Long-term Wave Analysis PDF

WAVE ANALYSIS Long-term analysis OUTLINE OUTLINE q Introduction q Introduction q Long term distributions for wave climate q Long term distributions for wave climate q Extreme value long term distributions q Extreme value long term distributions q AMEVA: Software to represent wave climate q...

WAVE ANALYSIS Long-term analysis OUTLINE OUTLINE q Introduction q Introduction q Long term distributions for wave climate q Long term distributions for wave climate q Extreme value long term distributions q Extreme value long term distributions q AMEVA: Software to represent wave climate q AMEVA: Software to represent wave climate MASTER Coastal Hazards-Risks, Climate Change Impacts and Adaptation (COASTHazar) 2 Short vs. long term wave analysis h WAVE CLIMATE SYSTEM Record of the sea surface vertical displacement PDF wave height P(H) t Sea state sea surface vertical displacement h H S(f) Wave spectrum t Hm0 SHORT TERM WAVE ANALYSIS The long-term prevalent wave conditions of a region / Characteristic state of the wave system Hm0 Hm0 CDF Hmo F(Hmo) Mean and extreme climate conditions F(H m0, Tp, etc..) WAVE CLIMATE SYSTEM Common variables of the wave climate: • Significant Wave height (Hs) • Peak-mean period (Tp-Tm) Wave conditions at a particular location over a long period of time ..conditioned by latitude, position relative to the continent, bathymetry, cyclonic storms.. ¿How long ? f t WAVE CLIMATE SYSTEM Components of the system: Atmosphere, hydrosphere, cryosphere, land surface, and biosphere. Sea state parameters time series (Hmo, Tp, etc.) LONG TERM WAVE ANALYSIS 3 Variables of the system: wind velocity, wind direction, wave height, mean period, wave direction, tide, mean sea level, surface currents, .. (from one month to many millions of years) Generally, at least 30 years… Dozens of years… As long as possible! Characteristics of the of the system can be defined by the averaged values and other statistics of the variables. WAVE CLIMATE SYSTEM WAVE CLIMATE SYSTEM everyday wave climate Funcionability, operability Getting started ... series of different wave climate parameters Getting started ... series of different wave climate parameters • Mean Wave direction (Q) Equilibrium beach plantform Variability through the year, how to spot storms events? regular wave condition of the whole year Agitation inside the harbor f(H,T,Q) Hs (also persistence, D) In the coastal area there are different systems: sand beach, port, sand dune system, etc zoomed-in, identify storms! Reliability, extreme conditions F(mean wave energy flux) regular and extreme wave, Hs Sediment transport In practice one recording of for instance 20 minutesis done every three hours. This record is thought to be representative for the entire time period of three hours. The duration of a storm is generally 6 to 8 hours duringwhich the conditions (mean wind speed) are more or less constant. Long term distributions OUTLINE Long term distributions Long term estatisitcal distribution of a sea state parameter over a period of time: Statistical distribution of sea states parameter over that time period. q Introduction q Long term distributions for wave climate Answers questions such: how many hours the wave climate exceeded threshold? q Extreme value long term distributions The most commonly used is the annual mean distribution of a sea state parameter: q AMEVA: Software to represent wave climate Statistical distribution of that parameter (Hs, Tp, …) over the year. Example: The annual distribution of Hs is represented by its distribution function defined by the probability that in any sea state of the year, the significant wave height, Hs is less than a given one, Hs0: Mean annual regime of Hs: P(� ≤ � ) in the year. The annual distribution of Hs can also be defined for time periods other than the year, e.g., January is the statistical distribution of Hs in the month of January. 10 Long term distributions Long term distributions Determination of the anual statistical distribution of Hs Time series of Hs values 1- Data base: Time series of Hs, for N data (ex. hourly) 2- Ordering from highest, Hs1, to lower, HsN 3- Assignment of probability to each data : Data i=1 Data i=2 … Data i=i … Data i=N Hs , T p y m: Long term distributions individual and join distributions Hs, Tp y m join distributions Frequency (probability of occurrence) Statistical distribution of Hs (Occurrence probability) Statistical distribution of Tp (Occurrence probability) P(Hs£Hs1) = 1 P(Hs£Hs2) = (N-1)/N … P(Hs£Hsi) = (N-i+1)/N … P(Hs£HsN)= 1/N 4- Fitting of a theoretical distribution, F(Hs) Joint average distribution and marginal de Hs, Tp (Frequency representation in %) Long term distributions Long term distributions Correlation Hs, and Correlation Hs, Tp m Long term distributions Fitting Hs long term distribution � � � � � � Hs- qm join distribution (In frequencies in %) Bar length: frequency Bar width: range of Hs Hs = 1 − ��� − � −� There are many distributions, which one to use? The one gives you the best correlation F(Hs = 2,6) = ??? P(Hs £ 2.6) = 0,95 It is generally assumed that the period has a normal statistical distribution, of parameters µT sT conditioned to Hs: � = � (� ) � = � (� ) 95% of the sea states of the year, Hs £ 2.5 m F(Hs) Long term distributions Long term distributions Fitting Hs long term distribution Fitting Hs long term distribution Most commonly used distributions: Weibull (2 or 3 par.) and Lognormal. PDF CDF Long term distributions F( H s ) = 2 é 1 æ ln 1 1 - Aö ù H exp ê - ç H s ÷ ú dH s ò0 s B ø úû 2p B Hs êë 2 è f ( H s )= Lognormal Variance 2 s C f ( H s )= C B H C -1 s [ ] C exp - (B H s ) ] 1 æ 1ö H s = G ç 1+ ÷ B è Cø Variance é æ 2ö 2 æ 1ö ù -2 s 2H = B ê G ç1+ ÷ - G ç 1+ ÷ ú Long term distributions Long term distributions Other representations used… Other representations used… N C Mean Range NNW 0 < B <¥ [ CDF Weibull -¥ < A < ¥ ; F( H s ) = 1 - exp - (B H s ) s Lognormal. 2 2A + B ( e B - 1) s 2H = e 0 < Hs < ¥ ; PDF Weibull (2par.) Box plot 2 é B ù H s = exp ê A + ú 2û ë Mean Range 2 é 1 æ ln 1 - Aö ù ÷ú exp ê- ç H s B ø ûú 2p B H s êë 2 è Other representations used… ë è Cø è Cø û July - june 0<H< ¥ NNE NW NE WNW ENE W 0% 10% 20% WSW E 30% ESE SW SE SSW S SSE Radius: occurrence frequency Colour: magnitude Hs (m) <=0.5 >0.5 - 1 >1 - 1.5 >1.5 - 2 >2 - 2.5 >2.5 - 3 >3 Radius: magnitude Colour: relative frequency Radius: magnitude Line: percentiles Extreme value analysis Event associated with certain conditions of the atmosphere-ocean system at its highest degree. Very intense, of the greatest severity. OUTLINE q Introduction q Long term distributions for wave climate q Extreme value long term distributions q AMEVA: Software to represent wave climate Coastal management Flooding risk 26 Maritime Works Extreme value analysis Extreme value analysis Extreme value analysis Hs Return period concept Extreme event --> a storm, what is a storm? We define it. Storm is created by atmospheric pattern For Peak over Threshold (POT) Method Why is it necessary to perform a special analysis for extreme events? How to define the threshold? For the timeseries data, try to find 10 occurrence of storms in a year How long the storms last? It depends, on the coast locations, period of meteorologicalatmospheric disturbance, The return period can be defined as an average time interval, usually expressed in years, after which an event of a certain magnitude will be equaled or exceeded. Maxima máximo H s Example: Some notes about Return Period special analysis for the extreme value -> no regular distribution fits the extreme probability f (Hs ) Hs Extreme Value Long term distribution Extreme value analysis Return period concept § This average time between independent events is what we call the "return period” and allows us to quantify the probability of the event, since if we are assuming that the event is exceeded once every 100 years, we can also assume that the probability of such an exceedance in any given year will be 1/100. t Hs Long term distribution Periodo de Probability (%)retorno Or Return period Extreme value analysis Independent events!!! Storm surge (cm) § In other words, the inverse of the return period turns out to be the annual probability of the event being exceeded. If it has a return period of 100 years, its average annual probability will be 1/100. Extreme value analysis Return period concept To study how probability "accumulates" over time, two details must be taken into account: hasil perkalian • For independent events, the joint probability of occurrence is the product of probabilities. • Since the occurrence will only happen once, we will study the joint probability of the years in which the event does not occur. Extreme value analysis Determination of the extreme value distribution of Hs Storm surge (cm) Time (years) Generalized Extreme Value distribution: A math probabilistic function which combines all the parameters in general distribution function (Weibull, Gumbel, Frechet) First year: § probability of exceedance is 1% (1/100) § probability of non-exceedance will be 1-1/100=0.99: 99%. • • • • • • Third year: § probability of non-exceedance for two (3) years: 0.99-0.99-0.99 = 0.993 = 0.9703 (97.03%) And for 100 years it will be: 4- Fitting of a theoretical distribution of extremes 0.99100 Time (years) = 0.366 (36.6%). That is, the probability of non-exceedance of the event in 100 consecutive years will be 36.6%. Subtracting 100% we will have the probability of overcoming, which will be 100% - 36.6% = 63.4%. Therefore, after 100 years, the probability of an event with a return period of 100 years is 63.4%. Extreme value analysis Scalar and directional annual extreme scalar and directional distribution of Hs • • • • • 1- Database: time series of Hs, for N years 2- Selection of the series of N annual maximum values of Hs 2- Ordering from highest, Hsmax1 to lowest, HsmaxN 3- Assignment of probability to each data : P(Hs£Hsmax1) = 1 P(Hs£Hsmax2) = (N-1)/N … P(Hs£Hsmaxi) = (N-i+1)/N … P(Hs£HsmaxN)= 1/N Return period concept Second year: § probability of non-exceedance for two (2) years in a row will be: 0.99*0.99 = 0.992 = 0.9801 (98.01%) • • Time (years) As an average, a storm surge event on the coast larger than 120 cm happens every 100 years Extreme value analysis Question: if the return period is the mean time between events and is related to the probability of each year, and what will happen after 100 years? What is the probability of overcoming a 100-year return period event in 100 years? Time (years) Storm surge (cm) Correlation Hs - Tp Regímenes extremales direccionales Ajustes de la media y la desviación típica Extreme value analysis Extreme value analysis Extreme value analysis Relationship between the annual extreme distribution of a parameter, the lifespan and the probability of failure of a structure Scalar and directional annual extreme scalar and directional distribution of Hs If the failure of a marine structure is associated exclusively with the exceedance of a certain value of a sea state parameter (e.g. Hso): PF = P(Hs >Hso) Extreme annual scalar distribution Probability that the structure does not fail in any given year : Extreme annual ENE distribution Cumulative Distribution Function (?) PNF1año = P(Hsmax anual £ Hso) = FRT (Hso) Probability that the structure does not fail in the V years of useful life, PNFV will be: Relationship between the annual extreme distribution of a parameter, the lifespan and the probability of failure of a structure If the failure is determined by exceeding a value of Hs, this value, called the calculation value, can be obtained by clearing Hs0 from the time regime in the above equation. FRT(Hso) = (1 – PFV)1/V The return period of a parameter X is the average time, measured in years between two exceedances of X. If the probability of exceedance of X in a year is 1 - FRT(X), the return period is : PNFV = PNFyear1 * PNFyear2 * …….PNFyearV = P(Hsmax anual £Hso)V = FRT(Hso)V The lifetime probability of failure, PFV, will therefore be : Parameters of the ParetoPoisson and GEV fits PFV = 1- FRT(Hso)V Extreme value analysis Relationship between the annual extreme distribution of a parameter, the lifespan and the probability of failure of a structure FRT(Hso) = (1 – PFV)1/V Distribution functions for extreme wave distributions: � �� Distribution functions for extreme wave distributions: (1) Generalized Extremes function (GEV). The GEV function is a family of continuous probability distribution functions developed by extreme value theory that combines the Gumbel, Fréchet and Weibull distributions, also known as type I, II and III extreme distribution functions, respectively. (1) Since the return period is the mean period mean period between two exceedances of Hs: Using the dimensionless variable : the GEV is given by : T(Hso) = 1 / (1 – FRT(Hso)) = 224 años Þ ; , , (2) Hs0 = 6,2 m �− Extreme value analysis For example, si PF = 0.2 y V = 50 years FRT(Hso) = (1 – PFV)1/V= 0,995547 �= example, Dutch build their sea dike with v = 10.000 years --> Pf = 0.00000.... Extreme value analysis Hs0 = � �� = ⁄ , the distribution function of − � � ; ≠ , > �/ µ: location parameter, s > 0: scale parameter, x: shape parameter. Extreme value analysis Extreme value analysis Distribution functions for extreme wave distributions: Distribution functions for extreme wave distributions: Distribution functions for extreme wave distributions: (1) Generalized Extremes function (GEV). (1) Generalized Extremes function (GEV). (1) Generalized Extremes function (GEV). -1/ x æ x - µ ö ù ïü ïì é F ( x;q ) = exp í - ê1 + x ç ú ý ÷ è y ø û þï îï ë µ ® location y ® scale x ® shape � > �: ���� � > �: �� � < �: � � Extreme value analysis -1/ x æ x - µ ö ù ïü ïì é F ( x;q ) = exp í - ê1 + x ç ú ý ÷ è y ø û ïþ ïî ë � > �: ���� � > �: �� � < �: � � ; , , For x > 0, la GEV is valid for Para x < 0, la GEV is valid for = = − � �� � � > � − �⁄ , ⁄ , ; ≠ , > �/ < � − �⁄ . In the first case, at the lower end the value is 0; in the second case, at the upper limit it is equal to 1. For x = 0, GEV is undefined and is replaced by its limit when x ® 0: ; , , = � � Extreme value analysis Extreme value analysis Distribution functions for extreme wave distributions: particular cases of GEV PDF � CDF Gumbel � � = = 1 Mean � − � = Variance � Range PDF − � − 6 −∞ < � < ∞; −∞ < Weibull (3par) o + 0,5772 = Distribution functions for extreme wave distributions: particular cases of GEV � − − Extreme value analysis FT-III C é æ - Aö F( H s ) = exp ê - ç H s ÷ êë è B ø CDF f ( H s )= Mean s < ∞; 0 < <∞ Extreme value analysis PDF ù ú úû C é æ - Aö ù exp ê - ç H s ÷ ú ëê è B ø úû 2 Hs 2ö é æ = B2 ê G ç 1 + ÷ - G2 è Cø ë A < Hs < ¥ ; (1) Peak Over Threslhod (POT). By restricting the database to a maximum per year, an important potential source of information, corresponding to other time periods, is discarded. POT Method The POT method is based on the use of all events in a time series that exceed a given threshold. The POT method incorporates the existing extreme value information in the database. Events are considered to be those extreme values that are separated from each other long enough to be considered independent (i.e.: minimum 5 days). Gros et al. (1994): Selection of a threshold that resulted in a number of annual exceedances on the order of 10 on average. Extreme value analysis POT Method Método (POT). 1- Definition of a threshold of Hs leaving an average, l, of about 10 storms per year. 2- Least-squares fit of the Weibull distribution of exceedances, obtaining the parameters a, b y g. =�− � � �� A æ Hsö ç ÷ Bè B ø -(A+1) -A é æ ö exp ê - ç H s ÷ êë è B ø Variance æ 2ö 2 æ 1ö ù 2é s 2H = B ê G ç 1 - ÷ - G ç 1 - ÷ ú è Aø è Aø û ë s Range 0 < Hs < ¥; -¥ < A < ¥ ; � = � � 0<B<¥ Extreme value analysis POT Method The probability that the largest storm occurring in a year has a significant height above a certain pre-established Hsr value is given by the expression: �− �� � = � > � � = �− � �� where ”λ” is the average number of storms occurring in a year, and Fw is the Weibull distribution of the exceedances whose expression is : � = �− �� � and FRT(Hsr) is the value of the distribution function of the temporary regime at Hs = Hsr Extreme value analysis OUTLINE q Introduction q Long term distributions for wave climate q Extreme value long term distributions q AMEVA: Software to represent wave climate 3- Extreme value distribution is: �� ù ú úû æ 1ö H s = B G ç1 - ÷ è Aø Event: a time series in which one or more of its sea states exceed the threshold, in which case, only the maximum value of the significant height will contribute data to the POT base. Extreme value analysis f ( H s )= Mean 0<B<¥ Extreme value analysis Distribution functions for extreme wave distributions: CDF 1ö ù æ ç 1+ C ÷ ú è ø û 0 < A < ¥¥; -A é æ ö ù F( H s ) = exp ê - ç H s ÷ ú B è ø úû ëê Frechet 1ö æ H s = A + B G ç1+ ÷ Cø è Variance Range (C -1) C æ Hs - Aö ç ÷ Bè B ø Distribution functions for extreme wave distributions: particular cases of GEV �� 54 Extreme value analysis Extreme value analysis Extreme value analysis MCR Installer: http://www.mathworks.com/products/compiler/mcr/index.html IH-AMEVA AMEVA, a user friendly toolbox to analyze statistically environmental variables Análisis Matemático Estadístico de Variables Ambientales Extreme value analysis Extreme value analysis change the pathway - in a folder with: ameva_v1.4.2_pkg Files in the folder if the installation is completed: (with administrator permissions) 58 59 version 8.1 (R2013a)