Expected mea i a evirometal oise measuremet ad its related ucertaity M. Paviotti ad S. Kephalopoulos Europea Commissio, via e. fermi,, 00 Ispra, Italy marco.paviotti@jrc.it 85
I the cotext of the implemetatio of the Evirometal Noise Directive 00/49/EC a study o oise measuremet ucertaity was performed. Averagig over differet samples of oise measuremets, there might be assumptios over the distributio ad idepedecy of the samples. I the cotext of evirometal oise, this might be the case of a series of measuremets of a costat oise source like a idustrial plat or a fluctuatig oise source like road traffic. Usig a series of 5 miutes L Aeq samples, the average of these values is usually cosidered as the expected mea, however, the error caused by the specific selectio of the samples is ot usually evaluated. Statistically speakig, before establishig a average value, at least the distributio of the samples ad the effect of addig-up several ucertaities should be evaluated. This article focuses o the mathematical formulas which could be used ad discusses the differeces i assessig the expected mea for ormally distributed values, or for log-ormally distributed, ad fially suggests a approach to properly addigup all ucertaities related to a log-term evirometal oise measuremet campaig. Itroductio A still usolved problem i evirometal oise measuremets is how to correctly measure a log-term oise idicator such as the Europea L de level. A good statistical aalysis should be performed o the measured values to allow for the best estimatio of the L de. L de is a oise idicator that requires averagig the source variatios over oe year ad the meteorology that affects the propagatio of the soud over several years.. This article describes possible approaches to be followed for assigig correct umbers to the aforemetioed oise idicators, although o defiitive solutio is give sice some problems remai still to be solved. Fig. Distributio of 5 mi L Aeq. Descriptio of the data characteristics The preset study was performed usig data recorded durig a log-term evirometal oise measuremet campaig i a urba situatio ad close to a road with 5000 vehicles pass-bys o average per day. Data were recorded ext to the road, to assess the road oise levels, ad the at approximately 00m away from the road, ext to a house, where some other local oise sources were simultaeously preset. Give the large umber of values recorded, it was possible to see the differeces i the predictio of the log term oise levels, whe the calculatios were based o a large set of recordigs or just subsets of these recordigs. All values used are 5 miutes log L Aeq samples. I Fig. the distributio of the samples is give for a selected subset, that is a subset durig the same ight-time hours (from 0:00 to 05:00) durig workig days ad alog several weeks. A few 5 miutes L Aeq samples recorded durig the ight reported o pass-bys at all, ad therefore they were out of the average group of L Aeq. Nevertheless, it was decided to iclude them, otherwise a bias would have bee itroduced i the evaluatio. The same evaluatio is performed o the same recorded time periods at the house locatio, far from the major source. The results are plotted i Fig.. Fig. Distributio of 5 mi L Aeq. Both data distributios have a quite commo shape. These sets of data were therefore selected as a represetative sample of data for aalysig how to attribute ucertaities. 3 Aalysis of a sigle set of L Aeq The first step of the aalysis cosists i idetifyig the correct approach for evaluatig the true mea. For clarity, this is the mea that would have bee obtaied if a ifiite time would have bee allowed for measuremets uder the same coditios (e.g.: usig oly data over ight-times, ext to the road, ad durig workig days oly). Two radomly selected subsets were extracted ad the meas were compared (see Table ). 85
Set of samples Mea L Aeq All 65.4 /0 th subset 66.7 /0 th subset 65.3 Table. Example of how averagig over a radom subset of L Aeq values ca differ from the mea obtaied from a umber of samples 0 times larger. Lookig at the ormality of the distributio, it ca be esured that the set of values chose did ot ideed iclude strage samples ad it seems that a sigle class of evets was captured. I other words, it meas that the samples were correctly selected from the same populatio. It would ot make certaily sese to mix a road oise measuremet durig mid-day with a correspodig measuremet at midight, sice it is expected that the traffic will differ a lot betwee day ad ight, therefore the samples would come from two statistically differet populatios. 4 Aalysis of the best approach to the true value Before proceedig i evaluatig the ucertaities related to a log-term evirometal oise measuremet campaig, it is ecessary to uderstad the properties of the mea to be calculated. I evirometal acoustics, the mea value of oise levels is commoly give by: L log 0.* L 0 i () 0 0 ad ot by the geometrical mea of the L i samples although from a purely statistical poit of view the quatity uder assessmet is the oise level. Cocerig the oise source uder assessmet, all L i values should be equivalet levels recorded for the same time iterval ad uder the same operatig ad meteorological coditios. Also, the measuremets should be idepedet from each other, ad this might be possibly checked. However, the quatity that the istrumet feels is the soud pressure p, ad ot the level L i. Although the pressure is averaged keepig the root mea square (rms) of the p values alog a certai time, for each of the sampled periods it will be the p ad ot the L i the quatity measured. It is possible to use the same data set ad see how the p values are distributed (Fig.3). Lookig at Fig. 3 it ca be oticed that eve the skewed values of the origial distributio showed i Fig. ow seems to be properly described (the oe o the left side of the distributio) by a ormal distributio. This ca be possibly attributed to the fact that i real pheomea, the soud pressure varies evely aroud a certai value. Fig.3 Distributio of 5 mi soud pressure (p) values relative to the data i Fig.. Oe would coclude that it is therefore possible to calculate the mea of the samples as: p p i however, this will produce a differet result compared to the oe obtaied usig the stadard way to get the average soud pressure, i.e. by the rms of the pressure levels. Aother questio to be aswered is whether the average obtaied by meas of Eq. () or Eq. () is the correct oe for estimatig the evolutio ad fluctuatio of the evirometal oise pheomeo that is impossible to practically measure thoroughly over a very log time legth (i.e., oe year). The evirometal oise measuremets over short periods of time are always performed with the aim to assess what the oise level would be o average over a ifiite time period. Therefore the average oise level calculated as described above is ever the true mea, however, it should be as close as possible to the true mea. Sice the best way to be close to the true mea is to have the largest amout of as much as possible idepedet ad ubiased samples, ad give that Eq. () is the oly correct approach that cosiders the rms, there is o other solutio tha usig Eq. (). Usig this approach, evertheless, there is oe coflictig hypothesis: if the mea of the populatio is defied by Eq. (), this meas that the averaged values are 0^(0.*L i ) istead of L i (i.e., squared soud pressures istead of levels). Cosequetly, the ucertaity should also be expressed i terms of squared soud pressure istead of levels. Ufortuately, for calculatig the stadard deviatio of the oise levels, it is commoly used the followig Eq. (3): ( L i L) () σ (3) The reaso why Eq. (3) is used is that for most of the measuremets the levels are cosidered to be ormally distributed, ot the squared soud pressures, or the soud pressures. To follow a cosistet statistical approach the questio is therefore whether to accept Eq. () ad modify Eq. () or vice versa. 853
Give the computatioal reaso expressed (rms of p) ad sice the regulatios (e.g., ISO 996-) require performig a squared soud pressure averagig of the differet time periods (ad this reflects the physical pheomea) Eq. () caot be modified. Therefore, statistically speakig, the populatio of values must be the populatio of squared soud pressures, ad this seems to be log-ormally distributed. I the case of evirometal oise, sice the 0*log 0 (p /p 0 ) is used everywhere, it should be better to assume that the 0*log 0 (x) is ormally distributed (here x p /p 0 ad ot p because the rms of the x values is chose). Give this assumptio, the correspodig probability distributio fuctio (PDF) ca be defied as: [ 0 log ( x) μ ] 0 0 σ PDF ( x) e (4) π x l(0) σ where, μ ad σ are respectively the mea ad the stadard deviatio of the populatio of values expressed i levels (L0*log0(x)), or: L i μ (5) ( L i μ) σ (6) Cosiderig the st momet of x, it ca be show that the expected mea ca therefore be expressed as: [ ( x) ] + x PDF( x) dx e 0 l(0) l(0) σ μ+ 0 0 E (7) It is ot the itetio to go deeper i the aalysis cocerig ways for calculatig the possible meas, however, at least for the data set used as a example i this article, the mea values obtaied by usig the three differet approaches discussed above are preseted i Table. Mea type Mea L Aeq Followig Eq. () 65.4 Followig Eq. () 65. Followig Eq. (7) 65.6 Table. Meas of the set of L Aeq values preseted i Fig. ad Fig. 3 calculated by three differet approaches. The slight differeces i the calculated meas ca be explaied as follows: the log-ormal approach leads to slightly higher values sice with respect to the stadard mea calculated by Eq. () the oe calculated by Eq. (4) describes a slightly higher probability that high values will be ecoutered tha whe a ormal PDF is used. This is illustrated i Fig.4. Fig.4 Differeces betwee a ormal ad a logormal fittig of the (p/p o )^ values. Although the differeces betwee the meas are ot great, ad oe ca therefore coclude that this does ot costitute a major problem, the implicatios whe performig evirometal oise measuremets are relevat. 5 Stadard deviatio of a sigle set of values Whe a mea value is calculated, the stadard deviatio is commoly used to kow how the measured values are distributed aroud the true mea value (the calculated mea would coverge to the true mea if oise would have bee measured over a ifiite time period). It is commo, for ormally distributed data (ad ot for logormally distributed), to defie the ucertaity related to a give set of measuremets by addressig the stadard deviatio σ. I the case of a ormal distributio of data, the iterval μ±σ should statistically iclude the 65% of the measured values (oly for that specific set of data recorded, ot for the etire populatio ). Ofte, this result is used for improperly addressig the true mea ad the ucertaity with which this is kow. For example, if measurig road traffic oise 00m away from the road, techicias usually perform measuremets over oe or two weeks ad ifer the yearly average just from the mea calculated from the short term measuremets. Also, the stadard deviatio is sometimes used i evirometal oise studies to ifer how good that mea is, regardless of the fact that this gives iformatio oly about the distributio of the measured values ad ot o the probability these values beig close to the true mea. Ideed, measuremets aim at revealig a good approximatio of the true mea ad provide iformatio o the reliability or distace of that evaluatio from the true mea. Give oly a small umber of samples from the etire populatio (e.g.: i case of road traffic oise the etire populatio are daily levels all over the year) it should be possible to use statistics to address the true mea (e.g.: the yearly average level). The icorrect approach is therefore takig σ to represet the ucertaity with which the true mea could be kow. A correct approach is give by the DIN 55 303 part [,] where the exact approach for iferig the true mea is described usig the Studet distributio for ormally distributed data. This orm implies the use of the squared soud pressure measured values (e.g.: oe sample of p /p 0 each day durig a few days alog the year i our case) as if the samples would have bee take from the etire populatio of values (365 daily samples over oe year), ad usig these values as they would have bee ormally distributed. The oly error associated to this approach is 854
that the populatio is ot ormally but log-ormally distributed. Based o this assumptio, ad cosiderig Eq. () for the mea μ as well, the followig expressios for σ ad the cofidece iterval for the true mea (μ mi < μ true < μ max ) are used: 0.* L ( 0 i μ) σ (8) max σ μ + * t μ (9) mi σ μ * t μ (0) The objectio for usig Eq. (8), (9) ad (0) is that the 0^(L/0) values used are log-ormally ad ot ormally distributed. So, the μ mi ad μ max are ot strictly applicable (ideed, i case of a large spread of values, egative μ mi values could be obtaied, which are ot possible sice the log fuctio is oly defied for positive real values). Mea type μ-σ Mea L Aeq Followig Eq. () Followig Eq. () Followig Eq. (7) μ+σ 65. 65.4 65.7 6.7 65. 67. 9.5 65.6 9. 6 Addig up stadard deviatios If the formula for ormal distributio of data is ayway used, it is the possible to add up the several stadard deviatios that are calculated for a real oise measuremet campaig. Basically, two coditios may exist: L average L sum 0.* L i 0 log0 ai0 0 log0 aixi ai ai () 0.* L i () Li 0 log0 0 0 log0 xi i i A matrix that cotais the a i coefficiets is built up ad solved, to have a sigle expressio of the L de as a fuctio oly of the iput L i values. This is essetial for the correct estimatio of the ucertaity. I the past, the authors themselves erroeously added more times the same stadard deviatio, sice for example a level was recorded for ighttime road traffic oise, ad the this level was combied with differet trasfer fuctios from the road to the house (receiver poit). 7 Calculatio of the overall stadard deviatio i a evirometal oise measuremet A procedure to measure the average L de was developed i the cotext of the Europea IMAGINE project based o the GUM ad o the ISO 996- [3, 4, 5]. I this procedure, geeral techiques to measure log-term oise values are described. The basic idea is that for each assessmet positio oise levels are recorded durig periods (shortterm or log-term) which do ot coicide with the etire year. Statistics is the used to describe the mea ad the ucertaity associated to the measuremet. The mea is obtaied as a combiatio of differet short term measuremets uder differet source coditios (differet traffic or operatig coditios) ad mixed with the occurreces of the differet meteorological propagatio coditios. I geeral, the source is measured close eough to avoid ay meteorological effects, the the propagatig part is measured, ad fially correctios are applied, like the correctio for the residual oise ad for the positio i frot of the façade, or correctio for the atmospheric absorptio [6, 7]. Classes of emissio ad propagatio are filled i with the correspodig log-ormal meas ad mea boudaries ad afterwards added up. A very simplified example of a combiatio of these measuremets is give i figure. I a real oise measuremet like the road traffic oise measuremet campaigs performed withi the IMAGINE project, the umber of classes was as follows: Source class: 5 classes from Moday to Friday, 5 classes Saturday, 5 classes Suday; Propagatio: 4 meteorological classes i summer time ad 4 i witer time; a i coefficiets (% time) 0 percetages (5 traffic X 8 propagatio) Eq. () is used each time a weighted average level is eeded, for example a weekly average, where it should be cosidered that i a give week from Moday to Friday the traffic is high, whereas for Saturday ad Suday is lower (e.g.: L Mo-Fr is weighted for 5/7 ad L Sa-Su is weighted /7 ad the two values are added up to have the weekly average). Aother example is the L de idicator. Eq. () is used whe the level at the receiver is obtaied as the sum of the source level plus a trasfer fuctio (e.g.: L source 70 db, L trasfer 0 db, the L receiver L source -L trasfer ). 855
Source class % time - Refereces sou, + sou, - sou, Propagatio prop + prop - prop time +/- time % time - +/- [] DIN 55 303 - Statistical iterpretatio of data; tests ad cofidece itervals relatig to expectatios ad variaces, May 984. Propagatio % time - [] VDI 373 Awedug statisticher Methode bei der Kezeichug schwakeder Gerauschimmissioe, May 993. time +/- time Source class sou, + sou, - sou, prop + prop - prop % time - +/- [3] GUM, Guide for the expressio of ucertaity i measuremet (GUMBIPM, IEC, IFCC, ISO, IUPAC, OIML) Overall L at receiver averaged durig the week Overall corrected L at receiver averaged durig the week Correctio Residual oise Correctio positio soud meter +/- soud meter atm abs +/- atm abs Fig.5 Flow chart (partial) of the combiatio of the differet source ad propagatio classes. The calculatio chart illustrated i Fig. 5 cosidered the followig ucertaities: - source daily variatio; - source seasoal variatio; - propagatio meteorological day/eveig/ight variatios; - propagatio meteorological seasoal variatios; - meteorological parameters ucertaity; - residual oise at the receiver; - residual oise at the source; - positio of the microphoe i frot of the façade; - microphoe class ucertaity. [4] H. Joasso et al., IMAGINE report, IMA3TR- 04050-SP0 Determiatio of L de ad L ight usig measuremets, 006 [5] M. Paviotti, S. Kephalopoulos, IMAGINE report IMA30TR-0606-JRC0, Test of the measuremet stadard cocerig the determiatio of L de ad L ight, 006 [6] S. Kephalopoulos, M. Paviotti, D. Kauss ad M. Béregier, Ucertaities i log-term road oise moitorig icludig meteorological iflueces, Noise Cotrol Eg. J. 55 () 007 [7] M. Paviotti, S. Kephalopoulos, A. Iacopoi, Techiques to measure the Europea L de oise idicator for major sources i complex urba eviromets, ICSV3 Viea (Austria), -6 July 006 8 Coclusio This article explored the differet approaches to use for assessig a log-term evirometal oise level. Differet meas are discussed as a fuctio of the distributio of values ad the requiremets of the Evirometal Noise Directive (00/49/EC). A methodology was the preseted to address the overall stadard deviatio. Some questios still remai ope, amely cocerig the correct approach for calculatig the mea, the approach for calculatig the sigle stadard deviatio ad for addig up the several stadard deviatios, therefore o defiitive solutios are give. This work is still uder developmet ad it is expected to be cocluded i the ext moths after havig reached wide scietific cosesus o the methodology to be fially adopted. 856