Estiation of the iniu integration tie for eterining the equialent continuous soun leel with a gien leel of uncertainty consiering soe statistical hypotheses for roa traffic S. R. e Donato a) (Receie 006 May 9; reise 007 Septeber 6; accepte 007 Septeber 6) In this work a relation is obtaine for calculating the iniu tie necessary for easuring the hourly equialent leel, with preset uncertainty on the L eq in the case of noise prouce by roa traffic uner ifferent statistical hypotheses for ehicular flow. A siple equialent leel preiction oel is use as reference. Soe specific relations between acoustic power an ehicle spee are ipleente in this oel. Through the application of the classic theory of errors, the expression for the uncertainty on the L eq is obtaine with reference, in particular, to arious ehicle istributions: unifor (rectangular), triangular, noral an Poisson that, accoring to the aailable inforation, can be applie for escribing traffic flow. Uncertainties oer the istance source/receier an spee of the ehicles are also taken into consieration in the calculation of uncertainty on L eq. The iniu easureent tie is obtaine fro the expression of the error associate with the L eq, accoring to the hourly nuber of ehicles, so that the uncertainty on the L eq stays within a preset alue. In this case too, the eterination of the iniu tie refers to the arious preiously entione hypotheses with respect to ehicle istribution. It is shown that it is possible to obtain a correct escription of roa traffic noise, within a preeterine uncertainty on the hourly L eq, by easuring oer ties consierably shorter than an hour. 007 Institute of Noise Control Engineering. Priary subject classification: 5.1; Seconary subject classification: 5.3 1 INTRODUCTION Definition of the uncertainty associate with enironental noise leels is currently the subject of particular attention since requests for assessing noise in urban areas 1 4 are on the increase. The nee to contain uncertainty on the easure alue ust howeer be in harony with the necessity to reuce the tie an resources to be use at the iniiual site, also in orer to increase the nuber of easureent points an consequently iproe the significance of the spatial analysis. To that en, use can be ae of teporal sapling techniques through which the long-ter alue of enironental noise can be estiate starting fro easureents on shorter tie interals 5 7. Since roa traffic represents the greatest source of noise in urban areas, a real opportunity to optiise resources is gien by the eelopent of relations that will allow a) Regional Agency for Enironental Protection (ARPA), Via Gabalunga 83, 47900 Riini, Italy. Electronic ail: seonato@arpa.er.it. shortening of sapling ties for eterining the hourly equialent leel prouce by roa traffic with a preeterine uncertainty on the L eq. In this work, starting fro a siple oel for preicting the L eq prouce by ehicular traffic, a relation is obtaine for calculating the uncertainty on the hourly L eq an fro this is obtaine the iniu integration tie that guarantees a preeterine uncertainty on the L eq itself. The relation obtaine allows easureents to be ae oer ties that are consierably shorter than an hour, accoring to the hypotheses on traffic flow istribution. As regars ehicular flow soe statistical istributions are taken into consieration. If no hypothesis can be forulate with regar to the istribution of ehicles, a unifor (rectangular) istribution is inicate; alternatiely triangular, noral or Poisson istributions can be consiere. The uncertainty on the aerage power leel of the ehicles is estiate consiering traffic flows ae up of ifferent percentages of light an heay ehicles, while for the ariables istance of the traffic line fro 56 Noise Control Eng. J. 55 (6), 007 No-Dec
Fig. 1 Geoetry of a ehicle traelling as a oing point source, S, past a receier, R, at a spee /s. the receier an spee of the ehicles, unifor istributions are assue for greater generality. DESCRIPTION OF THE MODEL With reference to Fig. 1, a ehicle oes along a straight roa at an aerage spee (/s). The obseration point R is at a istance () fro the roa an at a istance r (t) fro point S. Consiering a single ehicle as a source point, the groun as a perfectly reflecting surface an ignoring the attenuation cause by air absorption, the expression of the L eq can be obtaine by integrating the soun energy prouce by the passage of the source uring tie interal T. Consiering the syetry of the situation: L eq = 10 log 1 T p t T 0 p t 0 = 10 log 1 T WS 0 T 0 W 0 4r t t 1 where p t= Wc, p 4r t 0= 10 5 N/, W is the eitte soun power, suppose constant, W o is the reference power 10 1 W, S 0 =1, c is the characteristic ipeance of the eiu, an the factor before W takes account of the fact of the sei-spherical eission of the source. We hae: t = x = r cos where is the angle of iew between an r(t), an therefore Eqn. (1) becoes: W L eq = 10 log T 1 W 0 T 0 r t rt cos = L w + 10 log 1 3 T 0 with being the angle of iew of the section traelle in tie T. rt cos Also, obsering that: = r cos 4 fro Eqn. (3) we obtain: L eq = L w + 10 log 5 T where L w =10 log W W 0. In first approxiation, if ehicles pass by in the interal of tie T, it is necessary to ultiply the logarith arguent by, consiering at the sae tie the ean alue of the power of the ehicles: L eq = L w + 10 log 10 log 10 log T 10 log where: + 10 log 6 =nuber of ehicles in T =T (s) =aerage spee of the ehicles (/s) =angle of iew to the tie T (ra). =arctant/, / for T/, but 10 log/ 10 log0.1 for T/ 30. In practical situations can be assue to be a constant equal to /. =istance between the easureent point an the line of traffic flow () L w =ean acoustic power of the ehicles (B(A)) For eery i-a category of ehicles (light, heay, etc.) the soun power leel can be calculate by relations of the type: L wi = a i + b i log 7 where =aerage spee of the ehicles. Values for the coefficients a i an b i are gien in the Refs. 8 11. A coparison between soun power leels calculate fro soe relations of the type in Eqn. (7), is shown in Ref. 1. Defining the aerage power /10 leel as: k k n L w = 10 log i i=1 10L wi = 10 log i 10 L wi /10 p i=1 where: k=nuber of ehicle categories p i = n i =fraction of i-a category ehicles (between 0 an 1). Finally, the L eq can be expresse as: 8 Noise Control Eng. J. 55 (6), 007 No-Dec 57
L eq = L w + 10 log 10 logt 10 log 10 log 3 9 3 HYPOTHESIS ON THE STATISTICAL DISTRIBUTION OF TRAFFIC FLOW Let us suppose that we count the passage of ehicles in tie T 1 h. Let us also suppose that we interpret as the aerage alue of a syetrical istribution about with lower liit an upper liit -, with falling between 0 an 1. In general, can be set accoring to the inforation that can be obtaine with reference to the change in. On an hourly basis it can be estiate that N ehicles pass, where: N = 3600/T = k with k = 3600/T 10 The analogous istribution for the nuber of ehicles about N will therefore hae k as a lower liit an -k as an upper liit. 3.1 Case of Unifor Distribution for If nothing is known about the istribution of alues of within the interal of the liits, reference can be ae to a unifor istribution in which the probability that the true alue falls within the interal is 100% an the alues of are equiprobable. For a unifor istribution of liits u + an u the uncertainty u is calculate as 13 : For it will therefore be: = + 3 with 0 1. u = u + u 3 = 3 = 1 3 3. Case of Triangular Distribution for 11 1 If it can be consiere that the alues near the centre of the liits are ore likely than alues close to the liits, reference can be ae to a triangular istribution. In this situation, for a ariable u with liits u + an u the uncertainty u is calculate as: Therefore for it will be: u = u + u 6 = 1 6 13 14 3.3 Case of Noral Distribution for If it can be consiere that the ariable is istribute accoring to a noral istribution an that the upper an lower liits efine the aplitue interal 3 incluing 99.73% of the alues, then the uncertainty on can be obtaine fro: = 1 15 9 3.4 Case of Poisson Distribution for Various authors 14 17 hae referre to this istribution to escribe roa traffic an this case is therefore of particular interest. In particular it can be consiere that the Poisson istribution is applie for flows of less that about 600 ehicles/h. The uncertainty on for this istribution is gien siply by. 4 CALCULATION OF UNCERTAINTY ON THE L EQ Using the classic theory of errors on Eqn. (9) we hae 13 : L eq = in which: L eq = c L eq L L w w + L eq + L eq + L eq L eq L w =1; with that represents,,, an 16 10 c = 1n10 Recalling Eqn. (8) for L w it is possible to obtain the uncertainty L w, expresse in ters of uncertainty of the ean 13, fro: k L w = 1 10 0.1L wip i L wi + 10 logep i i=1 10 0.1L = w 17 The aerage power leel is always calculate as starting fro groups of objects: in this sense the uncertainty ust be expresse as uncertainty of the ean iien for. In Eqn. (17) Lwi represents the uncertainty of the single power leels for the i-a category of ehicles at a gien constant spee. 58 Noise Control Eng. J. 55 (6), 007 No-Dec
Fig. Variation in iniu easureent tie, T inutes, an traffic flow, N eh/h, for alues of =0.3, 0.5 an 0.7 an L eq =.5 BA. The uncertainty on L eq can be therefore be expresse as: L eq = c + + c + c 18 in which, assuing unifor istributions for an we hae: = +, = + 3 an = 1 3 n with n =3,6 or 9 for unifor, triangular or noral istribution. For Poisson istribution. =. The expressions for the L eq therefore becoe: L eq = c c + c1 n + + 19 with n =3,6 or 9 respectiely for unifor, triangular or noral istribution an, L eq = + c in the case of Poisson istribution. c + + c 5 DETERMINATION OF THE MEASUREMENT TIME 0 Inerting Eqns. (19) an (0), keeping the alue L eq fixe an consiering Eqn. (10), the alue of T is obtaine accoring to N (ehicles/h) an : 3600 T = N + 1 L eq 1 c n + 1 for unifor, triangular an noral istributions with the sae conention for n as before; an T = 3600 + c L eq c N + 1 in the case of Poisson istribution for. In Eqs. (1) an () tie T ust be interprete as the tie necessary to obtain a preset uncertainty on the L eq gien a certain nuber N of ehicles/h associate with a gien istribution. 6 RESULTS By way of exaple, Fig. shows the easureent ties so that the uncertainty on the L eq stays below.5 BA for arious alues of in the case of unifor istribution for N, an with: L wl =71.9+3.8 log, L wh =84.5+18.9 log, L wl =L wh =4 BA where l, h=light (weight 1.5 t) an heay ehicles (weight 1.5 t) [Ref. 8]; =13 /s; = =0.; p l=0.76; p h =0.4; p l =p h =0.. Assuing =0.5 an the sae ata of the preious exaple, Fig. 3 shows the easureent ties so that Noise Control Eng. J. 55 (6), 007 No-Dec 59
Fig. 3 Variation in iniu easureent tie, T inutes, assuing unifor, Poisson, triangular an noral istribution in traffic flow, N eh/h, for alues of =0.5 an L eq =.5 BA. The relationship erie by Fisk 5 is inclue. the iprecision on the L eq stays below.5 BA for the arious istributions consiere in relation to the nuber of ehicles/h. Obiously, it is possible to use ifferent relations to calculate L wl an L wh other then those preiously consiere 8. For exaple, relations use in the ASJ 10 etho gie soun power leels of ehicles lower than those of Ref. 8 for 13.9 /s. Howeer, in the sae conitions of the exaple, the ifferences between the ties calculate are always less than 0.9 inutes (ean of the ifferences=0.1 inutes). On the other han, using the relations gien in Ref. 9 for L wl an L wh which always gie ehicle soun power leels greater than Ref. 8, ifferences in tie easureent of less than 0. inutes are obtaine for all spees consiere (ean of the ifferences=0.03 inutes). Figure 3 also shows a cure obtaine fro the following reworking of the relation obtaine by Fisk 5 for the error on the L eq in conitions of low traffic ensity: T = 197136 NL eq with T in s. 4 Equation (4) takes into account only the hourly traffic flow an ientifies significantly shorter ties to guarantee the sae L eq. 7 CONCLUSIONS Starting fro a siple oel for preicting the L eq, expressions hae been obtaine for the error associate with the oel an for the iniu easureent tie for containing the L eq within preset alues. Through the use of relations that link soun power to the spee of the ehicles, the oel allows the arious situations of ixe traffic to be ealuate. Howeer, the oel oes not take eteorological conitions into consieration. Since the influence of weather increases with the istance, the oel can be applie within few tens of eters fro the line of traffic where ariations in such ariables can still be consiere ery sall. Furtherore, ariation ue to instruent uncertainty 1 has been consiere negligible copare to the other principal factors of uncertainty which hae been taken into consieration. Variables such as roa graient or roa surface, not expresse in the oel, can neertheless be taken into consieration by the use of opportune coefficients in Eqn. (7) 11.In this regar it shoul be unerline that application of ifferent relations of the type L wi =a i +b i log has not proe to be critical in the calculation of the iniu integration tie. Various statistical hypotheses hae been propose for escribing ehicular flow: unifor, triangular, noral an Poisson. Also, a suitable coefficient allows oulation of the liits of the statistical istributions 530 Noise Control Eng. J. 55 (6), 007 No-Dec
consiere. The results obtaine can also be suarize accoring to the rate of ehicles per secon =N/3600, respectiely as: 1 T = eq L c + 1 n + 5 for unifor, triangular or noral ehicle istributions (n=3, 6, 9 respectiely), an T = + c L eq c + 1 6 for Poisson istribution. In particular, the relations obtaine in this work also allow ealuation of the uncertainties regaring the istance of the obserer fro the line of traffic an the spee of the ehicles. The iportance of these ariables can be quantifie by analysing the ter on the enoinator of Eqn. (6) for which L eq c +, so leaing ieiately to L eq 1. BA for = =0.. Also introucing other paraeters relate to the istribution of (Eqn. (5)) we hae L eq c 1 n + + which for =0.5, n=3 (unifor istribution) an = =0., gies L eq 1.75 BA. The calculation carrie out by way of exaple in stanar conitions for ariables an, shows that it is possible to eterine the hourly L eq with an associate iprecision of L eq =.5 BA, with easureents lasting less than 15 inutes for hourly traffic flows (N) greater than 60 ehicles/hour. These ties are further shortene to alues of less than 5 inutes for N170 ehicles/hour. 8 REFERENCES 1. H. G. Jonasson, Uncertainties in easureents of enironental noise, Proc. International INCE Syposiu, (005).. W. Probst, Uncertainties in the preiction of enironental noise an in noise apping, Proc. International INCE Syposiu, (005). 3. A. Heiss, The natural uncertainty of escriptors for continuously fluctuating soun leels an its easureent, Proc. International INCE Syposiu, (005). 4. G. Zabon, A. Bisceglie an S. Raaelli, Error s ealuation in the estiate of the noise fro the roa traffic, Proc. International INCE Syposiu, (005). 5. D. J. Fisk, Statistical sapling in counity noise easureent, J. Soun Vib. 30(), 1 36, (1973). 6. D. Skarlatos an P. Drakatos, On selecting the iniu obseration tie for eterining the L eq of a rano noise with a gien leel of confience, J. Soun Vib. 15(1), 141 148, (199). 7. A. E. Gonzalez, M. Gairono Carozo, E. Pérez Rocaora an A. A. Bracho, Urban noise: Measureent uration an oelling of noise leels in three ifferent cities, Noise Control Eng. J. 55(3), 367 37, (007). 8. R. R. K. Jones an D. C. Hothersall, Effect of operating paraeters on noise eission fro iniiual roa ehicles, Appl. Acoust. 13, 11 136, (1980). 9. T. Suksaar, P. Sukase, S. M. Tabucanon, I. Aoi, K. Shirai an H. Tanaka, Roa traffic noise preiction oel in Thailan, Appl. Acoust. 58, 13 130, (1999). 10. C. Steele, A critical reiew of soe traffic noise preiction oels, Appl. Acoust. 6, 71 87, (001). 11. H. G. Jonasson, Acoustical source oeling of roa ehicles, Acta. Acust. Acust. 93, 173 184, (007). 1. E. Walerian, R. Janczur an M. Czechowicz, Soun leels forecasting forcity-centers. Part II: effect of source oel paraeters on soun leel inbuilt-up area, Appl. Acoust. 6, 461 49, (001). 13. ENV 13005, Guie to the expression of uncertainty in easureent, (1995). 14. S. R. De Donato an B. Morri, A statistical oel for preicting roa traffic noise base on Poisson type traffic flow, Noise Control Eng. J. 49, 137 143, (001). 15. Wu Shuoxian, A siple etho for preicting kerbsie L 10 leel fro a free ulti-type ehicular flow, Appl. Acoust. 0, 15, (1987). 16. D. Skarlatos an E. Manatakis, Noise Probability Density Functions for Poisson Type Traffic Flow, Appl. Acoust. 7, 47 55, (1989). 17. Z. Jiping, A stuy on the highway noise preiction oel applicable to ifferent traffic flow, Noise Control Eng. J.Noise Control Eng. J. 41(3), 371 375, (1993). Noise Control Eng. J. 55 (6), 007 No-Dec 531