(Extremes and related properties of) Stochastic asymmetric Lagrange waves

Transcription

1 (Extremes and related properties of) Stochastic asymmetric Lagrange waves Georg Lindgren and Finn Lindgren Lund University and University of Bath 8 2 March 23 Recent advances in Extreme value Theory honoring Ross Leadbetter Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 / 32

2 Contents I Introduction A long long time ago A long time ago 2 Gauss versus Lagrange More recent 3 The stochastic 3D Lagrange model The Gaussian generator The free Lagrange model 4 Front-back asymmetry The modified Lagrange model 5 The 3D-model and the generalized Rice formula Explicit definitions Multiple crossings Generalized Rice formula 6 Example Six directional spectra Asynchrone sampling Synchrone sampling at crossings Slope depends on main heading 7 References Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 2 / 32

3 Introduction A long long time ago An experiment 62 years ago JOURNAL OF THE OPTICAL SOCIETY OF AMERICAV VOLUME 44, NUMBER NOVEMBER, 954 Measurement of the Roughness of the Sea Surface from Photographs of the Sun's Glitter CIIARLES COX AND WALTER MUNK Scripps Institution of Oceanography,* La Jolla, California (Received April 28, 954) November 954 MEASUREMENT OF THE SEA SURFACE 839 A method is developed for interpreting the statistics of the sun's glitter on the sea surface in terms of the statistics of the slope distribution. The method consists of two principal phases: () of identifying, from geometric considerations, any point on the surface with the particular slope required for the reflection of the sun's rays toward the observer; and (2) of interpreting the average brightness of the sea surface in the vicinity of this point in terms of the frequency with which this particular slope occurs. The computation of the probability of large (and infrequent) slopes is limited by the disappearance of the glitter into a background consisting of () the sunlight scattered from particles beneath the sea surface, and (2) the skylight reflected by the sea surface. The method has been applied to aerial photographs taken under carefully chosen conditions in the Hawaiian area. Winds were measured from a vessel at the time and place of the aerial photographs, and cover a range from to 4 m sec~'. The effect of surface slicks, laid by the vessel, are included in the study. A two-dimensional Gram-Charlier series is fitted to the data. As a first approximation the distribution is Gaussian and isotropic with respect to direction. The mean square slope (regardless of direction) increases linearly with the wind speed, reaching a value of (tanl6 )2 for a wind speed of 4 m sec-'. The ratio of the up/ downwind to the crosswind component of mean square slope varies from. to.9. There is some up/downwind skewness which increases with increasing wind speed. As a result the most probable slope at high winds is not zero but a few degrees, with the azimuth of ascent pointing downwind. The measured peakedness which is barely above the limit of observational error, is such as to make the probability of very large and very small slopes greater than Gaussian. The effect of oil slicks covering an area of one-quarter square mile is to reduce the mean square slopes by a factor of two or three, to eliminate skewness, but to leave peakedness unchanged. FIG.. Glitter patterns photographed by aerial camera pointing vertically downward at solar elevation of k=7. The superimposed grids consist of lines of constant slope azimuth a (radial) drawn for every 3, and of constant tilt /3 (closed) for every 5. Grids have been translated and rotated to allow for roll, pitch, and yaw of plane. Shadow of plane can. INTRODUCTION barely be seen along a= 8 within white cross. White arrow shows wind direction. Left: water surface covered by all yielding maximum slopes of 25. Hulbert 2 natural slick, wind.8 m sec', rms tilt o=.22. Right: clean surface, wind 8.6 m sec', =-.45. The vessel Reverie is demonstrates by this method that the maximum slope in the within white circle. THE purpose of this study was to make quantita- T tive measurements 2. pertaining THE OBSERVATIONS to the roughness of North was toward Atlantic the sun. increased An attempt from was 5 made at a 3-knot wind to Aerial Observations 25 at an 8-knot wind. Shuleikin 3 to avoid Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange cloud shadows waves and atmospheric haze. In most March cases took the a 23 long series 3 / 32

4 Introduction A long long time ago The Cox and Munk experiment on wave asymmetry compared to Lagrange asymmetry: 2 2 Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 4 / 32

5 Introduction A long time ago Some good literature Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 5 / 32

6 Gauss versus Lagrange More recent Why using a stochastic Lagrange model? The (linear) Gaussian wave model allows exact computation of wave characteristic distributions (the WAFO toolbox) produces crest-trough and front-back stochastically symmetric waves The modified stochastic Lagrange model can produce (26) crest-trough asymmetric waves (2D) (29) front-back asymmetric waves (2D) (2) front-back asymmetric waves (3D) still allows for exact computation of wave characteristic distributions Software exists for calculation of statistical wave characteristic distributions (crest, trough heights, period, steepness etc) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 6 / 32

7 The stochastic 3D Lagrange model The Gaussian generator Gaussian generator and the orbital spectrum In the Gaussian model the vertical height W (t, s) of a particle at the free surface at time t and location s = (u, v) is an integral of harmonics with random phases and amplitudes: W (t, s) = κ = (κ x, κ y ) = κ(cos θ, sin θ) ω = ω(κ) = gκ tanh κh ω= π θ= π e i(κs ωt) dζ(ω, θ) with S(ω, θ) = the orbital spectrum and ζ(ω, θ) is a Gaussian complex spectral process. Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 7 / 32

8 The stochastic 3D Lagrange model The free Lagrange model The stochastic Lagrange model Describes vertical and horizontal movements of individual surface water particles. Use W (t, s) = e i(κs ωt) dζ(κ, ω) for the vertical movement of a particle with (intitial) reference coordinate s = (u, v) and write Σ(t, s) = Make Σ(t, s) stochastic. ( ) X (t, s) = horizontal location at time t Y (t, s) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 8 / 32

9 The stochastic 3D Lagrange model The free Lagrange model with horizontal Gaussian movements Use the same Gaussian spectral process as in W (t, s) to generate the horisontal variation Fouques,Krogstad,Myrhaug,Socquet-Juglard (24), Gjøsund (2,23) Aberg, Lindgren, Lindgren (26,27,28,29,2), Guerrin (29) ( ) X (t, s) Σ(t, s) = = s + H(θ, κ) e i(κs ωt) dζ(κ, ω) Y (t, s) where the filter function H depends on water depth h: H(θ, κ) = i cosh κh sinh κh ( ) cos θ sin θ Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 9 / 32

10 The stochastic 3D Lagrange model The free Lagrange model The stochastic Lagrange model The 3D stochastic first order Lagrange wave model is the triple of Gaussian processes (W (t, s), Σ(t, s)) = (W (t, s), X (t, s), Y (t, s)) All covariance functions and auto-spectral and cross-spectral density functions for Σ(t, s) follow from the orbital spectrum S(ω, θ) and the filter equation. Space wave : keep time coordinate fixed Time wave : keep space coordinate (X (t, s), Y (t, s)) fixed This will (later) lead us to a "Palm type" problem what are the distributions of (W (t, s), Σ(t, s)) when (X (t, s), Y (t, s)) = (, ), say. Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 / 32

11 The stochastic 3D Lagrange model The free Lagrange model Lagrange 2D space waves, time fixed A Lagrange space wave at time t is the parametric curve (2D) or surface (3D) L(x) : s (X M (t, s), W (t, s) 5 Depth = 4m Depth = 8m Depth = 6m Depth = 32m Depth = Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 / 32

12 The stochastic 3D Lagrange model The free Lagrange model Ocean waves need a direction The free Lagrange waves are peaked but they have no preferred direction! Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 2 / 32

13 Front-back asymmetry The modified Lagrange model Front-back asymmetric Lagrange waves To get realistic front-back asymmetry one needs a model with external input from wind, for example, by a parameter α: 2 X (t, s) t 2 = 2 X M (t, s) t 2 + αw (t, s) The filter function from vertical W (t, u) to horizontal X (t, u) is then H(ω) = i cosh κh sinh κh + α ( iω) 2 = ρ(ω) eiθ(ω), Implies an extra phase shift (θ = π/2 in the free model) X (t, u) = u + e i(κu ωt+θ(ω)) ρ(ω), dζ(ω, κ) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 3 / 32

14 Front-back asymmetry The modified Lagrange model Particle path depends on wavelength Wavelength: m (blue), 5m (green), 5m (red) Depth = 25meter Depth = 2meter Depth = 25meter Depth = 2meter α = X(t) α = X(t) α = 5 5 X(t) α = 2 2 X(t) α = X(t) α = X(t) α = 2 2 X(t) α = X(t) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 4 / 32

15 Front-back asymmetry The modified Lagrange model Space and time waves 5 A Lagrange space wave x (m) A Lagrange time wave t (s) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 5 / 32

16 Front-back asymmetry The modified Lagrange model For 3D waves the filter function needs to take directional spreading into account An example: H(θ, κ) = α ( (iω) 2 cos 2 (θ) cos(θ) cos 2 (θ) sin(θ) sign(cos θ) to take care of wind blowing in positive x-direction. ) cosh κh + i sinh κh ( ) cos θ. sin θ Front-back asymmetry depends on directional spreading in the orbital spectrum Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 6 / 32

17 The 3D-model and the generalized Rice formula Explicit definitions Explicit definitions I The model (W (t, s), Σ(t, s)) = (W (t, s), X (t, s), Y (t, s)) is an implicitly defined model. The space and time models (for time = t ) can be made explicit by s = Σ (t, (x, y)) equal to the reference point that is mapped to the observation point (x, y). Note: There may be many solutions to Σ(t, s) = (x, y). Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 7 / 32

18 The 3D-model and the generalized Rice formula Explicit definitions Explicit definitions II Space wave L(t, (x, y)) = W (t, Σ (t, (x, y))) Time wave L(t, (x, y )) is the parametric curve: t W (t, Σ (t, (x, y ))) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 8 / 32

19 The 3D-model and the generalized Rice formula Explicit definitions Wave slopes in space (Cox/Munck) and time The Cox/Munck experiment observed wave slopes at fixed time needs space derivatives: ( ) ( ) ( ) Wu Xu Y = u Lx () W v X v Y v L y An oil platform experiences wave slopes at fixed location needs time derivatives W t = L t + ( ) ( ) L X t Y x t L y L t = W t ( ) ( ) ( ) X W u W u X v Xt v Y u Y v Y t (2) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 9 / 32

20 The 3D-model and the generalized Rice formula Multiple crossings Asynchronous sampling in space = Cox/Munch, t = To find the distributions of the slopes L x, L y at a fixed point, say (, ), invert () and find the distribution of (W u, W v, X u, X v, Y u, Y v ) at reference point s under the condition that (X (s), Y (s)) = (, ) 25 2 Contour lines prefer to cross under straight angles; perpendicularitybias Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 2 / 32

21 The 3D-model and the generalized Rice formula Generalized Rice formula Standard Rice formulas The original Rice formula gives the expected number of level crossings of level u per time unit in a stationary process: E(#{t [, ]; X (t) = u, upcrossing}) = E(X () + X () = u) f X ()(u) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 2 / 32

22 The 3D-model and the generalized Rice formula Generalized Rice formula Marked Rice formula The Rice formula for marked crossing gives the expectede number of level crossings per time unit in a stationary process, at which a "mark" has a certain characteristic. The "mark" can be e.g. the derivative at the upcrossing. E(#{t [, ]; X (t) = u, upcrossing, and "mark" B}) = E(X () + "mark" B X () = u) f X () (u) Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

23 The 3D-model and the generalized Rice formula Generalized Rice formula Multivariate Rice formula A generalized (multivariate) Rice formula gives the expected number of simultaneous crossings with marks, for example with marks mark:(w u, W v, X u, X v, Y u, Y v ) with simultanous crossings: (X (s), Y (s)) = (, ) Reference: Azaïs and Wschebor: Level sets and extrema of random processes and fields, Wiley, 29. Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

24 The 3D-model and the generalized Rice formula Generalized Rice formula Application for slope distribution with asynchronous sampling Consider u-level crossings at (, ) in a specified direction. Slope is defined as a function of (W u, W v, X u, X v, Y u, Y v ). With N (A) = #{s R 2 ; Σ(s) = (, ), slope A} ( ) E(N (A)) = E det Σ (s) R 2 }{{} slope A Σ(s) = (, ) f Σ(s) (, ) ds {}}{ perpendicularitybiascorrection Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

25 The 3D-model and the generalized Rice formula Generalized Rice formula Some old literature still very useful Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

26 The 3D-model and the generalized Rice formula Generalized Rice formula Lesson to be learned The ratio between the expected values of marked and unmarked crossings is equal to the empirical observable distribution. Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

27 Example Six directional spectra Example specta - Pierson-Moskowitz with different spreading Directional Spectrum 9 2 Level 2curves at: Directional Spectrum 9 2 Level 2curves at: Directional Spectrum 9 2 Level 2curves at: Directional Spectrum 9 2 Level 2curves at: Directional Spectrum 9 2 Level 2curves at: Directional Spectrum 9 2 Level 2curves at: Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

28 Example Asynchrone sampling Slope PDF across wind and along wind, asynchrone Top: Large spreading Bottom: Little spreading Left: Along wind Right: Across wind; cf. Cox and Munk 4 pdf 3 2 pdf slope slope pdf 2 pdf slope slope Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

29 Example Asynchrone sampling The Cox and Munk experiment on wave asymmetry compared to Lagrange asymmetry: 2 2 Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32

30 Example Synchrone sampling at crossings Slope CDF for different directional spreading Left: slope at upcrossings Right: Slope CDF at downcrossings Moderate linkage parameter: α = cdf.4.2 PM spectrum, slope at upcrossings α =.4, h = 32m solid curves: m =, 2, 5,, 2, 2 dashed curve: unidirectional spectrum cdf.4.2 PM spectrum, slope at downcrossings α =.4, h=32m solid curves: m=, 2, 5,, 2, 2 dashed curve: unidirectional spectrum slope slope Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 3 / 32

31 Example Slope depends on main heading Slope CDF for different heading; Strong linkage: α = 2 Left: slope at upcrossings Right: Slope CDF at downcrossings θ = π/2.8 θ = π/4.8 θ = π/2 θ = π/4 cdf.6 θ =.4 θ = 3π/4 θ = π cdf.6.4 θ = θ = π θ = 3π/4.2 CDF for slope at upcrossings PM spectrum with m = α = 2., h = 32m θ =,..., π dashed curve: unidirectional spectrum slope.2 CDF for slope at downcrossings PM spectrum with m = α = 2., h = 32m θ =,..., π dashed curve: unidirectional spectrum slope Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March 23 3 / 32

32 References References Aberg, S. (27). Wave intensities and slopes in Lagrangian seas. Adv. Appl. Prob., 39 Lindgren, G. (26). Slepian models for the stochastic shape of individual Lagrange sea waves. Adv. Appl. Prob. 38 Lindgren, G. (29). Exact asymmetric slope distributions in stochastic Gauss-Lagrange ocean waves. Appl. Ocean Res, 3 Lindgren, G. (2). Slope distribution in front-back asymmetric stochastic Lagrange time waves. Adv. Appl. Prob., 42 Lindgren, G. and Aberg, S. (28). First order stochastic Lagrange models for front-back asymmetric ocean waves. J. OMAE, 3 Lindgren, G. and Lindgren, F. (2). Stochastic asymmetry properties of 3D- Gauss-Lagrange ocean waves with directional spreading. Stochastic Models, 27 WAFO-group, (2). WAFO a Matlab Toolbox for Analysis of Random Waves and Loads; Tutorial for WAFO Version 2.5, URL Lindgren and Lindgren (LU and BATH) Asymmetric Lagrange waves March / 32