Measurement of air-gun bubble oscillations

Transcription

1 GEOPHYSICS, VOL. 63, NO. 6 NOVEMBER-DECEMBER 1998; P , 14 FIGS., 1 TABLE. Measurement of air-gun bubble osillations Anton Ziolkowski ABSTRACT In this paper, I provide a theoretial basis for a pratial approah to measuring the pressure field of an air gun array present an algorithm for omputing its wavefield from pressure measurements made at known positions in the viinity of the gun ports. The theory for the osillations of a single bubble is essentially a straightforward extension of Lamb s original paper provides a ontinuous, smooth transition from the osillating wall of the bubble to the far-field, preserving both the fluid flow the aousti radiation, all to the same auray valid for bubbles with initial pressures up to about 00 atm 3000 psi or 0 MPa. The simplifying assumption, based on an argument of Lamb, is that the partile veloity potential obeys the linear aousti wave equation. This is used then in the basi dynami kinemati equations to lead, without further approximations, to the nonlinear equation of motion of the bubble wall the wavefield in the water. Given the initial bubble radius, the initial bubble wall veloity, the pressure variation at any point inside or outside the bubble, the algorithm an be used to alulate the bubble motion the aousti wavefield. The interation among air-gun bubbles the resultant total wavefield is formulated using the notional soure onept, in whih eah bubble is replaed by an equivalent notional bubble obeying the same equation of motion but osillating in water of hydrostati pressure, thus allowing the wavefields of the notional bubbles to be superposed. A separate alibration experiment using the same pressure transduers firing the guns individually allows the initial values of the bubble radius bubble wall veloity to be determined for eah gun. An appendix to the paper provides a test of the algorithm on real data from a single gun. INTRODUCTION This paper presents an algorithm for omputing the wavefield of an array of air guns from the minimum number of pressure measurements made at known positions in the viinity of the gun ports. Under normal firing onditions, an air-gun bubble exps to several hundred times the volume of the air gun Ziolkowski, At its maximum size, the bubble is still small ompared with the wavelengths of the pressure wave it generates in the water, it an therefore be treated as a spherially symmetri soure. It is well known that lose to the bubble the main effet is the osillation of the fluid as if it were almost inompressible, while far from the bubble the linear aousti approximation desribes the aousti radiation. The problem is that the inompressible fluid approximation forbids aousti radiation the linear aousti approximation forbids osillations of the bubble, so it is not obvious how to derive the aousti radiation from the bubble osillations that ause it. The radiated pressure from a spherial osillating bubble has been studied sine Rayleigh s paper 1917 has been of interest to exploration geophysiists, espeially for understing the pressure field of air guns. In 193 Lamb wrote, The aurate equations of motion of spherial waves in a ompressible medium are easily formulated, although a solution seems at present hopeless. Two elements of the solution are ritial for our appliation: 1 the equation of motion of the bubble the equations desribing the wavefield outside the bubble. Numerial solutions for these elements have been proposed by a number of authors, but every approah involves several approximations that lead to solutions whose auray is diffiult to quantify. Herring 1941 derived the exat wave equation first derived by Lamb then found the equation of motion of the bubble, whih is nonlinear, by an iterative approah of suessive approximations, starting with the inompressible ase, following the work of Rayleigh Kirkwood Bethe 194 derived the same exat wave equation postulated that the propagation veloity in the moving medium is equal to the speed of sound plus the Manusript reeived by the Editor February 7, 1996; revised manusript reeived Marh 3, University of Edinburgh, Dept. of Geology Geophysis, Grant Institute, West Mains Rd., Edinburgh EH9 3JW, Sotl, UK. anton. ziolkowski@glg.ed.a.uk Soiety of Exploration Geophysiists. All rights reserved. 009

2 010 Ziolkowski partile veloity v. They remarked, An exat determination of or even a rigorous proof that possesses the properties hypothetially asribed to it appears to be prohibitively diffiult. Gilmore 195 used the Kirkwood Bethe hypothesis to derive the equation of motion of the osillating bubble very similar to Herring s, but with orretions to eah term. He also developed a numerial sheme whih, unfortunately, requires further appproximations to alulate the wavefield propagating away from the bubble. Thus the omputed wavefield is of lower auray than the equation of motion of the bubble the Kirkwood Bethe hypothesis itself. All these referenes are ontained in a very omprehensive review paper by Flynn Ziolkowski 1970 uses Gilmore s sheme in the first attempt to model the sound wave produed by an air gun. Keller Kolodner 1956, in a paper that forms the basis of muh work on air-gun bubbles by geophysiists e.g., Shulze- Gattermann, 197; Safar, 1976; Johnston, 1980, quote Lamb s equation in their introdution, attributing it to Herring 1941; but they do not attempt to solve it. They begin their analysis with the assumption that the partile veloity potential obeys the linear aousti spherial wave equation but give no justifiation or referene for this. They then make further approximations that are not quantified to arrive at a nonlinear equation of motion almost idential with the one derived by Herring. Flynn 1975 gives a mathematial formulation of the bubble osillations but uses Herring s 1941 approah of iteration by suessive approximations to derive the equation of motion of the bubble. Vokurka 1986, in a omparison of the different bubble models, onludes that Gilmore s adaptation of the Kirkwood Bethe approximation remains irreplaeable in the studies of shok waves strong pressure pulse propagations. This paper presents a new algorithm that is essentially a straightforward extension of Lamb s original 193 paper hinted at by Lamb whih provides a ontinuous, smooth transition from the osillating wall of the bubble to the far field, preserving both the fluid flow the aousti radiation all to the same auray valid for bubbles with initial pressures up to about 00 atm 3000 psi or 0 MPa. Given the initial bubble radius, the bubble wall veloity, the pressure variation at any point inside or outside the bubble, the algorithm an be used to alulate the bubble motion the aousti wavefield. The interation between air-gun bubbles the resultant total wavefield is formulated using the notional soure onept of Ziolkowski et al The immediate ommerial appliation of this algorithm is the omputation of the wavefield of an air gun array from measurements of the pressure field made with gun-mounted sensors lose to the gun ports. The method of Ziolkowski et al. 198 for the omputation of the wavefield using near-field hydrophones plaed in the linear part of the pressure field is shown to be a speial ase of this more general approah. A sientifi appliation of the algorithm is to look inward to inrease understing of the thermodynamis of osillating bubbles. METHOD Lamb s exat wave equation for the partile veloity potential is derived. This has no known analytial solution, so a numerial solution must be found. Following the analysis of Lamb himself 193, 1945, it is shown that this exat wave equation differs from the linear wave equation by two additional terms that are always very small for normal air-gun bubbles. It follows that, to a very good first approximation, the partile veloity potential an be onsidered to obey the linear wave equation for air-gun bubbles, as in Keller Kolodner This approximation is used to alulate the nonlinear seond-order terms in the basi dynami kinemati equations in the water without further approximation unlike Keller Kolodner, leading to a new formulation of the equation of motion of the bubble an algorithm to alulate the pressure partile veloity everywhere. Thus, the pressure inside the bubble of a single air gun is related to the bubble radius by the initial onditions the equation of motion of the bubble. The pressure partile veloity outside the bubble are related to the onditions at the bubble wall by the basi dynami kinemati equations in the water. The interation among air-gun bubbles ours as hanges in the water pressure on the outside of the bubbles. Thus, a bubble osillating near to another bubble does not osillate in the same manner that it would if the other bubble were not there. Therefore, the pressure wave it generates is different from the wave it generates when it is osillating on its own. The nonlinear osillations are oupled. In an array of air-gun bubbles, eah bubble osillates in the pressure field of the other bubbles, any hange in one bubble affets all the others. Ziolkowski et al. 198 argue that the effet on the bubble osillations of the dynami interation pressure on the outside of eah bubble ould be made by an equal opposite pressure on the inside of the bubble, the notional bubble osillating in water of hydrostati pressure therefore behaving independently of the other bubbles. Sine all these notional bubbles are still small ompared with a wavelength, the pressure wave generated by eah bubble is spherial. The pressure field in the water is then simply the superpositon of the spherial waves from all the notional bubbles plus the refletions of these wave from the water surfae. The notional soure onept thus allows the priniple of superposition to be used even though the equation of motion of eah bubble is nonlinear. MODEL OF A SINGLE BUBBLE The model of a single bubble is illustrated in Figure 1. The osillating bubble is onsidered to be spherial at all times t, with radius Rt internal pressure Pt. The effets of gravity the effets of the free surfae of the water are negleted. The flow in the water is assumed to be invisid purely radial, all quantities in the water are funtions only of time radial distane r from the enter of the bubble. Inside the bubble the pressure is onsidered to be uniform; outside, the pressure is governed by the bubble motion the wave propagation in the fluid. At the bubble wall the pressure is ontinuous, but the pressure gradient is disontinuous, as indiated by the instantaneous pressure profile in Figure 1. BASIC EQUATIONS Following Lamb 193, the water is assumed to be invisid to have pressure p, whih is a definite funtion of the density ρ, suh that the enthalpy h may be desribed as p dp ρ h = p ρ = dρ ρ ρ, 1

3 Measurement of Air-Gun Bubble Osillations 011 where ρ is the undisturbed density the speed of sound is defined by = dp dρ. Sine the flow is irrotational, the partile veloity v is the gradient of a partile veloity potential φ: The equation of motion is v = φ. 3 h = Dv Dt, 4 in whih D/Dt is the total, or Lagrangian, derivative following the partile motion. Equation 4 an integrated to give the dynami equation h = φ v, 5 in whih v is the magnitude of the vetor v. The equation of ontinuity, or kinemati equation, may be written as 1 Dρ ρ Dt From equations 1 6 it follows that Dh Dt = ρ = φ. 6 Dρ Dt = φ. 7 LAMB S EXACT WAVE EQUATION IN SPHERICAL COORDINATES Now, imposing spherial symmetry eliminating h between equations 5 7 yields φ 1 v + φ 1+ r φ 1 φ = 0. 8 r r r r This equation was first derived by Lamb 193 has no known analytial solution. To find a suitable solution requires approximations. The approah adopted here is to preserve all essential features of the fluid flow while maintaining the linear aousti approximation at large distanes from the bubble. The linear aousti approximation is disussed first, then the inompressible flow approximation. It is shown below that the partile veloity potential obeys the linear aousti wave equation to a good first-order approximation for our problem. This solution is then used to alulate the seond-order terms required to speify the fluid flow. LINEAR ACOUSTIC APPROXIMATION When the problem is ompletely linearized the distintion between the material spatial desriptions of the motion vanishes altogether Ahenbah, Thus the total time derivative D/Dt redues to the partial time derivative /, the dynami equation 5 beomes simply h = φ, 9 the kinemati equation 7 beomes h = φ. 10 Eliminating h between equations 9 10 yields the linear aousti wave equation, φ = φ, 11 whih, with spherial polar symmetry, has the form φ r + φ r r 1 φ = 0 1 with the well-known solution φ = 1 r f t r. 13 The exat wave equation 8 is similar to the linear aousti wave equation 1 but ontains two extra terms. The first of these terms is learly small if the partile veloity is small ompared with the speed of sound. Lamb 193 showed this is also true of the seond term. To follow his reasoning, it is neessary to onsider the approximation of inompressible flow. INCOMPRESSIBLE FLOW APPROXIMATION FIG. 1. Model of a single bubble with its instantaneous pressure profile. A good desription of the fluid flow an be obtained by assuming the water to be inompressible. This is partiularly important lose to the bubble where the partile veloity is highest the nonlinear terms are most important.

4 01 Ziolkowski The density is onstant; therefore, from the kinemati equation 7, Dρ = 0 = φ. 14 Dt With spherial polar symmetry this may be written as φ = 1 r φ = 0, 15 r r r whih an be integrated to give integrated again to give φ r = ġt = v = R Ṙ 16 r r φ = ġt r = R Ṙ, 17 r with the potential at infinity being zero, ġt being the total time derivative of a time funtion gt, Ṙ being the partile veloity at the bubble wall r = R. By integrating equation 17, it is seen that the funtion gt is related to the instantaneous volume V B t of the bubble as follows: gt = R3 3 = V Bt 4π. 18 The hanges in the partile veloity potential φ propagate instantaneously in an inompressible fluid with finite veloity in the linear aousti ase. Using equations 1, 16, 17, the dynami equation 4 for the inompressible fluid may be written as p p ρ = φ v = R R + R Ṙ R4 Ṙ r r, 19 4 whih is the equation of motion of the water. At the bubble wall this beomes P p = R R + 3 ρ Ṙ, 0 in whih Pt is the pressure in the bubble. Equation 0 is the equation of motion of the bubble wall was first derived by Rayleigh LAMB S ANALYSIS OF NONLINEAR TERMS IN EXACT WAVE EQUATION For an inompressible fluid, equation 14 applies equation 8 may be written in the following way: [ φ r + φ r r = 1 φ φ ] φ φ r r + φ = 0. r r 1 Assuming an adiabati equation of state for the gas in the bubble, Lamb 193 found an analytial solution of the equation of motion 0 for the speial ase of the ratio of the speifi heats γ = C p /C v being equal to 4/3. He then used equation 17 ombined with his analytial solution for R to find how big the seond third nonlinear terms in the square brakets of equation 1 ould get. He found these terms are proportional to the square root of the initial pressure would be less than 1% for initial pressures up to about 1000 atm, when the maximum partile veloity would be 145 m/s or about one-tenth the speed of sound. Lamb wrote 193, however, The argument is of ourse far from rigorous, but it is of a kind to whih we are often onstrained to have reourse in mathematial physis. Sine air guns operate at pressures of normally not more than 00 atm, it is probably safe to assume that the partile veloity potential an be assumed to obey the linear aousti wave equation with negligible error. DERIVATION OF THE EQUATION OF MOTION OF THE BUBBLE IN THE NONLINEAR ACOUSTIC APPROXIMATION The goal of this paper is to find a numerial solution to the problem with a single approximation that is valid for a known range of pressures throughout the water. I all this the nonlinear aousti approximation. The partile veloity potential φ is now assumed to obey the linear aousti wave equation 1, with the solution given by equation 13. This solution is now substituted into the basi dynami equation 5 to yield φ = h + v = 1 r f t r, in whih the prime denotes differentiation with respet to the argument. This is the ruial step in deriving the nonlinear equation of motion. Arguing as Gilmore 195 did, it follows that the quantity r[h + v /] propagates outward at veloity without hange in amplitude. Therefore, r [ r ] h+ v = 1 [ r h+ v ] = 1 f t r Carrying out the differentiation in equation 3 yields h + v + r h v + rv r r = r h rv. 3 v. 4 To get at the equation of motion, following the partile paths, the partial derivatives h/ r, h/, v/ r, v/ in equation 4 need to be expressed in terms of total derivatives. First, with spherial polar symmetry the equation of motion 4 beomes h r = Dv Dt, 5 in whih D Dt = + v r. 6 Then, from equations 5 6, it follows that h From equations 3 7 it is found that From equations 6 8, v = Dh Dt + v Dv Dt. 7 v r = 1 Dh Dt v r. 8 = Dv Dt + v Dh Dt + v r. 9

5 Measurement of Air-Gun Bubble Osillations 013 Substituting these expressions into equation 4 rearranging yields r Dv 1 v + 3v 1 4v = h + r Dh v v 1 +, Dt 3 Dt 30 in whih v / is learly negligible sine the peak value of v/ is less than This is the equation of motion of the fluid. At the bubble wall this beomes R R 1 Ṙ + 3Ṙ 1 4Ṙ = H + R Ḣ 1 Ṙ, 3 31 in whih Ht is the enthalpy at the bubble wall, applying equation 4, is given by Ht = φ Ṙ. 3 It is seen from equation that the speed of sound beomes infinite if the water is inompressible, equation 31 then redues to equation 0, as it should. COMPARISON WITH KIRKWOOD BETHE APPROXIMATION It is of interest, given the history of this problem, to ompare equation 31 with the equation of motion that Gilmore derived using the Kirkwood Bethe approximation. Kirkwood Bethe 194 proposed that the partile veloity potential propagates with a speed + v. Equation 3 thus beomes [ ] r h + v r = 1 + v [ r h + v ], 33 whih, on arrying out the differentiation using the same expressions for the partial derivatives h/ r, h/, v/ r, v/ as above, yields an equation of motion in the water. At the bubble wall this equation is R R 1 Ṙ = H 1 + Ṙ + 3Ṙ 1 Ṙ 3 + R Ḣ 1 Ṙ. 34 This is the equation of motion Gilmore derived. It of ourse ontains the same four main terms, but it differs from equation 31 in the first-order orretions, of order Ṙ/, as expeted. The diffiulty with this approximation is the omputation of the wavefield in the water, as mentioned in the introdution explained in the next setion. PRESSURE AND PARTICLE VELOCITY IN THE WATER Using solution 13 in equation 3, the partile veloity in the water an be expressed as vr, t = φ r = 1 r f t r + 1 r f t r, 35 while the enthalpy is obtained by using solution 13 in equation 5 to give hr, t = 1 r f t r v. 36 The pressure must be a known funtion of the density. It an then be extrated from the enthalpy. The Tait equation is used by Gilmore 195 Ziolkowski This ould be done here, but it is probably not neessary. For the expeted pressure variations in the water, it is reasonable to assume the density is onstant in equation 1. This an be seen by examining equation. For an instantaneous peak pressure p of 0 MPa 3000 psi, ρ beomes about 9 kg/m 3 ρ/ρ is <0.01. This is the maximum possible instantaneous error that an be made. For most of the osillation, the error is muh less than this at the bubble wall dereases approximately inversely with the distane away from the bubble. With this approximation equation 36 beomes pr, t p ρ = 1 r f t r v r, t. 37 It is not ritial to the argument to assume the density is onstant. It is done here only to avoid ompliations that are not required for this level of auray. Given f t its first derivative f t, the partile veloity pressure an be found anywhere in the water from equations The remainder of this paper is onerned with developing a sheme to determine f t f t from measurements. In addition, the time-varying pressure volume funtions of the bubble an be found. It is interesting again to ompare this approah with the Kirkwood Bethe approximation, whih begins with the assumption that the partile veloity potential obeys the equation φ = 1 r f K B t r + v. 38 The partile veloity v is the gradient of this funtion, whih is a funtion not only of the independent variables r t but also of v itself. Gilmore 195 found it neessary to make further approximations to avoid this diffiulty; thus, the pressure partile veloity in the water are not alulated to the same preision as the equation of motion 33. This shortoming is repeated in my own paper Ziolkowski, It is exatly this additional level of approximation that this paper sets out to avoid. RECURSIVE ALGORITHM TO DETERMINE WAVE FUNCTION FROM PRESSURE MEASUREMENT IN WATER Suppose there is a pressure measurement in the water at a known distane r from the enter of the bubble, mr, t = pr, t p, 39 onsisting entirely of the outgoing wave from the bubble no refletions from the sea surfae or sea floor. Its derivatives m r, t m r, t an be omputed. Suppose the value of vr, t is known at some time t = τ. Then the following values of the funtions f t f t an be found from equations 37 35: f τ r [ ] mr,τ = r + v r,τ 40 ρ

6 014 Ziolkowski f τ r [ = r vr,τ mr,τ ] v r,τ. 41 ρ By suessive differentiation of equations 37 35, expressions for f t, f t, v t, v t an be found whih, for t = τ, are as follows: m vr,τ f τ r r,τ r + f τ r = ρ r 1 vr,τ, 4 v r,τ = 1 r f τ r + 1 r f τ r, 43 f τ r = m r,τ r ρ νr,τ f τ r + v r,τ + r 1 νr,τ, 44 v r,τ = 1 r f τ r + 1 r f τ r. 45 Then an estimate of vr,τ + τ an be made using the trunated Taylor expansion, vr,τ + τ = vr,τ + v r,τ τ + v r,τ τ. 46 Equations 40 to 46 represent a reursive sheme to determine f t f t from the pressure measurement mr, t an initial value of the partile veloity at the measurement position. From f t f t the pressure partile veloity an be omputed everywhere in the water, inluding the bubble wall. The initial value of the partile veloity needs to be known to start the reursion. It an be found by trial--error if there are two pressure measurements at different distanes, for instane, at r 1 r. The initial value of the partile veloity at r 1 is assumed, the pressure measurement at r 1 is used to ompute the pressure at r. The assumed value is adjusted until the measured alulated pressures math. This is demonstrated in the Appendix with real measurements made on a single air gun. The sheme an be rearranged in a number of ways. It is probably more onvenient to assume the initial value of f t rather than the partile veloity at some arbitrary point in the water beause this is related diretly to the motion of the bubble. RECURSIVE ALGORITHM TO DETERMINE PRESSURE AND VOLUME OF BUBBLE FROM PRESSURE MEASUREMENT IN WATER Suppose the volume of the bubble is known at some time t = τ. The bubble radius is then 1 3VB τ 3 Rτ =. 47 4π Using equations for r = R, the partile veloity aeleration at the bubble wall an then be expressed in terms of the wave funtion as follows: Ṙτ = 1 Rτ f τ + 1 Rτ f τ 48 Rτ = 1 Rτ f τ + 1 Rτ f τ, 49 in whih the time delay R/ is negleted beause the bubble is small ompared with all wavelengths of interest. Equation 37 applied at the bubble wall gives the pressure in the bubble Pτ = p + ρ [ 1 Rτ f τ Ṙ τ ]. 50 The bubble radius at time τ + τ may be estimate using the following trunated Taylor expansion: Rτ + τ = Rτ + Ṙτ τ + Rτ τ. 51 Equations 47 to 51 represent a reursive sheme to determine the internal pressure volume of the bubble as a funtion of time. The sheme requires the initial value of the bubble volume to start it off. This an be found by trial-error, for example, by omparing the omputed value of the pressure inside the bubble with a measurement made inside the bubble. This is demonstrated in the Appendix with measurements made on a single air gun. RECURSIVE ALGORITHM TO DETERMINE WAVE FUNCTION AND PRESSURE AND VOLUME OF BUBBLE FROM PRESSURE MEASUREMENT MADE INSIDE BUBBLE Suppose the bubble radius R bubble wall veloity Ṙ are known at some time t = τ there is a pressure measurement Mt = Pt p 5 made inside the bubble. The instantaneous pressure inside the bubble at time τ is assumed to be uniform is simply Pτ = Mτ + p. 53 The time derivatives of the measurement an be omputed. Then from equation 1, negleting the very small hanges in density aused by the ompressibility of the water, the instantaneous enthalpy of the water at the bubble wall is Hτ = Mτ ρ. 54 Hene, Ḣτ = M τ ρ 55

7 Measurement of Air-Gun Bubble Osillations 015 Ḧτ = M τ. 56 ρ Applying equations at the bubble wall negleting the time delay R/ yields the first derivative of the wave funtion the wave funtion itself: [ f τ = R Hτ + Ṙ τ ] 57 [ f τ = R Ṙτ f ] τ. 58 R The equation of motion 31 an be rearranged to give the bubble wall aeleration: Rτ = Hτ + RτḢτ 1 Ṙτ Rτ 1 Ṙτ 3Ṙ τ 1 4Ṙτ It is neessary to ompute Ṙ.., of ourse equation 59 may be differentiated. However, Ṙ.. is needed only to alulate a seond-order term in a Taylor series below, so it needs to be aurate only to first order. It is known from the nonlinear aousti analysis above that the fators in brakets in equation 59 are almost equal to one. An approximate first derivative of equation 59 is, therefore, Rτ = Ḣτ Rτ HτṘτ + Ḧτ R τ 3Ṙτ Rτ Rτ + 3Ṙ3 τ R τ. 60 APPLICATION TO ARRAYS OF AIR GUNS WITH PRESSURE MEASUREMENTS AT GUN PORTS An array of air guns an be monitored studied using pressure transduers mounted on the guns lose to the ports. The measurements would also give all neessary information on gun timing for tuning the array. In what follows, the bubbles are assumed to be entered initially on the gun ports to rise to the surfae at onstant veloity Parkes et al., 1984; Ziolkowski Johnston, 1997 while the air-gun array hydrophones move forward with the vessel. [This assumption is not essential to the argument. Reviewer Martin Lrø has kindly pointed out that the bubble rise veloity is not onstant. Herring 1941 shows it is v z t = dz dt = g t R 3 τ dτ, Rt 3 0 in whih z is depth, g is the aeleration from gravity, Rt is the bubble radius at time t. Sine Rt is alulated in the sheme, this formulation for nononstant veloity v z an easily be inorporated by replaing onstant veloity terms of the form v z t with t v 0 zτ dτ.] The origin of the right-h Cartesian oordinate system is at the surfae of the water, with the x-axis pointing diretly behind the vessel the z-axis pointing downward, as shown in Figure. A new origin is taken at eah shotpoint. Consider an array of n air guns, eah of whih has a pressure transduer, or hydrophone, mounted within entimeters of the gun ports. The guns may be as lose as 1 m within a luster but normally are meters apart. The guns are assumed to be arranged suh that no two guns are so lose that their bubbles oalese in the time interval of interest. As desribed above, there is interation among the osillating bubbles. The notional soure onept of Ziolkowski et al. 198 regards eah osillating bubble as an aousti monopole beause it is small ompared with a wavelength. The pressure ating on the outside of any bubble is hydrostati pressure plus all the ontributions from the other bubbles their sea surfae refletions. The The bubble wall veloity radius an then be estimated at time τ + τ using the trunated Taylor series: Ṙτ + τ = Ṙτ + Rτ τ + Ṙ.. τ τ 61 Rτ + τ = Rτ + Ṙτ τ + Rτ τ. 6 Equations 53 to 6 represent a reursive sheme to obtain the pressure radius of the bubble as a funtion of time as well as the propagating wave funtion in the water, given a measurement of the pressure inside the bubble initial values of the bubble radius bubble wall veloity. As before, these initial values an be obtained by trial--error, provided a measurement of the pressure in the water is available. This pressure an be found from equations using the funtions f t f t obtained with the above sheme. FIG.. Coordinate system, showing the ith air gun its pressure transduer.

8 016 Ziolkowski notional soure onept treats all these interation effets as if the bubbles were monopoles. Let the oordinates of the enter of the ports of the ith gun be xg i, yg i, zg i the oordinates of the pressure transduer on the ith gun be xh i, yh i, zh i at time t = 0. Let the time at whih the ithe gun fires be τ i. At time t the oordinates of the kth pressure transduer are xh k v x t, yh k, zh k, in whih v x is the forward veloity of the vessel relative to the water. The oordinates of the enter of the ith bubble are xg i v x t + ν x τ i, yg i, zg i ν z τ i, in whih v z is the bubble rise veloity. At time t the distane from the enter of the ith bubble to the kth transduer is r ik t = [ xg i + v x τ i xh k + yg i yh k + zg i v z τ i zh k ] 1 63 the distane from the enter of the ith virtual bubble the image of the ith bubble in the sea surfae to the kth hydrophone is rg ik t = [ xg i + v x τ i xh k + yg i yh k + zg i v z τ i + zh k ] Using the notional soure onept equation 35, the partile veloity at a distane r ik t from the ith bubble is v i r ik t = 1 rik t f i t r ikt + 1 r ik t f i t r ikt, 65 in whih f i t is the wave funtion of the ith bubble. From equation 36 the pressure at a distane r ik from the ith bubble is p i r ik t = 1 r ik t f i t r ikt v i r ikt. 66 Air guns normally operate at depths of several meters. The nonlinear terms an therefore be negleted in the interation effets of the virtual soures. Thus, the pressure ating at a distane rg ik t from the ith virtual soure is simply p i rg ik t = 1 rg ik t f i t rg ikt, 67 in whih the sea surfae refletion oeffiient is assumed to be 1. At the instant of firing, the air begins to esape from the guns the bubbles begin to form. Eah gun-mounted pressure transduer is about to be enveloped by the bubble from its gun. In pratie the guns do not fire simultaneously but in a sequene, whih may vary slightly from shot to shot. Suppose the jth gun fires first. Then the jth hydrophone is the first to register any signal, until any other signals arrive at this hydrophone, the measured pressure is ph j t = p j r jj t. 68 Given the initial bubble radius bubble wall veloity of the jth bubble, the wave funtion bubble motion an be alulated aording to the shemes desribed earlier [equations equations 47 51, respetively]. This sheme may be repeated at eah hydrophone until it begins to reeive signals from other guns. At this point the pressure at, say, the kth hydrophone is the superposition of the pressure fields of all the notional bubbles their virtual images: ph k t = p i r ik t + p i rg ik t. 69 The ontribution to this pressure by the diret wave from the kth gun is not known at time t, but the ontribution from its virtual soure is known from previous time steps. Similarly, the ontributions at time t from all the more distant guns are known from previous time steps. Therefore, the ontribution of the kth gun an be omputed by subtration: p k r kk t = ph k t p i r ik t + p i rg ik t, i k 70 its wave funtion bubble motion an then be omputed as desribed earlier. This sheme holds as long as the hydrophone is outside the bubble, that is, if r kk t > R k t. 71 When the transduer beomes enveloped by the bubble wall a different sheme operates. Consider a pressure measurement ph k t on the inside of the kth bubble made with the kth pressure transduer. The pressure within the bubble is regarded as uniform. The pressure measurement is related to the absolute pressure P k t inside the bubble as ph k t = P k t p, 7 where p is hydrostati pressure. The pressure ating on the outside of the kth bubble an be expressed as 1 pw k t = p + p i r ik t + p i rg ik t. 73 i k Therefore the pressure differene between the inside of the bubble the outside of the bubble is Pd k t = P k t pw k t = ph k t p i r ik t + = P k t i k p i r ik t + i k p i rg ik t p i rg ik t p. 74 Thus, the pressure differene aross the wall of the kth bubble is the same as the pressure differene aross the wall of a notional bubble whose internal pressure is P k t n p i r ik t + n i k p irg ik t whose external pressure is p. The nonlinear equation of motion of the kth bubble is obtained from

9 Measurement of Air-Gun Bubble Osillations 017 equation 31 as in whih R k R k 1 Ṙ k = H k + RḢ k + 3Ṙ k 1 Ṙk 1 4 R k 3, 75 H k t = ph kt pw k t p ρ 76 Ḣ k t = ph k t pw j t ρ 77 the prime indiates differentiation with respet to the argument. The wave funtion of the kth bubble is obtained from equations as f k t = R kt [ H k t + Ṙ k t ] 78 [ f k t = Rk t Ṙ k t f k t ]. 79 R k t To ompute the bubble motion wave funtion of eah notional bubble, it is essential to use the priniple of ausality. Wave propagation is ausal. The interation terms in the summations of equations are obtained from the wave funtions of the bubbles known at previous times. At eah time step the proedure is to update all the quantities that need to be omputed for eah bubble then go to the next time step. The algorithm ontains, in an inner loop over the gun index i, equations 65 to 75 an outer loop over the time index. The algorithm is initialized using the initial values of the measured pressure the initial values of the bubble radius bubble wall veloity of eah bubble. These an be obtained in a separate alibration experiment, whih may be arried out at the start of eah line, as desribed in the next setion. CALIBRATION FOR INITIAL VALUES The initial value of the bubble radius bubble wall veloity need to be known for eah gun to start the reursion. As with the single gun ase, these values an be found by trial--error if there are at least two pressure measurements at different distanes for eah gun firing on its own. To obtain the required data, it is neessary to fire eah gun independently at the run-in to eah line to reord the signals on all the gun-mounted pressure transduers. The signature for eah gun lasts no longer than one seond. For a typial array of 30 guns, the time needed to make the required alibration measurements is therefore of the order of one minute. The initial values of the bubble radius bubble wall veloity for the ith gun are assumed, the pressure measurement at the ith transduer is used to ompute the pressure at transduers on nearby guns. The assumed values of bubble radius bubble wall veloity are adjusted until the measured alulated pressures math. The proedure is repeated for all guns until all required initial values of bubble radius bubble wall veloity are known. COMPUTING A FAR-FIELD SIGNATURE FROM COMPUTED WAVE FUNCTIONS We now need to be able to ompute the signature of the air gun array at some point x, y, z in the water that is more than 1 m from any of the guns in the array is more typially 100 m below the air gun array. Sine this point is not in the nonlinear part of the pressure field of any of the soures, we an neglet the nonlinear term in equation 66. Using the notional soure onept Ziolkowski, 198, the pressure at this point is the superposition of the linear omponents of the spherial waves generated by all the bubbles their virtual images, or in whih px, y, z, t = 1 r i t f i t r it 1 rg i t f i t rg it, 80 r i t = [ xg i + v x τ i x + yg i y + zg i v z τ i z ] 1 81 is the distane from the ith bubble to this point rg i t = [ xg i + v x τ i x + yg i y + zg i v z τ i + z ] 1 8 is the distane from the ith virtual bubble to this point. This omputation is essentially idential to that proposed by Ziolkowski et al. 198 Parkes et al The differene is that in this ase the funtions f i t are obtained from pressure measurements made anywhere preferably with transduers plaed at the gun ports, whereas the method of Ziolkowski et al. 198 requires the pressure measurements to be made in the linear pressure field i.e., approximately 1mormore from the enter of any bubble. The method of Ziolkowski et al. 198 is therefore a speial ase of this method. CONCLUSIONS Given the initial bubble radius, the initial bubble wall veloity, the pressure variation at any point inside or outside the bubble, an algorithm has been found to alulate the bubble motion the aousti wavefield. The partile veloity potential of an osillating air-gun bubble obeys a wave equation that has no analytial solution. A numerial solution has been found by using the linear aousti approximation for the partile veloity potential substituting this into the basi kinemati dynami equations for fluid flow, leading without further approximations, to a new nonlinear equation of motion for the bubble wall the wavefield in the water. The interation among air-gun bubbles the resultant total wavefield is formulated using the notional soure onept, in whih eah bubble is replaed by an equivalent notional bubble obeying the same nonlinear equation of motion but osillating in water of hydrostati pressure, thus allowing the wavefields of the notional bubbles to be superposed. A separate alibration experiment using the same pressure transduers firing the

10 018 Ziolkowski guns individually allows the initial values of the bubble radius bubble wall veloity to be determined for eah gun. The Appendix provides a test of the algorithm on real measurements from a single gun demonstrates that the pressure field may be alulated in the far-field, in the near-field, inside the bubble from a single pressure measurement made at 1m. As far as the algorithms are onerned, you an start with a pressure measurement at any known point in the water or in the bubble. With the test data it proved to be impossible to start with the measurement made losest to the gun ports only 16 m away beause the Bolt stainless steel sensor that was used for the measurement had a frequeny-dependent response that was known at only a few frequenies. Ideally, this measurement should be made with a devie that has a onstant known sensitivity over the frequeny bwidth of interest, for then no deonvolution would be required. The immediate ommerial appliation of the algorithms presented here is the omputation of the wavefield of an air gun array from measurements of the pressure field made with gun-mounted sensors lose to the gun ports. The method of Ziolkowski et al. 198 for the omputation of the wavefield using near-field hydrophones plaed in the linear part of the pressure field has been shown to be a speial ase of this more general approah. A sientifi appliation of the algorithm is to look inward to inrease understing of the thermodynamis of osillating bubbles. ACKNOWLEDGMENTS I thank Martin Lrø for a very helpful remark before the paper was first submitted for several suggestions on review of the paper. I also thank Stephen L. Rohe the anonymous reviewer who pointed out where the original version laked larity. I am very grateful to Paul Rihards who made helpful omments remarks on the original version. My olleagues Sergei Zatsepin David Taylor have read the manusript in all its previous versions have always found points in the mathematis physis that needed better supporting arguments; it is a privilege to work with them. I thank the Petroleum Siene Tehnology Institute now Centre for Marine Petroleum Tehnology for their support for funding my hair in petroleum geosiene at the University of Edinburgh from April 199 to May Rodney Johnston I worked together to ode the algorithms presented here, whih fored me to think muh harder about the theory. I thank Rodney for his input. Finally, I am also very grateful to Bolt Tehnology Corporation for providing the data for the test in the Appendix. REFERENCES Ahenbah, J. D., 1990, Wave propagation in elasti solids: North Holl Publ. Co. Flynn, H. G., 1964, Physis of aousti avitation in liquids, in Mason, W. P., Ed., Physial aoustis priniples methods, Vol. 1, Part B: Aademi Press In. 1975, Cavitation dynamis I: A mathematial formulation: J. Aoust. So. Am., 57, Gilmore, F. R., 195, Collapse of a spherial bubble: Hydrodynamis Laboratory, California Institute of Tehnology, Report No Herring, C., 1941, Offie of Sientifi Researh Development Report No. 36 NDRC C4-sr Johnston, R. C., 1980, Performane of airguns: Theory experiment: Geophys. Prosp., 8, Keller, J. B., Kolodner, I. I., 1956, Damping of underwater explosion bubble osillations: J. Appl. Phys., 7, Kirkwood, J. G., Bethe, H., 194, Progress report on The pressure wave produed by an underwater explosion I : Offie of Sientifi Researh Development Report No Lamb, H., 193, The early stages of submarine explosion: Phil. Mag., 45, , Hydrodynamis: Dover Publ. In. Parkes, G. E., Ziolkowski, A. M., Hatton, L., Haugl, T., 1984, The signature of an air gun array: Computation from near-field measurements inluding interations pratial onsiderations: Geophysis, 49, Rayleigh, Lord, 1917, On the pressure developed in a liquid during the ollapse of a spherial avity: Phil. Mag., 34, Safar, M. H., 1976, The radiation of aousti waves from an air-gun: Geophys. Prosp., 4, Shulze-Gattermann, R., 197, Physial aspets of the airpulser as a seismi energy soure: Geophys. Prosp., 0, Vokurka, K., 1986, Comparison of Rayleigh s, Herring s Gilmore s models of gas bubbles: Austia, 59, Ziolkowski, A., 1970, A method for alulating the output pressure waveform from an air gun: Geophys. J. Roy. Astr. So., 1, Ziolkowski, A. M., Johnston, R. G. K., 1997, Marine seismi soures: QC of wavefield omputation from near-field pressure measurements: Geophys. Prosp., 45, Ziolkowski, A. M., Parkes, G. E., Hatton, L., Haugl, T., 198, The signature of an air gun array: Computation from nearfield measurements inluding interations: Geophysis, 47, APPENDIX MEASUREMENT OF AIR-GUN BUBBLE OSCILLATIONS: EXPERIMENT TO TEST THE METHOD ON REAL DATA Anton Ziolkowski Rodney Johnston EXPERIMENTAL SETUP AND MEASUREMENTS The objetive of the experiment desribed here is to demonstrate that measurements made in the nonlinear near-field in the bubble itself may be used to ompute the pressure field at any distane from the bubble. Bolt Tehnology Corp. reorded the data presented here in Lake Senea, New York. The setup is depited in Figure A-1. A single Bolt air gun was deployed at a depth of 6 m below the water s surfae. The near-field radiated pressure in the water was measured with two sensors: a onventional hydrophone 1 m above the enter of the gun ports a small erami sensor, known as the Bolt stainless steel BSS sensor, built into the gun 16 m from the enter of the gun ports. This small sensor is normally used to measure the time break. A far-field hydrophone was also deployed 110 m below the gun. All measurements were reorded using a Hewlett Pakard HP 35665A dynami signal analyzer. The lake is approximately 00 m deep. Figure A- shows the first 500 ms of the signals reeived at these three sensors for a single gun shot. The signals were reorded at a sample interval of s to give 048 samples per seond. The two hydrophones were alibrated, eah one has a response that is essentially flat over the bwidth of interest; the sensitivity is independent of frequeny. The BSS sensor is linear, but its sensitivity is frequeny dependent. The frequeny-dependent sensitivity of this partiular sensor was

11 Measurement of Air-Gun Bubble Osillations 019 measured subsequently in a separate experiment, the values are presented in Table A-1. The impulse response of the BSS sensor is not known. We found a plausible impulse response for this sensor, inluding the reording system, by fitting an amplitude spetrum through the measured amplitude values inluding an antialias filter, as shown in Figure A-3a; the phase spetrum was onstrained to be minimum phase is shown in Figure A-3b. Both the amplitudes the phases agree with the measured values given in Table A-1. The estimated impulse response of the sensor, inluding the reording system shown in Figure A-4, is omputed by finding the inverse Fourier transform of the funtion shown in Figure A-3. This is not ideal, but it is the best we ould do with the data available. DATA PROCESSING AND RESULTS We would have liked to use the BSS sensor measurement to ompute the far-field signature ompare it with the hydrophone measurement, using the near-field hydrophone to alibrate the initial values of bubble radius bubble partile veloity. Sine the response of the BSS sensor is not flat over the bwidth of interest, we would have had to deonvolve the BSS sensor measurement for the BSS sensor impulse response FIG. A-1. Experimental setup, Lake Senea. Table A-1. Measured sensitivity of the BSS transduer in the range 6 40 Hz. Frequeny Sensitivity Phase lag Hz volts/psi degrees FIG. A-. Measurements for one shot: a pressure at near-field hydrophone, b voltage output from BSS sensor, farfield hydrophone signal.

12 00 Ziolkowski to determine the true pressure at the sensor. We tried to do this with the estimated minimum-phase impulse response of Figure A-4 but were unable to obtain a satisfatory result. Instead, we began with the near-field hydrophone measurement, estimated the pressure at the BSS sensor, onvolved this estimate with the estimated impulse response of Figure A-4, ompared the result with the BSS sensor measurement. Computation of a far-field signature from a near-field measurement made in the linear part of the pressure field 1 m from the bubble is almost trivial, following Ziolkowski et al. 198 as we demonstrate below. The real test of theory is prediting the pressure at the BSS sesnor, whih initially is outside the bubble in the nonlinear zone then beomes enveloped by the bubble. The data proessing steps are desribed below. Resampling to smaller sampling interval FIG. A-3. Fitted transfer funtion of BSS transduer solid line, ompared with laboratory measurements squares: a amplitude spetrum; b phase spetrum. The proessing sheme is essentially a reursive integration that requires the measured pressure signal to be differentiated twie uses trunated Taylor series. This guarantees aumulation of errors. The errors in both differentiation integration are redued if the sampling interval is redued. To minimize these errors, we resampled the data by transforming to the frequeny domain, padding with zeroes to inrease the Nyquist frequeny, transforming bak to the time domain. In this ase inreasing the Nyquist frequeny by a fator of four was suffiient to redue errors to the point where they were insignifiant. The quality of the seond derivative of the pressure signal an be used as the riterion for this resampling. We found it neessary to smooth the data very slightly before resampling beause the energy in the reording was not zero at the Nyquist frequeny. FIG. A-4. Estimated impulse response of BSS sensor: the inverse Fourier transform of the transfer funtion shown in Figure A-3.

13 Measurement of Air-Gun Bubble Osillations 01 Removal of the soure ghost All the measurements inlude a refletion of the radiated wave from the sea surfae. The theory requires that the pure outgoing pressure signal be known, unontaminated by this refletion, or ghost. Equation 37 desribes the outgoing pressure wave. At 1 m the nonlinear seond-order term is negligible. The pressure at the hydrophone is therefore pt = V pt h = ρ rt f t rt ρ rgt f t rgt, A-1 in whih V p t is the reorded signal the numbers on tape, h is the overall sensitivity of the hydrophone the reording system, rt is the time-dependent distane between the hydrophone the enter of the rising bubble, rgt is the timedependent distane between the hydrophone the enter of the sinking virtual bubble, the sea surfae refletion oeffiient is taken to be 1. The bubble rise veloity is assumed to be onstant, suh that rt = r0 v z t tdelay rgt = rg0 v z t tdelay, A- A-3 in whih r0 is 1 m, rg0 is 11 m, tdelay is the time the gun fires after the start of reording, v z is known Parkes et al., 1984; Ziolkowski Johnston, 1997 to be about 1 m/s. The seond term in equation A1 is the ghost. The ghost-free soure funtion f t is obtained reursively from the measurement by rearranging equation A1 to give f t rt = rt ρ rt pt + rgt f t rgt. A-4 Removal of the ghost is the addition of the seond term on the right side of equation A4. The values of this term are obtained by linear interpolation of previously omputed values of the funtion f t. Normally, linear interpolation would be inadequate; but we have already oversampled by a fator of four, so at this point it is aeptable. For t < tdelay + rg0, A-5 the ghost has not arrived. The time delay between the arrivals of the diret refleted waves is rgt rt τt =, A-6 whih in general orresponds to a noninteger number of samples, hene, the need for interpolation, as shown in Figure A-5. The measured pressure signal at 1 m the deghosted FIG. A-5. Illustration of linear interpolation proedure to estimate the ghost.

14 0 Ziolkowski measurement essentially f t multiplied by the density of water are shown in Figure A-6. Integration sheme onstants We begin with the deghosted pressure measurement shown in Figure A-6, whih is the quantity mt in equation 39 required by the theory. The reursion for the determination of the wave funtion from a pressure measurement in the water is desribed earlier. It requires the first seond derivatives of mt, shown in Figure A-7, the initial value of the partile veloity at the hydrophone. This defines the onstant of integration required to integrate f t to obtain the wave funtion f t. We found a value of 0.13 m/s for this veloity. The wave funtion its first derivative are shown in Figure A-8. The onstant an be heked by omparing the predited pressure with the BSS sensor measurement. The sensor is initially in the water only 16 m from the enter of the gun ports learly in the nonlinear zone where v / term is not negligible. To ompute this term requires equation 35, in whih both f t f t need to be known. Of ourse the bubble is exping; after a few milliseonds it envelops the BSS sensor, whih then measures the pressure inside the bubble. To determine the time at whih this ours, it is neessary to ompute the bubble radius as a funtion of time, using the reursion desribed above, beginning with the initial value of the bubble radius. At the instant the bubble wall reahes the BSS sensor, the pressure in the bubble is given by equation 50 the dynami pressure at the bubble wall is given by the term [ ] 1 Pt p = ρ Rt f t Ṙ t, A-7 FIG. A-6. a Pressure measurement at 1 m; b same as a, but with soure ghost removed. The slight redution in amplitude of the first peak is a result of smoothing before interpolation. in whih Ṙt is the bubble wall veloity. The initial bubble radius is a seond onstant that needs to be found. We found a value of m. Figure A-9 shows the time-varying bubble wall veloity, the bubble radius, the absolute pressure inside the bubble. Figure A-10 shows the omputed pressure at the BSS sensor, Figure A-11 ompares the BSS sensor reording with the pressure funtion of Figure A-10, onvolved with the estimated impulse response of Figure A-4. As an be seen, the agreement between the measured omputed signals is good. For ompleteness in Figure A-1 we show the measured farfield signature ompared with the far-field signature omputed FIG. A-7. a First derivative of deghosted near-field pressure signal shown in A6b; b seond derivative of same signal. FIG. A-8. a The wave funtion f t; b its first derivative f t.