by
Rex Kelley A ndrew
B .S.. U niversity of W a sh in g to n . 1981 M .S .E .E ., U niversity o f W a sh in g to n . 1987 D issertation S u b m itte d in P a rtia l Fulfillment o f th e
R equirem ents for th e D egree of D O C T O R O F P H IL O S O P H Y
in th e D ep artm en t o f Electrical an d C o m p u te r E ngineering We accept this d isse rta tio n as conform ing to th e required s ta n d a rd
—
---Dr. Lynn ^irlhv, C o-supervisor (D e p t, of E lectrical a n d C o m p u te r Engineering)
___________________________________________________________ Dr. David M. Farm er. C o-supervisor (D e p t, of Elec. &: C om p. E ng.)
---Dx<-nale Shpak, D ep arjro en tal M em ber (D e p t, o f Elec. & C om p. E ng.)
Dr. C harles K onzelm an. O utside M em ber (D e p t, o f M echanical E ng.)
G . Lueck. Outside^^^i€hiber (School o f E a rth a n d O cean Sciences)
Dr. Michael .1. Buckingham . E xternal E xam iner (M a rin e Physical L ab o rato ry . Scripps In stitu te of O ceanography)
0 R e .x Kelley .Andrew, 1997 University of V ic to ria
All rights reserved. This d isse rta tio n m ay not be produced in w hole or in p a rt, by- photocopying or o th e r m eans, w ith o u t th e perm ission o f th e a u th o r.
A B S T R A C T
. \ n acoustic a rra y w as deployed in th e n earsurface lay e r in Saanich In le t. B .C .. to im age break in g waves using only th e n a tu ra lly occurring aco ustical rad ia tio n from the b rea k in g region over th e band [160 H z, 2000 Hz]. T he 1.5-element array was configured as a h o riz o n tal cross w ith an 8 m a p e r tu r e , b o tto m -m o o re d , and positioned nom inally 3 m b e n e a th the surface.
D ue to sensor sp arsen ess, th e a rr a y P S F at any p a rtic u la r frequency w as badly c o n ta m in a te d by g ra tin g lobes. .A novel b ro ad b a n d schem e was devised to com bine in fo rm a tio n at m ultiple in d ependent frequencies to yield unam biguous im ages w ith res o lu tio n o f a b o u t 0 . 2 m a t th e sea su rfa c e.
T h e b ro a d b a n d schem e assum ed sp a c e -tim e sep arab ility in th e source m u tu a l spec tra l d en sity . This is o nly considered vabd for breaking waves above a b o u t 400 Hz. N o n s ta tio n a rity and tim e -b a n d w id th c o n s tra in ts yielded a t m ost six in d ep e n d e n t fre q u en cy b a n d s w ithin th e system p a s sb a n d .
p a ra m e tric im age analysis show ed th a t the im ages align closely w ith th e wind and can be observed m oving dow nw ind w ith a speed a b o u t tw o-thirds the p h a se speed of th e d o m in a n t c o m p o n en t of th e w ind waves.
.A.bsolute power levels were found to be consistent w ith previously published results. T h e a b s o lu te power levels were p a ra m e te riz e d by w here So = 1 /iP a ^ /H z and A (/) is w ell-described by a sim ple first o rd e r relation X = bo + bi I n / , w here bo varied d e p e n d in g on th e size o f th e wave b u t 6% ap p e are d to be a m ore universal c o n sta n t e s tim a te d a t -4..5.5 ± 0.47.
T he source m echanism for frequencies below about, 400 Hz w as modeled tw o ways: ( I ) as a point source (w hich would follow if an acoustically c o m p act “collective os cillation" region had fo rm ed ), and (2 ) as d u e to oIf-peak s p e c tra l co n trib u tio n s from bubbles reso n an t a t 400 Hz. Neither m odel achieved a s a tis fa c to ry fit to th e observed d a ta . T his seems to im ply th a t the m echanism below a b o u t 400 Hz was acoustically extended a n d rad ia tin g as energetically as any reso n a n t bubbles.
Dr. R , 'C ^ i n ^ K ^ ü î ^ C o -supervisor (D e p t, o f E lectrical and C o m p u te r Engineering)
Dr. D avid M. Farm er. C o-supervisor (D e p t, of Elec. & C om p. E ng.)
__________________________________________________________________________________
Dr. TÎale S hpak. D e p a rtm e n ta l M em ber (D e p t, of Elec. & C om p. Eng.)
Dr. C harles K onzelm an. O utghfe M em ber (D e p t, o f M echanical E ng.)
olf G . Lueck. O u tsid e M em ber (School o f E a rth and O cean Sciences)M^iftb
Dr. M ichael .1. B uckingham . E xternal E x a m in e r (M arin e Physical L aboratory. Scripps In stitu te o f O ceanography)
C o n ten ts
A b s t r a c t ii A c k n o w le d g e m e n t s x iv F r o n t is p ie c e x v i L ist o f A c r o n y m s x v ii E p ig r a p h x v iii 1 I n t r o d u c tio n 1 1 . 1 B a c k g r o u n d ... 1 1.2 This W o r k ... 5 1.3 S u p e r r e s o lu tio n ... 6 1.4 Thesis O rg an izatio n ... 8 2 S o u r c e P h y s ic s 10 2.1 B u b b l e s ... 11 2.1.1 B irth in g W a i l s ... 1 2 2.1.2 .\d u lt S c r e a m s ... 132.1.3 G ang Rum bles (C ollective O scillations) ... 14
2 . 2 Breaking W a v e s ... 2 1 2.2.1 Spiliers versus P l u n g e r s ... 21 2.2.2 W h ite c a p E v o l u t i o n ... 23 2.2.3 B ubble Size D i s tr i b u ti o n ... 25 2.3 T he S o u r c e ... 26 3 S ig n a l P h y s ic s 29 3.1 E xtended S o u r c e s ... 30 3.2 Source In fo rm atio n P r o p a g a t i o n ... 32 3.3 P ro p a g atio n I s s u e s ... 38 3.3.1 G reen F u n c t i o n s ... 38
4 I m a g e R e c o n s t r u c t io n 4 6
4.1 I n tr o d u c tio n ... 46
4 .2 R esolution a n d S u p e r r e s o l u t i o n ... 49
4.2.1 F a r-Field Im aging ... 50
4.2.2 N earfield Im aging ... 53
4.3 B eam form ing I m a g i n g ... 54
4.4 .A.n .A lternative P ro b ab ilistic .A p p r o a c h ... 63
4..5 ( N o n )P a ra m e te riz in g the S o l u t i o n ... 6.5 4.5.1 .V onparam etric I m a g e s ... 6 8 4.5.2 P a ra m e tric I m a g e s ... 70 5 T h e E x p e r im e n t 72 5.1 SU R FE X III D e p l o y m e n t ... 72 5.2 D a ta D e s c r i p t i o n ... 77 6 S ig n a l P r o c e s s in g 85 6.1 D a ta R ecord I d e n tif ic a tio n ... 8 6 6.2 CSDM E s t i m a t i o n ... 8 8 6.3 P O C S ... 91 6.4 .A W ell-behaved I m a g e ... 93 6.5 W h a t C an Clo W rong ... 97 7 D a t a A n a ly s is 101 7.1 Event M o t i o n ...101 7.1.1 W ave W 3 ... 102 7.1.2 W ave W 5 ... 109 7.1.3 M otion . A n a l y s i s ... 109 7.2 Inferences on W in d D i r e c t i o n ... 112 7.3 Source S t r e n g t h ...114 7.3.1 D a ta R e d u c t io n ...114
7.3.2 Source S tre n g th .Analysis ... 118
7.4 M odeling Low -Frequency ( / < 400 Hz) E m is s io n s ...122
7.4.1 Point S ou rce Model ... 122
7.4.2 R a d ia te d Low -Frequency S p e c t r u m ...132
7.4.3 Low -Frequency M odeling C o n c l u s i o n s ...135
8 C o n c lu s io n s 137 8.1 R e s u lts ...137 8.2 A c h ie v e m e n ts ... 138 8.3 R eco m m en d atio n s ...140 B ib lio g r a p h y 146 A W a v e F o r e c a s t 158 B S im u la t io n s w i t h E x t e n d e d S o u r c e s 162
C N o n p a r a m e t r i c A l g o r i t h m D e t a il s 1 6 9 C l " C o s t” Function ... 169 C.2 " G ra d ie n t” F u n c t i o n ... 173 C.3 I m p le m e n ta tio n ... 174 D M a x i m u m L ik e lih o o d D e t a i l s 1 7 9 D .l “‘C o s t” Function ...179 D.‘2 “‘G ra d ie n t” F u n c t i o n ... 181 D.3 I m p le m e n ta tio n ... 182 D.4 S im u la tio n ... 184 D.5 E rro r E s t im a t e s ... 187 E U L S A S S y s t e m D e s c r i p t i o n 191 E .l S ystem Block D i a g r a m ... 191 E.2 A rray S u b s y s t e m ... 197 E.2.1 A ttitu d e S e n s o r s ... 198 E.3 A nalog S u b s y s t e m ... 207
E.3.1 G ain and T H D ... 209
E .3. 2 System Noise F l o o r ... 209
E .3 .3 C r o s s t a l k ... 210
E .l D igital Subsystem ... 210
E.4.1 .Analog to d ig ita l p a t h ... 210
E.4.2 M UX ... 21.5 E .4.3 System C o n tro lle r ... 215
E.5 G l o s s a r y ... 217
F I n - S i t u A m b i e n t S o u n d C a l i b r a t i o n 2 1 9 F .l I n tr o d u c ti o n ... 219
F.2 C a lib ra tio n E q u a t i o n s ...222
F.3 Sound Fields N ear th e Surface ... 223
F.4 E s tim a tin g th e O u tp u t . A u t o s p e c t r a ... 228 F.5 C a lib ra tio n P r o c e s s i n g ...230 G H y d r o p h o n e M o u n t s : V i b r a t i o n T r a n s m i s s i b i l i t y 2 3 3 H H y d r o p h o n e M o u n t s : D i f f r a c t io n 2 4 4 H .l I n tr o d u c tio n ... 244 H.2 B E M ... 246 H.3 R e s u l ts ...249 I W a v e P r o c e s s i n g S u m m a r y 2 5 1 I.1 W ave D a t a ... 251 1.2 W ave M odel P a r a m e t e r s ...“262 1.3 .A dditional P rocessing D e t a i l s ...265
S o u r c e S t r e n g t h C o n s id e r a t io n s 2 6 9
J . l P o in t S o u r c e s ... 269 J.2 S ource L a y e r ... 273 J.3 R a n d o m Source L a v e r ... 273
L ist o f F ig u res
1.1 A typical a m b ie n t noise s p e c t r u m ... 4
2 . 1 R esonant freq u en cy versus bu b b le r a d i u s ... 1 2 2.2 T h e bubble co lu m n e x p e r i m e n t ... 16
2.3 P ro p a g a tio n o f a W ave into a S w arm of B ubbles... 16
2.4 Effective S o u n d speed ... 19
2.5 B ubbly F luid a n d its Sound Speed A n a l o g ... 20
2.6 T h e bubble co lu m n ex p erim en t a n d its sound sp eed a n a l o g ... 20
2.7 P rim a ry a n d s e c o n d a ry ra d ia tio n fo r a breaking w a v e ... 28
2.8 Bulk Sound S peed A nalog for a B re a k in g W a v e ... 28
3.1 R a d ia tio n fro m a source elem ent ... 33
3.2 .Actual p ro p a g a tio n p r o b l e m ... 40
3.3 Idealized p ro p a g a tio n pro b lem ... 40
3.4 M a g n itu d e o f th e k space G reen f u n c t i o n ... 44
4.1 Schem atic d ia g ra m o f a 1-D im a g in g p r o b l e m ... 47
4.2 C o n cep tu al d ra w in g o f “re so lu tio n ” ... 50
4.3 G eo m etrical co n sid e ratio n s for d efin in g the far field o f an a r r a y ... 51
4.4 P S F h alf-w id th ( “reso lu tio n ” ), S U R F E X I I I ... 54
4.5 ULSAS se n so r c o n ste lla tio n , p lan view , SU R F E X I I I ... 56
4.6 P S F a t 603 H z ... 57 4.7 P S F a t 861 H z ... 57 4.8 P S F a t 1120 H z ... 58 4.9 P S F a t 1378 H z ... 58 4.10 P S F a t 1637 H z ... 59 4.11 P S F a t 1895 H z ... 59 5.1 D eploym ent g e o g ra p h y , S U R F E X I I I ... 74
5.2 D eploym ent s ite , S U R F E X I I I ... 75
5.3 M ooring sch em e for S U R F E X I I I ... 76
5.4 V ector-averaged w ind speed, S U R F E X III, 4 / 1 3 / 9 5 ... 77
5.5 A lr-Sea T e m p e ra tu re Difference, S U R F E X III, 4 / 1 3 / 9 5 ... 78
5.6 W ind D irec tio n , S U R F E X III, 4 / 1 3 / 9 5 ... 78
5.8 A rra y d e p th , S U R F E X III, 4 /1 3 /9 5 ... 79
5.9 A rra y b earing, S U R F E X III. 4 /1 3 /9 5 80
5.10 E x a m p le d a ta fro m ch annel 5 from th e w ind e v e n t of 4 / 1 3 / 9 5 ... 82
5.11 N arro w b a n d levels versus tim e for a sam ple b rea k in g event ... 83
5.12 B ro a d b a n d a c o u stic signals p e r channel for a ty p ical w a v e ... 84
6.1 W ave W 3: 15:24:01 P D T ... 87
6.2 T a p e rs A: = 1, . . . , 5 ... 90
6.3 Field in ten sity v e rsu s sen so r p o sitio n , x-axis, w ave W 3 ... 92
6.4 S equence of P O C S so lu tio n s for th e Field in te n sity , x-axis. w ave W 3. . . 93
6.5 N o n p a ra m e tric s o lu tio n . W ave W 3 ... 95
6 . 6 P a ra m e tric s o lu tio n . W ave W 3 ... 96
6.7 F ield in ten sity v e rsu s sen so r po sitio n , wave W6 ... 99
6 . 8 .Acoustic p ressu re, channels 6. 5 and 3. wave W6 ... 100
. 1 S h o rt-tim e s e g m e n ta tio n o f W ave W 3 ... 102
.2 N o n p a ra m e tric s o lu tio n . W ave W 3, segm ent .A ...103
.3 N o n p a ra m e tric s o lu tio n . W ave W 3. segm ent B ...104
.4 N o n p a ra m e tric s o lu tio n . W ave W 3, segm ent C ...105
.5 N o n p a ra m e tric s o lu tio n . W ave W 3, segm ent D ...106
. 6 S equence of s o lu tio n s. W ave W 3 ...107
.7 S h o rt-tim e s e g m e n ta tio n o f W ave W 5 ... 109
. 8 S equence of so lu tio n s. W ave W 5 ...I l l .9 S equence of in fe rre d wind d i r e c t i o n s ...113
.10 H isto g ram of re p lic a te s ... 114
. 1 1 S o u rc e stre n g th p a r a m e t e r s ... 115
.12 H isto g ra m of 6% r e p l i c a t e s ... 119
.13 C o m p a riso n o f W 3 so u rce level versus a m b ien t a n d Knudsen levels . . . 121
.14 C o m p ariso n o f W 3 im age an d finite dipole low -frequency so lu tio n s . . . 126
.15 C o m p a riso n of W 5 im ag e and finite dipole low -frequency so lu tio n s . . . 127
.16 C o m p a riso n o f W 7 im age an d finite dipole low -frequency so lu tio n s . . . 128
.17 C o m p a riso n o f W 1 3 im age and finite dipole low -frequency so lu tio n s . . . 129
.18 C o m p a riso n o f W 2 2 im age and finite dipole low -frequency so lu tio n s . . . 130
.19 C o m p a riso n o f W 23 im age a n d finite dipole low -frequency so lu tio n s . . . 131
. 2 0 C o m p a riso n o f b u b b le sp e c tru m versus W.3 a m b ie n t and signal levels . . 134
.21 C o m p a riso n o f W 3 im age and finite dipole low -frequency so lu tio n s. / = 245 H z ...136
B .l S im u la tio n s o u rc e -a rra y g e o m e t r y ... 163
B.2 E igenvalue Aj v e rsu s n u m b e r o f p o int s o u r c e s ...165
B.3 E igenvalue Aj v e rsu s n u m b e r o f p o in t s o u r c e s ...165
B.4 E igenvalue A3 v e rsu s n u m b e r o f p o in t s o u r c e s ...166
B.5 E igenvalue A4 v e rsu s n u m b e r o f p o in t s o u r c e s ...166
E .l ULSAS system d i a g r a m ... 192
E.2 ULSAS sy stem block d i a g r a m ...193
E.3 ULSAS shore-end sy ste m ...196
E.4 ULSAS la b o ra to ry play b ack s y s te m ...197
E.5 H y d ro p h o n e a rra y plan v i e w ... 199
E. 6 .\ r r a y ‘‘a rm s ” ...200
E.7 Excess h y d ro d y n am ic pressu re signal due to su rface w a v e s ... 203
E. 8 P re ss u re sensor c a li b r a t io n ... 203
E.9 P re ss u re sensor h y s t e r e s i s ... 206
E.IO .A^nalog signal c o n d itio n in g schem atic d i a g r a m ...207
E .l 1 S y stem tra n s fe r f u n c t i o n ...209
E. 1 2 S y stem noise floor, all c h a n n e l s ...2 1 2 E.13 D igital su b sy stem block d i a g r a m ... 214
E.14 F in ite S ta te M achine for th e m u lt ip l e x e r ...215
E.15 Level 0 c o n te x t d iag ram for th e system c o n t r o l l e r ...216
F .l E ssential Signal C o n d itio n in g C om ponents for H ydrophone C a lib ra tio n . 222 F.2 G e o m e try for N ear-S urface Sound Field C a l c u l a t i o n s ...224
F.3 R eflection g eom etry for a nearsurface s e n s o r ... 225
F.4 V a ria tio n o f a u to s p e c tra l level w ith d e p t h ...227
F.5 V ariatio n o f a u to s p e c tra l level w ith d e p th . d B ... 227
F. 6 H isto g ram o f channel 5 o u tp u t power l e v e l s ... 231
G .l ULS.A.S a rra y h y d ro p h o n e m o u n t ... 235
G. 2 Sim plified h ydrophone m o u n t ... 236
G.3 Sim plified 1-D sp rin g -m a ss-d a sh p o t m ount m o d e l ...239
Ci.4 T ran sm issib ility functio n s for th e I T C lO O l...241
G.5 T ran sm issib ility fu nctions for th e ITC4127 ... 241
G. 6 R e so n a n t "ringing” in th e d a t a ... 242
H .l E lem ent m esh for h y d ro p h o n e m o u n t ...*248
H.2 S c a tte re d field versus f r e q u e n c y ...250
I.1 W ave W l : 15:20:01 P D T ... 252 1.2 W ave W 2:..15:22:46 P D T ...252 1.3 W ave W 3:..15:24:01 P D T ...252 1.4 W ave W 4: 15:24:56 P D T ... 253 1.5 W ave W 5: 15:25:16 P D T ...253 1.6 W ave W6 :..15:27:04 P D T ... 253 1.7 W ave W 7: 15:28:07 P D T ... 254 1.8 W ave W S: 15:31:52 P D T ... 254 1.9 W ave W 9: 16:02:05 P D T ... 254 1.10 W ave W IO : 16:05:03 P D T ...255 1.11 W ave W l l : 16:05:52 P D T ... 255 1.12 W ave W 12: 16:07:37 P D T ...255
1.13 VVaveVVIS: 16:12:22 P D T ...256 1.14 Wave W 14: 16:14:30 P D T ... 256 1.15 W ave W 15: 16:21:42 P D T ... 256 1.16 Wave W 16: 16:22:43 P D T ... 257 1.17 Wave W 17: 16:22:57 P D T ... 257 1.18 Wave W 18: 16:23:31 P D T ... 257 1.19 Wave W 19:.16:36:51 P D T ...258 1.20 W ave W 20:.16:38:20 P D T ...258 1.21 Wave W 21: 16:46:53 P D T ...258 1.22 Wave W 22: 16:48:33 P D T ...259 1.23 Wave W 23: 17:02:33 P D T ...259 1.24 Wave W 24: 17:02:49 P D T ...259 1.25 Wave W 25: 17:03:40 P D T ...260 1.26 Wave W 26: 17:05:05 P D T ...260 1.27 Wave W 27: 17:05:13 P D T ...260 1.28 Wave W 28: 17:05:35 P D T ...261 1.29 Wave W 29: 17:05:47 P D T ...261 1.30 Wave W 30: 17:07:40 P D T ...261
L ist o f Tables
2.1 S u m m a ry of c h a ra c te ristic fe a tu re s o f M o n a h a n 's bubble fo rm a tio n s . . 25
3.1 Rayleigh roughness p a r a m e t e r s ... 42
4.1 O nset o f far field, —3 dB p o in t a n d h alf fo o tp rin t size ... 53
6 . 1 CSDM e s tim a tio n p a r a m e t e r s ... 89
6 . 2 T aper e ig e n v a lu e s ... 90
6.3 Source s tre n g th solution, w ave \ V 3 ... 97
6.4 P a ra m e te r solution for wave \V3 97
7.1 P a ra m e te r evolution for wave W 3 ... 108
7.2 Source s tre n g th evolution o f w ave \ V 3 ...108
7.3 P a ra m e te r evolution for w ave W 5 ... 110
7.4 Source s tre n g th evolution o f w ave W 5 ... 110
7.5 Source p a ra m e te r regression, each w a v e ...117
7.6 Fit o f L F finite-dipole m odel to several w a v e s ...125
.A..1 E stim a te d W avefield P a ra m e te rs for P a tric ia B ay S i t e ...161
C .I B enchm arks for cost and g ra d ie n t f u n c t i o n s ... 176
D .l Sim ulation resu lts, shape p a r a m e t e r s ... 188
D.2 Sim ulation resu lts, source s tr e n g th p a ra m e te rs ...188
D.3 S ta n d a rd e rro r e stim ates for w ave VV3 shap e p a r a m e t e r s ...190
D.4 S ta n d a rd e rro r e stim a tes for w ave W 3 source s tr e n g th p a ra m e te rs . . . 190
E .l Gain an d T H D p e r c h a n n e l ... 211
E. 2 DC bias a n d b ro ad b a n d noise d e n sity per c h a n n e l...2 1 1 E.3 ULSAS C r o s s t a l k ... 213
E.4 Digital d a ta f o r m a t ...215
E.5 A n n o ta tio n f o rm a t... 217
F .l .\u to s p e c tra l Levels versus D e p t h ... 228
F.2 C a lib ra tio n values for each c h a n n e l ...232
G .2 M o u n t E ig e n f r e q u e n c ie s ... 239 1.1 M L p a ra m e te rs for w a v e s h a p e s ... 262 1.2 M L p a ra m e te rs fo r wave source s tre n g th p a r a m e t e r ...264
A ck n o w led g em en ts
I would like to acknow ledge th e su p p o rt of my tw o co-supervisors D r. K irlin. for in v it ing me to th e U niversity of V ic to ria, and Dr. F a rm e r, for pro v id in g tru ly wonderful facilities. I m u st th a n k both D rs. Kirlin and F a rm e r for p u ttin g th e ir heads to g eth e r a n d com ing w ith , w h at was for m e a t least, a very challenging p ro je c t. I would also like to th an k th e m em bers of my c o m m ittee for p u ttin g up with me: D r. K onzelm an, for seem ingly endless patience in h an d lin g the sam e old acoustics q u e stio n s over and over; D r. Lueck, for a fun oceanography course w ith lo ts o f neat-o h a rd w a re asides: and D r. Shpak for s u p p o rt and help on questions reg a rd in g optim izers. I also acknow ledge th e financial s u p p o rt o f th e Office o f Naval Research.
No p ro je c t like this is a o n e-m an show, an d I am extrem ely g ra te fu l to th e m any people who helped along the way. Forem ost a m o n g th ese would be th e g u y s a t O ceanetic M easurem ent. L td ., who held my h a n d and m ade E v e ry th in g C om e O u t .A.11 R ight: Mike Dempsey, D avid S p ear, Kim W allace. Don L apshinoff and D ennis H edji. Hey. nothing would have w orked w ithout OML!
I also would like to th an k th e s ta ff a t the A co ustical O cean o g rap h y R esearch G roup for providing assistan ce, advice, an d exceptional co m p u tin g facilities, in p a rticu la r G race K a m itak ah ara-K in g , Ron Techirob, W illi W eichselbaum er. A lan A d rian , and a whole sq u ad o f technicians who would periodically have to com e to m y rescue: Nick HaU -Patch, Reo Phillips, .\n d re w Sinclair and Reece H asam en. P a tr ic ia K im ber also helped w ith g rap h ic s su p p o rt.
I also w ant to th an k Li D ing, Vadim Polonichko. Ron Kessel. Wei.xiu Du, Randy Howell. David C aughey, Jo h a n n es G em m rich a n d Y unbo Xie for s tim u la tin g conver satio n s d u rin g my to u r, and D r. Richard Paw low icz for talk in g tech on th e way to clim bing venues.
This m o n stro u s array would have rem ained a p a rk in g lot p ro je c t if it had not been for R /V S tr i c k l a n d and its m a s te r. Don Horn. Don show ed g reat p a tie n c e d u rin g tedious dockside h a rd w a re debugging cycles by lending th e m ain hoist on th e S tric k la n d as a su rro g a te for th e dockside cran e, which had gone dow n. Q uite a few fro sty m ornings it was ju s t m e a n d Don and T h e A rray. T h e S tric k la n d was also th e key c ra ft in o u r deploym ent an d recovery plans.
I also w ant to acknowledge th e efforts of G a ry D uncan. R oger K elly an d A1 Keddy a t UVic and R ichard O u terb rid g e a t lO S /C E O R for m ain ta in in g excellent co m p u ter netw orks. I rare ly saw them b u t I used their sy ste m s every day.
I m u st also ad m it a huge debt to th e legions o f people w ho have c o n trib u te d to the in tellectu al in fra stru c tu re o f the In te rn et. T h ere is no way I can even begin to list the people in Usenet n ew sgroups such as s c i . m a t h . n u m - a n a l y s i s . a l t . s c i . a c o u s t i c s , s c i . m a t h . s t a t c o m p .s o f t - s y s .m a t l a b and so on w ho have taken th e tim e to reply to my q uestions. T his list also includes th e u n c o u n tab le m asses of people an d in s titu tions th a t provide, m a in ta in and develop freew are for n u m erical processing, electronic d o c u m e n ts, co m p u ter services and word processing. T h a t I can see any d ista n c e a t all beyond m y own nose is d u e prim arily to th e fact th a t I h av e sto o d on th e sh o u ld ers of m illions.
I would be rem iss if I d id n 't also acknow ledge th e s u p p o rt provided by S haron M oulson (form erly E C E g ra d se cre ta ry ), T erry Russell (C E O R ), Vicky S m ith (E C E g rad se c re ta ry ) and Pip Sum sion (.A .0R G /10S ). who tirelessly s tra ig h te n e d o u t every a d m in is tra tiv e snafu I could devise. Gold s ta rs all a ro u n d .
L astly, I would also like to th an k : G ary and L arry a n d Michelle a t th e B ody B arn for ru n n in g a friendly place w here I was able to w ear o u t m y fru s tra tio n s a n d keep the ship to g e th e r during to u g h tim es; my M o th er and F a th e r, w ho have alw ays s u p p o rte d me even th o u g h I never seem ed to know w here 1 was going; a n d my best friend M arilee. who was m arried , e x tra c te d from a lucrativ e jo b . and y an k ed in to a foreign c o u n try , all w ithin a few m onths, a n d w ho still stuck aro u n d to s u p p o rt me.
B ro a d b a n d A c o u stic a l
S u p erreso lu tio n Im a g in g o f
B rea k in g O cean W aves
Dipole Source Strengths
ii 90 m 80 603 861 1120 1378 1637 1895 amplitude 0.0215 0.0186 0.0156 0.0127 0.0097 0.0068 '
Broadband Acoustic S h ap e Function
-1 0 1
array x axis [m]
L ist o f A cro n y m s
Acronym E xpansion D efined
AORG A coustical O c e an o g rap h y R esearch G roup S e c tio n E.5
CSDM C ro ss-S p e c tra l D ensity M a trix pg 61
HW HM H alf W id th a t H alf M axim um Pg 49
lOS I n s titu te o f O cean Science S ectio n E.5
M CF M u tu a l C oherence Function pg ;jO
M EM M axim um E n tro p y M ethod pg ~
MSD M u tu a l S p e c tra l D ensity pg 30
NAFI Nearfield A co ustical H olography pg 43
PO C S P ro je c tio n O n to Convex Sets pg 91
P S F P o in t Spread F unction pg 49
SU R FE X SU R Face E x p e rim e n t pg 6
O ften, th e things which a re m ost fam iliar to us tu rn o u t to be th e h a rd e st to u n d e rs ta n d . — E dw in T. J a yn es
T ake w h a t is useful and develop from there. — Bruce Lee
2 + 2 = 5 for sufficiently larg e values o f 2. N et.w isd o m fr o m U senet newsgroup s c i . m a t h . n u m - a n a l y s i s
If we d o n 't succeed, we run th e risk o f failure. — D an Quayle
Daddy. 1 overdid it again! — his son
In tr o d u c tio n
1.1
B ack grou nd
B reaking wind waves are now believed to play a fu n d a m e n ta l rôle in th e cou p lin g o f the a tm o sp h e re a n d th e ocean. They a re th e p rim a ry m ec h a n ism for tra n s fe r rin g a t m ospheric m o m e n tu m into surface c u rre n ts (M elville [96]), w hich in tu r n d riv e ocean circulation p a tte rn s . B reaking waves, b o th large a n d sm a ll, p ro vide a " d a m p in g fac tor" th a t d issipates m echanical energy and m o d era te s t h e g ro w th of waves a g a in s t th e influence of wind forcing a n d wave-wave in te ra c tio n (K o m en et al.. [69]). B re a k ers en tra in a tm o sp h e ric gases in to subsurface bubbles, p ro d u c in g an in h o m o g en eo u s layer of bubbly fluid, which m ay affect sound p ro p a g a tio n (F a rm e r a n d Lem on [46]) a n d m ay significantly increase aco ustic b a c k sc a tte r from th e su rfa c e a t n e a r-g ra z in g an g les ( Mc D onald [93]). T h e tu rb u le n c e g e n e ra te d by b reak in g w aves is su sp ec te d o f e n h a n c in g th e d issip atio n in th e near-su rface sublayer (C ra ig and B a n n e r [26]) which in tu r n g re a tly enhances h eat a n d gas tra n s fe r from th e b o tto m of th e a tm o s p h e ric b o u n d a ry lay er into the o cean (T h o rp e
[131])-O u r ab ility to m o n ito r th e co n trib u tio n o f breakers to th e coupling p ro cess requires m easu rem en ts d u rin g w indy an d often s to rm y co n d itio n s, w hich a re not id ea l a n d som e tim es h a z ard o u s for h u m an observers. For th ese re a so n s, o c e a n o g ra p h e rs a re seeking
a lte rn a tiv e m e th o d s for re m o te ly m o n ito rin g th e b rea k in g a c tiv ity o f th e sea s u rfa c e in all possible sea s ta te s .
One o f th e m o re accessible aspects o f b re a k in g w ind-driven w aves is th e “c ra sh in g " so u n d th ey ra d ia te . E a rlier th is century, d u r in g th e d ev elo p m en t o f m ilita ry so n a r sy ste m s, th e so u n d ra d ia tin g from break in g w aves was c o n sid ered “noise", a n d h a d a significant a lth o u g h d u b ious rôle as a d e g ra d in g fa c to r in s o n a r sy ste m p e rfo rm a n c e.
.More recently, how ever, we have g re a tly e x p a n d e d o u r a b ility to use th e so u n d ra d ia te d by b rea k in g waves to a ctu ally in te r p r e t th e s ta te o f th e m arine b o u n d a ry layer.
Early d e scrip tio n s of th e d e ep -w ater a m b ie n t noise field a sc rib e d a d o m in a n t c o m p o n en t o f th e a m b ie n t noise to a con tin u o u s s u rfa c e “s h e et" o f dipoles ( U rick [137]. c h a p te r 7). T h e dipole sh eet w as. of co u rse, a s ta tis tic a l m odel for a ran d o m ly rough sea surface ran d o m ly p o p u la te d by ra n d o m -siz ed b reakers.
L ater. .A.nderson [I] used a high-gain a c o u s tic a rra y m o u n te d on an u n d e rw a te r blim p to acq u ire d ire c tio n a l se a surface noise. . \ co m p lic a ted s ta tis tic a l a rg u m e n t was th e n used (.\n d e rs o n [Ij. S h a n g a n d .\n d e rs o n [120]) to infer t h a t th e sound so u rc e s a t th e sea surface were a c tu a lly d iscrete so u rces, a s o p p o se d to a c o n tin u u m .
This m ore-or-less in tu itiv e finding was verified by F arm er a n d V'agle [47] w ho used sim u ltan eo u s a u d io a n d visual o b servations to sh o w th a t a n in d iv id u a l b reak in g w ave indeed ra d ia te s a b ro a d b a n d aco u stic signal in to th e w ater.
This ra d ia te d signal was su b seq u en tly used by D ing a n d F a rm e r [37] to lo c a te a n d tra c k breaking w aves. U sing a four-elem ent a rr a y , th ey d e te rm in e d th e p o sitio n for in d iv id u al sources (i.e .. b re a k e rs) by c ro ss-c o rre la tin g th e signals received on each se n so r to c o m p u te th e tim e-difference o f arrival ( F a rm e r a n d Ding [4-5]).
Th^jse resu lts, com bined w ith recent a d v a n c e s m a d e in th e la b o ra to ry , show ed t h a t th e sound ra d ia te d by su rfa c e sources, specifically b rea k in g w aves, could in fact b e used to m easure sea su rface p ro p e rtie s . T h e m e c h a n ic a l energy d iss ip a te d by a b re a k in g w ave a p p e ars to be d ire c tly re la te d to th e in te n s ity o f th e r a d ia te d sound (L oew en a n d
M elville [82]): th is a d m its th e p o te n tia l for in fe rrin g th e surface w ave field d iss ip a tio n via an a co u stical m easu re. T h e ra d ia te d a u to s p e c tr u m also a p p e a rs to be re la te d to sim ple m odels o f th e b u b b le size d istrib u tio n w ith in th e breaking region (L oew en a n d Melville [83]. M edw in a n d D aniel [95]). possibly pro v id in g an inverse p ro b lem for th e bu b b le size d is trib u tio n a n d hence th e volum e o f gas e n tra in e d by th e b rea k e r. T h e s p a tia l s ta tis tic s o f waves tra c k e d across th e o c e a n surface provide key p a ra m e te rs needed in m odels o f su rface w ave field g row th (D in g a n d F arm er [36]. Phillips [111]). T h e sea su rface a m b ie n t so u n d a n d w h itecap p in g a c tiv ity also c o rre la te well w ith w ind s tre n g th (K n u d se n et al. [65]. M o n a h an and O 'M u irc h e a rta ig h [102]) a n d now a m b ie n t so u n d is used to m o n ito r o c e an surface wind s p e e d (K e rm a n et al. [64]).
These im p o r ta n t a c o u stic a l m easures of b re a k in g waves, and th e inferences th e y p ro vide on th e s ta te o f th e m a rin e b o u n d a ry layer, h a v e all been achieved w ith rela tiv e ly sim ple sensors. O u r u n d e rs ta n d in g o f sea su rfa c e processes, how ever, is far from c o m p lete . and we m u st th ere fo re tu r n to m ore s o p h is tic a te d in stru m e n ts . . \ s an e x a m p le , if breaking waves can be d e te c te d , m easured a n d tra c k ed by th e n a tu rtilly -o c c u rrin g so u n d they ra d ia te , can th e ir individual s p a tia l dim ensions also be im ag ed by th e ir ra d ia te d sound? If so. w h at w ould this tell us?
To d a te , tw o e x p e rim e n ts have been c o n d u c te d to im age b rea k in g w aves. T h e first ( C ro w th e r a n d H an sla [29] ) utilized a seven-beam s o n a r system co nfigured like a n in se c t- eye. b o tto m -m o o re d in 85 m o f w ater. T he se v en beam s provided seven o v e rla p p in g fo o tp rin ts a t th e su rfa c e, w ith fo o tp rin t d ia m e te r a p p ro x im a te ly 9.7 m a t 24 kHz. T h e y used a tw o -to n e im age m odel a n d an e n tro p y -ty p e function to reg u la riz e th e in verse problem a n d p ro d u ced im ages o f acoustically loud s h a p e s m oving acro ss th e a c o u stica lly q u iet background field-of-view . S im ultaneous v isu a l observations identified w h ite c a p s m ore-or-less w ith in th e p e rim e te rs of these s h a p e s. T h e ir results show ed th e a c o u stica lly a c tiv e region to be c o n sid e rab ly larg er th a n th e a p p a re n t visual size o f th e w h ite c a p .
T he second e x p e rim e n t (E p ifa n io and B u c k in g h a m [44]) used a s o n a r sy stem c o n ta in in g 126 beam s a t a sla n t ran g e from th e s u rfa c e o f 45 m. o p e ra tin g in th e ra n g e
mechanism ?
60
i
mechanism understood
1 0"frequency [Hz]
Fi g u r e 1.1: .A. t y p ic a l a m b i e n t noise s p e c t r u m .S kHz to 80 kHz. E ach b eam had a fo o tp rin t d iam e te r o f a p p ro x im ate ly 0.7-5 m at SO kHz. T h e se results show th a t the s p a tia l s tr u c tu r e of th e acoustically a c tiv e region evolves q u ite rapidly, on tim e scales a t least as sh o rt as 40 m s.
B oth o f these e x p e rim e n ts imaged th e b reak ers a t frequencies well above th e peak of th eir ra d ia te d s p e c tra l sig n a tu re . T h e p o rtio n of th e b ack g ro u n d a m b ie n t sound due to n a tu r a l surface sources has a w ell-know n shape as show n in figure 1.1. T he a u to s p e c tru m has a b ro a d peak a t a b o u t 200 - 500 Hz. and a slope o f a b o u t —5 o r — 6 dB per o c ta v e above. M easu rem en ts have gen erally shown t h a t th e ra d ia te d sig n a tu re s of in d iv id u al breakers te n d , although lo u d er, to have th e sam e sh ape. T h e source m echanism for frequencies above the peak is fairly w ell-established to be th e in co h e ren t "ringing" o f individual b u b bles g enerated by e n tra in m e n t in th e b reaker. T his region is sh ad ed in figure 1.1.
T h e so u rce m echanism for sp ectral en erg y below th e p e a k is not as w ell-established. T here a re . in th e a u th o r 's opinion, no e n tirely convincing th eo ries a t th e p resen t tim e.
One of th e c e n tra l p ro b le m s is th e lack o f q u a lity d a ta a t th ese frequencies. T h e tw o im aging e x p e rim e n ts d e s c rib e d above, for in s ta n c e , were c o n d u c te d a t frequencies well- above he p eak .
In su m m a ry , th e r a d ia te d break er s ig n a tu re is d o m in a te d by s p e c tr a l c o n trib u tio n s from th e d e c ad e below a n d th e decade a b o v e th e peak, a n d th is is th e signal t h a t appears to be directly c o rre la te d with several o f th e m ost in te re s tin g o c e an o g ra p h ic p a ra m e te rs, such as m ec h a n ic a l wavefield d issip a tio n a n d to ta l g a s e n tr a in m e n t.
In this p ro je c t, we a t t e m p t to image b re a k in g waves at th e se frequencies n e a r th e peak of th e a m b ie n t so u n d a u to s p e c tru m .
1.2
T h is W ork
T he im aging problem a t th e s e low -audio frequencies is form idable. T h e w av elength A a t •500 Hz is a p p ro x im a te ly 3 m . an d a conventional im ag in g sy stem will have a diffraction- lim ited s p a tia l reso lu tio n b o u n d e d by the “ R aleigh resolution” o f a b o u t A /2 . In p ra c tic e , this im plies t h a t th e s u rfa c e fo o tp rin t of a co n v en tio n al im ag in g s y s te m , positio n ed dozens o f m e te rs below th e su rfa c e as in th e tw o previous im aging e x p e rim e n ts, will be far too larg e to resolve th e b reaking region o f all b u t the m ost g a r g a n tu a n wave. T h e sp atial reso lu tio n av ailab le th ro u g h such an a p p ro a c h is u tte rly in a d e q u a te for th e size o f w hitecaps typically e n c o u n te re d in co astal lo ca tio n s.
.A. principle fe a tu re o f th is work is th erefo re th e design an d d e v e lo p m e n t of a te c h nique t h a t w ould p rovide s a tis fa c to ry sp atial reso lu tio n a t these low er a u d io frequencies. N um erous trade-offs were req u ired . T he h y d ro p h o n e a rra y was m oved fro m the far-field to the n earfield, a t th e c o st o f su b je c tin g th e sy ste m to th e h a rs h e r n e a rs u rfa c e e n v iro n m ent an d th o ro u g h ly c o n ta m in a tin g the d a ta w ith m ooring noise. T h e sy s te m 's u p p e r frequency w as set a t 2 kH z. yielding a sp a tia l reso lu tio n on th e s u rfa c e o f less th a n 0.2 m. at th e co st o f a u n ifo rm ly spaced sensor p a tt e r n . In fo rm atio n a c ro ss tw o o c ta v e s in frequency w ere com bined to e lim in ate sp a tia l aliasin g due to th e se n so r sp a rse n e ss, a t
t h e cost of fo reg o in g tru e single frequency im ages.
T he sy ste m w as deployed th re e tim es, in e x p e rim e n ts labeled herein as S U R F E X I. II a n d III. T h e first tw o d e p lo y m e n ts w ere essen tially "shake-dow n" tricds: th e im a g in g d a t a come from S U R F E X III.
T he e x p e rim e n t u ltim a tely y ield ed never-before-seen im age-like in fo rm a tio n a b o u t b reak in g regions a t frequencies n e a r th e peak of th e ra d ia te d a u to s p e c tru m . T h is in fo rm a tio n clearly show s an a c o u stic a lly a c tiv e sh a p e w ith roughly th e sa m e e lo n g a tio n displayed by w h ite c a p s . m oving d o w nw ind, a n d o rie n te d co nsistently w ith th e w ind d i rec tio n . T h e te c h n iq u e used h e re also provides a n e s tim a te o f the fre q u e n c y -d e p e n d en t so u rc e s tr e n g th levels of th e b re a k in g region, w hich h a d not been achieved in p rio r im ag in g e x p e rim e n ts.
T he tec h n iq u e, however, falls s h o rt of im aging a t frequencies below th e p eak o f th e rad iated a u to s p e c tru m . T h a t p roblem is really to u g h ! .Adequate s p a tia l reso lu tio n sim p ly could n o t be achieved a t w avelengths m any tim es g re a te r th a n th e e x p e c te d dim ensions o f fe a tu re s of in te re s t. T h e d a ta can be used, however, to m ake som e inferences re g a rd in g th e poorly u n d e rs to o d source m ech an ism in this regim e
1.3
S u p er reso lu tio n
T h e term " su p e rre so lu tio n " in th e title is c e rtain ly an e y ecatch er. b u t it also carries th e c a ch e t of a b u z z w o rd . It is th e re fo re p ru d e n t to arg u e w hy its usage here is a p p ro p ria te .
.A. quick su rv e y o f th e lite r a tu r e does not provide a precise definition o f s u p e rre so lu tio n . It a p p e a rs to have been used originally to d e scrib e th e process o f a n a ly tic a lly e x te n d in g th e F o u rie r tra n sfo rm o f a b a n d lim ite d im age beyond th e b a n d lim its ( Black- ledge et al. [15]). T h e im aging tec h n iq u es p u rsu ed here do not fall u n d e r t h a t s tr ic t d efin itio n . T h e te r m " su p e rre s o lu to n " . how ever, h as com e to signify a m uch w ider class of im age e n h a n c e m e n t tec h n iq u es. R ichardson a n d M arsh ([115]) sug g est t h a t s u p e rre so lu tio n defines any im a g in g process t h a t yields a " b e tte r" im age th a n som e
c o rre sp o n d in g “co n v e n tio n al" im ag in g process. T he im aging techniques used in this p ro je c t satisfy th is c rite r ia th re e w ays:
1. P ositioning th e a rr a y so t h a t sources a re located in th e nearfield o f th e array. .\s discussed in c h a p te r four, th e p o in t spread function (P S F ) o f th e nearfield a rra y in S U R F E X III has b e tte r resolution th a n th e PS F o f th e a r r a y used in S U R F E X II a n d provides m uch b e tte r resolution th a n would an a r r a y deployed a t th e c o n v e n tio n al d e p th s used by D ing and F arm er [.37]. C ro w th e r a n d H ansla [29] or E pifanio a n d B uckingham [44] .
T h u s, th e choice alone of a nearfield a rra y over a far-field a rra y confers a b e tte r resolving c a p ab ility .
2. Using im age p rio rs. T h e use o f im age priors is described m ore fully in c h a p te r four. b u t. briefly, an e n tro p y p rio r ( t h a t is. a d a ta -in d e p e n d e n t m o d el enforcing a m onotone im a g e ) p erm its th e reso lu tio n of closely spaced point so u rc e s which can n o t o th erw ise be resolved using conventional d iffraction-lim ited tech n iq u es. R eco n stru ctio n s using e n tro p y ( th e “m axim um e n tro p y m eth o d " o r M E M ) seem to be a u to m a tic a lly qualified for su p e rre so lu tio n s ta tu s (P ress et al. [114]). M EM rec o n stru c tio n s have been very successful in radio astronom y.
It is not clear, th o u g h , th a t e n tro p y is th e best prior for sm o o th e x te n d e d sources t h a t lack th e v a st differences in lu m in o sity th a t a re in h ere n t in a s tro n o m y . Very recently. B riggs [21] has su g g ested th e use of non-negative le a st-sq u a re s - i.e.. essentially a p rio r th a t enforces n o n -n eg a tiv ity - cis an a d e q u a te p rio r for high- fidelity im ag in g o f ex ten d ed so u rces. Inasm uch as his results a re c o m p a ra b le to M EM im ages in te rm s of reso lu tio n , no n -n eg ativ ity c o n s tra in ts , w hich a re used in this p ro je c t, m ay provide im ages w ith “resolution" higher th a n t h a t available w ith co n v en tio n al processing.
3. Using a p a ra m e tric schem e t h a t p e rm its a “b e tte r fittin g " im age so lu tio n th a n can be achieved o th e rw ise . H aacke. L iang a n d Izen [.55] d e m o n s tra te d t h a t a piecewise
lin ear m odel achieved a b e tte r fit to a piecew ise linear source th a n a c o n v e n tio n a l ( n o n p a ra m e tric ) m odel. T h e im p rovem ent was clearly re la te d to th e u se o f a m odel t h a t m odeled th e source very well. T his was also one o f few a p p ro a c h e s th a t defined reso lu tio n in term s o f an overall fit to an e x te n d e d im age. In th 's sense, a source is b e tte r resolved by a p a ra m e tric im age w hen th e m o d el is an a c c u ra te re p re se n ta tio n o f th e original so u rce. .A. p a ra m ete riz e d source m o d e l is used to a c cu m u la te so u rc e configuration s ta tis tic s in c h a p te r seven.
1.4
T h e sis O rga n ization
T h e essential background for th e im aging a lg o rith m used in th is w ork is d ev e lo p e d over th ree c h a p te rs . C h a p te r tw o se ts fo rth the n a tu r e o f th e p h en o m en a to be im a g e d , a n d reviews th e accep ted a n d sug g ested source m echanism s. C h a p te r th re e e s ta b lis h e s a m a th e m a tic a l m odel for th e source, including four key assu m p tio n s, a n d briefly c o n s id ers the signal p ro p a g a tio n issues. C h a p te r four m erges the source a n d signal p r o p a g a tio n m odels to a rriv e at a b ro a d b a n d im aging s ta te m e n t for producing resolved u n a m b ig u o u s im age-like in fo rm atio n ov er th e frequency b a n d 600 Hz to 2000 Hz. C h a p te r fo u r also has a brief discussion of c o n v en tio n al im aging techniques and why th ese te c h n iq u e s fail in this p a rtic u la r case.
C h a p te r five provides a b rief description o f th e salient fea tu re s o f th e S U R F E X III d ep lo y m en t. .-Attention is d ire c te d tow ards th e d a ta collected d u rin g a d ay-long s to rm on 4 /1 3 /9 5 .
C h a p te r six outlines th e basic signal processing needed to g e n e ra te b ro a d b a n d im ages a n d provides ex am p les from th e d a ta set o f 4 /1 3 /9 5 . C h a p te r seven e x te n d s th e analysis o f c h a p te r six by in te rp re tin g th e im ages in term s of o ce an o g ra p h ic p a r a m e te r s such as w ind d irection a n d b reak er source s tr e n g th level. Som e a d d itio n a l m o d e ls o f th e low er-frequency m echanism a re considered.
T h e re axe n u m e ro u s a p p e n d ic e s, involving m o stly signal p ro cessin g o r h a rd w a re d e ta ils. O f n o te is a p p e n d ix I. w hich co n tains a “g a lle ry " o f s trip c h a rt-lik e plots a n d a co m p en d iu m o f im a g in g resu lts for every wave a n a ly z e d in c h a p te r seven.
C h a p te r 2
S ou rce P h y sic s
Before tac k lin g th e im aging p ro b le m , it would be p ru d e n t to consider th e phenom ena we expect to ob serv e via th e im aging process. T h e m a th e m a tic a l m ach in ery of p a s sive im aging req u ires sources: upon fu rth e r e x a m in atio n , however, th e definition o f a “source” proves to be an elusive m a tte r . T his is not an issue in e le c tro m ag n e tic rad io m et ry or sp e ctro sc o p y , w here r a d ia n t sources have been used for m ore th a n a century and are quite well u n d e rsto o d . In c o n tra s t, th e acoustical rad ia tio n from breaking waves has a ttr a c te d significant research in te rest only w ithin th e last decad e. F u rth erm o re, th e use of a c o u stic a l ra d ia tio n to s tu d y th e breaking process itself is even m ore recent. .A.nd while th e acou stical p ro p e rtie s of th e breaking region are a n tic ip a te d to be related to the h y d ro d y n am ics of th e b rea k in g process, this relatio n sh ip is still, d esp ite a flurry o f recent in v e stig a tio n , not y e t fully und ersto o d .
T his th e re fo re m otivates a b rie f review of the sound g e n eratio n m echanism of b rea k ing waves. T h e rôle of b u b b les is addressed in m ore d e ta il in section 2 .1, including an e x am in atio n o f “collective o sc illa tio n s” . In section 2 .2, th e key fea tu re s o f th e breaking process t h a t a p p ly to sound g e n e ra tio n a re reviewed. In section 2.3, th e essential fea tu res of b re a k in g waves and su b su rfa ce bubbles are com bined to p ro d u c e a definitive pictu re of th e so u rce physics — th a t is, th e principle n a tu r e of th e p h en o m e n a to be im aged.
2.1
B u b b les
Bubbles play an essen tial rôle in virtually ev e ry aspect of sea su rfa c e sound: th e re fo re , any c o n sid eratio n of sources m u st necessarily involve a c o n sid e ratio n o f th e b u b b les them selves.
An individual bu b b le a c ts like a simple m ass-sp rin g oscillator, w ith th e highly co m pressible in te rn a l gas a c tin g like th e spring, a n d th e fluid e x te rn a l to th e b u b b le p ro viding th e m ass. If som ehow “stru c k " by a n im pulsive force, th e bubble will s e ttle into free decaying radial (o r volum e) oscillatio n s a t a resonant frequency /o inversely related to its quiescent ra d iu s Rq. A rguing t h a t th e internal gas behaves a d ia b a tic a lly .
M innaert [99] derived th e relatio n sh ip
^ ° " 2xiüoV
which is now generally know n as th e M in n a e rt form ula (L eig h to n [78]. section 3 .2 ). Here. 7 is th e po ly tro p ic gas c o n s ta n t. For o rd in a ry w ater (p = 1000 k g /m ^ ) a n d a tm o sp h eric pressure P,o, th is e q u a tio n specifies th e relationship show n in figure 2.1.
T he d am p in g forces t h a t cause bubble o scillations to decay w ere originally inves tig a te d by Devin ([33]) w ho show ed the d a m p in g coefficient to b e th e sum o f viscous d issip atio n , acoustic ra d ia tio n resistance a n d th e rm a l dissipation. E x p e rim en ta l o b s e r vations by S tra sb e rg [125] a n d UpdegrafF [134] su ggested decay c o n s ta n ts o f o rd e r o f a t m ost tens o f m illiseconds for th e sizes of b u b bles ordinarily e n c o u n te re d u n der n a tu r a l conditions.
W hile bubbles v ib ra tin g in th e radial o scillatio n m ode — w hich is th e d o m in a n t ra d ia tin g m ode — can th e re fo re be identified a s sim ple m onopole sources o f d a m p e d outgoing spherical harm o n ic so u n d waves, it is a lso relevant to c o n sid er how su ch b u b bles are in itially forced. T w o significant ways a r e discussed in sectio n s 2.1.1 a n d 2.1.2 below. .A rela te d problem is th e acoustical in te ra c tio n of m u ltip le bubbles in. e .g .. a sw arm . In th is case, th e b u b b les no longer o sc illa te independently. T h is issue c o n s tra in s th e choice o f analysis b a n d a n d is therefore o f d ire c t concern to th is w ork, a n d so is
N X >. Ü
c
Œ) 3cr
0c
CBc
o
« 0 q u i e s c e n t b u b b le r a d iu s [m]Fi g u r e 2.1: T h e M in n a e rt re la tio n sh ip . R esonant frequency v ersus b u b b le ra d iu s. 1 a tm o sp h e re , n o m in a l den sity of w a te r.
tre a te d in m ore d e ta il in section 2 .1.3.
2.1.1
B irth in g W ails
M in n aert and la te r S tra s b e rg p rovided e x p e rim e n tal c o n firm atio n o f E q .(2 .1 ) by lis te n ing to the so u n d p ro d u ce d by o sc illa tin g bubbles. M oreover, by u sin g a c o m b in a tio n o f high-speed p h o to g ra p h y a n d a c o u stic m ea su rem e n ts. S tra s b e rg w as a b le to infer t h a t , a t th e m om ent o f pinch-off (i.e., w h en th e bubble d e ta c h e s from th e in fla tio n nozzle to becom e an in d e p e n d e n t b o d y ), uneven surface forces over th e b u b b le w all a re su d d e n ly freed to induce a variety of v ib ra tio n a l m odes. He th e n show ed t h a t th e z e ro th -o rd e r o r radial m ode w as th e m ost efficient, so th a t th e b u b b le will e sse n tia lly “rin g " a t th e frequency p re d ic te d by E q.(2.1).
T h e a c o u stic s ig n a tu re o f a new ly -created (o r “n e w b o rn ") b u b b le is now g e n e ra lly know n as the “b irth in g wail" (C ru m [30]).
governed by th e in h om ogeneous wave e q u a tio n I
V ‘ p { T . t ) - = f ( t ) é { T - T o ) (2 .2 )
w here cq is th e p h a se speed in the s u rro u n d in g fluid. E q .(2 .2 ) is valid e v e ry w h e re excep t w ithin so m e sm all neighborhood o f ro (i.e .. except w ith in the b u b b le its e lf). T h e inhom ogeneous te rm , w hich includes t h e so u rce “a m p litu d e " / ( ( ) . re p re s e n ts th e forcing due to th e bubble.
2.1.2
A d ult Scream s
.A.ccording to L ight hill ([80]. section 1.10), th e u n s te a d y p ressu re flu c tu atio n s c h a ra c te ristic of tu rb u le n c e are also sources o f s o u n d . T h ese flu c tu a tio n s a p p e a r as a n in h o m ogeneous te rm in th e governing wave e q u a tio n a n d are m a th e m a tic a lly e q u iv a le n t to a volum e d is tr ib u tio n of q u a d ru p o le so u rc e s. C rig h to n an d Ffowcs-VVUliams [27] su g g e ste d th a t th e presen ce o f bubbles can "‘e n h a n c e " (i.e., am p lify ) th e ra d ia te d so u n d : th e q u a d ru p o le m echanism is in h erently in efficient, b u t m ight couple energy in to th e bu b b les, which th e n ra d ia te using the m o re efficient m o n o p o lar m ode. T h e o re tic a lly , th e bubbles could e n h a n ce th e tu rb u le n t p re s s u re a u to s p e c tru m by a factor of (c o /c e ff )"*. w here cq is th e s o u n d speed in th e pure liq u id a n d Cgg is th e effective or “b u lk " so u n d sp eed in th e b u b b ly tw o -phase flow (see se c tio n 2 .1 .3 ). As an ex am p le, a 1% volum e
fractio n of air (i.e ., a void fractio n of 1%) w o u ld am plify th e p re s s u re a u to s p e c tr u m by •50 dB .
In v estig a to rs (K o rm a n . Roy and C ru m [71], H ughes an d K o rm a n [61], K o rm a n a n d C ru m [70], K olaini a n d G oum ilevski [6 6], K o lain i a n d H obbs [67], Kolaini, p riv a te co m m u n ic a tio n ) h ave so u g h t to verify this p re d ic tio n by using s u b m e rg e d jets in la b o r a to r y ta n k s . .A h y d ro p h o n e m easures th e p ressu re a u to s p e c tr u m r a d ia te d from th e j e t region before a n d a fte r bub b les a re released in to t h e flow. R esults in d ee d in dicate t h a t th e r a d ia te d a u to s p e c tr u m is am plified by b u b b le s , a lth o u g h not to th e ex ten t e x p e c te d .
" a d u lt" , b u b b le s, w hich have existed long en o u g h for th e ir b irth in g wails to d ie a w ay ( requiring a t m o st ten s of m illiseconds), m ay be stim u la te d in to re-em ission by t h e local pressure flu c tu a tio n s c h a ra c te ristic of a tu rb u le n t flow. C ru m [30] has te rm e d th e s e acoustic s ig n a tu re s "ad u lt sc re a m s". T h e r a d ia te d acoustic field p ( r . t ) of a tu r b u le n t tw o-phase flow is th e n governed by [49]
V ~ p { r . t ) j ^ ^ p ( r . t ) = - m ( r . t ) — d { r . t ) — - — ( 2. 3)
eg o t - a x i d x j
T he in h o m o g en eo u s term s a re : m (r. t). th e m o nopole sources, rep resen tin g th e b u b b le s . th e d ip o le sources, a n d d ^ T , , j / d x i d x j . th e q u a d ru p o le te rm , involving th e excess m o m en tu m flux te n so r r ,.j. T h e m onopole so u rces, how ever, overw helm the d ip o le a n d q u ad ru p o le so u rc e s, so th a t th e equation governing the aco u stic field is e sse n tia lly th e sam e as t h a t in section 2.1.1.
2.1.3
G an g R um bles (C ollective O scillation s)
W hile a th e o ry involving th e " b irth in g wails" o f individually re so n a tin g n e w ly -c re a te d bubbles seem s to account satisfacto rily for th e so u n d g e n e ra te d by breaking w aves a t frequencies a b o v e a b o u t 400 Hz. it appears in a d e q u a te a t lower frequencies: m e a s u re m ents of th e rad ii o f bubbles produced in spilling flows in la b o ra to ry tan k s, u n d e r g e n tle w aterfalls a n d a t-s e a generally do not show evidence of bubbles w ith radii large e n o u g h to possess a re s o n a n t frequency below a b o u t 400 H z .' T w o possibilities exist:
• L arge b u b b les of these sizes are in fact c re a te d du rin g e n tra in m e n t but have th u s far ev ad ed detectio n :
• Som e a lte rn a tiv e m echanism is g e n eratin g so u n d below a b o u t 400 Hz.
T h e first p ossibility has largely been d isc o u n te d because th e rise tim es o f la rg e bubbles a re q u ite s h o rt, im plying th a t they w ould not be su b m e rg e d long e n o u g h to
b u b b le w ith a re s o n a n t fre q u e n c y n ear 200 Hz, for e x a m p le , w ould h a v t by E q .(2 .1 ) a r a d i u s o f 16 m m : n e a r 100 H z, a ra d iu s o f 32 m m .
m ake a m ajo r c o n trib u tio n to the ra d ia te d sound. In a d d itio n , b ro a d b a n d m e a su re m e n ts by HoUett [60] o f a c o u stic a u to s p e c tra ra d ia te d from break in g w aves show t h a t th e low-frequency c o n trib u tio n reaches its ma_ximum la te in th e dev elo p m en t s ta g e o f th e break er, and th is to o is inconsistent w ith th e s h o rt e x p e c te d lifespans of larg e b u b b les.
.A. popular th e o ry for an a lte rn a tiv e m echanism involves collective o s c illa tio n s of th e entrained bu b b le cloud. In this th eo ry , the in d iv id u al bubbles in a c o m p a c t region o f high bubble d e n sity oscillate in -p h ase (i.e.. “collectively" I. b u t a t a m uch lower
frequency then th e n a tu r a l resonant frequencies o f th e individual b u b b les. In th e sp irit o f earlier n o m en c latu re, th is acoustical sig n a tu re m ight be term ed a “g a n g ru m b le ".
W hile this th e o ry seem s to a c co u n t for a n u m b er o f features o b se rv e d in low- frequency acoustic ra d ia tio n , it m akes th e identification o f th e “so u rc e " a bit m ore su b jectiv e. .And since th e prim ary g oal of a passive im aging sy ste m is to iden tify th e
source, it is p erh ap s w orthw hile to ex a m in e the problem o f collective o scillatio n s a bit
m ore closely.
T h e b u b b l e c o l u m n e x p e r i m e n t Yoon et al. [143] p erform ed th e p a ra d ig m a tic collective oscillation e x p e rim e n t: a re g u la r colum n o f b ubbles was g e n e ra te d in a w a te r ta n k by pum ping a ir th ro u g h a n u m b er o f hollow needles a t th e b o tto m o f th e ta n k , as show n in figure 2.2. T h e radii of th e individual bubbles were m e a su re d o p tically . .A sensor located in th e bubble-free fluid o u tsid e th e bu b b le colum n m e a su re d a ra d ia tin g sound a u to s p e c tru m w ith resonances well below th o se o f th e in d iv id u al bub b les — th e colum n was “ru m b lin g " . .Analysis show ed th a t th e ru m b le resonances c o rre sp o n d e d to th e eigenm odes o f an aco u stic s ta n d in g wave b o u n d ed by (i.e.. tra p p e d in) th e a p p ro x im a te geom etric volum e o f th e bubble colum n. Indeed, w hen m oving a n in te rn a l sensor vertically along th e axis o f th e colum n. Yoon et al. w ere able to p ro d u c e th e definitive p a tte rn s of acoustic s ta n d in g wave m odes a t the ru m b le frequencies.
W a v e p r o p a g a t i o n i n a b u b b l y m e d i u m Since th e bubbles a re ra n d o m ly lo c a te d in th e bubble cloud, th e behavior o f th e acoustic wave in th e b u b b ly region m ay be
— internal p o in t % •o , J O oo OO 3 lll llll llll ll field point
Fi gL'R E 2.2: T h e b u b b le colum n e x p erim en t. B ubbles a re released from a n a rr a y o f
hollow needles show n a t th e b o tto m c e n te r o f th e ta n k . T h e so u n d speed in th e ta n k is Cq. T h e e x p e rim e n t is m o n ito re d with tw o h y d ro p h o n es, o n e o u tsid e th e b u b b le colum n m e a su rin g th e ra d ia tin g a c o u stic field, a n d a n o th e r inside th e colum n for s a m p lin g th e in te rn a l aco u stic field.
%
m
: ; :
F i g u r e 2.3: P ro p a g a tio n o f a plane wave in to a sw arm o f bu b b les. Left: a n in cid en t
plane wave p ,. rep re sen te d by rays, p ro p a g a te s to th e rig h t th ro u g h a fluid w ith sound sp eed Cq. R ight: th e c o h e re n t wave (p) inside th e b u b b ly fluid has th e s a m e ra d ia n frequency b u t a different w avelength th a n th e incident p la n e w ave p,.
viewed as wave p ro p a g a tio n in a ra n d o m m edium . C o n sid e r th e p ro p a g a tio n o f a plane wave from a bubble-free fluid halfspace in to a sw arm o f b u b b le s, as show n in figure 2.3. Each bubble wUl s c a tte r a p o rtio n o f th e acoustic field in cid en t on it. T h e form al solution for th e aco u stic field p ( r) a t a p o in t r is (M orse a n d In g a rd [103], section S.2)
p(r) = p . ( r )- r ^ y y ( p(ro)Vo5(ro. r) - 5 (ro, r) Vop( ro)) • dSo (2.4)
w here 5 , is th e surface of th e b u b b le, i = 1 .. ..V. T h is e q u a tio n is valid ev ery w h ere, in th e p u re fluid o r bu b b ly fluid, e x c e p t inside or on th e su rface o f a b u b b le. T h e te rm <7( r o .r ) is the G reen fu nction sa tisfy in g th e asso c ia te d inhom ogeneous e q u a tio n
V^g(To. T) -t- Argg(ro.r) = - ^ ( r o - r)
a n d the Som m erfeld ra d ia tio n b o u n d a ry condition a t infinity. T h e te rm fcQ = ~j/cq.
Eq.(2.4) is an inhom ogeneous in te g ra l equation; th e inhom ogeneous te rm is pi ( r ) .
rep resen tin g a field lau n ch ed by a so u rc e lo cated a t n e g a tiv e infinity. C o n sid e rin g only th e bubbly region. E q .(2 .4 ) c o rre sp o n d s to th e problem o f solving for th e field o beying th e hom ogeneous H elm holtz e q u a tio n
V^p( r ) 4- k^pi r ) = 0 (2.3)
w ith prescribed b o u n d a ry c o n d itio n s on th e surface o f e a ch bubble.
For th e lim iting case w here th e b u b b les can be c o n sid ered isotropic p o in t s c a tte re rs . a n d for in terb u b b le d ista n c es m uch less th a n an aco u stic w avelength. F oldy [50] showed t h a t th e ensem ble average o f th e su m o f sc a tte re d c o n trib u tio n s from a la rg e n u m b e r of ran d o m ly located s c a tte rs yields a “c o h e re n t wave" ( p (r)) in th e bubbly region o beying
V 2 (p { r))4 -fc 2 ^ (p (r)) = 0 (2.6 )
w here k ^ ^ is th e effective w av en u m b er given by k^g = ■ U sing term in o lo g y
from M orse an d In g a rd [103]. Peff is th e effective d e n sity a n d is th e effective com pressibility of th e bubbly fluid. T h e se a re essentially local p ro p ertie s a v e ra g e d over th e
e n tire bu b b le sw a rm . For tw o -p h ase flows w ith a void fra c tio n o f ab o u t L% o r less.
peff % pq. th e d en sity o f th e p u re fluid, b u t K^ff
kq-T h e physical p ic tu re is as follows: a d e te rm in istic wave incident on a ran d o m b u b b ly m edium s c a tte rs so u n d from every b u b b le. T h e bubbles a re n o t them selves p r im a r y sources o f so u n d , b u t secondary sources: th e s c a tte re d ra d ia tio n is therefore se c o n d a ry r a d ia tio n . T hese s c a tte re d w aves in te rfe re d e stru c tiv e ly w ith in th e random regim e in all d ire c tio n s except one: in th e fo rw ard d ire c tio n , th ey in terfere c o n stru ctiv ely a n d sum to p ro d u c e a sto c h a s tic w ave p {r). W h en a p p ro p ria te ly av erag ed . (p (r)) a p p e a rs to be a wave w ith th e s a m e fre q u e n c y as th e d e te rm in istic incid en t w ave, b u t a low er p h ase sp eed, a n d th ere fo re a s h o rte r w avelength.
N ote t h a t n e ith e r E q .(2 .5 ) nor E q .(2 .6 ) have inhom ogeneous te rm s — th e re a re no sources in th e bu b b ly region.
W hile th e aco u stic field in th e b u bble sw arm is a c tu a lly a sto c h a s tic wave, in p ra c tic e th e finite surface a re a o f an a c tu a l h y d ro p h o n e provides a degree o f sp a tia l a v e ra g in g th a t yields an a p p ro x im a te m ea su re Pmeasuredlf ) o f th e co h eren t field (p (r)).
T h e e flfe c tiv e s o u n d s p e e d T h e bulk o r effective so u n d speed properties o f a two- phase flow were d escrib ed by W ood [140]. w ho show ed th a t th e presence of th e b u b bles induces a significant increase in th e “bulk co m p ressib ility " over th e co m p ressibility of th e p u re fluid. U sing a sim ple iso th e rm a l re la tio n for th e bulk d e n sity p^ff, he d eriv ed th e rela tio n
1 d’Pfff ( 1 — d , Pod( 1 - d ) , ,
w here d is th e I'oid frac tion, th e volum e fra c tio n occupied by th e gas a n d c , is th e so u n d sp eed in th e gas. T h is is now know n as W o o d 's e q u a tio n . As an e x a m p le, th e effective so u n d sp eed for various void fra c tio n s is show n is figure 2.4 for one a tm o s p h e re , s ta n d a r d se aw a te r density, a n d cq = 1-500 m s~U T h e effective so u n d speed can clearly
be m uch less th a n th a t in th e p u re fluid — in fa c t, less th a n th e sound speed in th e p ure g as too!
(O TJ 0 m Q. U1 T3 c 3 O CO
1600
1400
1200
1000
800
600
400
200
v oid fra c tio n
Fi g u r e 2.4: E ffe c tiv e so u n d s p e e d ( W o o d 's e q u a tio n ) .
T his effective so u n d speed is significant when c o n sid e rin g p ro p a g a tio n c o n d itio n s in th e ocean because void fractions can range from very sm all ( 1 0“ ^ - 1 0“ *. see ta b le 2.1) in th e background b u b b le layer to q u ite high (up to 100% . L am arre a n d M elville [73. 74]) in th e core o f b rea k in g waves.
Since in p rac tic e th ere is little difference betw een ( p (r )) and Pmeasuredf r)- it is cus to m a ry when d ealin g w ith problem s o f p ro p a g a tio n in a bubbly fluid to consider th e "analogous sound speed pro b lem ". In th e analo g p ro b le m , th e tw o -p h a se flow is re placed by a pure fluid possessing th e bulk so u n d sp e ed as in figure 2.5. T his sim plifies th e tr e a tm e n t to th a t o f a first-year physics p roblem in w ave p ro p a g a tio n .
S o w h e r e ’s t h e s o u r c e ? T he bubble colum n e x p e rirn e n t can th e re fo re be an alyzed by considering th e so u n d speed a n alo g problem , a s in figure 2.6 : th is is th e m eth o d used by Yoon et al. T h e y solved for th e acoustic field s u b je c t to
V * p (r ) -t- kopi r) = 0 e x t e r n a l fluid ( 2 .8 a )
a n d
