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Curve fitting the probability distribution of acoustic noise from freely flowing traffic
Tnwpn Rer..
Vol. I?. pp. I I I-I II.
Pergamoa Press 1978. Printed in Grca~hilain
CURVE FITTING THE PROBABILITY DISTRIBUTION OF ACOUSTIC NOISE FROM FREELY FLOW-ING TRAFFIC DENNISE. BLUMENFELD Transport StudiesGroup. University College London, London WClE 6BT. England and
GEORGEH. WEISS National Institutesof Health, Bethesda,MD 20014. U.S.A. (Received
22 February
1977; in revised form
10 September
1977; received for publication
8 November
1977)
Abstrati-The theory of sound intensity or acoustic noise from freely flowing trafBc has been investigatedby severalauthors.It is possibleto calculatethe cumulantsof soundintensityexactly for the commonlyused model as well as the characteristic function,but an exact inversionto obtain the probability density function does not appear possible.Several approximationsto the full distributionof soundintensityhave been proposed. We have applied the Pearson curve methodologyto determine the full pdf and find that the best fit is given by a beta distribution. This is true with and without excessattenuationdue to atmosphericor groundabsorption and when a mixture of vehicle tvnes -. is .nresent. The exact and annroximatedistributionswere compared indirectly by comparinghigher momentsnot usedto fix parameters. ..
1.INTRODUCTlON A theory of acoustic noise from freely flowing traffic on an infinitely long road has been developed recently by several authors, (Weiss, 1970; Kurze, 197la, b, 1974; Marcus, 1973, 1975; Blumenfeld and Weiss, 1975a,b). In the simplest case in which headways between successive vehicles are independent, identically distributed, random variables with a negative exponential distribution, expressions can be given for the cumulants to all orders analogous to the theory of shot noise, (Cox and Miller, 1%5). From these one can find the Laplace transform of the probability density of the acoustic noise exactly. Unfortunately it is very difficult to invert these transforms numerically because the inversion involves the calculation of Bessel functions with complex arguments (Weiss, 1970). Hence simple approximations that make use of moment information would seem to be of considerable interest. Kurze (1971a) has used a GramCharlier expansion to derive approximations to the probability density function of noise from infinite stretches of road. He has also suggested approximations based on Daniels’s saddle point method (Daniels, 1954) applied to the known cumulant generating function for trallic with a single vehicle type (Kurze, 1974). ZBASICMODEL
In this note we consider t&c on an infinitely long highway and derive an approximation to the probability density function of the A-weighted squared sound pressure amplitude, or sound intensity which will be denoted by I. It will be assumed that the headways between successive vehicles on the road are independent random variables with probability density function p exp (- px) where p = number of cars/unit distance, and x = the distance measured along the highway. The approximation to be used here is that of choosing the appropriate one of the
Pearson family of distributions (Elderton and Johnson, .1971).The Pearson curve methodology uses the first four moments to choose an appropriate one of seven distributions which include the commonly used normal, beta, and gamma distributions. Let D be the distance of closest approach of an observer from the highway, and let g(Xi) be the squared sound pressure at the observation point from a car located at position xi along the road (the origin being at the point of closest approach). The total intensity is I = 2 g(xi) = Im g(X) dN(x) i--m
(I)
where dN(x) is the number of cars in (x,x +dx). The assumption of the negative exponential headway distribution allows us to write for the nth cumulant (Cox and Miller, 1%5), Kn = p
I-
00 g”(x)dx.
(2)
When excess attentuation can be neglected, g(x) is gt(x) = 91(x2+ D?, and when it cannot, gdx)= Q exp I- ad/(x2 + D?1/[x2 + 021. In these equations Q/D’ is the squared sound pressure without absoiption from a vehicle at the point of closest approach to the observer. It should be noted that we have assumed a negative exponential form for the excess attenuation. This function is not universally accepted because of a paucity of data and because ground effects can be important, but will be used, for mathematical convenience, as it is by other workers (see for example Sutherland, 1975,and the review article by Percy, Embleton and Sutherland, 1977). When there is a mixture of vehicles on the road one can adopt a suggestion by Marcus (1975). and assume that the headway
111
112
D. E.
BLUMENFELDand G. H. WESS
distribution is independent of vehicle type, and that successive vehicle types are independent random variables. In that case the parameter Q is taken to be a random variable, and the expression for K. in eqn (2) is to be averaged with respect to the distribution on the Q’s.
3.APPROXlMATlONFORUNlFORMTRAFFIC
In the case of uniform traffic (i.e. consisting of a single type of vehicle) we can consider the distribution of the normalized sound intensity I(x) = g(x)/g(O) from each vehicle, thereby eliminating the parameter Q. The resulting sound pressure level will be denoted by J, i.e. J=
2
J(Xi).
i--w
Using eqn (2) to calculate the moments, we have identified the appropriate Pearson distribution for a wide range of parameters, 0, p and a, and in each case the beta density is the one selected from the set Pearson curves. That is, if p(J) is the pdf of normalized sound intensity, the best Pearson fit to the moments has the form T(a+ +2) 1 P(J) = I.ta + ,& + 1) (,, _ ,o).+B+r(J -Jowl for jolJSl,,
sound intensities are not negligible increases and the beta distribution becomes a more diffuse function of J. Notice that since a is negative for 8 co.659 there can be two qualitatively different shapes to the curve of p(l) as a function of J, one of which goes to 0 as J + 0 and the other goes to infinity as J+O. However, it is known from the exact expression for the cumulant generating function corresponding to p(J) that it goes to zero at J = 0 like [0/(&*)] exp [- @*/(&)I (Weiss, 1970) so that neither eqn (3) or (4) can reproduce the small (J/Jo) behaviour of the pdf. This is quite reasonable since this portion of the curve contributes only a small amount to the moments. In Fig. 1 we show curves of p(J) and the comparable pdf, q(L), expressed in terms of decibels where L = IOlog,,J, for different values of 0. These illustrate the increased spread of the pdf as either D,the distance from the highway, or p, the traffic intensity, increases. The resulting curves are qualitatively reason-
- JIB (3)
= 0 otherwise. The parameters a, B, JOand I, can he calculated in terms of the tirst four moments. The parameter 114 was generally small. In the cases we ran when Jo was negative, implying an unsatisfactory physical model since J should always be non-negative, the value of l.Jd was of the order of 0.02 or less and lJ&J, was always less than 0.01. When JO was positive, for all cases with realistic parameters both JO and Jo/J1 were of the same orders of magnitude. Consequently we always set J,=O, for the single vehicle type case, allowing us to work with the simpler three parameter model,
Ua+B+2)
‘(‘) =T(a + I)rQ3
4
Fig. I(a).
6
Beta approximations to p(J) for 0 = 0.5, 1.X2.5.
1
0.16
II
for 0535/,,
--..___ .
J
‘(Ly(l_i)
+ 1) II
5
0.16
t 0.14
(4)
= 0 otherwise.
t a12
When there is only a single vehicle type on the road and excess attenuation can be neglected one can write the parameters a, j3 and Jr in terms of 0 = lrpD as
QKJ 0.10
I
a=(168*-60-3)/(108+3) /L?= (64fY2+ 308 + 3)/(loe + 3)
(5)
I, = (108 + 3)/2. Witbin the framework of the approximation of eqn (4). all three parameters a, j3, J, are increasing functions of 8 for B > 0. Hence as either the distance from the highway or the traffic density increases, the range over which the
Fig.
I(b). The pdf of noise intensity on a decibel scale for the same parameters.
Curveof acoustic noise from freely flowing t&k
able. When 6 is small the dominant contributions to p(l) are from low I; only vehicles nearest the observation point make an appreciable contribution. As 8 increases the sound from more vehicles are significant so that one gets more of an incoherent contribution. It is difficult to evaluate the accuracy of the beta approximation without calculating the exact probability density. However, one comparison that can be made without too great difficulty is that between the moments given by eqns (3) or (4) and the moments calculated from the exact cumulants. Such a moment comparison for eqn (4) is made for several values of g, in Table 1. Since the first three moments agree exactly, these are omitted. The pattern of relative error is the same for each value of 8, being less than 10% in absolute magnitude for the fourth through the sixth moment and increasing thereafter. An increase in B favours the accuracy of the beta approximation. A theoretically improved approximation can be made by expanding p(J) in terms of the polynomials orthogonal to the ,distribution in eqn (3) (which happen to be the Jacobi polynomials, (Abramowitz and Stegun)). These force agreement with successive moments, i.e. if we use the first Jacobi polynomial the moments or order l-4 will agree exactly, but then agreement with higher moments worsens and ~the approximation to p(/) does not remain positive for all /. Hence we recommend only the use of the simplest beta approximation. Table I. Relative errors in higher moments calculated from the three parameterbeta density for differentvalues of 0 Moment order 4 5 6 7 8 9 IO
6 0.4 -
0.084 0.047 0.054 0.227 0.350 0.477 0.682
I.0
1.6
0.054 - 0.025 - 0.046 -0.118 -0.190 - 0.275 - 0.380
0.037 -0.013 - 0.029 -0.066 -0.110 -0.166 - 0.233
2.0 0.029 - 0.009 - 0.022 - 0.047 - o.lMO -0.123 -0.174
4. APPItoxnUrIoN FoR IutxD TRMFlc It is also possible to take into account a mixture of vehicle types provided that two assumptions are valid: (1) headways are independent of vehicle type, and (2) the different vehicle types occur independently on the road. Let us assume that each vehicle type has exponentially distributed headways, and that there are n vehicle types, a given vehicle being of type i, with probability pi. The total sound pressure I can be written as the sum of contributions, I, from each vehicle type. Since these are assumed independent, the cumulants of the ZI are additive and we therefore have for the rth cumtdant K,(l) = $, Kr(&)= /J $, pi j-- gr’(x) dz.
113
In this case also it is convenient to define dimensionless intensities A = 1,/f where 1 is
(8) When these are introduced the cumulantSare given by where
c
= z p,Q,
(9)
differing from the cumulants for the single vehicle type case only by the factor (Q’/Q’). We have calculated the best Pearson curve fit for the n = 2 case over a wide range of QJQ,, (2-500), and p’s, and in each case the appropriate Pearson distribution was found to be the beta, although it is necessary to use the four parameter density given in eqn (3). In one or two cases the beta distribution of the second kind was found to be the best fitting Pearson curve. Table 2 gives some Table 2. Relative errors in using a four parameter beta approximation to the pdf for a mixture of vehicle types (n = 2,
6 = I, p = proportion of noisier vehicles) QJQ, = 2
Moment order
p = 0.0
p = 0.05
p = 0.10
p = 0.20
5 6 7 8 9 10
- 0.0% - 0.004 - 0.050 - 0.091 -0.146 - 0.228
- 0.005 - 0.002 - 0.044 -0.081 -0.130 -0.201
- 0.004 -0.001 - 0.039 - 0.074 -0.119 -0.185
- 0.004 - 0.002 - 0.041 - 0.078 -0.127 -0.195
5 6 7 8 9 10
- 0.006 - 0.004 - 0.050 -0.091 -0.146 - 0.228
G/Q, = to 0.022 - 0.254 0.053 0.012 - o.on - 0.200
- 0.008 - 0.335 -0.132 - 0.253 - 0.393 - 0.533
- 0.020 - 0.250 -0.173 - 0.2% - 0.430 - 0.559
typical values of the relative error using the four parameter beta approximation for QJQ, = 2 and 10 and different values of p2= p, the proportion of noisier vehicles. For &/Q, = 2. the relative errors are not a sensitive function of p. but at the higher value of QJQ, the fit for larger p is not very good. 5. EXCESS ATrRNuATloN
*
Finally, we consider the situation in which excess attenuation occurs, allowing one vehicle type. In this case the expression for the nth cumulant of the normalized sound pressure (J = D21 e&/Q) is
(6)
In particular, when excess attenuation can be neglected
Techniques for evaluating the integral are discussed by Blumenfeld and Weiss (1975a). A comparison of exact moments as calculated from this last equation and those
D. E. BLUMENFELD and G. H. WEISS
114
Table 3. Relative errors from a three parameter beta distribution approximation when excess attenuation is important, a= 0.001 ft-’
Moment order
p(ft_‘) = l/l056
D= 1OOft p(ft--‘) = p(ftt’) = l/528 l/352
p(ft_‘) = l/264
4 5 6 7 8
- 0.031 -0.115 -0.159 - 0.400 - oss2
- 0.024 - 0.086 -0.139 - 0.306 - 0.437
-0.017 -0.061 -0.109 - 0.229 - 0.339
- 0.012 -0.044 - 0.085 - 0.174 - 0.266
4 5 6 7 8
- 0.030 -0.101 -0.165 -0.341 - 0.477
D=2OOft - 0.015 - 0.052 - 0.099 -0.198 - 0.2%
- 0.008 - 0.030 - 0.062 -0.123 -0.193
- 0.005 -0.018 -0.041 - 0.081 -0.132
derived from a three parameter fit by a beta density, is made in Table 3. The numbers are presented as a function of the density of cars, p, and of D, for a, the attenuation parameter, equal to 0.001 ft-‘. To give some feeling for the meaning of the values of p. if trathc moves at a uniform speed of 50 mph the p = 1/1056ft-’ corresponds to a flow of 250 cars/hr and p = 1/264ft-’ corresponds to a flow of 1000 cars/hr. We see from Table 3 that also in this case the beta approximation is increasingly accurate as the flow increases and as the distance from the highway increases, i.e. as the sound becomes more incoherent. 6. CONCWSIONS
We have therefore found that the beta approximation to the pdf of sound pressure is useful for a variety of models of trafiic flow. The principal limitation of this study must be borne in mind-our criterion of error was chosen to be the relative error of the higher moments predicted by the beta approximation. Since the relative error increases’ with increasing order of moments we infer that the tail of the density is not accurately reproduced and on theoretical grounds we know that the small I portion of the density function cannot have a beta form. Since the predicted higher moments are not totally incorrect as they are given by the beta approximation we feel that the central part of the pdf is described accurately, and that this part is of greatest interest. This conclusion can only be verified completely by comparison with a numerical inversion of the characteristic function. However, the lower cumulants are sensitive to the maximum of unimodal pdf. Since we have fitted these exactly we feel that justified in claiming good reproduction of that region of the pdf. We have
tried to add higher corrections by expanding the pdf in terms of the Jacobi polynomials that are orthogonal to the beta function, but these caused the higher moments not used to fix parameters to diverge more from the exactly calculated moments than those from the uncorrected beta approximation. We have also tried fitting the pdf of sound pressure to a function of the form
suggested’by Kurze (1974), but his results when used to reproduce higher moments calculated theoretically, were not as good as those produced by the fit by a beta density. This is not too surprising since Kurze’s approximation uses two parameters whereas we use three or four. Although we have achieved some degree of success with the use of the Pearson methodology, we still feel that a more efficient use of the exactly calculated moments represents a challenge in this research area. Acknowledgemenr-We are grateful to the referee for his thoroughand careful review of the first draft of this paper.
REFERENCES
Abramowitz M. and Stegun I. A. (1964) Handbook of Marhemotical Functions, p. 774. U.S. Government Printing Office, Washington,D.C. Blumenfeld D. E. and Weiss G. H. (1975a) Attenuation effects in the propagationof traffic noise. Transp. Res. 9, 103-106. Blumenfeld D. E. and Weiss G. H. (1975b) Effects of headwav distributions on second order properties of traffic noise. ;. Sound and Vib. 41.93-102. Cox D. R. and Miller H. D.
(1%5) The
Theory
of Stochastic
Processes,p. 347. Methuen, London. Daniels H. E. (1954) Saddle point approximations in statistics. Ann. Math. Stat. 25.631-650.
Elderton W. P. and Johnson N. L. (1%9) Systems of Frequency Curves. Cambridge University Press, London. Kurze U. J. (1971a) Statistics of road traffic noise. 1. Sound and Vib. 18, 171-195. Kurze U. J. (197lb) Noise from complex road traffic. 1. Sound and Vib. 17, 167-177.
Kurze U. J. (1974) Frequency curves of road traffic noise. J. Sound and Vib. 33, 171-185. Marcus A. H. (1973) Traffic noise as a filtered Markov renewal model. 1. Appl. Prob. 10,377-386.
Marcus A. H. (1975) Some exact distributions in traffic noise theory. Adv. Appl. Prob. 7,593-606. Percy J. E., Embleton, T. F. W. and Sutherland L. C. (1977) Review of noise propagation in the atmosphere. J. Acousr. Sot. Am. 61, 1403-1418. SutherlandL. C. (1975)Ambient noise level above a plane with a continuousdistribution of noise sources. /. Acoust. Sot. Am. 57, 1540-1542.
Weiss G. H. (1970) On the noise generated by a stream of vehicles. Transp. Res. 4, 22%233.
Vol. I?. pp. I I I-I II.
Pergamoa Press 1978. Printed in Grca~hilain
CURVE FITTING THE PROBABILITY DISTRIBUTION OF ACOUSTIC NOISE FROM FREELY FLOW-ING TRAFFIC DENNISE. BLUMENFELD Transport StudiesGroup. University College London, London WClE 6BT. England and
GEORGEH. WEISS National Institutesof Health, Bethesda,MD 20014. U.S.A. (Received
22 February
1977; in revised form
10 September
1977; received for publication
8 November
1977)
Abstrati-The theory of sound intensity or acoustic noise from freely flowing trafBc has been investigatedby severalauthors.It is possibleto calculatethe cumulantsof soundintensityexactly for the commonlyused model as well as the characteristic function,but an exact inversionto obtain the probability density function does not appear possible.Several approximationsto the full distributionof soundintensityhave been proposed. We have applied the Pearson curve methodologyto determine the full pdf and find that the best fit is given by a beta distribution. This is true with and without excessattenuationdue to atmosphericor groundabsorption and when a mixture of vehicle tvnes -. is .nresent. The exact and annroximatedistributionswere compared indirectly by comparinghigher momentsnot usedto fix parameters. ..
1.INTRODUCTlON A theory of acoustic noise from freely flowing traffic on an infinitely long road has been developed recently by several authors, (Weiss, 1970; Kurze, 197la, b, 1974; Marcus, 1973, 1975; Blumenfeld and Weiss, 1975a,b). In the simplest case in which headways between successive vehicles are independent, identically distributed, random variables with a negative exponential distribution, expressions can be given for the cumulants to all orders analogous to the theory of shot noise, (Cox and Miller, 1%5). From these one can find the Laplace transform of the probability density of the acoustic noise exactly. Unfortunately it is very difficult to invert these transforms numerically because the inversion involves the calculation of Bessel functions with complex arguments (Weiss, 1970). Hence simple approximations that make use of moment information would seem to be of considerable interest. Kurze (1971a) has used a GramCharlier expansion to derive approximations to the probability density function of noise from infinite stretches of road. He has also suggested approximations based on Daniels’s saddle point method (Daniels, 1954) applied to the known cumulant generating function for trallic with a single vehicle type (Kurze, 1974). ZBASICMODEL
In this note we consider t&c on an infinitely long highway and derive an approximation to the probability density function of the A-weighted squared sound pressure amplitude, or sound intensity which will be denoted by I. It will be assumed that the headways between successive vehicles on the road are independent random variables with probability density function p exp (- px) where p = number of cars/unit distance, and x = the distance measured along the highway. The approximation to be used here is that of choosing the appropriate one of the
Pearson family of distributions (Elderton and Johnson, .1971).The Pearson curve methodology uses the first four moments to choose an appropriate one of seven distributions which include the commonly used normal, beta, and gamma distributions. Let D be the distance of closest approach of an observer from the highway, and let g(Xi) be the squared sound pressure at the observation point from a car located at position xi along the road (the origin being at the point of closest approach). The total intensity is I = 2 g(xi) = Im g(X) dN(x) i--m
(I)
where dN(x) is the number of cars in (x,x +dx). The assumption of the negative exponential headway distribution allows us to write for the nth cumulant (Cox and Miller, 1%5), Kn = p
I-
00 g”(x)dx.
(2)
When excess attentuation can be neglected, g(x) is gt(x) = 91(x2+ D?, and when it cannot, gdx)= Q exp I- ad/(x2 + D?1/[x2 + 021. In these equations Q/D’ is the squared sound pressure without absoiption from a vehicle at the point of closest approach to the observer. It should be noted that we have assumed a negative exponential form for the excess attenuation. This function is not universally accepted because of a paucity of data and because ground effects can be important, but will be used, for mathematical convenience, as it is by other workers (see for example Sutherland, 1975,and the review article by Percy, Embleton and Sutherland, 1977). When there is a mixture of vehicles on the road one can adopt a suggestion by Marcus (1975). and assume that the headway
111
112
D. E.
BLUMENFELDand G. H. WESS
distribution is independent of vehicle type, and that successive vehicle types are independent random variables. In that case the parameter Q is taken to be a random variable, and the expression for K. in eqn (2) is to be averaged with respect to the distribution on the Q’s.
3.APPROXlMATlONFORUNlFORMTRAFFIC
In the case of uniform traffic (i.e. consisting of a single type of vehicle) we can consider the distribution of the normalized sound intensity I(x) = g(x)/g(O) from each vehicle, thereby eliminating the parameter Q. The resulting sound pressure level will be denoted by J, i.e. J=
2
J(Xi).
i--w
Using eqn (2) to calculate the moments, we have identified the appropriate Pearson distribution for a wide range of parameters, 0, p and a, and in each case the beta density is the one selected from the set Pearson curves. That is, if p(J) is the pdf of normalized sound intensity, the best Pearson fit to the moments has the form T(a+ +2) 1 P(J) = I.ta + ,& + 1) (,, _ ,o).+B+r(J -Jowl for jolJSl,,
sound intensities are not negligible increases and the beta distribution becomes a more diffuse function of J. Notice that since a is negative for 8 co.659 there can be two qualitatively different shapes to the curve of p(l) as a function of J, one of which goes to 0 as J + 0 and the other goes to infinity as J+O. However, it is known from the exact expression for the cumulant generating function corresponding to p(J) that it goes to zero at J = 0 like [0/(&*)] exp [- @*/(&)I (Weiss, 1970) so that neither eqn (3) or (4) can reproduce the small (J/Jo) behaviour of the pdf. This is quite reasonable since this portion of the curve contributes only a small amount to the moments. In Fig. 1 we show curves of p(J) and the comparable pdf, q(L), expressed in terms of decibels where L = IOlog,,J, for different values of 0. These illustrate the increased spread of the pdf as either D,the distance from the highway, or p, the traffic intensity, increases. The resulting curves are qualitatively reason-
- JIB (3)
= 0 otherwise. The parameters a, B, JOand I, can he calculated in terms of the tirst four moments. The parameter 114 was generally small. In the cases we ran when Jo was negative, implying an unsatisfactory physical model since J should always be non-negative, the value of l.Jd was of the order of 0.02 or less and lJ&J, was always less than 0.01. When JO was positive, for all cases with realistic parameters both JO and Jo/J1 were of the same orders of magnitude. Consequently we always set J,=O, for the single vehicle type case, allowing us to work with the simpler three parameter model,
Ua+B+2)
‘(‘) =T(a + I)rQ3
4
Fig. I(a).
6
Beta approximations to p(J) for 0 = 0.5, 1.X2.5.
1
0.16
II
for 0535/,,
--..___ .
J
‘(Ly(l_i)
+ 1) II
5
0.16
t 0.14
(4)
= 0 otherwise.
t a12
When there is only a single vehicle type on the road and excess attenuation can be neglected one can write the parameters a, j3 and Jr in terms of 0 = lrpD as
QKJ 0.10
I
a=(168*-60-3)/(108+3) /L?= (64fY2+ 308 + 3)/(loe + 3)
(5)
I, = (108 + 3)/2. Witbin the framework of the approximation of eqn (4). all three parameters a, j3, J, are increasing functions of 8 for B > 0. Hence as either the distance from the highway or the traffic density increases, the range over which the
Fig.
I(b). The pdf of noise intensity on a decibel scale for the same parameters.
Curveof acoustic noise from freely flowing t&k
able. When 6 is small the dominant contributions to p(l) are from low I; only vehicles nearest the observation point make an appreciable contribution. As 8 increases the sound from more vehicles are significant so that one gets more of an incoherent contribution. It is difficult to evaluate the accuracy of the beta approximation without calculating the exact probability density. However, one comparison that can be made without too great difficulty is that between the moments given by eqns (3) or (4) and the moments calculated from the exact cumulants. Such a moment comparison for eqn (4) is made for several values of g, in Table 1. Since the first three moments agree exactly, these are omitted. The pattern of relative error is the same for each value of 8, being less than 10% in absolute magnitude for the fourth through the sixth moment and increasing thereafter. An increase in B favours the accuracy of the beta approximation. A theoretically improved approximation can be made by expanding p(J) in terms of the polynomials orthogonal to the ,distribution in eqn (3) (which happen to be the Jacobi polynomials, (Abramowitz and Stegun)). These force agreement with successive moments, i.e. if we use the first Jacobi polynomial the moments or order l-4 will agree exactly, but then agreement with higher moments worsens and ~the approximation to p(/) does not remain positive for all /. Hence we recommend only the use of the simplest beta approximation. Table I. Relative errors in higher moments calculated from the three parameterbeta density for differentvalues of 0 Moment order 4 5 6 7 8 9 IO
6 0.4 -
0.084 0.047 0.054 0.227 0.350 0.477 0.682
I.0
1.6
0.054 - 0.025 - 0.046 -0.118 -0.190 - 0.275 - 0.380
0.037 -0.013 - 0.029 -0.066 -0.110 -0.166 - 0.233
2.0 0.029 - 0.009 - 0.022 - 0.047 - o.lMO -0.123 -0.174
4. APPItoxnUrIoN FoR IutxD TRMFlc It is also possible to take into account a mixture of vehicle types provided that two assumptions are valid: (1) headways are independent of vehicle type, and (2) the different vehicle types occur independently on the road. Let us assume that each vehicle type has exponentially distributed headways, and that there are n vehicle types, a given vehicle being of type i, with probability pi. The total sound pressure I can be written as the sum of contributions, I, from each vehicle type. Since these are assumed independent, the cumulants of the ZI are additive and we therefore have for the rth cumtdant K,(l) = $, Kr(&)= /J $, pi j-- gr’(x) dz.
113
In this case also it is convenient to define dimensionless intensities A = 1,/f where 1 is
(8) When these are introduced the cumulantSare given by where
c
= z p,Q,
(9)
differing from the cumulants for the single vehicle type case only by the factor (Q’/Q’). We have calculated the best Pearson curve fit for the n = 2 case over a wide range of QJQ,, (2-500), and p’s, and in each case the appropriate Pearson distribution was found to be the beta, although it is necessary to use the four parameter density given in eqn (3). In one or two cases the beta distribution of the second kind was found to be the best fitting Pearson curve. Table 2 gives some Table 2. Relative errors in using a four parameter beta approximation to the pdf for a mixture of vehicle types (n = 2,
6 = I, p = proportion of noisier vehicles) QJQ, = 2
Moment order
p = 0.0
p = 0.05
p = 0.10
p = 0.20
5 6 7 8 9 10
- 0.0% - 0.004 - 0.050 - 0.091 -0.146 - 0.228
- 0.005 - 0.002 - 0.044 -0.081 -0.130 -0.201
- 0.004 -0.001 - 0.039 - 0.074 -0.119 -0.185
- 0.004 - 0.002 - 0.041 - 0.078 -0.127 -0.195
5 6 7 8 9 10
- 0.006 - 0.004 - 0.050 -0.091 -0.146 - 0.228
G/Q, = to 0.022 - 0.254 0.053 0.012 - o.on - 0.200
- 0.008 - 0.335 -0.132 - 0.253 - 0.393 - 0.533
- 0.020 - 0.250 -0.173 - 0.2% - 0.430 - 0.559
typical values of the relative error using the four parameter beta approximation for QJQ, = 2 and 10 and different values of p2= p, the proportion of noisier vehicles. For &/Q, = 2. the relative errors are not a sensitive function of p. but at the higher value of QJQ, the fit for larger p is not very good. 5. EXCESS ATrRNuATloN
*
Finally, we consider the situation in which excess attenuation occurs, allowing one vehicle type. In this case the expression for the nth cumulant of the normalized sound pressure (J = D21 e&/Q) is
(6)
In particular, when excess attenuation can be neglected
Techniques for evaluating the integral are discussed by Blumenfeld and Weiss (1975a). A comparison of exact moments as calculated from this last equation and those
D. E. BLUMENFELD and G. H. WEISS
114
Table 3. Relative errors from a three parameter beta distribution approximation when excess attenuation is important, a= 0.001 ft-’
Moment order
p(ft_‘) = l/l056
D= 1OOft p(ft--‘) = p(ftt’) = l/528 l/352
p(ft_‘) = l/264
4 5 6 7 8
- 0.031 -0.115 -0.159 - 0.400 - oss2
- 0.024 - 0.086 -0.139 - 0.306 - 0.437
-0.017 -0.061 -0.109 - 0.229 - 0.339
- 0.012 -0.044 - 0.085 - 0.174 - 0.266
4 5 6 7 8
- 0.030 -0.101 -0.165 -0.341 - 0.477
D=2OOft - 0.015 - 0.052 - 0.099 -0.198 - 0.2%
- 0.008 - 0.030 - 0.062 -0.123 -0.193
- 0.005 -0.018 -0.041 - 0.081 -0.132
derived from a three parameter fit by a beta density, is made in Table 3. The numbers are presented as a function of the density of cars, p, and of D, for a, the attenuation parameter, equal to 0.001 ft-‘. To give some feeling for the meaning of the values of p. if trathc moves at a uniform speed of 50 mph the p = 1/1056ft-’ corresponds to a flow of 250 cars/hr and p = 1/264ft-’ corresponds to a flow of 1000 cars/hr. We see from Table 3 that also in this case the beta approximation is increasingly accurate as the flow increases and as the distance from the highway increases, i.e. as the sound becomes more incoherent. 6. CONCWSIONS
We have therefore found that the beta approximation to the pdf of sound pressure is useful for a variety of models of trafiic flow. The principal limitation of this study must be borne in mind-our criterion of error was chosen to be the relative error of the higher moments predicted by the beta approximation. Since the relative error increases’ with increasing order of moments we infer that the tail of the density is not accurately reproduced and on theoretical grounds we know that the small I portion of the density function cannot have a beta form. Since the predicted higher moments are not totally incorrect as they are given by the beta approximation we feel that the central part of the pdf is described accurately, and that this part is of greatest interest. This conclusion can only be verified completely by comparison with a numerical inversion of the characteristic function. However, the lower cumulants are sensitive to the maximum of unimodal pdf. Since we have fitted these exactly we feel that justified in claiming good reproduction of that region of the pdf. We have
tried to add higher corrections by expanding the pdf in terms of the Jacobi polynomials that are orthogonal to the beta function, but these caused the higher moments not used to fix parameters to diverge more from the exactly calculated moments than those from the uncorrected beta approximation. We have also tried fitting the pdf of sound pressure to a function of the form
suggested’by Kurze (1974), but his results when used to reproduce higher moments calculated theoretically, were not as good as those produced by the fit by a beta density. This is not too surprising since Kurze’s approximation uses two parameters whereas we use three or four. Although we have achieved some degree of success with the use of the Pearson methodology, we still feel that a more efficient use of the exactly calculated moments represents a challenge in this research area. Acknowledgemenr-We are grateful to the referee for his thoroughand careful review of the first draft of this paper.
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