Short-Term Wave Analysis PDF

Hm0 h h t t Mean and extreme climate conditions F(Hm0, Tp, etc..) LONG TERM WAVE ANALYSIS Sea state parameters time series (Hmo, Tp, etc.) SHORT TERM WAVE ANALYSIS Sea state sea surface vertical displacement Record of the sea surface vertical displacement under a sea state Short vs. long...

Hm0 h h t t Mean and extreme climate conditions F(Hm0, Tp, etc..) LONG TERM WAVE ANALYSIS Sea state parameters time series (Hmo, Tp, etc.) SHORT TERM WAVE ANALYSIS Sea state sea surface vertical displacement Record of the sea surface vertical displacement under a sea state Short vs. long term wave analysis (Jackson and Short, 2021) Short vs. long term wave analysis Introduction to gravity waves t Hm0 F(Hmo ) S(f) P(H) Hm0 CDF Hmo f Wave spectrum H PDF wave height 8 parameters of the sea state: significant wave height, peak period, average Surface elevation vs. a wave: How to represent waves? Introduction to gravity waves q Introduction also refers to spectral analysis Properties of water waves q Directional Wave Spectra q Parametric Expressions for Wave Spectra q Frequency Domain Analysis q Probability Domain Analysis q Time Domain Analysis OUTLINE A wave can be defined as the profile of the surface elevation between two successive downward zero-crossings of the surface elevation (a wave cannot be negative) 4 MASTER Coastal Hazards-Risks, Climate Change Impacts and Adaptation (COASTHazar) WAVE ANALYSIS Short-term analysis Surface elevation is the instantaneous displacement of the sea surface relative to a sea level reference Surface elevation vs. a wave Introduction to gravity waves Still water level Variables -> direction... They are gravity waves, since the disturbance of the wind on the sea surface causes the water to tend to its state of equilibrium again through the action of gravity. : Succession of waves on the surface of the water. Generally generated by the action of the wind and comprised between periods of 2-30s. Introduction to gravity waves 3 2 17 16 18 A stationary ergodic process implies that the random process will not change its statistical properties with time and that its statistical properties can be deduced from a single, sufficiently long sample (realization) of the process. In the ocean, ensemble sampling is a complicated and expensive problem. It is very important that ensemble sampling can be replaced by single point sampling. A stochastic process is ergodic when the statistical properties of the process can be obtained from a single time series taken at one location. • Ergodic process Hypothesis: Why do we need ergodicity? Hypothesis: Gaussian distribution j h ( x, y, t ) = å a j cos í ìw 2j ü ï ( x cosq j + ysenq j ) - w j t + e j (w j ,q j )ïý ïg ï î þ If waves are a of zero mean and variance � , any random variable of the wave can be expressed as , equally distributed, of zero mean and sigma standard deviation. Hypothesis: gaussian stochastic process is a stochastic . 14 formed by the values of each of the series at instant tj, are called the set sample and can be considered as a random sample of size n. The set of random variables {1x(tj), 2x(tj), 3x(tj), ......, nx(tj)} 15 Stochastic process, x(t): Family of random variables, function of two arguments: time and spatial sample. The wave is analyzed as a Hypothesis: stochastic process At present, sea states of are used (the lower limit for statistical information is usually about 20 minutes from buoy records). The duration of the sea states is a compromise between a time short enough for the process to be weakly stationary and long enough to contain the necessary statistical information of the process. The time intervals in which the process can be considered weakly stationary are called . However, the wave is the response of the sea surface to non-stationary atmospheric conditions. In order to assume stationarity, it is necessary to work with "snapshots" of the process of such a duration that the variation of the process statistics is imperceptible. Simplification of the analysis usually requires the process to be stationary or (i.e. constant mean and variance throughout the time). Sea state concept 13 In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process. In probability theory, a process which exhibits both stationarity and ergodicity. iv) Narrow band spectrum ii) Maxima are independent ii) Gaussian process of zero mean i) Stationary and ergodic stochastic process Hypothesis: general Spectral analysis (frequency domain). Spectral sea state parameters are obtained (Hmo, Tp, qm,…). From them some statistical properties can be derived. Fitting of theoretical spectral and statistical models. Zero-crossing analysis (time domain). Statistical sea state parameters are obtained (Hs, Hmax, Tz, …). Fitting of theoretical statistical models. 4. Record splitting in sea states (i.e. 1 h duration) 3. Measurement and record of one or several kinematic variables of waves. 2. 4 3 1. 2 1 Short term wave analysis approach Short vs. long term wave analysis Short vs. long term wave analysis Login to the Moodle course website Donwload the files included in the “Buoy Data” folder Donwload the Matlab® script “upcrossing.m” in the “Matlab® Scripts” folder Copy of the files and scripts in the same folder § H(mean) = 0.82 m; H_rms = 0.93 m; H_s = 1.31 m; H_max = 2.65 m § Record_4.txt § H(mean) = 3.46 m; H_rms = 3.88 m; H_s = 5.44 m; H_max = 8.19 m § Record_3.txt § H(mean) = 1,21 m; H_rms = 1.35 m; H_s = 1.88 m; H_max = 3.41 m § Record_2.txt § Do the same with the other records Assignment 1: Examples: Let’s practice! § Calculate H (mean), H_rms, H_s, H_max Solution: H(mean) = 3.09 m; H_rms = 3.44 m; H_s = 4.84 m; H_max = 7.03 m (Check if H_s = 2/sqrt(2) * H_rms) Visualize individual waves >> plot(H,T,'.') >> xlabel('H (m)') >> ylabel('T (s)') Visualize waves by means of a histogram >> hist(H,30) >> xlabel('H (m)') >> ylabel('# of samples’) § Sort the waves in a descendent mode >> HH = sort(H,'descend’); 27 26 Example for the zero upcrossing method: Hmax THmax Hmax THmax Hmax THmax t t t • • • • • H 1/ n = é1 H rms = ê ëN i=1 N i=1 SH N/n û ù 2 iú i=1 N (i ) 1/2 S Hi SH n N Hm= Commonly Used Wave Height Parameters: 1 N (trough): Minimum vertical displacements. They can be positive or negative. = min Tc: Time interval between two consecutive crests. (crests): Maximum vertical displacements. They can be positive or negative. = max Regardless of the method, it is also determined: A: Wave amplitude: Maximum positive vertical distance between two consecutive upcrossing points T: Wave period: Time interval between two consecutive upcrossing points H: Wave height: Maximum vertical distance between two consecutive upcrossing points A discrete time series of wave heights and periods, Hi, Ti, i=1 to N, is obtained where N is the number of waves in the record. Example 1 (cont.): Tmax Tmax Tmax Time Domain Analysis 25 h Time Domain Analysis Examples: Let’s practice! § Use the script upcrossing.m to get the individual waves from the free surface time history recorded by the buoy using an upcrossing method. >> [T,H]=upcrossing(t,sup); § Load the file “Record_1.txt” in Matlab® “Record_1.txt” has to columns: Time, Free surface Elevation >> A=load('Record_1.txt'); >> t=A(:,1); >> sup=A(:,2); Example 1: 1. 2. 3. 4. SETTING-UP STEPS: Examples: Let’s practice! 23 22 is to analyze the water surface record The recorded water level must first be converted into a discrete time series of the fluctuation about the mean level by subtracting the mean water level from the record. The goal of and to get wave variables. Time Domain Analysis q Directional Wave Spectra q Parametric Expressions for Wave Spectra q Frequency Domain Analysis q Probability Domain Analysis q Time Domain Analysis q Introduction OUTLINE • Ergodic process: a process in which its statistical properties can be deduced from a single, sufficiently long, random sample of the process. Any collection of random samples from a process must represent the average statistical properties of the entire process. • Wave train / Sea state: a group of waves whose parameters remain constant. • Wave superposition: superposition may be applied to waves travelling through the same medium at the same time. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements. Basic concepts: Recapitulation 21 20 19 2.680 2.503 2.359 2.206 2.157 2.085 2.042 1.984 1.800 1.591 1.416 0.886 500 200 100 50 40 30 25 20 10 5 3 1 3.023 2.823 2.662 2.488 2.435 2.353 2.303 2.239 2.030 1.795 1.597 1.000 H 1/N / H ò p( H )dH H* 7.580 7.078 6.671 6.239 6.099 5.895 5.775 5.609 5.090 4.499 4.004 2.505 H 1/N / h rms = Quantile 1/ 3 35 36 1.075 1.087 1.099 1.115 1.123 1.131 1.138 1.146 1.186 1.254 1.351 Significant wave height ------ Mean wave height H1/N / Hqn remember this H1/ 3 = 1.416 H rms H1/ 3 = ¥ H* ¥ ò p( H )HdH ò p( H ) HdH H* ¥ q=1/n is the proportion of waves higher than a threshold value Hq H1/ N / Hrms N Wave Height Distribution Choose which one to use C is the Euler constant (C=0.57722). Wave Height Distribution Wave Height Distribution 2H H 2rms 2 æ H2 ö exp çç - 2 ÷÷ ; è H rms ø 2 H Hs 2ö æ exp çç - 2.005 H 2 ÷÷ ; Hs ø è H ³0 H ³0 2 æ H ö F(H) = 1 - exp çç - 2.005 2 ÷÷ ; Hs ø è f(H) = 4.01 æ ö F(H) = 1 - exp çç - H2 ÷÷ ; è H rms ø f(H) = Wave Height Distribution 2 2 ö æ p exp çç - H 2 ÷÷ ; 2 H è 4 H ø p H H ³0 H ³0 æ p H2 ö ÷ ; F(H) = 1 - exp çç 2÷ è 4 H ø f(H) = H ³0 H ³0 33 32 Water surface is assumed to be a superposition of infinite number of small waves æ H 2q ö q = 1 - F( H q ) = exp çç - 2 ÷÷ ; è H rms ø h rms 1 é h2 ù exp ê- 2 ú 2p ëê 2h rms ûú Hence, the distribution of (e.g. zero-downcrossing) wave heights can be represented by the Rayleigh distribution. This feature has been shown theoretically and verified empirically. Research has shown that for practically all locations the wave height distribution is reasonably close to Rayleigh distribution Probability domain analysis f( h ) = Water surface is Gaus The resulting sea surface is the sum of a large number of statistically independent processes Probability domain analysis 31 q Directional Wave Spectra q Parametric Expressions for Wave Spectra q Frequency Domain Analysis q Probability Domain Analysis q Time Domain Analysis q Introduction OUTLINE Probability domain analysis H ³0 H ³0 2 æ H ö F(H) = 1 - exp çç - 2 ÷÷ ; è H rms ø H rms 2 H ³0 f(H) = 2 ö æ exp çç - H2 ÷÷ ; è H rms ø 2H Probability domain analysis 30 29 28 q Directional Wave Spectra q Parametric Expressions for Wave Spectra q Frequency Domain Analysis q Probability Domain Analysis q Time Domain Analysis q Introduction OUTLINE § Plot H versus T. § Change the number of waves in the simulation. Check the values obtained for H_s, H_max and H_max/H_s ratio. § Open the file “HT_wave_emulator.m” in Matlab® Run it! Example 2: Numerical simulation of individual waves in a sea state 1. Login to the Moodle course website 2. Donwload the Matlab® scripts “HT_wave_emulator.m”, ”LH.m” and “Jonswap.m” in the “Matlab® Scripts” folder 3. Copy of the scripts in the same folder SETTING-UP STEPS: Examples: Let’s practice! § Why are there so many differences in wave statistics when changing the number of simulated waves? § Change the number of waves in the simulation. Check the values obtained for H_s, H_max and H_max/H_s ratio. § Open the file “Raleigh_wave_simulator.m” in Matlab® Run it! Example 1: Numerical simulation of individual waves in a sea state 1. Login to the Moodle course website 2. Donwload the Matlab® scripts “Raleigh_wave_emulator.m” and ”Raleigh.m” in the “Matlab® Scripts” folder 3. Copy of the scripts in the same folder SETTING-UP STEPS: Examples: Let’s practice! 47 44 43 Ha = H/Öm0 4n [ ] 1 m m ; n 2= 0 2 2 - 1 2p 1+ (1 +n 2 )-1/2 m1 how the period is being distributed (corespond) for certain wave height CDF for Raleigh (Hs) F(H > 6m) = ... In a Hrms = 3.12 m sea state, what is the probability of observing waves higher than 6 meters? (Sol.: 2.48%) § § What is the maximum wave height in the previous sea state if the storm is 3 hlong and the mean wave period is 12 s? (Sol.: 9.19 m) § Given a sea state for which H1/3 = 5 m, what is the probability of observing waves higher than 6 meters? (Sol.: 5.57%) Exercise ü ï ý ï þ Wave height and period joint density function for two different spectral widths according to Longuet-Higgins Wave height and period joint distribution CL= 2 2 ì 2 é æ ö 1 æ 1 ö ù ï ÷÷ ú f( H a , T a ) = C L çç H a ÷÷ exp í- H a ê1 + 2 çç 1 8 n è T a ø úû è Ta ø ï ëê î The mean period is associated to the mean frequency �� = �/� Given in terms of nondimensional variables Joint distribution for narrow banded spectrum (SWELL) n2 £ 0.36 (1975, 1983) Wave height and period joint distribution 42 41 40 2.280 2.427 2.609 2.738 2.862 2.932 2.980 3.017 3.130 H max , N H rms Probability domain analysis • • • s 0.02 0.04 0.06 0.08 0.1 0.12 0.14 max , N H 1.611 1.751 1.843 1.934 2.022 2.071 2.105 2.131 2.211 H Probability domain analysis 100 200 500 1000 2000 3000 4000 5000 10000 Number Of Waves Wave Height Distribution 0 0.5 2.146 2.302 2.493 2.628 2.757 2.830 2.880 2.918 3.035 ê Hrms h=0.4652 1.516 1.626 1.761 1.857 1.948 1.999 2.034 2.062 2.144 ~ H max , N Hs 1 1.5 2 Lab data: HR Wallingford ~ H max , N H rms 39 2.5 38 37 m_0 unit --> [m^2/s] Frequency Domain Analysis Frequency Domain Analysis Frequency Domain Analysis corresponds to maximum peak frequency (the spectrum which has the largest energy) 54 s-1 (6.3 s) SEA 190º, fp = 0.23 s-1 (4.3 s) SWELL 290º, fp = 0.10 s-1 (10 s) SWELL 315º, fp = 0.16 SEA and SWELL axis = wave frequency SWELL SWELL: Waves propagating away from the generation area Frequency Domain Analysis Swell: mar de fondo waves that generated far away character: similar period of wave (narrow range), energy is being concentrated in small band developed in extra-tropical cyclone SEA: Waves in the wind-generation area SEA and SWELL Frequency Domain Analysis waves formed by the wind blows on sea surface on the current location (or very close location/generation area) character: chaotic, short-crested, developed in hurricane, cyclone, Sea: mar de viento under Frequency spectrum (S(f)) and direction spectrum (S(θ)) can be obtained as: where is the expected value of the variance and is the amplitude, the underline __ indicates that the amplitude is treated as a random variable. The directional wave spectrum (S(f;θ)) provides a better understanding of the wave behaviour. The directional wave spectrum refers to the distribution of wave elevation variance as a function of both, wave frequency (f ), and wave direction (θ). Frequency Domain Analysis this process is transformed using fourrier analysis the larget energy of the wave would be where wp located (x axis could be w or f) Frequency Domain Analysis Frequency Domain Analysis 50 exceeding the limit, the wave will break (can be identified by white capping) The equilibrium/saturation range corresponds to frequencies ω>ω0 The magnitude of the spectral density function in the equilibrium range represents an upper limit of the wave energy in this frequency band. Parametric Expression for Wave Spectra Parametric Expression for Wave Spectra q Directional Wave Spectra q Parametric Expressions for Wave Spectra q Frequency Domain Analysis q Probability Domain Analysis q Time Domain Analysis q Introduction OUTLINE 65 64 63 n2 = m0 m2 -1 m12 Battjes and van Vledder(1984) Spectral Width: other expressions Frequency Domain Analysis Spectral Width: other expressions Frequency Domain Analysis 0.3 is an standard value of ν Spectral width: Spectral parameters Frequency Domain Analysis typical typhoon v > 2 -> wide banded 62 61 60 Frequency Domain Analysis stands for: zero moment, 2nd order Frequency Domain Analysis H S » 4 m0 Frequency Domain Analysis only sometimes 59 57 Parametric Expression for Wave Spectra q q q q q q OUTLINE Parametric Expression for Wave Spectra 73 72 It is the one with the highest practical use � ;� � ;� > ≤ ö ÷ ÷ ø -4 gd d : spectrum width in the peak region. g : peak parameter gd : peak enhancement factor. Parameters: a, wp, g, s01, y s02 ; � = ∗ 71 1 q US developed parameterization valid for swell - TMA, Bouws et al. (1984). Intermediate and shallow waters North Sea of Europe (Norwegian sea to Dutch sea) developed in deep water, not really valid for shallow w. - JONSWAP Hasselmann et al. (1973) - Pierson Moskowitz (1964) Deep water waves Spectral models . .∗ . σ :� > � σ :� ≤ � . . . (1 − 7) SI (Similarity Index, Garcia Gabin, 2015) � � � −� � SS (Skill Score, Gallager, 1998) �� = 1 − • �� = • Fit goodness parameters: −1.25.∗ � ∗ � ∗ 1.094 − 0.01915.∗ ��� γ σ = 0,07; σ = 0,09 .∗ . = 3.3, ������� � ∗ ��� ∗� JONSWAP can be fitted from the bulk sea state parameters by minimizing the RMSE between the raw frequency spectrum and the idealized formula (free degree: γ ) σ= α= � = ���� ��� = � � � � ℎ γ = ���� ℎ� � � � � S � = α∗� ∗� Goda (1988) w p = 0.879 × g U 2 -5 4 æ w ö ÷ -1.25×ç ç ÷ èwp ø ×e æ g ö ÷÷ - 0.74çç èw U ø S (w ) = 0.0081 × g × w Expression results: Peak frequency: U: Wind speed at 19.5 m height. S (w ) = 0.0081 g 2w - 5 e -q TMA for different water depths. Ochi (1998) -4 67 JONSWAP evolution with fetch. Massel (1996). Pierson y Moskowitz (1964), OTD: ∗γ - Parametric Expression for Wave Spectra æ p ö ÷÷ èB qø w p = çç æ 1 ö æ p - r -1ö mr = A × B (r - p +1) / q × ç ÷ × Gç ÷ q ø è qø è JONSWAP, ∗ 69 Peak frequency: Spectral moments: A, B, p and q are free parameters S (w ) = Aw - p e - Bw Parametric Expression for Wave Spectra gd Parametric Expression for Wave Spectra g = 3.3, s01 = 0.07, s02 = 0.09 Mean JONSWAP : �=� S (w ) = a g 2w e æ w -1.25ç ç wp -5 è JONSWAP Spectrum (1973) Parametric Expression for Wave Spectra peak • a P-M spectra multiplied intensification factor • by Swells under development are characterized • (Joint North Sea Wave Program) S (w ) = a g 2w e -4 General form of the spectral density function: ö ÷ ÷ ø JONSWAP Spectrum æ w -1.25ç ç wp -5 è Parametric Expression for Wave Spectra Parametric Expression for Wave Spectra (Jackson and Short, 2021) statisticla representation Recap 81 projection of horizontal plane, visualize frequency also direction Wave data information Parametric Expression for Wave Spectra Parametric Expression for Wave Spectra 80 79 78 , = , , , =1 Parametric Expression for Wave Spectra Sections of Longuet Higgins directional spreading function Parametric Expression for Wave Spectra ∫ Condition that the directional spreading function must fulfil: D(q, w): directional spreading function For wave analysis it is assumed: �� = ∫� Frequency or scalar wave spectrum, S( ) Parametric Expression for Wave Spectra 76