# Deterministic and statistical methods for the characterisation of
poroelastic media from multi-observation sound absorption measurements

Jacques Cuenca1, Peter Göransson2, Laurent De Ryck1, Timo Lähivaara3

1 Siemens Industry Software, Interleuvenlaan 68, BE-3001 Leuven, Belgium
2 Department of Aeronautical and Vehicle Engineering, KTH Royal Institute of
Technology,
Teknikringen 8, SE-10044 Stockholm, Sweden
3 Department of Applied Physics, University of Eastern Finland,
P.O. Box 1627, FIN-70211 Kuopio, Finland

### Abstract

This paper proposes a framework for the estimation of the transport and
elastic properties of open-cell poroelastic media based on sound absorption
measurements. The sought properties are the Biot-Johnson-Champoux-Allard model
parameters, namely five transport parameters, two elastic properties and the
mass density, as well as the sample thickness. The methodology relies on a
multi-observation approach, consisting in combining multiple independent
measurements into a single dataset, with the aim of over-determining the
problem. In the present work, a poroelastic sample is placed in an impedance
tube and tested in two loading conditions, namely in a rigid-backing
configuration and coupled to a resonant expansion chamber. Given the non-
monotonic nature of the experimental data, an incremental parameter estimation
procedure is used in order to guide the model parameters towards the global
solution without terminating at local minima. A statistical inversion approach
is also discussed, providing refined point estimates, uncertainty ranges and
parameter correlations. The methodology is applied to the characterisation of
a sample of melamine foam and provides estimates of all nine parameters with
compact uncertainty ranges. It is shown that the model parameters are
retrieved with a lower uncertainty in the multi-observation case, as compared
with a single-observation case. The method proposed here does not require
prior knowledge of the thickness or any of the properties of the sample, and
can be carried out with a standard two-microphone impedance tube.

##  1 Introduction

The estimation of material and geometrical properties of mechanical and
acoustical systems is central to many areas of engineering, yet numerous
methodological challenges exist. One such example, of particular interest
here, is the measurement of transport and elastic properties of poroelastic
media. These materials exhibit intricate and complex fluid-structure
interactions related to visco-thermal exchanges and visco-elastic energy
dissipation, and as a result are often modelled with a large number of
parameters. The estimation of a complete set of these parameters, which
consistently and coherently satisfy a comprehensive constitutive model, has
been a topic of increasing interest, for instance for its use in simulation
tools, as computational power has grown over the last decades.

###  1.1 Direct methods for individual parameters

A variety of methods for the characterisation of poroelastic media have been
developed, including direct, indirect and inverse methods. Direct methods
involve dedicated experimental rigs providing individual estimations of a
subset of properties, e.g. porosity, tortuosity, thermal and flow
permeability, viscous and thermal characteristic lengths, Young’s modulus and
mechanical loss factor [37, 38, 39, 47, 43, 49, 48, 50, 52]. These approaches
are reliable and powerful in that a potentially high degree of accuracy is
reached thanks to an appropriate control of experimental conditions and
isolation of sources of error. Nevertheless, using individually measured
parameters in a comprehensive simulation model may lead to inconsistencies.
For instance, separate samples are often required for the different dedicated
setups, thereby introducing variability due to inhomogeneity across samples.
Also, the availability and maintenance of a full suite of test rigs is often
impractical. As an alternative, indirect and inverse methods have been
proposed, of which a selection is recalled below. An exhaustive review of all
available methods is out of scope here and the reader is referred to the
recent review by Horoshenkov [33] and references therein for a complementary
overview.

###  1.2 Indirect methods

Indirect analytical methods rely on derivations of the constitutive properties
from the intrinsic equivalent acoustical properties of the porous sample,
namely its frequency-dependent equivalent density and bulk modulus, which can
be measured using a three- or four-microphone transmission impedance tube, or
a dual-load two-microphone tube. First proposed for the estimation of viscous
and thermal dissipation properties by Panneton and Olny [59, 58], the
methodology has been reported by Bonfiglio and Pompoli [6] to provide all 5
parameters in the Johnson-Champoux-Allard (JCA) model [44, 11] and recently by
Jaouen et al. [42] for the measurement of all 6 parameters of the Johnson-
Champoux-Allard-Lafarge (JCAL) model [44, 11, 46]. Such an indirect
methodology is robust and mature, and allows for a comprehensive estimation of
the full set of transport properties and their uncertainties. Nevertheless,
these estimations are inherently dependent on a number of assumptions as well
as on the accuracy of the measured intrinsic acoustical properties [59].
Additionally, a minor drawback is that the estimation of the porosity and flow
resistivity rely on the manual identification of frequency ranges where the
intrinsic acoustical quantities exhibit an asymptotic behaviour. Although this
could be automated, practical considerations such as frequency resolution
could influence the frequency ranges and potentially lead to an
underestimation of parameter uncertainties. Complementary analytical
approaches have been proposed, for instance by Groby et al. [30] for
estimating transport properties from ultrasonic reflection and transmission
measurements. The ultrasonic range inherently provides a direct comparison
with the high-frequency approximation of the JCA model but does not provide
access to the flow resistivity, thus requiring it to be measured separately.

###  1.3 Deterministic inverse methods

Inverse methods stand as an appealing alternative to direct and indirect
methods in that the material properties are obtained as the set of model
parameters yielding the best fit of a simulation based on the constitutive
model. This guarantees that the solution is a feasible set of parameters that
automatically satisfies the model, without the need for approximations or
additional steps. A vast number of deterministic inverse methods have been
developed and validated. For instance, methods have been proposed to estimate
all, or a subset of, the transport parameters in the JCA or JCAL models using
the measured complex characteristic impedance as the target data [4, 6, 69,
29], which can be carried out in an impedance tube with transmission
capabilities. Methods using a two-microphone tube have also been investigated.
For instance, Dossi et al. [20] have proposed the additional estimation of the
elastic parameters of polyurethane foams by combining datasets from specimens
of different thicknesses. A method for estimating the 5 JCA parameters and 3
elastic properties has been proposed by Verdière et al. [68] relying on a
numerical model accounting for intentional circumferential air gaps and using
sound absorption measurements, assuming prior knowledge of the sample
dimensions and density. A numerical model was also used by Vanhuyse et al.
[67] to account for intentionally clamped circumferential boundaries. The
retrieval of transport and elastic parameters from the reflection coefficient
was numerically validated therein. In addition to impedance tube methods,
deterministic inverse methods for the estimation of anisotropic transport and
elastic properties [19, 18, 15] as well as methods relying on ultrasonic
measurements [22, 17] have been studied. In particular, Ogam et al. [57]
proposed an ultrasonic time-domain method for the estimation of porosity,
tortuosity, viscous characteristic length, density, Young’s modulus and
Poisson ratio.

###  1.4 Statistical inverse methods

Fully deterministic approaches lack however the ability to estimate
uncertainty ranges for the model parameters. Chazot et al. [12] proposed a
Bayesian framework for the estimation of 8 properties of poroelastic media,
including mass density, 5 transport parameters and 2 elastic properties,
including estimated confidence intervals for all the parameters. As such, the
obtained solutions were not unique, and the mean estimates presented large
deviations from reference values in general, the reported elasticity showing
the largest estimation errors in particular. Nevertheless, the method
demonstrates the feasibility of fully characterising poroelastic media using
an impedance tube. Niskanen et al. [55] studied deterministic and statistical
inverse methods for the estimation of the 6 parameters of the JCAL model. The
method uses a transmission tube setup and provides point estimates, credible
ranges and correlations between parameters using the equivalent density and
bulk modulus as the target data. A two-microphone tube with a rigid-backed
sample was investigated as an experimentally simpler alternative, where the
difficulty of avoiding local minima was reported, as well as a higher degree
of inter-parameter correlation in the obtained solution. In further work,
Niskanen et al. [54] discussed the estimation of transport and elastic
properties of poroelastic media using reflection and transmission coefficients
in the ultrasonic range. The work numerically demonstrates the feasibility of
such an approach and allows for the estimation of shear properties by exciting
the material in oblique incidence. Other notable studies include the works of
Roncen et al. [62, 63, 64], who investigated the statistical inference of the
transport properties of rigid porous media in the time domain, including a
high order model of viscous effects. Fackler et al. [21] used a Bayesian
framework for the characterisation of multi-layer rigid porous media in terms
of the porosity, flow resistivity, tortuosity and thickness of the layers.

###  1.5 Motivation for the proposed method

The use of round robin tests has shown that acoustic measurements on porous
media exhibit relatively poor inter-laboratory reproducibility [35, 60]. The
reported measurements included frequency-dependent characteristic impedance,
complex wavenumber and sound absorption coefficient. Nevertheless, up to 3
times better reproducibility was reported for sound absorption measurements in
comparison to surface impedance [35], suggesting that the use of sound
absorption in combination with an inverse method can potentially reduce
uncertainties in the estimated parameters. A closed-form description of the
behaviour of a rigidly backed isotropic poroelastic layer exists [2]. Assuming
that this model applies as well to samples with longitudinally-sliding
boundaries in a rigidly backed tube, it forms the basis for the model
inversion in the present work. In this respect, the reduced amount of data
inherent to a purely energetic quantity, as compared to complex quantities,
could possibly add to the above-mentioned problems with parametric
insensitivity and the associated convergence of the solution to a unique set
of parameters. This is particularly true in the case of coupled resonant
systems, whose inverse problems are subject to multiple local minima due to
the non-monotonic nature of their frequency response.

Recent work by the authors proposes an incremental method for solving inverse
problems in coupled resonant systems from the knowledge of their transfer
function [28]. The method consists in solving the inverse problem in a series
of steps, starting with a sub-problem in a frequency domain where the observed
system exhibits an asymptotic behaviour, then gradually increasing the
frequency span towards that of the full problem. Such an incremental method
was shown to guide the design variables towards the global solution, while
avoiding local minima. This approach has also been applied in preliminary
numerical investigations aiming to estimate the properties of porous and
poroelastic media [14, 13], laying the ground for the present work.

###  1.6 Outline of the paper

This paper proposes an inverse method for the simultaneous estimation of a set
of 9 properties of a poroelastic material sample using a two-microphone
impedance tube. The target unknowns are the 5 transport parameters and 2
elastic properties within the Biot-Johnson-Champoux-Allard (Biot-JCA) model
[2, 44, 11], the material density and the sample thickness. Here the
previously mentioned incremental estimation method is augmented with a multi-
observation approach, which is introduced with the aim of over-determining the
problem. The multi-observation dataset consists of measurements of the sound
absorption coefficient of a rigidly backed sample in two different loading
conditions, namely with and without an expansion chamber. In order to ensure a
gradual increase of the problem complexity, the sequence of sub-problems is
defined by splitting the measured data into alternating monotonic segments
[7]. In the present paper, the multiple sub-problems are solved in a
deterministic sense using non-linear numerical optimisation. In order to
assess the uncertainty of the parameter estimation, a statistical method
within the Bayesian framework is then applied to the full problem as a means
to obtain credible ranges and refined point estimates for the different model
parameters.

The paper is organised as follows. Section 2 presents the proposed load cases,
the model of the corresponding sound absorption coefficient and the
experimental setup used. Section 3 formalises the multi-observation model and
proposes a deterministic incremental inversion method as well as a statistical
inversion method. The results of the approach are presented in Sec. 4 and a
summary of the findings is given in Sec. 5.

##  2 Multi-observation problem

The aim of the work is to estimate the properties of a poroelastic material
(PEM) sample from the knowledge of its sound absorption coefficient in one or
various loading conditions. In the following paragraphs we propose a model of
the sound absorption coefficient in the different loading conditions as well
as an experimental implementation.

###  2.1 Model

The starting point is the model of the sound absorption coefficient in the
different load cases as a function of the unknown material parameters of
interest. As the objective is to estimate these parameters based on
experimental data, the dimensions of the loading elements, which are subject
to uncertainties, are considered unknown as well. Furthermore, visco-thermal
losses at the tube walls are modelled using the approach in Ref. [8].

The two loading conditions considered in this work are depicted in Fig. 1.
Case (A) consists of a rigidly-backed poroelastic sample at the end of a tube
of length $l_{\text{a}}$. In case (B) the sample is loaded with three adjacent
cylindrical duct elements of lengths $l_{\text{b}}$, $l_{\text{e}}$ and
$l_{\text{c}}$ forming an expansion chamber of radius $r_{\text{e}}$, as
illustrated in the figure. The tube of length $l_{\text{c}}$ between the
expansion chamber and the foam acts as a coupling element that can be tuned so
as to maximise the influence of the elasticity of the foam on the overall
sound absorption.

(a)

(b)

Figure 1: Schematic representation of the impedance tube setup in the two
loading conditions. (a) Load case (A): rigid-backed material sample; (b) load
case (B): rigid-backed material sample coupled to expansion chamber.

The coupled system is modelled as a multi-layer system using the transfer
matrix method [2]. The rigid-backed poroelastic material sample is modelled
using the normal incidence impedance derived in [2, 16], here denoted

$Z_{\text{PEM}}(\mathbf{x}_{\text{PEM}},\omega),$ (1)

where $\omega$ is the circular frequency and

$\mathbf{x}_{\text{PEM}}=\left\\{h,\rho,\phi,\sigma,\alpha_{\infty},\Lambda,\Lambda^{\prime},E,\eta\right\\}$
(2)

is the set of parameters governing the behaviour of the poroelastic sample,
with $h$ the thickness, $\rho$ the mass density, $\phi$ the porosity, $\sigma$
the static flow resistivity, $\alpha_{\infty}$ the high-frequency limit of the
tortuosity, $\Lambda$ the viscous characteristic length, $\Lambda^{\prime}$
the thermal characteristic length, $E$ the Young’s modulus and $\eta$ the
viscoelastic damping ratio. The estimation of these 9 parameters is the main
objective of the paper. For the purposes of the present paper, the explicit
expression of Eq. (1) will not be repeated here, for which the reader is
referred to Refs. [2, 16]. It is worth noting that due to the one-dimensional
nature of the problem, the Young’s modulus and the Poisson ratio cannot be
measured as two independent variables. Therefore lateral deformation is here
ignored by setting the Poisson ratio to zero.

For convenience in modelling the sudden area changes in load case (B), the
transfer matrix method is here formulated in terms of the acoustic pressure
and flow. The transfer matrices of the individual duct elements are given by
[53]

$\mathbf{T}_{n}=\begin{bmatrix}\cos(k_{0}l_{n})&\dfrac{iZ_{0}}{\pi
r_{n}^{2}}\sin(k_{0}l_{n})\\\ -\dfrac{\pi
r_{n}^{2}}{iZ_{0}}\sin(k_{0}l_{n})&\cos(k_{0}l_{n})\end{bmatrix},\quad
n=\text{a},\text{b},\text{e},\text{c},$ (3)

where $k_{0}=\omega/c_{0}$ is the wavenumber in air and $Z_{0}=\rho_{0}c_{0}$
is the characteristic impedance of air, with $c_{0}$ the speed of sound and
$\rho_{0}$ the mass density of air. The terms $l_{n}$ and $r_{n}$ denote the
length and radius of the various duct elements, with
$r_{\text{a}}=r_{\text{b}}=r_{\text{c}}$. For load case (A), the transfer
matrix of the tube between microphone 2 and the sample is denoted

$\mathbf{T}^{\text{(A)}}(l_{\text{a}})=\mathbf{T}_{\text{a}}.$ (4)

For load case (B), the joint transfer matrix of the tube section, the
expansion chamber and the cavity is

$\mathbf{T}^{\text{(B)}}(\mathbf{x}_{\text{load}})=\mathbf{T}_{\text{b}}\mathbf{T}_{\text{e}}\mathbf{T}_{\text{c}},$
(5)

where

$\mathbf{x}_{\text{load}}=\left\\{l_{\text{b}},l_{\text{e}},r_{\text{e}},\eta_{\text{b}},\eta_{\text{e}},l_{\text{c}}\right\\}$
(6)

are the parameters governing the behaviour of the expansion chamber and
coupling tube. The complete set of variables of interest for load cases (A)
and (B) can be denoted

$\displaystyle\mathbf{x}^{(A)}$ $\displaystyle=\mathbf{x}_{\text{PEM}},$ (7)
$\displaystyle\mathbf{x}^{(B)}$
$\displaystyle=\left\\{\mathbf{x}_{\text{PEM}},\mathbf{x}_{\text{load}}\right\\},$
(8)

which consist of 9 and 15 unknown parameters, respectively. The impedance of
the coupled system is given by

$Z_{n}(\mathbf{x}^{(n)},\omega)=\dfrac{Z_{\text{PEM}}T^{(n)}_{11}+T^{(n)}_{12}S}{Z_{\text{PEM}}T^{(n)}_{21}/S+T^{(n)}_{22}},\quad
n=\text{A},\text{B},$ (9)

where $S=\pi r_{\text{a}}^{2}$ is the cross-section of the impedance tube and
$T^{(n)}_{ij}$ are the components of the transfer matrix of load (A) or (B).
The reflection coefficient of the compound system is then

$R_{n}(\mathbf{x}^{(n)},\omega)=\dfrac{Z_{n}(\mathbf{x}^{(n)},\omega)-Z_{0}}{Z_{n}(\mathbf{x}^{(n)},\omega)+Z_{0}},$
(10)

yielding the absorption coefficient as

$\alpha_{n}(\mathbf{x}^{(n)},\omega)=1-\left|R_{n}(\mathbf{x}^{(n)},\omega)\right|^{2}.$
(11)

###  2.2 Experimental setup

This section describes the experimental setup and provides the nominal values
of the dimensions as depicted in Fig. 1, on which the model depends. Fig. 2
shows the impedance tube in load case (B) and the foam sample of interest.

(a)

(b)

Figure 2: Experimental setup and material sample. (a) Impedance tube with
expansion chamber and poroelastic material termination; (b) melamine foam
sample.

The experimental setup relies on a Spectronics impedance tube with two
G.R.A.S. 46BD 1/4” microphones with nominal sensitivity $1.45\,$mV/Pa, spaced
by $29.21\,$mm, and the acquisition and processing are carried out via a
Simcenter SCADAS Mobile 05 in Simcenter Testlab 17 “Sound absorption testing
using impedance tube”, following the procedure described in the international
standard [37]. The calibration transfer functions and all subsequent
acquisitions are performed under broadband random noise excitation using 100
non-overlapping Hann window averages. The frequency resolution of the
measurements is $0.78125\,$Hz.

In order to achieve the loading condition illustrated in Fig. 1(a), the sample
is placed in a secondary tube connected to the primary tube section. The
loading condition illustrated in Fig. 1(b) is achieved by inserting an
expansion chamber between the primary and secondary tube sections. The
expansion chamber is made of aluminium, with acrylic lateral walls.

The sample of melamine foam was cut by hand to a thickness of $h=24\pm 1\,$mm
and attached to a rigid piston with double-sided tape. The piston comprises a
seal around its circumference and is allowed to slide along the tube. The tube
radius is provided by the manufacturer as $r_{\text{a}}=17.43\,$mm and here
assumed to be exact. The distance from microphone 2 to the modelled system was
measured as $l_{\text{a}}=80\pm 5\,$mm in load case (A) and nominally as
$\bar{l}_{\text{b}}=100\pm 5\,$mm in load case (B). The inner dimensions of
the expansion chamber were measured as $r_{\text{e}}=63\pm 2\,$mm and
$l_{\text{e}}=227\pm 2\,$mm.

The nominal length of the coupling tube between the expansion chamber and the
foam is $\bar{l}_{\text{c}}=40\pm 1\,$mm. Because of the sudden area changes
at the connection of this tube with the expansion chamber, a length correction
[61, 36, 41] is required in order to account for inertial effects. The
operational length of the coupling tube is thus estimated as
$l_{\text{c}}=51.7\pm 3.6\,$mm, where the lower and upper limits are estimated
in a conservative manner, corresponding to a tube of length
$\bar{l}_{\text{c}}$ connected to an expansion of radius $r_{\text{e}}$ and to
a baffled tube of length $\bar{l}_{\text{c}}+h$, respectively. A similar
reasoning for the length between microphone 2 and the expansion chamber yields
$l_{\text{b}}=111.7\pm 7.6\,$mm.

Further required parameters are those related to the properties of air, whose
values are here considered exact, namely the speed of sound
$c_{0}=343\,$m$\cdot$s-1, the mass density $\rho_{0}=1.204\,$kg$\cdot$m-3, the
atmospheric pressure $p_{0}=101320\,$Pa, the dynamic viscosity
$\mu_{0}=1.84\cdot 10^{-5}\,$kg$\cdot$m${}^{-1}\cdot$s-1, the ratio of
specific heats $\gamma=1.4$, and Prandtl’s number Pr$=0.71$.

##  3 Model inversion method

This section proposes an observation model and presents the different
inversion methods here developed. In the present application, individual
observations consist of a measurement of the frequency-dependent sound
absorption coefficient in one of the two loading conditions (A) or (B),
denoted $\alpha^{\text{meas}}_{n}$, with $n=1$ (load case (A)) or $n=2$ (load
case (B)). In this section, a generic notation $g^{\text{meas}}_{n}$ with
$n=1,\dots,N$ is preferred for the sake of generality.

###  3.1 Observation model

The experimental observation for a given load case $n$ at circular frequency
$\omega_{m}$ is denoted $g^{\text{meas}}_{n}(\omega_{m})$ and assumed real-
valued. It is further assumed that observations for different load cases are
independent, that is, the outcome of any of the observations does not affect
the others. It is then possible to construct a compound measurement
$\mathbf{g}^{\text{meas}}$ by concatenating individual observations, as

$\mathbf{g}^{\text{meas}}=\begin{bmatrix}\begin{bmatrix}g^{\text{meas}}_{1}(\omega_{1})&g^{\text{meas}}_{1}(\omega_{2})&\cdots&g^{\text{meas}}_{1}(\omega_{M})\end{bmatrix}^{\text{T}}\\\
\begin{bmatrix}g^{\text{meas}}_{2}(\omega_{1})&g^{\text{meas}}_{2}(\omega_{2})&\cdots&g^{\text{meas}}_{2}(\omega_{M})\end{bmatrix}^{\text{T}}\\\
\vdots\\\
\begin{bmatrix}g^{\text{meas}}_{N}(\omega_{1})&g^{\text{meas}}_{N}(\omega_{2})&\cdots&g^{\text{meas}}_{N}(\omega_{M})\end{bmatrix}^{\text{T}}\end{bmatrix},$
(12)

where $M$ is the number of frequency lines and $N$ is the number of
observations, yielding a dataset of size $(MN\times 1)$. It is assumed that
the experimental observation differs from the model prediction by an error,
such that

$\mathbf{g}^{\text{meas}}=\mathbf{g}(\mathbf{x}_{0})+\varepsilon,$ (13)

where $\mathbf{g}(\mathbf{x}_{0})$ is the mathematical model, evaluated at the
true set of parameters $\mathbf{x}_{0}$, and $\varepsilon$ is a random
variable representing the modelling and measurement uncertainties. Here, we
model $\varepsilon$ as Gaussian, with zero mean and a diagonal covariance
matrix $\mathbf{\Gamma}_{\varepsilon}=\sigma_{\varepsilon}^{2}\mathbf{I}$,
where $\sigma_{\varepsilon}^{2}$ is the error variance and $\mathbf{I}$
denotes the identity matrix of size $(MN\times MN)$.

Owing to the existence of the error $\varepsilon$, the true set of model
parameters $\mathbf{x}_{0}$ is inaccessible with complete certainty. The
following paragraphs present different model inversion paradigms aiming at
obtaining an estimate $\widetilde{\mathbf{x}}_{0}$ of the model parameters
$\mathbf{x}_{0}$.

###  3.2 Deterministic inversion framework

A deterministic estimate of the model parameters can be obtained by minimising
the difference between the experimental observation and the model prediction,
as

$\widetilde{\mathbf{x}}_{0}^{\text{(det)}}=\arg\min_{\mathbf{x}}f_{\text{obj}}(\mathbf{x}),$
(14)

where

$f_{\text{obj}}(\mathbf{x})=\dfrac{1}{MN}\sum_{n=1}^{N}\sum_{m=1}^{M}\left(g^{\text{meas}}_{n}(\omega_{m})-g_{n}(\mathbf{x},\omega_{m})\right)^{2}$
(15)

is the objective function, here defined as the residual sum of squares. The
normalisation factor ${1}/{MN}$ is added in order to enable comparing
residuals $f_{\text{obj}}(\widetilde{\mathbf{x}}_{0}^{\text{(det)}})$ for
observations comprising different numbers of frequency lines $M$ or load cases
$N$.

The minimisation of the objective function may be performed with any general
purpose optimisation tool. Two solvers are used in the present work, namely a
sequential quadratic programming algorithm (SQP) [56] and the globally
convergent method of moving asymptotes (GCMMA) [66], as detailed later-on in
Sec. 4. As a convergence and stopping criterion, tolerance factors
$\delta_{f_{\text{obj}}}$ and $\delta_{\mathbf{x}}$ are set for the relative
variation of the objective function and variables over a number
$n_{\text{it}}$ of consecutive iterations.

###  3.3 Incremental inversion

The above deterministic framework is here used within the incremental method
developed by the authors in [28, 14, 13]. The model inversion procedure
therein relies on a gradual complexification of the problem, starting from an
asymptotic observation and terminating at the full desired problem. At each
stage of the approach, the corresponding sub-problem is solved and the
solution is used as the starting point for the next sub-problem. For
frequency-dependent observations, the approach starts with a narrow low-
frequency observation whose upper frequency limit is incrementally raised.

A common measure for the complexity of a real function is the number of
alternating strictly monotonic segments [7]. In the present work, in order to
ensure that the series of sub-problems yield a gradual increase of the
complexity of the inverse problem, the frequency range is split into segments
where the measured data is monotonic. The problem is solved from the lowest
measured frequency to the upper limit of each monotonic segment sequentially,
until reaching the full frequency range. In the case of a multi-observation
dataset, the upper limits of all monotonic segments across the different
observations are used.

In the present work, the different steps in the incremental inversion are
performed using the deterministic inversion framework detailed in Sec. 3.2. In
addition, in order to capture the low-frequency asymptotic behaviour of the
system, an initial frequency range is added as half the frequency span of the
first monotonic section. Also, in order to ensure convergence of the estimated
set of parameters, a final sub-problem is appended, where the tolerance
factors are refined by a factor 10 and considered over a larger number of
iterations.

###  3.4 Statistical inversion framework

In order to obtain a measure of the uncertainty on the model parameters of
interest, the unknown quantities are modelled as random variables and the
problem is formulated using a Bayesian framework [45, 9]. The latter allows to
evaluate the set of unknowns in a probabilistic sense, that is, by estimating
their conditional probabilities for a given observation
$\mathbf{g}^{\text{meas}}$. In this work, the unknown quantities of interest
are the model parameters $\mathbf{x}$. In addition, the standard deviation of
the error, $\sigma_{\varepsilon}$, is considered unknown, and modelled as a
random variable as well. The conditional probability density of the set of
unknowns for a given observation is given by Bayes’ formula [45],

$P(\mathbf{x},\sigma_{\varepsilon}|\mathbf{g}^{\text{meas}})\sim
P(\mathbf{g}^{\text{meas}}|\mathbf{x},\sigma_{\varepsilon})P(\mathbf{x},\sigma_{\varepsilon}),$
(16)

where $P(\mathbf{x},\sigma_{\varepsilon}|\mathbf{g}^{\text{meas}})$ is also
referred to as the posterior density and
$P(\mathbf{g}^{\text{meas}}|\mathbf{x},\sigma_{\varepsilon})$ is the
likelihood of the observation given the unknowns. The prior
$P(\mathbf{x},\sigma_{\varepsilon})$ is here assumed uniform across a
sufficiently wide range within the admissible domain of the model parameters
and of the standard deviation of the error. Owing to the assumption of a
Gaussian error $\varepsilon$ in the observation model (13), the likelihood can
be expressed as

$\displaystyle P(\mathbf{g}^{\text{meas}}|\mathbf{x},\sigma_{\varepsilon})$
$\displaystyle=$ $\displaystyle
P_{\varepsilon}(\mathbf{g}(\mathbf{x})-\mathbf{g}^{\text{meas}})$ (17)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\left(2\pi\right)^{MN}\det\left(\mathbf{\Gamma}_{\varepsilon}\right)}}\exp\left(-\frac{1}{2}\left(\mathbf{g}^{\text{meas}}-\mathbf{g}(\mathbf{x})\right)^{\text{T}}\mathbf{\Gamma}_{\varepsilon}^{-1}\left(\mathbf{g}^{\text{meas}}-\mathbf{g}(\mathbf{x})\right)\right),$

where $P_{\varepsilon}$ is the probability density of the error. Furthermore,
the assumption that the covariance matrix of the error is diagonal yields

$P(\mathbf{g}^{\text{meas}}|\mathbf{x},\sigma_{\varepsilon})=\frac{1}{\left(\sqrt{2\pi}\sigma_{\varepsilon}\right)^{MN}}\exp\left(-\frac{1}{2\sigma_{\varepsilon}^{2}}\sum_{n=1}^{N}\sum_{m=1}^{M}\left(g^{\text{meas}}_{n}(\omega_{m})-g_{n}(\mathbf{x},\omega_{m})\right)^{2}\right).$
(18)

In the present work, the posterior density is sampled by means of a Markov
Chain Monte Carlo method, using the Metropolis-Hastings algorithm with an
adaptive proposal scheme [31, 32]. The purpose of such a procedure is to visit
the admissible domain of the unknowns with a probability
$P(\mathbf{x},\sigma_{\varepsilon}|\mathbf{g}^{\text{meas}})$, where the jumps
between successive points of the chain are determined by an adaptive proposal
distribution. In practice, $\sigma_{\varepsilon}$ is estimated together with
the model parameters $\mathbf{x}$, such that the complete set of unknowns can
be denoted

$\mathbf{y}=\left\\{\mathbf{x},\sigma_{\varepsilon}\right\\}.$ (19)

The proposal distribution is here defined as Gaussian, where its covariance
matrix is initialised as diagonal and then adapted as a function of the
covariance matrix of past samples [32], as

$\mathbf{C}^{(j)}=\gamma_{j}\left(\text{cov}\left(\mathbf{y}^{(0)},\dots,\mathbf{y}^{(j-1)}\right)+\delta\mathbf{I}\right),\quad
j\geq J,$ (20)

where $J$ is an arbitrary jump number at which the adaptation is started,
$\delta$ is an arbitrarily small number and $\mathbf{I}$ denotes the identity
matrix of size $(D\times D)$, $D$ being the dimension of the problem, i.e.
$D=10$ for the single-observation case (load (A)) and $D=16$ for the dual-
observation case (loads (A) and (B)). The scaling parameter $\gamma_{j}$ is
initialised at [24]

$\gamma_{J}=2.38^{2}/D$ (21)

and then updated following guidelines by Andrieu and Thoms [3] in order to
reach the desired acceptance rate $a_{*}$. Here we choose

$\gamma_{j+1}=\gamma_{j}\mu_{j+1}^{a_{j}-a_{*}},\quad j\geq J,$ (22)

where $a_{j}$ is the acceptance rate at jump $j$ and $\mu_{j}$ is an arbitrary
strictly decreasing sequence converging to 1.

The solution provided by the deterministic inversion procedure,
$\widetilde{\mathbf{x}}_{0}^{\text{(det)}}$, is a feasible set of model
parameters and therefore used as a starting point for the chain. This has the
advantage of reducing the required number of iterations in the burn-in phase.
Accordingly, an initial estimate of the standard deviation
$\sigma_{\varepsilon}$ of the error is obtained from Eq. (13) as

$\widetilde{\sigma}_{\varepsilon}=\sqrt{\frac{1}{MN}\sum_{n=1}^{N}\sum_{m=1}^{M}\left(g^{\text{meas}}_{n}(\omega_{m})-g_{n}\left(\widetilde{\mathbf{x}}_{0}^{\text{(det)}},\omega_{m}\right)\right)^{2}}.$
(23)

Alternatively, such an initial estimate can be obtained from the spread of a
series of repeated measurements, as suggested in Ref. [55]. However, as this
does not quantify modelling uncertainties it is not suitable for the present
study, and in particular for load case (B), where the one-dimensional
approximation of the expansion chamber is known to have limitations [53, 1].

In the present work, the aim is to obtain point estimates and uncertainty
ranges for all unknowns. We use the maximum a posteriori estimate (MAP), which
maximises the posterior probability density, the conditional mean (CM) of the
unknowns, and the narrowest 95% credible intervals [45].

##  4 Results

This section presents the results of the parameter estimation for the melamine
foam sample of interest, introduced in Sec. 2.2. The target experimental data
consists of a single-observation dataset and a dual-observation dataset. The
single-observation dataset consists of the sound absorption coefficient of the
foam in load case (A), and the dual-observation dataset is composed of both
the sound absorption coefficients corresponding to load cases (A) and (B),
concatenated as specified in Eq. (12).

The following four paragraphs respectively present the incremental inversion
results, a robustness study, the statistical inversion results and a
comparison of multi-observation vs. single-observation estimations.

###  4.1 Incremental model inversion

The full frequency range of interest for load case (A) is here defined as
$f\in[200,4500]\,$Hz. The low and high frequency limits are respectively
chosen so as to avoid uncertainties in the inter-microphone phase and to limit
the influence of the non-planar modes of the tube, whose first transversal
resonance corresponds to the first azimuthal mode, i.e. (1,0) [53], at circa
$5.7\,$kHz for the present setup. For load case (B), the main tube column
imposes a non-zero transversal velocity along the central line of the
expansion chamber and therefore azimuthal modes are not excited. The second
transversal resonance of the expansion chamber occurs at the first radial
mode, i.e. (0,1), at circa $2.65\,$kHz here, and therefore the upper frequency
limit is chosen at $2\,$kHz.

Table 1 shows the frequency ranges obtained for the single observation and for
the dual observation using the procedure described in Sec. 3.3.

Table 1: Frequency intervals and tolerance criteria for the incremental
inversion. (a) single observation; (b) dual observation.

Step | $f_{\text{min}}$ |  $f_{\text{max}}$ (Hz) | $\delta_{f_{\text{obj}}}$ | $\delta_{\mathbf{x}}$ | $n_{\text{it}}$
---|---|---|---|---|---
1 | $200$ | $669.53$ | $10^{-2}$ | $10^{-3}$ | 2
2 | $200$ | $1139.06$ | $10^{-2}$ | $10^{-3}$ | 2
3 | $200$ | $1351.56$ | $10^{-2}$ | $10^{-3}$ | 2
4 | $200$ | $3964.06$ | $10^{-2}$ | $10^{-3}$ | 2
5 | $200$ | $4500$ | $10^{-2}$ | $10^{-3}$ | 2
6 | $200$ | $4500$ | $10^{-3}$ | $10^{-4}$ | 4
(a)

Step |  $f_{\text{min}}$ (Hz) |  $f_{\text{max}}$ (Hz) | $\delta_{f_{\text{obj}}}$ | $\delta_{\mathbf{x}}$ | $n_{\text{it}}$
---|---|---|---|---|---
1 | $200$ | $469.53$ | $10^{-2}$ | $10^{-3}$ | 2
2 | $200$ | $739.06$ | $10^{-2}$ | $10^{-3}$ | 2
3 | $200$ | $889.06$ | $10^{-2}$ | $10^{-3}$ | 2
4 | $200$ | $1114.06$ | $10^{-2}$ | $10^{-3}$ | 2
5 | $200$ | $1339.06$ | $10^{-2}$ | $10^{-3}$ | 2
6 | $200$ | $1526.56$ | $10^{-2}$ | $10^{-3}$ | 2
7 | $200$ | $1901.56$ | $10^{-2}$ | $10^{-3}$ | 2
8 | $200$ | $3964.06$ | $10^{-2}$ | $10^{-3}$ | 2
9 | $200$ | $4500$ | $10^{-2}$ | $10^{-3}$ | 2
10 | $200$ | $4500$ | $10^{-3}$ | $10^{-4}$ | 4
(b)

Figures 3 and 4 show the measured and estimated sound absorption coefficient
for the single- and dual-observation cases, respectively, using a GCMMA
implementation of the proposed incremental inversion approach with a random
starting set of parameters. A good approximation of the measurement is
achieved with the proposed model in both cases, with minor discrepancies at
the low-frequency limit. These may be attributed to the model’s inability to
capture low-frequency dissipation effects in the impedance tube, or to the
known limitations of the porous material model here used [46]. The figures
depict as well the objective function as a function of the incremental
inversion steps, and show generally increasing values due to the increasing
sub-problem complexity.

(a)  (b)

Figure 3: Deterministic model inversion solution for a single-observation
measurement of a rigid-backed poroelastic material in loading condition (A).
(a) Sound absorption coefficient,  measurement,  model; (b) objective function
history. The vertical lines represent the upper limits of the successive model
inversion steps.

(a)  (b)

Figure 4: Deterministic model inversion solution for the dual-observation
measurement, including poroelastic material in loading condition (A) and
poroelastic material coupled to expansion chamber in loading condition (B).
(a) Sound absorption coefficient,  measurement,  model; (b) objective function
history. The vertical lines represent the upper limits of the successive model
inversion steps.

The figures clearly show the effect of the elasticity of the frame on the
sound absorption coefficient, observed for load case (A) as a dip at the
elastic resonance of the material at $1351.56\,$Hz. The elastic effects also
play an important role in load case (B), and arise as a peak at $1114.06\,$Hz.
This is interpreted as a coupling effect between the expansion chamber and the
material frame via the coupling tube. It is worth noting that a sufficient
degree of sensitivity of load case (B) to the properties of the foam is
crucial for the over-determination of the problem, as will be further explored
in Sec. 4.4.

###  4.2 Robustness study

A multi-start approach [26] is here used in order to evaluate the robustness
of the incremental search method, regardless of the initial guess of the
parameter values. The procedure consists in repeating the model inversion
process a number of times, here using a uniform random distribution for the
parameter values as the starting point. Figures 5 and 6 show the distribution
of the parameter values obtained for 1000 runs of a full-range search versus
the incremental search, for the single- and dual-observation cases
respectively, and presented in the form of histograms. In addition, the
distribution of objective function residuals and the solution time are also
represented as histograms. An implementation based on SQP was preferred for
this iterative procedure as the algorithm involves a lower computational
overhead than the authors’ implementation of GCMMA.

(a)

(b)  (c)

Figure 5: Normalised histograms of parameter values, residual values and
solution time for 1000 runs of the gradient-based single-observation case.
Light grey, Full-range search; blue, incremental search.

(a)

(b)  (c)

Figure 6: Normalised histograms of parameter values, residual values and
solution time for 1000 runs of the gradient-based multi-observation case.
Light grey, Full-range search; blue, incremental search.

It can be observed that the incremental parameter search performs more
consistently than the full-range search, especially in the dual-observation
configuration. The appearance of multiple solutions elucidates the existence
of local minima, which in turn results in a widening of the distribution of
the objective function residuals. Nevertheless, the proposed incremental
approach comes at a non-negligible computational cost. In the present case the
expected solution time is approximately fivefold with respect to the full-
range search.

###  4.3 Statistical model inversion

This paragraph presents the results of the Bayesian inversion approach,
consisting in sampling the posterior density, see Sec. 3.4. For the present
work, the target acceptance rate is set to $a_{*}=0.2$. The Markov Chain Monte
Carlo sampling algorithm is here initialised at the solution obtained from the
incremental inversion. Although such a starting point is a feasible solution,
a burn-in phase is considered in order to gather a sufficient number of
samples for a reliable initial estimate of the covariance matrix of the
unknowns. After the burn-in phase, the total number of samples kept for the
analysis is set to $2\cdot 10^{6}$. For the purposes of the problem of
interest, two point estimates are extracted, namely the maximum a posteriori
(MAP) and the conditional mean (CM). In addition, the 95% credible ranges of
each model parameter are extracted.

Figure 7 shows the pairwise 2D marginal densities for the 9 parameters of the
poroelastic sample. The figure also includes the limits of the 95% credible
ranges as well as the 1D marginal densities of the individual parameters. In
addition, a measure of the inter-parameter correlation is provided thanks to
the Pearson correlation coefficient, represented with the symbol $\mathrm{P}$.
It can be observed that the support of the densities is compact and presents a
unique maximum point. Indeed, thanks to the incremental inversion solution
being used as a feasible starting point, the mapping of other regions of the
parameter space is avoided, and accordingly no multi-modal behaviour is
observed.

Figure 7: Pairwise posterior marginal densities for the single-observation
case.  individual marginal densities;  credible ranges;  MAP estimate;  CM
estimate; $\mathrm{P}$, Pearson correlation coefficient. Bottom left:
Posterior marginal density of the error $\varepsilon$.

A number of notable correlations among the parameters can be observed. For
instance, the sample thickness exhibits a higher correlation with the
acoustical transport properties than with the elastic properties. Indeed, the
elasticity of the frame does not contribute to a global increase or decrease
of the sound absorption coefficient, but locally via the frame resonance.
Furthermore, the Young’s modulus $E$ and the mass density $\rho$ are
positively correlated, which is expected due to their joint control of the
elastic resonance. The flow resistivity $\sigma$, the tortuosity
$\alpha_{\infty}$ and the viscous characteristic length $\Lambda$ are strongly
correlated, as small variations in their values contribute to sound absorption
in a comparable manner. In fact, a similar trend in the correlation of
$\alpha_{\infty}$ and $\Lambda$ has been reported [12]. In addition to these
correlations, the thermal characteristic length $\Lambda^{\prime}$ and the
Young’s modulus $E$ exhibit a strong correlation. Indeed, for the melamine
foam here tested, the thermal and elastic effects are both dominant over a
common mid-frequency range.

These correlations show the existence of a local interdependence of the model
parameters. Several authors have reported on the scaling laws governing the
constitutive properties of poroelastic media over small variations of the
parameters [25, 27, 51, 10]. For instance these scaling laws predict a
positive correlation between the density and the mass density, between the
flow resistivity and the tortuosity, and between the tortuosity and the
viscous characteristic length, such as observed here. In addition to being
intrinsic to the constitutive model, these parameter correlations may also be
exacerbated by the choice of the sound absorption coefficient as the target
data, as reported by Niskanen et al. [55].

The correlations observed here suggest that the model parameters are not
independent of one another. As a matter of fact, a set of 3 independent
parameters has been shown by Horoshenkov et al. [34] to be sufficient to
correctly represent the acoustical behaviour of porous media, classically
modelled using the 5-parameter JCA model herein [44, 11], or the 6-parameter
Johnson-Champoux-Allard-Lafarge (JCAL) model [44, 11, 46].

Figure 8 shows the pairwise marginal densities for the dual-observation case.
The additional parameters exhibit compact marginal densities and unique maxima
as well, and introduce additional correlations. For instance, a high degree of
correlation can be observed between $l_{\text{b}}$ and $\eta_{\text{b}}$,
which respectively account for low-frequency and broadband dissipation in the
impedance tube. Similarly, $r_{\text{e}}$ and $\eta_{\text{e}}$ respectively
account for absorption peak depth and broadband dissipation in the expansion
chamber. Overall, a low correlation between the marginal densities of the
poroelastic material parameters and those of the expansion chamber is
observed. However, the coupling tube length $l_{\text{c}}$ exhibits a somewhat
higher correlation with $E$ and $\rho$. This bears witness to the fact that
the coupling tube strongly interacts with the elastic deformation of the foam.

Figure 8: Pairwise posterior marginal densities for the dual-observation case.
individual marginal densities;  credible ranges;  MAP estimate;  CM estimate;
$\mathrm{P}$, Pearson correlation coefficient;  delimitation of pairwise
densities pertaining to the sample only, the load only, and the coupling of
the two. Bottom left: Posterior marginal density of the error $\varepsilon$.

###  4.4 Multi-observation vs. single-observation estimations

Table 2 displays the parameter values obtained for the deterministic
incremental inversion as well as the MAP and CM estimates and the 95% credible
intervals, for both the single- and dual-observation cases. In addition, Fig.
9 shows the marginal densities for all individual parameters, superimposed for
the single- and dual-observation cases, as well as the credible intervals and
MAP estimate values.

Table 2: Parameter values obtained from the deterministic incremental
estimation and from the statistical inversion, including MAP estimate, CM
estimate and 95% credible intervals.

Parameter | Unit | Determ. | MAP | CM | Cred. low | Cred. high
---|---|---|---|---|---|---
$h$ | mm | $23.14$ | $23.14$ | $23.15$ | $23.01$ | $23.29$
$\rho$ | kg$\cdot$m-3 | $11.865$ | $11.958$ | $11.994$ | $11.562$ | $12.449$
$\phi$ | - | $1.0000$ | $0.9996$ | $0.9966$ | $0.9906$ | $0.9998$
$\sigma$ | kN$\cdot$s$\cdot$m-4 | $18.594$ | $18.709$ | $18.748$ | $18.516$ | $19.044$
$\alpha_{\infty}$ | - | $0001$ | $1.001$ | $1.008$ | $1$ | $1.023$
$\Lambda$ |  $\mu$m | $69.53$ | $70.93$ | $71.61$ | $69.09$ | $74.82$
$\Lambda^{\prime}$ |  $\mu$m | $158.66$ | $159.07$ | $159.46$ | $155.92$ | $163.24$
$E$ | kPa | $194.199$ | $196.697$ | $196.715$ | $190.276$ | $203.756$
$\eta$ | % | $4.618$ | $4.696$ | $4.671$ | $4.364$ | $4.982$
$\sigma_{\varepsilon}$ | $10^{-3}$ | $6.364$ | $6.447$ | $6.492$ | $6.099$ | $6.916$
(a)

Parameter | Unit | Determ. | MAP | CM | Cred. low | Cred. high
---|---|---|---|---|---|---
$h$ | mm | $23.17$ | $23.18$ | $23.18$ | $23.08$ | $23.29$
$\rho$ | kg$\cdot$m-3 | $11.766$ | $11.821$ | $11.846$ | $11.512$ | $12.190$
$\phi$ | - | $1.0000$ | $0.9998$ | $0.9980$ | $0.9942$ | $0.9999$
$\sigma$ | kN$\cdot$s$\cdot$m-4 | $18.594$ | $18.681$ | $18.694$ | $18.529$ | $18.899$
$\alpha_{\infty}$ | - | $0001$ | $1.001$ | $1.005$ | $1$ | $1.015$
$\Lambda$ |  $\mu$m | $69.91$ | $70.65$ | $71.22$ | $69.38$ | $73.47$
$\Lambda^{\prime}$ |  $\mu$m | $159.07$ | $159.75$ | $159.62$ | $156.75$ | $162.62$
$E$ | kPa | $193.228$ | $194.352$ | $194.792$ | $189.792$ | $200.025$
$\eta$ | % | $4.662$ | $4.716$ | $4.702$ | $4.456$ | $4.950$
$l_{\text{b}}$ | mm | $118.07$ | $118.13$ | $118.13$ | $115.33$ | $121.00$
$l_{\text{e}}$ | mm | $225.41$ | $225.42$ | $225.41$ | $225.38$ | $225.45$
$r_{\text{e}}$ | mm | $84.62$ | $84.60$ | $84.63$ | $84.27$ | $84.99$
$\eta_{\text{b}}$ | % | $0.11$ | $0.11$ | $0.11$ | $0.10$ | $0.12$
$\eta_{\text{e}}$ | % | $0.143$ | $0.143$ | $0.143$ | $0.136$ | $0.151$
$l_{\text{c}}$ | mm | $52.61$ | $52.68$ | $52.68$ | $52.38$ | $52.98$
$\sigma_{\varepsilon}$ | $10^{-3}$ | $6.091$ | $5.224$ | $5.235$ | $5.003$ | $5.477$
(b)

Figure 9: Individual marginal densities for all model parameters.  Single-
observation,  dual-observation. The horizontal error bars represent the
credible ranges and the crosses denote the MAP estimate.  Manual length
measurement.

The obtained point estimates provide reasonable values for the different
parameters of the poroelastic material model. Moreover, these are consistent
with those reported elsewhere for melamine foams [55, 19, 15, 18, 60, 23, 43,
5].

The most notable result of the present work is that the credible intervals
obtained by the proposed method are narrower in the dual-observation case than
in the single-observation case. This demonstrates that performing the
measurement in two independent load cases effectively over-determines the
inverse problem, in turn yielding an estimation with narrower parameter
uncertainty ranges.

As a complement to the estimated model parameters, the ratio of characteristic
lengths can be obtained from their sampled values. For the single observation,
the MAP estimate is $\Lambda^{\prime}/\Lambda=2.241$ and the 95% credible
interval is $\Lambda^{\prime}/\Lambda\in[2.122,2.326]$. The dual observation
yields a MAP estimate of $\Lambda^{\prime}/\Lambda=2.246$, and a narrower 95%
credible interval, $\Lambda^{\prime}/\Lambda\in[2.162,2.316]$. It is worth
noting that the results with both observation approaches respect the well-
established condition $\Lambda^{\prime}/\Lambda\geq 2$ [2].

In addition to the poroelastic material properties, the dimensions of the
expansion chamber are obtained as a by-product of the method and can serve as
part of the validation. In the present work we have chosen to introduce these
geometrical properties as additional unknowns, but alternatively they could be
assumed at their nominal values, albeit at the risk of introducing a bias in
the values of the properties of the sample. Indeed, the present analysis shows
that the geometrical parameters of the expansion chamber and coupling tube
differ from their nominal values. For instance, the radius $r_{\text{e}}$ of
the expansion chamber presents an apparent overestimation. In the opinion of
the authors this arises as a compensation of one or both of the limitations of
the model in this respect. The first possible cause is the well-known
intrinsic inability of the classical transfer matrix method to correctly
represent three-dimensional effects at sudden expansions [53, 1], and the
second is the unaccounted-for finite stiffness of the expansion chamber. It is
worth noting that the uncertainty ranges in the manual measurements are
directly derived using the precision of the length readings. This is naturally
different from the statistical definition of 95% credible intervals, which
prevents a rigorous comparison. Nevertheless, the measurements of lengths $h$,
$l_{\text{b}}$, $l_{\text{e}}$ and $l_{\text{c}}$ overlap with the model
inversion results. In particular, the length correction procedure correctly
predicts the length of the coupling tube $l_{\text{c}}$ and that of the
primary tube section $l_{\text{b}}$.

##  5 Summary of the findings

Based on the results above, the primary findings of this work can be
summarised as follows. First, the proposed transfer matrix model of the system
provides an accurate representation of the behaviour of the material sample
under both loading conditions (A) and (B), thus enabling a good fit onto the
experimental acquisitions while exhibiting sufficient sensitivity to all model
parameters.

The proposed incremental inversion method has been shown to guarantee that the
global solution of the problem is obtained independently from the initial
guess, albeit at a significant computational cost as compared to a full-range
search. This has been illustrated for both a single observation in load case
(A) and a dual observation combining load cases (A) and (B).

Furthermore, the proposed framework for statistical model inversion provides
marginal posterior densities for all unknowns of the problem. Accordingly,
this enables a reliable extraction of point estimates and credible intervals.
A notable demonstration of the robustness of the method is that the credible
intervals for the single- and dual-observation cases overlap. More
specifically, the point estimates for the single-observation case are
systematically contained in the credible intervals of the dual-observation
case, and vice versa.

The central result of this manuscript is the observation that the credible
intervals are narrower in the case of the dual observation. This shows that
the use of a multi-observation dataset provides an over-determination of the
inverse problem, thereby yielding refined uncertainty ranges on the unknowns.

A triple-observation model inversion was also performed, using an additional
load case with a coupling tube of nominal length $\bar{l}_{\text{c}}=30\,$mm.
However, the results were comparable to the dual-observation case and no
improvement was observed, presumably due to the fact that the additional load
case did not significantly over-determine the inverse problem. These results
are not reported here.

The application example shows that the method is capable of providing the full
set of properties required to model macroscopically-homogeneous poroelastic
media under the Biot-Johnson-Champoux-Allard model, with the exception of the
Poisson ratio, from two-microphone impedance tube measurements. In the opinion
of the authors, this achievement is of substantial value as previous methods
rely on the assumption that one or more of these parameters are initially
known, therefore removing the need for multiple test rigs.

Finally, in addition to point estimates and credible intervals, the
statistical inversion framework allows for correlations among the parameters
to be observed. Using the present results, these can be tied to a certain
extent to the one-degree-of-freedom scaling laws for poroelastic media [25,
27, 51, 10].

##  6 Conclusion

A multi-observation approach is here proposed, based on two consecutive
measurements of the sound absorption coefficient of a rigid-backed sample of a
poroelastic material, where in the second the sample is coupled to an
expansion chamber. The methodology is shown to provide all 9 parameters of a
comprehensive poroelastic material model, including the sample thickness and
density, 5 transport parameters and 2 elastic properties. This is achieved
through a combination of an incremental method designed to increase the
complexity of a deterministic inverse problem gradually, and a statistical
inversion framework providing the posterior probability density of the model
parameters, from which point estimates, credible intervals and correlations
are inferred. The results show reasonable credible intervals for all material
parameters in accordance with values reported for melamine in the literature.
Furthermore, the over-determination induced by the use of a multi-observation
approach is shown to reduce the uncertainty in the parameter estimation.
Additionally, the correlations observed among the porous material properties
support the claim that the classical parametrisation of the acoustical
behaviour of open-cell porous media is somewhat redundant and can be replaced
with a model with fewer parameters [34].

The method has been here illustrated for a melamine foam sample, for which the
dimensions of the expansion chamber and coupling tube were suitable for
enhancing fluid-structure interaction with the frame’s elasticity. Applying
the method to other materials may require different dimensions of the
expansion chamber loading in order to promote a suitable degree of coupling.
In future work, further over-determining the inverse problem could be achieved
for instance by varying the expansion chamber length, which controls the major
resonances of the coupled system.

In light of the results obtained by Niskanen et al. [55], the present method
could be further applied to measurements in a three-microphone impedance tube
with a rigid-backed sample, for instance using the processing proposed in
Refs. [40, 65]. This would enable the additional measurement of the equivalent
density and bulk modulus, while ensuring a proper control of the boundary
conditions.

It is worth noting that the proposed incremental inversion procedure is
applicable to a wider variety of problems so long as modelling complexity can
be treated gradually, for instance exploiting an asymptotic behaviour in high
frequencies, or along other conditional variables of interest such as
temperature or mechanical stress, for example.

## Acknowledgements

This article is based upon work initiated under the support from COST Action
DENORMS CA-15125, funded by COST (European Cooperation in Science and
Technology). The work has been supported by the Academy of Finland (Finnish
Centre of Excellence of Inverse Modelling and Imaging and project number
321761) and by the Centre for ECO2 Vehicle Design at KTH, Vinnova Grant No.
2016-05195.

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