Uncertainty of decibel levels Open Access

Assume that repeated measurements give the observed levels L₁, L₂,…, L_n. The levels may correspond to sound exposure, sound power, or an A-weighted maximum level as discussed in more detail by concrete examples in Sec. 3. In most cases the observed levels will all be different, and this variation is associated with the statistical component of the measurement uncertainty. The aim of this note is to present a method for the estimation of this component.

In some cases it is correct to state $L ¯ = ( L 1 + ⋯ + L n ) / n$ as the estimated level from the experiment. The standard uncertainty of the mean is $u ( L ¯ ) = S L / n$⁠, where $S L 2 = [ ( L 1 − L ¯ ) 2 + ⋯ + ( L n − L ¯ ) 2 ] / ( n − 1 )$ is the estimated variance. The expanded uncertainty is given by $U ( L ¯ ) = k u ( L ¯ )$⁠, where the coverage factor k can be calculated as explained in the Guide to the expression of Uncertainty in Measurement (GUM).¹ 

In many cases, however, it is necessary to calculate the mean differently as follows. Let L_i be as above and define $W i = W 0 10 L i / 10$⁠, where W₀ is a reference value. The mean is calculated as $L W ¯ = 10 lg ( W ¯ / W 0 )$⁠, where $W ¯ = ( W 1 + ⋯ + W n ) / n$⁠. The mean $L W ¯$ is here referred to as the energy mean as opposed to the mean $L ¯$⁠.

The reader may find it noteworthy that this uncertainty, which is stated on a decibel scale, is calculated without taking the logarithm of a corresponding sound power uncertainty. It is simply given by the relative sound power uncertainty directly, but scaled by the factor 10/ln(10).

A warning regarding the notation can be given. The GUM notation $u ( L W ¯ )$ is not interpreted as a function u evaluated at the point $L W ¯$⁠. In Eq. (1) the right-hand side cannot be written as a function of $L W ¯$⁠. The notation u(⁠ $L W ¯$⁠) denotes the uncertainty associated with the experiment leading to the calculated value $L W ¯$⁠.

Which formula is best? This depends on the unknown statistical distribution, and cannot be decided in general. The formula for $u F ( L W ¯ )$ has the advantage that it will not give unrealistically high values for the measurement uncertainty when $u ( L W ¯ )$ is not small. We will, in particular, show that the formula for $u ( L W ¯ )$ is questionable if the resulting $u ( L W ¯ )$ value is not much smaller than 4 dB. This will be discussed in some more detail, including examples in Sec. 3, but the next step is to present the theoretical justification for these formulas and claims. The paper ends with some concluding remarks in Sec. 4.

Without further knowledge regarding the unknown probability distribution the best estimator for μ_W is $W ¯ = ( W 1 + ⋯ + W n ) / n$⁠. It is, in particular, unbiased since $E ( W ¯ ) = ( E ( W 1 ) + ⋯ + E ( W n ) ) / n = n E ( W 1 ) / n = μ W$⁠. This unbiasedness is of central importance in many cases met with in acoustics, and examples are given in Sec. 3.

The standard uncertainty is given by $u ( W ¯ ) = S W / n$ as before. The law of large numbers ensures that $W ¯$ is asymptotically distributed like a Gaussian variable with mean μ_W and a standard deviation well approximated by $u ( W ¯ )$ which converges to zero as the sample size n goes to infinity.

The above explains why $L W ¯$ is used instead of $L ¯$ as an estimator of the decibel level in many cases. The problem now is to find the corresponding standard uncertainty $u ( L W ¯ )$ of $L W ¯$⁠. This can be done as explained more generally next.

The validity of the previous argument depends on $u ( W ¯ ) / W ¯$ being small compared to 1, or more simply that the resulting standard uncertainty $u ( L W ¯ )$ is small compared to 10/ln(10) dB. This gives the 4 dB requirement as stated in Sec. 1.

An alternative to formulas (1) and (2) is given by a Monte Carlo estimate u_MC(⁠ $L W ¯$⁠) of the uncertainty. This would typically depend on a necessary choice of a parametric statistical model, and this choice will be important for the conclusion. The choice in our examples is to assume that the observed L₁,…, L_n is a random sample from a Gaussian distribution. The parameters are then given by the mean μ and the standard deviation σ. The variance of $L W ¯$ can then be found by Monte Carlo simulation by assuming that the estimates $L ¯$ and S_L for μ and σ are the true values, and this determines u_MC(⁠ $L W ¯$⁠). For the purposes here the simulation was made by simulating 10⁵ repetitions of the experiment by use of a random number generator. In the nonparametric case, as assumed initially here, it is unclear if the Monte Carlo method would be an improvement, but it is used in some of the examples below for comparison purposes.

Five examples are presented below. The first and the last treat maximum levels from a passing snowmobile and music festival participants. It is explained that it is not correct to use the energy mean for these two examples as opposed to the three other examples: Pass-by noise exposure for a single snowmobile, pass-by noise exposure for a population of road vehicles, and a sound intensity measurement. The latter example illustrates that Eqs. (1) and (2) can be used also in cases without independent repeated measurements.

The pass-by maximum noise level was measured on several snowmobiles and under different driving conditions in Norway in March 2015. The measurement distance was 7.5 m from the center of a 60 m straight driving path with a microphone at 1.2 m height. The resulting A-weighted maximum levels with fast time averaging were 66.97, 66.44, 67.10, 66.62, 66.96, and 66.91 (dB) for a particular snowmobile driving at 30 km/h.

These measurements can be summarized by the mean 66.8(1) dB and the standard deviation 0.25(8) dB, where the numbers in parentheses correspond to the standard uncertainty. The standard uncertainty 0.08 dB for the standard deviation was calculated by the formula $u ( S L ) = S L 1 − [ 2 / ( n − 1 ) ] × [ Γ ( n / 2 ) / Γ ( n − 1 ) / 2 ] 2$⁠. This formula is justified by the corresponding formula for the variance of the empirical standard deviation S_L assuming that the levels follow a Gaussian distribution. This assumption also allows for the calculation of any other statistic of interest such as the upper 95% percentile or the probability of the noise exceeding a given level. The standard uncertainty of the statistic of interest can be estimated based on the standard uncertainties $u ( L ¯ ) = 0.1$ dB and u(S_L) = 0.08 dB.

The above takes only the statistical variation of the given observations into account. There are other sources of errors that must be considered: Accuracy of the sound level meter, uncertainty in measurement geometry, meteorological conditions, and possibly other sources of uncertainty inferred from the specific field work or previous experience with the measurement method. An overall uncertainty of $u ( L ¯ ) = 2$ dB would come as no surprise, and if so the above statistical contribution to the uncertainty is insignificant. It is, however, important to be able to calculate the statistical part of the uncertainty in order to reach this conclusion! The focus above, and in the following, is on this statistical part of the measurement uncertainty. The reader should, however, keep in mind that all contributions to the measurement uncertainty must be taken into account in any given concrete measurement.

In the experiment just described the noise exposure was also measured. The resulting A-weighted sound exposure levels were 77.97, 78.69, 78.39, 78.55, 78.08, and 78.97 (dB) for the same particular snowmobile passing by at 30 km/h. In this case the result is the energy mean 78.5(2) dB, where the uncertainty $u ( L W ¯ ) = 0.1539 ≈ 0.2$ dB has been calculated by Eq. (1). The uncertainty $u F ( L W ¯ ) = 0.1512$ dB from formula (2) is very close to $u ( L W ¯ )$ for this example. Incidentally, the uncertainty $u ( L ¯ ) = 0.1537$ dB is also a good approximation of the uncertainty of $L W ¯$ in this case.

Measurements of pass-by noise exposure from passenger cars were made at Bratsberg in Trondheim in August 2000.³ Inversion of the sound propagation problem, including, in particular, the ground effect,^4,5 gives the estimates 87.3, 85.5, 83.0, 116.7, 79.5, 84.2, 79.2, 79.9, 77.3, 78.1, 78.3, 87.9, 77.7, 87.3, 87.6, 87.0, 79.3, 83.7, 85.5, 82.1, 81.2, 80.9, 84.7, 86.2, 79.7, 84.2, 84.3, 84.6, 78.9, and 81.2 (dB) for the sound power in the 100 Hz third-octave frequency band for light vehicles driving at 55 km/h.

The method used for the estimation of the sound power will not be explained here, but it is based on an array of microphones as explained in the Nordtest NT ACOU 116 method.⁶ This method could also have been used to determine the sound power of the snowmobile considered above. An important difference from the snowmobile case is that the data were sampled from a population of vehicles, and not only for one particular vehicle. Both cases correspond to well defined statistical ensembles, but the spread in the data were naturally larger in the case of a population of different vehicles.

The result of these measurements is the energy mean 102(3) dB, where the uncertainty $u F ( L W ¯ ) = 2.98 ≈ 3$ dB has been calculated by Eq. (2). The expanded uncertainty is 6 dB corresponding to 95% confidence. This includes only the uncertainty from the variation in the data, and additional sources of uncertainty must be added in a complete analysis. In this case, as opposed to the snowmobile case, the statistical component of the uncertainty is highly significant, but it can be reduced by increasing the sample size.

The uncertainty $u ( L W ¯ ) = 4.28$ dB from Eq. (1) and the uncertainty $u ( L ¯ ) = 1.29$ dB are both different from $u F ( L W ¯ )$ for this example. The Monte Carlo estimate is $u MC ( L W ¯ ) = 2.07 ( 1 )$ dB, where 0.01 dB is the Monte Carlo error corresponding to the finite Monte Carlo sample size 10⁵.

This indicates that the finite difference uncertainty $u F ( L W ¯ )$ is a better estimator for the uncertainty than the GUM uncertainty $u ( L W ¯ )$ in this case. The assumption $u ( L W ¯ ) ≪ 4$ dB is not satisfied, and the failure of the GUM uncertainty estimate is as expected. The use of $u ( L ¯ )$ as an indicator of the uncertainty of $L W ¯$ seems on the other hand to be too optimistic. In this case $u F ( L W ¯ )$ seems to be on the safe side, and is recommended.

This example illustrates that the formulas can be used also for cases where the problem is not given by a sample of identically distributed decibel values. All that is required is an estimate of $u ( W ¯ )$ and an estimate of $W ¯$⁠, or equivalently $L W ¯$⁠.

Sound power can be estimated by measurements of the sound intensity on a surface enclosing the source. The resulting uncertainty will then depend on the uncertainty of the sound intensity measurement. This problem, and the choice between a scanning technique versus discrete spatial sampling, was discussed by Jacobsen.⁷ He considered, in particular, a measurement where a loudspeaker was placed less than 1 m from several reflecting surfaces, and the intensity probe was placed between the loudspeaker and the surfaces.

From Jacobsen's Fig. 1 it can be estimated visually that the sound intensity in the 1 kHz third octave band is about 60(2) dB relative to 1 pW per square meter with a relative uncertainty of 0.17(2) on the energy scale. The numbers in parentheses indicate the uncertainty resulting from the authors’ manual reading off from Fig. 1 of Jacobsen. These errors will be ignored here.

Formula (1), with W replaced by I, can be used to translate the relative uncertainty of 0.17 into the uncertainty estimate $u ( L I ¯ ) = ( 10 / log ( 10 ) ) ⋅ 0.17 = 0.73$ dB. Equation (2) gives the alternative $u F ( L I ¯ ) = 0.68$ dB. It can be concluded that the sound intensity is 60.0(7) dB with an expanded uncertainty of 1.4 dB corresponding to 95% confidence. We prefer this statement of the uncertainty in favor of stating that the relative uncertainty of the intensity is 0.17.

The World Health Organization (WHO) recommends that the A-weighted equivalent level over 4 h should not go above 100 dB at music events. The equal energy hypothesis assumed by the WHO states that an equal amount of sound energy always has the same damaging potential. Tronstad and Gelderblom⁸ investigate this and related issues by measurements at two Norwegian music festivals.

The Hove music festival lasted for 5 days. Four persons equipped with personal noise dose meters gave 17 maximal four-hour equivalent levels 93.2, 97.8, 98.7, 99.0, 93.4, 90.9, 93.7, 93.1, 95.1, 96.9, 97.6, 98.9, 104.0, 103.0, 102.0, 101.0, and 102.0 dB. Each level corresponds to a maximum 4-h equivalent level for one person during one day of concerts, but every person did not attend every festival day.

These measurements can be summarized by the mean 97.6(9) dB and the standard deviation 3.9(7) dB, where the numbers in parentheses correspond to the standard uncertainty. The analysis is as for the maximum pass-by level for vehicles. One day of concerts for one person corresponds to a vehicle pass-by, but the fast time averaging has been replaced by the extremely slow time averaging given by a 4-h time window.

Even though it is the noise dose which is the focus here it would be wrong to calculate the energy mean $L W ¯$⁠. It is the distribution of the levels itself which is of interest, and which gives for instance the probability of exceeding the 100 dB limit of the WHO for similar festivals. The distribution is completely specified by the mean and standard deviation assuming a Gaussian distribution.

This short note has presented two alternative formulas for the calculation of the standard uncertainty of the energy mean decibel level. The first, formula (1), is as suggested by application of the GUM. The validity of the derivation of the formula requires $u ( L W ¯ ) ≪ 4$ dB. The second, formula (2), has a more direct interpretation, and is more generally valid. Both formulas can alternatively be seen as simply convenient translations of the relative uncertainty on the energy scale into an uncertainty on the decibel scale. Formula (2) is the recommended general choice.

The authors have discussed the substance of this paper with several acousticians for a long period of time. At the risk of omitting some relevant names, we would like to especially thank our colleagues Tron Vedul Tronstad, Peter Svensson, Tor Erik Vigran, Ulf Kristiansen, and Erlend Magnus Viggen from the Acoustics Research Centre at NTNU and SINTEF.