Adaptive Noise Parameter Determination Based on a Particle Filter Algorithm

Due to the growing number of vehicles using the national road networks that link major urban centers, traffic noise is becoming a major issue in relation to the transportation system. Thus, it is important to determine noise model parameters to predict road traffic noise levels as part of an environmental assessment, according to traffic volume and pavement surface type. To determine the parameters of a noise prediction model, statistical pass-by and close proximity tests are required. This paper provides a parameter determination procedure for noise prediction models through an adaptive particle filter (PF) algorithm, based on using a weighin-motion system, which obtains vehicle velocities and types, as well as step-up microphones, which measure the combined noises emitted by various vehicle types. Finally, an evaluation of the adaptive noise parameter determination algorithm was carried out to assess the agreement between predictions and measurements.


Introduction
Roadway network expansion and urbanization have brought about not only increased traffic volume but also higher emitted noise levels.Thus, it is important to determine the parameters of traffic noise for models according to various pavement surface types, vehicle types, and vehicle speeds [1][2][3][4].Based on a particle filter algorithm, the findings of this study can be used to determine accurate parameters for a traffic noise model that predicts specific environmental impact noise assessments for given traffic volume conditions, average vehicle speeds, and pavement surface types.
According to the Acoustic Society of Japan (ASJ) model [5,6], four vehicles were selected for the measurements: a large vehicle, a medium vehicle, a light vehicle, and a car.For pavement types, a Portland cement concrete (PCC) pavement and a stone mastic asphalt (SMA) surface were chosen.Two kinds of PCC pavement tining patterns were evaluated in terms of the characterization of traffic noise (Figure 1).The surface textures in the PCC pavements consisted of 30-mm center-to-center transverse tining and 18-mm center-to-center longitudinal tining.For an asphalt concrete (AC) pavement section, stone mastic asphalt (SMA) was evaluated with respect to traffic noise.
Regarding the influence of the pavement surface and its model characterization, several researchers have made important contributions with regard to modeling characteristics of the noise emitted by the tire-pavement interaction and the power-train [7][8][9][10][11][12].For example, Waters [7] found that tire noise was generated by the impact between the tire tread and road surface, including a significant noise parameter determination for a diesel-engine truck.Sandberg [9] provided a mechanism for tire-road noise generation and its relationship to pavement surface characteristics using the surface texture profile and a spectral analysis of the profile curve.McNerney et al. [12] showed a test procedure with a set-up of road-side microphones and microphones mounted on a test trailer to analyze the recorded data to evaluate differences in tire-pavement noise.This paper proposes a particle filter-based estimation of the noise parameters in terms of different road surface types, such as PCC and asphalt concrete pavements.Section 2 Center-to-center length describes the noise model and vehicle types.Section 3 explains the particle filter-based algorithm.Section 4 uses the PF method to estimate the noise parameters, and Section 5 presents the conclusions of this research.

Noise Model
To consider the variation in noise level as a vehicle's speed increases and the noise characteristics of vehicles on various pavement surfaces, a test method [13] was appropriately used to determine sound power levels.The vehicle's power pressure level,  , , which represents the noise emitted by the vehicle as it runs on various road surface types, can be denoted by where the subscripts,  and , represent -weighting and a frequency band, respectively, omitting the subscript, , represents the overall value,  eq, is the square mean noise level, and  , is the correction factor used for converting the sound pressure level into the sound power level.Details of sound power level determinations can be found in [13].
The following sound power level regression equation, which is based on several noise prediction models [1][2][3][4][5][6], can be determined: where  , and  , are given for the octave band center frequency and  is the vehicle driving speed.
The noise prediction method is based on the attenuation of sound during propagation outdoors from all the significant mechanisms found in the method issued from ISO 9613-2 [14].In this standard, the equivalent continuous downwind octave band sound pressure level at a receiver,  , , is calculated for each point source, for its image sources, and for the eight octave bands with nominal frequencies from 63 Hz to 8 kHz, by the following equation: where  , is the octave-banded sound power level, in dBA, radiated by the sound source relative to a reference sound power of one picowatt (1 pW),  , is the directivity correction, in dB, in the direction from the center of the source to the receiver, and  , is the combined attenuation, in dB, that is due to geometrical divergence ( div, ), atmospheric absorption ( atm, ), ground effect ( gr, ), a barrier ( bar, ), and miscellaneous other effects ( misc, ).The equivalent sound power level,   , can be calculated with the following equation: Once the equivalent sound power level of LW is calculated, the sound power level emitted by a type of vehicle needs to be determined.The sound power level during a specific time duration emitted by a type of vehicle moving on a road can be calculated with the following equation: where  , is the equivalent sound power level (dB) emitted by a specific vehicle type shown in Table 1, Δ is the length of the road segment, in meters,  is the mean speed of a vehicle type (km/h), and  is the hourly traffic flow of a vehicle type (vehicles/h).The subscript, , represents a vehicle type.Furthermore,  , is the -weighted sound power level of each vehicle type; for example,  ,1 ,  ,2 ,  ,3 , and  ,4 are the -weighted sound power levels of a large vehicle, a medium vehicle, a light truck, and a car, respectively (Table 1).The totally equivalent sound power level,   , emitted by all the vehicles moving in the road can be calculated as follows:

Particle Filter-Based Algorithm
The particle filter (PF) method is a useful state estimation for a nonlinear model with non-Gaussian noise and is used widely in the field of prognostics [15,16].Additionally, the PF method represents a posterior distribution by an expression as a number of particles and their weights [15,16].The update process of the PF method algorithm consists of a state model and measurement function, as follows: where  is the step index, x  is the state vector,   is the state process noise,   is the step-dependent state transition function,   is the measurement, V  is the measurement noise, and ℎ  is the step-dependent measurement function.The noise model parameters can be estimated using an algorithm of a PF-based method by updating the noise model functions with the discrete measurement data that represent the vehicle and pavement surface types as well as the traffic volume.
A noise model form of the log function is expressed by the following equation: where the sound power level,  The procedure of the PF-based method is based on Bayes' theorem, as follows: where   (x  |   An updating step of the likelihood function is followed, using a log-normal distribution, as follows: where Thus, the initial distribution of parameters and the likelihood function are normal and log-normal distributions, respectively. In the following procedure, the posterior can be calculated with (9).The probability density function (PDF) value of   at the given th samples of the unknown parameters x  corresponds to the weight of the th samples; the weight is proportional to the magnitude of the PDF value, which is expressed as the length of the vertical bar in Figure 2. Finally, the samples with high or low weight are duplicated or eliminated, respectively, at the resampling step (e.g., the th step).
Among several methods available, the inverse cumulative density function (CDF) method [16,17] was used.Thus, a CDF was constructed from the likelihood function in (10).Next, a random value was generated from the uniform distribution of (0, 1) between 0 and 1, which becomes a CDF value.Thus, a sample of the parameter having the CDF value was found.By repeating this process  times,  samples are obtained.
Because samples exist in a discrete form, the sample having the closest value to the CDF value is selected.Consequently, the resampled results become the posterior distribution   (x  |  1: ), which corresponds to the posterior distribution at the current step and is also used as the prior distribution at the next ( + 1) step.
The implementation of sequential importance sampling helps in reducing the number of samples required to approximate the future state probability distributions, increasing the computational efficiency with respect to other classical Monte Carlo methods [17].The PF-based algorithm is capable of controlling the uncertainty associated with long-term predictions by exploiting the system state model.The Bayesian solution to the problem of estimating the dynamic state (e.g., x  ), given the measurements   up to time , is sought in terms of the probability density function   (x  |  1: ).This PDF contains all the information about the state x  , which is inferred from the measurements,  1: = {  ,  = 1, . . ., }, and the initial distribution of the system state   (x 1 ) is assumed to be known.Assuming that a set of random samples (particles) x −1 ,  = 1, 2, . . ., , at the system state at the time  − 1 is available as a realization of the posterior probability   (x −1 |  1:−1 ), an approximation of the posterior distribution   (x  |  1: ) can be obtained based on the procedure shown in Figure 1.
Caution in the implementation of PF is necessary due to the degeneracy problem; as the algorithm evolves in time, the weight variance increases and the importance weight distribution becomes progressively skewed, until all but one particle have negligible weights.As a result, the approximation of the target distribution (x  |  1: ) becomes very poor and significant computational resources are spent trying to update particles with minimum relevance.
To avoid this problem, one can proceed to resampling a new swarm of realization x   from the approximate posterior distribution, constructed on the weighted samples previously drawn.All particles thereby generated are assigned equal weights,    = 1/.One has to resample a new swarm of points x   from the posterior distribution.

Results and Discussion
To evaluate the three different pavement surface types, the 30mm transverse tining surface, 18-mm longitudinal tining surface, and stone mastic asphalt, as shown in Figure 1, comparison studies were performed, based on observations between measurements and predictions.First, in terms of applying the PF method, the vector x  = [ 1,  2,  3,  4, ]  , which consists of the model parameters, is incrementally improved through the posterior probability,   (x  |  1: ), based on the noise measurements,  1: = {  ,  = 1, . . ., }. Figure 3(a) shows that the initial parameters can be generated randomly and used for the prior,   (x  ), based on a normal distribution, given the mean and standard deviation.Then, the final parameters can be obtained from the PF method using the likelihood function and posterior, as shown in Figure 3(f).
To evaluate the PF algorithm for determining the model parameters, the input values, which are the average speed and number of vehicles in terms of the four different vehicle types, were measured for the parameter determination of the chosen noise model in this study.As shown in Figures 4-6, the surfaces of 30-mm transverse tining concrete, 18mm longitudinal tining concrete, and SMA pavement were used for the algorithm evaluation.Furthermore, the weighin-motion system can be used for obtaining the average speed Through Figures 7-9, a comparison analysis between noise measurement and prediction was carried out, resulting in a relative error of less than 1% in the final time.Furthermore, we can observe that the initial prediction results show a large difference, as compared to the initial measurement; however, the prediction accuracy is improved with convergence through increased measurement time, based on the comparison between the measurements and predicted noise levels.
Thus, the number of various vehicles and average traffic speeds with respect to time history are available as inputs used for predicting the parameters of noise models, depending on the pavement surface types.Because the estimated model   parameters are obtained through each PF process, which is based on prior (e.g., prediction), likelihood (e.g., update), and posterior (e.g., resampling) as shown in Figure 2, the parameters can be finally converged in terms of the prediction accuracy as shown in Figure 3, resulting in no discrepancy between measured and predicted noise levels.Therefore, the PF algorithm based on the Bayesian statistical predictions can be used to superimpose the data of vehicles and speeds to update the model parameters in order to obtain the accurate noise prediction of various pavement surface types.

Conclusions
In this study, a PF method was used for determining the parameters of a traffic noise prediction model, based on  the inputs of vehicle types, average vehicle speeds, number of vehicles, and pavement surface types.Finally, a proposed algorithm of the PF method can be applied effectively not only to predict noise but also to consider various road surface types as part of the prediction process through the determination of noise parameters.Additionally, an evaluation of the adaptive noise parameter determination algorithm of the PF method was evaluated, showing good agreement between predictions and measurements.

Figure 2 :
Figure 2: Procedure of the PF-based algorithm.

Figure 3 :
Figure 3: Parameter convergence in the case of a 30-mm transverse tining surface: (a) 1st particle filtering data set, (b) 2nd particle filtering data set, (c) 5th particle filtering data set, (d) 10th particle filtering data set, (e) 20th particle filtering data set, and (f) 30th particle filtering data set.

Figure 4 :
Figure 4: Vehicle speeds and passing vehicles on the surface of the 30-mm transverse tining concrete pavement.

Figure 5 :Figure 6 :
Figure 5: Vehicle speeds and passing vehicles on the surface of the 18-mm longitudinal tining concrete pavement.

Figure 7 :
Figure 7: Comparison between noise measurements and predictions with respect to the 30-mm transverse tining concrete pavement.

Figure 8 :
Figure 8: Comparison between noise measurements and predictions with respect to the 18-mm longitudinal tining concrete pavement.

Figure 9 :
Figure 9: Comparison between noise measurements and predictions with respect to the SMA pavement.
*  is a velocity.
1: ) is the posterior,  1 is the normalizing constant, (  | x  ) is the likelihood function representing the conditional probability of the measurement data   under the condition of vector of model parameters x  , and   (x  ) is the prior.As shown in Figure1, the viscoelastic parameters, such as x  = [ 1,  2,  3,  4, ] , can be determined through a PF-based method.The notation [⋅]  denotes the transpose matrix; thus, x  is a column state vector.At the first step, that is,  = 1,  samples of the four parameters are generated randomly from the prior distribution, based on the Gaussian distribution of specified means and standard deviations.