DE3014897A1 - METHOD AND DEVICE FOR DETERMINING THE SEISMOGRAPHIC POSITION - Google Patents

Description

Antoni Marjan Ziolkowski, London SW 15
and

Seismograph Services Limited, Keston (Great Britain)

Method and device for seismographic
Orientation

The present invention relates to a method for determining the position of underground interfaces or boundary layers and / or for determining their acoustic properties
by emitting acoustic waves and analyzing the reflected seismic signals.

There is an established technique using such seismic lessons that uses a sound source on or near the surface of the earth to emit an acoustic shock wave at a predetermined time. As this sound wave travels through the earth, it hits the interfaces between the various subterranean layers. At, - each of these interfaces becomes a fraction of the
Sound wool is transmitted, and the other fraction is reflected. A receiver on or near the surface of the earth in the immediate vicinity of the sound source registers the reflected sound waves that arrive one after the other.

The receiver takes a recording (a seismogram) of these sound waves, this recording is then evaluated, the amplitudes and arrival times of the individual reflections

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to evaluate functions. These can then be used to the position of the rock strata or geological formations within the earth or their acoustic properties (the Rock layers) on both sides of an interface.

The accuracy of such an analysis depends on how far one uses a method that is able to identify the individual Separate reflections of the individual layers from one another. This is not trivial as it is extremely difficult to generate a pure short-term shock wave. Namely, the sound waves generated by most seismic sound sources have a duration which is longer than the actually possible temporal resolution of the recording system. Different In other words, the sequence of yeast lessons that arrive at the recipient is not the pulse sequence that one actually wants to know, namely a sequence of impulses that the Specifies reflection distances, i.e. the temporal reflection sequence; rather, it is a series of mutually overlapping wave trains. The procedure that is used to eliminate the influence of the sound source from the recorded signal is used to determine the actual reflection sequence winning is commonly referred to as deconvolution.

The generally accepted description of a seismic signal assumes that the propagation of seismic waves is linear and elastic, the seismic signal X ₁ Ct) from a convolution (superposition, folding) of the impulse response of the earth g (t) with the emitted Wave train s (t.) Of a point transmitter. In addition, there is usually background noise, so that this relationship can be described as follows:

X ₁ Ct) = s (t) * g (t) + U ₁ OO CD

where * indicates convolution. The aim now is to find the impulse response g (t) from this connection with s (t) and n

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to solve to gain the reflection distances. However, n ^ (t) is usually not known and s (t) can often be are not measured or determined, must therefore also be regarded as unknown.

Since the functions s (t), g (t) and n ^ (t) are all unknown, the above problem arises, the function.g (t) (the impulse response of the earth) from the measured seismic signal x ^ (t) essentially poses the problem of solving an equation with three unknowns. This is natural not possible. Even if the background noise can be neglected, the main difficulty remains namely the separation of the functions s (t) and g (t). Without the knowledge of s (t), the function g (t) can only take a series be determined by assumptions and assumptions.

The noise n ^, (t) can be compared to the other equation term s (t) * g (t) can be assumed to be small if there is sufficient signal energy. In order, therefore, a reasonable one To achieve the signal-to-noise ratio, it is sometimes necessary to repeat the measurement several times at the same point, total p ~ times. using the same sound source or identical sound sources. The sequence of the seismic signals x ^ Ct), Xg (t) .... X-jCt) is added up and gives a composite signal x (t), where

This equation (1) becomes

x (t) = p.s (t) * g (t) + n (t), (1a)

where n (t) is a resulting noise signal represented by the function

n (t) =

i = 1

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This summation is known as a "vertical stack", where ρ is an integer greater than or equal to 1. The same or a similar result can be achieved by generating ρ identical sound waves at the same time. If these sound waves do not affect each other, each will produce the same output wave s (t), and the received one seismic signal x (t) is described by equation (1a).

Over a period of more than 20 years is a lot of effort has been used to figure out methods to solve equation (1) or (1a), assuming as much as possible should be realistic. The fact remains, however, that these assumptions are purely for mathematical reasons are required and concrete information cannot replace.

The best known example of such a method is an industrial process C ^ Qast-squares time-domain inverse filtering "), for whose validity the following conditions must be met:

(1) s (t) is a stationary, "white" random sequence of pulses;

(2) s (t) has minimal phase and has the same form throughout Seismogram;

(3) there is no absorption.

All of these assumptions are very significant, and they must all hold true simultaneously if this method is to produce reliable results should deliver. This condition is very difficult to meet, especially since the existence of these assumptions does not exist mutually supported. For example, trying to meet the condition "g (t) stationary" leads to a species of spherical divergence correction, which must first be carried out «, this in turn leads to a distortion of s (t) in the seismogram, which obviously contradicts the assumption that the shape of s (t) is constant; moreover it will

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reinforces a tendency that s (t) does not have a minimum phase in the early part of the seismogram.

In contrast, it is the object of the invention to provide a method and an arrangement that are based on assumptions and assumptions this scope is not required.

This object is achieved by the invention according to the characterizing part of claim 1 and claim 11, respectively.

In the present invention, the first and second sound sources may be or may be individual point sound sources can have a number of identical non-interactive point sources which generate a seismic signal with a better signal-to-noise ratio. Alternatively, can the receiver can be equipped to sum up a sequence of identical seismic signals transmitted by a repeated generation of identical sound waves can be caused by one or more identical sound sources.

The term "point source" used in the present description is used, is intended to mean that a sound source is used, the maximum extent of which is small Compared to the shortest wavelength of the usable radiation it generates. If such a sound source housed in a homogeneous, isotropic, elastic medium, it generates at intervals greater than approximately a wavelength a spherically symmetric radiation. In this area (far-field) there are therefore deviations from spherical symmetry of the wave field of this point source only at high frequencies that are outside the practically usable bandwidth lie.

An embodiment of the method according to the invention will now be explained in more detail with reference to figures. Show it:

Fig. 1 is a schematic representation of two sound sources of different energy and the generated by them

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begot scarf! wave,

2 shows the course of a calculated puncture s (t) for different radiation energies,

Fig. 3 an assumed impulse response (reflection distances) g (t),

4 shows the punctures assumed according to the figures 2 and 3 resulting seismic signals x (t) and x '(t),

5 shows the results from the predetermined signals x (t), x '(t) according to FIG. 4 function g (t) obtained by the method according to the invention.

The method according to the invention is suitable for subterranean point sources, both in the ground and in water. None of the The conditions or assumptions described above are required. In particular, nothing is said about the phase relationships of the Spectra of s (t) and g (t) assumed. The present invention is based on the fact that between two wave trains different energy content there is a scale relation of the following kind:

In this equation, the parameters are TT. and T "p both nearly equal to the value T" = t- ^r / c, where t is the time measured from the ignition time of the sound source, r is the distance from the sound source to a point in the spherically symmetrical range of sound radiation and c is the speed of sound in the relevant area Medium; s (TT ^) is the train of sound waves from a sound source, similar to the sound source that ^{generates the signal 3} CTT * -]), but which has oC ^ times as much energy. FIG. 1 shows schematically how this scale relationship describes the wave train in the spherically symmetrical region of the point source.

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That such a hate-rod relationship actually exists for a large number of point sources, there are excellent ones experimental evidence. This relationship can be derived immediately, for example for explosive charges, if the following assumptions are made:

i) that the acoustic radiation from the source is spherically symmetric; this applies to most waterborne sound sources to, such as a single air blast, a single water blast, or a sea blast, such as it is available under the name "Maxipuls" or a Seescha.llq.ueHe using pressure steam, which produces an implosion, as it is available under the name "Vaporochoc", or a spark discharge or explosives below the surface of the earth. However, this may not apply to sound sources on the surface because their radiation field is not spherically symmetrical is constructed;

ii) that fraction of the total energy stored within the explosive that is in acoustic radiation is transformed is a constant for a particular type of explosive and medium;

iii) that the volume of the explosive versus the volume of the area of inelastic deformation caused due to the explosion, can be neglected;

iv) that the elastic radiation from the explosion could be obtained by removing the area of the inelastic Deformation replaced by a cavity inside to which a time-dependent pressure function P (t) is applied and that P (t) is independent of the mass of the explosive material and constant for explosives of the same chemical composition is in the same medium and

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ν) that ΊΓ ^ | for the explosion of a first mass and ^ for an explosion of a second mass can be roughly equated and equal to T ". This is sufficiently accurate if the time interval Δ T7 between T ^ and T ^ cannot be observed within the frequency band used, ie ΔΤ should be less than one measurement interval.This approximation applies to values of © C above approximately 5.

To evaluate this scale relationship, a seismic signal x (t) is generated as described by equation (1a). This process is then repeated at the same point with a sound source of the same type, only that this sound source oC - has times as much energy. This leads to the following seismogram:

X ¹ Ct). qs ¹ (t) * g (t) "+ n ¹ (t) (3)

where s (t) is the emitted wave train of the sound source according to equation (2); g (t) has the same function as in equation (1), because it represents the impulse response of the earth to an impulse at the same point that the noise signal η (t) can be different from n (t) of equation (1a); q is a known integer with a value greater than or equal to 1, the can be different from the number ρ in equation (1a).

If one neglects the noise signal, these equations can be written as follows:

x (t) «ps (t) * g (t) (4)

x ¹ (t). qs ¹ (t) * g (t) (5)

s ¹ (t) - ^ s (t / oO. (2)

In these three independent equations there are now three unknowns, namely the functions s (t), s (t) and g (t). Therefore it is basically possible to exactly match this system of equations for all three unknown functions

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solve if one only assumes that the noise signal is negligible is small.

By forming the Fourier transform and by appropriate Conversion we get the following equation:

qS (oCf) = ~ S (f) .R (f) (6)

This equation (6) leads to a recursion formula of the following kind:

η = 1, 2, ... ₉ N (7)

wob & i IT given by the highest frequency of interest and the recursion method must be started with an estimated value at the frequency f. If oC ^ d, allows equation (7) to process the spectrum, where-

p at then the values at the frequencies o-Cf-. » c4 f _o ?

TI OO

oC f can be calculated based on an estimate at the frequency f.

In order to calculate values at frequencies below f _Q au, equation (6) can be changed over as follows

pS (f) = qoc S (oCf) / R (f)

so that we get the following edification rule?

° / ^ 2 ο / o / .n-iw ,, / ο /, SC J ^ ) / RC / oC

, 11 = 1, 2, ..., H (8)

where M is given by the lowest frequency of interest. This now enables us to calculate the values for the frequencies £ _ / © £ ₉ f _ft / cC, .... «, £ _ / © (. See below =

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From the recursion rules of equations (J) and (8)

/ M / 11-1 we can find values at the frequencies f _Q / oL , ^f ₀ / ^ '° · - · »

f ₀ / oL »f _o > ο ^ ί _ο , ...... · μΑ _ο received.

We can now use an interpolation method to find a value at another specific frequency, for example f "and use the heursion tools,

ι 2

around the corresponding V / er at the frequencies <^ f ^, c < f ^, etc. to be determined. This process is repeated until sufficient values have been calculated. Once S (f) has been calculated, s (t) can be calculated using the reverse Fourier transform received back

It should be noted here that the quantities used in this algorithm are complex «, one can either calculate with the magnitude (amplitude) and the angle (phase) or with the real and imaginary part. In the example is the real and imaginary parts have been used, as these are the most fundamental components of complex numbers in one Computers are treated, whereas the magnitude and angle (amplitude and phase) of these basic quantities are real and real Imaginary part are »

The initial approach

The algorithm is started with an initial approximation. If this initial approximation is wrong, the end result will be wrong too »The approximation at frequency f is a complex number that in all likelihood does not give the true value at frequency f. In fact, the approximation ß _G (f _Q ) is correlated with the true value S (f) as follows:

S _G (f ₀ ) = re ^ie S (f ₀ ) (9)

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where re ^{σ is} an unknown complex error factor. If this error factor is not taken into account, these values are produced:

η = 1, 2,, If (10)

which, with sufficient interpolation, lead to the function S _r (f) for the frequency range o / oC ^ f ^ f. This range can be extended to zero by defining S ~ (o) = o, which _{is compatible with a time curve s &} (t) with! Zero amplitude.

The effects of the estimation error can be estimated by inserting equation (9) into equation (10), this results in:

It is evident that the error factor is constant for all values derived with this algorithm. So far has this algorithm thus allows to calculate the following function:

S _G (f) = re ⁱ⁰ S (f), (Oifijg (12)

where the frequency f has been assumed to be positive.

Now there are two problems: First, the transformation can be completed by adding values of S ^ (f) to negative Frequencies are generated.

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Second, the error factor must be found in order to function Calculate S (f) from equation (12). Both problems can be solved by looking at the physical properties of the function s (t) are taken into account, which consequently lead to correlated properties of the frequency spectrum S (f).

Ss is known that the function s (t) is a real function is, and therefore the wave train obtained by estimation should also be real. This restriction requires that the frequency spectrum S (f) has Hermitian symmetry. That in turn means that the real part and the imaginary part of the function S (f) each consist of even and odd functions must exist. From this it follows in turn that if the function β (f) is known for positive frequencies, then too the function S (f) can easily be determined for negative frequencies.

On the other hand, only the contactor spectrum S ~ (f) is known, that with an error in the form of a phase shift θ and a proportionality fcsfector r compared to the actual Spectrum is affected. The pronortionality factor r is insignificant as it has no effect on the form of the function s (t) and therefore not the intended one Estimate about the shape of g (t) can affect. This proportionality factor r can therefore not be taken into account stay"

However, the error angle θ cannot be neglected, because this makes the function S "(t) non-causal and it is known that s (t) is causal. This means that s (t) is also zero for times t less than zero »in the frequency representation the causality requires that the even and odd parts of the Fourier transform Hilbert transform must be. It can be shown that this relationship exists only when the error angle θ Is zero.

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This consideration leads to a method for improvement the estimate of the puncture s (t). This is done as follows:

1. Calculate the "frequency spectrum S ^ (f) with an initial estimate" at the frequency f as described above, taking into account equation (12) for the functions S _G (f) and S (f).

2. Multiply the estimated frequency spectrum S _&

with a correction factor e ~ ° ^ö _s where θ ^ is an estimated value.

3o Require Hermitian symmetry of S (f) »

4-; "check the ICausality condition o If the wave train s (t) obtained in this way is non-causal, return to step" 2 and use a different estimate for 0q. This procedure is repeated ₃ until the causality condition is met

The equations can therefore be solved in frequency space using the algorithm described above, the recursion rule _s , whereby two restrictions can be used which follow from the physical properties of the function s (t), namely that this function is real and causal in the sense explained above must be “The approximate value of s (t) finally reached will only differ from the actual curve of this acoustic wave train due to the proportionality factor r, which is irrelevant here. Once a satisfactory estimate of the function s (t) is made i'st ₉ , the impulse response g (t) of the earth can be determined from equation (1) using standard methods.

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The algorithm described includes a complex division in the frequency domain. There are two problems associated with thisί First, the break becomes unstable at frequencies at which the amplitude of the connoisseur is too small ο Second, when the Denominator out-of-minimum phase includes components that are not are contained in the numerator, the quotient also becomes unstable, in this sense that it cannot be calculated »

To solve the first problem, it is customary to add a small noise spectrum (white noise) to the denominator, to prevent division by zero or values close to zero from occurring., An alternative, but temporal The more elaborate grade is to look for very low values of the denominator and to add them to small positive values to replace"

Finding the inverse function of non-minimum phase wave trains is a known problem. However, this problem can be circumvented simply by applying an exponential attenuation function of the form e ö to both the function x (t) and the function χ (t). By choosing ^ G ^ oß enough, the quotient R (f) can be made stable, but then the estimates of the functions s (t), s (t) and g (t) will be skewed. In practice, the distortion can be eliminated simply by applying the inverse attenuation factor e R to these functions »

In the case of the presence of noise, the problem is to obtain a reliable estimate of the ratio spectrum R (f) to get, since in this case the hatred relation and the recursion algorithm are also applied can be used to determine the function S (f) as described above «

From equation (6) we define R (f) in the absence of noise as follows %

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x'Cf) 0.S ¹ Cf)

(13)

X (f) pS (f)
It follows that

s ¹ (t) = r (t) * s

where rCt) is the reverse Fourier transform of RCf) and, since sCt) and s (t) are both real and causal, consequently rCt) must also be real. On the other hand, rCt) will not be causal, unless sCt) has a minimum phase. Therefore, both sCt) and s (t) must be brought into minimum phase, by applying the exponential attenuation function as described above to the functions x (t) and χ Ct)

applies. Under these conditions rCt) becomes real and be causal.

Im! "All that there is no interference also applies

X ¹ Ct) «rCt) * x (t) (15)

from which it follows that r (t) is simply a! tilter, which transforms the function χCt) into the function χ Ct), provided that the correct exponential attenuation function has been applied. If there is noise, the Estimate of rCt), and this can can easily be achieved by minimizing the root mean square (N. Levinson in N. Wiener, 19 ^ 7; extrapolation, Interpolation and smoothing of Stationary Time Series, Wiley, New York). From this one can find a filter r Ct) that for an output signal x (t) results in an output signal that is the better approximation in terms of a minimization in the quadratic mean to the function χ Ct). This filter function r (t) represents the best estimate of the function r (t).

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In other words, in the case of a house r (t) can be calculated in the period with standard programs, and then the Fourier transform can be formed,

whereas s (t) etc. can be calculated as described above =

Although the problem so far is related to the scaled energies of the elastic radiation of the sound sources Has been discussed, the particle velocities or the sound pressures emitted by the sound source generated, registered and recorded using either a geophone or a hydrophone «

It should also be pointed out that the individual components of the device according to the invention can be freely selected to suit the particular purpose for which they are required, \ vie ζ 0B ₀ air explosives, water explosives, etc. _o can be used as sound sources ο It wirü assumed that <sL has a value from 1.1 to 5? in particular from 1 ₅ 5 to 3 may have _o

In the following, the algorithm described above is applied to a given example

The starting point are two sonic wave trains, as shown in the figure, This is a calculated curve, based on a model as described in the Geophysical Journal of the Royal Astronomical Society, 21.0.137-161 j for a signal such as that generated by an air blast in water ο This model is based on the non-linear oscillations of a spherically symmetrical bubble in water and also takes into account non-linear elastic effects in the un-

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indirect environment of the air bubble. This model allowed the computation of waveforms which correspond very closely to corresponding results.

The wave train s (t) in the upper half of Figure 2 was for an explosive charge with a volume of 10 cubic inch that was at a depth of 30 feet and one Fire pressure of 2,000 p.s.i. exhibited as well as a range 500 feet from the explosive charge. No surface reflection of the army was included. The wave train s (t) in the lower half of FIG. 2 was generated using the same Gomputerprogramins calculated, with only the The volume of the explosive charge was 00 cubic inches, but otherwise the same depth, same fire pressure and same range were used. In other words, only the volume has been changed.

These two wave trains are now also artificial Impulse superimposed at tvi location g (t), as shown in Figure 3 is shown. The result of these superimpositions is shown in Figure 4-. That emerges from the superimposition of this The seismic signal x (t) resulting from both functions in the upper half of FIG. 4- accordingly shows the superposition from g (t) with s (t), the seismic signal χ (t) in the lower Half of Figure 4- shows the superposition of the functions

si

nen g (t) with the wave train s (t) of Figure 1. These two reflected seismic wave trains are therefore mathematical have been constructed independently of each other without using equation (2).

To check the method according to the invention it was now assumed that these two seismic signals from one Measurement were obtained of which one only knows that it was made at the same point and that one signal with an explosive charge of 10 cubic inches, the other signal with an explosive charge of 80 cubic inches at the

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same depth and pressure.

Since only the volume of fire was changed, equation (2) can be applied, in which case <? C = 8 is, and therefore o4 = 2.

the recursion algorithm is now used, as described above, to obtain the functions s (t) and g (t) from the equations (4 ·)> (5) and (2) with ρ = q = 1 and oL = 2, the result is shown in FIG. 5: The upper half of FIG. 5 shows the course of the function s (t) as it was calculated using the method according to the invention; the lower half shows the impulse response g (t ) shown how it was calculated with the method according to the invention. If one now compares these wave trains with the curves of FIG. 2 and FIG. 3 on which the fictitious seismic signal is actually based, one recognizes a very satisfactory correspondence. Small differences between the calculated spectra and the original spectra are mainly due to computer errors.

This fictitious example, based on calculated functions for the recovery of the entered functions, shows that the method according to the invention is fundamentally his Purpose served

Therefore, the method according to the invention can provide the impulse response Detect g (t) exactly in the absence of thimbles. In the case of the presence of noise, it can be stable Approximation to this impulse response gCt) can be achieved, the accuracy of which depends on the strength of the noise.

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Claims (12)

30U897 claims: 1. A method for determining the location of subterranean interfaces or boundary layers and / or for determining their acoustic properties by emitting acoustic waves and analyzing the reflected seismic signals, characterized in that at least one first point sound source and at least one second point sound source are used as a transmitter that the energy of the radiation of the two sound sources or groups of sound sources are selected in a known ratio to one another and that the reflected seismic signals are correlated with one another in order to carry out the position determination. 2. The method according to claim 1, characterized in that a plurality of identical signals are summed, to generate the first of the two seismic signals. 3. The method according to claim 2, characterized in that the plurality of identical seismic signals are thereby obtained is that a series of identical sound waves generated by means of one or more identical sound sources will. 4. The method according to claims 1-3 »characterized in that that a plurality of identical seismic signals are summed to obtain the second seismic signal. 5. Method according to claim 4, characterized in that that this plurality of identical seismic signals is obtained in that a series of identical Sound waves is generated by means of one or more identical sound sources. / 030045/0722 ■ ι- 6. The method naoh claim 1, characterized in that a number of identical, non-interacting sound sources simultaneously generating the first sound wave to be used. 7. The method according to claim 1 or 6, characterized in that that a plurality of identical, non-interacting sound sources for generating the second sound wave can be used at the same time. 8. The method according to any one of claims 1-7 »characterized in that that every sound source is an air charge, a water charge, an underwater or subterranean one Contains an explosion generator, an underground implosion source, or a spark discharger. 9 · The method according to claim 1, characterized in that the ratio factor is selected between 1.33 and 125. 10. The method according to Anspructui and 9 »characterized in that that the known ratio factor between 3 »375 and 27 is selected. 11. Apparatus for performing the method according to claim 1, characterized in that it has at least one first point source of sound and at least one second Contains point source of sound that generate the first and second sound waves in the earth that carry the energy of the radiation emitted by the first sound source by a known ratio fairbor to the energy of that of the second sound source emitted radiation differs that receivers are provided to the reflections of the first and secondly to register sound waves from inside the earth in order to generate the first and second seismic signals, and that devices are provided to the first 030045/0722 and to compare and analyze second received seismic signal » 12 = device according to claim 11, characterized in that the first sound source consists of one or more identical sound sources "which are connected in such a way that they generate a series of identical sound waves and
that the receiver contains devices to this series
of identical sound waves to sum up to the first
receive seismic signal = 13 ° device according to claim 11 or 12, characterized in that that the second sound source includes one or more identical sound sources which are connected so that eie generates a series of identical sound waves and that the receiver contains devices to generate this series of identical Sum up sound waves to form the second seismic Generate signal » 'U (-o device according to claim 11, characterized in that that the first sound source contains a plurality of identical, non-interacting point sound sources which are switched in this way are that they 'generate the first sound wave at the same time " 15 «Device according to claim 11 or 14, characterized in that the second sound source has a plurality of
contains identical, non-interacting sound sources which are connected in such a way that they simultaneously generate the second
Generate sound wave » 16o device according to claim 11, characterized in that that the "known ratio factor is chosen between 1.33 and 125 ο 17 »Device according to claim 11 or 16, characterized in that that the known ratio factor between 3.375 and 27 is chosen «, * "* & S S ^ f \ Q if D f \ ⁵ ^? ^ ^%