Stochastic swell events generator

Transcription

1 Stochastic swell events generator K. A. Kpogo-Nuwoklo,, M. Olagnon, C. Maisondieu, S. Arnault 3 Laboratoire Comportement des Structures en Mer, Ifremer, Brest, France. Institut für Meteorologie, Freie Universität Berlin, Berlin, Germany. 3 LOCEAN/IPSL, Université Pierre et Marie Curie, Paris, France. Workshop on Stochastic Weather Generators Vannes, 7- May 6 K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

2 Context Long-term sea states statistics for engineering applications Offshore oil exploitation platform Coastal erosion Design of coastal and offshore structures Coastal erosion Wave energy harvesting K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

3 Sea state characterization: conventional approach sea state = description of the sea surface (wind sea + swell) sea state duration = from 3 min to 3h (in general) Characterization: Time-series of sea surface motion ({x(t), y(t), z(t)}) = Directional spectrum Directional spectrum = H s, f p, θ p, P w, etc. K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 3 / 8

4 Introduction Data Identification Swell Event Model (SEM) Stochastic generator Conclusion Sea states off West Africa K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 4 / 8

5 proposed approach: events based approach Dec :: swell.5.5. S(f) wind sea f (Hz).5 swell events wind sea events f (Hz).5 // // //3 date (year/month/day) Individual omnidirectional spectrum (3h) Time series of omnidirectional spectra Goals: Events based approach for swell climate modeling Stochastic generator of swell events K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 5 / 8

6 Adopted scheme K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 6 / 8

7 Data in situ Buoy {x(t), y(t), z(t)} Girassol: depth 45 m duration 3 years 3 o N 5 o N Akpo: depth 37 m duration an o Akpo 43 Girassol hindcast directional spectra (Wave Watch III) duration years 8 o W o 8 o E 36 o E 54 o E 5 o S 3 o S 8 63 K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 7 / 8

8 Events identification K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 8 / 8

9 Time-domain Expansion of Spectral Partitionning technique (TESP) smoothing S(t, f,θ) extraction (from S(t, f,θ)) 3 parameters estimation (H s, f p, etc.) 4 classification (swell or wind sea) K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 9 / 8

10 TESP result.3 f (Hz)...5 //5 // //5 // //5 //3 wind sea H s (m).5.3 //5 // //5 // //5 //3 f p (Hz).. //5 // //5 // //5 // θ p ( ) 5 5 //5 // //5 // //5 //3 date(year/month/day K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

11 Swell Event Model K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

12 Swell Event Model Characteristic sea state parameters Wave power (Energy flux) (P w ) Energy frequency (f e) Mean direction (θ m) typical swell event 4 P w (kw/m) f e (Hz) θ m ( ) time (h) K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

13 Swell Event Model (SEM) Power = Gamma distribution function: P w(t) = Frequency = linear: f e(t) = g 4πd t + T max Direction = linear: θ m(t) = ωt +θ E τ α Γ(α) tα exp ( ) t τ P w (kw/m) θ f (Hz) m ( ) e obs model time (h) 7 parameters: E = t f t i α = shape τ = scale P w(t)dt = energy d= travelled distance T max = maximal period ω = direction slope θ = initial direction K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

14 Result K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 3 / 8

15 Stochastic swell events generator Goals: Joint distribution of SEM parameters (E, α, τ, d, T max, ω, θ ) Occurrence process of swell events K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 4 / 8

16 Joint distribution of SEM parameters Adopted scheme Seasonality + Dependence between some parameters E (kwh.m ) d (Km) E(kWh.m ) K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 5 / 8

17 Joint distribution of SEM parameters Adopted scheme Seasonality + Dependence between some parameters E (kwh.m ) d (Km) E(kWh.m ) K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 5 / 8

18 Joint distributions of {E, d} et {ω,θ } using copula x x sim x x sim.8 F(x).6 u.4..8 F (x).6 u sim.5 u.4..5 u sim u u Marginal distributions validation: Kolmogorov-Smirnov test Choice of copula: BIC and AICc Goodness-of-fit test for copulas: Cramer-Von Mises (S n) and Anderson-Darling (A n) statistics K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 6 / 8

19 Joint distributions of {E, d} et {ω,θ } using copula Marginal fit (case of parameter E: Gamma) 4 Jan 4 Feb 4 Mar 4 Apr empirical E 3 empirical E 3 empirical E 3 empirical E 3 4 theoretical E May 4 4 theoretical E Jun 4 4 theoretical E Jul 4 4 theoretical E Aug 4 empirical E 3 empirical E 3 empirical E 3 empirical E 3 4 theoretical E Sep 4 4 theoretical E Oct 4 4 theoretical E Nov 4 4 theoretical E Dec 4 empirical E 3 empirical E 3 empirical E 3 empirical E 3 4 theoretical E 4 theoretical E 4 theoretical E 4 theoretical E K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 7 / 8

20 Joint distributions of {E, d} et {ω,θ } using copula Testing different copulas (August) F(d) F(d) Gaussian F(E) Clayton F(E) F(d) empirical model.8.6 F(d) Gumbel F(E) Frank F(E) parameter (ρ).66 Gaussian BIC -97. AICc - parameter(θ).68 Gumbel BIC AICc -78. parameter(θ).4 Clayton BIC AICc parameter (θ) 5.8 Frank BIC -95. AICc -98. K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 8 / 8

21 Joint distributions of {E, d} et {ω,θ } using copula Testing different copulas (August) F(d) F(d) Gaussian F(E) Clayton F(E) F(d) empirical model.8.6 F(d) Gumbel F(E) Frank F(E) parameter (ρ).66 Gaussian BIC -97. AICc - parameter(θ).68 Gumbel BIC AICc -78. parameter(θ).4 Clayton BIC AICc parameter (θ) 5.8 Frank BIC -95. AICc -98. Goodness-of-fit test of the Gaussian copula Jan Fev Mar Avr mai Jun Jul Aou Sep Oct Nov Dec S n (.89) (.4) (.6) (.65) (.67) (.74) (.66) (.9) (.) (.5) (.8) (.43) A n (.4) (.87) (.7) (.56) (.76) (.5) (.65) (.3) (.3) (.65) (.65) (.3) K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 8 / 8

22 Joint distributions of {E, d} et {ω,θ } using copula d(km) d(km) E (kwh.m ) 3 4 E (kwh.m ) Figure: Joint distributions of {E, d}: reference (left) and simulation (right). Histograms correspond to the marginal distributions K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 9 / 8

23 Distributions of α, τ, T max using GLM Distributions: Gamma Link function: Xβ = µ g(e(y)) = g(µ) = X β K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

24 τ τ τ τ τ τ τ τ τ τ τ τ Distributions of α, τ, T max using GLM 5 5 Jan sim ref 5 5 Feb 5 5 Mar 5 5 Apr α May 5 3 α Jun 5 3 α Jul 5 3 α Aug α Sep 5 3 α Oct 5 3 α Nov 5 3 α Dec α 3 α 3 α 3 α Simulation of (E, α, τ, d, T max, ω et θ ) = individual swell event simulation K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

25 Occurrence process of swell events Inter-arrival time (I) P w (kw.m ) 5 I(day) 5 4 I time (h) E (kwh.m ).5 Empirical Exponential 9 8 D * = empirical I(day) Gamma GLM : g(e(i)) = β +β E +β d +β 3 T max K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator / 8

26 Occurrence process of swell events Inter-arrival time (I) Reference Simulation 8 8 I (Days) 6 4 I (Days) d(km) d(km) Figure: Joint distribution of {d, I}:reference (left) and simulation (right) (case of August). Contours are spaced. from. to.4 K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 3 / 8

27 Simulation of swell events sequence 6 5 P w (kw.m ) 5 8/ 8/7 8/ 8/7 8/ 8/7 4.5 P w (kw.m ) 3 f e (Hz)..5 8/ 8/7 8/ 8/7 8/ 8/7 3 θ m ( ) /8 4/9 5/9 7/8 9/6 /6 /5 month/day One year of swell events 8/ 8/7 8/ 8/7 8/ 8/7 month/day August Validation: Monthly distribution of instantaneous H s Weather windows K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 4 / 8

28 Instantaneous H s 3 Jan 3 Feb 3 Mar 3 Apr simulation.5 simulation.5 simulation.5 simulation reference reference reference reference 3 May 3 Jun 3 Jul 3 Aug simulation.5 simulation.5 simulation.5 simulation reference reference reference reference 3 Sep 3 Oct 3 Nov 3 Dec simulation.5 simulation.5 simulation.5 simulation reference reference reference reference quantile-quantile plot of simulated H s against reference H s K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 5 / 8

29 Weather windows 6 4 H s < m, 4h Window reference simulation number of Access Days Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Number of access days per month K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 6 / 8

30 Conclusions We proposed a new approach (events based approach) to model swell climate. We developed a new method for swell and wind sea events identification: Time-domain Expansion of Spectral Partitionning technique (TESP). We proposed a model for an individual swell event: Swell Event Model (SEM). We constructed a stochastic generator of swell events. K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 7 / 8

31 Perspectives Extreme sea states Interannual variability Coupling swell and wind sea Other geographical zones Other met-ocean variables (current, wind, etc.). K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 8 / 8

32 Thank you for your attention K. A. Kpogo-Nuwoklo et al. Stochastic swell events generator 9 / 8