100-year and 10,000-year Extreme Significant Wave Heights How Sure Can We Be of These Figures?

Transcription

1 100-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

2 Wave crest elevations: BP s EI 322 A Statoil photograph

3 Wave breaking: M/T Prestige A Statoil photograph

4 Wave Crest Elevations Predicting 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 2 nd order theory, which is the one currently used by the oil industry.

5 Non-Exceedance Probability D 2 Some evidence of rogue waves 1 0 higher than Forristall distribution (Sterndorff et al. C max / HOMAE s 2000) Field Measurements Gumbel fit of Measurements C max / H s

6 Recent example with C/H s = 1.6

7 Strongly-nonlinear crest behaviour A Statoil photograph

8 Observations from Dale Princess A Statoil photograph

9 Explanation for violent breaking particle escape (Rainey J.Eng.Maths 2007) P jp Px jp Dx tr 0.8

10 Wave Crest Elevations Predicting 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 2 nd order theory, which is the one currently used by the oil industry.

11 Probability Empirical Distribution Function probability smallest ann. max Hs (metres)

12 The Extremal Types Theorem

13 Bayesian method for estimating parameters

14 Markov Chain Monte Carlo (MCMC)

15 Options for priors

16 Our initial choice of priors

17 P (m u ) P (s i g m a ) mu sample: sigma sample: mu sigma

18 P (e ta ) eta sample:

19 Maximum Hs Return levels: 1 to 1000 years years

20

21 P (e ta ) eta sample: eta

22 Maximum Hs Return levels to yrs: eta -ve years

23 Can we do any better? Use information on boundedness of Hs Use more of the data Seasonality Thresholds Clusters

24 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.

25 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, 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 Davison, A.C. and Smith, R.L.(1990) Models for exceedances over high thresholds. J.Roy.Stat.Soc B.Vol.52, No 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 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 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, 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 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, Tawn, J.A. (1992) Estimating probabilities of extreme sea-levels. J.Roy.Stat.Soc.C. Vol.41.No 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,