Bayesian estimations of dissipation, sound speed, and microphone positions in impedance tubes

: Sound speed, microphone positions, and tube wall dissipation are critical parameters for absorption and impedance measurements using the transfer-function method in an impedance tube. This work applies a Bayesian method, based on a reﬂection coefﬁcient model of an air layer and a boundary layer dissipation model, to estimate the values of these parameters for tube measurements. This estimation is based on experimental measurements obtained in the empty impedance tube with a rigid termination. Analysis results demonstrate that this method is able to accurately estimate the dissipation coefﬁcient, the sound speed, and the microphone positions for highly accurate tube measurements. V C 2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) .


Introduction
Since Chung and Blaser (1980) pioneered a two-microphone transfer-function method for impedance tube measurements, this measurement method has widely been accepted in material research (Begum et al., 2022;Stender et al., 2021) and application (ISO 10534-2, 1998), including noise control engineering, physical acoustics, and architectural acoustics.Bod en and A ˚bom (1986), Chu (1986), and Katz (2000) have also investigated measurement uncertainties due to microphone phase-mismatches and positional errors of the two-microphone method in impedance tubes.Their findings suggest both the microphone phase-mismatches and the microphone position errors negatively impact the measurement accuracy.To avoid the microphone phase-mismatches, Chu (1986) suggested one single microphone channel using a correlation measurement technique, such as maximum-length sequences, to accomplish the two-microphone transfer-function measurements in a sequential manner.Katz (2000) inserted an additional microphone flush-mounted in the rigid backing at the tube termination, which allows the accurate estimation of the microphone positions through calculating the null frequencies.Note that Katz (2000) determined the null frequencies for determining the equivalent centers of the microphones.The null frequencies are by definition the singular ones, which need to be avoided in an actual impedance tube measurement.
This work also employs sequential measurements at two microphone positions yet applies a Bayesian probabilistic inference to estimate the sound speed, the tube wall dissipation, and the microphone positions based on a hypothetical air layer in front of the rigid termination at the tube end.Bayesian analysis has recently been applied in porous material research by other researchers (Chazot et al., 2012;Niskanen et al., 2017;Roncen et al., 2022).Their studies distinctly differ from the current effort.This work focuses on the standard tube measurement using the two-microphone transfer-function method (Chung and Blaser, 1980;ISO 10534-2, 1998), and it employs the reflection coefficient of the air layer as the prediction model, while the other authors engage the porous material models, such as the Johnson-Champoux-Allard-Lafrage model, to estimate intrinsic properties of porous media but not the tube dissipation and the sound speed.Also, the primary method used in their experimental effort is not a two-microphone transfer-function method (Chung and Blaser, 1980) but rather three position measurements (Roncen et al., 2022) or the four-microphone tube method (Song and Bolton, 2000).The tube measurements using a two-microphone transfer-function method require accurate values of the sound speed and the microphone positions for determining the sound absorption and the surface impedance of the material under test.The microphone positions only need to be calibrated once, but the sound speed of medium air is changing constantly, depending sensitively on the ambient environment.Therefore, this work parameterizes the well-known equation to calculate the reflection coefficient such that the sound speed along with the microphone positions become pending parameters.These are simultaneously estimated within the same model-based framework.
Another influencing parameter within this measurement technique is dissipation due to the medium air and a so-called boundary effect of the tube interior walls.Although the air dissipation may not be always noticeable when measuring materials under certain upper limit frequencies, the boundary dissipation is an issue that depends on the tube materials and the ambient environment.As the preliminary results of this work demonstrate, the reflection coefficient measurement of a hypothetical air layer will more accurately match the well-established model when the boundary dissipation of the tube interior walls is included in the calculations.
The significance of this work is that the critical parameters for the measurements are directly determined from acoustical measurements over the valid frequency range.They are dictated by the dimensions of the impedance tube and microphone separations.Among the critical parameters, the dissipation due to the tube wall boundary is accurately estimated.To the best of the authors' knowledge, the dissipation has not yet been sufficiently investigated in the published literature, and the model-based Bayesian method has not yet been applied in estimating these critical parameters.The estimation method presented in this paper may also be relevant to other tube measurements, such as the three-or fourmicrophone methods (Salissou and Panneton, 2010;Song and Bolton, 2000).
The rest of this paper is structured as follows.Section 2 formulates the air layer prediction model and details the parameterization of the reflection coefficient calculation of the two-microphone transfer-function method.Section 3 introduces the model-based estimation method using Bayesian inference.Section 4 discusses preliminary results.Section 5 further discusses a few issues related to increased accuracy of critical parameter estimation.

Model formulation
This section parameterizes the reflection coefficient calculation used for the two-microphone transfer-function method.When the tube does not contain absorbing materials under test, an air layer in front of the rigid termination can be considered as a hypothetical material under test.The reflection coefficient of this hypothetical air layer serves as the theoretical prediction model of the reflection coefficient.

Reflection coefficient measurement using a transfer function
Figure 1 presents an impedance tube setup used for the measurement.Note that k in Fig. 1 denotes the wavelength of the sound wave.A plane wave travels along a long narrow tube, and the material under test is placed at the other end.With the microphones embedded in the wall of the tube measuring the standing waves, the acoustical properties of the material can be calculated through the microphone signals containing the incident and reflected wave components.The inner wall of the tube should be as smooth as possible to avoid any disruptions of the plane waves, including the microphone membrane/protection grid having to be placed flush to the inner wall (see Fig. 1).
With the measured responses using the microphones at positions "1" and "2" in Fig. 1, the separation between the two microphones s, and the distance from the position 1 to the frontal surface of the material under test L, the complex-valued reflection coefficient can be determined as where P i ; P r represents the incident and reflected sound pressures in the frequency domain at the front surface of a material under test.H 12 ¼ P 2 =P 1 is the transfer function between the Fourier transforms of the microphone signals at positions 1 and 2. b ¼ x=c is the propagation (phase) coefficient of the sound wave in the air, x is the angular frequency, and c is the sound speed of air.Equation ( 1) is valid for calculating the reflection coefficient in lossless media for f < c=2s.However, due to the dissipation in the medium air and the boundary effect of the tube walls (Xiang and Blauert, 2021), there is always some energy loss during the sound propagation in narrow tubes.For weak losses, the phase coefficient b in Eq. ( 1) can be replaced with a complex-valued propagation coefficient c, (2) with Fig. 1.Two-microphone impedance tube measurement setup using the transfer-function method.
ARTICLE asa.scitation.org/journal/jel where a f is the damping coefficient, being a function of frequency.Accounting for the dissipation due to boundary effect of tube walls, it can be predicted through a so-called "wide tube" dissipation model, contingent upon an unknown parameter f, where U is the circumference of the tube, A is the cross-sectional area of tube, and r is the radius of the circular tube [see Eq. (11.9) in Chap.11 of Xiang and Blauert (2021)].This work additionally parameterizes the dissipation model with a scalar parameter f to include possible variations due to different tube wall materials, and it also copes with ambient environment changes.

Hypothetical air layer model
In front of the rigid termination, an air layer of thickness d is hypothetically assumed, and the surface impedance Z s and reflection coefficient R M of this air layer incorporating the dissipation can be written (Xiang and Blauert, 2021) as e cd À e Àcd : (5) The surface reflection coefficient of the air layer backed by a rigid termination can be expressed as with q c being the characteristic resistance of medium air.The surface reflection coefficient R M of the air layer in Eq. ( 6) is considered as a prediction model in the following discussion.Figure 2 illustrates the air layer model for different layer thicknesses.R D in Eq. ( 2) represents the (experimental) data when the transfer-function H 12 is experimentally measured in the tube assuming microphone positions were known via s, L.

Model-based Bayesian estimation
This work applies Bayes's theorem to estimate the parameters encapsulated in Eqs. ( 2 where I includes the background information that "the prediction model is able to describe the data well."h ¼ ½s; L; c; f collectively represents all the parameters to be estimated.The denominator is a normalization constant in the current context and can therefore be ignored so that the posterior is proportional to the likelihood multiplied by the prior.Unlike the conventional Bayesian parameter estimation (Xiang, 2020), the following estimation process affects both the model R M and data R D .This is also expressed in Eq. ( 7), for example, by the likelihood pðR D ; R M jh; IÞ, which represents the probability that both a particular data set R D and the reflection coefficient model R M would be observed given the model parameters h and the background information I.
The prior distributions encode initial knowledge of the sound speed, the microphone positions, and the scale factor of the damping coefficient.The microphone centers are already known to be located within the vicinity of the microphone diaphragm.The sound speed is influenced by the ambient environment, such as the temperature.To avoid imposing any bias for certain values of each parameter, the prior probabilities of the parameters are assigned to be uniform distributions over individual parameter ranges based on the principle of maximum entropy (Xiang, 2020).
The likelihood function pðR D ; R M jh; IÞ represents the probability of the difference between the measured and modeled data, which is so-called residual error.In this work, the measured data are the reflection coefficient R D in Eq. ( 2) of the air layer backed by a rigid termination, and the modeled data are the complex-valued reflection coefficient R M in Eq. ( 6) calculated from the thickness of a hypothetical air layer.Based on the available information, applying the principle of maximum entropy assigns the likelihood function to be a Student's t-distribution (Xiang, 2020), where CðÁ Á ÁÞ is the standard gamma function, and k is the residual error at a certain frequency point k.
As previously mentioned in Sec. 3, the parameters in this case are the sound speed, the scale factor of the dissipation coefficient, and the locations of the equivalent acoustic center positions for the microphones.The only prior knowledge on the equivalent acoustic centers of the microphones is that they are within the physical range of the microphone diameter around the microphone locations.This work uses nested sampling to estimate the parameter h (Skilling, 2004).

Experimental results
The measurement is carried out in a long narrow tube made of polyvinyl chloride (PVC) with a wall thickness of 0.64 cm (0.25 in.) and an inner tube diameter of 3.81 cm (1.5 in.).A solid metal plate of 2 cm thickness is tightly attached to the tube end as the rigid termination.Microphone position 1 is 11.4 cm (4.5 in.) away from the rigid end, while microphone position 2 is 10.2 cm (4 in.) away from the rigid end.A 1/4 in.microphone (PCB Piezotronics, Depew, NY) is placed in a sequential manner into the tube (see Fig. 1) to measure individual sound pressure impulse responses at the two microphone positions.This approach helps avoid phase-mismatches between different microphones.
As discussed in Sec. 3, the prior probability density functions of the parameters are assigned to uniform distributions as listed in Table 1.This assignment is based on the available knowledge of the pending parameters, including the dissipation factor, the microphone positions, and the sound speed (Xiang, 2020), where short-hand notation PðhÞ ¼ pðhjIÞ is used for simplicity.The prior range of the microphone positions can be set within broad ranges as listed in Table 1.Before the measurement, broad prior ranges of the sound speed and the unknown dissipation factor are also listed.After the rigid termination is measured, the hypothetical air layer model is involved.The uniform prior assignment of relevant unknown parameters as in Table 1 is to impose no prior preference to any possible values of these unknown parameters.
The reflection coefficient model in Eq. ( 6) hypothetically allows an arbitrary thickness d of the air layer to be chosen.Different choices of d would lead to a straightforward modification in the distance L from microphone position 1 to the surface of the air layer (in Fig. 1). Figure 3 compares the estimated results of the measured and modeled reflection coefficients of the air layer of different thicknesses.Experimentally measured reflection coefficient R D without dissipation is also calculated to compare with the one with dissipation determined using Eq. ( 4) through the estimated dissipation factor f. As shown in the figure, the model match to the experimental data is more accurate with the tube wall dissipation taken into consideration, especially the imaginary part of the reflection coefficient within the frequency range 2.5-4.2kHz, the valid range (ISO 10534-2, 1998) as expected for the tube diameter and the microphone separation used in this measurement setup.As presented in the third row of Fig. 3, the residual absorption errors are drastically reduced after incorporating the estimated parameters, particularly the dissipation coefficient.The residual absorption error is equal to the difference between the measured absorption coefficient and the theoretical absorption coefficient.This quantity is the Table 1.Prior probability assignment for the four parameters.
PðcÞ ¼ uniform(340, 350) m/s PðLÞ ¼ uniform(10.5, 12.5) cm PðfÞ ¼ uniform (3,8) absorption error of the tube without any material under test.The mean residual absorption reduces from 0.13 down to 0.013.Table 2 lists the estimated values of the parameters for two different groups of measurements.One group was measured during a hot summer day in 2021 (temperature 22.3 C, humidity 69%), while another was measured on a cold winter day of 2021 (temperature 18.7 C, humidity 22%).Both the sound speed and the dissipation coefficient depend on the temperature.The estimated sound speed increases with the increasing temperature, while the estimated dissipation factor decreases.According to Xiang and Blauert (2021), the dissipation is intrinsically related to thermal energy transfer into the thermally conducting tube walls [see Chap.11 of Xiang and Blauert (2021)].At a lower temperature, the dissipation due to the boundary layer becomes larger.

Discussion
The Bayesian inference applied in this work directly relies on experimental data to update the experimenter's prior knowledge on these parameters.After involving the measurement data, the model-based Bayesian estimation leads to the updated knowledge encoded in the parameter estimates as listed in Table 2, which are derived from extremely sharply peaked posterior distributions.To be more precise, the experimenter's (posterior) knowledge on these parameters becomes tremendously sharpened.Note that some other approaches would also serve the goal of estimating the necessary parameters, yet the Bayesian probabilistic approach presented in this work also provides uncertainty estimates in addition to the mean parameter values.It is apparent that the microphone positions do not drift when the ambient environment changes, but the sound speed and dissipation do.The model-based Bayesian estimation method presented in this work can converge to a fixed set of positional parameters, including the microphone separation and the distance to the frontal surface of the material under test.In further applying the Bayesian estimation method, the well-converged set of positional parameters can then be fixed, so that the parametric model includes only the sound speed c and the dissipation factor f.
The method discussed in this work based on a rigid backing can be used immediately before or after the actual material measurements in the same tube to calibrate the critical parameters that enter into the acoustical property calculations since the environmental conditions are constantly drifting.The preliminary results as listed in Table 2 indicate that not only the sound speed, but also the dissipation coefficient, change with the ambient environment.This is largely due to the boundary effect of the tube walls, which is influenced by the thermal energy transfer in the tube walls (Xiang and Blauert, 2021).

Concluding remarks
This work proposes a model-based Bayesian method to estimate the sound speed and the dissipation coefficient along with the microphone positions.A hypothetical air layer backed by a rigid termination is considered as the material under test; its prediction model is used inversely to estimate the critical parameters for accurate tube measurements.The twomicrophone transfer-function method for the impedance tube measurement requires accurate values of the sound speed, positional parameters of the microphone used, and the dissipation coefficient during the measurements of sound absorbing materials.Incorporating these estimated parameters into Eq.( 2) eventually results in reduced residual absorption errors, with an average reduction from 13% down to 1.3% as illustrated in Figs.3(g)-3(i).The preliminary results indicate that the model-based Bayesian method is able to estimate the accurate dissipation coefficient, the sound speed, and the microphone positions through the measured and modeled data.The accurate estimation of these parameters serves the calibration strategy for measuring other acoustic properties of absorbing materials in the same impedance tube.Future efforts will be dedicated to systematic investigations over even wider frequency ranges, including experiments over different tube diameters and the influences of different tube materials on the dissipation coefficient.The sound speed drifts with slow changes in ambient conditions, including temperature and humidity; their influencing effects need to be further explored, which is beyond the scope of the current work.Another line of research is to investigate this estimation framework in the three-or four-microphone methods using impedance tubes.
) and (6).The parameter set h collectively contains four parameters: the microphone separation s, the distance from microphone position 1 (in Fig.1) to the frontal surface of the material under test L, the sound speed c, and a scale parameter f of the damping coefficient a f .Given the model R D and the data R M , Bayes's theorem is represented as pðhjR D ; R M ; IÞ zfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflffl ffl{ posterior ¼ pðR D ; R M jh; IÞ zfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflffl ffl{ likelihood Â pðhjIÞ zffl ffl}|fflffl{ prior pðR D ; R M jIÞ ; (7)

Fig. 2 .
Fig. 2. Air layer model prediction of complex-valued reflection coefficient for four different layer thicknesses.

Table 2 .
Predicted parameters from Bayesian parameter estimation.