A new trial to estimate the noise propagation characteristics of a traffic noise system

Transcription

1 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) A new trial to estimate the noise propagation characteristics of a traffic noise system Mitsuo Ohta*, Kazutatsu Hatakeyama*, Tsuyoshi Okita**, and Hirofumi Iwashige* *Faculty of Engineering, Hiroshima University, 3-8-2, Senda-machi, Hiroshima, 730 Japan **Faculty of Engineering, Yamaguchi University, Tokiwadai, Ube, Yamaguchi, 755 Japan (Received 1 May 1979) Generally speaking, a level fluctuation of road traffic noise is brought on by various causes. It is obvious that the ultimate causes are due to a variety of noise sources, i.e., uncertain behavior of individual cars, and to some effect of the noise propagation characteristics affected by reflections and/or absorptions owing to surrounding buildings and their topographical locations. By dividing the road into a suitable number of blocks, and paying close attention to the mean value of noise intensity in each block, the mean value can be proportional to the number of cars in the block. In this paper, on the basis of the additive property of sound energy, the unified method has been proposed to estimate the inherent characteristics of noise propagation in each block in the form of a synthetical evaluation with the number of cars for each car-type, also considering the entire background noise. We have confirmed the validity of our theoretical results, not only by means of digital simulation, but also by road traffic noise data experimentally observed near Hiroshima City. The experimental results clearly show a good agreement with the values recently reported by other official groups. PACS number: Qp 1. INTRODUCTION In recent years, environmental pollution is not only gradually injuring our psychological and emotional well-being, but also hazarding our health and safety. One of the worst pollutions is road traffic noise. Generally speaking, a level fluctuation of road traffic noise is brought on by various causes. It is obvious that the ultimate causes are a variety in noise sources, i.e., uncertain behavior of individual cars, and some effect of the noise propagation characteristics affected by reflections and/or absorptions owing to surrounding buildings and their topographical locations. It is necessary for engineers to obtain a better knowledge about future nuisance levels caused by road traffic noise before improving traffic control systems. It seems that the problem of describing relationships governing such a traffic noise system has not yet been solved in principle, even though a priori information of passing cars is partially available. Noise propagation characteristics must be studied. This situation occurs mainly because the noise propagation paths between noise sources and an observer are too complex to be expressed in the form of deterministic equations. By dividing the road into a suitable number of blocks, and paying close attention to the mean value of the noise (sound) intensity in each block, the mean value can be proportional to the number of cars in the block. This averaged relationship is generally true, based upon an adequate number of passing cars. In this paper, at first, this propor-

2 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) tional relationship is shown to be a logical result based upon the additive property of energy quantity or noise intensity generated by the road traffic. And a so-called proportional parameter, which relates the number of passing cars to the sound intensity received, is interpreted as the average product of the sound power generated by each car and the noise attenuation factor along all of the sound propagation paths. We recognize these parameters as noise propagation characteristics in a wider sense, which are indices characterized by reflections and/or absorptions owing to surrounding buildings and their topographical locations. The problem of describing the traffic noise system is now reduced to one of estimating the unknown characteristics of unknown noise propagation in each block. Thus, the traffic noise must be controlled based upon the estimates of these characteristics. In this paper, on the basis of the additive property of sound energy, some unified method has been proposed to estimate the inherent characteristics of noise propagation in each block in the form of synthetical evaluation with the number of cars for each car-type, including a consideration of the entire background noise. This procedure can be done independent of the surroundings and topography. In a more concrete form, well-known recurrence algorithms such as the Kalman filtering technique1) and the least-square method were applied to estimate the noise propagation characteristics when the observed values of sound intensity were available. The algorithms based upon the extended Kalman filtering technique2) or the stochastic approximation method3) were used to overcome the difficulty of logarithmic type non-linearity when the observed values were the db levels of sound intensity. In view of the arbitrariness of observed traffic noise, and the complexity of mathematical expressions and their statistical treatment, it can also be argued that a method of digital simulation is indispensable not only to an experimental confirmation but also to a theoretical calculation. We have confirmed the validity of our theoretical results, not only by means of digital simulation, but also by road traffic noise data experimentally observed near Hiroshima City. The experimental results clearly show a good agreement with the values recently reported by other official groups.4) The statistical treatment in this paper is appli- Fig. 1 A model for the road traffic noise system. cable not only to a field such as noise and vibration control, but also to other engineering fields, because of the conservative property of energy, and of the general nature of the estimation methods. 2. THEORETICAL CONSIDERATIONS 2.1 Modeling of A Road Traffic Noise System The large-scale use of motor vehicles as general means of surface transportation in the Twentieth Century has given rise to several environmental problems, in which the road traffic noise is the most troublesome one. Although our analytical treatment in this paper is widely applicable, we restricted our interest to the road traffic noise. In order to establish a mathematical model for a road traffic noise system affected by the statistical property of noise propagation paths under various conditions, in terms of the flow details, several fundamental facts with respect to the road traffic noise under the consideration are listed as in the following: 1) Many kinds of units evaluating the effects of road traffic noise, in particular Lr, LNP and ECPNL, are closely related to the average value of sound energy, i.e. Leq. In other words, what in fact being proposed, is that the "average noise" should be used as a criterion for planning. 2) Although there are many kinds of physical properties, such as force, momentum and velocity, to describe the internal or external relationships in practical systems, what is conservative is the average energy which has a scalar property. Then the analysis based upon the average energy could be applicable to any problem in principle, even if energy changes its form. 3) The average energy has the additive property in itself. This makes it possible to analyze easily the internal structure of physical systems. 4) So far as the instantaneous values of road traffic noise are concerned, they consist of various

3 M. OHTA et al.: ESTIMATION OF NOISE PROPAGATION CHARACTERISTICS causes for fluctuating randomly with respect to time and space. It is reasonable to describe such a complex system with the aid of the concept of the hierarchical structure. Being based on this concept, the road traffic noise observed at a fixed point near the road was considered as the total effect of the noise generated by the cars passing in each spacially divided interval of road. 5) The measurement values of physical quantities are always smoothed out according to the characteristics of measurement equipments. In fact, the measurement values of road traffic noise are averaged according to the characteristics of the sound level meter. Considering these fundamental facts with respect to the road traffic noise, we will establish a model to give an insight into the properties of the traffic noise field. A heterogenous stream of acoustic point sources moving along a straight road is shown in Fig. 1. A microphone placed at a fixed point, shown as O, near the flow detects the resultant sound intensity, whose value varies stochastically with time on account of a variety of different sources and noise propagation characteristics. By dividing the road under consideration into a suitable number of blocks P, denoting each block by the suffix p, the instantaneous sound intensity at the fixed observation point O, generated by all the cars of the j car-type passing in the spatially divided interval of the p-th block is expressed by Eq. (1) at any instant of time K based on the facts mentioned above as follows: locations of noise sources contained in the p-th from the mean value. If passing cars can be made up of a finite number of car-types, say J car-types altogether, the additive property of energy implies that the resultant sound intensity, Y(K), recorded at O will be the sum of the instantaneous noise generated by passing cars of each car-type contained in each block, i. e., where, with the notation W (K) which is called the background noise by definition, we express the fluctuation factor including not only an additional noise, independent of the traffic noise under con- general the resultant sound intensity recorded at time K, Y(K), is affected by its past values in the situations that reverberations and sound propagations have enough time to be heard. But we consider that noise intensity usually decays by a negligible value in adequate sampling interval of time. Let us denote the time-averages of <Qij(K) gp(d, Zi(K); K)> and W(K) by the constant parameters apj and W respectively. Their fluctuations, due to the non-stationary behavior of spatially distributed variables contained in each block and in the background noise, can be expressed as governing these variables are expressed as: where d denotes a perpendicular distance from the observation point O to the center of road, npj(k) is the number of j-type cars in the p-th block at time K, and gp(d, Zi(K); K) is an attenuation factor with By definition, apj means the average propagation respect to an acoustic power Qij(K), generated by characteristics of road traffic noise generated by the i-th car of the j car-type located at a point Zi(K) in the p-th block, at the observation point passing cars of the j car-type in the p-th block, and W means the averaged intensity of background noise. Usually we denote apj as the noise propagation that the noise intensity from the p-th block is characteristics which are indices character- interpreted simply as a proportional quantity to the ized by reflections and/or absorptions owing to total number of cars in each block. According to surrounding buildings and their topographical other papers, the relationship holds true under the locations. Using the notations introduced in Eqs. assumption of a suitable number of cars contained (3) and (4), the traffic noise observed at the fixed in each block. In Eq. (1), < > denotes the point near the road at time K is expressed as: spatial averaging operation with respect to the (2) 89

4 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) where Eq. (6) as the observation equation for the road traffic noise system, the estimate is calculated in the form of the following recurrence algorithm: (6) and T denotes the transposition of matrices and vectors. This equation can be considered to show the proportional relationship between the number of cars passing in each block and the observed noise at O. This proportional relationship exactly coincides with the empirical relationship reported in the previously mentioned papers. In addition, it should be noted that the relationship holds true for a general road traffic noise system including the case of a homogeneous loss-free atmosphere. In order to complete the description of the statistical model for the road traffic noise, we must determine the unknown values of noise propagation characteristics in Eq. (5) based on the road traffic noise data. The noise propagation characteristics are the constant-valued parameters of a road traffic noise system, and are not dependent upon the transition of time. Now for the convenience of establishing the recurrence algorithms to estimate the noise propagation characteristics in Eq. (5), the unknown parameters containing, not only the noise propagation characteristics but also the average background noise, should be expressed with a notation a(k), which is always equal to a in spite of the time-dependent suffix K. As a(k) is the constant-valued unknown parameter, the following equation holds: Of course, a validity of the above estimation algorithm is guaranteed under the assumption of some white Gaussian properties with respect to the algorithm is applicable in the case that any information about the distribution of these fluctuations is available a priori. If this information is not available, an algorithm based on the well-known least-square method can be used to give an estimate of the noise propagation characteristics. By minimizing the following, the estimate is obtained by the algorithm: (7) In the following section, we will consider the problem of how to estimate the unknown noise propagation characteristics, considering that Eq. (7) is the system equation with unknown state a(k), and Eq. (5) is the observation equation for the road traffic noise system under consideration. 2.2 Estimation of the Noise Propagation Characteristics Based on the Observed Noise Intensity When the observed values of noise intensity at the fixed point are available, a recurrence algorithm to estimate the noise propagation characteristics can be derived from the direct use of the Kalman filtering technique1) in the field of estimation theory. Considering Eq. (7) as the system equation, and (11) 2.3 Estimation of the Noise Propagation Characteristics Based on the Observed Values of Noise Levels As the observed data of road traffic noise is, in common, obtained through a sound level meter, it is more convenient to estimate the noise propagation characteristics by using the A-weighted sound level (noise level in JIS) as the unit of measurement. When the road traffic noise is observed in this unit, the observation Eq. (5) must be converted into the following expression: 90

5 M. OHTA et al.: ESTIMATION OF NOISE PROPAGATION CHARACTERISTICS The observation equation is expressed in terms of the logarithmic type non-linear function. Due to the non-linear transformation, the observed data Z(K) does not have a proportional relationship to hibits, generally, non-gaussian properties of distribution. In order to overcome the analytical difficulty owing to these properties it can be argued that more flexible algorithm must be used instead of the algorithms mentioned in the previous section. Among several analytical methods in the field of estimation theory, the applications of the extended Kalman filtering technique and of the stochastic approximation method are recommended, because of their flexibility for applications. The algorithm based on the extended Kalman filtering technique is derived after linearizing the non-linear function given. Thus, the estimate used with the algorithm must contain the estimation error due to the linearization. The algorithm based on the stochastic approximation method is expected to have a consistent estimation in any case concerning arbitrary non- Gaussian and non-linear systems. In the remainder of this section, the noise propagation characteristics for the road traffic noise system are estimated in terms of these two algorithms, considering Eq. (12) as the observation equation Algorithm based on the stochastic approximation method The algorithms in this section are based on the stochastic approximation method proposed by Robbins and Monro.3) The method is a stochastic analogy of the simple gradient algorithm for finding the unique root of an equation, or the regression function. It is required that the regression function is bounded on either side of a true solution by straight lines; however, this requirement is not especially rigid. Consider a mean-square derivation with respect to the observed value Z(K) such as: unknown parameter a to be estimated. This dependence indicates that the estimate obtained by minimizing J(a) cannot coincide uniquely with the noise propagation characteristics of the road traffic noise system. Then we pay close attention to a condition- Then, we adopt the following regression function M(a) to derive a recurrence algorithm based on the stochastic approximation method.

6

7

8 Any noise intensity from a road traffic flow usually decays by a negligible value in adequate sampling interval of time, and an auto-correlation of the noise intensity is very poor. This fact permits us to consider that the second term in Eq. (30) is the observation white Gaussian noise. From this point of view, we can apply the extended Kalman filtering technique to the problem of estimating the unknown parameter a(k), considering Eq. (30) as the observation equation. The algorithm based on the filtering technique is written as follows: J. Acoust. Soc. Jpn. (E) 1, 2 (1980) The validity of approximation in the algorithm is confirmed by a digital simulation shown in the following section. 3. EXPERIMENTAL CONSIDERATIONS From the viewpoint of the arbitrariness in observed traffic noise pattern, and the complexity of mathematical expressions and their statistical treatment, we have confirmed the validity and effectiveness of our theoretical results, not only by means of digital simulation, but also by road traffic noise data experimentally observed on the Nishi-Hiroshima Bypass near Hiroshima City. 3.1 Experimental Confirmation by Means of Digital Simulation In order to investigate concretely how noise propagation characteristics are estimated by the estimation algorithms described in the previous chapter for the road traffic noise system, it is theoretically convenient to consider an idealized special case for the road traffic noise system. We adopted the following assumptions with respect to the idealized road traffic noise system in a homogeneous, loss-free atmosphere. (i) The road under consideration is divided into three blocks, where each block has the same interval 2d (see Fig. 1). (ii) The relative location of passing cars is dis- (a) Fig. 2 A comparison between true values and estimates of a/(q1/d2) by using the observed data of noise intensity in terms of the error function es.

9 M. OHTA et al.: ESTIMATION OF NOISE PROPAGATION CHARACTERISTICS (36) Figure 2-a shows an comparison between true values and estimates of normalized noise propagation characteristics. The former was described in Eq. (34) and the latter was calculated by the algorithm based on the Kalman filtering technique and data of noise intensity. In this figure, the behavior of these estimates is totally evaluated in terms of the following error function as: where denotes the norm with respect to a vector. It is obvious that the successive addition of road traffic noise data makes the estimates for the noise propagation characteristics and average background noise closer to the true values, or makes es smaller. Figure 2-b shows the results which were calculated by algorithm based on the least-square method and the data of noise intensity. The results the extended Kalman filtering technique and the data of noise levels are shown in Fig. 3-a. We can observe that our theoretical method can be effec- tive for an estimation with non-linear observation equation. Finally, Fig. 3-b shows the results which were calculated by the algorithm based on the stochastic approximation method and the data of noise levels. At the same time, the figures show the results calculated by exchanging a sequence of gains in the same way as described in section In each case, we can observe that the estimate becomes closer to the true values of noise propaga- tion characteristics in accordance with an increase in the number of noise data. In particular, Fig. 3-b shows a much more effective improvement of the estimates in spite of the non-linear property of the observation. This indicates that the algorithm based on the stochastic approximation method 3.2 Application to an Actual Road Traffic Noise In this case, the average of the background noise is We confirmed the validity and effectiveness of assumed to be equal to Q1/d2. our theoretical results by the road traffic noise The following two cases are adopted as weighting functions. data experimentally observed on the Nishi-Hiroshima Bypass near Hiroshima City. The road under which were calculated by the algorithm based on seems appropriate enough to be applied to empirical data of road traffic noise. consideration was divided into three blocks as shown in Fig. 4. As information with respect to

10 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) (a) (b) Fig. 3 A comparison between true values and estimates of a/(q1/d2) by using the observed data of noise level in terms of the error function es. Fig. 4 An empirical road model on the Nishi-Hiroshima Bypass. the fluctuation was not available, the noise propagation characteristics were estimated by the algorithm based on the stochastic approximation method. The method is accompanied by the simultaneous estimation of ƒó(n(k), a), using the instantaneous readings of a sound level meter. When K=2800, the noise propagation characteristics and the average background noise were estimated by adopting the weighting function (35) (s=2.0) as: car-types, and attenuation with distance, calculated by estimated values, agreed with the results calculated for the acoustic power of cars reported by other official groups, under the assumption of a free sound field. Furthermore, we considered a prediction of the road traffic noise levels, with aid of the estimates and a priori information with respect to the number of cars, as: When we predicted the levels in time region from K=2801 to 2831, the expectation, standard deviation and median value with respect to the predicted levels agreed with the result with respect to the experimentally observed levels as shown in Table 1 (see Fig. 5). Finally, we drew the cummulative distribution curve of Z(K) with the direct use of the work5) such as: The ratio of acoustic power between two different (40) where M denotes the smallest level of the road 96

11

12 J. Acoust. Soc. Jpn. (E) 1, 2 (1980) 6) M. Ohta, T. Okita, K. Hatakeyama, and M. Nishimura, "A new trial to estimate the noise propagation characteristics of a traffic noise system and its application," Report to the 1st System Symposium (1975), pp (in Japanese). 7) M. Ohta, T. Okita, and K. Hatakeyama, "A new trial to estimate the noise propagation characteristics of a traffic noise system and its application," Report to the International Federation of Automatic Control Symposium on Environmental Systems-Planning, Design and Control (1977), pp