The Fundamental Elements in Certain Inverse Acoustic Problems: Their Roles and Interactions

Vincent Martin The Fundamental Elements in Certain

Vincent Martin Inverse Acoustic Problems: their CNRS [email protected] Roles and Interactions Institut Jean le Rond d'Alembert UMR CNRS/UPMC 7190 Acoustic holography and holophony, wave field synthesis and active noise control are based 75005 Paris, France on common elements which are causality, model, objective, and regularization. In the frequency domain (putting causality aside), a simple formulation states the influence – not the interaction – of errors of the model and objective and of regularization of the results. However, it does not give either an understanding or any relation of cause to effect. When the objective can be reached using the available model, regularization is not needed and the information liable to be extracted from this determined problem is poor, unlike in the overdetermined case when the model does not allow the objective to be reached. The geometrical interpretation of the over-determined problem written in the least-mean square sense could be a tool to enlighten the influences and interactions in question. After having shown the interest of the geometrical interpretation, a pseudo-analytical inverse problem in spherical holophony and a numerical problem in plane holography provide particular illustrations. From among the properties accessible, one is highlighted: in the case of a perfect objective but inaccurate model, its adaptation brings a decrease in the amount of regularization required and an improvement in the results. Keywords: inverse problems, regularization, acoustics

Introduction 1 with the primary ones at the origin of the given pressure field and they cannot radiate the opposite field (except in the very particular Acoustic holography (Metherel et al., 1967; Maynard, Williams 1D case of controlling plane waves in ducts); in holography primary and Lee, 1985, for the early works), wave field synthesis (Berkhout, and secondary sources are often the same, but here an erroneous de Vries and Vogel, 1993, for the early work), holophony, and measured field has no reason to be accessible. The best we can do is active noise control (Nelson and Elliott, 1992; and Elliott, 2000, to reach the projection of the given field on the set of fields the textbooks presenting a synthesis) are inverse acoustic problems secondary sources can radiate, leading to the geometrical point of sharing common elements such as causality, transfer function or view. The same will occur when the model is erroneous, and model, objective or goal to be reached and regularization. In the therefore, not liable to generate the objective. To generalize, it will frequency domain, the question of causality is put aside. be said that the model is not exact when incapable of radiating the In these problems, one or a few sources are required to radiate a exact objective (which does not mean that the radiation itself is given acoustic pressure field (for an extensive overview of the past erroneous), while the definition of an erroneous or an exact three decades with 170 references, see Wu, 2008). Indeed, sound objective is kept natural. field synthesis consists in determining driving signals to feed several So much for the objective space. Now what about the solution loudspeakers in order to radiate a particular wavefront, a particular space (that of driving signals or source velocities)? To go from one directivity in holophony; active noise control is linked to sound space to the other, the model and its inverse are needed, and it is in synthesis indirectly as the goal is to oppose a so-called acoustic the nature of things that the inverse of the direct operator is ill- primary field at the control microphone locations; holography is conditioned (to the point that the differentiation between the two is concerned with reconstructing a field in a whole domain from through conditioning) which means that errors, whatever their pressure measurements at a finite number of locations, this reason for being, are greatly amplified in the solution and in the reconstruction often resting on the finding of the source strength at reconstructed field. The regularization acts on the inversion to limit the origin of the measured field. the amplification and also plays a preponderant role in the spatial With less sources to control than given pressures to reach, the resolution of the solution obtained, as was underlined by Nelson and problem is overdetermined and rests on the least-square (LS) Yoon (2000), by Yoon and Nelson (2000), by Nelson (2001), by method, while with more sources than pressures leads to Kim and Nelson (2003). Regularization acts first in the solution underdetermined problems with the least-norm (LN) method (the space; in the objective space, only the consequences can be smallest solution from the infinity due to underdetermination, i.e. observed and this will be proved. Besides, we have noticed that the dealing with the LS method plus a constraint). Emphasized by Hald adaptation of the model to better radiate the perfect objective (the (2009), these solutions seem to coincide when regularization is measured pressure without any errors) is accompanied by a need for involved. Limiting the investigation to overdetermined problems, less regularization and a better reconstructed field due to a better the geometrical interpretation accompanies naturally the LS method determination of the solution (driving signal or source velocities). with the Euclidian distance. The use of this interpretation has proved This could help in finding the relation between solution and to be a good tool for investigating some properties such as the objective spaces through regularization and perhaps make use of the guaranteed quality of the procedure for reaching the objective (the geometrical interpretation. given pressure field). Having this research in mind the formulated framework is The geometrical view is also closely linked to the description of given, and then the geometrical interpretation in the objective inverse problems with the vocabulary associated with sets, as space is described. Its usefulness will be shown through an presented by Kirsh (1996). For example, the given pressure field application. Contrary to the demonstration of a probable could be inaccessible for various physical reasons, all related to the impossibility to describe the model regularization in the objective pressure field not belonging to the set of fields the sources can space, a demonstration of the relation between adaptation of a generate. In active control, the secondary sources are not merged model and the amount of regularization has still not been found and the difficulties arising therein will be shown. Applications in Paper received 21 April 2012. Paper accepted 23 August 2012.

584 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

+ far-field acoustic holophony and in nearfield acoustic holography R = space of positive real numbers illustrate the above properties. Open questions conclude this Tr = trace of a matrix paper, devoted to an attempt to gather a variety of acoustic inverse v = vector of the source velocities or source controls problems with their uncertainties. V = space of source velocities or source controls Y = spherical harmonics Nomenclature β = acoustic admittance a = scalar ε = regularization coefficient a = vector ψ = angle between the objective vector and the a% = erroneous vector hyperplane originating from the transfer matrix E σ an = nominal value of a = radiation efficiency opt ϑ = angle between the erroneous and the exact a or aLS = least mean square solution obtained via the objective vectors minimization of a functional; aopt is also for the ANC = active noise control case of optimality in general A = matrix BIEM = boundary integral equation method CSLA = compact spherical loudspeaker array A% = erroneous matrix ESM = equivalent source method a* , A * = transpose conjugate of a , of A NAH = nearfield acoustic holography −1 A = inverse of A SONAH = statistically optimized nearfield acoustic holography A† = pseudoinverse of A WFS = wave field synthesis a , A = L2-norm of a vector, of a matrix Framework δ a , δ A = perturbation (variation) on a , on A Cm = complex space of dimension m The source or sources whose velocities are sought are called “secondary source(s)” to differentiate their role from the so-called = minimal distance between two vectors in the L2 sense dmin primary source(s) at the origin of the radiated pressure. They differ min in sound field synthesis, holophony and active noise control while dmin = minimum possible value of dmin they are merged in holography. Thus, in synthesis and control, max dmin = maximum possible value of dmin secondary sources cannot radiate the primary field accurately in the 1D, 3 D = one-dimensional, three-dimensional whole 3D domain (contrary to cases related to 1D propagation). e = relative error on the objective As the primary field does not belong to the space of the fields liable to be generated by the secondary sources, it is said that the e = minimal relative error on the objective min radiation model cannot be exact, with a clear sense in mathematical E = transfer matrix or propagator set terms. In contrast, there might be such an exact model in F = efficiency of the inversion procedure holography. Here a perturbed propagator, which is thus not exact, h = Hankel function arises not from conceptual reasons but because physical data like I = identity matrix sound speed, source and microphone locations are only approximated and/or because the measurements and calculations are Im = identity matrix of dimensions mxm 2 not exact. J = functional made up of the square of the L norm of Concerning the objective, it is said to be exact if it perfectly the distance between two vectors describes the field to be generated. On a microphone array, it is J0 = initial value of J necessary to have a sufficient number of sensors with a well-chosen J = part of J the minimization procedure makes it distance between them according to the sound spectrum under att 0 study, and also strictly no measurement errors on the recorded and possible to reach signal-processed pressures. J res = residual part of J0 after its minimization The above problems always rest on equation K = space of the image of the space of the source velocities or control through the transfer matrix E p = Ev (1) Log = logarithm with base 10 LN = least norm where the vector p is the objective made up of the measured pressures at the M microphones of the antenna, the velocity or LS = least square command vector v is sought at N points or on N sources, and p = vector of measured acoustic pressure at the E is the transfer matrix or propagator or model of dimensions M x microphones (objective vector) N (M ≥ N) with rank E ≤ N. Quite often, the inverse problem is , p = projection of the exact nominal pressure vector to solved in the least-mean square sense to lead to the optimal velocity n * (or driving signals). Writing H = E E , where the asterisk the erroneous nominal pressure vector indicates the transpose-conjugate, we have: Q = quality of the inversion procedure opt = -1 * Qg = guaranteed quality of the inversion process v( Ep , ) HEp (2) R = regularized matrix which would be the exact solution insofar as E and p are exact Rm = real space of dimension m and also with H well-conditioned.

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 585 Vincent Martin

Very simple manipulations of Eq. (2) result in the perturbation point of view as this operation can be seen as the projection of the δv = vopt - v due to perturbation on the exact model E, on the objective p% on the hyper-plane spanned by the columns vector of exact objective p and on the numerical inversion of H. From now matrix E% . The minimization leads to the minimal distance on, v is the exact vector solution. %= % opt − dmin (Ep ,% ) Ev p% which is the square root of the residual Let us consider the perturbed model E% = E + δ E and the * part J of J after the operation. Let us define J as the value of perturbed objective p% = p + δ p . As long as H% = E % E % is res 0 = 2 = − perfectly invertible i.e., as long as the strict equality J when the velocity is zero, that is J 0 p% and Jatt J0 J res -1 opt -1 * HH% % - I = 0 holds, then v( Ep% , %) = HEp % % % according the part of J that has been attained thanks to the calculation of to Eq. (2). This leads to opt opt 2 v . It appears that J = Ev% . These definitions allow us to att δv = vopt ( Ep% , %) - v = HEp %%-1 * δ + ( HEEv %% -1 * - v ) (3) read symbolically the projection operation in Fig. 1 where the objective does not belong to the hyper-plane due to a perturbation in p , in E , or in both. In these conditions there exists a non-zero where H%-1 E % * approximates the “inverse” of the rectangular matrix angle ψ% (written ψ for the exact p ) such that E% . Equation (3) is simple but with the drawback that it requires previous knowledge of E and v , making this equation d (p% ) uninteresting in the real world but a helpful first approach from the sin ψ% = min , (6) theoretical point of view. In fact Eq. (3) will become an equation p% about the error δ v when prior additional information on δ E δ differing from the general cross-validation parameter defined as (see and on p gives an upper bound. Hansen, 1998): When matrix H% is either not invertible or poorly inverted, because ill-conditioned, the pseudo-inverse H%-1 E % * of E% is d 2 min . % ε 2 approached by a matrix Rε where the parameter can be adjusted Tr (I− EE% % *) to solve at best the problem while still avoiding unacceptable % The efficiency is defined by solutions. The matrix or operator Rε is called regulator, the most famous being that of Tikhonov. If the numerical calculation error is J negligible, we have: F(p% )= cos 2 ψ % = att . (7) p% 2 δ=opt % = % + δ v vEp (, %) - v Rpε ( p ) - v (4) % % % = Rεδ p + ( R ε Ev - v ) Therefore, given p% and E , the die is cast for the optimal value of the velocity and for the efficiency. To improve the efficiency ψ which is similar to Eq. (3). there is no choice but to reduce angle % (through dmin ) hoping to How Ev% opt tends towards p or p% or, when a calculation is better approach the true value v under some assumptions. This opt opt angle has proved to be efficient for classifying various propagators carried out with the adjusted value of ε named ε , how Ev% ε opt (transfer matrices) in the automotive industry by Le Bourdon, tends towards p or p% or how ε opt depends on δ p or δ E , are Picard, Martin (2009). the questions addressed in this paper. The geometrical interpretation of the inverse problems dealt Error of the objective with previously by the author could constitute a tool to contribute to In general, the vector of the acoustic pressure is measured data answering the questions. and, for this reason, inevitably erroneous, while being the only data available. It is said "of reference" or "nominal" and written p . The Geometrical Interpretation n true pressure p is somewhere around this nominal value and is of The raw material of this section is to be found in Martin and the form p= p + δ p . For the time being and for the sake of Cariou (1997), in Le Bourdon (2009), and in Martin, Le Bourdon n = + δ and Arruda (2012). general notation let us write p%n p n p with p% n seen as a

perturbed field relatively to pn , underlining that the unknown true Inverse problem and projection pressure results from a variation in the available nominal pressure. In presence of perturbation on the objective and on the model, The error in the objective is defined by solution vopt ( E% , p% ) in Eq. (2) results from the algorithm p% − p = n n e(p% n ) (8) 2 p minJ (v )= min Ev% − p% (5) n v v with the L2 -norm in the objective space. This Euclidian (Hermitian norm in the complex field) norm leads naturally to a geometrical

586 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

% = p% and p J0 max dmin ( e min ) pn % = pn dmin J res ′ p% n

ψ dmin (pn ) space originating from E% % θ ψ opt %′ E% v space originating dmin (pn )

opt % ψ min and % = from E E v Jatt n dmin ( e min )

Figure 1. Projection operation related to the minimization of functional J.

As emphasized by Martin and Gronier (1998) and by Martin, Le Bourdon and Arruda (2012), this quantity may constitute a vicinity Figure 3. Cone related to defining a vicinity of the given nominal field . around the reference field containing all the perturbed fields such that their relative error is less than or equal to a given value of e . p%n  Ideally, a vicinity around p should correspond to a vicinity around A point of coordinates emin (p%n ),  can be n d (p% )  = min n the efficiency Fn F (p n ) in order to establish a connection associated to each perturbed field in the vicinity of the reference between efficiency and error in the objective and, by so doing, to be % ' able to see how the efficiency diverges from that of reference when p% pn field . Let us notice that n = where is the the error in the reference field increases. In fact, the previous ' dmin (p%n ) dmin (p%n ) definition of the vicinity around pn , expressed by e , is such that to projection of on . For a set of perturbed fields sharing the a given efficiency corresponds an infinite number of different values min + same minimal relative error, the values dmin( e min ) and of e spread over the interval [0 ,+∞ ( . Indeed the efficiency dmax ( e ) (see Fig. 3) bound the range of d (p% ' ) for any field defined by the angle ψ% associated with p% is the same for an min min min n n n considered in that set. Figure 4 illustrates the theoretical result. infinite number of collinear fields p% n , and, therefore, to an infinite number of values of e(p%n ) . On the contrary, for all these collinear Guaranteed quality of the inverse process in the objective space fields sharing the same efficiency, a single minimum value For a given reference objective , the reference or nominal belonging to interval [0,1] was defined by Martin and Gronier quality of the process is defined by the quantity (1998). The following definition is thus preferred:

= J0 (pn ) 2 = 1 = 1 p% *. p Qn or Qn or Qn (10) n n − J (p ) 1− F sin ψ p% n p n res n n n p% 2 = n emin (p% n ) (9) p p% ' n n ()% ' dmin pn % ' pn Moreover it has been shown that a given value of emin corresponds min () dmin e min to a single value of the guaranteed efficiency when the perturbed field p% belongs to the vicinity defined by e . n min % ' pn θ max () It appears that emin is the sinus of angle shown in Fig. 2 1 dmin e min sin ψ and therefore, is a measurement of the aperture of the cone centered n on the reference objective (Fig. 3).

1 p%n emin 0 θ θ= ψ 1 emin ( for n ) dmin (p%n ) p%′ n Figure 4. Theoretical form of the bounds of the cloud of points dmin (pn ) (e(),p%p % / d ()p % ) . pn dmin (p% 'n ) minn n min n θ

space originating ψ Were we working in active noise control, Q would be closely n n from E% linked to the predicted attenuation. Now let us envisage a perturbation of the nominal objective such that it has a minimal

error, of less than or equal to a value emin . Geometrically speaking, θ with = , this means that, as seen in Fig. 3, the true pressure is Figure 2. Angle related to . located somewhere close to the measured one, in or on the cone centered on the reference objective and of aperture angle . If, in

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 587 Vincent Martin

the geometrical representation, pn ' is the projection of the Model Adaptation versus Regularization: Observations, reference pressure on the edge of the cone located farthest from the Generalization, Difficulties model hyper-plane, the quality of inverse process is always better than that defined by When the model and/or the objective is/are not exact, there is a need for regularization such as Tikhonov's or of another type, for example by discarding the small eigenvalues of the propagation J (p ') Q = 0 n , (11) matrix. When only the model is not exact, the operation of adapting g the model consists in reducing the angle ψ to the exact objective. Jres(p n ') Thanks to case studies, it has been observed on the one hand that the ψ said to be the guaranteed quality. Therefore, it is straightforward regularization has no effect on and on the other hand that the that the guaranteed quality is given by smaller the angle ψ , the less optimal regularization needed and the better the results. While we have found a demonstration for the first 1 1 observation, there is still research to be done to find how Q = = g θψ+ θψ + θψ regularization and errors are related. sin(n ) sin cos n cos sin n or 1 = Error of the objective and of the model eF+1 − e2 1 − F minn min n In very general terms and before speaking of exact or perturbed objective and model, the question is to see if the writing of Ev= p = 1 Qg . is meaningful and if there is a solution. Let V be the solution space −+2 −+ − 2 − ≡ 1FeFnnmin (2 1) 2 e min (1 eFF min ) nn (1 ) and let K be the image of V through the model E : E(V) K . (12) When p ∈ K , the writing Ev= p is meaningful and the LS solution v= (*) EE−1 Ep * leads to the exact value v . Indeed Equation (12) was found by Martin and Gronier (1998) but via an LS =−1 = − 1 = analytical approach far more complicated than this simple vLS (*) EE Ep * (*) EE EEvv * . geometrical way. The analytical demonstration is still interesting but When p∉K , the writing Ev = p has no sense for v ∈V (if no longer as for the result more directly obtained today. ∈ ∈ Graphs showing, in logarithmic scale, the guaranteed quality v V , then p K ) and either v exists outside V , or it does not =−1 ∈ against the reference quality for various values of emin are given in exist. If the LS solution vLS (*) EE Ep * V , it would Fig. 5. The main point here is that the graphs present a plateau for generate p' ∈K , necessarily different from p . When p∈K but high nominal quality leading us to conclude that, for a given value E(V)≡ K' where p is not located, the same remark can be made. of e , an increase in the nominal quality is not rewarded by a min In the case of errors in the objective only, the problem to solve significant increase in the guaranteed quality. When the nominal is more precisely written Ev= p% which has no sense as it is quality is obtained by adapting a model to make the hyperplane of unlikely for p% ∈K . For a well-conditioned matrix H the optimal draw nearer to , this information gives the limit of the effort beyond which there is no need to go further. solution vopt ( Ep , %) = HEp-1 * % differs from the exact solution v

associated with the exact pressure by

vopt ( Ep , % ) - v= HE-1 * δ p (13)

as stated by Eq. (3) with E in place of E% . When H is ill- conditioned the regularized Tikhonov solution is

opt −1 vε =(* EE + ε IEp ) * % (14)

which differs from the exact solution by

opt −1 − 1 vε − v =(* EE+Iε )* E δ p + ( (* EE+I ε )* Epv − ) (15)

Figure 5. Guaranteed quality against nominal quality for various errors in in conformity with Eq. (4) with E in place of E% . the objective. In the case of errors in the model only, the problem to solve is written Ev% = p . The regularized Tikhonov solution is It has thus been shown in this section that the geometrical interpretation has proved to be a useful tool for presenting the opt =% % + ε −1 % guaranteed quality of the process of inversion, and also to reveal a vε (* EE IEp ) * (16) cost function for adapting the model. However this tool operates only in the objective space. It differs from the exact solution by (see Eq. (4) with δ p = 0 ).

588 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

opt −=% % +ε −1 % − ε opt = Λ† vε v(* EE I ) EEvv * (17) For a given , v ε V ΛΛ ε Up * arises from minimizing † −1 J (v ) with ΛΛΛ =()Λ* Λ + ε I Λ * . % → ε ε n It is clear that if there exists, when E E (in a sense to be + defined), an optimal value ε opt of ε that tends towards zero in , % = Λ Λ R 3. Let us write Eε UΛΛ ε V * not yet having defined ΛΛε opt then vε → v (in a sense to be defined). So far, the adaptation of (but the access to which is not difficult) and let us consider % ' = % 2 ε the model carried out to obtain E→ E rested on the minimization J ε(v ) Ev-p ε , the minimization of which for a given ψ of angle . And, effectively, the process was accompanied by the ' opt ' opt= opt provides vε . It appears vε v ε . So, for the time being, opt observation of ε → 0 . It was tempting to assume that the 2 2 2 functionals J (v ) = Ev% - p + ε v and J ' (v ) = Ev-p% share regularization modified the model in order to reduce ψ . The ε ε ε observation countered this assumption and we demonstrate below in the same objective p and the same optimal value of their variables. four steps the independence of angle ψ from the coefficient ε with 4. But we have the consequence that the effect of regularization cannot be described easily in the objective space (Martin and Le Bourdon, 2010), Ev%' opt= Ev % opt = ( UVVUpΛΛΛ *) ΛΛΛ† * perhaps a natural conclusion in retrospect since E% is not acted ε ε εε ε ε %−1 % % † . upon, contrarily to its "inverse" H E * now written E . In 0  opt = U  U* p = Ev% 0 0  Independence of the angle ψ from ε 2 Without jeopardizing generality, suppose only the model is not In these conditions, functionals J (v ) = Ev% − p and exact. ' 2 2 J (v ) = Ev% − p share the same objective and the same minimal 1. Let us consider minJ (v )= min Ev% - p with ε ε v v distance d and this whatever ε . Therefore angle ψ such that + min E%:VC⊂n → E % ( VKC ) =⊂ m and J: Cn → R where m> n . %opt % ' opt Ev Eε v ε The space V could contain for example only those variables the cos ψ = = does not depend on the Tikhonov norms of which are bounded by a given maximal value. With in p p place of , Fig. 1 above has given the Euclidian representation with coefficient ε . In the case of the Euclidean configuration shown in V⊂ R 2 and E% (V ) = K ⊂ R 3 , and in the case where p ∉ E% (V ) . Fig. 1 this is true for both functionals. % The singular value decomposition (SVD) of E is such that 5. Finally, the Tikhonov regularization does not seem to have % = ΛΛΛ = = = = ψ E{ U{ { V { * with U* U UU * I m and V* V VV * I n . an influence on angle in space of the objectives, but it plays m, nm, m diagonal n, n m , n a role in space V of the solutions. Therefore, we cannot expect to Matrix ΛΛΛ is diagonal in its upper part and zero in its lower part. represent easily (if indeed it is possible) the regularization process in Matrix U is made up of base vectors of the objective space the objective space. where the data p is located, and matrix V is formed with the base Optimal regularization and illustration on a simple example vectors of space V of variable v . When p∉ E % (V ) , one cannot find opt −1 % v ∈V (one cannot invert E% ) and E% is replaced by the term called Going back to vε ( E , p ) , let us notice that, when solving % † −1 opt 2 pseudo-inverse of E that is E%= () EE %%* E % * . This operation min vε − v with the L -norm in the solution space, the results { { ε ≥0 n, m 14243 n, m n, n observed in examples (therefore without generalization yet) is leads to the solution vopt ∈V such that it generates the nearest ε opt = 0 whatever δ E and therefore, vopt − v cannot be used p'∈ E (V ) = K of p in the sense of the Hermitian norm in . ε opt As vopt = E% † p , it results in vopt = VΛΛΛ† Up* with for finding ε (leaving aside that v is not known). Were we to accept negative values of ε , there could be a value to make − † * 1 † In 0  Λ= () ΛΛ Λ * . We have also ΛΛΛΛΛΛ = leading to opt − opt % %−1 % { { {   vε v zero. Indeed vε −= v(* EE +ε I ) EEvv=0 * − n, m 14243 n, m m, m 0 0  n, n %= % % + ε % δ ε when EEv* (* EE Iv ) i.e. E* E v =- v . This can be Ev%opt = EV % Λ† Up* = ( UVΛΛΛ *) V Λ † Up * or true only if E% *δ E v is collinear with v , a very strong constraint. εopt = −−1 + δ δ Its LS value would be LS (*)vv v *( E * E *) E v opt † In 0  Ev% = UΛΛΛΛΛΛ UpU* =   Up * . δ 0 0  where it is true that the dependence on E is made explicit, but opt − this value does not minimize vε v and moreover, what has been 2. Now let us consider the functional of Tikhonov which observed concerns εopt ( ψ ) , not εopt ( δ E ) . helps in approximating the inverse matrix E% −1 and is a opt 2 2 In fact, the optimal value ε of ε is such that the objective is regularization process: J (v ) = Ev-p% + ε v with ε K V best reached while the amplitude of the velocity is kept reasonable. n → + ε ∈ + Jε : C R and R . This qualitative definition has received a heuristic representation through the parametric so-called L -curve which gives the value of

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 589 Vincent Martin

Log vopt against Log Ev% opt − p (Kirsh, 1996; Hansen, 1998). 17.  ε ε   opt these conditions, the perfect objective is = . Let us The optimal value ε corresponds to the compromise located at p 17.5    the “corner” of the L -curve where the curvature is maximum. 37.  However, the idealized L -curve has not always so clear a form and perturb the term (1,2) of E to obtain E% with a law such that ε opt % =+ −+ the location for obtaining should be determined with care. E1,2 E 1,2 (1. 0.5 Random [ Real, 1, 1]) , i.e. with unstructured Let us say that ε opt has been defined by Steiner and Hald uncertainties. First, calculate the angle ψ (calculated through (2001), as depending on the signal to noise ratio (SNR) seen at the % opt − objective and due to all sorts of reasons (measurements, errors in the Ev p sin ψ = ) , then draw the L -curve, and in the case of a model, calculation approximations). The SNR is defined by rule of p thumb and is not seen as model dependent. However, Hald (2003) opt wrote with the present notations that “re-scaling of the matrix relatively visible “corner”, find the ε , and finally calculate E%* E % in turn necessitates a re-scaling of the regularization opt − ε opt vε opt v . It must be said that the value of retained is such parameter ε opt ”. And again to emphasize the state of the art opt2 opt 2 opt that the quantity (LogEv% − p )( + Log v ) is minimal, concerning the dependence of ε on the model, let us refer to ε ε Gomes and Hansen (2008) where, according to the type of model which is never far from the maximum curvature but not always at chosen – which could be that of Statistically Optimized Nearfied exactly the same location. Figure 6 shows two types of L -curve Acoustic Holography (SONAH) to replace the first version of NAH opt retained during the process and Fig. 7 shows ε( ψ ) , and from Maynard et al. (1985), or that of the Boundary Integral opt Equation Method (BIEM) as developed by Bai (1992) or described vε − v against ψ . by Augusztinovicz and Tournour (1999), or that of Equivalent

Source Method (ESM) – , there is a change in the best regularization ε opt parameter. The work of Gomes and Hansen is an echo of the earlier 3.5 work carried out by Williams (2001). 3.0

2.5 1.2 opt Log vε 2.0 1.0 1.5

0.8 1.0

0.6 0.5 ψ in degrees

0.4 1 2 3 4 5 6 7

0.2 opt % opt Log Evε − p 12 − vε opt v

0.2 0.4 0.6 0.8 1.0 1.2 1.4 10

0.8 opt 8 Log vε

6 0.6

0.4 2 ε opt 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Log Ev% opt − p ε εopt ψ opt − ε opt Figure 7. ( ) above, vopt v against below. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ε Figure 6. Two types of L-curve for the numerical example dealt with .

This simple numerical experiment shows that the decreasing of ε opt ε opt ψ opt − ε opt Is the observation concerning the decreasing of with that of with and that of vε opt v with could be a general ψ made when solving physical inverse problems closely linked to property rather than an observation deriving from the physical nature the underlying physics, or is it a more general property? To try to of the inverse problem dealt with in acoustics. However in this answer that question, let us take a very arbitrary matrix arbitrary example, we see a jump of εopt ( ψ ) and a corresponding 1. 2.    −7.  void in the curve of error in the velocity. Moreover, if several E = 1.1 2.1 and an arbitrary perfect velocity v =   . In components of matrix E are submitted to variations, the graphs are   12.  5. 6.  of far greater complexity as clouds take the place of the curves.

590 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

Beyond the observation made on this very simple calculation centre of the primary source(s). This difference P− S ' will play the opt − role of regularization. experiment, how can it be demonstrated that vopt v decreases ε with the angle ψ ? What are the difficulties? Indeed, let us write

opt−= opt − opt + opt − vεopt vv ε opt v v v =% − %%−1* + %% − 1* − − 1* Rεopt p HEp HEp HEp (18) ≤−% %%−1* + %% − 1* − − 1* (Rεopt HEp )( HE HEp )

Figure 8. Illustration of the exterior spherical acoustical holophony. where it can be demonstrated (via the SVD of E ) that

% %−1 % * (Rε − HE ) p → 0 when ε → 0 , where it is natural that The primary harmonic pressure is developed in the form − − (HE%1* % − HE 1* ) p → 0 when (H%−1* E % − H − 1* E ) → 0. But how NP n = θ ϕ can the dependency on ψ be expressed? To this end, the pp(/)x O∑ ∑ C nmn ()()(,) d hkrY nm (19) n=0 m =− n opt opt expression of ε( ψ ) will be needed. By accepting that ε is the

% opt2 opt 2 where hn are the Hankel functions and Ynm are the spherical solution of min (LogEvε− p ) + ( Log v ε ) , the expression ε = + 2 harmonics. There are P( N p 1) terms; however, the Ev% opt − p in terms of ψ such that sin ψ = does not seem an axisymmetry of the pressure field reduces the number of terms to = + = p P( N p 1) as m 0 . For example, the field shown in Fig. 9 is easy task. such that there is a need for n >1 to radiate it, whatever O (i.e., While the following case studies as well as the above calculation P ≥ 3). The coefficients C can be determined almost exactly example have shown that the adaptation of a model using the nm ∀ parameter ψ could be an interesting direction to follow, the d providing N p is sufficiently large. In the example chosen, = = = definitive interest ought to depend on the demonstration sought and N p 6 , therefore P 49 in general and P 7 here. In other this question seems to be still open. words, the number of modes with non-zero (or negligible) coefficients depends on the location of point O. Case Studies

Semi-analytical spherical holophony in the far field In Pasqual and Martin, 2012, a primary field due to a source (or Γ sources) in a sphere c of centre C is given on the spherical surface Γ c (Fig. 8). This area is located in the far field relative to the source. The given pressure is expanded in a series of P independent functions composed of spherical harmonics considering the coordinate center O. The number of « modes » to describe the primary pressure with a good approximation depends on the coordinate center O. The minimum value of P occurs when O is at Γ the so-called acoustic centre, in general unknown, even when the Figure 9. Axisymmetric primary field on c . The shape and the color map geometrical centre of the primary source is given. indicate, respectively, the amplitude and phase (in degrees) of pp(x c ) . A secondary source is required to radiate the given primary pressure as well as possible. This source, within the sphere but also far from its surface, is constrained to radiate a controllable given 2 After linear indexing such that i= n ++ n m + 1 and with the number S of « modes » with S≤ P ; some of them not radiating vector s( x ) made up of components s()x = h () kr Y (,)θ ϕ and efficiently, leaving S'(< S ) efficient controllable modes. i n nm vector c with components c= C , the primary pressure is also Where should it be located to reproduce the primary field as i nm well as possible? The problem is equivalent to that of finding the = t expressed by pp (/x O ) sxc (). . As the number of components location of the secondary source centred at O. It should be said that in the vectors depends on the location of point O, their dimension is the problem of reproducing a field by a spherical source has also kept the same but the number of non-zero components varies. The been investigated by Peleg and Rafaely (2011). identification of the coefficients in vector c is obtained through the We are facing an inverse acoustic problem where the model of the secondary radiation has to be adapted through the secondary LS-method, but to overcome a difficulty due to the ill-conditioned γ source location to reproduce at best the primary field. Due to the matrix in the inversion, it is practical to deal with coefficients i (r ) secondary source constraints, there are P− S ' « modes » missing γγγ = such that p H. c where components Hii of diagonal matrix H to reach the goal, a number that depends on O. Often, but not are . In these conditions: always, the minimal difference occurs when O is at the acoustic hn ( kr )

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 591 Vincent Martin

γ σ≥ σ  s, i if n min t −1 γ = = γγγ s, regul , i  (23) pp(/)x O sxH (). . p (20) σ< σ  0 if n min

These new coefficients make up the spherical wave spectrum of the Let us define the regularization quality factor λ by Γ primary pressure evaluated at c . The identification of γγγ leads to their value γγγˆ and, in the γ− γ p p s, regulˆ p λ =1 − (24) present situation, P is sufficient to ensure a good if not exact γγγˆ reconstruction of p(x / O ) , whatever O. p p Similarly the secondary harmonic field is expressed by which is the inverse of the amount of regularization. Clearly, when γ→ γ s, regulˆ p thanks to seeking the best location for the secondary NS n = θ ϕ ps(/)x O∑ ∑ DhkrY nm n ()(,) nm (21) source, then the amount of regularization decreases as well as the n=0 m =− n residue R =p − p . s p written

=t = t −1 γγγ ps ()x sxd (). sxH (). . s (22)

Simply enough, our goal would be perfectly achieved with γ= γ sˆ p if the secondary source were located at the acoustic centre of the primary source and able to radiate S= P modes. However, the secondary source constraints with S< P have been mentioned and moreover, from among the S modes, a number do not radiate efficiently. Indeed, the radiation efficiency of a compact spherical loudspeaker array (CSLA) has the form shown in Fig. 10. Asking of such a source to radiate in the low frequency range with a large n leads to far too high a level of driving signal or velocity. It was decided to discard these disturbing high values of n and consider σ only those n associated with a reasonable minimum value min of the efficiency. To illustrate, working with ka = 0.3 and considering σ> σ = −5 = = only those modes with min 10 , only n 0 and n 1 are = = left, therefore, N s 1 and S 4 or 2 for axisymmetric pressure distribution. Let us recall that 3≤P ≤ 7 in the case studied. It appears that 4 sources of the CSLA, placed at the so-called extremal points for hyperinterpolation distributed over the spherical box can be driven to radiate efficiently 4 modes (and 2 sources for 2 modes). Figure 11. Norm of the optimal radial velocity without and with regularization by discarding. Results as function of for the primary field shown in Fig. 9. Only the graphs are considered in this paper.

In the numerical simulation, the primary field of Fig. 9 was chosen because it cannot be produced by a multipolar source of order n ≤1. It was obtained by using Eq. (19) with d = 0 and the wave spectra = 0.2, = 1, = 1. Due to the axisymmetry of pp ( x ) , we only deal with points O located on the = z axis. Therefore only positions given by d (0,0,zs ) are considered here. The values of the parameters are those given in the = = = text above added to rc 20 a and thus kr c 6 since ka 0.3 . Figure 10. Radiation efficiency curve for a spherical source with radius a Moreover, the spherical area was sampled with 300 elements and for n ≤ 3 . each elementary area weighted in order not to privilege densely sampled regions on the sphere. The position of the secondary

source is assumed to be within the range −2a ≤ z ≤ 2 a to ensure This discarding plays the role of the regularization as it s facilitates access to the solution of the inverse problem, and the best far-field conditions. Figure 11 shows the norm of the optimal radial = t −1 γ velocity without and with regularization; only graph e is considered solution in the LS-sense for ps()x s (). xH . s leads to here. The results are given as function of z . It is clear that γ= γ in the following way: s s s, regul regularization is needed to prevent the loudspeakers from being overloaded. Figure 12 shows the normalized residual norm and the

592 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

How to take into account the radiation behind this front plane due to regularization quality factor as functions of zs (only graph e is of the vibration of the rear face and the thickness? This problem opt > interest here). It appears that R( z s ) 0 because the primary field originates from the acoustic holography of a rolling wheel cannot be accurately decomposed into spherical harmonics of order embedded in a car body. Despite the slightly different goals, let us n ≤1. However R is greatly reduced if the secondary source is quote the work of Schuhmacher et al. (2003), on a quite similar placed at the optimal location. subject.

z Γ =Γv ∪Γ r 2 2 2 Ω

y x Γ 3 Γ =Γr ∪Γ v 1 1 1 Γ Γ Figure 14. A thick disc, with vibrating front face 1 , rear face 2 , thread Γ Ω 3 , radiates in the unbounded 3D space .

Presently the radiation model is obtained through the boundary integral method that has the continuous form

pQ() = iρ ω vPgPQP () (, )d ∫ Γ 1 1 2 Figure 12. Above, normalized residual norm R()/ z sp p ( x c ) and below regularization quality factor function of (primary field in Fig. 9). + ikβ pPg ( ) ( PQP , )d (25) ∫ Γ 1 1 Graphs are considered in this paper. 1 2 + ikβ c pPg ( ) ( PQP , )d ∫ Γc 1 1 1 2

leading to the discrete form

-1 =ρωtc ββ ββ c  + t p i( ikc (,)-11 IA ik (,) 11  Bdv ) = β β c or p E(,1 1 ) v (26)

β β c where 1 and 1 are the unknown admittances (in a particular Figure 13. Sound pressure field produced by a spherical secondary sense) respectively on Γ and Γc . source placed at the optimal position for reproduction of the primary field 1 1 in Fig. 9. The shape and color maps indicate respectively the amplitude For the numerical simulation, the given modal forms on the and phase (in degrees). faces and on the thickness are

The results presented show, in an almost natural way, that the r- r π  vr()(),θ= a sin l θ cosmin π -  adaptation of the model (here through the source location) is i r- r 2 accompanied by a reduction in the regularization needed and improves max min  (27) the reconstruction. The synthesized field is given in Fig. 13. ()()θ= θy π  vy, ai sin l cos   e  Numerical nearfield holography β β c In Martin, Le Bourdon and Pasqual (2011), a thick vibrating disc Admittances 1 and 1 are local reactions and depend on the (Fig. 14) radiates sound in the unbounded 3D space. The acoustic point x ∈ Γ . Therefore, there are an infinite number of variables pressure is measured on a plane array of microphones parallel to one s side of the disc, called the front face. to identify and it is impossible to find the true propagator in these From the inverse procedure called here holography, the type of conditions. However, it has been observed that the admittance on vibration on the front face is deduced from the measured pressure at Γ tends toward zero when the radial distance tends towards the microphone array. It is supposed that the disc is embedded in a s infinity. Beyond a certain distance, the admittance is negligible and Γ Γc total front plane made up of 1 and its complementary part 1 . makes finite meshing of the source plane possible. The number of

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 593 Vincent Martin unknowns in the propagation model is directly linked to the number Conclusion and Open Questions of points in the meshing of the source plane. It appears unrealistic for computation time reasons to identify each local admittance. To The present paper has given very general equations linking reduce the number of unknowns, the admittance is approached by a errors in models as well as in objectives and errors in the solution of known function based on its expected form, and now only a small certain inverse problems. Some interactions between models, number of coefficients of the function must be identified. In the case objectives, errors, and regularizations are not visible in these equations while their consequence on the solution is well described. of the thick disc, the following approximation β% has been chosen: The current research attempts to contribute in finding the actions and interactions at play. c r  c2 r ++5 + ∀∈ [] The geometrical interpretation has proved useful to show very (cccc1e 3) i( 4 e 6) rr 0, max β% =  (28) easily the guaranteed quality of the inverse procedure, and also to c8+ c 10 ∀∈+∞[ (  crcr7i 9 rr max , define a cost function (angle ψ ) to adapt the model in cases where the objective is exact. It has been demonstrated that the ∈ regularization does not act on ψ in the objective space (it acts in where the coefficients ci() i = 1,10 R . Adapting the propagator consists now in identifying the 10 coefficients. However this the solution space) and that a smaller cost function leads to a lesser simplification is at the price of information about the function. optimal regularization parameter and gives a better solution. These Using the genetic algorithm with the cost function made up of links have been seen in a very simple calculation example outside the angle ψ , the coefficients have been obtained approximately and all physical meaning. However no demonstration has yet been found. the results are given in Fig. 15. The question of error in the objective with an exact model has not been addressed here, except through the guaranteed quality. Nevertheless angle ψ exists for an exact model and a perturbed 0.25 actual β 0.2 R objective. However, the aim here, not worked on yet, is to adapt the actual β pressure to reach the model, not the opposite. 0.15 I βopt Now it must be underlined that the adaptation of the model has 0.1 R βopt been carried out via the reduction of angle ψ from the exact 0.05 I

objective. In the presence of perturbation on the objective, the same 0 β ψ+ θ -0.05 procedure would be followed by reducing angle without

-0.1 being able to distinguish ψ from θ . If the adaptation works well,

-0.15 the model will reach the perturbed objective and thus be far from the -0.2 exact objective by angle θ . If θ> ψ , the cure is worse than the -0.25 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 disease! How should the problem, clearly defined in geometrical r (m) terms, be tackled? Figure 15. Admittance on the front face, expected and reconstructed. Another question concerns the geometrical representation of the best location of the sensors and their robustness. It was at the center of preoccupations in active control in the 2000’s, as the paper by Baek and Elliott can testify. In this field, Martin and Gronier (2001) showed, first, that it is always possible to obtain the nominal quality of the inverse process by filtering a small number of components of the reference objective and, second, that the robustness of the filtered sensor configurations was not identical. It could be that the first property may be understood thanks to geometrical

representation in the objective space (moreover with an understanding of one of the constraints associated with the assertion) and that the robustness may be understood by looking in the direction of the projection of the cone on the subspace to which the filtered components of the objective belong.

Acknowledgements Figure 16: Optimal Tikhonov regularization parameter and error in the reconstructed velocity versus the cost function. The author thanks J.R. Arruda, Professor at the University of Campinas, Brazil, and R. Sampaio, Professor at the Pontifical Catholic University, Rio de Janeiro, Brazil, for the interest they Table 1 shows the vibrating form obtained for the front face showed to the present work. and Fig. 16 presents the same results in another form, namely the optimal value of the Tikhonov regularization parameter against angle ψ , itself depending on the admittance on the front face References Γ ∪ Γ c , as well as the error in the reconstructed velocity against Augusztinovicz, F., Tournour, M., 1999, “Reconstruction of source 1 1 strength distribution by inversing the boundary element method” in O. Von angle ψ . Estorff (Ed), Boundary Elements in Acoustics: Advances and Applications, These results illustrate again the interaction between the The Bristish Library, The world’s knowledge, pp. 243-284 (Chapter 8). adaptation of the model and the value of the regularization Baek, K.H., Elliott, S.J., 2000, “The effect of plant and disturbance parameter with the consequence on the reconstructed velocity. uncertainties in active control systems on the placement of transducers”, Journal of Sound and Vibration , 230(2), pp. 261-289.

594 / Vol. XXXIV, Special Issue 2, 2012 ABCM The Fundamental Elements in Certain Inverse Acoustic Problems: their Roles and Interactions

Table 1. The vibrating form obtained for the front face of the disc without regularization and with optimal Tikhonov regularization versus the angle ψ playing the role of the cost function to optimize the admittance on the front face. ψ (degrees) Γv ε = 0 ε = εopt Actual normal velocity on 1

15 ε = 0 ε = 20

6 ε = 0 ε = 17

4 ε = 0 ε = 13

2 ε = 0 ε = 10

0.9 ε = 0 ε = 8

Bai, M.R., 1992, “Application of BEM (boundary element method)- Le Bourdon, T., 2009, “Interprétation geometrique de l'holographie based acoustic holography to radiation analysis of sound sources with acoustique et usage pour l'adaptation du propagateur”, Thèse UPMC. arbitrarily shaped geometries”, Journal of the Acoustical Society of America , Le Bourdon, T., Picard, C., Martin, V., 2009, “A parameter to classify 92, pp. 199-209. different propagators for identifying acoustic source strengths”, in the Berkhout, A.J., de Vries, D., Vogel, P., 1993, “Acoustic control by wave Proceedings of Novem 09 (Noise and Vibration: Emerging Methods), field synthesis”, Journal of the Acoustical Society of America , 93(5), pp. Oxford-UK. 2764-2778. Martin, V., Cariou, C., 1997, “Improvement of active attenuation by Elliott, S.J., 2000, “Signal Processing for Active Control”, Academic Press. modifying the primary field”, in the Proceedings of the Congress Active 97, Gomes, J., Hansen, P.C., 2008, “A study on regularization parameter Budapest, Hungary. choice in nearfield acoustical holography”, in the Proceedings of Euro Noise, Martin, V., Gronier, C., 1998, “Minimum attenuation guaranteed by an Paris-France, pp. 895-900. active control system in presence of errors in the spatial distribution of the Hald, J., 2003, “Patch near-field acoustical holography using a new primary field”, Journal of Sound and Vibration , 217(5), pp. 827-852. statistically optimal method”, in the Proceedings of Internoise 2003, Korea, Martin, V., Gronier, C., 2001, “Sensor configuration efficiency and pp. 2203-22210. robustness against spatial error in the primary field for active sound control”, Hald, J., 2009, “Theory and properties of statistically optimized near- Journal of Sound and Vibration , 246(4), pp. 679-704. field acoustical holography”, Journal of the Acoustical Society of America , Martin, V., Le Bourdon, T., 2010, “Indépendance de l’adaptation du 125(4), pp. 2105-2120. propagateur et de la régularisation en holographie acoustique”, in the Hansen, P.C., 1998, “Rank-deficient and discrete ill-posed problems”, SIAM. Proceedings of the 10 ième Congrès Français d’Acoustique, Lyon-F. Kim, Y., Nelson, P.A., 2003, “Spatial limits for the reconstruction of Martin, V., Le Bourdon, T., Pasqual, A.M., 2011, “Numerical acoustic source strength by inverse methods”, Journal of Sound and simulation of acoustic holography with propagator adaptation; application to Vibration , 265, pp. 583-608. a 3D disc”, Journal of Sound and Vibration , 330, pp. 4233-4249. Kirsh, A., 1996, “An introduction to the mathematical theory of inverse Martin, V., Le Bourdon, T., Arruda, R., 2012, “Geometrical problems”, Springer. interpretation of acoustic holography: adaptation of the propagator and

J. of the Braz. Soc. of Mech. Sci. & Eng. Copyright  2012 by ABCM Special Issue 2, 2012, Vol. XXXIV / 595 Vincent Martin minimum quality guaranteed in the presence of errors”, Journal of Sound Peleg, T., Rafaely, B., 2011, “Investigation of spherical loudspeaker and Vibration , 331, pp. 3493-3508. array for local active control of sound”, Journal of the Acoustical Society of Maynard, J.D., Williams, E.G., Lee, Y., 1985, “Nearfield acoustic America , 130(4), pp. 1926-1935. holography: 1. Theory of generalized holography and the development of Schuhmacher, A., Hald, J., Rasmussen, K.B., Hansen, P.C.., 2003, NAH”, Journal of the Acoustical Society of America , 78(4), pp. 1395-1412. “Sound source reconstruction using inverse boundary element calculations”, Metherel, A.F., El-Sum, H.M.A., Dreher, J.J., Larmore, L., 1967, Journal of the Acoustical Society of America , 113(1), pp. 114-127. “Introduction to acoustical holography”, Journal of the Acoustical Society of Steiner, R., Hald, J., 2001, “Near-field acoustical holography without America , 4, pp. 733-742. the errors and limitations caused by the use of spatial DFT”, International Nelson, P.A, Elliott, S.J., 1992, “Active control of sound”, Academic Journal of Acoustics and Vibration , 6(2), pp. 83-89. Press, London. Williams, E.G., 2001, “Regularization methods for nearfield Nelson, P.A., 2001, “A review of some inverse problems for acoustics”, acoustical holography”, Journal of the Acoustical Society of America , International Journal of Acoustics and Vibration , 6(3), pp. 118-134. 110(4), pp. 1976-1988. Nelson, P.A., Yoon, S.H., 2000, “Estimation of acoustic source strength Wu, S.F., 2008, “Methods for reconstructing acoustic quantities based by inverse methods - part I: conditioning of the inverse problem”, Journal of on acoustic pressure measurements”, Journal of the Acoustical Society of Sound and Vibration , 233(4), pp. 643-668. America , 124(5), pp. 2680-2697. Pasqual, A.M., Martin, V., 2012, “Optimal secondary source position in Yoon, S.H., Nelson, P.A., 2000, “Estimation of acoustic source strength exterior spherical acoustical holophony”, Journal of Sound and Vibration , by inverse methods - part II: experimental investigation of methods for 331, pp. 785-797. choosing regularization parameters”, Journal of Sound and Vibration , 233(4), pp. 669-705.

596 / Vol. XXXIV, Special Issue 2, 2012 ABCM