Travel time Sensitivity Kernels
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- Ἀλκαῖος Μελετόπουλος
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1 Trave time Sensitivity Kernes in ocean acoustic propagation Manois Skarsouis Institute of Appied & Computationa Mathematics Foundation for Research & Technoogy - Heas Herakion, Crete, Greece In coaboration with Bruce Cornuee, Matt Dzieciuch Scripps Institution of Oceanography, University of Caifornia, San Diego, La Joa, Caifornia, USA FORTH - IACM - Wave Propagation Group - Underwater Acoustics
2 pressure Source Muti path propagation Receiver temperature sound speed range Typica measurement Arriva structure time (sec) time
3 Ray arrivas 250 c c + Δc τ = Γ 1 c () d τ τ +Δτ t Typica measurement depth d Γ c(z) range Time (sec) 1 Δ τ = Δcd () 2 c () Γ Ray theoretic trave time sensitivity kerne
4 Ocean Acoustic Tomography Low frequency O(100 Hz) transmissions / ong range O(1000 km) propagation Ray theory, commony used for data anaysis, is a highfrequency asymptotic approximation How good are ray theoretic (high frequency) methods for the anaysis of ow frequency data? How do sensitivity kernes of ow frequency trave times compare with the corresponding ray theoretic kernes?
5 Outine Wave theoretic propagation modeing + notion of peak arrivas Finite frequency trave time sensitivity kernes (TSKs) Effects of increasing frequency / range on TSKs 3D vs. 2D kernes Vertica trave time sensitivity kernes (VTSKs) Effects of increasing range on VTSKs Concusions
6 Propagation modeing The pressure at the receiver in the time domain p r + 1 i t pr(; t c) = ω Ps( ω) G( xr xs; c; ω) e dω 2π P ( ω; c) Ps( ω ) :the emitted (source) signa in the frequency domain Gx ( x; cω ; ) : the Green s function of the acoustic channe s r ω cx ( ) x, x s r : the circuar frequency : the sound speed distribution : the source / receiver ocations
7 Green s function Hemhotz equation (in cyindrica coordinates), c=c(z) 2 2 G 1 G G = r2 r r z2 cz () 2 ω δ δ G 2 ()( r z zs) 2πr Boundary interface radiation conditions Pressure reease (Dirichet) condition at the sea surface Continuity of pressure & vertica veocity at the interfaces Radiation condition (outgoing waves at infinity)
8 Norma mode representation of the Green s function (far fied) Grz (, z) s e iπ /4 ϕ ( z ) ϕ ( z) M m s m = 8π m= 1 kr m e ik r m k m, ϕ m( z) : rea eigenvaues and associated eigenfunctions of the Sturm Liouvie probem d ϕ ( z) ω + ϕ = ϕ dz c ( z) 2 2 m m( z) km m( z) with appropriate boundary interface conditions
9 Trave time modeing Peak arriva times (oca maxima of arriva pattern atc (; ) = p(; tc) : a& ( τ ; c) = 0 Arriva time perturbations (due to sound speed perturbations): a& ( τ +Δ τ ; c+δ c) = 0 r a(t;c) τ τ c c + Δτ + Δc t Linear approximation: Δ τ = u& Δ u + u Δ u& + w& Δ w + wδw& 2 2 u& + u u&& + w& + ww&& p = u+ iw Δ p =Δ u+ iδw u = u( τ ; c) where,,, r r
10 Born approximation r r r r r r r Δcx ( ') r Δ Gx x = Gx x Gx x dvx 2 ( r s) 2 ω ( ' s) ( r ') ( ') 3 c ( z') V The perturbation sequence Δ c Δ G Δ p,i.e.( Δu, Δ w ) Δτ Trave time sensitivity kerne Representation of Δτ in terms of r r r r r r Δ τ = K ( x' x ; x ; c) Δ c( x') dv( x') V s Δc r Wave theoretic trave time sensitivity kerne
11 Trave time sensitivity kerne { r r r u iw r r r iωτ K ( x' xs; xr; c) =R iωq( x' xs; xr; ω; c) e dω 2π b u& iw& r r r iωτ + Qx ( ' xs; xr; ω; ce ) dω 2π b } & && & && 2 2 b = u + uu + w + ww where and 2 2 ω Ps ( ω) s r = 3 s r r r r r r r r Qx ( ' x; x; ω; c) Gx ( ' x; ω; cgx ) ( x'; ω; c) c ( z')
12 Numerica resuts Ray theory 150 m Source 1503 m/s Receiver 2.5 km 1547 m/s 52 km
13 Numerica resuts Ray theory 150 m Source 1503 m/s Receiver 2.5 km 1547 m/s 52 km
14 Numerica resuts Wave theory (400 Hz) 150 m Source 1503 m/s Receiver 2.5 km 1547 m/s 52 km
15 Numerica resuts Wave theory (100 Hz) 150 m Source 1503 m/s Receiver 2.5 km 1547 m/s 52 km
16 Effects of increasing range (100 Hz)
17 Cross section at mid range (100 Hz)
18 3D vs. 2D TSK G 3D e (, r z z ) = s iπ /4 M ϕm( zs) ϕm( z) ikmr e G2 D 8π m= 1 kr m 3D iπ /2 M e ϕm( zs) ϕm( z) ( x, z zs) = e 2 k m= 1 m ikmx 2D
19 3D vs. 2D TSK iπ /4 M ϕm( zs) ϕm( z) ikmr 3D(, r z zs) = e 8π m= 1 kr m Horizonta cross range G e m s m m G2 D( x, z zs) = 3D 2 m= 1 km 2D margina of the 3D TSK e iπ /2 M ϕ ( z ) ϕ ( z) e ik x
20 3D vs. 2D TSK (Cross range marginas)
21 3D vs. 2D TSK (Horizonta marginas)
22 Vertica sensitivity kerne (VTSK) Δ τ = D ( z) Δc( z) dz c c + Δc Ray arrivas dz 1 Δ τ = Δczdz ( ) 2 cz (( z) )sin( ψ (( z )) )) Γ τ τ +Δτ 0 t Source Γ depth z s ψ(z) z r Receiver c(z) range R D ( z)
23 Vertica sensitivity kerne (VTSK) Δ τ = D ( z) Δc( z) dz Peak arrivas a(t;c) atc (; ) = p(; tc) r c c + Δc + 1 iωt pr(; t c) = Ps( ω) Gsr( ω; c) e dω 2π τ τ +Δτ t u& Δ u + u Δ u& + w& Δ w + wδw& Δ τ = 2 2 u& + u u&& + w& + ww&& Norma mode representation ( ; ; ) ( ; ; ) Gsr ( ω; c) = e iπ /4 M e ϕn ω zs c ϕn ω zr c 8 π n= 1 kn ( ω; c) R ik n ( ω; ) c R
24 Vertica sensitivity kerne (VTSK) Eigenvaue/eigenfunction perturbations 2 ω Δ k = n h Δcz ( ) 2 ϕn( z ) dz ( ) 3 kn c z 0, Δ ϕ ( z) = 2ω n h 2 0 Δc( z ) ϕ ( z ) ϕ ( z ) ϕ ( z) dz c z k k M n m m ( ) m= 1 n m m n Green s function perturbation iπ /4 h M M 2 e Unmϕn ( z ) ϕm( z ) Δ G = ω ir U ( ) 2 2 nnϕn z 2π R n= 1 m= 1 kn km kn 2kn m n e kc ik R n n 3 Δcz ( ) dz ( z ) where U nm ϕn( zs) ϕm( zr) + ϕm( zs) ϕn( zr), m n = 1 ϕn( zs) ϕn( zr), m= n 2
25 Vertica sensitivity kerne (VTSK) iπ /4 1 e iωτ D( z) = R ( u iw ) L( z; ) e d 3 & & ω ω bc ( z) 2π R 2π + + iπ /4 e iωτ + ( u iw ) iωl(; z ω) e dω 2π 2 M M ikmr 2 ϕm( z) ϕn( z) U mn 1 Ummϕm ( z) e Lz (; ω) = ω Ps ( ω) ir + + m= 1 n= 1 Λmn 2km k m kmr n m { ( τ ; )} u = R p c r { ( τ ; )} w = I p c r b u& u u&& w& w w&& 2 2 = Λ = k k 2 2 nm n m U nm ϕn( zs) ϕm( zr) + ϕm( zs) ϕn( zr), m n = 1 ϕn( zs) ϕn( zr), m= n 2
26 North Pacific environment SD=RD=1100 m
27 Wave theoretic VTSK (100 Hz)? : Wave theoretic VTSK : Ray theoretic VTSK D ( z)/ R (smoothed)
28 Effects of increasing range (100 Hz) : Wave theoretic VTSK : Ray theoretic VTSK D ( z)/ R (smoothed)
29 Effects of increasing range (100 Hz) : Wave theoretic VTSK : Ray theoretic VTSK D ( z)/ R (smoothed)
30 Long range asymptotics M + 2 iπ /4 2 ( ) = R ( ) ( ) 3/2 3 & & 3/2 ( 2 π ) bc ( z) m= 1 km 1 ( ) D z u iw e S e d Ummϕm z i( ωτ kmr ω ω ) ω ( ) M + 2 iπ /4 3 Ummϕm ( z) i( ωτ kmr) + ( u iw ) e iω S( ω) e dω 3/2 m= 1 ( km )
31 Long range asymptotics Stationary Phase Approach i( km ) ( ) ( ) iδmπ/4 ωτ ω R 2π e ˆ ωτ Χm ω e dω = Χ ( ˆ m ωm) e 2 2 r d k ( ˆ ω ) dω m m ( ( ˆ ω ) ) i k R m m m dkm R ( ˆ ωm) = τ d ω Stationary point for mode m : δ ( 2 ( ˆ ) 2 m = sign d km ωm dω ) ˆ D z u iw e M 2 2 ˆ π 1 ˆ ωmsm ˆ ϕm( z) ˆ ϕm( zs) ˆ ϕm( zr) i ˆ ωτ m ikmr i( δm 1) ( ) ( ) 4 = R 3 & & 4 πbc ( z) 3/2 2 2 m= 1 kˆ ˆ m d km dω ˆ ( u iw ) e M 3 2 ˆ ˆ π ˆ ωmsm ˆ ϕm( z) ˆ ϕm( zs) ˆ ϕm( zr) iωτ m ikmr i( δm+ 1) 4 3/2 2 2 m= 1 kˆ ˆ m d km dω
32 Long range asymptotics Stationary Phase Approach : Wave theoretic VTSK : Ray theoretic VTSK : Stationary Phase approximation
33 Long range asymptotics Stationary Phase Approach : Wave theoretic VTSK : Ray theoretic VTSK : Stationary Phase approximation
34 Long range asymptotics + 2 mm ( ωϕ ) m ( z; ω) i( ωτ km ( ω) R) 3/2 km ( ω) 2 U I( z) = ω S( ω) e dω
35 Long range asymptotics Condition for SP approximation: 2 (; z ) smooth function of, ϕ ω ω m compared to i( km ( ) R e ωτ ω ) ϕm ( z) 2 2 ϕm ( z) << ϕm ( z) τ ω e km R ω e R=50 km, T=33.52 sec, f=100 Hz
36 Long range asymptotics Condition for SP approximation: 2 (; z ) smooth function of, ϕ ω ω m compared to i( km ( ) R e ωτ ω ) ϕm ( z) 2 2 ϕm ( z) << ϕm ( z) τ ω e km R ω e R=50 km, T=33.52 sec, f=100 Hz
37 R=100 km, T=67.04 sec, f=100, f=100 Hz Hz R=200 km, T= sec, f=100 Hz
38 Concusions The wave theoretic VTSK approaches the ray theoretic VTSK as the range increases The stationary phase approximation confirms ong range asymptotic behavior The condition for vaidity of the stationary phase approximation can be used as a measure of wave/ray theoretic proximity. From eigenfunction/eigenvaue characteristics a minimum range for ray theoretic behavior can be inferred on.
39 References H. Marquering, Dahen, Noet, Three dimensiona waveform sensitivity kernes, Geophys. J. Int., Vo. 132, pp , Marquering, Noet, Dahen, Three dimensiona sensitivity kernes for finitefrequency trave times: the banana doughnut paradox, Geophys. J. Int.,Vo. 137, pp , Skarsouis, Cornuee, Trave time sensitivity kernes in ocean acoustic tomography, J. Acoust. Soc. Am., Vo. 116, pp , Skarsouis, Cornuee, Dzieciuch, Trave time sensitivity kernes in ong range propagation, J. Acoust. Soc. Am., Vo. 126, pp , Skarsouis, Cornuee, Dzieciuch, Second order sensitivity of acoustic trave times to sound speed perturbations, Acta Acustica, Vo. 97, pp , 2011.
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