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# Statistics& Applications CEE 3770

GPA 3.94

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This 0 page Class Notes was uploaded by Coty Von on Monday November 2, 2015. The Class Notes belongs to CEE 3770 at Georgia Institute of Technology - Main Campus taught by Francesco Fedele in Fall. Since its upload, it has received 25 views. For similar materials see /class/234173/cee-3770-georgia-institute-of-technology-main-campus in Civil and Environmental Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

me VAKIABLEs 94 kW M f 506 AM a u M aw W was Hunt 439 Ma 39lsa w quot My EMXamJ 1 m on Most is A X43 gt w lt I we w W gfwcfkm WEEKW1 u m W L39s 5 5mm Xquot Mg ma at BM 0 g X 15 oqus1 i39 8 5 h 91 PfxtlSXj 1 s Squot 55 2 i o 1 S a W Lislha sq llS 0 0221 r7JC 7 4 l5 1 50 0145 395 4x H U 7 13L Lair 1 b 3 was Hrs lie S s a a S m a 4 D A Z 0er awnmay 1 45 m y m 393 JwLpC i f 9095 Cp fluquot I r 0 4 5 Li 45 2 I5 4 b LLPS K 15j o K 4 J4 JmQ14 gth04x 3 ifrs Gt qlsf Ax 5 s MXqQ gt X3 skewness Cgt A37 41 W o it 9J0 ltXlt 1W2 ltXltEj SW ML 4de 19a GMEENAN IAEIABLES 4L 1 1 2 7Q mag r 39K Ema mohsv IX 1 91 3 ex a i 481 o x t m tewWI L m w 1 3 1 L ve x 4 Z 1quot WW M Am7 A 4573 J a L 3 01 ROGUE EM QCEANIC TURBULENCE WORHCES EM MSYMMIEITRIC TURBULENT PIPE FLOJWz AN EHREME VIEW FRANCESCO FEDEEI G H mm it eo Ia 11831 M a Awmmpm gm QTrgmmmw Savannah ATMOSPHERIC BOUNDARY LAYER amp WINDWAVE INTERACTION STEREOVIDEO IMAGERY amp HOTWIRE AN EMOMETRY EXPERIMENTS GLOBAL TEAM F Fedele GATECH Savannah Civil Engineering 39 A Yezzi GATECH Atlanta Electrical Engineering N J Nelson Skidaway Oceanographic Institute K N K 39 P BOCCOtti University Mediterranea ITALY F M J A Tayfun University of Kuwait A Benetazzo University of Padua ITALY G Forristall Forristall Ocean Engineering Inc USA ROGUE WA VES HURRICANE WA VES GIANT WA VES FREAK WA VES Mam2539 Ragae wane53 Giant WaVzas i a a We wam39 Freak W M RAUPNER EVENHT JANUARY 1995 Time History Recorded at Draupner Platform on 010195 152009 A freak wave in the central North Sea 7 l l 0 U l surface elevation m ill 800 1000 1200 Hmax256 m Extremely rare event according to Gaussian model Probability lt 10396 ll But they still occur in open ocean ROGUE WA VES Rare events of a normal population or typical events of a special population do we need new physics OCEANIC TURBULENCE OF ZAKHAROV weak wave turbulence NLS turbulence Concept of STOCHASTIC WAVE GROUP my contribution I thw39hWWVrM Hwy InkHanan 1 Nil h t lh NVquot 394 Aygqhh 5444952 Himhi 7 aflrhup39uw ADVqu H y m dim Hrmz39fln len39l zjunhmm 93quot M m 3quot quotquot a n14 lullm fth quotquot3 Wailm W u 17quot r 53m rain nib W quot A j J wm WTwwyr 39 4 quotMn 4 Mquot quot quoti 47 mm urnfwmwun quot fth luvNu fry 9quot V gt quotWham ght 1 TURBULENCE Uriel Frisch the Kortewegde Vries Equation Quantum version of the The Nonlinear Schrodinger NLS equation 11 Turbulence and symmetries In Chapter 41 of his Lectures on Physics devoted to hydrodynamics and ambulance Richard Feynman 1964 observes this in neople in some unjusti ed fear of physics say you can write an eqi In As a matter qfcht we very possiny airsa Jzion ta 1 suf cient approximation when we write the equm ion Qf39nzumum l jmnit s nlil Of course if we only had this equa in without detailed observation of biological phenomena we would be u able to reconstruct them Fe5mm behaves and this author shares his vie point that an analogous SitIkx u prevails in turbulent ow of an inompressible uid The equassa generally referred to as the Navieritokes equation has been quotnew since Xavier 1823 0vD Vv Vv0 notation cousin of START WETIHJ NAWE RST KES EQUATIQINJS T0 IMJQIDEL WAVE DYKNNMJECS M 621 621 2 0 6x2 622 61 61 acp a a acp V2 Vx 7 7217 2 a 7a a xl7 52 a 2 2 Inviscid irrotational l 2 a 2 2 ax 62 g 7 277 Z A boundary conditions 1 th m X g x gt gt VTquot dz d dx dx T and by mulituiplje scene pemrlbatdion method you get the Zalrdnarov model for WAVE TURBULENCE Third order effects 1 a FOURWAVE RESONANCE my 2 2 W Bncos1n 1 gong WAVE TURBULENCE II n 2g k Quartet interaction dB gtxlt liq d laan 127pqr5npqBquBr KP 19 193 1 p5q5r k Conserved quantities H B 3 I 1 T a 3 3 B B I Hamiltonian gmquot quot quot anr npqr np qir n p q r Wave action Wave momentum A 2 B 23 r M anBntBt Chaotic behavior of a sea of weakly dispersive nonlinear waves J mereQVer tar I Iallfrr Wball ldJ waves the Zakhamw equatth reduces m m deep Waite NLS Exact analytical solutions via mm mm mm the Inverse Scattering Transform 39 Technique NLS solitons and KdV Cnoidal waves chaotic behavror clue to nonlinear 1 1 X l 1 Interaction of waves and solitons b1 2 52 bzt l b3 bl I smiliw b3 chipWynn 211217 bjlb2 cschiW 911Wlmn mmi cothWEr 1 NON LIN EAR FOURIER ANALYSIS PERIODIC ORBIT THEORY bl mm b 1x425 tnbl bzn f naJJFT cmh Turbulence walk throuqh a repertoire 39 bzbl M W m of recurrent patterns Cvtanovic39 GA TECH 2 LINEAR WAVES GAUSSIAN N 77xt 2a 003ijajt8j j1 1 00 Eada 32 af yT J EacosaT da 1 0 130 77x t W M Gaussian emp l nc amp stajt maw I m Mm m pmcess ftc me Time covariance 39 wads H mm has on W Spectrum l 10 surface wavs T A o 5 V 05 L i f 0 75 711 0 0 annual V a 2 6 4 00 2 0 an 60 NECESSARY AND EMEEIC TLENJT CONDITE DJNS FOR THE CURRE1NJCE W A HIGH WAVE IN TIME T 0Hquot Theory of quasideterminism Bocoot P Wave Mechanics 2000 Elsevier What happens in the neighborhood of a point xo if a large crest followed by large trough are recorded in time at xo What is the probability that 77X0 XtO T e uudu conditioned to 77XOJO H27 77X02t0 T2 2 H2 LP SPACETIME covariance A random residual h Rayleigh variable dominant wave directiu stodmastdlc wave group 3 NQNILIINJEAR RANM SEAS Second order effects BOUND WAVES L IN EAR TERM h vv LINEAR amp NONLINEAR TERMS VV NON LINEAR WAVE A vv Crest trough asymmetry skewnessgt0 Wave height almost linear Effects on Short time scale wave period Third order effects FOURWAVE RESONANCE Cresttrough symmetry kurtosisgt3 BenjaminFeir Index BFIsteepnessbandwitdh Modulation instability Fermi UlamPasta recurrence Effects on slow time scale wave periodsteepnessAZ DOMINANT ONLY IN UNIDIRECTIONAL NARROWBAND SEAS Weak turbulence Linear conditional process Gaussian group Nonlinear Conditional process NonGaussian group Fedele F 2008 Rogue waves in oceanic turbulence Physica D in press ProbabYi39ty 0f exeeedanee for crests the generalized Ta un distribution PrcrestheightgtZeXp 2 12 1 12uZ2 1Z4 8Z2 8 4 Ta yfun distribution SECOND ORDER EFFECTS THIRD ORDER EFFECTS nonresonant interactions resonant interactions steepness BFIBenjaminFer Index PZ number of waves with crest greater than Z total number of waves AAAMAVAVANAAAA VVVVV VVVVUV t WAVE FLUMJE TA CCJMJPARESQNJS Onorato et al 2005 3 Wave tank experiments 10quot GeneralZed Ta yfun unidirectional narrowband seas Onorato et all 2005 unrealistic ocean conditions 5 10 THIRD ORDER SECOND ORDER EFFECTS Tayfun BOTH DOMINANT 5 mom 1 39 230 0225 2140 0800 giving Rayleigh R BenjamnFeir Index BFIJ 4 mi quot quot f U 1 2 3 Steepness u0 075 y WHATABOUT REALISTIC OCEANIC CONDITIONS Fedele F 2008 Rogue waves in oceanic turbulence Physica D in press VARIATIONAL WAVE ACQUISITION STEREO SYSTEM MS s 4 OCEANIC DATA collected at Tern platform in North Sea 1 Tern93 a 0449 rads 1 2 3 4 5 6 7 8 9 time h BF Benjamin Feir Index resonant interactions steepness parameter scaled elevauons 5 20 45 40 5 o 5 10 15 0 time from largest Cl CSl s 2 5 time h usteepness nonresonant Interactions cxcccdancc probzib OCEANIC DATA collected at Tern platform in North Sea Dominant 4 876 waves OrdEF p 000 3 000 nonresonant rm 0699 7 Interactions Generalized Tayfun Ta un 10 are the same 2 E39 fr E 39 U 2 mo 1 3394 5 6 7 s scaled heights 2 a 77 i 771 Linearized Surface Displacement l m l scaled elevations 1o 5 u 5 in tune from largest crest s a00377 v 10 4379 Waves 1 oo99 A 30010 9 10 x mo699 E a x 2 1 Q 3 Z r I 3 10 a J u 39 u K v i I g a l l 10 II E E E 10 o 7 is 7 B 9 10 3 4 S 6 scaled amplitudes amp heights Nonlinear Surface Dispacement TW uansients m Exact vaellmg 7 gt r V V quot19 HOW Wave mum Mmmhn m m mm mm w m a m Wm expand as P ZAHQJNHO chaotic behavior due to nonlinear interactions of Cnoidal waves Incoherence Coherence Cnoidal wave group streamwise direction more work to be done EWHAVAVAVAVAVYAVM 535 140 0 4o WAVAVAVAVAVAVAVAVAVA 140 2o 0 20 4o 140 2 0 6 26 40 OM Cnoidal 1 v waves 140 20 o 20 40 05 A i o v Cnoidal 40 2o 0 o 40 streamwise direction SOIltonS Toroidal vortex tube Modulated by Cnoidal waves in the streamwise direction ROGUE WAVES IN OCEANIC TURBULENCE Francesco Fedele 56700 of Civil amp En vironmenta Engineering Georgia Institute of Technology USA h l Georgia mg w z Georgia A fTechtmH y TECH M Savannah 1 The Acqua Alta Team GLOBAL TEAM F Fedele GATECH Savannah Civil Engineering A Yezzi G Gallego GATECH Atlanta Electrical Engineering 0 A Benetazzo PROTECNO srl ITALY Orb1 A Boscolo Phoenix srl ITALY G Z Forristall Forristall Ocean Engineering USA Aziz Tayfun Kuwait University L Cavaleri ISMAR CNR Venice ITALY I l Geor ia tm tui Georgla n Nil Tehtm y Tech amp Savannah OUTLINE EITECHNOLOGY 4DVar Wave Acquisition Stereo System EIIMAGE PROCESSING THEORY EIOCEAN SCIENCES Variational methods for partial wave turbulence of Zakharov differential equations nonlinear random wave fields El APPLICATIONS FOR WAVE MECHANICS 4D wave patterns hydrodynamic effects amp Rogue waves 0 CHEVRON PROJECT in Northern Adriatic Sea ITALY o MOSE PROJECTin the Venice Lagoon A NATURAL BE4UTY Freak waves Rogue waves DRAUPNER EVENT JANUARY 1995 Time History Recorded at Draupner Platform on 010195 152009 A freak wave in the central North Sea 3 x 0 0 surface elevation m U 0 800 1000 1200 Extremely rare event according to Gaussian model But they still occur in open ocean ROGUE WA VES nu events of a normal population bad day at the tower or events of a special population 7 need afIew physics ANSWER hidden in the statistical structure of weakly nonlinear random waves MAIN RESULT Crest exceedance u weakly nonlinear random new COROLLARY the Generalized Tavfun distribution GT for the oceanic turbulence of Zakharov MY CONTRIBUTION CONCEPT OF NONLINEAR STOCHASTIC WAVE GROUP Nonlinear Slepian model Fedele F 2008 Rogue waves in oceanic turbulence Physca D Fedele F amp Tayfun 2009 Nonlinear Wave groups and crest statistics Journal ofFud Mechanics Ty f MNAIA5EM39 A ma x mm v r a 39 39 huh vh Ann r 39 gquot M b39v oMJ 3 39V nm w 3quot M mama If V Aquot Il39olf m JPNY m 1 Ma o d39 quotin 115 p39 o dudclonrdgm V N N 333321 593 MA Q Wm39W WJ Mu A 39 we 4 Mi quotin Aiwufmianuar aural nayJul fir 3H i dunquot M3315 A no N 4 IiiHP me thrillquot 21 TURBULENCE Uriel F risch 1i 7 Quantum version of the 101 0224 ku2u The Nonlinear Schrodinger NLS equation 6 2 6amp2 cousin of 3 the Kortewegde Vries Equation 61 a kua MZ a a a 11 Turbulence and symmetries In Chapter 41 of his Lectures on Physics devoted to hydrodynamics and turbulence Richard Feynman 1964 observes this people in some unjusti ed fear of physics say you can39t write an equation fly life Well perhaps we can As a matter of fact we very possibly already hare die equation to a su cient approximation when we write the equation of quantum mimics 11 Of course if we only had this equa 1n without detailed observance of biological phenomena we would be u able to reconstruct them Feynman believes and this author shares his vie point that an analogous situation prevails in turbulent ow of an in mpressible uid The equation gnerally referred to as the Navieretokes equation has been know since Navier 1823 L 6tvvVv VpvV1n 12 V v O It must be supplemented by initial and boundary talc as the vanishing of v at rigid walls We sh come back Later to the choice of notation 0 0 S734R739 WIT H STOKES EQUATIONS T0 MODEL WA 5 DYNAMICS and by multiple scale perturbation method amp MUCH MORE you get the WA 5 TURBULENCE onakharov i can go ggm 2 nm cosl n C0quot F92 cos1nm 2 52 cos1nm n Br Bm nm m dB ioan i ngqr5npqerBqBr dt Paqal Qua rTet interaction 1 31 gq 5 moreo ver for deep water narrowhano waves the Zakharov equation reduces to the MS equation 2 ia ula Lku2u 0 at 284C Nonlinear interaction of w and soiitons Turbulence walk through a repertoire of N O N LI N EAR FOU RI E R ANALYSIS recurrent patterns C Vitanovic GA TECH Oceanic turbulence of Nonlinear interactions of Za kharov stochastic wave groups Let s start from scratch GAUSSIAN SEAS 77 X2 a COS k 0Xa9jt8j j1 77xt Gaussian ergodic amp stationary process of time amp space RANDOM FIELDS 77xtlrx0to hhWxx0t t0d 1 I SPACETIME covariance A random residual la Rayleigh variable stochastic wave group 9113 Boccotti 1989 C33 Slepian model Lindgren 1972 Adler 1981 Generic weakly nonlinear eld n 111 fn1 f0 nonlinear 08T 082T Nonlinear Conditional process Difficult to solve Easy nonlinear function of a Gaussian stochastic group NonGaussian stochastic group Fedele F 2008 Rogue waves in oceanic turbulence Physca D Fedele F amp Tayfun 2009 Nonlinear Wave groups and crest statistics Journal ofFud Mechanics EXmax z 39 k r 7 expected number of maxima with amplitude z jig I I NONLINEAR EFFECTS TAKEN INTO ACCOUNT Second order effects BOUND WAVES Third order Effects 1 FREE WAVES mum mm HNEAR N NHNEAR TERM WW NDNLWEARWAVE W Cresttrough symmetry kurtosisgt3 BenjaminFair Index BFIsteepnessbandwitdh Crest trough asymmetry DOMINANT ONLY IN UNIDIRECTIONAL NARROWBAND SEAS skewnessgt0 The generalized Tayfun GT distribution Prcrestheigh2 gt Z eXp Ta yfun distribution GT similar to the GRAMCHARLIER GC approximation of Tayfun amp Fedele OE 2007 GT stems from the Zakharov model with no priori assumptions on the statistical structures of the solution as in GC models steepness kurt sis 1 cos Aco3 4t 1 0C J512k1 k2 k1 k2 140 0C JK1335152S3 T m z ul zs 12 Fedele F 2008 Rogue waves in oceanic turbulence Physica D WAVE FLUME DATA COMPARISONS Onorato et al 2005 7Z3ny Probablllty P u 0075 7 Ma 300 Rayleigh WHATABOU Fedele F 2008 Rogue waves in oceanic turbulence Physca D in press BenjaminFeir Index B 4 Steepness u0075 L OCEANIC CONDITIONS 7 MARINTEK WAVE FLUME DATA COLLABORATORS Z Cherneva C Guedes Soare Instituto Superior T cnico Lisbon Portugal M Aziz Tayfun Kuwait University Kuwait 3 W t 2 2 jaw 2 771337717il ax 2 at k Ale 0 77 771 7712712 MARINTEK DATA 20 Draupner 15 quotNew Year Wavequot Elevations m N2560 cX2981 2 0c001 6384 cX1329587 X 41227 x30x1o055052 77 Linearized Surface Displacement Nonlinear Surface Dispacement Hand A 008 006 004 Process 1110 500 1000 1500 2000 Distance in m 2500 3000 w and vm 030 032 7 WU 7 028 7 7 V0 024 7 w 7 M O 39 500 1000 500 2000 2500 3000 1 Distance in m Proc 933 nm 7L4Utheory 05 7 39 v vv I D 047 I l lt I 390 I C 03 7 I 7 U I l S 39 V I39 k 0 0 099008 lt 02 I of 39 I39 o O094864 I 0 024365 01 V I I l l l O l w w w w w 0 500 1000 1500 2000 2500 3000 Distance in m 140 OCJK1335123 12kmo2 2 140 x V 0 1 005Ak135 24x 34 Akl2 x 12ia zarcsm T 51324dk23 1 L Im 6J J Wave Generator Amplitudes and Heights scaled with 6 L 80m B 6 I I 50m 8 I I qoaeg Buiqmsqv Amplitudes and Heights scaled with G Exceedance Probability O l I l l l 61 A I 2 l l s QGC y 10 12 Exceedance Probability adquot 35 3 h am a O l l l l l l 2 l 4 C c CRESTfl39ROUGHWAVE HEIGHT DISTRIBUTIONS VVHVED ARE 3D WAVES IN SPACE X Y WAVES IN SPACETIME XYT Excuvsmn Set for h i feivcvvv in 2n 3 4 5n 7 7n u an inn an 10 EU 4 20 DD BEYOND WAVES amp SPECTRA Euler Characteristic of excursion sets 771e geometry of random elds Adler 1981 Adler Taylor amp Worsley 2007 WAVE ACQUISITION and ANALYSIS STEREO SYSTEM WASS E ii 9 Image acquisition BiTrinocular Synchronized digital cameras e Image processng Epipolar Variational Stereo method VARIATIONAL WAVE ACQUISITION STEREO SYSTEM VWASS In ut stereo air imaoes The rectanoular domain 8 In X 87 In The height of the waves is in the range 02 cm Reconstructed wave surface 2 centimeters V meters PRELIMINARY RESULTS wave statistics and spectra Ky 1I39naters u 2 Kx 1melers PRELIMINARY RESULTS Probability density function of wave surface 10 i i i i i i 7 PnZ iiii i Gaussian 7 Charier 7 pdfTayfun 10391 7 7 10392 1039s 7 10394 4 3 2 1 0 1 2 3 4 5 Generalized Tayfun Tayfun SECOND ORDER EFFECTS DOMINANT BEYOND WAVES amp SPECTRA Euler Characteristic of excursion sets 7776 geometry OfIaIuulll elds 39 Adler 1981 Adler Idylor 8L Worsley 2007 Excursion Set in h 2 EC connected components tunnels Hollows Excursion Set m h 1 Excursion Sal m EC counts number no large maxima Euler characteristic EC of nonlinear wave fields PiterbargTayfun model VIDEO DATA VIDEO DATA quotquot quot Gaussian EC Euler Characteristic EC Euler Characteristic EC Prlrggsxmm gt th ECh CONCLUSIONS am open ocean rouge waves appear to be simply rare events of normal populations oFor special wave conditions undirectional waves in wave flumes third order resonant interactions are also dominant RECONSTRUCTION OF THE WATER 2D SURFACE FROM IMAGES Variational stereo solution Philosoghy adjust the 3D model to the 3D world represented by the data images so that an energy is minimized Explicit amp deformable 3D model of surfaces Deform surface ant7 quotbest match Forward prOJ39eCtiOH v quot is achieved by energy mhmzaton Left image Right image EC counts also large 3D Upcrossings Onetoone correspondence between large maxima amp 3D upcrossings as in one dimensional stochastic processes Eu er Charmanm EC A E E G fun L 000 X quotrbumber 0 heads FrEX EXHU3amp HXZUI x4um uUXl000 frx k x5m r er X1 10003 m g a n pr I FrDFO UX U x2ux33 P QmUkn x2Ux31 2 I39moeowa Wr k k 100000 ZJISM0 1000ng amok 2 gt2 r 2 39 3 3939 339233i25 X 1 L555Xv392 3 Z 39P 500 039 W x52 X So 3 T 3 z M amp s L39P 5 0 50KVP 5 go z I 8 a gtiw 58 PrEZ 2 3153 Jam E i z ja 8sz M 92quot lo 9 U39 exf i T rim 2 Mos 9305 39 24 fr as 2 am a m a 3 57 3amp5 slnswee 2140961371 T quot w ffmg 39hm J 1755 TM 30 ml 3 gm i06 5301 MD M 1 Mat w J30 5 f 0 30 R 0T 3 0 i a 13 Ag39o 05154310 30 Fi m 91 zjwam 95E qu Hu DAL OuJIWs jam Jr gm 9 Ha W h 5 gt2 wr stuck B 1ZWI6389 to We 932014 2 3 30 56 g S Ak kl18 55 AK kLIB kfrIXikfr D inDriJ 3 6 E rm W U W0 u1nuzleuznp 1 i2 3636 36 HM 1326 f4 z 6270Wgt U D4 mm W 940924 a s L v IS 35 LJHHLL w r I P U 41 m1le W23 ijmpmm 11mm mmm NEH 0 may mum 1399Um 2m y 3 SLLJ A MLimuJMmm LLlLiLJ k O AM ltoct 0 1 5amp9 619 3 2 as 0 Swaps mm mgr 5er 7b 321 h m 4 H erca LHLBIBELU 39z f M fgnflz 1 5 W W rt F 4 F7Ik4 33SF z quot an SZ Z5 OJ WVMQ UB X 394 S In 41 90 008 nal mic SvdA WV 93 5 LL pa 4 X ZWquot xO 7 z M7L 5mm 1 ftfx6W r0 m b Ax 1 511 52b 390 4 123 ther 0 3a WHO the I 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