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Wave Analysis
Short Term and Long Term Analysis
Haryo Dwito Armono. PhDUpdated 4 March 2009
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Introduction
n Short term analysis: analyse of wave
that occurs withinone wave train
n Long term analysis derivation of
statisticaldistributions thatcover manyyears
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(m)
www.aviso.oseanobs.com
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Why collect wave data?n Monitoring of coastal processes
such as beach erosion andsediment transport.
n Baseline design statistics forcoastal projects.
n Operational assistance in coastalconstruction projects.
n Monitoring of severe weatherconditions.
n Oceanographic research. (Manly Hydraulic Lab : http://mhl.nsw.gov.au/)
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How to measured wave?n Use wave recorder : i.e:
Wave staffsn e.g: www.oceansensorsystem.com
Wave rider / wave buoyn e.g: www.datawell.nl
Pressure sensorn e.g : www.civiltek.com
Satellite imagesn e.g : GFO (Geosat Follow On)
n Place wave recorder in deepwater (>0.5L)
n Record wave height, period and direction (duration 15 60)n Links http://cdip.ucsd.edu, http://mhl.nsw.gov.au/,
http://www.coastal.udel.edu/coastal/comps.html, etc
n Assignment 2 :
n Find more info on wave recorder (product detail, vendors, measurement and analysismethods, etc)
n
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Wave Animation
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Typical wave recorder
http://www.scienceprog.com/wp-content/uploads/2007i/Ocean_embedded/wave_heigh_measurement.jpg
www.datawell.nl
www.civiltek.com
www.oceansensorsystem.com
TRITON-ADVwww.sontek.com
AWACwww.nortek.com
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Typical recorded samples
File created by:Device: Model
Serial No: 1752File Type: WAV
Operating Mod
Contract Refere
Site Reference:
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Triton Webinar
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Termsn Realization
Representative recordat particular timerange
n Ensemble Compilation of several
realization Each ensemble has
parameters such asmean, standarddeviation, skewness,
kurtosis, etcn
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0
0.01
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0 5 10 15 20 25 30
Time (sec)
WaterLevel(m
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(
Ensemble of Three Realizations
1
2
3
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Termsn Stationary
If none of theensemblesparameter (z) vary intimeexp:
2a= 2b
=
2c a = b = c 3a = 3b =
3c 4a = 4b =
4c
n
n ErgodicTime
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(m
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
WaterLevel(
a
b
c( )
( ) ( ) ( )
1 1
3
1 1
k a k b k c
k
K N
k k jk j
z z z
zharus sama
z zK N = =
+ +
= =
Assume ergodicity in wave recordsKamphuis, Intro to Coastal Eng & Management
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Short Term Analysis
n Height of waves ( ) are randomn It is impossible to predict exact value of at any
time or locationn Probability that has a certain value is called
PDF (Probability Density Function), p( ).n p( ) can be described by normal distribution.
n
22 2
21
1 1( ) exp ,
22
N
j
j
pN
=
= =
Ntt1 1
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Short Term Analysis contd
n Assumed that H = 2 max , then PDF for H(Probability that H has a certain value)
n
n Rayleigh Distribution
n
n
n
2 2
2 2( ) exp
4 8
H Hp H
=
=
=
===
j
j
tt
tR
t2
Ndt
t
1=
R
R
1
2
0
22 1lim
=
=N
j
jN 1
22 1
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Short Term Analysis contd
n The Cumulative Distribution Function (CDF) of waveheight: probability that any individual wave height H isless than a specified wave height H
n
n
n
n The Probability of Exceedance: the probability that any
individual wave of height H is greater than a specifiedwave height H :
n
n
2 2 2
2 2 2
0
( ' ) exp 1 exp4 8 8
HH H H
P H H dH
< = =
2
2( ' ) 1 ( ' ) exp 8
H
Q H H P H H
> = < =
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N NE E SE S
Total % 15,51 5,90 6,85 5,82 10,33
6.0 - 6.5 0,01 0,01 0,01 0,01
5.5 - 6.0 0 01 0 02 0 06 0 10
Omnidirectional Percentage Excee
Annual Directional Perc
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Short Term Analysis contd
0
0.1
0.2
0.3
0.4
0.5
0.60.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
H/Sigma
(H).p(H),P,Q
(H).p(H)
P
Q
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Short Term Analysis contd
n Wave height with Probability of Exceedance Q:
n
n 2
2 2
1
18 ( ln ) 2 2 ln
1
Q
N
j
j
H QQ
N
=
= =
=
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Short Term Analysis contd
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Short Term Analysis contdn Example 3.1 Calculation of Short-Term Wave Heightsn
n To analyze a wave record it must be stationary. Hence, it is normalto record waves for relatively short time durations (10 to 20minutes). A longer record would not be stationary because windand water level variations would change the waves. Thus it is
usual to record, for example, 15 minutes every three hours. It issubsequently assumed that the 15 min. record is representativeof the complete three hour recording interval.
n
n Suppose the analysis of such a record yieldsn
n
n We want to calculate significant wave height Hs, average waveheight , average of the highest 1% of the waves , and themaximum wave height in the record.
mandT 0.1sec10 ==
01.0HH
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Short Term Analysis contd
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Long Term Analysis
n Purpose :To organise wave height data
To extrapolate data set to extreme valuesof wave height occuring at low probabilityof exceedance
n Methods
Statistical Analysis of grouped wave dataExtreme Value Analysis from ordered data
n
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Nov 01 - Nov 30, 1983
0
0.51
1.5
2
2.5
3
3.5
4
4.5
0 100 200 300 400 500 600 700 800
Time (hrs from Nov 01, 0:00, 1983)
WaveHeight(m)
Ht = 1.5 m
(2)
(4)
(3)
(1)
(6)
(5)
(8)(7)
Grouped wave data
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Grouped wave data contd
Data in the left table obtainedfrom 34,9 years of record.
= number of data points / yr
= 282306 / 34.9 = 8089
= 2738 / 34.9 = 78.45
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Grouped wave data contd( ' )
1019( ' 1.75) 0.372
2738
1019 549( ' 2.00) 0.573
2738
1019 549 382( ' 2.25) 0.712
2738
( ' ) 1
P P H H
P H
P H
P H
Q Q H H P
=
= =
+ = =
+ + = =
= > =
Curveline is difficult to interpolate! transformed into straightline
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Curve Transformation
n Normal Probability Distribution
n Transferred to:
Log-Normal Probability
Distribution
Gumbel Distribution
Weibull Distribution
Cumulative Distribution Function
Normal Distribution
Log Normal Distribution
Gumbel Distribution
Weibul Distribution
y=A.x + B
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Distribution Models
sH = standard deviation,
H = mean wave height and = Weibull and Gumbel Parameter
= lower limit of H = threshold value in a Peak over Threshold data set
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Probability Table
z = -3.4 P = 3.37 x 10 -4
z = +3.4 P =1 - 3.37 x 10-4
= 0.999663
In Excel : NORMINV
z =0
z = -3.4
Normal Distribution Curve
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(1) (2) (3) (4) (5) (6) (7) (8) (9)
H N P Q z ln H G W
=0.8W
=1.31.75 1019 0.372 0.628 -0.326 0.560 0.012 0.384 0.555
2.00 549 0.573 0.427 0.183 0.693 0.584 0.816 0.883
2.25 382 0.712 0.288 0.560 0.811 1.081 1.316 1.184
2.50 254 0.805 0.195 0.859 0.916 1.528 1.848 1.459
2.75 174 0.869 0.131 1.119 1.012 1.959 2.421 1.723
3.00 113 0.910 0.090 1.339 1.099 2.359 2.996 1.964
3.25 81 0.939 0.061 1.550 1.179 2.772 3.627 2.210
3.50 60 0.961 0.039 1.766 1.253 3.232 4.366 2.477
3.75 40 0.976 0.024 1.976 1.322 3.713 5.176 2.7504.00 27 0.986 0.014 2.190 1.386 4.244 6.105 3.044
4.25 19 0.993 0.007 2.442 1.447 4.916 7.326 3.406
4.50 10 0.996 0.004 2.683 1.504 5.611 8.638 3.769
4.75 4 0.998 0.002 2.849 1.558 6.122 9.632 4.031
5.00 2 0.99854 0.00146 2.976 1.609 6.528 10.436 4.234
5.25 1 0.99890 0.00110 3.063 1.658 6.816 11.014 4.377
5.50 2 0.99963 0.00037 3.378 1.705 7.915 13.276 4.9105.75 0 0.99963 0.00037 3.378 1.749 7.915 13.276 4.910
6.00 1 1.00000 0.000
Total 2738
X in Distribution Models : [1], [6]
Y in Distribution Models : [5], [7], [8], [9]
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Extrapolation
n The Exceedence Probability of one event in TR
years :
n
n
n
n
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Extreme Value Analysis
n If only few major events are known
n
n Limited number of extreme eventsn
n
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Method
n Rank the data in decreasing ordern Calculate Probability of Exceedence (Q)
n
n
i = ranking of the data point N = total number of points
c1, c2 = constants for unbiased plotting positionn Calculate Probability (P)n Calculate Reduced Variate (z, W, G)
n
2
1
cN
ciQ+=
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Constant for Unbiased Plotting
Distribution c1 c2Normal 0.375 0.375Log Normal 0.250 0.125Gumbel 0.440 0.120Weibull 0.2 + 0.27 0.2 + 0.23
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Distribution Models
sH = standard deviation,
H = mean wave height and = Weibull and Gumbel Parameter
= lower limit of H = threshold value in a Peak over Threshold data set
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Probability Table
z = -3.4 P = 3.37 x 10-4z = +3.4 P = 1- 3.37 x 10-4
= 0.999663
In Excel : NORMINV
Example: =NORMIV(J38,0,1)
z =0
z = -3.4
Normal Distribution Curve
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i H Q W i H Q W
1 5.95 0.010 6.675 23 4.22 0.505 0.621
2 5.38 0.033 4.642 24 4.21 0.527 0.572
3 5.26 0.055 3.775 25 4.20 0.550 0.526
4 5.03 0.078 3.227 26 4.20 0.572 0.482
5 4.82 0.100 2.832 27 4.17 0.595 0.441
6 4.75 0.123 2.524 28 4.17 0.617 0.402
7 4.71 0.145 2.274 29 4.16 0.640 0.365
8 4.68 0.168 2.064 30 4.16 0.662 0.330
9 4.63 0.190 1.884 31 4.14 0.685 0.297
10 4.54 0.213 1.727 32 4.14 0.707 0.266
11 4.49 0.235 1.588 33 4.13 0.730 0.236
12 4.43 0.258 1.463 34 4.09 0.752 0.208
13 4.40 0.280 1.351 35 4.09 0.775 0.182
14 4.38 0.303 1.250 36 4.08 0.797 0.156
15 4.36 0.325 1.157 37 4.07 0.820 0.13316 4.35 0.348 1.071 38 4.07 0.842 0.111
17 4.34 0.370 0.993 39 4.06 0.865 0.090
18 4.33 0.393 0.920 40 4.05 0.887 0.071
19 4.29 0.415 0.852 41 4.04 0.910 0.053
20 4.25 0.437 0.788 42 4.04 0.932 0.036
21 4.24 0.460 0.729 43 4.03 0.954 0.022
22 4.23 0.482 0.673 44 4.01 0.977 0.009
2
1
cN
ciQ
+
=
1
1ln
=
QW
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Plot Extreme Distribution
y = 3.395x - 13.704
R2 = 0.995
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
4 4.5 5 5.5 6 6.5
Wave Height - H (m)
W
eibullReduced
Variate-
Weibull Distribution for Ordered Data Set ( =0.8).
y=A.x + B
A = 3.395
B = - 13.704
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Plot Extreme Distribution
Log-Normal Distribution for Ordered Data Set
y = 5.270x - 7.180R
2= 0.979
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
1.4 1.5 1.6 1.7 1.8 1.9
ln H
ReducedVar
iate-
y=A.x + B
A = 5.270
B = - 7.180
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Plot Extreme Distribution
Gumbel Distribution for Ordered Data Set
y = 2.211x - 8.591
R2
= 0.992
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
4 4.5 5 5.5 6 6.5
Wave Height - H (m)
G
umbelReduced
Variate- y=A.x + B
A = 2.211
B = - 8.591
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Distribution Models
sH = standard deviation,
H = mean wave height and = Weibull and Gumbel Parameter
= lower limit of H = threshold value in a Peak over Threshold data set
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Return Period Prediction
Return Period (Yrs)
N 20 50 100 20044 1.26 0.80 0.29 3.97 5.22 5.68 6.05 6.43
Return Period (Yrs)
N 20 50 100 20044 1.26 0.45 3.87 5.31 5.73 6.04 6.36
Return Period (Yrs)
N Hln s 20 50 100 200
44 1.26 1.36 0.19 5.44 5.86 6.16 6.45
Weibull
Gumbel
Log Normal