A Method for Estimating Sound Speed and the Void Fraction of Bubbles from Sub-Bottom Sonar Images of Gassy Seabeds. T.G. Leighton

Transcription

1 A Method for Etimating Sound Speed and the Void Fraction of Bubble from Sub-Bottom Sonar Image of Gay Seabed T.G. Leighton ISVR Technical Report No 320 December 2007

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3 A Method for Etimating Sound Speed and the Void Fraction of Bubble from Sub-bottom Sonar Image of Gay Seabed T G Leighton ISVR Technical Report No. 320 December 2007

4 UNIVERSITY OF SOUTHAMPTON INSTITUTE OF SOUND AND VIBRATION RESEARCH FLUID DYNAMICS AND ACOUSTICS GROUP A Method for Etimating Sound Speed and the Void Fraction of Bubble from Sub-bottom Sonar Image of Gay Seabed by T G Leighton ISVR Technical Report No. 320 December 2007 Authorized for iue by Profeor R J Atley, Group Chairman Intitute of Sound & Vibration Reearch

5 ACKNOWLEDGEMENTS Thi work i funded by the Engineering and Phyical Science Reearch Council, Grant number EP/D000580/1 (Principal Invetigator: TG Leighton). The author i grateful to Jutin Dix for providing the background data to figure 3 (water depth, and the denity and ound peed of the bubble-free ediment) and to Gary Robb for ueful comment on the manucript. ii

6 CONTENTS ACKNOWLEDGEMENTS... ii CONTENTS...III ABSTRACT... iv LIST OF FIGURES... V 1 INTRODUCTION EFFECTIVE MEDIUM MODEL USE OF THE EFFECTIVE MEDIUM MODEL TO INTERPRET SUB-BOTTOM PROFILES RESULTS CONCLUSIONS APPENDIX REFERENCES iii

7 ABSTRACT There i increaing interet in the effect of bubble in gay ediment. Thi i, firt, becaue of the impact thoe bubble have on the tructural integrity and load-bearing capabilitie of the ediment; econd, becaue the preence of bubble can be indicative of a range of biological, chemical or geophyical procee (uch a the climatologically-important flux of methane from the eabed to the atmophere); and third, becaue of the effect which the bubble have on any acoutic ytem ued to characterie the ediment. For thi reaon, a range of method have been invetigated for their ability to etimate the bubble population in the eabed. Within uch a range, there will a mix of advantage and limitation to given technique. Thi report outline a very baic method by which an obervation which have already been taken for other purpoe (ub-bottom profile) may be ubjected to a rapid analyi to obtain an etimate of the effect of bubble on the ound peed in the ediment, and from there to provide a rapid preliminary etimate of the void fraction of bubble preent (auming quaitatic bubble dynamic). Thi approach i not meant to compete with large-cale field trial which deploy pecialit equipment to monitor ga bubble in ediment, but rather to provide a method to exploit archived ub-bottom profile, or to urvey a large area rapidly with commercial equipment from a mall veel, in order to obtain an etimation of the local void fraction preent, and their location and extent in three dimenion. iv

8 LIST OF FIGURES Figure Page Figure 1. Schematic of the effect on ound peed of variou monodipere bubble population of air in water (all bubble are aumed to have an equilibrium bubble radiu of 0.1 mm). Thi chematic i generated through qualitative conideration of the form of equation (9), auming that all the non-gay medium contain only water at 1 atmophere tatic preure. 4 Figure 2. (a) The meaured bubble ize ditribution found in the ocean [Phelp and Leighton 1998], from which the author (and tudent SD Meer) calculated (b) the phae peed and (b) the extra aborption which the addition of bubble generate. 5 Figure 3. A chirp onar image, howing a cro-ection of the eabed (maximum penetration approximately 20 m) in Strangford Lough, Northern Ireland. Reproduced by permiion of Southampton Oceanography Centre (J.S. Lenham, J.K. Dix and J. Bull). The two-way travel time from the ource to the top of the eabed wa 20 m, from which the depth of the water wa etimated to be 15.5 m (the ource wa ~0.5 m below the water urface). 7 Figure 4. Schematic repreentation howing power in returned ignal (the darker grey, the more power) a a function of time (the datum correponding to tranmiion of the pule). 8 Figure 5. Reproduction of Figure 3 with location label added (ee text for detail). 11 Figure 6. The etimated ound peed at the location labelled on Figure 5. Serie1 refer to the ound peed averaged between the top of the eabed and layer 1 (aumed to have a contant depth of 5.1 m). Serie2 refer to the ound peed averaged between the top of the eabed and layer 2 (aumed to have a contant depth of 7.4 m). 11 v

9 Figure 7. The etimated void at the location labelled on Figure 5. Serie1 refer to the void fraction vertically-averaged between the top of the eabed and layer 1 (aumed to have a contant depth of 5.1 m). Serie2 refer to the void fraction vertically-averaged between the top of the eabed and layer 2 (aumed to have a contant depth of 7.4 m). 13 Figure 8. Greycale chematic of the time hitory of the acoutic return from a layered 16 gale eabed. The echo which i received at time t b occur from a layer (termed b ) which i at depth d b below the eabed, and the echo which i received at time t c occur from a layer (termed c ) which i at depth d c below the eabed, in ga-le condition. Table Table 1. Etimated parameter for location A to G in layer 1 and layer vi

10 1 Introduction Marine ediment containing ga bubble occur at many location [Judd and Hovland, 1992; Fleicher et al., 2001]. They are important, firt, becaue of the impact thoe bubble have on the tructural integrity and load-bearing capabilitie of the ediment [Wheeler and Gardiner, 1989; Sill et al., 1991]; econd, becaue the preence of bubble can be indicative of a range of biological, chemical or geophyical procee (uch a the climatologically-important flux of methane from the eabed to the atmophere [Judd, 2003]); and third, becaue of the impact which the bubble have on any acoutic ytem ued to characterie the ediment [Robb et al., 2006, 2007a, 2007b; Leighton et al., 2007a, 2007b]. When driven by an acoutic field, a ga bubble urrounded by a uitable hot material act a a nonlinear ocillator (which tend to linear dynamic at low pulation amplitude). It exhibit a pronounced breathing-mode reonance uch that, when driven at frequencie much le than thi reonance, it repone i tiffne-controlled, and the preence of bubble reduce the ound peed (tending to quai-tatic condition at very low driving frequencie). When driven at frequencie much greater than reonance, the bubble repone i inertiacontrolled, and the preence of bubble tend to increae the ound peed, the effect decreaing with increaing frequency [Leighton, 1994]. Whilt there i a coniderable body of work in the literature on the theory of acoutic propagation in marine ediment, the incorporation of ga bubble into uch theorie i done with the incluion of aumption which everely limit the applicability of thoe model to practical ga-laden marine ediment. A a reult, uch theorie are limited in term of which component of the above behaviour they can decribe [Leighton et al., 2004]. The theorie mot frequently ued include modified verion of the Biot-Stoll Theory [Biot, 1956a, 1956b; Stoll, 1974] and an approach developed by Anderon and Hampton [1980a, 1980b]. The Biot model aume that the bubble doe not affect the ediment tructure (i.e. it only affect the pore fluid propertie). Mot manifetation of the Biot model aume quai-tatic bubble repone [Domenico, 1976, 1977; Andreaen et al., 1997; Hawkin and Bedford, 1992, Gregory, 1976; Herkowitz et al., 2000; Minhull et al., 1994; Smeulder and Van Dongen, 1997]. The aumption of quai-tatic ga dynamic limit the applicability of the reulting theory to cae where the frequency of inonification i very much le than the reonance of any bubble preent. It alo eliminate from the model all bubble reonance effect, which often of are overwhelming practical importance when marine bubble population are inonified. Thi limitation become more evere a ga-laden marine ediment are probed with ever-increaing frequencie [Leighton et al., 2007a, 2007b]. -1-

11 Some verion of the Biot model include a imple harmonic ocillator term for the compreibility of the fluid, which incorporate the inertia, tiffne and damping term relevant to the bubble [Biot, 1962; Stoll and Bautita, 1998]. The acoutic theory of Anderon and Hampton [1980a, 1980b] imilarly aume that only linear, teady tate bubble pulation occur. A a reult, neither cla of theory i applicable to the propagation of field which are ufficiently high amplitude: the ubiquitou aumption of linear bubble pulation become increaingly quetionable a acoutic field of greater amplitude are ued to overcome the high attenuation, and the reulting poor-ignal-to-noie ratio, that are often encountered in marine ediment. Furthermore the aumption of monochromatic teady-tate bubble dynamic i inconitent with the ue of hort acoutic pule to obtain range reolution. A further complication which limit the applicability of model of the dynamic of ga bubble in ediment, i the bubble may not be pherical at all time. It i well-known that there are clae of bubble in ediment which do not behave in thi way (e.g. thoe which bear a cloer reemblance of lab of ga and ga-filled crack, than they do to ga-filled phere [Hill et al., 1992; Anderon et al., 1998; Reed et al., 2005]). Thi report outline the ue of a very imple theory, which model the bubble a noninteracting linear ocillator. An effective medium approach i ued to generate a form of Wood equation. It then ue that theory to demontrate a imple method of etimating the bubble void fraction, which i valid in condition where the bubble are inonified at frequencie much le than the general reonance of the bubble population. 2 Effective medium model Conider a volume V eff of eabed which i conidered to be an effective medium to which parameter pertinent to the ound peed can be aigned. It i conidered to conit of two contituent effective medium: ga, and non-gay material (water plu olid). The volume V eff contain a volume V of non-gay material, and a volumev g of ga (ditributed amongt a population of bubble). Conervation of volume give: Veff = Vg + V. ( 1) -2-

12 Ma conervation i imply expreed by multiplication of the volume with the repective denitie (of effective medium, ρ eff ; and of ga, ρ g ; and where ρ i the patially-averaged denity of all the non-gay component). Ma conervation give: ρ V = ρ V + ρv. ( 2) eff eff g g Under the aumption that each of the three media (ga, non-gay media, and the effective medium of the eabed) conerve ma eparately, the differential of Eq. ( 2) with repect to the applied preure P i, of coure, zero. In an infinite body of either water or ga that contain no diipation, ound peed ( c g and c, repectively) may be defined according to: ( ρ S) P, 2 B ε ε ρε ρ c = = = ε ( ε,g), where S i the entropy and the ubcript ε can refer to application to non-gay medium ( ) or the ga ( g ). Similarly, differentiation of Eq. ( 1) with repect to the applied preure give, with Eq. ( 3), the relationhip between the bulk moduli of the effective medium ( B ga ( B g ), and the bulk modulu of the non-gay material ( B ): eff ( 3) ) and the 1 V 1 V ( 1 β) B V B V B B B g = + = β +, eff eff g eff g ( 4) where β = Vg Veff i the void fraction. Let u define a function ζ eff (which i not an inherent property of the bubble cloud in the thermodynamic ene), equal to the root of the ratio of the bulk modulu of the bubbly cloud to it denity [Leighton et al. 2004], which with Eq. ( 4) give: ζ eff B eff V V eff g V ( 5) = = + ρ eff ρgvg + ρv Veff Bg Veff B 1 β 1 β = + ρβ+ ρ β B B ( 1 ) g g 12 β 1 β = ( ρβ g + ρ ( 1 β) ) + Bg B 12 Equation (5) for the ound peed in a two-phae medium i alo known a Wood equation (Wood 1964), which applie to a upenion (uch a mineral particle in water) or to any medium lacking rigidity, in term of weighted mean of the denitie and the compreibilitie of the two contituent of the material. Clearly it i only a limited model of the real ituation in gay ediment, but it erve for the imple method of interpretation which will be ued in thi paper. -3-

13 Auming that the volume of ga i much le than the volume of non-gay component, equation ( 5) implifie through binomial expanion a follow: ζ 1 β B + eff 2 2 c cbg 1 2 uing ( 3), which through binomial expanion reduce to ζ eff β B c 1 2B g. Furthermore from ( 3), c = B / ρ, ( 8) 2 and P P g 3 / Bg = Vg = V R R Becaue the phae of the ocillation depend on the bubble equilibrium ize and the inonifying condition, the gradient of R / P i a function of the bubble ize, for given inonification condition [Leighton, 2004; Leighton et al., 2004]. The general cae will therefore require that ζ eff in ( 7) be evaluated through an integration over the bubble ize ditribution. For the imple purpoe of the inverion required in thi report, the inverion will be implified through the ue of quai-tatic aumption. ( 6) ( 7) ( 9) 1600 Sound peed (m/) ,000 bubble per cubic metre 10,000 bubble per cubic metre 1400 Bubble-free water Frequency (khz) Figure 1. Schematic of the effect on ound peed of variou monodipere bubble population of air in water (all bubble are aumed to have an equilibrium bubble radiu of 0.1 mm). Thi chematic i generated through qualitative conideration of the form of equation (9), auming that all the non-gay medium contain only water at 1 atmophere tatic preure. The net effect of thi can be een in Figure 1. In quai-tatic condition (near DC), bubble are more compreible than water, and the effect on compreibility i greater than the effect on -4-

14 the denity in determining the ound peed. At DC, a poitive preure caue a decreae in bubble volume, o that R / P in equation ( 9) i negative, cauing a decreae in ound peed. Thi effect increae a the bubble ize approache reonance. In the figure thi occur around 30 khz. However like any ocillator the bubble undergo a phae change of Pi a the frequency increae through reonance (taking them from a tiffne-controlled regime to an inertia-controlled regime). Figure 2. (a) The meaured bubble ize ditribution found in the ocean [Phelp and Leighton 1998], from which the author (and tudent SD Meer) calculated (b) the phae peed and (b) the extra aborption which the addition of bubble generate. In the inertia-controlled regime, the bubble are expanding during the compreive halfcycle of the acoutic pule, and o R / P in equation ( 9) i poitive, cauing an increae in ound peed. At the highet frequencie the acoutic cycle change from compreion to rarefaction at a rate o much fater than the repone time of the bubble (approximately of the order of the period of it natural frequency) that the bubble pulation i very low amplitude, and the effect of the bubble on ound peed i minimal. Whilt the above calculation were for monodipere bubble population (where all bubble have roughly the ame ize), imilar effect can be een in polydipere bubble population (containing a wide range of bubble ize). Conider the figure 2. Part (a) how a bubble ize ditribution, meaured in the oceanic water column, along with the aociated ound peed (part (b)) and the component of attenuation for which bubble are reponible (part (c)). Although a wide range of bubble ize are preent (from at leat micron to -5-

15 millimetre) in the ocean, the population a a whole tend to impart to the ocean characteritic uch that, for frequencie below about 20 khz, the bubble reduce the ound peed to le than that of bubble-free water (~1480 m -1 ), However, for frequencie above about 40 khz, the bubble tend to increae the ound peed (part (b)), returning to the ound peed of bubble-free water at the highet frequencie. The magnitude of the change to ound peed increae the cloer the inonifying frequency i to the critical khz range. The additional attenuation caued by bubble (over and above that which occur in bubble-free water) alo peak in thi range (part (c)). Given thee conideration, therefore, let u return to conideration of how the bubble population can have attributed to it a erie of aumption imple enough to allow a ready inverion, to obtain an etimation of the void fraction from the bubble-induced ound peed perturbation. If the inonification frequency i ufficiently le than the main reonance of the bubble preent (noting from Figure 2 that even the broad ditribution of Figure 2(a) exhibit a main reonance in Figure 2(b)), then R / P doe not vary greatly between the variou bubble in the population [Leighton, 2004; Leighton et al., 2004]. In the linear limit of mallamplitude bubble pulation we have: dr 1 =, dp R ρ (( ω ω ) + 2i β ω) where tot ( 10) β tot i a damping parameter of dimenion time -1, and ω 0 i the circular pulation reonance frequency. [Leighton et al. 2004]. If all the bubble in the population were the ame ize, of radiu R 0, and undergoing linear pulation, then ubtitution of ( 8), (9) and (10) into ( 7) would give: ζ 3βB R 3β B ρ c 1+ c 1 ω ω β ω eff R P 2 R0 ( 0 ) + 2i tot, ( 11) which, when the frequency of inonification tend to much le than the reonance of the bubble preent 1, tend to ζ 3β c eff c 2 2 R0ω0, ( ω << ω ) 0. ( 12) If the bubble reonance frequency can be aumed to reemble the equivalent Minnaert frequency, i.e. 3κ p ω, R0ρ ( 13) 1 The inonification frequency of the chirp ued to obtain Figure 3 ranged from 2 khz to 8 khz in a linear weep of 32 m duration, which therefore will mean that, whilt it i certainly poible that thi condition wa met for mot of the bubble preent, it i unlikely that it wa met for all. -6-

16 where p 0 i the tatic preure at the poition of the bubble, ρ i the equilibrium denity in the effective medium, and κ i the polytropic index of the ga within the bubble, then ( 12) reduce to: ζ eff c βc ρ 2 1 2κ p0 ω << ω0 ( ) ( 14) 3. Ue of the effective medium model to interpret ub-bottom profile Conider Figure 3. It i a chirp onar image, howing a cro-ection of the eabed (maximum penetration approximately 20 m) in Strangford Lough, Northern Ireland [Lenham et al., 1998]. The dark line, which i uually 8-10 m from the top of the frame, indicate the ea floor. Hence the labelled feature are beneath the eabed. Thee include hallow ga depoit in the underwater ediment. The onar cannot penetrate thee, a the majority of the ound i cattered from the ga bubble. A a reult, very little information i obtained from beneath the ga layer. The range i calculated by auming that the ound peed in the water column velocity wa 1480 m -1, and for thi Strangford ection ediment package the ound peed wa 1600 m -1. However before the geological layering feature on either ide of the ga pocket become obcured, they appear to dip to greater depth. If it i aumed that in fact thee feature in actual fact remain at roughly contant depth, then thi perceived dipping could be attributed to a reduction in the ound peed. Figure 3. A chirp onar image, howing a cro-ection of the eabed (maximum penetration approximately 20 m) in Strangford Lough, Northern Ireland. Reproduced by permiion of Southampton Oceanography Centre (J.S. Lenham, J.K. Dix and J. Bull). The depth of the water wa etimated to be 15.5 m. -7-

17 Aume a layer feature i at depth d below the eabed, which i itelf at a depth d w below the onar ource (which i uually cloe to the ea/air interface). The ound peed in the ga-free region of the eabed i c, and in the water column it i c w. From Figure 4, the twoway travel time for an echo from the layer feature i: t = 2( d / c + d / c ) 2 w w ( 15) The term dw and cw can be aigned value with relative eae (a procedure common in bottom profiler), ince the two-way travel time for an echo from the top of the ediment (Figure 4) i: t = 2 d / c 1 w w ( 16) Figure 4. Schematic repreentation howing power in returned ignal (the darker grey, the more power) a a function of time (the datum correponding to tranmiion of the pule). In order to etablih the depth below the eabed at which a trongly reflecting layer generate an echo, the time of interet i that delay from the echo which correpond to the top of the eabed, to the echo from ome region at point, i.e. t t = 2 d / c 2 1 ( 17) uch that the depth of that feature below the eabed i d = c ( t t )/2 2 1 ( 18) Now imagine that the eabed contain a population which reduce the ound peed in the eabed from c to c eff. The two-way travel time from the monotatic ource to that reflecting layer i now: t = 2( d / c + d / c ) 3 w w eff ( 19) -8-

18 and the delay between the echo from the top of the eabed, and the echo from the layer, arriving back at the poition of the ource i: t t = 2 d / c 3 1 eff ( 20) Therefore the perceived depth of that layer below the eabed will now be: d = c ( t t )/ ( 21) Therefore the perceived change in depth of that later i: d d = c ( t t )/2 c ( t t )/2 = c d / c d ( 22) eff The ound peed change in the gay ediment can therefore be calculated from the perceived depth change to be: 1 c = c 1 + d d = c d 2 eff d d2 Equation ( 23) indicate, for example, that if one were monitoring a large-cale ga blanket, then the depth of the perceived top of that blanket may be inaccurately calculated a being deeper than it i in reality, becaue of the preence of a more pare bubble population at hallower depth which reduce the ound peed, but whoe preence i overwhelmed in the onar profile by the catter from the top of the dene ga blanket. Thi effect could be teted by weeping the frequency to tet whether the perceived depth of the top of the ga blanket i frequency dependent, ince at ome frequencie it i poible for bubble to increae the ound peed and o make thi feature appear more hallow than it truly i (an event which would be valuable in etimate the bubble ize ditribution. Thi paper utilie a different cenario, where the image include the region between the edge of a ga layer and a bubble-free region of ediment. The approach provide a quick firtorder technique, but the implicity of it ue i offet by limitation. The aumption of quaitatic bubble dynamic may be violated if the bubble are ufficiently large or the inonification frequencie are ufficiently high. The bubble are furthermore aumed to be pherical and bubble-bubble interaction are neglected. Any bubble-mediated change in the ound peed profile are aumed to occur uniformly along a vertical line in the ediment: change in thi perturbation occur only in the horizontal. Thi will not be a realitic aumption if the ga population varie in the vertical direction (a i almot certain) a well a the horizontal direction, along lengthcale of an acoutic wavelength or greater. The technique can of coure be adapted to account for vertical variation in the bubble population through ue of a varying ound peed along any vertical, although at the cot of adopting aumed characteritic for that variation. ( 23) -9-

19 4 Reult Conidering the label on Figure 5, whereby a equence of point on the plot are indicated with labelled arrow. From the reaoning given earlier in thi report, it i aumed that, at the point labelled B1, C1, D1, E1, F1 and G1, the preence ha ga ha made the perceived depth of layer 1 greater than the actual depth; and at the point labelled B2, C2, D2, E2, F2 and G2, the preence ha ga ha made the perceived depth of layer 2 greater than the actual depth. From thi hypothei, the problem cenario i a follow. It i aumed that the ediment at the line joining A1 to A2 contain no bubble, a not unreaonable aumption given the near-horizontal nature of both of the layer ( 1 and 2 ) at thee location. It i further aumed that the layer labelled A1, B1, C1, D1, E1, F1, G1 i in reality at a contant depth (indicated by the poition of A1 ) below the top of the eabed, but that the perceived dipping of thi layer i caued by a reduction in ound peed in the ediment a a reult of the preence of bubble. Comparing the depth of A1 to each in turn of B1, C1, D1, E1, F1, G1 allow value of d / d 2 to be calculated for the location of each lettered label (Table 1). Similarly, it i aumed that the layer labelled A2, B2, C2, D, E2, F2, G2 i in reality at a contant depth (indicated by the poition of A2 ) below the top of the eabed, but that the perceived dipping of thi layer i caued by a reduction in ound peed in the ediment a a reult of the preence of bubble. Again, comparing the depth of A2 to each in turn of B2, C2, D2, E2, F2, G2 allow value of d / d 2 to be calculated for the location of each lettered label (Table 1). Thi then allow the etimated peed in the gay ediment at that horizontal coordinate to be calculated (Table 1) through ceff = cd / d2 (equation (24)). The reult are hown in Figure 6. If the aumption (ection 3) that the perturbation in ound peed i contant for any given horizontal coordinate were not to be true, the value of ound peed in erie 1 for a particular letter would differ from that etimated in erie 2 for the ame letter (ee the Appendix). Figure 5. Reproduction of Figure 3 with location label added (ee text for detail). -10-

20 Vertically-averaged ound peed (m/) Serie1 Serie2 A B C D E F G Horizontal location Figure 6. The etimated ound peed at the location labelled on Figure 5. Serie1 refer to the ound peed averaged between the top of the eabed and layer 1 (aumed to have a contant depth of 5.1 m). Serie2 refer to the void fraction vertically-averaged between the top of the eabed and layer 2 (aumed to have a contant depth of 7.4 m). Figure 6 how that for the uncertaintie aociated with thi data, it i not poible to prove uch a violation of thi aumption. A the Appendix how, fact that the pacing between the two layer 1 and 2 in Figure 5 i expanded in proportion to the increae in depth, ugget that the ga layer extend from the urface to at leat a deep a layer 2. The Appendix how that, if the ga layer extended imply to a deep a layer 1, but the ediment were ga-le between layer 1 and 2, then the pacing between layer 1 and 2 would remain contant, although both would dip down to greater depth in the eabed a a reult of the change in ound peed which occur between the top of the eabed and layer 1. Having etimated the ound peed at any particular horizontal coordinate, equation ( 14) can be rearranged to etimate the void fraction at that coordinate: 2κ p β c c 1 ( ω << ω0 ) 0 eff 2 ρ c ( 24) Thee void fraction can now be calculated, and will be done o to obtain etimate of the vertically-averaged void fraction between the top of the eabed and each of the layer (1 and 2) in turn. At the time the meaurement of Figure 3 were taken, technique for meauring the denity and ound peed in the ediment were not a advanced a they are today. The value -11-

21 for the denity of the aturated ga-free ediment will be ued for ρ, although clearly with new data it would be better to meaure high. A value of ρ directly, particularly when the void fraction i ρ =2300 kg m -3 for the ilt and clay which are typical of the area wa taken a a firt etimate, uitable for thi preliminary analyi [Richardon & Brigg, 1993]. The atmopheric preure i taken to be 103 kpa (the data were recorded in the lat few day in May 1997). The hydrotatic head of the 15.5 m water column would add to thi a further 152 kpa contribution to the tatic preure. Auming (from the above dicuion) a denity of ρ =2300 kg m -3 kg m -3 for the ediment/water mixture found beneath the eabed, the contribution of the ediment to the tatic preure at the bubble i etimated to be ρgh1 115 kpa at layer 1 (which i aumed to be at depth h 1 =5.1 m below the top of the eabed) and ρ0gh2 167 kpa at layer 1 (which i aumed to be at depth h 2 =7.4 m below the top of the eabed) where g i the acceleration due to gravity (Table 1). Since the inverion will be undertaken to obtain the vertically averaged void fraction between the top of the eabed and each layer in turn, the hydrotatic preure to ue will be that found half-way between the top of the eabed and the repective layer. Therefore the value ued for p 0 when etimating the average void fraction between the top of the eabed and layer 1, will be ( /2) = kpa. Similarly the value ued for p 0 when etimating the average void fraction between the top of the eabed and layer 2, will be ( /2) = kpa. The ound peed c i taken from the ga-free meaurement in Figure 6, which naturally reflect the value of the 1600 m -1 ound peed which had been aumed for thee frequencie for ga-free ediment ued in converting the time erie of the echo into Figure 3. The polytropic index i aumed to be κ =1.3, a value typical for bubble containing the ga methane which are aumed to pulate adiabatically. The reulting void fraction are calculated in Table 1 for the coordinate in quetion, and plotted on Figure 7. The error bar in Figure 6 have not been tranlated onto Figure 7 becaue the aumption in the ediment parameter introduce an unknown uncertainty. -12-

22 Vertically-averaged void fraction (%) Serie1 Serie2 A B C D E F G Horizontal location Figure 7. The etimated void at the location labelled on Figure 5. Serie1 refer to the void fraction vertically-averaged between the top of the eabed and layer 1 (aumed to have a contant depth of 5.1 m). Serie2 refer to the void fraction vertically-averaged between the top of the eabed and layer 2 (aumed to have a contant depth of 7.4 m). Horizontal location Perceived depth of layer 1 (m) Perceived depth of layer 2 (m) Average ound peed between top of eabed and layer 1 (m/) Average ound peed between top of eabed and layer 2 (m/) Average void fraction between top of eabed and layer 1 (%) A Average void fraction between top of eabed and layer 2 (%) B C D E F G Table 1. Etimated parameter for location A to G in layer 1 and layer

23 5 Concluion Thi report outlined a very imple cheme for aeing the ound peed perturbation induced by bubble in the eabed, and for etimating the void fraction and extent of the bubble layer (in term of it penetration depth into the eabed and it horizontal extent in the profile) uing that cheme. If there i a location in an image where it i geologically reaonable and accurate to make the aumption that two layer hould be at contant depth (or, if not, that the lope i known and contant from other data), then the vertically-averaged ound peed perturbation and bubble void fraction between thoe two layer can be etimated. With ufficient layer the eabed may be divided into vertically tacked layer, and the void fraction in each can be etimated, ince the ound peed perturbation i imple the ratio of the actual eparation of thoe layer to the perceived eparation. A uch, given ufficient layer, the profile of ound peed perturbation can be determined at a glance, a can it horizontal variation. The principle of the approach can be extent to three-dimenional profile [Bull et al., 2005a, 2005b; Gutowki et al., 2008]. The implicity of the cheme i bought at a price, in term of the wide ranging aumption that are made about the bubble and ediment. Mot of thoe aumption will be violated to a greater or leer extent by the environmental and inonification condition. Neverthele, the eae with which firt-order environmental data can be gained at little extra effort uing exiting technology and through examination of hitorical record of ub-bottom profile, offer the poibility of making rapid progre rapidly. Thi i ignificant given: (i) the uual interpretation when ga pocket of the ort hown in Figure 4 are viible in a ub-bottom profile i that, other than indicating the preence and location 2 of the pocket, that the preence of ga o dirupt the ub-bottom profile that it everely hinder the ability to analye it at the location of the ga pocket; (ii) the complexity of the acoutical interaction generated by ga pocket in ediment mean that mot experimental meaurement of thee require very complex equipment (including difference frequency onar and CT canner etc.) [Wilken and Richardon, 1998; Anderon et al., 1998; Boudreau et al., 2005; Otrovky et al., 2005]). Thi tudy i of coure in no way intended to complete with the innovative and largecale field trial deigned to meaure at-ea bubble population in ediment. Rather it i a method of exploring the value of retropectively analying pat ub-bottom profile, and 2 Note furthermore that the extent of the ga, a indicated by the void fraction hown in Figure 7, i much greater than the extent of the hadow in Figure 4 which, by viual inpection, one might conider to be the location and extent of the ga pocket. -14-

24 aking what might be determined from routine ub-bottom meaurement which are not pecifically deigned a one-off large deployment. Of coure the aumption in thi model will limit it accuracy. Geological expertie will be required in each cae to ae the likelihood that a layer i in fact horizontal, and that the perceived dipping i due to ound peed perturbation. In the model ued here, the material parameter of the ediment enter only through the term c, and other than thi and the material denity there i no reflection of the complexity of propagation that can occur in uch material (ee Section 1). However becaue the method relie on ound peed perturbation, it i not a enitive to incluion of ome parameter a would be one baed on aborption. Furthermore, whilt improved model for ound peed will be available for ubtitution into thi cheme (and whilt the aumption of quai-tatic dynamic can be replaced uing a more ophiticated inverion routine), the importance of thi report lie in expreing uch a imple cheme for obtaining the void fraction and extent (in the vertical and horizontal) of bubble population in marine ediment. -15-

25 6 Appendix Thi Appendix conider the effect of the violation of the aumption that, at a given horizontal coordinate, the bubble population (when averaged over lengthcale of the order of an acoutic wavelength) i uniform with depth throughout the meaured profile. Thi aumption wa employed throughout the body of thi report, and yet it clearly will be violated in many practical cenario. Figure 8. Greycale chematic of the time hitory of the acoutic return from a layered gale eabed. The echo which i received at time t b occur from a layer (termed b ) which i at depth d b below the eabed, and the echo which i received at time t c occur from a layer (termed c ) which i at depth d c below the eabed, in ga-le condition. Conider Figure 8, which add a extra layer to the chematic hown in Figure 4. In ga-le condition, the two-way travel time to receive an echo from the top of the eabed (at depth d w ) i t = 2 d / c a w w ( 25) Similarly, in ga-le condition, the two-way travel time to receive an echo from the top of the eabed (at depth t = 2( d / c + d / c ) b w w b d w ) a layer at depth d b i ( 26) and the two-way travel time to receive an echo from a layer at depth d c i t = 2( d / c + d / c ) c w w c. Uing an algorithm which produce the onar profile by auming a contant ediment ound peed of c, then from equation ( 26) the depth of the layer ( 27) d b below the eabed i given by: -16-

26 d = c ( t /2 d / c ) b b w w ( 28) The ame algorithm would of coure calculate the depth of the layer (equation ( 27)) a: d = c ( t /2 d / c ) c c w w. However if the region between the top of the eabed and the layer at depth uch that the ound peed there i c eff d c below the eabed ( 29) db contain ga, then the two-way travel time to receive an echo from the top of the eabed (at depth t = 2( d / c + d / c ) ' b w w b eff d w ) a layer at depth d b i ( 30) and the two-way travel time to receive an echo from a layer at depth d c i t = 2( d / c + d / c + ( d d ) / c ) ' c w w b eff b c ( 31). Conequently, if the algorithm which produce the onar profile were to aume a contant ediment ound peed of c, then the perceived depth of layer b ( d ) would be found by replacing t b from ( 30) in equation ( 28) by ' tb to give: d = c ( t /2 d / c ) = c ( d / c + d / c d / c ) = c d / c ( 32) ' ' b b w w w w b eff w w b eff ' However the perceived depth of layer c ( d c ) would be found by replacing t c from ( 31) in equation ( 29) by ' tc to give: ' ' dc = c( tc /2 dw / cw) = c( dw / cw + db / ceff + ( db dc)/ c dw / cw) = c( db / ceff + ( db dc)/ c) = ( db dc) + cdb / ceff. That i to ay, that the depth of the layer i no longer increaed by the ame multiplicative factor c / c eff a before (equation ( 23)). Rather, the depth interval between layer b and c i the ame a it would be in the bubble-free condition, but layer c ha been tranlated downward by the ame abolute ditance a wa layer b. ' b ( 33) -17-

27 Reference Anderon, A. L., and L. D. Hampton (1980a), Acoutic of ga-bearing ediment I. Background. J. Acout. Soc. Am., 67(6), Anderon, A. L., and L. D. Hampton (1980b), Acoutic of ga-bearing ediment II. Meaurement and model. J. Acout. Soc. Am., 67(6), Anderon, A. L., F. Abegg, J. A. Hawkin, M. E. Duncan, and A. P. Lyon (1998), Bubble population and acoutic interaction with the gay floor of Eckernforde Bay, Cont. Shelf Re., 18(14-15), Andreaen, K., P.E. Hart and M. MacKay (1997), Amplitude veru offet modelling of the bottom imulating reflection aociated with ubmarine ga hydrate, Marine Geology, 137(1-2), Biot, M. A. (1956a), Theory of Propagation of Elatic Wave in Fluid-Saturated Porou Solid: 1. Low-Frequency Range, J. Acout. Soc. Am., 28(2), Biot, M. A. (1956b), The theory of propagation of elatic wave in a fluid-aturated porou olid: 2. Higher frequency range, J. Acout. Soc. Am., 28(2), Biot, M.A. (1962), Generalized Theory of Acoutic propagation in porou diipative media, J. Acout. Soc. Am., 34(9), Boudreau, B. P., C. Algar, B. D. Johnon, I. Croudace, A. Reed, Y. Furukawa, K. M. Dorgan, P. A. Jumar, A. S. Grader and B. S. Gardiner, (2005), Bubble growth and rie in oft ediment. Geology, 33(6), Bull, J. M., M. Gutowki, J. K. Dix, T. J. Hentock, P. Hogarth, T. G. Leighton and P. R. White, (2005a) 3D chirp ub-bottom imaging ytem: deign and firt 3D volume. Proceeding of the International Conference on Underwater Acoutic Meaurement, Technologie and Reult, J.S. Papadaki and L. Bjorno, ed. (Crete, 2005) Bull, J. M., M. Gutowki, J. K. Dix, T. J. Hentock, P. Hogarth, T. G. Leighton and P. R. White, (2005b) Deign of a 3D Chirp ub-bottom imaging ytem. Mar. Geophy. Re., 26, 2005, Chapman, R. B., and M. S. Pleet, (1971), Thermal effect in the free ocillation of ga bubble. Tran ASME D: J Baic Eng., 93, Domenico, S. N. (1976), Effect of brine-ga mixture on velocity in an unconolidated and reervoir, Geophyic, 41(5),

28 Domenico, S. N. (1977), Elatic propertie of unconolidated porou and reervoir, Geophyic, 42(7), Fleicher, P., T. H. Ori, M. D. Richardon, and A. L. Anderon (2001), Ditribution of free ga in marine ediment: a global overview, Geo-Marine Letter, 21, Foldy, L. L. (1945), The multiple cattering of wave, Phy. Rev., 67, Gregory, A. R. (1976), Fluid aturation effect on dynamic elatic propertie of edimentary rock, Geophyic, 41(5), Gutowki, M., Bull, J. K. Dix, T. J. Hentock, P. Hogarth, T. Hiller, T. G. Leighton and P. R. White, (2008), 3D High-Reolution Acoutic Imaging of the Sub-Seabed. Applied Acoutic (in pre). Hawkin, J.A., and A. Bedford (1992), Variational theory of bubbly media with a ditribution of bubble ize. 2. Porou olid, Int. J. Engineering Science, 30(9), Herkowitz, M., S. Leviky, and I. Shreiber (2000), Attenuation of ultraound in porou media with dipered microbubble, Ultraonic, 38, Hill, J. M., J. P. Halka, R. Conkwright, K. Koczot, and S. Coleman (1992), Ditribution and effect of hallow ga on bulk etuarine ediment propertie, Cont. Shelf Re., 12(10), Judd, A. G. (2003), The global importance and context of methane ecape from the eabed, Geo-Marine Letter, 23, Judd A. G., and M. Hovland (1992), The evidence of hallow ga in marine ediment, Cont. Shelf Re., 12(10), Leighton, T. G. (1994), The Acoutic Bubble, Academic Pre, London. Leighton, T. G. (2004), From ea to urgerie, from babbling brook to baby can: The acoutic of ga bubble in liquid, Int. J. Modern Phyic B, 18(25), Leighton T G, Meer S D and White P R (2004), Propagation through nonlinear timedependent bubble cloud, and the etimation of bubble population from meaured acoutic characteritic. Proceeding of the Royal Society A, 460(2049), Leighton T G, (2007a), Theory for acoutic propagation in marine ediment containing ga bubble which may pulate in a non-tationary nonlinear manner, Geophyical Reearch Letter, 34, L17607, doi: /2007gl Leighton T G. Theory for acoutic propagation in olid containing ga bubble, with application to marine ediment and tiue, ISVR Technical Report No. 318 (2007b) -19-

29 Leighton, T. G. (2007c), What i ultraound?, Progre in Biophyic and Molecular Biology, 93(1-3), Lenham, J. M., J. M. Bull and J. K. (1998), A marine geophyical urvey of Strangford Lough. Archaeology Ireland, 11, Minhull, T. A., S. C. Singh, and G. K. Wetbrook (1994), Seimic velocity tructure at a ga hydrate reflector, offhore wetern Colombia, from full waveform inverion, J. Geophy. Re., 99(B3), Phelp A D and Leighton T G (1998), Oceanic bubble population meaurement uing a buoy-deployed combination frequency technique. IEEE Journal of Oceanic Engineering, 23(4), Otrovky L. A., S. N. Gurbatov and J. N. Didenkulov, (2005), Nonlinear acoutic in Nizhni Novgorod (A review). Acoutical Phyic, 51(2), Properetti, A. and Y. Hao (1999), Modelling of pherical ga bubble ocillation and onoluminecence, Phil. Tran. R. Soc. Lond. A, 357, Reed, A. H., B. P. Boudreau, C. Algar, and Y. Furukawa (2005), Morphology of ga bubble in mud: A microcomputed tomographic evaluation, Proc. 1t Int. Conf. Underwater Acoutic Meaurement: Technologie and Reult, Heraklion, Crete (2005). Richardon, M. D., and K. B. Brigg (1993), On the ue of acoutic impedance value to determine ediment propertie, Proc. Intitute of Acouic, 15(2) pp Robb, G. B. N., T. G. Leighton, J. K. Dix, A. I. Bet, V. F. Humphrey, and P. R. White (2006), Meauring bubble population in gay marine ediment: A review, Proc. Int. Acout., 28(1), Robb G B N, Leighton T G, Humphrey V F, Bet A I, Dix J K and Kluek Z (2007a), Invetigating acoutic propagation in gay marine ediment uing a bubbly gel mimic. Proceeding of the Second International Conference on Underwater Acoutic Meaurement, Technologie and Reult, J.S. Papadaki and L. Bjorno, ed. (Heraklion, Crete) 2007, Robb G B N, Leighton T G, Humphrey V F, Bet A I, Dix J K and Kluek Z (2007b), Invetigating acoutic propagation in gay marine ediment uing a bubbly gel mimic, ISVR Technical Report 315, Univerity of Southampton. Sill, G. C., S. J. Wheeler, S. D. Thoma, and T. N. Gardiner (1991), Behaviour of offhore oil containing ga bubble, Geotechnique, 41(2),

30 Smeulder, D. M. J., and M. E. H. Van Dongen (1997), Wave propagation in porou media containing a dilute ga-liquid mixture: theory and experiment, J. Fluid Mech., 343, Stoll, R. D. (1974), Sediment Acoutic, Lecture Note in Earth Science 26, Springer-Verlag, New York. Stoll, R. D., and E. O. Buatita (1998), Uing the Biot theory to etablih a baeline geoacoutic model for eafloor ediment, Continental Shelf Re., 18, Vorkurka, K. (1986), Comparion of Rayleigh, Herring, and Gilmore model of ga bubble, Acutica, 59, Wilken, R. H. and M. D. Richardon (1998), The influence of ga bubble on ediment acoutic propertie: in itu, laboratory, and theoretical reult from Eckernförde Bay, Baltic ea. Continental Shelf Reearch, 18(14-15), Wheeler, S. J., and T. N. Gardiner (1989), Elatic moduli of oil containing large ga bubble, Geotechnique, 39(2),