observation of significant wave height
and water level during a typhoon (station: Hualien)
significant wave height / water level (m)
12
10
Hualien
significant wave height
water level
8
6
4
2
0
5
10
15
20
25
30
data sequent number
Figure 3 The maximum significant wave heights (left line) and their
corresponding water level (right line) during all typhoons (Station: Hualien,
31 typhoons)
3. Frequency Analysis
Hydrologic systems are sometimes impacted by extreme events, such as severe
storms and floods. The magnitude of an extreme event is inversely related to its
frequency of occurrence. The objective of frequency analysis of hydrologic data is
to relate the magnitude of extreme events to their frequency of occurrence through
the use of probability distributions. Since the duration of field observation is short,
the data amount is always limited. Some approaches are developed for data
generation. Hawkes and Hague (1994) use Monte Carlo simulation to generate
large amount data for statistical analysis. Li and Song (2006) use wave and flow
models to generate long-term data. In this paper we focus on the joint probability
analysis of typhoon cases, so the data is insufficient. The data used for frequency
analysis are assumed to be independent, so the maximum significant wave height
and its corresponding water level of each typhoon are assumed as independent
annual data. More than 30 typhoons data observed at four stations around Taiwan
are used in this study. The objective of this section is to understand the design
value of wave and water level on various return periods by traditional frequency
analysis approach. The results are used for comparison with joint probability
method presented in next section.
3-1 Marginal distribution models and statistical tests
A probability distribution is a function representing the probability of occurrence of a
random variable. By fitting a distribution to the data, the probabilistic information in
the sample can be compactly summarized in the function. There are a lot of
distributions presented in the text book (Ang and Tang, 1975; Hahn and Shapiro,
1967). In this study, several common used distributions are adopted. These are
3-parameters Log-Normal distribution (LN3), Rayleigh distribution (RL),
3-parameters Weibull distribution (WB3), 3-parameters Gamma distribution (GM3)
and the the type I extreme value distribution (EV1). When the histogram of
maximum significant wave height data is plotted such as shown in Figure 4, the
above five distributions are failure to model the data. The exponential distribution
may be a good choose. The fitness to the model may be checked by applying
goodness-of-fit measures. In this paper, the Chi-square test (C-S test) and
Kolmogorov-Smirnov tests (K-S test) (Hahn and Shapiro, 1967) and the
non-dimensional root mean square error (RMSE) of the distribution fitting are used
to find the best model (Doong, 1996).
Figure 4 shows the histogram of the maximum significant wave height data and the
fitting distribution of parameter exponential distribution (EP1). The formula is listed
in Eq (1), where λ is the parameter. This is the result for the data from station
Longdong. There are the same results at stations Hualien, Eluanbi and Chiku. The
result of goodness-of-fit for station Longdong is shown in Table 2.
f ( H S ) = λ ⋅ e − λ ⋅H s
(1)
For the distribution analysis of water levels, it is shown that the type I extreme value
distribution (EV1) is the best fitting model, s shown in Figure 5 and Table 2. The
probability density function for Type I extreme value distribution is listed below,
where μ and σ are location and scale parameters, respectively.
(η − μ )
−
⎡ 1
⎤
f (η ) = exp ⎢− (η − μ ) − e σ ⎥
σ
⎣ σ
⎦
1
(2)
Table 2 Diagnosis of model fitting to the wave height and water level data
at station Longdong
significant wave height
water level
C-S
test
K-S
test
RMSE
C-S
test
K-S
test
RMSE
LN3
F
F
-
P
P
11.1%
EV1
F
F
-
P
P
8.5%
RL
F
F
-
P
P
13.1%
WB3
F
F
-
P
P
12.3%
GM3
F
F
-
P
P
10.4%
EP1
P
P
23%
-
-
-
F means failure, and P means pass.
3-2 Design value of return periods
The objective of frequency analysis is to evaluate the design value of various return
periods. The return period refers to the average period of time between
occurrences of a particular high value of that variable. This design value is useful
for design of dyke height on the purposes of flood protection at river or coast. The
design values of significant wave height and water level in this study are listed in
Table 3.
Table 3 Design value of significant wave heights (SWH) and water levels
(WL) by frequency analysis
Station
Longdong
Hualien
Eluanbi
Chiku
Content
Return period (years)
5
50
100
200
SWH
2.8
5.1
5.9
6.8
WL
3.2
3.6
3.7
3.8
SWH
2.0
4.1
4.7
5.9
WL
3.5
3.9
4.0
4.2
SWH
2.8
5.0
5.9
7.1
WL
3.18
3.58
3.63
3.7
SWH
1.8
3.2
3.7
5.6
WL
3.4
3.6
3.7
3.7
Unit: meter
probability (%)
40
30
Distribution model :
Exponential f ( x ) = λ ⋅ e − λx
20
10
0
0
2
4
6
8
significant wave height (m)
10
Figure 4 Distribution model fitting to data series of max. significant wave
height during each typhoons (Station: Longdong, Model: Exponentional)
probability (%)
30
Distribution model :
Extreme Value type I (EVI)
20
10
0
2
2.5
3
3.5
water lever (m)
4
4.5
Figure 5 Distribution model fitting to data series of water level
corresponding to the max. significant wave height during each typhoons
(Station: Longdong, Model: Extreme Value type I)
4. Joint Distribution Analysis
4-1 Joint Probability Method
When assessing the probability of failure for a single sea state variable, the events
which give failures are easily characterized as failures of the structure caused by
extreme value of the variable, i.e. failures occur whenever the variable exceeds
some level. The frequency analysis presented in last section is one of the
approaches. When the sea state variable is inherently multivariate, the joint
probability approach (JPM) is applied. JPM is an approach for estimating the
probability of a structure variable exceeding a critical level, based on the joint
analysis of the sea state variables. The joint probability typically refers to two or
more partially related environmental variables occurring simultaneously to produce
a response of interest, such as large wave heights and high water levels, large river
flows and high sea levels, large surges and high astronomical tidal levels.
The failure probability is critical in the evaluation of the reliability of a structure.
Traditionally, the probability failure of a coastal structure to waves or water level
(storm surge) can be obtained from a function (distribution) by a given value of the
variable (failure condition). For assessing the probability failure of multivariate
environment factors, the failure probability (exceedence probability of occurrence)
is determined by integrating the joint probability density function over the failure
region. As illustrated in Figure 6, with the same return period, the joint exceedence
probability may be smaller than the failure probability because the structure may
fail when one of the parameter (wave height or water level) is high but the other is
low.
Variable y
Failure region
Joint exceedence
probability
critical condition
Same return period
Safety region
Variable x
Figure 6 Illustration of the failure domain and the joint exceedance
probability plotted on the graph of probability density function
Structure fails when the wave height and water level falls within the ‘failure region’
with boundary b( x,ν ) = 0 , where x = ( x1 , x2 ) indicates the variables such as wave
height and water level, and ν denotes the structural parameters. The boundary
b( x,ν ) = 0 indicates the limiting state exceeding which the structure will be
incapable to resist the environmental loads or flooding. The boundary function is
different for different structures and for different locations. The failure probability is
given by F = ∫∫ f ( x1 , x2 )dx1dx2 in which f ( x1 , x2 ) is the joint probability density
function and the return period T = 1 F .
4-2 Results
The results of joint probability distribution analysis are shown in Figure 7. The
scatter diagrams in Figure 7 are the results of the wave height and water level data.
The curves of return period 5, 50, 100 and 200 years are plotted in the figure. The
curve means that the joint exceedance probability is 0.2, 0.02, 0.01 and 0.005
when the sea state is more severe than it occurs at the ‘critical condition’. If one
uses the critical condition as the ‘design value’, there are infinite solutions on the
curve. For example, in Figure 7(a) the exceedance probability is 0.01 (T=100) when
the water level and significant wave height are respectively (3.5, 3.5), (3.0, 5.4),
(2.5, 5.6)…and so on. However, the significant wave height for 100 years return
period from frequency analysis method is 5.9m as listed in Table3, and the wave
level for 100 years return period from frequency analysis is 3.7 m. The height
summation is therefore 9.6m on the consideration of water level plus wave impact.
Normally, the wave set-up and run-up heights should be added to be the design
value of sea dike. By joint probability analysis, we know there are infinite
combinations of wave height and water level, i.e. there are infinite design heights.
The combination is shown as the curve in Figure 8. It is shown that the height
summation of wave height and water level estimated by traditional frequency
analysis is higher than that estimated from the joint probability method. The
average falling percentage is about 54.5% for the example at Longdong.
Longdong
10
(a)
significant wave height (m)
significant wave height (m)
10
8
T=200
6
T=100
T=50
4
T=5
2
Eluanbi
(c)
8
T=200
6
T=100
T=50
4
T=5
2
0
1.5
2
2.5
3
3.5
4
0
1.5
4.5
2
2.5
water level (m)
Hualien
10
significant wave height (m)
(b)
8
6
T=200
T=100
4
2
T=50
T=5
3.5
4
4.5
Chiku
3.5
4
4.5
(d)
8
6
T=200
4
T=100
T=50
2
T=5
0
1.5
2
2.5
3
3.5
4
4.5
0
1.5
2
2.5
3
water level (m)
water level (m)
Figure 7 Diagrams of joint probability of wave height and water level
(a)Longdong (b)Hualien (c)Eluanbi (d)Chiku
12
summation height of
design wave height and water level (m)
significant wave height (m)
10
3
water level (m)
T=200 (FA)
10
T=100 (FA)
T=200(JPM)
T=50(FA)
T=100(JPM)
8
T=50(JPM)
6
T=5(FA)
T=5 (JPM)
4
2
high water level
low
low
high wave height
Figure 8 Comparison of summation of wave height and water level by
traditional FA (Frequency Analysis) and JPM (Joint Probability Method).
This is the example at Longdong.
5. Conclusion
The largest risk to coastal defense structures tends to occur at this time of
unusually high water levels combined with large waves. The overall effect of high
wave conditions on coastal processes is highly dependent on the water level. For
example, a severe typhoon coinciding with neap tide conditions would have less
impact on beaches than a more moderate event in conjunction with a large spring
tide. It has been shown that the waves and water levels should be jointly
considered. Reliable estimates of the probability of occurrence of such combined
conditions are given in this paper by joint probability method (JPM).
Distribution models are fitted to the data to find the best fitting function by statistical
tests. The design value of specific return period (year) is evaluated. The traditional
approach is an empirical model assuming that the wave heights and the water
levels are uncorrected. This paper presents the illustrative approach to find the joint
probability of two variables of significant wave height and water level. From the
analysis of joint probability method, infinite solutions are found for a specific failure
condition. They are dependent on the conditions of wave height and water level.
This is the advantage of the JPM method. One can consider which one is the
important factor to decide the best solution. This paper compares the results of
height summation of wave height and water level (design value) from traditional
frequency analysis and JPM. It is shown that the height summation of wave height
and water level by JPM is lower than that estimated by frequency analysis. The
average falling percentage is 54.5% for the study cases in this paper. This paper
presents a new idea on the estimation of design height of sea dike. It is also noted
primarily proofed that the joint probability analysis is worthy for further studies.
Acknowledgements
This work was supported by National Science Council of Taiwan (NSC
94-2625-X-006-006). The field data were provided by Central Weather Bureau and
Water Resources Agency in Taiwan. The authors would like to express their great
acknowledgements.
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