100-year and 10,000-year extreme significant wave heights – how sure can we be of these figures?
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100-year and 10,000-year Extreme Significant Wave Heights – How Sure Can We Be of These Figures?. Rod Rainey, Atkins Oil & Gas Jeremy Colman, Independent Consultant. Wave crest elevations: BP’s EI 322. A Statoil photograph. Wave breaking: M/T Prestige. A Statoil photograph. - PowerPoint PPT PresentationTRANSCRIPT
100-year and 10,000-year Extreme Significant Wave
Heights – How Sure Can We Be of These Figures?
Rod Rainey, Atkins Oil & Gas
Jeremy Colman, Independent Consultant
A Statoil photograph
Wave crest elevations: BP’s EI 322
A Statoil photograph
Wave breaking: M/T Prestige
Wave Crest ElevationsPredicting extreme crest elevations is a two-stage process:
1. Find extreme values of significant (4xRMS) wave height, from “hindcast” databases produced by calibrated meteorological computer models, which cover the last 60 years. These are in the public domain – the area West of Shetland is pertinent, as the stormiest in the oil industry.
2. Combine with the probability distribution of wave crest elevations, for given significant wave height. This is the Rayleigh distribution on linear theory, and the “Forristall distribution” on Stokes 2nd order theory, which is the one currently used by the oil industry.
Some evidence of “rogue waves” higher than Forristall distribution (Sterndorff et al. OMAE 2000)
0
1
2
3
4
5
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
0.2
0.4
0.6
0.8
1
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Cmax / Hs
No
n-E
xcee
dan
ce P
rob
abili
ty Field Measurements
Gumbel fit of Measurements
Recent example with C/Hs = 1.6
A Statoil photograph
Strongly-nonlinear crest behaviour
A Statoil photograph
Observations from “Dale Princess”
Explanation for violent breaking – “particle escape” (Rainey J.Eng.Maths 2007)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0
P jp
0.80.8 Px jp Dx tr
Wave Crest ElevationsPredicting extreme crest elevations is a two-stage process:
1. Find extreme values of significant (4xRMS) wave height, from “hindcast” databases produced by calibrated meteorological computer models, which cover the last 60 years. These are in the public domain – the area West of Shetland is pertinent, as the stormiest in the oil industry.
2. Combine with the probability distribution of wave crest elevations, for given significant wave height. This is the Rayleigh distribution on linear theory, and the “Forristall distribution” on Stokes 2nd order theory, which is the one currently used by the oil industry.
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Empirical Distribution Function
Hs (metres)
Pro
ba
bili
ty
probabilitysmallest ann. max
The Extremal Types Theorem
Bayesian method for estimating parameters
Markov Chain Monte Carlo (MCMC)
Options for priors
Our initial choice of priors
sigma sample: 100500
sigma
0.5 1.0 1.5 2.0 2.5
P(sigm
a)0.0
2.0mu sample: 100500
mu
9.0 9.5 10.0 10.5 11.0
P(mu)
0.0
2.0
eta sample: 100500
eta
-0.5 -0.25 0.0 0.25 0.5
P(eta)
0.0
2.0
4.0
2 5 10 20 50 100 200 500
10
15
20
25
30
35
40
Return levels: 1 to 1000 years
years
Ma
xim
um
Hs
eta sample: 100500
eta
-0.4 -0.3 -0.2 -0.1 0.1
P(eta)
0.0
10.0
5 10 50 500 5000
12
14
16
18
20
22
Return levels to 10000 yrs: eta -ve
years
Ma
xim
um
Hs
Can we do any better?
• Use information on boundedness of Hs
• Use more of the data– Seasonality– Thresholds– Clusters
Can we extend the analysis to individual wave heights – rather than Hs?
• Just give us the probability distribution for individual waves, given Hs
• BUGS will do the rest– Extracts the maximum possible information
from• The data• Your prior knowledge
– Expressed as probability distributions of the parameters of interest.
References
• Coles, S.G. and Tawn, J.A.(1996)A Bayesian analysis of extreme rainfall data. J.R.Stat.Soc. C.Vol.45,No.4,463-478• Coles, S.G. (2001).An introduction to statistical modelling of extreme values. Springer-Verlag.• Coles, S.G. and Powell, E.A.(1996)Bayesian methods in extreme value modelling: a review and new developments.
Int.Stat.Review.Vol.64.No.1.119-136.• Davison, A.C. and Smith, R.L.(1990) Models for exceedances over high thresholds. J.Roy.Stat.Soc B.Vol.52, No. 3
393-442• Dekkers, A.L.M. and De Haan, L. (1989). On the estimation of the extreme-value index and large
quantile estimation. Ann.Stat. Vol.17, No. 4.1795-1832• Eastoe, E.F. and Tawn, J.A. (2009) Modelling Non-stationary extremes with application to surface ozone.
J.Roy.Stat.Soc. C.Vol.58.No.1. 25-45.• Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997) Modelling Extremal Events Springer• Gilks, W.R. and Spiegelhalter, D.J.(1996) Markov chain Monte Carlo in practice. Chapman and Hall• Leadbetter, M.R., Lindgren, G., and Rootzén, H. (1983). Extremes and related properties of random
sequences and processes. Springer-Verlag.• Resnick, S.I. (1997). Heavy tail modelling and teletraffic data. Ann.Stat. Vol.25, No. 5, 1805-1849• Smith, A.F.M. and Roberts, G.O.(1993) Bayesian Computation via the Gibbs sampler and related Markov
chain Monte Carlo methods.J.Roy.Stat.Soc.B.Vol.55, No.1 3-23• Smith, R.L. (1989) Extreme value analysis of environmental time series: an application to trend detection in
ground-level ozone.Stat.Sci.Vol.4, No.4, 367-377• Tawn, J.A. (1992) Estimating probabilities of extreme sea-levels. J.Roy.Stat.Soc.C. Vol.41.No.1. 77-93• Wadsworth, J.L., Tawn, J.A. and Jonathan, P. (2010). Accounting for choice of measurement scale in extreme
value modelling. Ann. Appl. Stats,Vol. 4, No. 3, 1558-1578.