GB2320327A

GB2320327A - Determining the far field pressure signatures of air-gun arrays

Info

Publication number: GB2320327A
Application number: GB9624556A
Authority: GB
Inventors: Mo Safar
Original assignee: Individual
Current assignee: Individual
Priority date: 1996-11-26
Filing date: 1996-11-26
Publication date: 1998-06-17
Anticipated expiration: 2016-11-26
Also published as: GB2320327B; GB9624556D0

Abstract

An improved method is developed for the purpose of determining the far field pressure signature radiated by a marine seismic source which consists of an array of n air-guns. The improved method involves the measurement of the pressure field in the vicinity of each air-gun by n independent calibrated hydrophones with known positions when all the air-guns are fired individually and simultaneously. The far field pressure signature of the marine seismic source is obtained by ghosting and summing the recorded near field pressure signatures when all the air-guns are fired simultaneously afier removing the contributions from the neighbouring air-guns and the sea surface from the output of each near field hydrophone. The new method differs from the so-called notional source signatures method (US Patent No. 4,868,794) in that it does not involve the formation and solution of n simultaneous equations, which may give rise to instibility problem, and it requires a knowledge of the distance of each hydrophone from the air-gun immediately below it only but not its distances from the neighbouring air-guns and consequently there is no need to calculate the air-bubbles rise velocities and the relative horizontal velocities. One important feature of the new method is that the near field pressure signatures, obtained when all the air-guns are fired individually, are used to calculate not only the geometry of the air-gun-hydrophone system but also the depth of each air-gun and consequently, the depth transducers normally used for determining the depths of the air-guns are not required.

Description

METHOD FOR DETERMINING THE FAR FIELD PRESSURE SIGNATURES OF AIR-GUN ARRAYS The present invention relates to an improved method for determining the far field pressure signatures radiated by air-gun arrays.

One essential part of the marine seismic acquisition system used when carrying out marine seismic surveys is the marine seismic source which consists of an array of air-guns having different chamber sizes and fired simultaneously.

Each air-gun when fired releases an air-bubble which generates a seismic pressure pulse.

The effective seismic pressure pulse, which is referred to as the far field pressure signature of the array, radiated by the whole array is simply given by the sum of the seismic pressure pulses generated by the individual air-guns and their reflections by the sea surface.

Basically the air-bubble produced by an air-gun is a nonlinear resonant system which consists of a mass, a spring and some damping. The mass, referred to as the acoustic radiation mass, consists of a shell of water surrounding the air-bubble which acts as a spring. The damping of the air-bubble motion consists of two parts. One part accounts for the radiation of the acoustic energy away from the air-bubble and the other part accounts for heat losses. The damping of the air-bubble motion due to heat losses is considerably greater than that due to acoustic radiation. Consequently the air-bubble is extremely inefficient in generating acoustic energy.

For the case of an array of air-guns fired simultaneously, the acoustic radiations mass and resistance of each air-bubble will be modified. This is because of the presence of interaction between each air-bubble and the other air-bubbles released by the other airguns forming the array. Therefore because of the presence of interaction the period of the seismic pressure pulse, which is proportional to the square root of the acoustic radiation mass, of each air-bubble will be modified. This means that the far field pressure signature radiated by the array cannot be determined simply by summing the seismic pressure pulses and their reflection by the sea surface generated when the air-guns forming the array are fired individually.

According to the present invention an improved method for determining the far field pressure signatures radiated by air-gun arrays comprises; 1. measuring the acoustic pressure in the vicinity of each air-gun by an independent calibrated hydrophone when all the air-guns fired simultaneously and individually; 2. determining the periods of the pressure bubble pulses radiated by the air-bubbles from the measured acoustic pressure when all the air-guns are fired simultaneously and individually.

According to another aspect of the invention there is provided a method for determining the depth of each air-gun forming the array.

The theoretical background required for describing the invention is given below with reference to FIG1 which shows two hydrophones H1 and H2 placed in the vicinity of two air-bubbles 1 and 2. Hydrophone H1 is placed at a point with distances of r11 and r12 from the centres of air-bubbles 1 and 2 respectively. Hydrophone H2 is placed at a point with distances r21 and r22 from the centres of air-bubbles 1 and 2 respectively.

It can be shown from the theoretical results obtained by Safar (1976) in the frequency domain for the case of two identical air-bubbles, the acoustic pressure pnf(t) measured at a distance equal to the equilibrium radius a0 of the air-bubble is given by; pnf(t) = [#0/4#a0] v (t-a0/c) + [#0/4#D] v (t-D/c) (1) where the dots denote differentiation with respect to time t and; p0 is the density of the water surrounding the air-bubble c is the velocity of sound in water v(t) is the instantaneous volume of the air-bubble D is the spacing between the two air-bubbles For the case of two air-bubbles produced by two air-guns having different chamber volumes, eqn (1) becomes; pnf1(t) = [#0/4#a01] v1 (t-a01/c + [#0/4#D) v2 (t-D/c) (2) for the pressure measured by hydrophone H1 when placed at a distance aOi and; pnf2(t) = [#0/4#D] v1 (t-D/c) + [#0/4#a02] v2 (t-a02/c) (3) for the acoustic pressure measured by hydrophone H2 when placed at a distance a, where aOi and a02 are the equilibrium radii of the two air-bubbles.

For the case when the hydrophones H1 and H2 are placed at distances of r11 and r, from the centres of air-bubbles 1 and 2 respectively, then eqns (2) and (3) become; pnf1(t) = [#0/4#r11] v (t-r11/c) + [#0/4#r12] v2 (t-r12/c) (4) pnf2(t) = [#0/4#r21] v1 (t-r12/c) + [#0/4#r22] v2 (t-r22/c) (5) The far field pressure p(t,r) radiated by the two air-bubbles measured at a point r1 and r2 from the two air-bubbles is given by; p(t,r) = [#0/4#r1] v1 (t-r1/c) + [#0/4#r2] v2 (t-r2/c) (6) It is pretty obvious from eqns (4), (5) and (6) that the far field pressure signature radiated by an array of two air-guns can be determined by measuring p,(t) and pnf2(t) and solving eqns (4) and (5) to obtain v1(t) and v"2(t). This is in fact the basic principle of the so called notional source signatures method (US Patent No.4,868,794).

The notional source signatures method (US Patent No.4,868,794) was evaluated by Landro et al (1991) who found that the method suffers from a major shortcoming when dealing with compact air-gun arrays, namely the stability problem. Landro et al (1991) concluded that in order to avoid the stability problem, the near field hydrophones should be placed at distances from the air-guns closer than those proposed in the Patent.

So far I have neglected the pressure arriving at the hydrophones H1 and H2 after being reflected by the sea surface. When this is taken into account eqns. (4) and (5) become; pnf1(t) = [#0/4#r11] v1 (t-r11/c) + [#0/4#r12] v2 (t-r12/c) + Rc[#0/4#g11] v1 (t-g11/c) + Rc[#0/4#g12] v2 (t-g12/c) (7) pnf2(t) = [#0/4#r21] v1 (t-r21/c) + [#0/4#r22] v2 (t-r22/c) + Rc[#0/4#g21] v1 (t-g21/c) + Rc[#0/4#g22] v2 (t-g22/c) (8) where Rc is the reflection coefficient of the sea surface g11 & g12 are the distances of hydrophone H1 from the images of air-bubbles 1 & 2, respectively g21 & g, are the distances of hydrophone H2 from the images of air-bubbles 1 & 2, respectively A brief description of the proposed new method is given below. When only air-gun no.2 is fired eqn.(7) becomes; pnf1' = [#0/4#r12] v2 (t-r12/c) + Rc[#0/4#g12) v2 (t-g12/c) (9) where v'2 (t) is the instantaneous volume of air-bubble 2 when air-gun no.2 is operating alone which differs from that when air-guns nos. 1 & 2 are fired simultaneously. This is due to the presence of interaction between air-bubbles 1 & 2.

The volume acceleration vz(t) of air-bubble 2 when air-guns nos. 1 & 2 are fired simultaneously is obtained simply by stretching v'2(t) by the amount T2 - T'2 where T2 & T 2 are the periods of the bubble pulses generated by air-bubble 2 in the presence and absence of air-bubble 1, respectively Therefore, it follows that; [#0/4#r11] v1 (t-r11/c) + Rc[#0/4#g11] v1 (t-g11/c) (10) can be determined simply by subtracting p',(t) from eqn.(7) after being stretched by the amount T2 - T'2. In order to determine vl(t) of air-bubble 1, the second term of (10) is removed by carrying out a deghosting of (10).

Similarly v2(t) can be determined by subtracting ptn,2(t), obtained when air-gun no.1 is operating alone from eqn. (8) after being stretched by the amount Tl-T'l and carrying out a deghosting operation where Tl & T'l are the periods of the bubble pulses generated by air-bubble 1 in the presence and absence of air-bubble 2, respectively.

The procedure outlined above for determining the volume acceleration of two air-bubbles produced by two air-guns can be applied to air-gun arrays having n air-guns provided that the pressure in the vicinity of each air-gun is measured by an independent calibrated hydrophone.

I will demonstrate below how the depth of the air-bubbles released by an array of air-guns can be determined from the near field pressure measurements obtained when all the airguns are fired separately.

Consider FIG 1 the distance r21 is given by; r21 = r11 + (T21 - Tl1) x 1500 (11) where T,1 & T21 are the arrival times at hydrophones Hi & H2 of the primary of the pressure pulse generated by the air-bubble, respectively.

The distance g21 is given by; g21 = r11 + (T'21 - Tl1) x 1500 (12) where T'21 is the arrival time at hydrophone H2 of the primary of the pressure pulse generated by the air-bubble after being reflected at the sea surface.

The values of (T'2, - T1l) and (Tl2 - T11) can be obtained from the pressure signatures measured by hydrophones H1 & 112. If the value of r11 is given then the distances r2, & g21 can be calculated from eqns (11) & (12).

The distance between the centres of the air-bubble and its image, which is twice the depth d of the centre of the air-bubble, is given by; 2d = (g221 - r21 + r22)+ r, (13) In order to test the accuracy of the proposed new method, the far field pressure signature of M/V Geco Gamma air-gun array of Geco-Prakla was calculated from the measured near field signatures using the new method and compared with that determined by Geco Prakla using the notional source signatures method (US Patent No.4,868,794). The comparison between the calculated far field pressure signatures and their amplitude spectra are shown in Figures 2 & 3. The difference between their calculated amplitude spectra is shown in Figure 4. It is clear from Figures 2, 3 & 4 that the proposed new method predicts the far field pressured signature of air-gun array in the presence of interaction fairly accurately.

Claims

I claim; 1. A method of acquiring data for use in determining the far field pressure signature of an array of n air-guns comprising the steps of: recording the pressure field in the vicinity of each air-gun by a hydrophone with known sensitivity and distance from the air-gun immediately below it when all the air-guns are fired individually and simultaneously; subsequently processing the data by: determining the periods Tj and T'j of the pressure pulses radiated by the jth air-gun in the presence and absence of the other air-gun where j =1 ,2,---,n; sorting the single air-gun shot gathers h'jj(t) into common hydrophone gathers hil,ha,---,h,n(t) where i=1,2,---,n; stacking the common hydrophone gather either with or without stretching by the amount of CTj-T'i) to produce;

i=1,2,---,n where Si is the sensitivity of the ith hydrophone subtracting pi(t) from hi(t)/S1 to produce qi(t) where hi(t) is the output of the ith hydrophone when all the air-guns are fired simultaneously
2. A method as set forth in Claim 1 comprising the further step of generating the far field pressure signature of the array p(t) by stacking qi(t) and their ghosts with the appropriate time delays

where ri & gi are the distances of the point at which the far field pressure signature is calculated from the centres of the ith air-bubble and its image, respectively.
3. A method for determining the distances of each hydrophone from the neighbouring air-guns according to; rij = rij + (Tij - Tjj) x 1500 where rij is the distance of the ith hydrophone from the jth air-gun rij is the distance of the jth hydrophone from the jth air-gun Tb. is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the jth hydrophone Tij is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the ith hydrophone.
4. A method for determining the depth of each air-gun of an array consisting of n air-guns according to; 2dj = (gij - rij + rij) + rii where dj is the depth of the centre of the air-bubble produced by the jth air-gun when operating alone is is the distance of the ith hydrophone from the centre of the image of the air-bubble produced by the jth air-gun when operating alone and it is given by; gij = rij + + (T8 - Tü) x 1500 where Tij' is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the ith hydrophone after being reflected by the sea surface Amendments to the claims have been filed as follows I claim; 1. A method of acquiring data for use in determining the far field pressure signature of an array of n air-guns comprising the steps of: recording the pressure field in the vicinity of each air-gun by a hydrophone with known sensitivity and distance from the air-gun immediately below it when all the air-guns are fired individually and simultaneously; subsequently processing the data by: determining the periods Tj and Tj' of the pressure pulses radiated by the jth air-gun in the presence and absence of the other air-guns where j=1,2,---,n; determining the distances of each hydrophone from the neighbouring air-guns according to; rij = = rjj + (Tij - T) x 1500 where is is the distance of the ith hydrophone from the jth air-gun is is the distance of the jth hydrophone from the jth air-gun Tij is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the jth hydrophone Tij is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the ith hydrophone. determining the depth of each air-gun of an array consisting of n air-guns according to; 2dj = (g2ij - r2jj + r2 + r where dj is the depth of the centre of the air-bubble produced by the jth air-gun when operating alone gij is the distance of the ith hydrophone from the centre of the image of the air-bubble produced by the jth air-gun when operating alone and it is given by; = = rjj + (T'jj - Tj;) x 1500 where T'j is the arrival time of the primary of the pressure pulse generated by the jth air-gun when operating alone at the ith hydrophone after being reflected by the sea surface. sorting the single air-gun common shot gathers obtained when all the air-guns are fired individually into common hydrophone gathers consisting of the direct arrivals and their surface reflections generated by the air-guns which are not placed immediately below the common hydrophone. converting the common hydrophone gathers from voltage into pressure by dividing by the common hydrophone sensitivity; stacking each common hydrophone gather either with or without stretching by the amount Tj - Tj' where Tj & Tj' are the periods of the bubble pulses generated by the jth air-gun in the presence and absence of interaction respectively, to produce the contributions of (n-1) air-guns p (t) to the output of the ith hydrophone when all the air-guns are fired simultaneously; subtracting p (t) from the output of the ith hydrophone obtained when all the air guns are fired simultaneously to produce qj (t) which is the diverging spherical wave generated by the ith air-gun at im in the presence of interaction; 2. A method as set forth in Claim 1 comprising the further step of generating the far field pressure signature of the array p(t) by stacking qj(t) and their ghosts with the appropriate time delays

where r & g are the distances of the point at which the far field pressure signature is calculated from the centres of the ith air-bubble and its image, respectively.