Characterizing a porous road pavement using surface impedance measurement: A guided numerical inversion procedure €lle Benoita) Gae Laboratoire R egional des Ponts et Chauss ees de Blois, Centre d’Etudes Techniques de l’Equipement Normandie-Centre, 11 rue Laplace CS2912 41029 Blois Cedex, France
le Christophe Heinke Laboratoire R egional des Ponts et Chauss ees de Strasbourg, Centre d’Etudes Techniques de l’Equipement Est, 11, rue Jean Mentelin 67035 Strasbourg Cedex 2, France
Emmanuel Gourdon Laboratoire G enie Civil et B^ atiment LTDS Unit e Mixte de Recherche CNRS 5513, Ecole Nationale des Travaux Publics de l’Etat, rue Maurice Audin 69518 Vaulx-en-Velin Cedex, France
(Received 22 October 2012; revised 24 July 2013; accepted 5 August 2013) This paper deals with a numerical procedure to identify the acoustical parameters of road pavement from surface impedance measurements. This procedure comprises three steps. First, a suitable equivalent fluid model for the acoustical properties porous media is chosen, the variation ranges for the model parameters are set, and a sensitivity analysis for this model is performed. Second, this model is used in the parameter inversion process, which is performed with simulated annealing in a selected frequency range. Third, the sensitivity analysis and inversion process are repeated to estimate each parameter in turn. This approach is tested on data obtained for porous bituminous concrete and using the Zwikker and Kosten equivalent fluid model. This work provides a good foundation for the development of non-destructive in situ methods for the acoustical characterization C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4824971] of road pavements. V PACS number(s): 43.55.Ev, 43.58.Bh, 43.40.Le [KVH]
I. INTRODUCTION
Forty percent of European inhabitants are exposed to traffic-related noise; the cost of this exposure amounts to over 1 106 healthy years of life lost annually in this part of the world, according to World Health Organization.1 This is the reason that several strategies have been devised to reduce traffic noise, and these include sound-absorbing porous road pavements. Road surfaces are a specific type of porous material: Porosity is low compared with typical acoustical materials (maximum porosity of 30%), and few measurements or evaluation of other intrinsic parameters are available from the literature.2–4 Moreover, as the characteristics of porous road surfaces vary over time, it is of significant interest to be able to measure them in situ without coring the road. Berengier et al.2 showed the effect of clogging on the absorption coefficient of porous asphalt. It would also be interesting to be able to assess the effect of clogging also on parameters such as open porosity and static air-flow resistivity, especially on road asphalt as it clogs, in an intermediate state between open and closed pores. This paper therefore aims to offer a guided inversion method for assessing the intrinsic parameters of road pavement as a porous material from surface impedance measurements given that these measurements can be carried out in situ. a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
4782
J. Acoust. Soc. Am. 134 (6), Pt. 2, December 2013
Pages: 4782–4791
Used with porous materials, equivalent fluid models allow a relationship to be established between measurable acoustical quantities, such as surface impedance, and microstructure-related parameters, such as open porosity or static air-flow resistivity. Furthermore, these models also involve a limited number of parameters that makes them suitable for an inversion procedure that leads from measurement of acoustical quantities to assessment of the intrinsic parameters of road pavement. The issue consists in using a non-destructive method of measurement in situ, such as direct measurement of impedance using a PU-probe, or the transfer function method. This supposes that only one complex quantity is available, i.e., surface impedance. This sets up a different problem from the usual problems of characterization: Several characterization procedures using transmitted and reflected waves have been proposed.5–9 Another consequence of in situ problems is that an experimentally robust method is required. Indeed the measurement error may be important when carrying out in situ impedance measurements (see for example the work of Kruse et al.10 for an estimate of the measurement error with the transfer function method). This is especially true for measurements on materials with low absorption. This article describes a numerical inversion procedure for characterizing porous road pavements based on quantities available as a result of in situ measurements. Transmitted waves cannot be measured, and surface impedance is the only data accessible. To make the most of available data, therefore, sensitivity analysis is applied to the identification model, using an index based on a Bayesian approach. The
0001-4966/2013/134(6)/4782/10/$30.00
C 2013 Acoustical Society of America V
main idea is to analyze the model’s behavior with regard to its parameters before the former is inverted to determine the latter. This approach adds a criterion that provides both qualitative and quantitative information and that plays a decisive role in the numerical inversion in distinguishing the key parameters of the frequency range. In this way, numerical inversion of the model may be performed step by step to determine each parameter separately. In the first instance, measurements were carried out on samples with the impedance tube. This ensured that the method of measurement could be monitored and that access to reliable measurement data was available and also meant that the procedure could be compared with validated methods.6,7 The procedure was designed to take account of the constraints imposed by the use of in situ measurements with regard to the data available. The goal here is to design a numerical procedure for identification of parameters suitable for application in situ. Therefore the number of parameters required by the model has been reduced as much as possible. This initial investigation is thus carried out using an equivalent material model, the Zwikker and Kosten model11 with three parameters, i.e., flow resistivity r, open porosity U, and high frequency limit of tortuosity a1 . This is a similar approach to that described by H€ubelt et al.12 In their work, H€ubelt and colleagues use a numerical algorithm to assess porosity and thickness from surface impedance measured in an impedance tube. However, as they assume that flow resistivity and tortuosity may be expressed through porosity, they do not attempt to estimate them directly. Furthermore the innovation of the proposed approach relies on the individual evaluation of each parameter on targeted frequency ranges. Section II presents the sensitivity analysis method used and its results with the Zwikker and Kosten model; Sec. III describes the numerical inversion procedure and Sec. IV its application to a sample of porous bituminous concrete. II. SENSITIVITY ANALYSIS A. Method
Several methods may be applied to carry out a sensitivity analysis, including local methods such as one-at-a-time technique, and global methods, for example, the importance measure method. Here, the sensitivity analysis relies on the Sobol sensitivity index.13 The same index has been used recently by Ouisse et al.14 in their work relating to the sensitivity analysis of porous material models. It is especially interesting as it gives qualitative and quantitative information about the sensitivity of the model’s output to input parameters. Note that all parameters vary simultaneously with this approach. Let Y be the output of a model involving n input Xn . The sensitivity index of Y to parameter Xi is expressed as I Xi ¼
VðE½YjXi Þ ; VðYÞ
(1)
where V is the variance and E is the mathematical expectation. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
Through the expectation of the Bayesian quantity YjXi , this expression quantifies the degree of variance of the output resulting from the variance of the investigated input. Clearly, this index falls within the range [0,1] and the bigger the index IXi , the greater the influence of parameter Xi on output Y. This is the first-order sensitivity index. A second-order sensitivity index takes into account the sensitivity derived from the interactions between parameter Xi and another input Xj ; a third-order index may be defined taking into account interactions between three parameters, and so on. Last, a global sensitivity index, being the sum of these different order indexes, takes into account the total sensitivity of Y to parameter Xi . For example, for a three-parameter model, the global sensitivity index of parameter X1 is expressed as IX1 ; T ¼ IX1 þ IX1 ; X2 þ IX1 ; X3 þ IX1 ; X2 ; X3 ¼ 1
VðE½YjX6¼1 Þ : VðYÞ (2)
These elements require some comment (1) It should be noted that first-order indexes give only a partial indication of the influence of each parameter, and they may not enable conclusions to be drawn with regard to the influence of a particular parameter. This is especially true when the sum of the indexes for each parameter is less than 1, as they do not reflect the coupling effects between parameters. However, they already provide useful quantitative information at a much lower computation cost. This is why they have been investigated here. (2) Ouisse et al.14 showed the importance for parameter sampling of the choice of probability density function. In this case, as few measured values are available for intrinsic road surface parameters, a uniform probability density function has been selected. Ouisse et al.14 indicated in particular that for porosity values close to 1, firstorder and total indexes are similar when using uniform probability density function. This feature is the reason that only first-order indexes have been estimated. (3) Ouisse et al.14 also showed that the choice of relevant parameter variation ranges is crucial. In this exercise, the limits of the ranges were chosen using values found in the literature2–4 (obtained from identification or measurements). They were also selected to enable an understanding of porous road surfaces throughout their life cycle until such time as they may be considered non-porous: This is true for resistivity variation range in particular. We are confident in the limits of the variation range for porosity as this parameter can be directly linked to the void ratio, a parameter that is commonly measured for road surfaces during whole life cycle. From a numerical point of view, all these indexes were estimated using Monte Carlo method. Convergence was ensured by checking with a larger sample of parameters if the difference between the estimated indexes was less than 1%. Benoit et al.: Characterizing a porous road pavement
4783
B. Zwikker and Kosten model
With regard to the porous material model, it was decided to restrict the number of required parameters to three initially. The Hamet model3 for porous asphalts was considered, but it does not fit the high and low frequency limits with regard to frequency dependence. This is the reason that the Zwikker and Kosten model, using the same parameters, was selected. In addition, the Zwikker and Kosten model relies on parameters that form a subset of the parameters of models used for more traditional porous materials, like Johnson–Champoux–Allard15,16 and Johnson–Champoux— Allard–Lafarge,15–17 etc. It makes this model attractive should additional parameters be required to increase the accuracy of the model. Moreover, in their work, H€ubelt et al.12 mention that the lack of precision in the inversion might come from the Hamet model. It is therefore of interest to test another model. The Zwikker and Kosten model is one of the equivalentfluid material, which assumes that (1) Wave lengths are larger than pore sizes. Here the upper bound of the frequency range considered is 4000 Hz; lower wave lengths then have to be larger than 0.08 m, a value much higher than the pore sizes of road surfaces. (2) The material is considered as homogeneous. (3) The frame is assumed to be rigid and motionless. This assumption can be regarded as valid for porous road surfaces because of the high density of their frame.
Figure 1 presents surface impedance from the Zwikker and Kosten model for three sets of parameters found in the porous road surface literature;2,4 these sets are given in Table I. C. Results
This section presents the results of the sensitivity analysis of the Zwikker and Kosten model. It was carried out using the following parameter variation ranges: (1) Open porosity (referred to as porosity in the following) U: ½5%35%. (2) Static air-flow resistivity (referred to as resistivity in the following) r: ½1000 150 000 Nm4 s. (3) high frequency limit of tortuosity (referred to as tortuosity in the following) a1 : ½1 5. The thickness of the wearing course of a road surface varies from 20 to 60 mm. This is why the thickness was set to 40 mm in this example. The sensitivity indexes were estimated on sets of 5 105 realizations, ensuring good convergence, as indicated previously. Figure 2 presents the sensitivity indexes of normalized surface impedance to the three parameters of the model and the sum of these indexes (thicker line). It may be observed that all parameters are important, but on different frequency ranges. Flow resistivity is the most influential parameter on ~ 0 Þ at around 1000 Hz, then it quickly decreases and > ~ ðxÞ ¼ 0 FðkÞ; > q > < eq U x " pffiffiffiffiffiffiffi pffiffiffiffiffiffi #1 > cP Tð N k jÞ > > K~ eq ðxÞ ¼ 0 1 þ 2ðc 1Þ pffiffiffiffiffiffiPrffi pffiffiffiffiffiffi ; (3) > : U NPr k j where x ¼ 2pf is the pulsation, r is the flow resistivity, U is the open porosity, a1 is the high frequency limit of tortuosity, q0 is the air density, NPr is the Prandtl number, c is the adiabatic constant, P0 is the static pressure, j2 ¼ 1, F is the correction function for viscosity defined by Biot,18 T =Jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ J1p 0 is the ratio of Bessel functions of first kind and k ¼ 8a1 q0 x=rU. ~ charThese quantities relate to complex wave number k, ~ acteristic impedance Z c , and measured surface impedance Z~ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ~ > ~ eq ðxÞK~ eq ðxÞ; q ðxÞ ¼ Z > < c sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) ~ eq ðxÞ q > > ~ ; kðxÞ ¼ x > : K~ eq ðxÞ ~ ~ ZðxÞ ¼ jZ~c cotanðkeÞ;
(5)
where e is the thickness of the material sample, backed by a rigid impervious plane. 4784
J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
FIG. 1. (Color online) Surface impedance from the Zwikker and Kosten model for three sets of parameters (see Table I). Benoit et al.: Characterizing a porous road pavement
TABLE I. Surface impedance from the Zwikker and Kosten model for three sets of parameters. Parameters
U
r (Nm4 s)
a1
Solid line Dashed line Dotted line
0.10 0.15 0.24
50 000 15 000 4000
3 2 1
~ 0 Þ, porosity is the only influential parameter up to =ðZ=Z 800 Hz, then tortuosity gradually takes the lead. No precise conclusion can be drawn beyond 2000 Hz, as the sum of sensitivity indexes is much less than 1: Coupling effects should be investigated with total sensitivity indexes beyond that point. Some phenomena described in these results were already known thanks to Taylor series expansions [see, for example, the Ph.D. dissertation of Sellen19 for the sensitivity ~ 0 Þ to U at low frequencies], but the sensitivity analof =ðZ=Z ysis allows for the addition of quantitative indications with respect to frequencies at which these phenomena are valid, and parameter hierarchy. Regular oscillations may also be observed on the sensi~ 0 Þ to a1 . This could be explained by tivity index of =ðZ=Z ~ 0 Þ for tortuthe fact that the sensitivity maximum on =ðZ=Z osity matches the range where the quarter-wave frequency
FIG. 2. (Color online) Sensitivity indexes of surface impedance in the Zwikker and Kosten model for a sample of thickness 40 mm. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
fk=4 is located for this thickness in most of the configurations created by the parameter samples (see Fig. 3) fk=4 ¼
cmaterial cair ¼ pffiffiffiffiffiffi : 4e 4e a1
(6)
The results in relation to the dominance of parameters are slightly different from those obtained by Ouisse et al.,14 who concluded that the sensitivity analysis confirmed the predominant position of resistivity. This is not the case here; this could be explained by the range of variation that is different for porosity with values much lower than for typical acoustical porous materials, such as foams. Identifying the frequency ranges where one of the parameters is predominant will help guide the inversion procedure for this parameter. III. INVERSION STRATEGY A. Selected algorithm and associated parameters
Simulated annealing was selected for numerical inversions because of its effectiveness in finding the global minimum of an optimization problem. It is based on a random walk through parameter space in combination with the Metropolis criterion20 with the intention of finding the global minimum of challenging functions with many local minima. The initial model is drawn randomly from the predefined interval of each parameter I. Arbitrary temperature T is gradually reduced according to a cooling schedule; this allows for exploration of most of the space possibilities. Figure 4 shows a process map of the simulated annealing algorithm. Several researchers present techniques for finding the optimum cooling schedule21 and determining the perturbation size. In this paper, the parameters used to achieve convergence are: 1000 neighbors tested at each iteration on j and 100 external iterations on i (trials with 50 iterations gave unsatisfactory results and with 200 iterations, outcomes were exactly the same as with 100). Cost function Fcost was selected to normalize real and imaginary parts of a complex quantity, so that real and imaginary parts have equal consideration in the inversion process.
~ 0 Þ to a1 and distribution FIG. 3. (Color online) Sensitivity index of =ðZ=Z function of quarter-wave frequency fk=4 . Benoit et al.: Characterizing a porous road pavement
4785
Indeed, as absolute values of the imaginary part of surface impedance exceed those of the real part at low frequency, the cost function using Z~ would have given the same results ~ 0 Þ. ~ 0 Þ, skipping
le Christophe Heinke Laboratoire R egional des Ponts et Chauss ees de Strasbourg, Centre d’Etudes Techniques de l’Equipement Est, 11, rue Jean Mentelin 67035 Strasbourg Cedex 2, France
Emmanuel Gourdon Laboratoire G enie Civil et B^ atiment LTDS Unit e Mixte de Recherche CNRS 5513, Ecole Nationale des Travaux Publics de l’Etat, rue Maurice Audin 69518 Vaulx-en-Velin Cedex, France
(Received 22 October 2012; revised 24 July 2013; accepted 5 August 2013) This paper deals with a numerical procedure to identify the acoustical parameters of road pavement from surface impedance measurements. This procedure comprises three steps. First, a suitable equivalent fluid model for the acoustical properties porous media is chosen, the variation ranges for the model parameters are set, and a sensitivity analysis for this model is performed. Second, this model is used in the parameter inversion process, which is performed with simulated annealing in a selected frequency range. Third, the sensitivity analysis and inversion process are repeated to estimate each parameter in turn. This approach is tested on data obtained for porous bituminous concrete and using the Zwikker and Kosten equivalent fluid model. This work provides a good foundation for the development of non-destructive in situ methods for the acoustical characterization C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4824971] of road pavements. V PACS number(s): 43.55.Ev, 43.58.Bh, 43.40.Le [KVH]
I. INTRODUCTION
Forty percent of European inhabitants are exposed to traffic-related noise; the cost of this exposure amounts to over 1 106 healthy years of life lost annually in this part of the world, according to World Health Organization.1 This is the reason that several strategies have been devised to reduce traffic noise, and these include sound-absorbing porous road pavements. Road surfaces are a specific type of porous material: Porosity is low compared with typical acoustical materials (maximum porosity of 30%), and few measurements or evaluation of other intrinsic parameters are available from the literature.2–4 Moreover, as the characteristics of porous road surfaces vary over time, it is of significant interest to be able to measure them in situ without coring the road. Berengier et al.2 showed the effect of clogging on the absorption coefficient of porous asphalt. It would also be interesting to be able to assess the effect of clogging also on parameters such as open porosity and static air-flow resistivity, especially on road asphalt as it clogs, in an intermediate state between open and closed pores. This paper therefore aims to offer a guided inversion method for assessing the intrinsic parameters of road pavement as a porous material from surface impedance measurements given that these measurements can be carried out in situ. a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
4782
J. Acoust. Soc. Am. 134 (6), Pt. 2, December 2013
Pages: 4782–4791
Used with porous materials, equivalent fluid models allow a relationship to be established between measurable acoustical quantities, such as surface impedance, and microstructure-related parameters, such as open porosity or static air-flow resistivity. Furthermore, these models also involve a limited number of parameters that makes them suitable for an inversion procedure that leads from measurement of acoustical quantities to assessment of the intrinsic parameters of road pavement. The issue consists in using a non-destructive method of measurement in situ, such as direct measurement of impedance using a PU-probe, or the transfer function method. This supposes that only one complex quantity is available, i.e., surface impedance. This sets up a different problem from the usual problems of characterization: Several characterization procedures using transmitted and reflected waves have been proposed.5–9 Another consequence of in situ problems is that an experimentally robust method is required. Indeed the measurement error may be important when carrying out in situ impedance measurements (see for example the work of Kruse et al.10 for an estimate of the measurement error with the transfer function method). This is especially true for measurements on materials with low absorption. This article describes a numerical inversion procedure for characterizing porous road pavements based on quantities available as a result of in situ measurements. Transmitted waves cannot be measured, and surface impedance is the only data accessible. To make the most of available data, therefore, sensitivity analysis is applied to the identification model, using an index based on a Bayesian approach. The
0001-4966/2013/134(6)/4782/10/$30.00
C 2013 Acoustical Society of America V
main idea is to analyze the model’s behavior with regard to its parameters before the former is inverted to determine the latter. This approach adds a criterion that provides both qualitative and quantitative information and that plays a decisive role in the numerical inversion in distinguishing the key parameters of the frequency range. In this way, numerical inversion of the model may be performed step by step to determine each parameter separately. In the first instance, measurements were carried out on samples with the impedance tube. This ensured that the method of measurement could be monitored and that access to reliable measurement data was available and also meant that the procedure could be compared with validated methods.6,7 The procedure was designed to take account of the constraints imposed by the use of in situ measurements with regard to the data available. The goal here is to design a numerical procedure for identification of parameters suitable for application in situ. Therefore the number of parameters required by the model has been reduced as much as possible. This initial investigation is thus carried out using an equivalent material model, the Zwikker and Kosten model11 with three parameters, i.e., flow resistivity r, open porosity U, and high frequency limit of tortuosity a1 . This is a similar approach to that described by H€ubelt et al.12 In their work, H€ubelt and colleagues use a numerical algorithm to assess porosity and thickness from surface impedance measured in an impedance tube. However, as they assume that flow resistivity and tortuosity may be expressed through porosity, they do not attempt to estimate them directly. Furthermore the innovation of the proposed approach relies on the individual evaluation of each parameter on targeted frequency ranges. Section II presents the sensitivity analysis method used and its results with the Zwikker and Kosten model; Sec. III describes the numerical inversion procedure and Sec. IV its application to a sample of porous bituminous concrete. II. SENSITIVITY ANALYSIS A. Method
Several methods may be applied to carry out a sensitivity analysis, including local methods such as one-at-a-time technique, and global methods, for example, the importance measure method. Here, the sensitivity analysis relies on the Sobol sensitivity index.13 The same index has been used recently by Ouisse et al.14 in their work relating to the sensitivity analysis of porous material models. It is especially interesting as it gives qualitative and quantitative information about the sensitivity of the model’s output to input parameters. Note that all parameters vary simultaneously with this approach. Let Y be the output of a model involving n input Xn . The sensitivity index of Y to parameter Xi is expressed as I Xi ¼
VðE½YjXi Þ ; VðYÞ
(1)
where V is the variance and E is the mathematical expectation. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
Through the expectation of the Bayesian quantity YjXi , this expression quantifies the degree of variance of the output resulting from the variance of the investigated input. Clearly, this index falls within the range [0,1] and the bigger the index IXi , the greater the influence of parameter Xi on output Y. This is the first-order sensitivity index. A second-order sensitivity index takes into account the sensitivity derived from the interactions between parameter Xi and another input Xj ; a third-order index may be defined taking into account interactions between three parameters, and so on. Last, a global sensitivity index, being the sum of these different order indexes, takes into account the total sensitivity of Y to parameter Xi . For example, for a three-parameter model, the global sensitivity index of parameter X1 is expressed as IX1 ; T ¼ IX1 þ IX1 ; X2 þ IX1 ; X3 þ IX1 ; X2 ; X3 ¼ 1
VðE½YjX6¼1 Þ : VðYÞ (2)
These elements require some comment (1) It should be noted that first-order indexes give only a partial indication of the influence of each parameter, and they may not enable conclusions to be drawn with regard to the influence of a particular parameter. This is especially true when the sum of the indexes for each parameter is less than 1, as they do not reflect the coupling effects between parameters. However, they already provide useful quantitative information at a much lower computation cost. This is why they have been investigated here. (2) Ouisse et al.14 showed the importance for parameter sampling of the choice of probability density function. In this case, as few measured values are available for intrinsic road surface parameters, a uniform probability density function has been selected. Ouisse et al.14 indicated in particular that for porosity values close to 1, firstorder and total indexes are similar when using uniform probability density function. This feature is the reason that only first-order indexes have been estimated. (3) Ouisse et al.14 also showed that the choice of relevant parameter variation ranges is crucial. In this exercise, the limits of the ranges were chosen using values found in the literature2–4 (obtained from identification or measurements). They were also selected to enable an understanding of porous road surfaces throughout their life cycle until such time as they may be considered non-porous: This is true for resistivity variation range in particular. We are confident in the limits of the variation range for porosity as this parameter can be directly linked to the void ratio, a parameter that is commonly measured for road surfaces during whole life cycle. From a numerical point of view, all these indexes were estimated using Monte Carlo method. Convergence was ensured by checking with a larger sample of parameters if the difference between the estimated indexes was less than 1%. Benoit et al.: Characterizing a porous road pavement
4783
B. Zwikker and Kosten model
With regard to the porous material model, it was decided to restrict the number of required parameters to three initially. The Hamet model3 for porous asphalts was considered, but it does not fit the high and low frequency limits with regard to frequency dependence. This is the reason that the Zwikker and Kosten model, using the same parameters, was selected. In addition, the Zwikker and Kosten model relies on parameters that form a subset of the parameters of models used for more traditional porous materials, like Johnson–Champoux–Allard15,16 and Johnson–Champoux— Allard–Lafarge,15–17 etc. It makes this model attractive should additional parameters be required to increase the accuracy of the model. Moreover, in their work, H€ubelt et al.12 mention that the lack of precision in the inversion might come from the Hamet model. It is therefore of interest to test another model. The Zwikker and Kosten model is one of the equivalentfluid material, which assumes that (1) Wave lengths are larger than pore sizes. Here the upper bound of the frequency range considered is 4000 Hz; lower wave lengths then have to be larger than 0.08 m, a value much higher than the pore sizes of road surfaces. (2) The material is considered as homogeneous. (3) The frame is assumed to be rigid and motionless. This assumption can be regarded as valid for porous road surfaces because of the high density of their frame.
Figure 1 presents surface impedance from the Zwikker and Kosten model for three sets of parameters found in the porous road surface literature;2,4 these sets are given in Table I. C. Results
This section presents the results of the sensitivity analysis of the Zwikker and Kosten model. It was carried out using the following parameter variation ranges: (1) Open porosity (referred to as porosity in the following) U: ½5%35%. (2) Static air-flow resistivity (referred to as resistivity in the following) r: ½1000 150 000 Nm4 s. (3) high frequency limit of tortuosity (referred to as tortuosity in the following) a1 : ½1 5. The thickness of the wearing course of a road surface varies from 20 to 60 mm. This is why the thickness was set to 40 mm in this example. The sensitivity indexes were estimated on sets of 5 105 realizations, ensuring good convergence, as indicated previously. Figure 2 presents the sensitivity indexes of normalized surface impedance to the three parameters of the model and the sum of these indexes (thicker line). It may be observed that all parameters are important, but on different frequency ranges. Flow resistivity is the most influential parameter on ~ 0 Þ at around 1000 Hz, then it quickly decreases and > ~ ðxÞ ¼ 0 FðkÞ; > q > < eq U x " pffiffiffiffiffiffiffi pffiffiffiffiffiffi #1 > cP Tð N k jÞ > > K~ eq ðxÞ ¼ 0 1 þ 2ðc 1Þ pffiffiffiffiffiffiPrffi pffiffiffiffiffiffi ; (3) > : U NPr k j where x ¼ 2pf is the pulsation, r is the flow resistivity, U is the open porosity, a1 is the high frequency limit of tortuosity, q0 is the air density, NPr is the Prandtl number, c is the adiabatic constant, P0 is the static pressure, j2 ¼ 1, F is the correction function for viscosity defined by Biot,18 T =Jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ J1p 0 is the ratio of Bessel functions of first kind and k ¼ 8a1 q0 x=rU. ~ charThese quantities relate to complex wave number k, ~ acteristic impedance Z c , and measured surface impedance Z~ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ~ > ~ eq ðxÞK~ eq ðxÞ; q ðxÞ ¼ Z > < c sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) ~ eq ðxÞ q > > ~ ; kðxÞ ¼ x > : K~ eq ðxÞ ~ ~ ZðxÞ ¼ jZ~c cotanðkeÞ;
(5)
where e is the thickness of the material sample, backed by a rigid impervious plane. 4784
J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
FIG. 1. (Color online) Surface impedance from the Zwikker and Kosten model for three sets of parameters (see Table I). Benoit et al.: Characterizing a porous road pavement
TABLE I. Surface impedance from the Zwikker and Kosten model for three sets of parameters. Parameters
U
r (Nm4 s)
a1
Solid line Dashed line Dotted line
0.10 0.15 0.24
50 000 15 000 4000
3 2 1
~ 0 Þ, porosity is the only influential parameter up to =ðZ=Z 800 Hz, then tortuosity gradually takes the lead. No precise conclusion can be drawn beyond 2000 Hz, as the sum of sensitivity indexes is much less than 1: Coupling effects should be investigated with total sensitivity indexes beyond that point. Some phenomena described in these results were already known thanks to Taylor series expansions [see, for example, the Ph.D. dissertation of Sellen19 for the sensitivity ~ 0 Þ to U at low frequencies], but the sensitivity analof =ðZ=Z ysis allows for the addition of quantitative indications with respect to frequencies at which these phenomena are valid, and parameter hierarchy. Regular oscillations may also be observed on the sensi~ 0 Þ to a1 . This could be explained by tivity index of =ðZ=Z ~ 0 Þ for tortuthe fact that the sensitivity maximum on =ðZ=Z osity matches the range where the quarter-wave frequency
FIG. 2. (Color online) Sensitivity indexes of surface impedance in the Zwikker and Kosten model for a sample of thickness 40 mm. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
fk=4 is located for this thickness in most of the configurations created by the parameter samples (see Fig. 3) fk=4 ¼
cmaterial cair ¼ pffiffiffiffiffiffi : 4e 4e a1
(6)
The results in relation to the dominance of parameters are slightly different from those obtained by Ouisse et al.,14 who concluded that the sensitivity analysis confirmed the predominant position of resistivity. This is not the case here; this could be explained by the range of variation that is different for porosity with values much lower than for typical acoustical porous materials, such as foams. Identifying the frequency ranges where one of the parameters is predominant will help guide the inversion procedure for this parameter. III. INVERSION STRATEGY A. Selected algorithm and associated parameters
Simulated annealing was selected for numerical inversions because of its effectiveness in finding the global minimum of an optimization problem. It is based on a random walk through parameter space in combination with the Metropolis criterion20 with the intention of finding the global minimum of challenging functions with many local minima. The initial model is drawn randomly from the predefined interval of each parameter I. Arbitrary temperature T is gradually reduced according to a cooling schedule; this allows for exploration of most of the space possibilities. Figure 4 shows a process map of the simulated annealing algorithm. Several researchers present techniques for finding the optimum cooling schedule21 and determining the perturbation size. In this paper, the parameters used to achieve convergence are: 1000 neighbors tested at each iteration on j and 100 external iterations on i (trials with 50 iterations gave unsatisfactory results and with 200 iterations, outcomes were exactly the same as with 100). Cost function Fcost was selected to normalize real and imaginary parts of a complex quantity, so that real and imaginary parts have equal consideration in the inversion process.
~ 0 Þ to a1 and distribution FIG. 3. (Color online) Sensitivity index of =ðZ=Z function of quarter-wave frequency fk=4 . Benoit et al.: Characterizing a porous road pavement
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Indeed, as absolute values of the imaginary part of surface impedance exceed those of the real part at low frequency, the cost function using Z~ would have given the same results ~ 0 Þ. ~ 0 Þ, skipping
