CA1161539A - Seismographic method and apparatus - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
The specification describes a method of determining the location in the earth of sub-surface boundaries and/or the acoustic properties of sub-surface layers in the earth. An apparatus to effect the method is also described.

Description

S ~ ~

S P E C I F I C A T I O N

Title: S~ISMOGRAPHIC METHOD AND APPARATUS

D~SCRIPTION

This invention relates to a method of determining the location in the earth of sub-surface boundaries and/or the acoustic properties of sub-surface layers in the earth and to apparatus for this purpose.
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There is well-kno~ a seismic reflection technique ~ which employs a sound source at or near the earth's ; surface to emit an impulsive sound wave at a known time.
As this.sound-wave:passes through the~-earth it en~ounters ~ .
boundaries between:~the-.di-fferent sub-surface layers~:~At-;~
each boundary some:of.the-sound-is-.transmitted-and some.
i reflectsd. A re-ceiver at or-near--the-sur~aGe-close to-the-source detects the reflected waves which arrive at later and-later~times.

A record.(or-seismogram)-made.o~ th~.receiuer response.:--.
is then processed to determine the amplitudes and ar~i~a~ ;times- o~ t~-e individual reflections. ~hes~.ma~.. ..

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then be used to determine the locations of the xock boundaries within the earth and/or the acoustic properties of the rock either side of each boundary.

The accuracy of such analysis depends on the ability of the processing technique to separate the individual reflections from one another. One of the reason that this is a non-trivial task is that it is extremely difficult to generate a purely impulsive sound wave~
The sound wave generated by most seismic sound sources has a duration which is longer than the smallest separation time interval of which the recording system is capable. In other words, the series of reflections which arrives at the receiver is not the series of impulses one desires (the reflectivity series); it is a series of over-lapping wavelets. The processing step which is used to remove the effect of the source from the recorded signal in an attempt to recover the reflectivity series is usually known as deconvolution.

The conventional description of a seismic signal regards the propagation of seismic waves as a linear elastic process in which the signal xl(t) is obtained as the con-volution of the impulse response of the earth g(t) with a far field source wavelet s(t). Usually some additive noise is also present so that xl(t) = s(t) * g(t) + nl(t) (1)

- 2 ~

( ~) S~

where the asterisk * denotes convolution. One wishes to extract g(t) uncontaminated by either s(t) or n1(t). However, n1~t) is not normally known, and often s(t) cannot be measured or predicted and must also be regarded as unknown.

Since s(t), g(t) and n1(t) are all unknowns, the problem of finding g(t) from the measurable quantity x1(t) is basically that of solving one equation containing three unknowns. It cannot be done, of course. ~ven when the noise can be ignored the essen-tial difficulty remains; that of deconvolving s(-t) and g(t). Unless s(t) is known g(t) cannot be found without .a lot of assumptions.

The noise term ~ nl(t) will be ~ll compared wlth the signal term s(t) * g(t) provided there is enough signal energy~ In order to achieve an adequate signal-to-noise ratio it is sometimes necessary to repeat the experiment a number of times in the same place, up to a total of, say, p time~, using the same sound source, or identical sound sources. ~he series..of-received seismic signals x1(t), x2(t),.
...xp(t~ is summed to form-a composite-signal-x(t), where .

x(t) = ~ Xi t) i = 1 . ' ~quation(1 )then becomes x(t) = p.s(t) * g(t) + n(t), (1a) where n(t) is a composite noise signal, given by n(t) = ~ ni(t)~~'i ~his summation i~ known as a "vertical stack" and p i8 the "fold of stack"and i9 an integer greater than or equal to 1. ~he same, or similar,-~result may-some.times be obtained by generat~.ng p identical- -.... . . . . . . ............... - . . = . . .......... - - .
.. ~ .. ~ .. . .

(4) S~
impulsive sound waves simultaneously. If the sources do not inter-act, each one will generate an identical far field wavelet s(t), and the received signal x(t) will be described by equatioh (1a).

~or more than twenty years much ingenuity has been devoted to de-vising methods for solving equation (1) or (1a) using assumptions which are as realistic as possible. But the fact remains that these assumptions are made purely for mathematical convenience.
They are not substitutes for hard information.

The best known.example of such-a method is the least - squares time - domain inverse filtering method used throughout the industry.
For this method to be valid it is required that (1) g(t) be a stationar~, white, random sequence of impulses;-..
(2) ~(t) be minimum-phase.and have the same shape throughout the seismigram;

(3) there be no absorPtion.

All these as~umptions-æ e--very strong, and they must all be correct simultaneousl~^.if:the.,method-is to:work~ This condition-iæ very-- -difficult to-satisfy,..especially-since,the assumptions are not.,=~
mutually reinforcing. ~or example..in attempting to æatisfy,,th~ta-..,'' tionarity assumption,: some sort of spherical divergence--correction must-first be applied.-~his has the effect of.distorting s(t~ un--evenly down the seismogram whi,ch immediately invalidates the-assumption that-,the shape of s(t) remains constant;-it:also intro~
duces a tendency for s(t) to-be non-minimum-phase-in-the early part o~ the seismogram,.

, , ~ : J

(5 ~16~S3 According to a first aspect of the present invention there is provided a method of determining the location in the earth of sub-surface boundaries and/or the acoustic propertie~ of sub-surface features in the earth which method comprises employing one or more first and one or more second point sound sources to produce respectively first and second sound waves, the energy of the elastic radiation of the or each first source differing by a known factor from the energy of the elastic radiation of the or each second source, detecting reflections of said first and second sound waves from within the earth and generating therefrom respective first and second seismic signals and subjecting these two seismic signals to analysis and comparison.

According to a second aspect of the present invention there is provided apparatus for determining the locati.on in the earth of sub-surface boundaries and/or the acoustic properties of sub-surface features in the earth which apparatu~ comprises one or more first point sound sources and one or more second point sound sources adapted respectively to-produce first and-second sound - I
_....... !
waves in the earth, the energy of the elastic radiation of the. .
or each first source differing-by a known factor from the energy -. .
of the elastic radiation of the or each second source3 receiver.
means for detecting reflections of said first and se.cond sound waves from within the earth and generating therefrom respective first and second seismic signals, and means for analysing and comparing said first and second seismic signals.

In the present inventicn the first and second sound sources may be individual point sound sources or there may..be employed a :
--- plurality of identical non-interacting Point sound sources which =_ will produce a seismic signal having a greater signal-to-noise ratio-O~-l (5a) Alternatively the receiver means may be adapted to sum a series of identical seismic signals obtained by repeated production of identical sound waves by one or more iden-tical sound sources.

The term "point source"is employed throughout the specification to denote a source whose maximum dimension is small compared with the shortest wavelength of the useful radiation it generates. If this source is buried in an homogenous isotropic elastic medium it will generate spherically symmetric radiation at distances greater than about a wavelength. This is the far-field region in which any aspherical distortions of thewavefield from this point source will occur only at high frequencies outside the useful bandwidth, ~

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The method Or the present invention is suitable for buriea point sources, on land ana at sea. It requires ~one of the ~ssumptions demanded by known ~et}ods. In pa~ticular, nothing is assumed about differences in the a~plitude or phase spectra of s(t) and g(t). The present invention is based upon the fact that the wavelet obeys a scaling law of the t~pe:

s1~ 2) = c~ s (Cc~ ) (2) In this equation,~1 and ~2 are both very nearly equal to ~=t-r~ whe~e t is time ~easured from the shot instant, r is the distance from the sound source to a point in the far field, and c is the speed of sound in the medium; s1 ~ 2) is the far field wavelet of a source similar to that which ~enerates s(rj), but which contains o~3 times as much ener~y.
~iure 1 sho~s diacrz~m2tically how this source scaling 12w ,~
affects the far field ~avelet. ~;

There is excellent experi~e~tal evidence- for the existence ¦
of such a scaling law for a variety of point sources and -this law can readily-be derived for explosives,-for exa~ple~ ~
if the following assumptions-are ~ade:
i. that the elastic radiation from the source possesses---spherical symmetry; thus it will be applicable to ~ost marine sources such as a single air gunj a single water-gun~
a marine explosive such as that available under--the Trade ~rk "~laxipulse", a marine source emplo~ing high pressure steam to cause an implosion such as that available under the Trade ~ark "Vaporchoc"l ol a sparker and to;explosives buried on land but probably not to surface sources because~ --(7) their radiation is not spherically symmetric;

ii. that the fraction of the total available energy stored in the explosive which is converted into elastic radiation is a cbns~t for a given type of explosive and a given medium;

iii. that the volume of the explosive may be neglected relative to the ~olume-of the sphere--of anelastic defor-mation produced by the explosion, iv. that the elastic radiation produced by the explo-sion could be obtained by replacing the sphere of anelastic -~:' deformation by a cavity at the interior of which there is appl.ied a time-dependent pressl~e function P(t) and that 2(t) is independent of the mass of the e~plosive and is',constant for explosives of.the same chemical composition in the:-same---medium; and ~
._ .
~ v.: that ~r1 -.-.fo~ an--explosi~n-of a first mass and- ' ,., ~ .
~',for-an explosion-.:of.. a second-ma&s ~an be,taken-to-be: , approximately equal and equal to ~. Thi-s-is sufficientl~ --accurate: if the time-interval ~^lr ~etween~ and ~ is unobservable within:the frequency band--of~interest7-i.e.~
~ r =should--be less than about one sam~le interval. -This--approximation will suffice for values of C<up to about 5 or soc .. . .
.... . .
. . .
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To exploit the scaling law a seismic signal x(t) as described by equation (la) is generatedO The experiment is then re-peated in the same place using a source of the same type but containing a3 times as much energy. This will generate a seismogram:
xl(t) = qsl(t) * g(t) + nl(t) (32 where sl(t) is the far field wavelet of the source and is defined in equation (.2).; g(t) is the same as in equation (1) because it is the response of the earth to an impulse in the same placei th,ë. noise nl(.t) may be different from n(t) in equation (la),; q is a known integer greater than or equal to 1 and which may be different from p in the equation (la~

.
Let us consider these equations together for the case where the noise is negligibly small:
x(.t) = ps~t) * g'(t) (4~
x (t) = ~s (.t), * g(.t) (5) s (t), = ~s(t/~) (2) In these three independent equations there a~e three unknowns:
s(,t).~ sl(,t). and g(,t) r Therefore, in principle when the noise is negligi~ly small~ we can solve for all three exactly without maki~ng further assumptions.
.

By ta~ing Fourier transforms and by manipulation we obtain - the equation qS(af) = P2 S(f), R(f) (.62 a where R(,fl is defined as for equation 13 hereinafter.
, .
~ 8 -s~g Equation (6) suggests a recu~sive algorithm of the form:
9~5(C~r\4D ) '- ~ 5 ~ ) R (~
n ~ J ~
where ~ is dicated by the highest frequency of interest and the process must be initiated with a g~less at ~0. If b~ > 1 equation (7) enables us to worX up the spectrlLm calculating values at ~(fo , ~2 fo , ,..,.~. ~ fO
star~ing with a guess at fO.

To co~pute values at frequencies less than fO, equation (6) can be rearranged: /
p S (~ R (f ) ~ such that we obtain the recursion:

3 p ~

-~ where M is dictated by--the lowest frequency of interest.--This now enables the values-:at frequencies ~0 /~ f ~/~7 ' _ to be computed.

. . .
, : " , q (10) S~

~hus from the recursion scheme of equations ~7) and (8) we can obtain values at frequencies ~0/~ 7 f~
~/~ o , ~ f o ~ o( f o ~
We can now use an interpolation routine to find a value at another specified frequency, sa~ f1 and use the recursion to calculate-values-at-~f1, ~2f1 etc. ~his procedure is repeated until sufficient values have been computed. Once S(f~ has been calculated, s(t) is obtained by taking the inverse Fourier transform.

It should be noted that the quantities involved in the al~orith~ are complex. One can operate either with the ~odulus (amplitude) and arg~ment (phase) 3 or with the real and imaginary parts. The~eal and-imaginary parts have been used in the-example as.-these are considered to be the ~ost 'basic' components o~-the comp-lex.n~bers.in-a~-computer~ whereas ampl-itude:-and phase-are---ad~i~tures~~~o~~~
these quantities9 The Initial Guess The.algorithm.is initiated -with a guess. If this:.guess.is wrong, the final result will be wrong. The guess at fO ~s a complex number which, in all probabilitg, will not be the true value at f . In fact, the-guess SG(.~ -is-related -~, ~lS~9 to the true value S(f ) in the following way :

S ~ ' S ~
. (9) i where r e is.the u~known complex error factor. ~f this error-is not taken.into account tnere will be generated the values:

~ 5~ 2 ~ ~ l l ' 2) (10) ., which,--.with sufrisient interpolation-yiela~the functio~--~ SG:-(fO)~forh ~ ~ ...the range :ca~-:be~.extended-t`o~
: the origin by defining~ (o)~ D which:is-compatible.with.
a time-series-8G (t)-with-z-er-o mean.

~he ~ect--of.the-initial error ca~ be seen~by substituti~g :equation 9 into equation 10; thus:

~S4(~ ~o)=~

~"~2 S(~ 0 ) `
~` ~ '' ' . (11) "~

~ , . . .

~ 39 It is evident that the error factor is constant fpr ali the values deduced.from the algorithm. Thus far the algorithm has allowed computation of the function:

SG, ~ e, S~f), (O ~

(12) where ~0 has been assumed to be positive.

~wo problems now exist. ~irst-the transform must be co~pleted by generating values of SG(f) at ne~ative frequenciesO

Secondly the error factor must be found to obtain S(f) from equ~tion (12). Both these problems can be solved by consider-ation-of the physical-properties of s(t~, which impose con-strain~s on the properties-of S(f)...`.

It is known th~t S(t) is real, and therefore-the estimated wavelet should be real. This constraint imposes Hermitian symmetry on Stf). That-is, the real and imaginar~-parts.of S(f) must be eve~ ~nd odd functions, respectivelg. Thus if S(f) is known-for positive frequencies, S(f) can easil~ be computed for negative frequencies using this condition.

Howe~er, only SG(f) is known, which is in error by a phase shift 0 and a scale factor r. The scale factor is unimportant because it has no effect on the shape of s(t), and conse-quently cannot affect our estimate of the sampe of g(t). It can therefore be ignored.

However, the phase error ~ cannot be ignored, because this will make SG(tl non-causal~ and it is known that s(t? is causal. That is, s(,t) is zero for times t less than zero.
In the frequency domain causality imposes the condition that the odd and eyen parts of the Fourier transform are a Hilbert transform pair. It can be shown that t~is causal relation-ship is destroyed unless the phase error ~ is zero.

This consideration sug~ests a trial-and-error procedure for improving the estim,ate of s(t), This is as follows:

lo Compute SG(f) from an initial guess at fO as described above, notin~ that SG(f~ and S(f) are related as in ~12)~
2. Multiply S~(,f) by a correction factor e i~G
where~ Q~ is a guess.
3. Impose Hermitian symmetry.

4, Check for causality. If the recoYered wavelet is non-causal, return to step 2 and repeat, using a different HG. This procedure is repeated until the causality condition is met.

mab~c (14) ~ 6~ 5~9 Thus the equations may be solved in the.frequency domain using the algorithm described above and applying the con-straints which follow from two physical properties of s(t):
it is real and causal~ ~he final estimate of s(t) will be in error only by a scaling factor r, which is tri~ial.
Having obtained~a satisfactory estimate of s(t),g(t) can be obtained using e~uation (1), by standard methods~

The algorithm described depends on -a complex division in the frequency doLlain. :~here are two problems associated with this. First, the ratio will become unstable~-at any freouency at which the amplitude of the denominator is too s~all.
.~.condly, if the denominator contains non- minil~u~-pha.se components ~lhich are not conta~ned in the numerator then the quotient becomes-unstable in the sense that it is non-realisable.

To solve the ~irst problem-it is usu-al--to-add--a s~all threshold :.
of white noise to the denominator to negate-the possibilit~
of zero -or near zero division. An alternative but ~ore time--consuming method is ~o.search for -low values in the denominator -~nd to replace them with small-positive -values~ -Finding the inverse of non-minimum phase wavelets i9 a well-kn`own ~roblem. However, the problem can~be?a~oided~-simply b~

(15)

5~
applying an exponential taper of the form e ~t to both x(t) and x (t). By choosing ~ large enoughthe quotient R(f) can be forced to be stable, but then the estimates of s(t), s1(t) and g~t) will be distorted~ In practice the distortion may'be removed simply by applying the inverse taper e ~t to these functions.

In the presence of noise the problem is to obtain a reliable estimate of the ratio spectrum R(f), for then the scaling law and recursive algorithm can be used -to find S(f) as described above.

From equation (6) we define R(f) in the absence of noise as:
x1(f) qS (f) R(f) = = - ~ ' (1~) X (~) pS (f) It follows that s1(t) = r (-t) * s (t) (14) .
where r(t) is the inverse Fourier transform of R(f) and, since s(t) and s1(t) are both real ~nd causal, r(t) mu~t also be real.
However,~~'r(t) will not be causal unless s(t) is minimum-phase.
Both s(t')-and s1(t) must~~e forced to be minimum-phase by applying the exponential taper to x(t) and x1(t) as described above. Under these conditions r(t) will be real and causal.

.. ,; . . , . .- , -, .. ~ .. ..... . ~

(16) ~l ~l 5 39 In the noise-free case it is also true that , x1(t) = r(t) * x(t) (15) and it will be seen that r(t) is simply a one-sided filter which shapes x(t) into x1(t), provided the correct exponential taper has been applied. When noise is present the estimate of r(t) must be stabilized and this can easily be done using a least-square~
approach (N. ~evin~on, in N. Wiener, 1947; ~xtrapolation, Inter-polation and smo~thing of S-tationary ~ime Series, Wiley~ New York).
A ~hat is, a filter r1(-t) is found which,-for an ~ x(t)-wlll l'npu~
give an ~u~t which is the best fit in a least-squares sense to x1(t). This filter r1(t) will be the best estimate of r(t).

In other words, in the presence of noise r(t) can be calculated in the time domain using standard progra~s, and then its ~ourier transform taken, whence s(t) etc, can be found as described above. -It will be understood:that although~-the problem-has been discussed ~
.
: in terms of the scaled-energies of the elastic radiatlon-of the-=-sources, normally . . .

(17) the particle veloc~ies or the sound pressures generated by the source ma~ be detected and recorded using respectivel~
a geophone or a hydrophone as conventionally employed.

It will further be understood that the individual elements of the apparatus Or this invention ma~ .be chosen at will to be suitable for the particular purpose for ~hich they are required thus air guns, water guns,!IIaxipulse ~Vaporchoc Jsparkers etc.
~i ~ag be employed as the sources. Similarly any suitable ¦ analysers, receivers etc may be em~loyed as necessary.

¦- ~ It is believed that ~may have a value of.from 1.1 bo ~ore pre~erablg fromi1.5 to 3 I
~X~ ~LE

Applying the above-~avelet deconvolution-scheme---to:a -~-.. s~nthetic-example................................................. -: Two independent-synthetic far field source-wavelets as:sh~Q ~-: in ~i~ure 2 were generated. Each wavelet was-calculated:usi~g a ~odel as-describea in the Geoph~sical Journal of the~Royal hstronomical Soci-ety 21, 137-161, for the signal generated . by a~ air gun in water. The ~odel is based on the nonlinear . : oscillations of.a spherical.bubble-in wateriand itakes into - -.
~- .
. ,`''~ .;

~ S 3~

account nonlinear elastic effects close to the bubble. The model predicts waveforms which very closely match measure-ments.

The top wavelet s(t) of Figure 2 was computed for a 10 cubic inch gun at a depth of 30 feet/ a firing pressure of 2000 p.s.i. and a range of 500 feet from the gun. No sea'f sur-face reflection has been included. The bottom wavelet sl(t) was computed using the same computer program for a 80 cukic inch gun, at the same depth, firing pressure, and range.
In other words, only the volume was changed.

Secondly each of these wavelets was convolved with a syn-thetic reflectivity series g(t) shown in Figure 3. The result of performing these convolutions is shown in Figure 4. The top trace x(t) represents the convolution of g(t) with the upper wavelet s(t) of ~igure l; the bottom trace x (tl represents the convolution of g(t) with the lower wave-let sl(t) of Figure-l. Thus these two traces, x(t) and xl(t), were constructed entirely independently without any use of the scaling law.
, It was then assumed that these two traces had been obtained knowing ly-that they were from the same place and that the top one was made using a 10 cubic inch gun, while the bottom was made using an 80 cubic inch gun at the same depth and pressure.

mak/J t ( 1 9 ) 1~15~9 Since only the gun volume was changed, the scaling law can be invoked. In this case 3 = 8; therefore = 2.

Solving for s(t) and g(t) as described above using the set of simultaneous equations (4), (5) and (2) with p = q - 1, sub stituting ~ = 2, the recovered wavelet and reflectivity series are shown in Figure 5; they compare very well with the top wavelet of Figure 2 and the original reflectivity series of ~igure 3. ~he small difference between the recovered series and the original are attributed primarily to computer round-off error.

This example shows that the method is valid in principle.

~hus by means of the present invention in the absence of noise the impulse response of the earth can be obtained exactly. In the presence of noise a stable approximation to this impulse response can be obtainedj- the accuracy of which approximation^
is dependent on the noise level present.
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Claims (17)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A method of determining the location of sub-surface boundaries and/or the acoustic properties of sub-surface features in the earth, which method comprises employing a first point sound source and subsequently but at the same location as said first point sound source a second point sound source to produce first and second mutually non-interfering sound waves, the energy of the elastic radiation of the first source differing by a known factor from the energy of the elastic radiation of the second source and being such that the frequency spectra of the respective sound waves overlap, detecting reflections of said first and second sound waves from within the earth and generating therefrom respective first and second seismic signals and subjecting these two seismic signals to analysis and comparison to define the impulse response of the earth substantially in the absence of the far field source wavelet and noise.

2. A method according to claim 1, wherein a plurality of identical signals is summed to generate said first seismic signal.

3. A method according to claim 2, wherein said plurality of identical seismic signals is obtained by producing a series of identical sound waves by the use of one or more identical sound sources.

4. A method according to claim 1, wherein a plurality of identical seismic signals is summed to generate said second seismic signal.

5. A method according to claim 4, wherein said plurality of identical seismic signals is obtained by producing a series of identical sound waves by the use of one or more identical sound sources.

6. A method according to claim 1, wherein a plurality of identical non-interacting sound sources is employed simultaneously to produce said first sound wave.

7. A method according to claim 1 or 6, wherein a plurality of identical non-interacting sound sources is employed simultaneously, to produce said second sound wave.

8. A method according to any of claims 1 to 3, wherein each sound source comprises an air gun, a water gun, a marine or a sub-surface land explosion generator, an implosive marine source or a sparker.

9. A method according to any of claims 1 to 3, wherein said known factor is from 1.33 to 27.

10. A method according to any of claims 1 to 3, wherein said known factor is from 3.375 to 27.

11. Apparatus for determining the location in the earth of sub-surface boundaries and/or the acoustic properties of sub-surface features in the earth, which apparatus comprises a first point sound source and a second point sound source adapted respectively to produce first and second mutually non-interfering sound waves in the earth with said second point sound source being employed subsequently to said first point sound source but at the same location as said first point sound source, the energy of the elastic radiation emitted by the first source differing by a known factor from the energy of the elastic radiation emitted by the second source and being such that the frequency spectra of the respective sound waves overlap, receiver means for detecting reflections of said first and second sound waves from within the earth and generating therefrom respective first and second seismic signals, and means for analysing and comparing said first and second seismic signals to derive the impulse response of the earth, substantially in the absence of the far field source wavelet and noise.

12. Apparatus according to claim 11, wherein said first sound source comprises one or more identical sound sources arranged to produce a series of identical sound waves and the receiver includes means for summing said series of identical sound waves to produce said first seismic signal.

13. Apparatus according to claim 11 or 12, wherein said second sound source comprises one or more identical sound sources arranged to produce a series of identical sound waves and the receiver includes means for summing said series of identical sound waves to produce second seismic signal.

14. Apparatus according to claim 11, wherein said first sound source comprises a plurality of identical non-interacting point sound sources arranged to produce simultaneously said first sound wave.

15. Apparatus according to claim 11 or 14, wherein said second sound source comprises a plurality of identical non-interacting point sound sources arranged to produce simultaneously said second sound wave.

16. Apparatus according to claim 11 or claim 12, wherein said known factor is arranged to be from 1.33 to 125.

17. Apparatus according to claim 11 or claim 12, wherein said known factor is arranged to be from 3.375 to 27.