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Source Identification/Inverse Problems Inverse Problems in Sound Radiation Earl G. Williams Physical Acoustics, Code 7137, Naval Research Laboratory, Washington D.C. We present an introduction and overview of inverse problems in sound radiation. The typical inverse problem involves determining the pressure and/or velocity and/or surface intensity (the reconstructed fields) on a surface of interest (such as a vibrating body) from measurement of the pressure on an imaginary surface located a small distance away. This inverse problem is ill-posed, requiring special care in the solution for the reconstructed fields. Methods now exist which are extremely successful in the solution of this ill-posed problem for complicated geometries; for both exterior problems, such as the radiation from machines, and interior problems, such as interior noise in aircraft and automobiles. With these methods under the belt one can turn to post processing of the reconstructed data with the aim of source identification. A new and important source identification tool is wavenumber-filtered intensity which provides an unambiguous identification of the radiating areas of the vibrating source by filtering out small wavelength (poorly radiating) waves. INVERSE NAH PROBLEM Solution of the integral equations Nearfield acoustical holography (NAH) [1] has be- Discretization of the surface using boundary element come a very powerful tool throughout the world for the methods (BEM)[4, 5] leads to a linear matrix equation study of noise sources for both exterior problems and in- δ = δ. terior problems. NAH is usually based on inversion of p Gv (5) one or more of the following integral equations: δ where v represents a vector of length N of one of the δ unknowns p,vν,µ in Eqs. (1-3), p the vector of length N p(M)= GN(M|Q)vν(Q)dSQ, (1) of measured pressure and G the matrix resulting from the discretization. The superscript δ indicates a quantity with noise resulting from the measurement. α,δ ∂G One possible solution v to Eq. (5) is given in terms p(M)= D (M|Q)p(Q)dS , (2) ∂n Q of the regularized inverse Rα using Tikhonov regulariza- tion theory[6]; α,δ δ where vν is the normal surface velocity, p(Q) the sur- v = Rα p , (6) face pressure, G and G are the surface Neumann and N D where Rα is called the regularized inverse of G and Dirichlet Green functions known analytically for separa- ble surfaces, and derived numerically for non-separable α,1 1 1 1 H Rα = VF diag ,···, ,···, U . (7) surfaces, M is the closed measurement surface taken con- λ1 λi λN formal to the integration (reconstruction) surface Q; The filter factor Fα,1 (a diagonal matrix) is ∂GF p(M)= GF (M|Q) − i (M|Q))µ(Q) dSQ, (3) ∂n |λ |2 α,1 ≡ ··· i ···. F diag 2 (8) |λ |2 + α α where µ(Q) is the unknown source density[2, 3] and i α+|λ |2 ikR i GF = e /4πR (free-field Green function), R = M − Q. The inversion of these integral equations determines and U and V result from the singular value decomposi- = Σ H Σ = [λ ,···,λ ] the unknown in the integrand, i.e. p(Q), vν(Q) or µ(Q), tion (SVD) of G, G U V , where diag 1 N . α → α → ∞ which can then be used to predict the field on any other When 0 the £lter factor is unity, and when the surface P by replacing M with P in the associated integral factor goes to zero. The regularized inverse Rα is derived equation and recomputing the Green functions. Further- from the minimization of the Tikhonov equation: more, the velocity vector field v at P can be determined δ δ δ ||Gv − p ||2 + α||Lv ||2 (9) by taking the vector gradient of the integral equation. For example for Eq. (1): (||·|| represents the L2 norm) with L =(I −Fα)VH where 2 2 α λ λ iωρv = ∇P p(P)= ∇PGN(P|Q)vν(Q)dSQ. (4) F = diag 1 ,···, N (10) α + λ2 α + λ2 1 N is the £lter factor for the solution to Eq. (9) when L = NAH (b) vs Accel (r) α I. Note that (I − F ) is a high-pass £lter so that (I − 10 Fα)(VH vα,δ) passes the noisy high-wavenumber, small- H α,δ wavelength Fourier components V v while blocking 20 the relatively noise-free low wavenumber components. Thus the penalty function represented by the last term in 30 Eq. (9) does not include the lowest wavenumbers and is weighted towards the high wavenumber components. 40 At this point the value of α is still undetermined. The solution of the following discrepancy equation due to 50 Mobility m/s/N) (dB re Morozov provides an estimate of the value of α: 60 √ ||Gvα,δ − pδ|| = δ, δ = Nσ, (11) 70 500 1000 1500 2000 2500 3000 where σ is the variance of the noise (assumed Gaussian). Frequency (Hz) The variance is determined using FIGURE 1. Measured (from the impedance head) versus re- || H δ||/ ≈ σ ∈ Ω, constructed velocity (from NAH) at the drive point on a point- Uq p Q q (12) driven vibrating rectangular plate. where the norm is taken only over the set Ω containing the last Q eigenvectors of UH with the smallest eigenvalues, λq. ACKNOWLEDGMENTS One can view the first term in the L2 norm in the discrepancy equation, Eq. (11), as the predicted and Work supported by the Office of Naval Research. smoothed value pα,δ of the pressure (now depending on α); α,δ α,δ δ α,1 H δ p ≡ Gv = GRα p = UF U p , (13) REFERENCES derived from the reconstructed velocity. Fα,1 acts to 1. Earl G. Williams. Fourier Acoustics: Sound Radiation and smooth pδ, reducing the high spatial frequency oscilla- Near£eld Acoustical Holography. Academic Press, London, tions. Thus the best value of α corresponds to when the UK, 1999. £ltered pressure differs from the measured pressure by 2. Yu. I. Bobrovnitskii and T. M. Tomilina. General properties just the noise, a result we would expect if the £ltered and fundamental errors of the method of equivalent sources. pressure was identical to the actual pressure p (with no Acoustical Physics, 41:649–660, 1995. noise). This procedure provides a computationally robust 3. Gary H. Koopmann, Limin Song, and John Fahnline. A estimate of α. method for computing acoustic £elds based on the principle of wave superposition. J. Acoust. Soc. Am., 86:2433–2438, 1989. 4. Mingsian R. Bai. Application of bem (boundary element EXPERIMENT method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries. J. Acoust. A point-driven rectangular plate was scanned with a Soc. Am., 92:533–549, 1992. microphone to measure a 37x25 point planar hologram. 5. S. T. Raveendra, S. Sureshkumar, and E. G. Williams. Noise The surface velocity of the plate was reconstructed at source identi£cation in an aircraft using near£eld acoustical 2500 different frequencies using Equations (6-8). The holography. In Proceedings of the 6th AIAA, number AIAA- £gure shows the reconstructed velocity (actually mobil- 2000-2097, Lahaina, Hawaii, 2000. ity) at a point versus a surface mounted accelerometer at 6. Per Christian Hansen. Rank-De£cient and Discrete Ill-Posed the same point. The agreement is excellent. The Moro- Problems. Siam, Philadelphia, PA, 1998. zov discrepancy principle was successful at determining a value of α at every frequency. The signal to noise ra- (|| ||/σ) tio 20log10 p was estimated by Eq. (12) and varied from close to zero dB for the lowest frequencies to a max- imum 45 dB. ———————————————————- Optimal Conditioning of Inverse Problems in Acoustic Radiation P. A. Nelson and Y. Kim Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Solution of the discrete inverse problem in acoustics can yield estimates of acoustic source strength from measurements of acoustic pressure in the radiated field. In this paper the conditions are identified for which the inverse problem is optimally conditioned. This involves sampling both source and field in a manner which results in the discrete pressures and source strengths constituting a discrete Fourier transform pair. INTRODUCTION distribution. Now note that if the source distribution is assumed to be bounded and sampled It has been noted in a previous publication [1] that, spatially at N discrete points such that the for a particular 2D acoustic radiation problem, the continuous variable u z (x) is replaced by the matrix of Green functions relating the sampled far discrete variable uz(mx), then we may write field acoustic pressure to the strengths of a discrete array of elementary acoustic sources has unit M 1 jkˆmx condition number at a certain frequency. The p(kˆ) G (kR) u (mx)e (2) c z associated inverse problem is thus very easy to m 0 solve at this frequency and the strengths of the elementary sources can be reliably estimated from Now if the variable kˆ (which lies in the range –k the inversion of the Green function matrix [2, 3]. It to +k) is also divided into M equal increments is the purpose of this paper to explore this kˆ 2k / M , then we may write relationship further and extend the result of this potentially useful observation to the 3D case. M 1 ˆ j(ukˆ)(mx) p(u k) Gc (kR) u z (n x)e (3) THE 2D RADIATION PROBLEM m0 where u denotes the index associated with the The far field acoustic pressure radiated by 2D sampled pressure. velocity distribution u z (x) can be written as Note that this is accomplished by sampling the far field pressure at M equal increments of A sin [1].
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