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A UNIFIED APPROACH TO EXTREME VALUE
ANALYSIS OF OCEAN WAVES
Harald E. Krogstad
Stephen F. Barstow
TRONDHEIM, NORWAY
ISOPE, Seattle 2000
CONTENTS
• Multiple scale stochastic models
• Probability distributions for extreme values
• Short term parameterizations
• Analysis of storm histories
• The maximum significant wave height
• Dependence on short term distributions
• Bootstrapping and assessment of reliability
• Climatic variations
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A multiple scale process:
X s(t ) (t )
X is varying on a fast scale, whereas
the state s(t) is a slowly varying
process.
The process is locally stationary.
Examples:
X
Crest height
Wave height
Sign. wave height
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s
Sea state
Sea state
Seasonal/yearly variations
The extreme value distribution for a constant state:
Pmaxt X s (t )  x ; 0  t  D  Fs ( x)D / T ( s )
s(t)
Locally constant
t
Time history:
 D

dt

P(maxt X s ( t ) (t )  x | D )  exp  logFs ( t ) ( x )
 t 0
Ts ( t ) 


Distribution of states: P(maxt X (t )  x | D )  exp D logFs ( x )


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
s
( s )ds 

Ts 
Dependent on:
• The short term parametrization,
Fs (x)
• The time history or the distribution of states, s
• Suitable period parameter
The expressions tend asymptotically to extreme value distributions
(typically Gumbel distributions) since they are of the form
P(maxt X (t )  x | D )  G ( x ) N ,
N  D 1 / T (s) ,



(
s
)
G ( x )  exp  logFs ( x )
ds ,
s
T ( s) 1 / T ( s)


( s )
1 / T (s)  
ds.
s T (s)
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WAVE AND CREST HEIGHT


Fs ( x)  1  exp  (4 x / H s ) /  , s  H s , Tz ,,
T ( s)  TZ .
Crest height
Gaussian sea
Laser
Wave staff
Large buoy
Pressure Cell (Ekofisk)
Wave height
Gaussian sea
Rayleigh, narrow band
Forristall, buoy data
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Data set


~60
~30
~100
~30
2
2
1.99
2
2.05
2
2.5
2.2
1.85
1.79
-
2
2
2.13
4(1-min)
8
8.42
1.2
Mode of maximum crest height/Hs
Laser,WADIC,Tern
Marex radar
1.1
Wavestaff
1
Gaussian
Large buoy
Waverider
Presure cell
0.9
0.8
0.7
0.6
102
103
Number of waves, D/Tz
104
Various parametrisations illustrated by the mode of the non-dimensional
crest height as a function of the number of waves
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Time histories of storm events
Time history
Probability distributions
15
10
5
0.4
Probability density
Hs (m), Tz (s)
Hs
Tz
Expected value = 16.2 m
0.35
Expected value = 26.2 m
Observed value = 26.4 m
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Time (hours)
20
25
10
15
20
25
30
Maximum wave/crest height (m)
The Frigg storm (Courtesy of Elf Petroleum Norway)
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35
Dependence on the short term parametrization
  hs  0.7m 1.26 
,
P( H s  hs )  1  exp  
  2.14m  


1 / Tz ( H s )  0.234( H s m)0.37
From the left:
Pressure cell data
Gaussian sea
Laser data
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From the left:
Forristall analysis
Næss formula
Rayleigh
SIGNIFICANT WAVE HEIGHT
(Rice’s formula and Poisson up-crossing of high levels)


P(maxH s  h)  exp  aHs (h)FHs (h)D , h  h0
a H s ( h ) : Mean positive slope
Mean positive slope (m/h)
FH s ( h ) : Cumulative distribution of Hs
10 0
10 -1
10 -2
1
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2
4
Level (m)
6
8
10
The Y year maximum significant wave height
Engineering approach:
D/Y independent events (D = duration of event)
D
 1  FH s (maxH s ),
Y
Y
D

  h  H  
o
  .
P(maxH s  hs )  1  exp   s
  H c  




New formula:


P(maxHs  hs )  exp  aHs (hs )FHs (hs )Y , hs  h0
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Probability density of maximum significant wave height
for an exposed Norwegian Sea location
Engineering approach: Dotted line (depends on D)
New formula: Solid line
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Simplified formulae for the maximum wave height derived
from the full expressions:
(For typical North/Norwegian Sea long term Weibull distributions of Hs)
E( H max,100 y )  1.92  H s ,100 y ,3h ,
100y
 1  FH s ( H s ,100 y ,3h ).
3h
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RELIABILITY ASSESSMENT FOR EXTREME
WAVE AND CREST HEIGHTS
• The short term parametrizations are (almost) universal
• The long term statistics is site specific
Intrinsic uncertainty: Remaining stochastic variability when
the long term statistics is known.
Additional uncertainty: Caused by incompletely known long term statistics
Key problem: What is a certain long term data material worth?
Simplistic approach: Try to estimate the equivalent amount of
independent data and assess the reliability by bootstrapping
(simulations).
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Analysis of reliability by bootstrapping
Simulated variability of the expected maximum wave height (green)
when the long term statistics is based on 1000 independent
measurements (left) and 500 measurements (right).
Intrinsic variability (probability density) of Hmax red, dashed.
0.4
0.5
0.35
0.3
Probability density
Probability density
0.4
0.3
0.2
0.1
0.25
0.2
0.15
0.1
0.05
20
25
30
35
Wave height (m)
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40
20
25
30
35
Wave height (m)
40
Increased precision in the probability distribution for the maximum
wave height as the size of the long term Hs data base increases
(Expressed in terms of the number of independent measurements )
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Probability density for the 100 year maximum wave height based on
various sets of long term distribution data from Haltenbanken
(Confidence intervals for E(Hmax) based on bootstrapping)
Probabity density
0.3
Whole period
0.25
74-87
0.2
88-93
0.15
80 % conf. intervals
for E(Hmax )
0.1
74 -93
74 -87
88-93
0.05
0
26
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28
30
32
34
36
Maximum wave height (m)
38
40
CONCLUSIONS
• The relations for the extremes of multiple scale stochastic processes offer
a unified approach to estimation of extreme values of individual have and
crest height as well as the significant wave height.
• The results depend on the probability distributions and the mean
up-crossing frequencies for the process (that is, the mean storm duration
for significant wave height).
• Simplified methods may be derived and analyzed by means of the full
expressions.
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