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A statistical model of traffic noise
175
A STATISTICAL M O D E L OF T R A F F I C NOISE J. FoxoN* and F. J. PEAr,SON
Department of Applied Physics, Lanchester College of Technology, Covento" (Great Britain) (Received: 23 April, 1968)
SUMMARY
Traffic noise recordings have been analysed and presented in the form of graphs of the percentage of time during which the noise remains between certain levels against the levels involved. It has been established experimentally that observations presented in this form approach a Gaussian distribution. To account for this Gaussian distribution a statistical model of traffic noise has been evolved and used with reasonable success as a prediction mechanism to evaluate another random noise situation similar to that of traffic noise. The model also enables experimental data relating to noise surveys to be characterized by a limited number of parameters capable of describing a particular random noise situation in a physically meaningful manner. The article includes an account o f the various practical techniques used in the statistical analysis of traffic noise and a discussion of the extent to which the theoretical model is verified experimentally.
INTRODUCTION
It is significant that a number of surveys of traffic noise contain evidence that the proportion of time the noise remains between certain levels is normally distributed in terms of noise levels. This suggests that a statistical model of traffic noise might be developed which could account for this distribution in a physically meaningful manner. The object of the present research is to evaluate recordings of typical traffic noise situations in Coventry in terms of a putative normal distribution and to present a theory which accounts for the observed effects. * This work formed part of the author's thesis for the degree of M.Sc. of the University of Bristol in 1967. Applied Acoustics--Elsevier Publishing Company Ltd., England--Printed in Great Britain
176
J. FOXON, F. J. PEARSON PREVIOUS NOISE SURVEYS
A survey of city noise in the area around Chicago in the United States of America was initiated in 1947.1 The survey included noise measurements of traffic, industrial and residential noise. In addition to overall noise levels, a spectral analysis into octave bands was performed, the data being presented in the form of graphs of octave band level against octave bands. It was found in this survey that a plot of the levels in the 400-800 Hz band for 5-dB ranges approached a normal distribution. Similar distributions were obtained from the results of noise surveys in West Germany notably that by F. G. Meister in 1953. 2 A noise survey was conducted in New Jersey in 19643 and modal values of the background noise levels were measured. When the data were converted into a statistical frequency distribution a normal curve was obtained. An obvious extension of the above approaches was therefore undertaken in the course of a similar survey of traffic noise in the City of Coventry.
APPARATUS AND TECHNIQUES
The recording apparatus used for this purpose consisted of a portable tape recorder (E.M.I. Type RE 321) and a moving coil microphone (S.T.C. Type 4021 J), which, together with a suitable wind shield and an extension lead, were mounted on a small trolley for easy transportation to the recording site as shown in Fig. 1.
Fig. I.
STATISTICAL
~ "~ ~.__.
~..~
"a s
° "- o :
/ '
]ffgOTOtiiiOL
Applied Acoustics, I (1968) 175-188
NOISE
i ~ . s
]77
]
.p:
sqloll ~,.,
" ,i,a
'('--
OF TRAFFIC
°-~
.-.,,,
[
MODEL
~
'
/l
178
J. F O X O N , F. J. P E A R S O N
The recordings were replayed on a studio instrument (E.M.I. Type TR 52) whose output could be connected to either a High Speed Level Recorder (Dawe Type 1406 D) to obtain a trace variation in peak level or a Statistical Analyser (Dawe Type 1466 A) which indicated the number of times a certain level had been exceeded out of a certain total number of samples. When using the above equipment to record and analyse traffic noise a prerecorded white noise signal of 90 dBC was used for calibration purposes. The traffic noise was then recorded using the portable apparatus, the trolley being positioned so that the microphone was pointing in a direction normal to the main traffic flow. The site chosen for all the traffic noise recordings relating to the present considerations is illustrated on the map in Fig. 2. It consisted of a main thoroughfare (Jordan Well), together with two intersections. The nearest intersection to the recording position was a road which was only an inlet to the main thoroughfare (Bayley Lane) and the other intersection was a road used only as an outlet from the main thoroughfare (Much Park Street). The position was chosen since it represented many of the possible situations encountered in city traffic flow, i.e. freely-flowing traffic, moving at speeds up to 30 m.p.h., traffic slowing down or being brought to a halt and then pulling into or leaving the main traffic stream. Having chosen the recording position, recordings were made at various times of the day and for varying periods of time. No recordings were made at times when the traffic situation was unusually quiet (early morning or late evening) or when it was unusually noisy (morning and evening rush-hour periods). While each recording was timed using a stopwatch, the actual recording times cannot be regarded as being accurate to this standard since some allowance must be made for tape slack. After recording, the tapes were replayed on the studio recorder with the output of the tape recorder connected directly to either the High Speed Level Recorder or the Statistical Analyser. With the motor of the High Speed Level Recorder disengaged the output of the tape recorder was adjusted, using the calibration signal, so that 90 dB on the chart was represented by a line about 20 dB from the upper limit of the stylus movement. This allowed for a maximum sound level of 110 dB to be recorded and a minimum sound level of 60 dB. The chart speed chosen was 0.1 cm sec- 1 so that the longest continuous recording of about 20 min could be accommodated on a chart length of 130 cm allowing for the calibration signal and annotation. When the tape recorder output was connected to the Statistical Analyser, a valve voltmeter was interposed to monitor the voltage input to the analyser terminals because the analyser was used on a particular counting range which required input signals not to be greater than 1 volt. The setting of the tape recorder gain was, however, easily arranged to be at a suitable level after inspection of the Level Recorder trace.
179
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE PRESENTATION
OF D A T A
One of the objectives of the present work was to develop a statistical model of traffic noise which could account for a normal distribution of the proportion of the time the noise level lay between certain levels. By using the Statistical Analyser in the manner previously described the number of occasions, n, on which a certain level is exceeded during the sampling period may be obtained. The total number of samples, N, is related to the total time of the recording since a sampling rate of 0.5 samples/sec was used throughout all the present investigations. Each complete recording was divided into different periods of time from ~ min to 1043-min and thus an appropriate value of N emerged from each section of each recording. When n was expressed as a proportion of the total number of samples and plotted against the noise level a curve was produced which was of the general form of the function F(x) against x (dB) illustrated in Fig. 3(a). If the proportion f(x) of the time during which the noise level lies between two levels x and x + dx is plotted against level the curve of Fig. 3(b) is obtained, essentially by differentiating the F(x) curve. It is thisf(x) curve of apparently Gaussian form which has led to the development of the traffic noise model described later.
A ×
x b.
x
(aB)
x(dB)
Fig. 3(a).
Fig. 3(b).
STATISTICAL ANALYSIS
Sampling of the various recordings was accomplished by opening a gate in the statistical analyser at regular intervals. Consider a recording time of T sec which is sampled N times and assume that the level exceeds a certain value for a time t sec (t ~< T). The gate registers only one count regardless of the number of pulses of sound above the certain level which occurs during the period for which it is open. Applied Acoustics, 1 (1968) 175-188
180
J. FOXON, F. J. PEARSON
If n is the average number of counts o f a certain level it is easily shown from a theoretical consideration of the operation of the statistical analyser that the points on the F(x) curves are subject to a percentage error given by
--
x loo%.
_--_
100 R 20.9.66 I0 M mu1'es 90
80
70
60
o
5o
"5 g40 <
50
\
20
I0
I 75
I
80
I 90
85 Level
Fig. 4.
dB
i I_~-~--95
1 I00
S T A T I S T I C A L MODEL OF T R A F F I C NOISE
181
An example is given in Fig. 4 of a typical F ( x ) curve with the estimated errors indicated in an obvious manner for each experimental observation. We also note that in the present investigation 2N=
T.
T R A F F I C NOISE MODEL
Consider a traffic situation where on average there is a certain number of vehicles within earshot, all emitting noise. Let this average number of vehicles be called the " p o o l " of emitters and let the pool contain a range of different types of emitter, i.e. cars, lorries, buses, etc. Individual emitters of the various types join or leave the pool so that the overall noise level increases or decreases in discrete random steps. Let r = no. of emitters arriving and leaving in a specified time, i.e. the total no. of vehicle movements during the specified time. Also let r = n~ + n 2 when n2 = no. of arrivals and nl = no. of departures. Thus the increase in the number of sources
~--- /12 - - n l "
If the intensity I being emitted by the pool when the change nz - n~ occurs is large, it implies that the overall situation is noisy and the change n2 - n~ is likely to be due to a number of, for example, heavy lorries. Conversely, if I is small, n2 - n~ will be a number of quiet vehicles. Let A ! be the change in intensity produced by each of the vehicle movements and assume that at the time of these vehicle movements AI is proportional to I so that the changes in intensity actually occurring reflect the character of the existing noise situation at the time in the manner outlined above, i.e.
A I = ~1
where ct = a constant. I f Io = intensity of the virgin pool before any change occurs, it is easily shown that the resulting intensity after n2 arrivals is given by (1 + Ct)n~Io. When considering departures it must be borne in mind that the above initial intensities have been reckoned at the left of the interval. Thus when a source departs the intensity decreases by an amount proportional to what the intensity will be when it has left. On this basis it follows that the intensity after nl departures is thus = Io/(1 + ~)nl = / o ( 1 + 5) -/11. So intensity after n2 - nl changes = (1 + ct)¢n~-"lllo = fllo. Applied Acoustics, 1 (1968) 175-188
182
J. FOXON, F. J. PEARSON
The resulting change in sound pressure level arising from the above intensity changes may now be calculated using the relation I = Io eux, where x is in decibels measured from the mean (where I = I0) and /~ = 1/10 log e 10. N o w if I = f l ( " - ' - " l ) I o then x can be regarded as changing by n, - n~ equal steps of Ax from the mean. i.e.
therefore
I = fl("~ - "~ )I o = Io elL(''-' - "~ ) ~ ' , (/72
-
n l ) I n fl
so
=
//(/'/2
--
nl)Ax
Ax = In fl/la.
Hence the/12 - - /71 source changes (each producing equal percentage changes in the existing intensity) will produce a change from the mean level in decibels of (n2 - n l) Ax. Assuming that arrivals and departures are random, the probability of/72 arrivals and nl departures is given by the appropriate coefficient in the expansion o f (p + q)" where p = q = ½, assuming equal probabilities of arrival and departure. If these probabilities were unequal then the size of the pool would increase or decrease with time, but we do not wish to consider in detail at this stage the problem of time-varying mean levels. The probability of a change ( n 2 - - /71)Ax in decibels around a constant mean is therefore
(nl + n2)'(1) ("~+nO nl! n2! from which a Gaussian distribution is easily derived in the f o r m f ( x ) = A e - ~ ' - ~ " ) ~ where A and a are constants, xo is the mean level and a =
(1)
1 / 2 r ( A x ) 2.
F r o m the above theory nl and n 2 are the numbers of departures and arrivals of different sources of different strengths to and from the pool. Consider the case o f r = 2 where the following arrangement o f arrivals to and departures from the pool (of unspecified size) is possible. Binomial expansion n2
ii I
2 arrive
pool 0 leave
1 arrives first
pool
1 arrives second pool 0 arrive
term
p2
1 leaves second
pq
1 leaves first
qp
pool 2 leave
q2
S T A T I S T I C A L M O D E L O F T R A F F I C NOISE
183
The 2pq term is interpreted as two distinguishable situations or configurations in which the arrivals and departures occur in a different order. Thus it is suggested that the four situations outlined above correspond to four different distinguishable traffic configurations and that these different configurations occur at regular intervals of time on the average. Furthermore, it is well known that the sum of the above binomial coefficients is 2" which can be shown to be related in the present context to the total number N of samples taken throughout a period of observation. A relationship of the form
2" = N/k
(2)
is therefore suggested, where k is a scale factor. Equation (2) may be justified from the fact that the probability of obtaining the maximum deviation +rAx is (½)'. This can be regarded, by definition, as that particular pair of levels on either extreme of the mean which is observed the least out of 2' observations. Thus k may be regarded as the number of counts out of N for which the maximum and minimum levels are observed. Also, 2 r being the total number of distinguishable traffÉc configurations of the type discussed earlier, and also being equal to N/k according to eqn. (2), it follows that the average duration of one of these traffic situations will be given by T/Number of situations = 2k in the present investigation where the sampling rate is 0-5 sec- 1. This gives a simple physical meaning to the scale factor k as being directly proportional to the average duration of each distinguishable situation. It should perhaps be emphasized at this juncture that one of the essential features of the theory is the assumption that it is these distinguishable configurations which occur on the average at regular intervals of time and not necessarily the individual arrivals and departures. Whence, from eqns. (1) and (2), it follows that In N = In k + (In 2)/2a(Ax) 2.
(3)
A graph of In N against I/a should therefore be linear and yield values of k and Ax appropriate to the particular traffic situation observed for different lengths of time, thus leading to a confirmation of the theory.
EXPERIMENTAL
RESULTS
Values of In N and 1/a taken from a recording made on 20th September, 1966 are shown plotted in Fig. 5. This graph demonstrates the linear relationship between In N and 1/a. In this experimental verification of eqn. (3), the values of a were determined directly from the F(x) curves by a variety of methods, perhaps the Applied Acoustics, 1 (1968) 175--188
184
J. FOXON, F. J. PEARSON
R 20"9"66 In N,v.I/a. ,8 = 0 ' 0 7 3 ~ = 2"42
e
~
5
×
~
×
4 Z c
×
0
L I0
I 20
I 30
I 40
I 50
~o Fig. 5.
simplest being that illustrated in Fig. 6, where a curve of the F(x) type shows certain parameters do and D. A tangent, drawn to touch the curve at a point corresponding to the mean level Xo, for which F(x) = ½, when produced, intersects the x-axis at x o 4- do. The value of the tangent at the point is the maximum of the correspondingf(x) curve and can be shown to be f(X)max =
x/a/rr.
By inspection of Fig. 6 it follows that the tangent at x o is given by x o = 1/2do = f(X)m, x whence d o =
½x/n/a.
A value of a for each section of a particular recording can thus be obtained from the F(x) curves and used to plot a graph of In N against 1/a according to eqn. (3). From this graph values of k and Ax for the particular traffic situation can be evaluated from the gradient and intercept. A number of recordings have been made and analysed according to the methods just described, the values of k and Ax emerging as approximately 10 and 2 dB respectively for the particular traffic situations considered.
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE
185
\\
~t t.L
Fig. 6. The results show that values of k and Ax could be deduced in a relatively simple manner from the F(x) curves. Since the theoretical model was developed in terms of noise emitters without specific reference to the precise nature of the emitters it should also be possible to apply the model to analogous situations involving different types of emitter, and to predict the type of results that might be obtained. F r o m a consideration of the theory it can also be seen that increasing r (and hence N and the duration T of the analysis) increases the half-width of the corresponding normal distribution, a feature that was confirmed experimentally from an examination of compatible F(x) and f(x) curves for different durations. As an example the values of the half-widths of the f ( x ) distributions for the different durations into which a continuous recording of 21] min was divided are given in the table below. The duration time of each section of the recording is given together with the corresponding values of r and half-width.
Half-width (dB) Duration r (min) traffic pedestrian ~1~24} 5~ 10~" 214¢ Applied Acoustics, I (1968) 175-188
1 2 3 4 5 6
11'5 12.5 13'0 14.0 13"0 17'5
2"3 4'2 6"3 8"2 8-3 8"0
186
J. FOXON, F. J. PEARSON
Thus for a given maximum deviation, it may be seen that keeping r constant and decreasing Ax lead to steeper F(x) curves. By way of recapitulation, therefore, it can now be stated that the present considerations enable random traffic noise to be characterized by two parameters k and Ax, the former being proportional to the duration of a distinguishable traffic configuration, and the latter being related to the percentage change of intensity produced by a change in the number of emitters. This leads to the obvious prediction that for purely pedestrian noise k would be smaller than for traffic noise (because of the more rapidly changing configurations to be expected), hence r would be larger. Also Ax would be smaller since each noise emitter would be likely to produce a smaller proportional change in relation to the numerically larger pool than would be expected in the case of groups of pedestrians as compared with, for example, the size of the groups of heavy lorries likely to be encountered in practice. To confirm the possibilities of extending our model of random noise in the light of the above remarks, a number of recordings were then made in the pedestrian shopping Precinct in Coventry. This area is entirely free from motor traffic and the noise originates from footsteps and talking. The High Speed Level Recorder trace of this recording showed that the range of levels was less than the range of levels for traffic noise. The calculated values of k and Ax were 8 and 0.5 dB respectively showing that the predictions for the results of a pedestrian noise recording were substantially verified.
CONCLUSIONS
The range of levels over which the traffic noise spreads is 2(r + 2)Ax by reference to Fig. 6. If r remained constant and Ax decreased the range of levels would decrease and the F(x) curves would be steeper and produce f(x) curves with smaller halfwidths, as may easily be seen from the relationship between r and a given in eqn. (1). A small value of Ax would be expected from pedestrian noise since each of the noise emitters produces a smaller proportional change in intensity than would the corresponding emitters in a traffic noise situation. This assumption still applies even when it is realized that in any given situation there are likely to be more pedestrians than road vehicles. In this case the pedestrians may be considered in groups of more than one, each group counting as one emitter, each emitter producing less noise than one road vehicle. The effect of k on Ax is less obvious and in fact these two quantities need not be related since situations may be conceived where the value of Ax is large (heavy lorries) but nevertheless fast moving so 2k (duration of a particular configuration) will be small. In the case of the pedestrian noise recording which was considered
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE
187
above, Ax and k were both less than the corresponding traffic noise values. A pedestrian noise situation could be envisaged, however, where Ax was small and 2k was large. Such a situation is considered to be less likely since pedestrian noise is mainly due to footsteps which will produce a louder noise if moving quickly. Thus in such a pedestrian situation, if 2k were large, Ax would be even smaller than traffic noise values and may be due to other pedestrian noises than footsteps. A situation of this kind would be difficult to realize in practice due to the necessity of finding a location free from interfering background noise from industrial plant and distant traffic. It can be seen from the expression 2" = N / k that the probability of observing the lowest or the highest level is (42)'. The time during which this level is observable is (½)'T where T is the total recording time. This may be verified in practice by consideration of a specific traffic noise recording made on 2nd February, 1967. For illustration purposes let us accept that Xo ~ 88 dB, Ax ~ 2 dB and k ~ 10. Using 2" = N / k , with r = 6, it can be shown that N = 640. The total recording time in this case is 21] min. The maximum level observable should be 100 dB for a time of 1/64 x 21] = ½ min. Consideration of the appropriate F(x) graph (not illustrated) shows that 100 dB is observed for 2 % of the total time, i.e. for 1/50 x 21] ,~ 0.4 min, which is sufficiently close to the theoretical value. Furthermore, it is easily shown in more general terms that if observations continue for 2'k/30 min, the extreme value of Xo +_ t a x will be heard for k/30 min, although this general principle eventually breaks down under certain circumstances, one obvious one being when x o - t A x = O, i.e. when r = xo/Ax. In this case the noise level would have reached the threshold of hearing, which would be an unexpected situation. However, on the basis of the above theory it also follows that the greatest length of observation time throughout which a theory based on a constant mean level may be expected to apply is 2x"/aXk/30 min. Taking as typical values the above x o = 88 dB, Ax = 2 dB, and k = 10, this time is found to be 6 x 1012 min. In other words, the time taken for the predicted increasing spread of the noise levels about a constant mean to assume ridiculous proportions is so long that it therefore follows that it is justified to apply the theory to any reasonable periods of observation of random noise situations of the "arrival and departure" type. Throughout the whole of the present considerations, however, the noise situation has been characterized by a single constant mean level. This limitation arises out of the assumption that the probability of arrivals to, and departures from, the traffic pool is the same and equal to one-half. While the theory is therefore capable of further extension to the case of time-varying mean levels, the increased complexity appears to add little to the value of the present, more limited approach which lies in its essential simplicity, its ability to extract a physically meaningful picture in terms of the parameters k and Ax from the rather diverse range of data which almost inevitably arises from surveys of such a random quantity as traffic noise Applied Acoustics, 1 (1968) 175-188
188
J. FOXON, F. J. PEARSON
level, and its ability to yield useful predictions, as exemplified by the treatment of pedestrian noise in the Precinct.
ACKNOWLEDGEMENTS
The authors would like to thank Mr. E. Keeley and Mrs. C. M. Lovering of the Department of Applied Physics, Lanchester College, for assistance during the research and preparation of this article and Mrs. C. Gow, also of the above Department, for typing the manuscript.
REFEREN CES
1. G. L. BOUVALLET, J. Acoust. Soc. Am., 23 (1951) 435-39. 2. F. G. MEISTER,J. Acoust. Soc. Am., 29 0957) 81-84; F. G. MEISTERand W. RUHRBERG,Z. Vet. deut. lng., 95 (1953) 373. 3. R. DONLEYand P. B. OSTERGAARD,J. Acoust. SO¢. Am., 36 (1964) 409-13.
A STATISTICAL M O D E L OF T R A F F I C NOISE J. FoxoN* and F. J. PEAr,SON
Department of Applied Physics, Lanchester College of Technology, Covento" (Great Britain) (Received: 23 April, 1968)
SUMMARY
Traffic noise recordings have been analysed and presented in the form of graphs of the percentage of time during which the noise remains between certain levels against the levels involved. It has been established experimentally that observations presented in this form approach a Gaussian distribution. To account for this Gaussian distribution a statistical model of traffic noise has been evolved and used with reasonable success as a prediction mechanism to evaluate another random noise situation similar to that of traffic noise. The model also enables experimental data relating to noise surveys to be characterized by a limited number of parameters capable of describing a particular random noise situation in a physically meaningful manner. The article includes an account o f the various practical techniques used in the statistical analysis of traffic noise and a discussion of the extent to which the theoretical model is verified experimentally.
INTRODUCTION
It is significant that a number of surveys of traffic noise contain evidence that the proportion of time the noise remains between certain levels is normally distributed in terms of noise levels. This suggests that a statistical model of traffic noise might be developed which could account for this distribution in a physically meaningful manner. The object of the present research is to evaluate recordings of typical traffic noise situations in Coventry in terms of a putative normal distribution and to present a theory which accounts for the observed effects. * This work formed part of the author's thesis for the degree of M.Sc. of the University of Bristol in 1967. Applied Acoustics--Elsevier Publishing Company Ltd., England--Printed in Great Britain
176
J. FOXON, F. J. PEARSON PREVIOUS NOISE SURVEYS
A survey of city noise in the area around Chicago in the United States of America was initiated in 1947.1 The survey included noise measurements of traffic, industrial and residential noise. In addition to overall noise levels, a spectral analysis into octave bands was performed, the data being presented in the form of graphs of octave band level against octave bands. It was found in this survey that a plot of the levels in the 400-800 Hz band for 5-dB ranges approached a normal distribution. Similar distributions were obtained from the results of noise surveys in West Germany notably that by F. G. Meister in 1953. 2 A noise survey was conducted in New Jersey in 19643 and modal values of the background noise levels were measured. When the data were converted into a statistical frequency distribution a normal curve was obtained. An obvious extension of the above approaches was therefore undertaken in the course of a similar survey of traffic noise in the City of Coventry.
APPARATUS AND TECHNIQUES
The recording apparatus used for this purpose consisted of a portable tape recorder (E.M.I. Type RE 321) and a moving coil microphone (S.T.C. Type 4021 J), which, together with a suitable wind shield and an extension lead, were mounted on a small trolley for easy transportation to the recording site as shown in Fig. 1.
Fig. I.
STATISTICAL
~ "~ ~.__.
~..~
"a s
° "- o :
/ '
]ffgOTOtiiiOL
Applied Acoustics, I (1968) 175-188
NOISE
i ~ . s
]77
]
.p:
sqloll ~,.,
" ,i,a
'('--
OF TRAFFIC
°-~
.-.,,,
[
MODEL
~
'
/l
178
J. F O X O N , F. J. P E A R S O N
The recordings were replayed on a studio instrument (E.M.I. Type TR 52) whose output could be connected to either a High Speed Level Recorder (Dawe Type 1406 D) to obtain a trace variation in peak level or a Statistical Analyser (Dawe Type 1466 A) which indicated the number of times a certain level had been exceeded out of a certain total number of samples. When using the above equipment to record and analyse traffic noise a prerecorded white noise signal of 90 dBC was used for calibration purposes. The traffic noise was then recorded using the portable apparatus, the trolley being positioned so that the microphone was pointing in a direction normal to the main traffic flow. The site chosen for all the traffic noise recordings relating to the present considerations is illustrated on the map in Fig. 2. It consisted of a main thoroughfare (Jordan Well), together with two intersections. The nearest intersection to the recording position was a road which was only an inlet to the main thoroughfare (Bayley Lane) and the other intersection was a road used only as an outlet from the main thoroughfare (Much Park Street). The position was chosen since it represented many of the possible situations encountered in city traffic flow, i.e. freely-flowing traffic, moving at speeds up to 30 m.p.h., traffic slowing down or being brought to a halt and then pulling into or leaving the main traffic stream. Having chosen the recording position, recordings were made at various times of the day and for varying periods of time. No recordings were made at times when the traffic situation was unusually quiet (early morning or late evening) or when it was unusually noisy (morning and evening rush-hour periods). While each recording was timed using a stopwatch, the actual recording times cannot be regarded as being accurate to this standard since some allowance must be made for tape slack. After recording, the tapes were replayed on the studio recorder with the output of the tape recorder connected directly to either the High Speed Level Recorder or the Statistical Analyser. With the motor of the High Speed Level Recorder disengaged the output of the tape recorder was adjusted, using the calibration signal, so that 90 dB on the chart was represented by a line about 20 dB from the upper limit of the stylus movement. This allowed for a maximum sound level of 110 dB to be recorded and a minimum sound level of 60 dB. The chart speed chosen was 0.1 cm sec- 1 so that the longest continuous recording of about 20 min could be accommodated on a chart length of 130 cm allowing for the calibration signal and annotation. When the tape recorder output was connected to the Statistical Analyser, a valve voltmeter was interposed to monitor the voltage input to the analyser terminals because the analyser was used on a particular counting range which required input signals not to be greater than 1 volt. The setting of the tape recorder gain was, however, easily arranged to be at a suitable level after inspection of the Level Recorder trace.
179
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE PRESENTATION
OF D A T A
One of the objectives of the present work was to develop a statistical model of traffic noise which could account for a normal distribution of the proportion of the time the noise level lay between certain levels. By using the Statistical Analyser in the manner previously described the number of occasions, n, on which a certain level is exceeded during the sampling period may be obtained. The total number of samples, N, is related to the total time of the recording since a sampling rate of 0.5 samples/sec was used throughout all the present investigations. Each complete recording was divided into different periods of time from ~ min to 1043-min and thus an appropriate value of N emerged from each section of each recording. When n was expressed as a proportion of the total number of samples and plotted against the noise level a curve was produced which was of the general form of the function F(x) against x (dB) illustrated in Fig. 3(a). If the proportion f(x) of the time during which the noise level lies between two levels x and x + dx is plotted against level the curve of Fig. 3(b) is obtained, essentially by differentiating the F(x) curve. It is thisf(x) curve of apparently Gaussian form which has led to the development of the traffic noise model described later.
A ×
x b.
x
(aB)
x(dB)
Fig. 3(a).
Fig. 3(b).
STATISTICAL ANALYSIS
Sampling of the various recordings was accomplished by opening a gate in the statistical analyser at regular intervals. Consider a recording time of T sec which is sampled N times and assume that the level exceeds a certain value for a time t sec (t ~< T). The gate registers only one count regardless of the number of pulses of sound above the certain level which occurs during the period for which it is open. Applied Acoustics, 1 (1968) 175-188
180
J. FOXON, F. J. PEARSON
If n is the average number of counts o f a certain level it is easily shown from a theoretical consideration of the operation of the statistical analyser that the points on the F(x) curves are subject to a percentage error given by
--
x loo%.
_--_
100 R 20.9.66 I0 M mu1'es 90
80
70
60
o
5o
"5 g40 <
50
\
20
I0
I 75
I
80
I 90
85 Level
Fig. 4.
dB
i I_~-~--95
1 I00
S T A T I S T I C A L MODEL OF T R A F F I C NOISE
181
An example is given in Fig. 4 of a typical F ( x ) curve with the estimated errors indicated in an obvious manner for each experimental observation. We also note that in the present investigation 2N=
T.
T R A F F I C NOISE MODEL
Consider a traffic situation where on average there is a certain number of vehicles within earshot, all emitting noise. Let this average number of vehicles be called the " p o o l " of emitters and let the pool contain a range of different types of emitter, i.e. cars, lorries, buses, etc. Individual emitters of the various types join or leave the pool so that the overall noise level increases or decreases in discrete random steps. Let r = no. of emitters arriving and leaving in a specified time, i.e. the total no. of vehicle movements during the specified time. Also let r = n~ + n 2 when n2 = no. of arrivals and nl = no. of departures. Thus the increase in the number of sources
~--- /12 - - n l "
If the intensity I being emitted by the pool when the change nz - n~ occurs is large, it implies that the overall situation is noisy and the change n2 - n~ is likely to be due to a number of, for example, heavy lorries. Conversely, if I is small, n2 - n~ will be a number of quiet vehicles. Let A ! be the change in intensity produced by each of the vehicle movements and assume that at the time of these vehicle movements AI is proportional to I so that the changes in intensity actually occurring reflect the character of the existing noise situation at the time in the manner outlined above, i.e.
A I = ~1
where ct = a constant. I f Io = intensity of the virgin pool before any change occurs, it is easily shown that the resulting intensity after n2 arrivals is given by (1 + Ct)n~Io. When considering departures it must be borne in mind that the above initial intensities have been reckoned at the left of the interval. Thus when a source departs the intensity decreases by an amount proportional to what the intensity will be when it has left. On this basis it follows that the intensity after nl departures is thus = Io/(1 + ~)nl = / o ( 1 + 5) -/11. So intensity after n2 - nl changes = (1 + ct)¢n~-"lllo = fl
182
J. FOXON, F. J. PEARSON
The resulting change in sound pressure level arising from the above intensity changes may now be calculated using the relation I = Io eux, where x is in decibels measured from the mean (where I = I0) and /~ = 1/10 log e 10. N o w if I = f l ( " - ' - " l ) I o then x can be regarded as changing by n, - n~ equal steps of Ax from the mean. i.e.
therefore
I = fl("~ - "~ )I o = Io elL(''-' - "~ ) ~ ' , (/72
-
n l ) I n fl
so
=
//(/'/2
--
nl)Ax
Ax = In fl/la.
Hence the/12 - - /71 source changes (each producing equal percentage changes in the existing intensity) will produce a change from the mean level in decibels of (n2 - n l) Ax. Assuming that arrivals and departures are random, the probability of/72 arrivals and nl departures is given by the appropriate coefficient in the expansion o f (p + q)" where p = q = ½, assuming equal probabilities of arrival and departure. If these probabilities were unequal then the size of the pool would increase or decrease with time, but we do not wish to consider in detail at this stage the problem of time-varying mean levels. The probability of a change ( n 2 - - /71)Ax in decibels around a constant mean is therefore
(nl + n2)'(1) ("~+nO nl! n2! from which a Gaussian distribution is easily derived in the f o r m f ( x ) = A e - ~ ' - ~ " ) ~ where A and a are constants, xo is the mean level and a =
(1)
1 / 2 r ( A x ) 2.
F r o m the above theory nl and n 2 are the numbers of departures and arrivals of different sources of different strengths to and from the pool. Consider the case o f r = 2 where the following arrangement o f arrivals to and departures from the pool (of unspecified size) is possible. Binomial expansion n2
ii I
2 arrive
pool 0 leave
1 arrives first
pool
1 arrives second pool 0 arrive
term
p2
1 leaves second
pq
1 leaves first
qp
pool 2 leave
q2
S T A T I S T I C A L M O D E L O F T R A F F I C NOISE
183
The 2pq term is interpreted as two distinguishable situations or configurations in which the arrivals and departures occur in a different order. Thus it is suggested that the four situations outlined above correspond to four different distinguishable traffic configurations and that these different configurations occur at regular intervals of time on the average. Furthermore, it is well known that the sum of the above binomial coefficients is 2" which can be shown to be related in the present context to the total number N of samples taken throughout a period of observation. A relationship of the form
2" = N/k
(2)
is therefore suggested, where k is a scale factor. Equation (2) may be justified from the fact that the probability of obtaining the maximum deviation +rAx is (½)'. This can be regarded, by definition, as that particular pair of levels on either extreme of the mean which is observed the least out of 2' observations. Thus k may be regarded as the number of counts out of N for which the maximum and minimum levels are observed. Also, 2 r being the total number of distinguishable traffÉc configurations of the type discussed earlier, and also being equal to N/k according to eqn. (2), it follows that the average duration of one of these traffic situations will be given by T/Number of situations = 2k in the present investigation where the sampling rate is 0-5 sec- 1. This gives a simple physical meaning to the scale factor k as being directly proportional to the average duration of each distinguishable situation. It should perhaps be emphasized at this juncture that one of the essential features of the theory is the assumption that it is these distinguishable configurations which occur on the average at regular intervals of time and not necessarily the individual arrivals and departures. Whence, from eqns. (1) and (2), it follows that In N = In k + (In 2)/2a(Ax) 2.
(3)
A graph of In N against I/a should therefore be linear and yield values of k and Ax appropriate to the particular traffic situation observed for different lengths of time, thus leading to a confirmation of the theory.
EXPERIMENTAL
RESULTS
Values of In N and 1/a taken from a recording made on 20th September, 1966 are shown plotted in Fig. 5. This graph demonstrates the linear relationship between In N and 1/a. In this experimental verification of eqn. (3), the values of a were determined directly from the F(x) curves by a variety of methods, perhaps the Applied Acoustics, 1 (1968) 175--188
184
J. FOXON, F. J. PEARSON
R 20"9"66 In N,v.I/a. ,8 = 0 ' 0 7 3 ~ = 2"42
e
~
5
×
~
×
4 Z c
×
0
L I0
I 20
I 30
I 40
I 50
~o Fig. 5.
simplest being that illustrated in Fig. 6, where a curve of the F(x) type shows certain parameters do and D. A tangent, drawn to touch the curve at a point corresponding to the mean level Xo, for which F(x) = ½, when produced, intersects the x-axis at x o 4- do. The value of the tangent at the point is the maximum of the correspondingf(x) curve and can be shown to be f(X)max =
x/a/rr.
By inspection of Fig. 6 it follows that the tangent at x o is given by x o = 1/2do = f(X)m, x whence d o =
½x/n/a.
A value of a for each section of a particular recording can thus be obtained from the F(x) curves and used to plot a graph of In N against 1/a according to eqn. (3). From this graph values of k and Ax for the particular traffic situation can be evaluated from the gradient and intercept. A number of recordings have been made and analysed according to the methods just described, the values of k and Ax emerging as approximately 10 and 2 dB respectively for the particular traffic situations considered.
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE
185
\\
~t t.L
Fig. 6. The results show that values of k and Ax could be deduced in a relatively simple manner from the F(x) curves. Since the theoretical model was developed in terms of noise emitters without specific reference to the precise nature of the emitters it should also be possible to apply the model to analogous situations involving different types of emitter, and to predict the type of results that might be obtained. F r o m a consideration of the theory it can also be seen that increasing r (and hence N and the duration T of the analysis) increases the half-width of the corresponding normal distribution, a feature that was confirmed experimentally from an examination of compatible F(x) and f(x) curves for different durations. As an example the values of the half-widths of the f ( x ) distributions for the different durations into which a continuous recording of 21] min was divided are given in the table below. The duration time of each section of the recording is given together with the corresponding values of r and half-width.
Half-width (dB) Duration r (min) traffic pedestrian ~1~24} 5~ 10~" 214¢ Applied Acoustics, I (1968) 175-188
1 2 3 4 5 6
11'5 12.5 13'0 14.0 13"0 17'5
2"3 4'2 6"3 8"2 8-3 8"0
186
J. FOXON, F. J. PEARSON
Thus for a given maximum deviation, it may be seen that keeping r constant and decreasing Ax lead to steeper F(x) curves. By way of recapitulation, therefore, it can now be stated that the present considerations enable random traffic noise to be characterized by two parameters k and Ax, the former being proportional to the duration of a distinguishable traffic configuration, and the latter being related to the percentage change of intensity produced by a change in the number of emitters. This leads to the obvious prediction that for purely pedestrian noise k would be smaller than for traffic noise (because of the more rapidly changing configurations to be expected), hence r would be larger. Also Ax would be smaller since each noise emitter would be likely to produce a smaller proportional change in relation to the numerically larger pool than would be expected in the case of groups of pedestrians as compared with, for example, the size of the groups of heavy lorries likely to be encountered in practice. To confirm the possibilities of extending our model of random noise in the light of the above remarks, a number of recordings were then made in the pedestrian shopping Precinct in Coventry. This area is entirely free from motor traffic and the noise originates from footsteps and talking. The High Speed Level Recorder trace of this recording showed that the range of levels was less than the range of levels for traffic noise. The calculated values of k and Ax were 8 and 0.5 dB respectively showing that the predictions for the results of a pedestrian noise recording were substantially verified.
CONCLUSIONS
The range of levels over which the traffic noise spreads is 2(r + 2)Ax by reference to Fig. 6. If r remained constant and Ax decreased the range of levels would decrease and the F(x) curves would be steeper and produce f(x) curves with smaller halfwidths, as may easily be seen from the relationship between r and a given in eqn. (1). A small value of Ax would be expected from pedestrian noise since each of the noise emitters produces a smaller proportional change in intensity than would the corresponding emitters in a traffic noise situation. This assumption still applies even when it is realized that in any given situation there are likely to be more pedestrians than road vehicles. In this case the pedestrians may be considered in groups of more than one, each group counting as one emitter, each emitter producing less noise than one road vehicle. The effect of k on Ax is less obvious and in fact these two quantities need not be related since situations may be conceived where the value of Ax is large (heavy lorries) but nevertheless fast moving so 2k (duration of a particular configuration) will be small. In the case of the pedestrian noise recording which was considered
S T A T I S T I C A L M O D E L OF T R A F F I C NOISE
187
above, Ax and k were both less than the corresponding traffic noise values. A pedestrian noise situation could be envisaged, however, where Ax was small and 2k was large. Such a situation is considered to be less likely since pedestrian noise is mainly due to footsteps which will produce a louder noise if moving quickly. Thus in such a pedestrian situation, if 2k were large, Ax would be even smaller than traffic noise values and may be due to other pedestrian noises than footsteps. A situation of this kind would be difficult to realize in practice due to the necessity of finding a location free from interfering background noise from industrial plant and distant traffic. It can be seen from the expression 2" = N / k that the probability of observing the lowest or the highest level is (42)'. The time during which this level is observable is (½)'T where T is the total recording time. This may be verified in practice by consideration of a specific traffic noise recording made on 2nd February, 1967. For illustration purposes let us accept that Xo ~ 88 dB, Ax ~ 2 dB and k ~ 10. Using 2" = N / k , with r = 6, it can be shown that N = 640. The total recording time in this case is 21] min. The maximum level observable should be 100 dB for a time of 1/64 x 21] = ½ min. Consideration of the appropriate F(x) graph (not illustrated) shows that 100 dB is observed for 2 % of the total time, i.e. for 1/50 x 21] ,~ 0.4 min, which is sufficiently close to the theoretical value. Furthermore, it is easily shown in more general terms that if observations continue for 2'k/30 min, the extreme value of Xo +_ t a x will be heard for k/30 min, although this general principle eventually breaks down under certain circumstances, one obvious one being when x o - t A x = O, i.e. when r = xo/Ax. In this case the noise level would have reached the threshold of hearing, which would be an unexpected situation. However, on the basis of the above theory it also follows that the greatest length of observation time throughout which a theory based on a constant mean level may be expected to apply is 2x"/aXk/30 min. Taking as typical values the above x o = 88 dB, Ax = 2 dB, and k = 10, this time is found to be 6 x 1012 min. In other words, the time taken for the predicted increasing spread of the noise levels about a constant mean to assume ridiculous proportions is so long that it therefore follows that it is justified to apply the theory to any reasonable periods of observation of random noise situations of the "arrival and departure" type. Throughout the whole of the present considerations, however, the noise situation has been characterized by a single constant mean level. This limitation arises out of the assumption that the probability of arrivals to, and departures from, the traffic pool is the same and equal to one-half. While the theory is therefore capable of further extension to the case of time-varying mean levels, the increased complexity appears to add little to the value of the present, more limited approach which lies in its essential simplicity, its ability to extract a physically meaningful picture in terms of the parameters k and Ax from the rather diverse range of data which almost inevitably arises from surveys of such a random quantity as traffic noise Applied Acoustics, 1 (1968) 175-188
188
J. FOXON, F. J. PEARSON
level, and its ability to yield useful predictions, as exemplified by the treatment of pedestrian noise in the Precinct.
ACKNOWLEDGEMENTS
The authors would like to thank Mr. E. Keeley and Mrs. C. M. Lovering of the Department of Applied Physics, Lanchester College, for assistance during the research and preparation of this article and Mrs. C. Gow, also of the above Department, for typing the manuscript.
REFEREN CES
1. G. L. BOUVALLET, J. Acoust. Soc. Am., 23 (1951) 435-39. 2. F. G. MEISTER,J. Acoust. Soc. Am., 29 0957) 81-84; F. G. MEISTERand W. RUHRBERG,Z. Vet. deut. lng., 95 (1953) 373. 3. R. DONLEYand P. B. OSTERGAARD,J. Acoust. SO¢. Am., 36 (1964) 409-13.