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Uncertainty of an in situ method for measuring ground acoustic

impedance

David Ecotière

a)

Philippe Glé

b)

Center for Expertise and Engineering on Risks, Urban and Country Planning, Environment and

Mobility (CEREMA), DTer Est, ERA32, 11 rue Jean Mentelin, 67035 Strasbourg Cedex 2,

France.

Benoit Gauvreau

c)

LUNAM Université, French Institute of Science and Technology for Transport, Development

and Networks (IFSTTAR), AME, LAE, CS4, 44344 Bouguenais, France.

Régis Boittin

d)

Center for Expertise and Engineering on Risks, Urban and Country Planning, Environment and

Mobility (CEREMA), DTer Normandie-Centre, 11 rue Laplace CS2912, 41029 Blois, France.

Hubert Lefèvre

e)

Center for Expertise and Engineering on Risks, Urban and Country Planning, Environment and

Mobility (CEREMA), DTer Centre-Est, 8-10, rue Bernard Palissy 63017 Clermond-Ferrand

Cedex 2, France.

David Lunain

f)

Center for Expertise and Engineering on Risks, Urban and Country Planning, Environment and

Mobility (CEREMA), DTer Méditerannée, Pôle d'activités Les Milles, 30 avenue Albert Einstein

CS 70499, 13593 Aix en Provence Cedex 03, France.

a)

email: david.ecotiere@cerema.fr

b)

email: philippe.gle@cerema.fr

c)

email: benoit.gauvreau@ifsttar.fr

d)

email: regis.boittin@cerema.fr

e)

email: hubert.lefevre@cerema.fr

f)

email: david.lunain@cerema.fr

The ground acoustic impedance is a key parameter for outdoor sound propagation.

Laboratory approaches for measuring the impedance of outdoor ground are often less

interesting than in situ approaches because these kinds of ground often require non

destructive techniques. The chosen in situ method consists in fitting the transfer function

between the frequency spectra measured at two different microphones above the ground

with a theoretical model. The theoretical model is based on the Rudnick theory together

with an impedance model adapted to the ground studied (the Miki model or the Zwicker

and Kosten model are used here).

The uncertainty of the method has been estimated by organizing an inter laboratory

experimental campaign and by estimating the repeatability and the reproducibility

according to the ISO 5725. Five independent laboratories have tested five different types of

outdoor grounds (synthetic lawn, lawn, grass, slightly compacted soil, asphalt). It is shown

that the uncertainty for estimating the flow resistivity of the impedance model is increasing

with the flow resistivity value : the relative standard uncertainty for the flow resistivity

estimation is smaller than 20% for flow resistivity smaller than 5 000 kNsm-4 but can be

much higher for very reflective material.

1 INTRODUCTION

The ground acoustic impedance is a key parameter for outdoor sound propagation

1

.

Laboratory approaches for measuring the impedance of outdoor ground are often less interesting

than in situ approaches because these kinds of ground often require non destructive techniques.

We present here the estimation of the uncertainties of an in situ method for measuring the ground

impedance parameters.

2 EXPERIMENTAL ESTIMATION OF THE GROUND IMPEDANCE

PARAMETERS

2.1 Theory

The two microphones method

The in situ method

2,3

consists in fitting the difference of frequency spectra measured at two

different microphones above a ground, with a theoretical one.

Fig. 1 – General confguration of the impedance measurement

Acoustic pressures from a single source are recorded for two positions above the ground (Fig. 1),

which yields to a measured transfer function that can be set under the form:

=∆

2

1

log20 p

p

L

measured

(1)

The identification of the ground impedance parameters consists in computing the theoretical

transfer function

computed

L∆ and to minimize the following difference between these two functions

using the least square method on a given frequency range:

∑

∆−∆=

max

min

2

),(),()(

f

fmeasuredccalculated

ZfLZfLZF (2)

where Z is the acoustic surface impedance.

Propagation model

Ground and atmosphere are supposed homogeneous and separated by a plane boundary. In these

conditions, assuming that atmosphere and ground are two semi-infinite media, Rudnick’s model

4

can be used to describe the sound propagation over this plane. The time dependence

tj

e

ω

−

is

used. Thus, pressure at a position

i

in the atmosphere, resulting from a point source, can be found

using the following relationship:

ri

jkR

i

di

jkR

i

R

e

Q

R

e

p

ridi

+=

(3)

Where

R

di

and

R

ri

are respectively the lengths of the direct and and the reflected paths between

the source and the receiver

i

as described Fig. 1.

Q

i

is the complex reflexion coefficient in

spherical waves corresponding to the reflected path

i

and is related to the complex reflection

coefficient in plane waves

R

pi

by:

)()1(

ipipii

wFRRQ −+= (4)

with:

+

−

=

−

−

=

+=

∫

∞

−

−

−

0

0

0

2

00

²

2/1

cos

cos

²sin

²²

1

²sin)²1( 2

21)( 2/1

ZZ ZZ

R

k

k

Z

Z

RRk

jw

dueejwwF

i

i

pi

i

ipi

ri

i

jw

u

w

ii

i

i

θ

θ

θ

θ

(5)

In these equations, k and Z are respectively the wave number and surface impedance of the

ground while k

0

and Z

0

are the wave number and the characteristic impedance of the air.

θ

i

is the

angle between the incident wave and the normal to the ground.

Impedance model

These acoustical properties of the ground can be predicted by a number of models in the

literature, depending on the various mechanisms of dissipation taken into account and of the

microstructure description. Ground is usually considered motionless

5

, which enables to use the

rigid frame hypothesis, in such a way we can directly describe the ground with an equivalent

fluid model. Two models have been used here, by Miki

6

and Zwikker and Kosten

7

. Surface

impedance can be calculated in these conditions as a function of the characteristic impedance of

the material Zc, its wave number k, and the thickness e of the material:

)coth( jkeZcZ

−

=

(6)

Miki model:

Miki model is a correction of Delany and Bazley

8

empirical model initially developed for fibrous

materials whose porosities are close to 1, in the frequency range delimited by :

00.101.0 <<

σ

f (7)

The following relationships are used to compute the characteristic impedance and the complex

number of the material, as a function of the airflow resistance

σ

expressed in kN.s.m

-4

and the

frequency f :

+

+=

+

+=

−−

−−

618.0618.0

0

632.0632.0

0

41.1181.71

43.85.51

σσ

σσ

f

j

f

kk

f

j

f

ZZc

(8)

Zwikker and Kosten model:

This second model is a more general description of porous materials, on the basis of cylindrical

pores, leading to:

KZc

ρ

=

and

K

k

ρ

ω

=

(9)

With

1

0

1

0

)(

)(

2

1

−

∞

−

−

−

−= jJ

jJ

j

λ

λ

λ

φ

ρα

ρ

and

1

0

1

0

)Pr(

)Pr(

Pr

)1(2

1

−

−

−

−

−

+= jJ

jJ

j

P

K

λ

λ

λ

γ

φ

γ

(10)

σφ

αωρ

λ

30

10

8

∞

=,

J

0

and

J

1

are respectively the Bessel functions of order 0 and 1,

Pr

is the Prandtl

number of air, γ the ratio of the specific heats of the air, and

X

the conjugate number of

X

.

Three parameters are used in this model to describe the material, its open porosity φ, its

resistivity σ (expressed in kN.s.m-4) and its tortuosity

∞

α

.

2.3 Experimental set-up

The method described in Section 2.1 has been applied with a dedicated portable measurement

device presented

Fig. 2

and composed of:

- for the emission:

o a loudspeaker Navsound Fugue 70W,

o a power amplifier Pioneer GM-A3602,

- for the reception:

o Microtechgefell M370 class 1 microphones,

o MMF M32 conditioners,

o a Acoustics Engeneering Triton 2 channels acquisition card.

Fig. 2 – Experimental apparatus (left) and experimental set-up in situ (right)

This device is completely controlled via a laptop with the help of a Scilab program enabling the

execution of the different steps of the measurement:

- calibration of the acquisition card and microphones,

- emission of sound (white noise),

- recording, impulse response calculation and time windowing (10 ms windowing),

- result display and identification.

The performance of the card has thus been evaluated by looping inputs and outputs, and by

calculating the ratio between signals amplitudes as well as time delays for 1000 frequencies

between 10 and 20kHz. The characteristics are presented in Fig. 3. It appears that the difference

in attenuation between the two channels is lower than 0.1dB in this frequency range, and that the

attenuation can be assumed constant in the frequency range of interest for the measurement (100-

2000Hz) since variations are again lower than 0.1dB. Concerning time delays, the card shows a

quite constant evolution as a function of frequency, with delays of about 0.293s. The peaks

visible here are not related to a frequency behaviour but to a random delay in the computer.

Fig. 3 – Characteristics of the acquisition card: ratio between input and output (left) and time

delay (right) as a function of frequency

Finally, the following geometry has been chosen for the experimental setup:

- Source height (hs): 0.6m

- Source-microphones distance (d): 4m

- Microphone heights (hr1, hr2) : 0.06 and 0.6m

3 UNCERTAINTIES ESTIMATION OF THE METHOD

3.1 Round Robin Test

The uncertainties of the method are estimated with the reproducibility and the repeatability

thanks to a round robin test.

In order to get an estimate of the reproducibility, five independent laboratories have been

involved in an experimental campaign on a single site, at the same time, and with the same kind

of material (see section 2.3). The dependence between uncertainties and impedance parameters

have been investigated by testing five different types of outdoor grounds (synthetic lawn, natural

lawn, grass, slightly compacted soil, non porous asphalt : Fig. 4). For each ground, 3 different

close locations have been tested (except for grass : 6 locations). For each location and for each

laboratory, 5 repetitions of the measurement enable to estimate the repeatability of the method.

For each ground, the environmental conditions were considered to be stable enough between sets

of measurements to ensure the stability of the ground properties.

a) Synthetic lawn

b) Natural lawn

c) Grass

d) Compacted ground

e) Non porous asphalt

Fig. 4 – Ground samples

The Miki impedance model has been used for all ground, except for synthetic lawn for which the

Zwikker and Kosten model has been considered. The mean least square procedure of

minimization has been done for frequencies ranges from 100 Hz to 1500 Hz (2000 Hz for

synthetic lawn), excluding the frequency domain centred on the first dip interference +/- 100 Hz.

The tested flow resistivities range from 10 to 10

6

kNsm

-4

(5.10

5

kNsm

-4

for synthetic lawn) and

the tested ground thicknesses range from 5 to 50 mm. For synthetic lawn, the tortuosity has been

set to 1.

Two examples of the measured and of the fitted transfer function are presented on Fig. 5.

Fig. 5 – Example of fitted and measured transfert function for asphalt (left) and for synthetic lawn

(right).

3.1 Results

Although the method provides estimations of both parameters air flow resistivity and ground

thickness (for Miki model), only the uncertainties on the air flow resistivity parameter σ are

presented here.

The reproducibility variance

2

R

s is given by

10:

222 rLR

sss += (11)

where

2

r

s is the repeatability variance and

2

L

s is the interlaboratory variance. Both are calculated

from the round robin test results according to the ISO 5725-2

10

. These two quantities provide the

range of the uncertainties of the method: the repeatability can be interpreted as a minimum value

of the uncertainty as it only includes the random variations that remain when all influence factors

are under control (same apparatus, same operator, same sample) ; whereas the reproducibility

can be interpreted as a maximum value of the uncertainty because it includes not only the

repeatability, but also the variations between laboratories (different apparatus, different operator,

...).

For the five ground tested, the mean air flow resistivity ranges from around 50 kNsm

-4

, for

synthetic lawn, to around 20 000 kNsm

-4

, for asphalt (Fig. 6). We can notice very contrasted

dispersion of the flow resistivity depending on the ground type: the higher is the mean flow

resistivity, the higher is the dispersion. The synthetic lawn ground presents a very narrow

dispersion with quasi constant values, this may be due to the artificial nature of this material that

may probably provide very homogeneous properties of the ground.

Very contrasted values are also observed for reproducibility and repeatability that range from

around 10 to 15 000 kNsm

-4

for reproducibility, and from around 5 to 13 000 kNsm

-4

for

repeatability (Table 1).

Asphalt

Compacted

Lawn

Grass

Synth_lawn

10 50 100 500 5000 50000

Air flow resistivity

kNsm-4

Fig. 6 – Box plot of measured air flow resistivities for the 5 ground tested

Table 1 : Mean, reproducibility and repeatability (kNsm-4) of the air flow resistivity of the 5

ground tested.

Mean

R

s

r

s

L

s

Non porous Asphalt 19 609 15 070 13 065 7 510

Compacted ground 3 519 567 512 245

Natural lawn 1 154 248 217 119

Grass 277 34 34 6

Synthetic lawn 51 7 5 5

The dependence between reproducibility (resp. repeatability) and the air flow resistivity is

plotted on Fig. 7: a remarkable linear behaviour is observed for air flow resistivity smaller than 5

000 kNsm

-4

, whereas uncertainties drastically increase in a non linear manner for high value of

air flow resistivity (asphalt). For air flow resistivity smaller than 5 000 kNsm

-4

, some linear

approximations can be obtained for reproducibility and repeatability (Fig. 8):

1214,0 −=

σ

R

s with

4

0005

−

≤kNsm

σ

(12)

σ

06,0=

r

s with

4

0005

−

≤kNsm

σ

(13)

A statistical linear regression analysis indicates that those linear models are statistically

significant with R-squared >0,99 and p_value<10

-12

.

The linear approximations (12-13) show that the relative uncertainty of the method is lower than

20% (14%) when reproducibility conditions are assumed, whereas it is lower than 6% when

repeatability conditions can be assumed. Those values are low enough to give satisfactory results

in outdoor sound propagation applications. Nevertheless, the method appears to be less efficient

for more reflective surface : relative uncertainty can reach 100% for high values of flow

resistivity, as for asphalt here.

50 200 500 2000 10000

0 5000 10000 15000

Reproducibility

Air flow resistivity (kNsm-4)

sR (kNsm-4)

Asphalt

Compacted

Lawn

Grass

Synth_lawn

50 200 500 2000 10000

0 5000 10000 15000

Repeatability

Air flow resistivity (kNsm-4)

sr (kNsm-4)

Asphalt

Compacted

Lawn

Grass

Synth_lawn

Fig. 7 – Reproducibility (left) and repeatability (right) as a function of the air flow resistivity.

0 1000 2000 3000 4000

0 100 200 300 400 500

Reproducibility

Air flow resistivity (kNsm-4)

sR (kNsm-4)

Asphalt

Compacted

Lawn

Grass

Synth_lawn

sR = 0.14 sigma -12

R2=0.996

0 1000 2000 3000 4000

0 100 200 300 400 500

Repeatability

Air flow resistivity (kNsm-4)

sr (kNsm-4)

Asphalt

Compacted

Lawn

Grass

Synth_lawn

sr = 0.06 sigma

R2=0.99

Fig. 8 – Linear approximation of reproducibility (left) and repeatability (right) for

4

5000

−

≤kNsm

σ

(R2 is the squared regresion coefficient).

The standard deviation of reproducibility

R

s (or repeatability, depending of the measurements

conditions) can finally be used to provide a confidence interval associated with future

measurements obtained with the method:

];[

+−

=

σσ

IC

with

R

sd.

,

±=

+−

σ

. d is the coverage

factor that depends on the confidence level that is chosen and of the shape of the probability

density function of

σ

that is assumed. The value of d=1.95, relative to a normal distribution and

to a 95% confidence interval, is often considered.

4 CONCLUSIONS

A method for estimating ground impedance parameter from in situ measurements has been

presented. The method is based on the fitting of the transfer function of pressures measured and

calculated at 2 microphones above a ground. The uncertainty of the method for the air flow

resistivity estimation has been quantified by organizing an inter laboratory experimental

campaign and by estimating the repeatability and the reproducibility according to the ISO 5725.

The reproducibility (resp. repeatability) gives a relative uncertainty lower than 20% (resp. 6%)

for air flow resistivity smaller than 5 000 kNsm

-4

. For this range, a very satisfactory linear

approximation can be adopted to deduce the uncertainty from the air flow resistivity value. For

very reflective material, the uncertainty (reproducibility or repeatability) is no longer linear and

can reach 100% of the estimated value. The method is therefore more adequate for air flow

resistivity values ranging from 10 to 5 000 kNsm

-4

than for more reflective ground. This

restriction interval corresponds however to many common outdoor ground.

5 ACKNOWLEDGEMENTS

This research was supported by the French Ministry of Sustainable Development and of the

Energy (DGPR, DGITM) under the PLUME project. The authors thank the St Georges sporting

center of Blois for facilitating the conduct of the round robin test.

6 REFERENCES

1. K. Attenborough, K.M. Liand K. Horoshenkov, Predicting outdoor sound, Taylor & Francis,

London, 2007.

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In situ measurements of soil physical properties by acoustical techniques, Soil Sci. Soc. Am.,

54 (1990).

4. I. Rudnick, “The Propagation of an Acoustic Wave along a Boundary”, J. Acoust. Soc. of

Am., 19(2), (1947)

5. K. Attenborough, I. Bashir and S. Taherzadeh, “Outdoor ground impedance models”, J.

Acoust. Soc. of Am., 129(5), (2011)

6. Y. Miki, “Acoustical properties of porous materials. Modifications of Delany-Bazley

models”, J. Acoust. Soc. of Jap., 11(1), (1990).

7. C. Zwikker and C.-W. Kosten, Sound Absorbing Materials, Elsevier, New-York, (1949).

8. M.-E. Delany and E.-N. Bazley, “Acoustical properties of fibrous absorbent materials”, App.

Acoust., 3 (1970).

9. G. Dutilleux and D. Ecotière, “Automatic characterization of ground surfaces from in situ

measurements”, Acoustics’08, Paris, (2008).

10. Standard ISO 5725-2:1994, Accuracy (trueness and precision) of measurement methods and

results - Part 2: Basic method for the determination of repeatability and reproducibility of a

standard measurement method, 1994.