WO1994010585A1

WO1994010585A1 - A method for determining the far-field signature of a marine seismic source array

Info

Publication number: WO1994010585A1
Application number: PCT/NO1993/000156
Authority: WO
Inventors: Martin LANDRØ
Original assignee: Den Norske Stats Oljeselskap A.S.
Priority date: 1992-10-23
Filing date: 1993-10-22
Publication date: 1994-05-11
Also published as: NO176227C; NO924122L; NO176227B; AU5434794A; NO924122D0

Abstract

A method for determining the far-field signature of a marine seismic source array (2) from near-field measurements of acoustic signals from the seismic sources (6), wherein the near-pressure field is measured by means of a plurality of sensors (8) on a streamer means (3) located beneath the source array (2). The pressure field from each single source (6) is simulated by means of a computer on the basis of equations describing the pressure field, free adjusting parameters being used to see to it that the simulated pressure becomes as equal as possible to the pressure measured by the sensors (8), the free parameters being adapted to the measured pressure field by means of an inversion algorithm, so that the parameters are updated gradually until optimum accordance between the measured and the simulated pressure field is obtained. The far-field signature of the source array (8) thereafter is determined in that each single source (6) is simulated with the best determined parameters and the contribution from each single source is superposed in the far field.

Description

A method for determining the far-field signature of a marine seismic source array.

The invention relates to a method for determining the far-field signature of a marine seismic source array from near- field measurements of acoustic signals from the seismic sources, wherein the near-pressure field is measured by means of a plurality of sensors on a streamer means located beneath the source array.

Such a method is known from US patent specification No. 4 648 080.

The best known method for estimating or determining far-field signatures as a function of angle from a seismic source array is the so-called near-field to far-field extrapolation method which was first proposed by Ziolkowski et al. (see A.

Ziolkowski, G. Parkes, L. Hatton and T. Haugland: "The signature of an air gun array: Computation from near-field measurements including interactions", Geophysics, Vol. 47, No. 10, 1982, p.

1413-1421. By placing a hydrophone closed to each single gun in the array (typically at a distance of 1 m), so-called effective sources can be determined. The far-field signature in any direction thereafter can be found by summing the contribution from the effective sources.

This method has, however, some deficiencies, viz. that it is unstable for compact/large arrays, that it is sensitive for errors in source-hydrophone positioning, - that it is a problem that one does not know the movement of the air bubble, and the method is based on the assumption that the hydro¬ phones are situated in the linear zone around the source. The most serious weakness of this method is that it becomes unstable for arrays having many air guns (see M. Landrø, S. Strandenes, S. Vaage, 1991, "Use of Near-field Measurements to compute Far-field marine source signatures - Evaluation of the Method", First Break Vol. 9, No. 8, p. 375-385). An error in the source-hydrophone positioning of 10 cm typically will give an amplitude error of 10% in the effective source signature. Further, it is supposed that the air bubble rises with a constant velocity after the air gun has been fired, and possible deviat- s ions from this assumption will also deteriorate the result. The method is based on the fact that the near-field hydrophones are placed in the linear zone around each source, i.e. that possible non-linear effects will disturb the estimated effective source signature (see M. Landrø, 1988, "Higher Order Corrections in o Calculation of the Pressure Wave Field generated by Seismic Sources", Expanded Abstracts from the 58th annual SEG-meeting in Anaheim 1988).

Another method which -has been used for determining the far-field signatures of a seismic source array, consists in s measuring the source response of a signature streamer (mini- streamer) just beneath the source array. This method corresponds to the introductorily stated method and was proposed by Neil D. Hargreaves in 1984 (see said US patent and also N. Hargreaves, 1984, "Far-field Signatures by Wavefield Extrapolation", a paper o presented on the 46th EAEG-meeting in London, 1984). The far- field signature in an arbitrary point is determined by extrapola¬ ting the measured response with a two-dimensional migration algorithm. The drawback of this method is that it is based on a two-dimensional wave equation, and that aperture effects as a 5 result of a finite length of the streamer will deteriorate the result.

It is an object of the invention to provide an improved method for determining the far-field pressure of a marine source array in all directions, the method being more flexible and o accurate than the previously known methods.

The above-mentioned object is achieved with a method of the introductorily stated type which, according to the invention, is characterized in that the pressure field from each single source is simulated by means of a computer on the basis of equations describing the pressure field, free adjusting para¬ meters being used to see to it that the simulated pressure becomes as equal as possible to the pressure measured by the sensors, the free parameters being adapted to the measured pressure field by means of an inversion algorithm, so that the parameters are updated gradually until optimum accordance between the measured and the simulated pressure field is obtained, whereafter the farfield signature of the source array is determined in that each single source is simulated with the best determined parameters and the contribution from each single source is superposed in the far field.

The method according to the invention is based on the fact that the physical parameters describing each single source in the array are calculated by means of an inversion algorithm. After the pressure field of the seismic sources has been measured on the streamer which is situated just beneath the source array, effective source signatures for each single source are determined by means of the inversion algorithm. The inversion algorithm is based on the fact that the air bubble generated by each source can be modelled or simulated physically. Typical parameters which are determined by the inversion, are source depth, attenuation constants (which will be further described later), the reflection coefficient at the water surface and parameters describing the geometrical shape of the streamer. It is not necessary to know the exact depth of each single source, and neither the position of each single sensor or hydrophone in the streamer, the method also making possible a determination of these parameters.

The invention will be further described below with reference to the drawing the single figure of which shows a side view of an equipment used when carrying out the method according to the invention.

In Fig. 1 there is shown a vessel 1 towing a seismic source array 2 and a signature streamer 3 (mini-streamer) through a body of water 4 above a seabed 5. The source array 2 consists of a plurality of seismic acoustic sources 6 towed at a depth z_g beneath the ocean surface 7, and which for example may consist of air guns. The streamer 3 comprises an array of a plurality of sensors 8 which for example may be hydrophones having a suitable spacing, in the way it will be known to a person skilled in the art. The streamer 3 extends a distance ahead of and behind the source array 2 and is towed at a depth z₀ beneath the ocean surface, so that the streamer is situated just beneath the source array. The sensors 8 then sense the near-pressure field of the sources 6 when these are fired when practising the method. A description of the modelling or simulating theory underlying the invention will be given below.

In 1942, Kirkwood and Bethe made a very thorough theoretical study of the shock wave formed in an underwater 5 explosion (see J.G. Kirkwood and H.A. Bethe, 1942, OSRD Report No. 588). This theory also may be used to describe what happens when a seismic air gun is fired under water. This theory takes into account that the water is compressible. The most important result of the work of Kirkwood and Bethe is a non-linear equation o (see below) describing the movement of the air bubble under water. Gilmore in 1952 derived an expression for the acoustic pressure field which will be generated by an expanding air bubble in the far field of the bubble, which for an air gun means ca. 100 m from the bubble (see F.R. Gilmore, 1952, "Collapse of a s spherical bubble", Report No. 26-4, Hydrodynamics Laboratory, CALTEC, Pasadena, California). Ziolkowski in 1970 presented a method based on the works Kirkwood-Bethe and Gilmore, for estimating the pressure wave shape from a seismic air gun ( see A. Ziolkowski, 1970, "A Method for Calculating the Output o Pressure Waveform from an Air Gun", Geophys. J.R. Astr. Soc. , Vol. 21, p. 137-161).

When a seismic air gun is fired, an air bubble is produced in the water. The pressure within the bubble in the beginning is very high, ca. 100-200 bars, whereas the hydrostatic 5 pressure in the water typically is 1,5-2,0 bars (dependent on the depth). Because of this overpressure, the bubble will expand quickly in the beginning, and the pressure within the bubble after a while will be less than the hydrostatic pressure. As the bubble arrives at its maximum size, the pressure within the o bubble will be minimum, and the hydrostatic pressure will see to it that the bubble again starts contracting. This results in that the bubble movement gets an oscillating nature, and the bubble will remain oscillating quite to the point when it arrives at the surface of the water. Since the acoustic signal is due to volume 5 changes from the bubble, also the observed pressure field of a seismic air gun will be oscillating.

A main problem in the modelling theory of seismic air guns has been that the theoretically estimated pressure sig¬ natures have had a too small attenuation from one oscillation to the next one. This means that the air bubble gives off more energy to the surroundings than what is included in the theory of Kirkwood and Bethe. According to the present invention, in order to take this effect into account, there is used a modified Kirkwood-Bethe equation for the bubble:

wherein R is the bubble radius, C is the sound velocity in the water, H is the enthalpy on the bubble wall, α is an attenuation constant (α = 0 corresponds to a "true" Kirkwood-Bethe equation), and β is an empirical time constant influencing the bubble period. (Increased β gives an increased distance between two peaks in the signal) . The main point of introducing the addi¬ tional members α and β is to enforce a best possible matching to the measured near-field signature. Without these members there will be large deviations between theoretically modelled and measured source signatures from air guns. The attenuation constant α and the time constant β represent free adjusting parameters which are adapted to the measured pressure by means of the inversion algorithm.

The enthalpy H may be calculated from the so-called Tait equation which is a condition equation for the water and has the following form:

'Pl^rl_^) . (A. <2>

wherein p„ is the hydrostatic pressure, p is the density of the water, and p(r) is the pressure. B and ζ are empirical constants (B = 2500 atmospheres and ξ = 8). The enthalpy is defined as h = f ^P P . ( 3 )

^Jp- P

Use of the Tait equation now gives the following expression for the enthalpy on the bubble wall:

wherein p(R) is the pressure in the water at the bubble wall.

Mass transport of air from the air chamber of the air gun is described by the following equation:

A = — for 0<t<τ, (5)

wherein m₀ is the initial air mass and τ is the time needed for emptying the air chamber.

An equation for describing the temperature (T) within the air bubble may be derived, the starting-point being taken in a quasi-stationary process in an open thermodynamic system. In such a system, the first principle clause of the thermodynamics is modified to the following expression:

hdm + dQ - dU + pdV, (6)

wherein h is specific enthalpy, m is air mass, Q is heat transfer, U is internal energy, p is the pressure, and V is the volume. It is supposed that the heat transfer dQ is small as compared to the other members in the above equation. The internal energy and the enthalpy are given by

U - mc_vT, (7 ) h ^~ C_pT, (8) wherein c_υ and c_p are specific heat capacities with constant volume and pressure, and T is the air temperature in the bubble. Use of the equations 6, 7 and 8 gives

R_σTm - pV

T - —^~ — , tnc„ o:

wherein R_g is ideal gas constant, p is the pressure within the bubble, V is the volume of the bubble, m is the air mass, and c_u is specific heat capacity with constant volume. In the expression (9) one has utilized the circumstance that, for an ideal gas, the difference between c_p and c_υ is equal to the ideal gas constant

The equations 1, 5 and 9 now may be solved numerically to find bubble radius, bubble wall velocity, air mass in the bubble and the bubble temperature as a function of time.

The far-field pressure generated by an oscillating air bubble may be expressed by the enthalpy on the bubble wall, the bubble wall velocity and the bubble radius. Gilmore (1952) derived the following expression for the far-field pressure:

P_f - P. ^~ - * ^♦ f ^■ (10)

wherein p is the density of the water and r is the distance from the centre of the bubble to the far-field point.

For an array consisting of several air guns, the far- field pressure may be estimated by a simple superposition of the pressure contribution from each single source. However, the pressure field from the single sources is modified because of interactions between the sources. These interactions may be included by introducing a time-dependent member (see the aforementioned paper of Ziolkowski et al., 1982) in the expres¬ sion of the hydrostatic pressure in the water around each source: P_h("^t) = P- ^{+ Σ} pressure contributions from other sources and surface reflections (11)

This time-dependent hydrostatic pressure will affect the enthalpy (see above) which will in turn affect the bubble movement ( equation 1 ) . This means that the acoustic pressure from one source will affect the bubble dynamics of another source at a delayed time corresponding to the propagation time between the sources.

The far-field signature (s_n(t)) of a source n at a distance r_n now can be expressed as follows:

^- ^• -r_Ω ^~ - - } ^t - - ^t-^Ξ '- ^<12)

wherein κ₀ is the reflection coefficient at the water surface, and r'_n is the propagation distance for the reflected wave from the water surface (see the drawing figure). Here, p_n(t) is the farfield pressure given by equation (10), but with interaction effects. The pressure field at the streamer in the Figure now may be calculated from equation (12).

When practising the method according to the invention, the pressure field from each single source is simulated by means of a computer on the basis of the above stated equations describing the pressure field. As mentioned, free adjusting parameters are used to see to it that the simulated pressure becomes as equal as possible to the pressure measured by the sensors, the free parameters being adapted to the measured pressure field by means of an inversion algorithm. This means that the adjusting parameters are updated gradually, quite until optimum accordance is achieved between the modelled or simulated field and the pressure field at the streamer. The total pressure field of the array thereafter is determined by modelling each single source with the best determined adjusting parameters, and thereafter superposing the contribution from each single source in the far field.

The utilized inversion procedure is based on minimizing the following mismatching function: ^• t_j) )- ( 13 )

wherein s^ob'(x₁,t_j) is the pressure measured at the signature streamer in the drawing figure, N_r and N_t are the number of receivers and the number of time steps.

In order to avoid numerical instabilities as a result of differences in order of magnitude of the model parameters, the following scaling is used:

(14)

wherein H = J^TJ is an approximate Hess matrix and J is the Jacobi matrix. The Jacobi matrix is given as the derivative of the modelled signal with respect to the model parameters θ (e.g. source depth, attenuation constants, time constants asf.), i.e. J_±J = dSi/θθ_j, wherein i goes across data space and j across parameter space, g = J^τΔd is the gradient, where Δd = s-s^obs is the data residual vector. It is important to note that there are a source depth (z_g), an attenuation constant (α) and a time constant (β) which are to be determined for each source, so that the number of inversion parameters increases with the number of sources, attenuation constants, time constants.

As an optimalization method there is used an attenuated Gauss-Newton method. The scaled and updated model parameters thereby are given as :

Δδ = (H + iD ^g, (15)

wherein λ is the Levenberg-Marquardt attenuation parameter.

In practice it is difficult to know exact positions of all single hydrophones in the signature streamer. Possible positioning errors will affect amplitudes as well as propagation times. In order to make the method according to the invention more robust and flexible, the geometrical shape of the streamer in the water may be included in the inversion algorithm. This is done by assuming that the shape of the streamer can be expressed in the following manner:

z = z_n ( 16 )

wherein z₀ is average streamer depth, L is the streamer length and K is the number of Fourier coefficients which are necessary for representing the shape of the streamer. It is envisaged that only 4-5 Fourier coefficients will be necessary to describe a usable streamer shape.

It is well known that the reflection coefficient (κ₀) at the water surface may deviate from the ideal value (-1,0). This effect may be due to the fact that surface is rough, and that air all the time leaks out from the air guns. In order to takes this effect into account, also the reflection coefficient of the water surface may be included in the inversion procedure, the reflection coefficient being included in the above equation (12).

Claims

P a t e n t C l a i m s

1. A method for determining the far-field signature of a 5 marine seismic source array from near-field measurements of acoustic signals from the seismic sources, wherein the near- pressure field is measured by means of a plurality of sensors on a streamer means located beneath the source array, CHARACTERIZED IN that the pressure field from each single source is simulated o by means of a computer on the basis of equations describing the pressure field, free adjusting parameters being used to see to it that the simulated pressure becomes as equal as possible to the pressure measured by the sensors, the free parameters being adapted to the measured pressure field by means of an inversion s algorithm, so that the parameters are updated gradually until optimum accordance between the measured and the simulated pressure field is obtained, whereafter the far-field signature of the source array is determined in that each single source is simulated with the best determined parameters and the contribu- o tion from each single source is superposed in the far field.

2. A method according to claim 1, CHARACTERIZED IN that parameters for the geometrical shape of the streamer means in the water is determined by means of the inversion algorithm.

3. A method according to claim 1 or 2, CHARACTERIZED IN 5 that the reflection coefficient at the water surface is deter¬ mined by means of the inversion algorithm.

5