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# Special Topics ASEN 6519

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This 16 page Class Notes was uploaded by Laila Windler on Friday October 30, 2015. The Class Notes belongs to ASEN 6519 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/232170/asen-6519-university-of-colorado-at-boulder in Aerospace Engineering at University of Colorado at Boulder.

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Date Created: 10/30/15

Easlc Terms of Nonllnear Dynamlcs http Hmonet physlk unlbas chloelmedpendulumbterm htm w W M gt4 ng LAB M RM The u atlons of Osclllau39ons and Non ear Mlscell39aneous Lan and Motlon Reso lts Toplcs thelature l 39 Smblllq mi The Narme of M Ezgyconon cm Basic Terms of Nonlinear Dynamics H r terms of thls language All these terms Wlll be lllustrated at the pendulum Dynamical system Aparl l b ha l r lnn nlln ar dwaml by lts dgamlcs A pendulum ls an example for a dynamlcal system state F r the Mmirrven peruiulum the state ls unlquely de ned by the angle n and the angular veloclt w d Idt In the case of driving the drlvmg phase 5 ls also needed because the pendulum becomes a nonautonomous sygtem eld The yeloclty eld of a uid ls a weleown example Phase space tn l w ln tb case of The eld Dynamlts or equation of motion small Thus 39 39 dudr 1700 71 W do not necessarlly occur equldlstantly ally the order has to be the same That 15 n lt m lmplles r lt rm 39 Otherwlse lt ls nonlinear nal napp n ll Thus p forgotten or course thls ls a cruclal polnt whlle modelllng a realrllfe systems But beslde thls there are lof3 11609411FM Easlc Terms of Nonllnear Dynarmcs bttp Hmonet physlk unlbas chselmedpendulumbterm btrn ume Thus um ttannlt the statel 1 Hdr t where fls the dnylng frequency the angular drlylng frequency ls 2 7m usually unldnown Therefore the next state can not be deduced from the present one The deterrnlnlstlc rule phase space to be the next state Orbit or trajectory l e c r ln general the howrora nlte tune step ls not ldnown analyucally because thls would be equlyalent to have a solutlon of the equatlon of rnotlon angularyelnerty As an example the gure shows I ow of a h Leg end 013 uw pen 1 The null E1mE black arrows of stable tlxed pulnt the vector eld re tangentlal basln ut attraetlun atthe kalectur stable rnanltuld unstable rnanltuld plcture of the ow and the orblts rst H r the angle ls zero when the angular yeloclty ls zero The other null Clme ls a e w Dzsm w l y Ch these null Cllnes r n tl l Vector eld In the dlrectlon north east south east south west or north west The dlrectlon ls deterrnlned by the lcllnes ul l r u lor r Jet Poincar section and Poincar apt A carefully chosen curved plane In the phase space that ls crossed by almost Poincar all orblts lt ls a tool developed by Henn Polncare 18544912 for a section e ti o rn r n For secuon has one dlrnenslon less than the phase space The Poincare map maps the polhts of the Polncare sectlon onto ltself It relates two Zen 11609411FM Enstc Terms of Nonhnent Dynnnntcs http Hmonet phystk umbas chwlmedpendulumbterm hLm y consecuuve tntetseehon potnts Note that only those tntetseehon potnts orb counts whteh come from the same stde of the plane A Polncar map tums a eonttnuous dmmteat s tem tnto a dtsetete one It the Pomear seeuo ts Raw ar m n nfrom um1 In M Psndvlmm Tab seetton ts de ned by a serum phase of the tnepenodte dnvmg Nonwandering set m avors 1 39 39 F r pendulum W 0 dWdr o and W 1so dwdt 0 are xed potnts t the In M Psndumm Tab mtlo 0t undamped but dnven pendula 4 Chaotic orbits ts strong enough k A quotA Tn39hmanh In th sevenuest xed potnts t tn can nonrwandenng sets due to a bttutcatton Q E 9 gtevtous Log M 001 n V 3am 11609A11 PM Optical Remote Sensing with Differential Absorption Lidar DIAL Christoph Senff CIRES University of Colorado amp NOAAESRLCSDAtmospheric Remote Sensing Group httpwwwetlnoaaoovet2 Guest lecture for ASEN6519 Lidar Remote Sensing CU Boulder November 3 2008 Outline DIAL concept A short history of DIAL DIAL equation error analysis and system components DIAL systems at NOAAIESRLICSD Multiwavelength ozone DIAL Applications of airborne ozone DIAL Differential Absorption lear DIAL Concept Distributed Baa kseettering Medium Laser Transmitter Optical Receiver ell 39 0 Power Absorption n Received Cress Seetie39 g Pr quot alla l gquot39IZ I39ff AIlifln Wavelength Atmospheric gases measured with DIAL H20 03 SO2 N02 NO NH3 CH4 002 Hg VOCs Volatile Organic Compounds Toluene Benzene First DIAL measurements Richard M Schotland The father of DIAL 1964 Measured vertical profiles of water vapor by thermally tuning a ruby laser on and off the water vapor absorption line at 69438 nm Only 4 years after invention of ruby Iidar FADiGSDQE ALllile km a w Meme 5 Fig 420 Comparison of atmospheric water vapor vertical pro les impressed as dew point temperature measured b differential absorption lidai and radiosonde 482 Major milestones in the history of DIAL Maiman Invention Shumate amp Menzies Browell First autonomous DIAL LASE H20 DIAL on ER2 aircraft 1995 of the laser First Airborne DIAL column 03 1960 1964 1977 1978 1981 Schotland First H20 DIAL measurements DIAL Megie First Browell First airborne H20 o3 DIAL Present Spacebased DIAL DIAL equation Single scattering elastic backscatter LIDAR equation Nsm R NLl LB1RAR eXp 2525011er 771G1R N301 Ratio LIDAR equations for online and offline wavelengths Aon and Aoff Ns10 aRNBlo 7R NL10 77lo Glo 7R lo 7R Ns10naR NBMonaR NL10n7710nG10nR Mimi R Number density X eXP 2J0a10 a r Mona r 6 1quot of constituent C gtlt exp ZJXGC 10 r 0 C10n 1 er xexp 2Ji0Xilo r 039XilonrnXir dr DIAL equation cont d n 1 i n C 2A0CR dR NSxiO R NBxion 1 d 1nGxio R G 7 2A0CR GxionR xi R 1 i In off 2 on7R 1 or xi R 0 xi R E on of anX R nX R X MAR i1 quot quot with AO39CR O39CionR 0Cxio R G differential geometrical factor B differential backscatter E differential extinction X interfering constituents How to choose an appropriate absorption line for DIAL NSMOWR oc exp 2lf 00 Am r W dr Extinction of online wavelength clue to absorption by constituent C must be neither too small or too large Absorption too strong Aoff Rmax Best precision in nC when 2110 Rmax 2 J0 0C an rnC r dr 2 11 Example mro3 80ppb gt no3 2gtlt1018m3 Rmax 23km 039O3 r0nno3 R 211 gt 003 on183gtlt1022m2 max Precision of DIAL measurements Simple back of the envelope calculation 1 Nola RARN10nR 11C In With Nst NB 2A0CRAR N10nRAR N010 R 511 1 252 Nai R 1 5N 1 C 2A0CR AR 1 iiiiRjZ AJC AR N AGC AR SNR 5 70 1 3 SNR 1 nC TAR SNR TAR 511C nC Example TAR 005 5nCnc 5 gt SNR 400 Even modest precision of 5 requires high SNR SNR can be increased by averaging onoffline signals time and rangewise Poisson statistics 5 N N 0395 3 SNR N 0395 Since N oc AtAR SNR oc AtO39SARO395 and 511C oC At O39SAR l395 Accuracy of DIAL measurements quotC 2 1 ilnNS o 7RNB o 2AFAR 61R Ns1onaR NB10n 1 i 1nGoggle ZAGC R dR G10nR 1 i In MOM 2A0CR dR 30101213 1 m 0 390n7R a 390 R 1 m MAR gAaXmexR Accuracv affected by B E X gt How well is absorption cross section known gt Improper correction of signal offsets eg background light gt Geometrical factor different for Am and Ao 3000 I I I I 2500 2000 1500 Range m 1000 500 o39 50 o Effect of differential geometrical factor on O3 retrieval 50 OZONE ppbv gt Differential backscatter amp extinction not properly corrected gt Interfering species not taken into account Transmillance Narrow and broad absorption lines Narrow absorption line 081 I39 i l 06 1 CH4 Aoff 04 r 012 I A 7 Wlu leasuriameint 0n 7 Calculation 00 1347135 184710 1847115 1847213 Wavelength nml AA z 50 pm No correction for differential backscatter or extinction needed Transmit laser needs to be tunable High frequency stability amp spectral purity 2 10 24 m Ozone obs cross section Broad absorption feature Ozone absorption in the UV 100000 10000 39 Aon 289 nm 1000 299 nm 100 0 Aoff 10 01 1 240 260 280 300 320 340 360 Wavelength nm AA 10 nm Correction for differential backscatter or extinction necessary Fixed wavelength lasers OK High frequency stability amp spectral purity not needed

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