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

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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

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