Query: nearfield acoustical holography 

title: Time domain nearfield acoustical holography without wrap around error and spectral leakage for forward propagation.
abstract: Nearfield acoustical holography can be used to visualize time domain evolution of non stationary sound fields. It requires sampling of the pressure field with a microphone array to calculate the sound field on other parallel surfaces, either toward the source or farther away from it. This paper focuses on forward propagation of fields sampled with a planar array. Several formulations exist to perform such calculation; however, despite being mathematically well posed, they generate significant errors, especially for larger propagation distances. These errors are mainly due to wrap around and spectral leakage. It is shown in this paper that generation of leakage can be avoided by sampling the analytical Green's function in the time and space domains. Such formulation allows the application of linear convolution, which completely suppresses wrap around errors. The results obtained from the proposed formulation are compared to the analytical solution of a baffled piston with transient displacement. The average discrepancy is below 3.5% for propagation distances of 5 to 55 cm. In comparison, the relative error obtained with other propagators varies from 6% to 73% depending on the propagation distance and method used. Finally, the proposed formulation is applied to experimental data obtained from an impacted plate.

title: Reconstructing Excitation Forces Acting on a Baffled Plate Using Nearfield Acoustical Holography
abstract: This paper deals with reconstruction of excitation forces and analyses of the root causes of vibro acoustic responses of an elastic structure by using nearfield acoustical holography and modal expansion theory. Derivations of formulations for reconstructing excitation forces, including distributed, line, and point forces, acting on the back side of a rectangular thin plate simply supported on an infinite, rigid baffle, are presented. The reason for choosing a baffled plate is that analytic solutions to vibro acoustic responses are readily available, so the accuracy in reconstruction can be examined rigorously. For simplicity, the effect of fluid loading is neglected, and input data are assumed error free. Numerical examples of reconstructing excitation forces are presented, and results agree very well with benchmark values. The impacts of various parameters, such as the ratio of measurement aperture versus plate size, microphone spacing, standoff distance, the number of natural modes, etc. on reconstruct.

title: Cylindrical Nearfield Acoustical Holography: Practical Aspects and Possible Improvements
abstract: This chapter discusses Nearfield Acoustical Holography (NAH) for the characterization of cylindrical sources. Cylindrical NAH is an experimental airborne characterization technique, and it is suited for any type of cylindrical source. NAH allows to evaluate sound intensity, pressure level and particle velocity. Practical aspects of Nearfield Acoustical Holography such as positioning error, measurement noise, hologram distance and measurement aperture are investigated and discussed with the aid of numerical examples. Moreover, a technique referred to as compressive sampling (CS) is discussed, aiming to reduce the number of sensors required by the classical NAH in the high frequency range.

title: Acoustic Array System Near Field Acoustic Holography and Vibro Acoustic Field Reconstruction Paper ICA 2016 607 Localization of a defect in an impacted plate using time domain nearfield acoustical holography and time evolution of spatial features
abstract: In this proceeding, a method for localizing defects in a structure by analysing its radiated sound field is presented. The studied structure is a Plexiglas plate with a hole of 4.5 cm in diameter. The plate is impacted at its centre. The radiated pressure field is sampled using a microphone array located in the nearfield of the source. A formulation for time domain nearfield acoustical holography developed by the authors is used to calculate the pressure field at the source. An important feature of this formulation is that it considers three dimensional linear convolution to avoid wrap around errors. The determination of the back propagated pressure field is an illposed problem, and regularization is performed using Tikhonov's method and generalized cross validation. An image processing algorithm is then applied to the back propagated sound field to localize the defect. This algorithm assumes that the defect is small compared to the wave front and that it produces a discontinuity in the impedance of the structure. To pinpoint possible defect positions, the algorithm calculates and compares statistical features between pixels at the same radius from the impact point. Reflection of the sound pressure at the pinpointed locations is then used to confirm the actual position of the defect.

title: Cylindrical Nearfield Acoustical Holography

title: Transient nearfield acoustical holography
abstract: Transient Nearfield Acoustical Holography Sean F. Wu Department of Mechanical Engineering, Wayne State University, Detroit, MI, 48202 This paper presents the general formulations for reconstructing the transient acoustic field generated by an arbitrary object with a uniformly distributed surface velocity in free space. These formulations are derived from the Kirchhoff Helmholtz integral theory that correlates the transient acoustic pressure at any field point to those on the source surface. For a class of acoustic radiation problems involving an arbitrarily oscillating object with a uniformly distributed surface velocity, for example, a loudspeaker membrane, the normal surface velocity is frequency dependent but is spatially invariant. Accordingly, the surface acoustic pressure is expressible as the product of the surface velocity and the quantity that can be solved explicitly by using the Kirchhoff Helmholtz integral equation. This surface acoustic pressure can be correlated to the field acoustic pressur.

title: REGULARIZATION AND NEARFIELD ACOUSTICAL HOLOGRAPHY
abstract: The reconstruction of the pressure and normal surface velocity provided by nearfield acoustical holography (NAH) from pressure measurements made near a vibrating structure is a linear, ill posed inverse problem. Regularization provides a technique of overcoming the ill posedness to generate a solution to the linear problem in an automated way. Our objectives are (1) to develop an approach which formulates exterior and interior NAH problems in an identical way so that a single regularization theory may be applied to both these problems and (2) using model data, representative of a typical NAH measurement, we compare the errors and filter factors associated with four regularization schemes; Tikhonov, truncated singular value, conjugate gradient and Landweber iteration given various levels of signal to noise in the pressure data. We use the Morozov discrepancy principle for parameter estimation.  The theory of regularization for linear problems is a heavily researched area and many books  are available on the subject. However it is clear that there is no holy grail as too the best approach, and the success of any particular approach depends on the nature and the physics of the problem which is being solved. NAH is distinguished by the existence of evanescent waves which decay exponentially from the surface of the vibrator. This decay is the root of the ill posed nature of the inverse problem which turns these exponential decays into exponential amplifications reeking havoc due to noise contained in the measured data. With regard to regularization approaches one thing is clear, the singular value decomposition (SVD) of the system matrix plays a crucial role. We provide in Section 2 a formulation of the exterior, separable geometry NAH problem which casts it as a singular value decomposition so that it can be treated in the same way as the conformal exterior NAH and interior NAH problems, both which have been solved using the SVD. This formulation brings all of the NAH problems under the same umbrella so that they can be treated equally with a single regularization approach. In Section 3 we discuss four regularization schemes and in Sec. 4 apply them to some numerical data generated to be representative of the evanescent wave nature of the NAH problem. The reconstruction errors and reconstruction filters are compared. 2 SEPARABLE GEOMETRY NAH AND THE SVD Let wd be the discretized normal velocity of a separable (planar, cylindrical or spherical) vibrating surface S' and pd be the discretized hologram (measured) pressure on the surface S located at a constant distance d from S' The following relationship exists between them  pd F 1 d GNdFdwd (1) where Fd represents a two dimensional discrete Fourier transform. We call Fdwd and Fdpd the k space representations of the velocity and pressure, respectively. Here pd is a column vector pd C wd wd C a column vector of normal velocity and GNd GNd CMxN is the diagonal Neumann matrix. Since the operations given by F 1 d and Fd are just matrix multiplications we can define a spatial transfer function HMxN which relates directly the pressure vector to the velocity vector:  where H embodies the DFT operations in Eq. (1) H F 1 d GNdFd. Now we can consider the singular value decomposition (SVD) of the transfer function H in Eq. (2) given by H USV H (3) where U and V are left and right unitary (orthonormal) matrices, respectively, and S is a diagonal matrix of complex singular values. Comparison of Eq. (3) with Eq. (1) reveals that the SVD and Fourier transform decomposition are one in the same thing, as long as we allow the singular values to be complex numbers. Thus V H Fd, U F 1 d and S GNd. For example, the singular values for the planar geometry are just sij rck k2 kxi kyj e k2 kxi kyj(z z' with S diag [s11, s12 sij given the discretization of (kx, ky) (kxi, kyi) where diag is a diagonal matrix and (i,j spans all the k space components used in the discretization. When kcij kxi kyj k (kcij lies outside the radiation circle) the magnitudes of the singular values decay exponentially. 3 REGULARIZATION The literature for regularization of linear equations, like Eq. (2) is vast and many books have been written on the subject. Unfortunately, there appears to be no holly grail with respect to the best technique, since the characteristics of the inversion depend very much on the physics of the problem. Applications to NAH problems are just beginning to appear in the literature. We concentrate on regularization techniques based on a knowledge of the noise variance, and present a rather simple method of determining the noise from the data. We discuss four regularization techniques; the standard Tikhonov method, Landweber iteration, the conjugate gradient approach, and a modified truncated SVD approach. In all these approaches one must also apply a stopping rule or parameter selection method. We use the most popular one, the discrepancy principle of Morozov. To formulate the effects of noise in the hologram data pd, assume that spatially uncorrelated random noise e with zero mean and variance s is present in the measurement. We use a superscript d to indicate a quantity with noise. Thus pd is the pressure data with noise. We rewrite Eq. (2) including noise as (E is expectation) pd Hw d d, p d d pd e and E pd pd s M (4) 3.1 Tikhonov regularization Tikhonov regularization minimizes Hwd d pd 2 a wd d 2 represents the L2 norm) We use the discrepancy principal to determine the regularization parameter a. The solution using Eq. (3) is well known and is given by w d Rap d d where Ra is called the regularized inverse of H and Ra V Fdiag 1 s1 1 sN U F diag |s1| a |s1| |sN a |sN (5) F is the k space filter factor. The predicted value of the hologram pressure p d (now depending on a and the reconstructed velocity) is p d Hw d HRapd UFUpd (6) The Morozov discrepancy equation is Hwad d pd pad d pd d, d Ms (7) that is, the smoothed pressure should differ from the measured pressure by the noise variance. This mimics the condition shown in Eq. (4) for the noise variance. Using the SVD we can write this equation in k space as  Fh diag a a |s1| a a |sN (8) We recognize pd as the k space representation of the measured pressure. F a h is a high pass filter admitting only small singular values. We use this version of the discrepancy principle for all four regularization techniques presented here, with separate expressions for the filter factor and with a representing the corresponding regularization parameter of the particular method. The discrepancy equation is robust computationally; the left hand side represents a monotonic function in a. 3.2 Landweber iteration This popular technique [4] provides the solution (integer regularization parameter m) w Rmp V diag 1 1 b |s1| )m s1 1 1 b |sN )m

title: NEAR FIELD LIMIT IN POSITIONING THE MICROPHONE FOR PRESSURE MEASUREMENTS IN USING THE NEARFIELD ACOUSTICAL HOLOGRAPHY
abstract: The recently developed BEM based NAH (nearfield acoustical holography) is a useful technique for identifying the sound source of vibrating objects. The acoustic parameters of a sound source can be reconstructed by using the vibro acoustic transfer matrix, which is determined by means of BEM, and the sound pressure measured in the nearfield. Theoretically, one can come up with a very nice reconstructed result as the field plane gets near to the source surface. However, when a microphone is placed in the very close nearfield of the source surface, the scattering, reflection, or resonance in the gap between the source and the microphone can distort the acoustic field, and therefore, the measured field pressure would differ from the actual one in the absence of the microphone. In order to analyze this problem, the interference effect of the microphone is numerically calculated by using the non singular BEM that yields very small error in the nearfield. From this analysis, it is found that the prediction error of the field pressure decreases firstly and then increases as the microphone approaches the vibrating surface from the farfield to the close nearfield. It is noted that the microphone should be separated from the source surface by at least a diameter of the microphone for an error ratio less than about 2% This means that if one wants to put a microphone in the very close nearfield, a microphone with small diameter should be used 1 INTRODUCTION The BEM based NAH technique is known to be very powerful in dealing with the sources having irregular shaped boundaries [1 4] When the vibro acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the non propagating wave components. If the microphones are placed in a very close nearfield for reducing the reconstruction error, the scattering, reflection, or resonance in the space between the vibrating sound source and the microphone can distort the acoustic field. In this case, the measured field pressure can differ from the actual radiating pressure. In this paper, the interference effect of the microphone is analyzed in order to characterize this problem and to suggest a guideline to reduce the measurement error. 2 ACOUSTIC HOLOGRAPHY EQUATION Discrete form of acoustic boundary integral equation can be used in realizing BEM and its matrix vector form can be written as {p}f [D]f {p}s [M ]f {v} in the domain, (1a) s {v}s on the boundary. (1b) Here, {p}s, {v}s are the pressure and velocity vector on the surface, respectively, {p}f denotes the field pressure vector in the domain, [D]s, [M ]s mean the dipole and monopole matrices on the surface, and [D]f [M ]f are those corresponding to field pressures, respectively. From Eqs. (1a) and (1b) the following field pressure can be described only by the surface velocity provided [D] 1 s exists: {p}f [M ]f [D]f [D] 1 s [M ]s {v}s [G] {v}s (2) Here, [G is the vibro acoustic transfer matrix correlating the surface normal velocity and the field pressure that contains the geometric information of the system. If the field pressure at m points is known, the surface velocity at n m) nodes can be uniquely determined. By virtue of the leastsquared solution and the singular value decomposition (SVD) the inverse of Eq. (2) can be expressed as follows: {v}s [G] {p}f [G] [G] 1 [G] {p}f [W [L] 1 [U {p}f (3) Here, [G denotes the nxm pseudo inverse matrix of [G and the superscript H signifies the Hermitian operator. By the SVD, the transfer matrix [G can be decomposed by [G] [U [L] [W where [L] diag (l1, l2, ln) l1 l2 ln 0. Here, li denotes the singular values of [G The columns of [U and [W comprise the left and right singular vectors of [G Eq. (3) enables the reconstruction of the velocity field of source surface, in principle, if the near field pressures are measured and the transfer matrix is generated by BEM. Now, using {v}s, it is possible to determine other acoustic parameters on the surface or in the field such as the pressure, velocity, surface impedance, acoustic intensity or acoustic power distribution. 3 SINGULARITY IN THE NEARFIELD When the conventional BEM is utilized for NAH simulation, the field pressures can not be predicted accurately when the field points are located within about 20% of characteristic length of BEM model  This inaccuracy is generated from the singular behavior of Green function near the numerical integration points, and increases as the field points move close to the surface. Consequently, the transfer matrix cannot be built precisely in this range, and significant error can be included in the restored results. In order to realize the experimental source identification by using the BEM based NAH, the field pressures should be measured by using the microphones. Theoretically, the reconstruction error should be reduced with closing the field points to the surface because the ratio of measured field pressure to the background noise can be increased, the non propagating wave component can be included in the measurement, and the singularity of transfer matrix can be further reduced. Consequently, in order to collect such rich information on the sound signal from the source, the microphones should be placed as close as possible to the source surface. In this situation, the acoustic interference effect between the microphones and source surface becomes influential to the final reconstructed results. By employing the nonsingular BEM [5] the very accurate field parameters can be calculated in the nearfield. 4 INTERFERENCE EFFECT DUE TO THE MICROPHONE To analyze the interference effects of microphone for measuring the field pressure in the nearfield, the nonsingular BEM simulation was carried out. The simulation model is illustrated in Fig. 1(a) A baffled piston of 13.2 mm diameter was assumed as the sound source, and a 1/2 inch microphone was modeled by a cylinder of 13.2 mm diameter and 5 mm height. The maximum frequency limit of BEM model was about 20.8 kHz for l/10 criterion because the linear elements were adopted. Three sets of phase condition of the piston were considered as shown in Figs. 1(b) (d) For set I, the piston was vibrating with in phase at all nodes. For sets II and III, two and four segments of the piston were vibrating out of phase, respectively. If the pressures on the vertical axis of the piston are compared, the results cannot be observed because the on axis pressures are canceled by the opposite phase condition. In this analysis, when the origin is the center of the piston, the pressures calculated at (x, y, z (1, 1, 1) mm are compared in Fig. 2 for several different distances between the microphone and the source surface. When the microphone is located far enough from the source, the field pressure converges to the pressure level in the absence of the microphone. But, if the microphone is placed closely to the source, the pressure level is increased and the resonance occurs in the high frequency range. The pressure amplitudes of the out of phase conditions are reduced very much compared with those of in phase condition while the overall trend is the same. At 1 kHz, the error ratio of the field pressure is compared in Fig. 3. Note that  If one wants to place the microphone in the very close nearfield, the diameter of the microphone should be as small as possible.  (a) BEM model of a baffled piston and a microphone (diameter of microphone 13.2 mm, height of microphone 5 mm, 428 nodes, 818 linear triangular elements) and the phase sets of the piston are: (b) set I; (c) set II; (d) set III.  Amplitude of field pressure at (x,y,z (1,1,1) mm for a baffled piston by varying the distance of microphone from the source surface (z relative distance of the microphone, d diameter of the microphone) 5  In this study, the interference effect of the microphone is analyzed in the nearfield. When the microphone is placed in the very close nearfield of the source surface, the reflection or resonance between the vibrating sound source and the microphone can distort the acoustic field. It is found that the microphone should be placed at least its diameter away from the source surface in order to reduce the interference error of the measured field pressure less than 2% If one wants to measure the field pressure accurately in the very close nearfield, the microphone with small diameter should be employed. It is already shown in the vibroacoustic problems that the nearfield error caused by the singularity in conventional BEM can be greatly reduced by the use of nonsingular BEM. However, in the viewpoint of measurement, one should keep in mind that the meaningful measurement by the microphone placed in the nearfield is subject to the error due to the interference between the microphone and the source surface.  Error ratio of field pressure at (x,y,z (1,1,1) mm for a baffled piston by varying the distance of microphone from the source surface (1 kHz, z relative distance of the microphone, d diameter of the microphone)

title: Improved Statistically Optimal Nearfield Acoustical Holography in subsonically moving fluid medium
abstract: (SONAH) can be used to reconstruct three dimensional sound fields by projecting two dimensional data measured on a "small" aperture that partially covers a composite sound source in a "static" fluid medium. Here, an improved SONAH procedure is proposed that includes the mean flow effects of a moving fluid medium while the sound source and receivers are stationary. The backward projection performance of the proposed procedure is further improved by using a wavenumber filter to suppress subsonic noise components. Through numerical simulations at Mach 0.6, it is shown that the improved procedure can accurately reconstruct sound source locations and radiation patterns: e.g. the spatially averaged reconstruction errors of the conventional and improved SONAH procedures are 15.40 dB and 0.19 dB, respectively, for a monopole simulation and 21.60 dB and 0.19 dB for an infinite size panel. The wavenumber filter further reduces spatial noise, e.g. decreasing the reconstruction error from 1.73 dB to 0.19 dB for the panel simulation. An existing data measured in a wind tunnel operating at Mach 0.12 is reused for the validation. The locations and radiation patterns of the two loudspeakers are successfully identified from the sound fields reconstructed by using the proposed SONAH procedure.

title: Sound radiation from Caribbean steelpans using nearfield acoustical holography.
abstract: The sound field of a Caribbean low tenor steelpan is examined using nearfield acoustical holography (NAH) The particular instrument, a Bertrand Kelman low tenor, has been previously examined in a single plane using a four microphone sound intensity technique. This paper will examine the sound radiation in a cuboid space enclosing the instrument at three frequencies. Presented will be the results of sound pressure level (SPL) active acoustic intensity (AI) and reactive acoustic intensity (RI) obtained through application of the NAH technique. The limitations of NAH when applied to an irregularly shaped object such as a tenor steelpan will also be presented.