We have studied the effects of imperfections in spherical gravitational wave
antenna on our ability to properly interpret the data it will produce. The
results of a numerical simulation are reported that quantitatively describe the
systematic errors resulting from imperfections in various components of the
antenna. In addition, the results of measurements on a room-temperature
prototype are presented that verify it is possible to accurately deconvolve the
data in practice.Comment: 5 pages, 2 figures, to be published in Europhysics Letter

Finn L. S.

Lobo J. A.

Merkowitz S. M.

S. M Merkowitz

Velloso W. F.

W. W Johnson

Wagoner R. V.

English

arXiv

We have studied the effects of imperfections in spherical gravitational-wave antenna on our ability to
properly interpret the data it will produce. The results of a numerical simulation are reported that
quantitatively describe the systematic errors resulting from imperfections in various 
components of the antenna. In addition, the results of measurements on a 
room-temperature prototype are presented that verify it is possible to 
accurately deconvolve the data in practice

S. M. Merkowitz

W. W. Johnson

EDP Sciences OAI-PMH repository (1.2.0)

Deconvolving the information from an imperfect spherical 
gravitational-wave antenna

We have studied the effects of imperfections in spherical gravitational-wave antenna on our ability to properly interpret the data it will produce. The results of a numerical simulation are reported that quantitatively describe the systematic errors resulting from imperfections in various components of the antenna. In addition, the results of measurements on a room-temperature prototype are presented that verify it is possible to accurately deconvolve the data in practice

Merkowitz, S. M.

Johnson, W. W.

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 2-15-1998 Deconvolving the information from an imperfect spherical gravitational-wave antenna S. M. Merkowitz Louisiana State University W. W. Johnson Louisiana State University Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Merkowitz, S., & Johnson, W. (1998). Deconvolving the information from an imperfect spherical gravitational-wave antenna. Europhysics Letters, 41 (4), 355-360. https://doi.org/10.1209/epl/i1998-00156-5 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:gr-qc/9801027v1  9 Jan 1998Deconvolving the information from an imperfect spherical gravitational wave antennaStephen M. Merkowitz∗ and Warren W. JohnsonDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803(January 9, 1998)We have studied the effects of imperfections in spherical gravitational wave antenna on our abilityto properly interpret the data it will produce. The results of a numerical simulation are reported thatquantitatively describe the systematic errors resulting from imperfections in various componentsof the antenna. In addition, the results of measurements on a room-temperature prototype arepresented that verify it is possible to accurately deconvolve the data in practice.PACS numbers: 04.80.Nn, 95.55.YmA spherical gravitational wave antenna was first proposed by Forward in 1971 [1]. The large cross section and theability to determine the direction of a gravitational wave have been known for some time [2], but it was not untilrecently that its potential advantages have been actively studied [3–5]. For the past few years, several groups havebeen researching the possibility of constructing large scale spherical antennas [6].To realize the full potential of these detectors, we must first be able to understand their behavior and properlyinterpret the data they will produce. The technique needed will depend on the particular type and arrangement ofsensors used to detect the motions of the sphere. We have proposed and investigated the “truncated icosahedral”(TI) arrangement [3] for six identical radial motion resonant transducers. We have shown that this arrangement ona perfect sphere, but with equally noisy transducers, has the property that all five quadrupole components of thegravitational strain tensor are measured with equal sensitivity [7]. Other arrangements of transducers have beenproposed [8,9], but most of them do not share this property [10].The TI model, also called a truncated icosahedral gravitational wave antenna (TIGA), has been criticized on thebasis that normal (small) imperfections will break the perfect symmetry, possibly having drastic effects on such adegenerate system, thereby making it difficult to extract the desired information from the data. We report here thetheoretical and experimental results of an investigation into the effects of imperfections in a spherical antenna thatrefutes these criticisms and shows that the symmetry breaking can be easily handled.The normal modes of an uncoupled elastic sphere can be described by the spacial eigenfunctions ~Ψnℓm (r, θ, φ)(functions of the spherical harmonics) and the purely time dependent mode amplitudes anℓm(t) [7,11,12]. In generalrelativity only the ℓ = 2 quadrupole modes strongly interact with a gravitational wave so for the remainder of thisdiscussion we will assume n = 1, ℓ = 2 and drop these subscripts from our notation. We denote the orientationof the 5 quadrupole modes relative to the lab frame by 2 Euler angles βm and γm. If the quadrupole modes aredegenerate these angles are arbitrary (restricted only by orthogonality of the modes). Imperfections of the sphere canlift the degeneracy causing the modes to “fix” themselves in a particular orientation. This effect has been shown to bethe main result of symmetry breaking in an uncoupled sphere; the “shape” of the quadrupole mode remains mostlyunchanged [13,14].We consider that a number of small resonant transducers, with index j = 1 . . . J , are added to the surface of thesphere to mechanically transform the small motion of the surface into a large motion of the resonator. We assumethe resonators are constructed to obey a one-dimensional harmonic oscillator equation. The particular arrangementof resonators is indicated by their locations (θj , φj) on the sphere surface and their coupling to the surface motionin different directions ~ǫj . In the case of ideal radial resonators we would set ~ǫj = r̂. The relative amount of surfacedisplacement coupled to each resonator for a particular mode is specified by a “pattern matrix” defined asBmj ≡1α~ǫj · ~Ψm (R, θj , φj) , (1)where α depends on the material and the radius R of the sphere [2,7]. Again, we assume that the transducers stronglyinteract only with the quadrupole modes of the sphere, which is appropriate if the mass of a resonator is smallcompared to the mass of the sphere [8,15].The complete set of coupled equations of motion for the sphere and resonators can now be written [16][M s 0αMrBT M r] [ä(t)q̈(t)]+[Ks −αBKr0 Kr] [a(t)q(t)]=[I −αB0 I]f(t), (2)∗Present affiliation INFN-Laboratori Nazionali di Frascati, Via Enrico Fermi 40, I-00044 Frascati (Roma), ITALY1where we have defined 4 diagonal matrices whose elements are the mass m and spring constants k of both the spheremodes (superscript s) and the resonators (superscript r),M sjm ≡ δjmmsm, Mrjm ≡ δjmmrj , Ksjm ≡ δjmksm, Krjm ≡ δjmkrj . (3)The column vector a has five components, one for each sphere mode amplitude, and has a one-to-one correspondencewith the quadrupole components of the gravitational wave strain tensor [7,12]. The column vector q has J components,one for each resonator displacement, and are the observable quantity. The column vector f contains any forces (suchas a gravitational wave) acting on the sphere and resonators.A normal mode solution of these coupled equations is possible using standard techniques [16]. We find the responseof the normal modes η(ω) to any external forces, including gravitational waves, is given byη(ω) = G(ω)V −1[Ms 0αMrBT Mr]−1 [I −αB0 I]f(ω). (4)The matrix V is constructed from the eigenvectors of the normal modes, and the matrix G(ω) is constructed fromtheir eigenvalues. The matrix V can also be used to return to the original sphere and resonator coordinates[a(ω)q(ω)]= V η(ω). (5)V is always invertible thus it is possible to use the inverse of Eq. (5) to transform the data to normal coordinateswhere the frequency response is simple.Examining Eqs. (4) and (5), we see that in order to obtain all the information about an excitation we must haveknowledge of both the sphere a and resonator q modes even though only the resonator motion is observable. Asimple solution emerges if we limit ourselves to the case of “close” to the TI arrangement (six radial resonators placedat the center of six non-antipodal pentagon faces of an imaginary TI concentric to the sphere). The eigenfunctionsof an ideal TIGA are such that the motion of the resonators mimic the ellipsoidal (quadrupole) deformation of thesphere’s surface either in phase or anti-phase [7]. We can picture the collective motion of the six resonators to describesix “resonator ellipsoids”, five of which are mimicking the “quadrupole ellipsoids” of the sphere (the sixth resonatorellipsoid is just a “breathing” sphere: the six resonators are moving in unison with equal amplitude and the spheresurface does not move at all). Each individual resonator displacement qj represents a superposition of the pointradial deformation of the six resonator ellipsoids at a particular location. The transformation between point radialdeformations q and ellipsoidal amplitudes g is given by the pattern matrix B:g = B q. (6)The resonator ellipsoids g are proportional to the sphere modes a so we sometimes refer to them as “mode channels.”We also point out that Eq. (6) can be seen as a linear coordinate transformation between the measured point radialdisplacement and a 5-dimensional abstract vector space based on the spherical harmonics that has a one-to-onecorrespondence with the five quadrupole components of the gravitational strain tensor [7,15]. While not unique toTIGA, such a simple transformation is not possible for most other arrangements of transducers.We are now prepared to discuss how imperfections effect our ability to deconvolve the signal from the transducers.One source of error is the resonator ellipsoids may be deformed so that they no longer mimic the sphere surface.This could lead to errors in calculating both the matrix V and the mode channels g. Another source of error is theuncertainty of the values of all the system parameters. Most of the parameters can be measured, either directly orindirectly, but we will be limited by the accuracy of that measurement.The transformation matrix V can be measured by applying a continuous sinusoidal force anywhere on the sphere’ssurface at the frequency of one of the normal modes. (Note that this technique requires the normal modes to benon-degenerate, thus it is actually preferable to have a small amount of symmetry breaking.) The frequency responseof the resonators will be simple because they are being driven at a single frequency. From Eq. (5) we see that theamplitude and phase of their response make up a single column of V . By exciting each normal mode in turn, thecomplete V matrix can be measured. The only assumption made in calculating V is that the resonators are “close”to the TI arrangement so that an ideal pattern matrix B can be used and the sphere modes a can be replaced by themode channels g.The quadrupole mode orientation angles βm and γm can be measured directly only before the resonant transducersare attached [13]. This is also true for the the mass of the resonators mrj and the sphere modes msm, and theirrespective spring constants krj and ksm. Once the system is coupled together these parameters cannot be measured2independently. The positions of the resonators (θj , φj) can be easily measured, but precise values of their couplings~ǫj are much harder to determine.We developed a numerical simulation of a TIGA that calculates the error associated with a measurement due toimperfections and asymmetries as well as our uncertainty of the precise values of the above parameters. We simulateda gravitational wave burst excitation of an imperfect TIGA and used the above analysis techniques to estimate thedirection of the wave from the simulated output of the transducers. The analysis assumed parameter values of anideal TIGA and a measured V matrix. The “real” (perturbed) parameters of the imperfect TIGA were calculated byadding uniformly distributed random numbers to the ideal values. The range of values of the random number was afraction (tolerance) of the ideal value (2π for the angles).The results of the simulation are shown in Fig. 1. Plotted is the solid angle estimation error ∆Ω and the percentageerror σh/h for estimating the strain amplitude for a range of tolerances. Parameters not shown had errors close or equalto zero. Because the tolerance is expressed as a percent of the ideal parameter values, varying the magnitude of theideal parameters (for example doubling the mass of the sphere) produced identical results. Also varying the directionand polarization amplitudes of the gravitational wave produced similar results. From these results we conclude thatthe analysis becomes unreliable only after the tolerance of all the parameters exceeds about 3%. This is certainly anobtainable level of precision; one expects to set the tolerances on a real antenna to better than 1%.Also shown in Fig. 1 are the errors due to a finite signal-to-noise ratio, calculated in a similar fashion to the analysisof Zhou and Michelson [9], but using the direction finding algorithm used in our measurements [13]. Note that thisis a random error whereas the tolerance errors are systematic. Comparing the two sources of error, we find that onewould need a signal-to-noise ratio of about 1000 in energy before the tolerance errors become significant. While onemight hope to observe sources at this level, the most optimistic predictions lead to considerably smaller values [17].To further evaluate the feasibility of a spherical antenna, we constructed a 0.8 m diameter, Aluminum alloy,room-temperature prototype TIGA [13,16]. Six radial resonant transducers were attached to the surface in the TIarrangement: placed at the center of each of the six lower pentagon faces. The response of each resonator wasconverted to an electrical signal by a piezoelectric strain gauge, whose output was recorded on a high speed dataacquisition system. The apparatus is described in detail in Ref. [18].Since no laboratory source of gravitational waves exists, we chose to apply an impulsive force to the surface of theprototype to test the developed techniques. To quantify the test, we checked if the location of the impulse could bemeasured from the response of the resonators.We began by measuring the transformation matrix V as described above. Impulsive forces were then appliedby sending a short electrical pulse to a piezoelectric shaker attached to the surface of the TI. The data from thetransducers were recorded, transformed to normal coordinates, fit for phase and amplitude, and finally transformed tomode channels. From the mode channels the location of the impulse was calculated. The algorithm used to determinethe location of the impulse can also be used to calculate the direction of a gravitational wave [13,16]. The results ofthis analysis for the various impulse locations is shown in Fig. 2. From these results, we conclude that it was possibleto reconstruct the location of the impulses, within the accuracy of the experiment, without major difficulties.In summary, from the numerical and experimental results we conclude that imperfections in a spherical antennacan be easily handled and that the error associated with the asymmetries are small. We find that the TI arrangementis fairly robust to these normal imperfections. The techniques for deconvolving the data are mostly linear algebra,making their implementation simple in an automated data analysis system. The measurement of the transformationmatrix V takes into account most deviations from perfect symmetry and enables the data to be transformed to aspace where the frequency complications can be easily handled. The algorithms were tested on the prototype TIGAand found to be consistent with the measured results. Finally, since all the techniques described and tested can beapplied in situ, they are directly applicable for use on a real spherical antenna searching for gravitational waves.We thank W. O. Hamilton for essential advice and support. This research was supported by the National ScienceFoundation under Grant No. PHY-9311731.[1] R. Forward, Gen. Relativ. Gravit. 2, 149 (1971).[2] R. V. Wagoner and H. J. Paik, in Proceedings of International Symposium on Experimental Gravitation, Pavia (RomaAccademia Nazionale dei Lincei, Roma, 1976), pp. 257–265.[3] W. W. Johnson and S. M. Merkowitz, Phys. Rev. Lett. 70, 2367 (1993).[4] E. Coccia, J. A. Lobo, and J. A. Ortega, Phys. Rev. D 52, 3735 (1995).[5] G. M. Harry, T. R. Stevenson, and H. J. Paik, Phys. Rev. D 54, 2409 (1996).3[6] Omnidirectional gravitational radiation observatory: Proceedings of the first international workshop, edited by W. F.Velloso, O. D. Aguiar, and N. S. Magalhães (World Scientific, Singapore, 1997).[7] S. M. Merkowitz and W. W. Johnson, Phys. Rev. D 51, 2546 (1995).[8] J. A. Lobo and M. A. Serrano, Europhys. Lett. 35, 253 (1996).[9] C. Zhou and P. F. Michelson, Phys. Rev. D 51, 2517 (1995).[10] T. R. Stevenson, Phys. Rev. D 56, 564 (1997).[11] N. Ashby and J. Dreitlein, Phys. Rev. D 12, 336 (1975).[12] J. A. Lobo, Phys. Rev. D 52, 591 (1995).[13] S. M. Merkowitz and W. W. Johnson, Phys. Rev. D 53, 5377 (1996).[14] E. Coccia et al., Phys. Lett. A 219, 263 (1996).[15] J. A. Lobo and M. A. Serrano, Submitted to Phys. Rev. D.[16] S. M. Merkowitz and W. W. Johnson, Phys. Rev. D 56, 7513 (1997).[17] L. S. Finn, in Proceedings of the OMNI-1 workshop [6].[18] S. M. Merkowitz, Ph.D. thesis, Louisiana State University, 1995.0 1 2 3 4 510-810-610-410-2100θφ mrkrεθεφallTolerance (%)∆Ω (sr)101102103Energy Signal-to-Noise Ratio0 1 2 3 4 510-210-1100101102θφmsmrkskrallTolerance (%)101102103Energy Signal-to-Noise Ratioidealidealσ h/h (%)FIG. 1. The solid angle direction estimation error and the percentage error in estimating the strain magnitude as functionsof the tolerance on the system parameters of a TIGA and as functions of the signal-to-noise ratio for an ideal TIGA. Each solidline corresponds to the bottom axis and represents the results of a 200 trial simulation with the indicated parameter variedwithin the corresponding tolerances. The dashed lines corresponds to the top axis and represents the results of a simulation ofan ideal TIGA for a range of signal-to-noise ratios.4  22.5  45  67.5  90302106024090270120300150330180 0FIG. 2. Location of several impulses applied to the prototype TIGA. Each x marks the location calculated from the motionsensor data and the nearby o marks the location of the center of the shaker measured geometrically.5

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Deconvolving the information from an imperfect spherical gravitational wave antenna

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