INVESTIGATING PARENT DISTRIBUTION OF TYPHOON-GENERATED ANNUAL MAXIMUM WAVE HEIGHT AND SAMPLE DISTRIBUTION OF RETURN WAVE HEIGHT ON THE EAST CHINA SEA

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1 INVESTIGATING PARENT DISTRIBUTION OF TYPHOON-GENERATED ANNUAL MAXIMUM WAVE HEIGHT AND SAMPLE DISTRIBUTION OF RETURN WAVE HEIGHT ON THE EAST CHINA SEA Masataka YAMAGUCHI Graduate School of Science and Engineering, Ehime University, Japan

2 . Aim of the Study Identifying the parent distribtion of extreme waves may contribute to an improvement of statistical reliability of the estimated return wave height. From this point of view, by using a Monte-Carlo simulation model, we make a huge size sample of typhoongenerated(tg) annual maximum(am) wave height on the East China Sea with extensive shallow water area, where AM wave height is usually produced by a strong typhoon.

3 The extreme value analyses using the sample are conducted for two purposes. ) Estimation of a parent distribution of TG-AM wave height a TG-AM sample with size of 0,000 ) Investigation of sample distribution of return wave height 00 sets of a TG-AM sample with size of 0

4 . Description of the Method Model for Pressure Distribution in a Typhoon Elliptical distribution( considering decay of typhoon power) p=p c + (0-p c ) exp[-{(x/a) +(y/b) } -/ ] 0 y α x Y X c,y c,p c,θ,a,b b a b a ψ 6 parameters R=(a+b)/ : mean typhoon radius θ X (Xc, Yc)

5 Monte-Carlo Simulation Model consists of sub-models for ) stochastic generation model for parameters of a typhoon ) gradient wind model ) -G shallow water wave model ) LSM-based extreme value analysis model

6 Modeling Domain j(y) area 9. i(x) 7. ξ 6.7 ξ X=80km x ξ Wave computation is conducted on the East China Sea area enclosed by dotted line. 6

7 Stochastic Typhoon Model consists of sub-models for ) the annual occurrence rate : Poisson dist. ) generation of the parameters of a typhoon on the boundary ) change of the parameters with movement of a typhoon in the inner region

8 Flow for Sequential Generation of Typhoon Parameters start yearly loop seasonal loop typhoon occurrence rate Generation t=0 Poisson dist. init. position (X c0,y c0 ) cumul. dist. init. central pressure p c0, etc. spline + dev. iteration (every 6 hrs.) Development & t=i Movement position (X ci+,y ci+ ) i=i+ X ci+ =a r +b r X ci +dev. central pressure p ci+,etc. p ci+ =a r +b r p ci +dev. typhoon radius R i+ R i+ =a r +b r R i +c r init. typhoon radius R 0 R 0 =a r +b r p c0n +dev. radius ratio (b/a) 0 (b/a) 0 =a r +b r p c0 +c r R 0 +dev. p ci+ radius ratio (b/a) i+ (b/a) i+ =a r +b r (b/a) i +c r p ci+ +d r R ci+ inner area or p c <008hPa end

9 hist. simul. Tracks of historical and simulated typhoons in 7 years

10 LSM-Based Extreme Value Analysis Model Candidate distributions: a) Gumbel distribution F(x)=exp[-exp{-(x-B)/A}] b) Weibull distribution with 7 kinds of fixed shape parameter (k=0. - 0) F(x)=-exp[-{(x-B)/A} k ] where A : scale parameter, B:location parameter Criterion of the largest correlation coefficient

11 Conditions of Wave Computation on the East China Sea (0 km grid) ) about 0,000 typhoons simulated over 0,000 years ) strong historical typhoons in years from 98 to 998

12 . Estimation of a Parent Distribution a) Simulated typhoon case A sample of TG-AM wave height with size of 0,000 b) Historical typhoon case A sample of TG-AM wave height with size of

13 Gumbel distribution (A, B) is also represented by H 0 and (=H 0 /H 0 ) γ 0 γ 0 :spread parameter(goda) H r :r-year return wave height Weibull distribution (k, A, B) is also represented by k, H 0 and γ 0 (=H 0 /H 0 )

14 0 j(y) i(x) x = 0km 7 x 6 simul. typ. H0 m H0=m N 0 j(y) i(x) x = 0km 7 x 6 simul. typ. k.. G N G i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ. γ N γ 0=. γ H 0, k, 0 =H 0 /H 0 (simulated typhoon case)

15 0 0 0 i(x) j(y) 0 0 x = 0km 7 x 6 hist. typ. H0 m 6 N i(x) j(y) 0 0 x = 0km 7 x 6 hist. typ. Hσ 0 m j(y) i(x) 0 0 x = 0km 7 x hist. typ. γ N H 0, H σ0, γ 0 =H 0 /H 0 (historical typhoon case)

16 . Sample Distribution of Return Wave Height Extreme value analyses for 00 sets of a TG-AM sample with size 0 are conducted in a) Fixed Shape Parameter (FSP) case shape parameter is fixed to that estimated using a sample with size 0,000. and b) Variable Shape Parameter (VSP) case shape parameter is variable. (criterion of the largest correlation coefficient)

17 From a return wave height sample with size 00 in FSP or VSP case, statistics such as mean H r standard deviation H σr skewness α kurtosis β r r are calculated. r : return period

18 0 0 0 i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ H0 m H0=m fixed shape para. case variable shape para. case i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ. Hσ 0=.7m Hσ 0 Statistics of return wave height sample ( H 0, Hσ 0, ) in FSP or VSP case. VSP case > FSP case for, Hσ 0, fixed shape para. case variable shape para. case i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ H 0 γ 0.7 N γ 0= fixed shape para. case variable shape para. case γ 0 γ 0

19 The LSM Based Model is also applied for approximating a distribution of return wave height sample with size 00 in FSP or VSDP case.

20 H0cal m H00cal m 0 fixed k (,) Weibull 0 k0=.0 ρ 0 = fixed k (,) H0data m Weibull k00=.0 ρ 00 = H00data m variable k (,) Weibull k0=.0 ρ 0 = H0data m variable k (,) Weibull k00=. ρ 00 = H00data m Q Q plot to a return wave height sample in FSP or VSP case ρ r (FSP) > ρ r (VSP), k r (FSP) > k r (VSP), r =0,00

21 0 0 0 i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ. k0 G fixed shape para. case i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ. Shape parameter of Weibull distribution in FSP or VSP case k 0 (FSP) > k 0 (VSP) > k 00 (VSP) : smaller variability k 0 G G G variable shape para. case G i(x) j(y) 0 0 x = 0km 7 x 6 simul. typ. k00 G G G G G G G G variable shape para. case G G

22 0 0 0 i(x) j(y) j(y) 0 0 x = 0km 7 x 6 simul. typ. Hσ 0 x = 0km 7 x 6 simul. typ. Hσ Hσ 0=m.7 Hσ 0=.7m fixed shape para. case cal. data Comparison between sample statistics and calculated statistics in FSP or VSP case ( ) Hσ 0 variable shape para. case cal. data

23 0 0 j(y) 0 0 x = 0km 7 x 6 simul. typ. α 0cal α 0data j(y) 0 0 x = 0km 7 x 6 simul. typ. β 0cal /β 0data j(y) 0 0 x = 0km 7 x 6 simul. typ. α 0cal α 0data 0. 0 j(y) 0 0 x = 0km 7 x 6 simul. typ. β 0cal /β 0data i(x) N N N N fixed shape para. case fixed shape para. case 0. variable shape para. case 0.8 variable shape para. case 0.8 β 0 Comparison in FSP or VSP case ( skewness α0 0,kurtosis ) Lower moment statistics yields better agreement. Fitting is better in FSP case than in VSP case.

24 . Conclusions ) Parent distribution of typhoon-generated AM wave height is well-expressed by the Weibull distribution. ) Shape parameter and the other parameters of the parent distribution are space-dependent. ) Identification of the parent distribution of AM wave height contributes to more efficient estimation of return wave height. ) Sample of return wave height is subject to the normal distribution in a rough sense, but is well approximated by asymmetrical distribution such as the Weibull or Gumbel distribution in detailed aspects.

25 H0 m H0 fixed k (,) Hσ 0 m H0 m H0 variable k (,) Hσ 0 m α0 0 0 fixed k (,) Hσ 0 α β 0 α0 0 0 α 0 Hσ 0 variable k (,) 0 8 β N β 0 β N Variation of statistics with sample size N. Each statistics approaches nearly constant value.

26 A typhoon is represented with 6 parameters such as ) position of typhoon center (X c,y c ) ) central pressure p c ) inclination angle of ellipseθ ) typhoon radii (a, b), R=(a+b)/ : mean typhoon radius

27 Spatial variation of parameters of estimated parent distribution in simulated typhoon case. ) H 0 is more than 6 m on the southeastern area and reduces to 6-8 m towards the northwestern area due to the decay of typhoon intensity. ) Shape parameter k tends to decrease from on the central and southeastern areas toward around. on the northwestern area. ) Spread parameter γ 0 indicates almost opposite spatial distribution to the case of shape parameter. Frequent passage of typhoons on the southeastern area may bring about a sharper probability distribution of typhoon - generated AM wave height data.

28 Spatial distribution of return wave height statistics in historical typhoon case. ) Distribution of return wave height and spread parameter are similar to those in simulated typhoon case. ) Standard deviation from 0. to.m is much larger than the maximum value of 0.07m in the simulated typhoon case.

29 Spatial distribution of return wave height statistics( 0,, γ σ ) in FSP or VSP case H H 0 0 γ 0 ) Mean value H 0, in Fixed SP case almost completely agrees with H 0, γ 0 obtained form 0,000 sample respectively. ) Fixed Shape Parameter case gives a bit smaller H 0, smaller Hσ 0, and smaller spread parameter γ 0 than Variable Shape Parameter case.

30 In FSP case ρ r, k r is independent of return period. ρ r, k r is greater than in VSP case. better fitting of optimum distribution and smaller variability of return wave height. In VSP case ρ r, k r is smaller with increasing return period. lesser fitting and larger variability of return wave height. with increasing return period.

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33 Conditions of Wave Computation on the Northwestern Pacific Ocean(80 km grid) A. Typhoon case About 0,000 - typhoon simulations over 0,000 years yield * an AM sample with size 0,000 or * 00 sets of sample with size 0

34 . Analysis Method of Sample LSM-Based Extreme Value Analysis Model Candidate distributions: a) Gumbel distribution F(x)=exp[-exp{-(x-B)/A}] b) Weibull distribution with 7 kinds of fixed shape parameter (k=0. - 0) F(x)=-exp[-{(x-B)/A} k ] where A : scale parameter, B:location parameter to be estimated by the least square method Criterion of largest correlation coefficient

35 LSM Based Model is applied for extreme value analysis of typhoon - generated AM sample a) a sample with size 0,000 b) each of 00 sets of sample with size 0 Low case is the same as typhoon case. a) a sample with size 0,000 b) each of 00 sets of sample with size 0

36 . Sample Distribution of Return Wave Height ) Typhoon Generated Wave Case 0 i(x) j(y) 0 0. simul. typ. k= Shape parameter k (Weibull distribution) of a parent distribution Shape parameter tends to decrease from or on the southwestern area toward around. on the northwestern area and on the northeastern area.....

37 H0 m 6 Hσ 0 H0 fixed shape para. case Hσ 0 m 0 8 α 0 α0 0 β 0 β Variation of statistics associated with sample size Each statistics takes almost constant value for N > 00. N

38 . Conclusions Identification of the parent distribution of return wave height contributes to more efficient estimation of return wave height. Sample of return wave height is subject to the normal distribution in a rough sense, but is approximated well by asymmetrical distribution such as the Weibull or Gumbel distribution in detailed aspects. Shape parameter k R hardly depends on return period R in KPD case, but it decreases in UKPD case (growing variability of return wave height).