PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Evaluation of Measurement Uncertainties of the D2S in Open-PlanOffices
Lucas LENNE1,2; Patrick CHEVRET1; Étienne PARIZET21

Institut National de Recherche et Sécurité, France
2

Laboratoire Vibrations Acoustique, FranceABSTRACT
Measuring the acoustic performance of open-plan offices is an increasingly important issue for companies
and employees. Indeed, it has been highlighted in several studies that noise in such workplaces reduces
employees' performance and makes them be unsatisfied with their work environment. This is especially
important for conversational noise. The ability of a room and its layout to reduce conversational noise can beassessed using acoustic indicators and in particular the D2S (spatial decay of the A-weighted level of a speech
signal when doubling the distance from the source on a line passing over workstations). This indicator is now
widely used in the field because it is recommended by several national standards that give values to beachieved. However, to date, there are no studies that discuss the accuracy of a measurement in an office. Thepurpose of this presentation is to provide an analysis of the measurement uncertainties for this indicator. To
that end, the measurement principle of D2S will be presented as well as an analysis of uncertainties based on
their theoretical developments. This analysis will be conducted using real cases (measurements) andsimulated cases using a room acoustics software.
Keywords: Open-Plan Office, Uncertainties, D2S, Speech, ISO 3382-3

1. INTRODUCTION
Noise in open-plan offices represents one of the greatest issues for these workplaces. Indeed, it has
been highlighted that the acoustic environment of open-plan offices reduces the employees’
performance as well as their job satisfaction (1). Furthermore, among all noise sources, conversational
noise has the biggest impact on performance (2) and constitutes the biggest source of annoyancecaused by the acoustic environment (3,4). Therefore, it appears that the way the office (meaning theroom and the furniture) influences conversational noise plays a major role in its acoustic quality.
The ISO 3382-3 (2012) standard (5) introduced an acoustic index which characterizes the spatial
decay of speech across workstations: D 2S. More precisely, D 2S is defined as the decrease of theA-weighted level of a speech signal when doubling the distance from the source on a line passing overworkstations. D 2S has then been used in many national standards (6,7) which prescribe required values.
Yadav et al. (8) evaluated the reliability and repeatability of the D 2S, but, to our knowledge, nostudy was devoted to the measurement uncertainties of this index. Therefore, the goal of this study isto evaluate such measurement uncertainties and to determine their influencing factors.

2. MEASUREMENT OF THE D2S INDEX
2.1 Measurement principle
When measuring speech spatial decay, the first step is to determine a measurement path. This line
must run from workstation to workstation, each of these workstation s constituting a point ofmeasurement. Two examples of paths are represented on the left-side of Figure 1. TheISO 3382-3 (2012) standard proposes a method to evaluate the A-weighted level that would have beenmeasured at the different workstations if the noise emitted by the source had a speech-like spectrum(noted Lp,A,S in the standard but called Li thereafter). D2S is then calculated as the slope of the linear
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lucas.lenne@inrs.fr2423

regression (using the least mean square method) of the A-weighted level of speech as a function of thedistance from the source to the points of measurements (expressed on a logarithm scale – see theright-side of Figure 1). The calculation of D2S given in the standard can be performed using equation 1.

Figure 1: Examples of measurement paths from the ISO 3382-3 (left) and measurements realized in anopen-plan office (right)

D2,S = −N ⋅ ∑(L ⋅ log 2 (r)) − ∑ L ∑ log 2 (r)N ⋅ ∑ log 2 (r)2 − (∑ log 2 (r))

2(eq. 1)

2.2 Measurements used for the study
For this study, D 2S values were measured in 8 open-plan offices, resulting in 21 paths ofmeasurement. The 21 measured values are summarized in Figure 2. These values are comprised
between 3.8 dB(A) and 7.4 dB(A) and their paths were constituted of 4 to 10 points of measurementsin accordance with the ISO 3382-3 (2012) standard.

Figure 2: Measured values of the D2S

3. EVALUATION OF THE MEASUREMENT UNCERTAINTIESThe Joint Committee for Guides in Metrology issued guidelines for the evaluation of measurement
uncertainties (9). According to this document, the uncertainty made on the evaluation y of a measurand
Y is obtained by combining the uncertainties of each input used to evaluate Y. In the case of D 2S (for apath constituted of N workstations), these inputs are composed of measurements of distances andsound pressure levels, resulting in equation 2 which enables to estimate the measurement uncertaintyof the D 2S. The measurement uncertainty of the index depends on its first-order derivatives withrespect to distances ( ∂D2S ⁄∂ri ) and speech levels ( ∂D2S ⁄∂Li ) but also on the measurement
uncertainties of distances (σri ) and speech levels (σLi ).

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22NN
∂D2S
∂D2S2
⃗⃗)
σD2S = ∑ (⋅ σri ) + ∑ (⋅ σLi ) = σ2D2S (r⃗) + σ2D2S (L∂ri∂Lii=1i=1(eq. 2)

This expression being the result of a first-order Taylor series approximation, guidelines precise that
if the dependence of the measurand on inputs is significantly non-linear, it becomes necessary toinclude higher-order partial derivatives to the expression presented on equation (2).
3.1

Uncertainties due to the measurement of speech levels⃗⃗) in (eq. 2)), it is required toTo evaluate the uncertainties due to speech level measurements (σ2D2S (Lassess the uncertainties of speech levels measurements σLi but also the partial derivatives of the D 2S
with respect to Li .
The derivative of the D 2S with respect to speech level, which expression is given by equation 3,
does not depend on the level. Given the linearity of the D 2S with respect to speech level, the expression⃗⃗) is only composed of first-order derivatives and stays accurate regardless of theof σ2D2S (Luncertainties made in measuring the speech level.
∂D2S
N ⋅ log 2 (ri ) − ∑ log 2 (r)=−∂LiN ⋅ ∑ log 2 (r)2 − (∑ log 2 (r))2(eq. 3)

Speech levels are directly derived from measurements of octave-band levels. Therefore, theevaluation of their measurement uncertainties is based on the uncertainties of octave-band levelmeasurements. The IEC 61672-1 (2002) standard (10) defines two classes of sound level meters based,in part, on tolerance limits for the measurement uncertainties of third -octave band levels. The
measurement uncertainty of octave-band levels are defined from these limits, depending on the classof the sound level meter used to measure the D 2S. These limits are given in table 1.
Table 1 – Measurement uncertainties (σoct in dB) of octave-band levels for the two classesof sound level meters
Octave band

Class 1

Class 2125 Hz1.1

1.3

250 Hz1.0

1.2

500 Hz1.0

1.21000 Hz0.80.92000 Hz

1.1

2.0

4000 Hz

1.1

2.0

8000 Hz

1.8

3.1

The measurement uncertainties of speech level is then evaluated with the same method as
previously, resulting in the equation 4. In this case, the accuracy of the approximation has to beverified because the overall level is non-linear with respect to the octave-band levels.
2Li,oct

2⁄10
∂Li
10
σ2Li = ∑ (σoct ) = ∑ (⋅ σoct )Li∂Li,oct
10 ⁄10(eq. 4)

To assess the precision of this estimation, the result obtained using the equation 4 was compared tothe one obtained from a Monte-Carlo method (11) on 137 measurements realized in several open-planoffices. This method consists in adding a random error on measured octave-band level. The dispersionof the overall levels obtained from this error sampling is equal to the real uncertainty.

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Figure 3: Error in the evaluation of σL for the two classes of sound level meters
The result obtained for this comparison are presented on Figure 3, for the two classes of sound levelmeters.
For both class of sound level meters, the evaluation of measurement uncertainties of the speech
level made using (eq. 4) seems to be very accurate: the absolute value of the error was less than 2% for
the 137 measurements realized in various open-plan offices.
⃗⃗), some manipulation were made on equation 2.
In an effort to simplify the expression of σ2D2S (LEquation 5 was obtained, where Var and Cov represent respectively the variance and the covariance ofthe variable(s).
⃗⃗) =σ2D2S (L1
⋅ Cov(log 2 (r) − ̅̅̅̅̅̅̅̅̅
log 2 (r) , (log 2 (r) − ̅̅̅̅̅̅̅̅̅log 2 (r)) ⋅ σ2L )N ⋅ Var(log 2 (r))2(5)3.2 Uncertainties due to the measurements of distances
Evaluating the uncertainties in the D 2S evaluation caused by the measurements of distances σ2D2S (r⃗)
requires to assess the uncertainties of distance measurements σri but also the partial derivatives of theD2S with respect to ri .
The derivative of the D 2S with respect to the distance is obviously non-linear and its expression isgiven by equation 6.
∂D2S−1 N ⋅ Li − ∑ L + 2 ⋅ D2S (N ⋅ log 2 (ri ) − ∑ log 2 (r))=⋅[N ⋅ ∑ log 2 (r)2 − (∑ log 2 (r))2 ] ⋅ ri
∂rilog⁡(2)(6)

To assess the distance measurement uncertainty, two hypotheses are made. Firstly, the error isassumed to follow a normal distribution centered on the true value of the distance to be measured.
Secondly, the error is assumed to be independent of the distance. From these hypotheses, the distancemeasurement uncertainty is characterized by one parameter, the half-length of the 95% confidence
interval, thereafter called d, following the expression given by equation 7.
ddσri = σr =≈(7)
−1
√2 ⋅ erf (0.95) 1.96
As D2S is significantly non-linear with respect to the distances from the noise source, it is necessary
to evaluated the accuracy of the expression of σ2D2S (r⃗) for different values of d. The uncertainties
resulting from the distance measurement uncertainty were evaluated (using equations 2, 6 and 7) for 21
in situ measurements of D2S realized in various open-plan offices. These evaluations were comparedto the real measurement uncertainties, estimated using a Monte-Carlo method (consisting in adding arandom error to the measured distances). The result of this comparison presented on Figure 4.

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Figure 4: Error in the evaluation of σD2S (r⃗) for different dFigure 4 indicates that as long as d stays below 50 cm, the evaluation of σD2S (r⃗) remains accurate(absolute value of the error below 5%). This limitation is not very restrictive because it is consideredthat with the current measuring equipment (laser range finder), it is easy, in an open office, to estimate
the distance with an error of less than 50 cm.
In an effort to simplify the expression of σ2D2S (r⃗), some manipulations were made on equation 2.
The result is presented in equation 8, where Var and Cov represent respectively the variance and thecovariance of the variable(s).
σ2D2S (r⃗) =1
σ2rα−α̅⋅⋅ Cov (α − α̅, 2 )2N ⋅ Var(log 2 (r)) log⁡(2)r(8)

Where α = L + 2 ⋅ D2S ⋅ log 2 (r)4. MEASUREMENT UNCERTAINTY OF D2SFinally, the total measurement uncertainty of D 2S was estimated using the abovementioned
expressions for various values of d and for the two classes of sound level meters. Obtained values
approximatively correspond to the half-length of the 95% confidence intervals of the corresponding
measurement. The results are represented on Figure 5, in which each line correspond to a single D2S
measurement.
The principal observation that can be made from Figure 5 is that the expanded measurement
uncertainty of D 2S remains of reasonable value: for both classes of sound level meters, it stays below
1 dB as long as the error made on distance measurement is less than 25 cm.

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Figure 5: Expanded measurement uncertainty of D2S

5. CONCLUSION
In this paper, a theoretical expression of D 2S measurement uncertainty was derived, depending on
the measured speech levels and distances from the noise source. The validation of this expression wasrealized using a Monte-Carlo method on in situ measurements ranging from 3.5 dB(A) to 7.5 dB(A).
The first results indicate that the measurement uncertainty of D 2S is less than 1 dB(A) for bothclasses of sound level meters as long as the distance measurement uncertainty is less than 25 cm.
To complete this study, the expression of the D2S measurement uncertainty must be validated on awider range of D 2S and a parametric analysis will be carried out aiming to identify key factors thatstrongly influence the measurement uncertainty of the D 2S.

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