Parameter Estimation for Traffic Noise Models Using a Harmony Search Algorithm

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Introduction
It is important to evaluate the primary impact of traffic noise for several different road surfaces in order to predict the interactive noise between road surfaces and vehicle tires.A number of noise prediction models [1][2][3][4][5][6] have been developed for environmental estimation of traffic noise levels in terms of vehicle and pavement types.For example, the vehicle types in the ASJ model [1,2] are large vehicles, medium vehicles, light trucks, and cars, and the pavement types are dense-graded asphalt (DGA) and permeable asphalt (PA).Thus, the ASJ traffic noise prediction model is restricted to four vehicle types, varying vehicle speeds, and two pavement types.
In order to expand the applicability of the ASJ model, the present paper introduces a parameter estimation procedure based on a harmony search (HS) algorithm as follows: (a) the traffic noise for a targeted road surface is measured, (b) the parameters of the noise prediction model are estimated via the HS algorithm, and (c) the resulting coefficients are evaluated using another set of measurements (consisting of vehicle speeds and vehicle types) for a different traffic volume on the targeted road surface.To validate the proposed traffic noise prediction technique, traffic noise measurement sets from three different surface types were used in this study: stone mastic asphalt surfaces (SMA), 30 mm transversely tined Portland cement concrete surfaces (30 mm trans.), and 18 mm longitudinally tined Portland cement concrete surfaces (18 mm long.), as shown in Figure 1.The measurement site is a 7.7 km long.2-lane highway along the side of the south bound of Jungbu inland highway in South Korea.This measurement section includes both asphalt and Portland cement concrete pavements.
This paper is organized as follows.Section 2 describes the ASJ model and vehicle characterization.Section 3 explains the application of the HS algorithm to estimate the parameters of ASJ-based noise prediction models, and Section 4 presents the conclusions of this research.

ASJ Model
The Acoustic Society of Japan (ASJ) published the ASJ model [1,2] for calculating road traffic noise.The procedure involves calculating the noise level generated by traffic as well as the attenuation during noise propagation.The octave-band power spectrum for nominal midband frequencies (63 Hz to 8 kHz) can be generated according to the ISO 9613-2 standard [7].The ASJ model classifies vehicles into four types listed in Table 1.
The A-weighted overall sound power levels ( WA ) of the noise emitted by interactions between vehicles and pavement are listed in Table 2 for a dense-graded asphalt (DGA) surface.In terms of nominal midband frequencies, the individual Aweighted sound power level for each octave band ( WA, ) is calculated as follows: where  WA is the overall sound power level (dB) and Δ  is the relative level (dB) at the th nominal mid-band frequency given by ] , ( where   is the nominal mid-band frequency.Finally, Δ adj is the correction factor defined by

Equivalent Sound Power
Level for a Road.The A-weighted sound power level emitted by a specific type of vehicle moving along a road over a specified time period,  WAT , can be calculated via the following equation: where  WA is the basic sound power level (dB) emitted by the vehicle (listed in Table 2), Δ is the length of the road segment (in meters),  is the mean speed for the vehicle type (in km/h), and  is the hourly traffic flow for the vehicle type (in vehicles/h).The equivalent sound power level,  eq(WA) , emitted by all vehicles moving along the road can then be calculated as follows: where  WAT, is the A-weighted sound power level for each vehicle type.Here,  WAT,1 ,  WAT,2 ,  WAT,3 , and  WAT,4 are the A-weighted sound power levels for a large vehicle, a medium vehicle, a light truck, and a car, respectively, as listed in Table 2. Furthermore, the attenuation (e.g., geometrical divergence, atmospheric absorption, ground effect, and screening structures) was calculated based on ISO 9613-2 [7].

Application of the Harmony Search Algorithm
3.1.Harmony Search Algorithm.This section describes the procedure for estimating the parameters of noise prediction models, using a harmony search (HS) algorithm that employs a heuristic algorithm based on an analogy with natural phenomena [8][9][10][11][12].The detailed procedure for applying a harmony search consists of four steps as follows.
(1) The algorithm parameters are specified.These include the harmony memory size (HMS), initialized as the number of solution vectors in the harmony memory (HM), the harmony memory consideration rate (HMCR, between 0 and 1), the pitch adjustment rate (PAR, between 0 and 1), and the maximum number of improvisations (or stopping criterion), which terminates the HS program.The optimization problem is specified as follows: Minimize  () subject to   ∈   = 1, 2, . . ., , where where   eq(WA) is the predicted equivalent sound power level of ( 5), which can be calculated as follows: eq(WA) = 10 log 10 [10 0.1{ WAT,1 + WAT,2 + WAT,3 + WAT,4 } ] , where  WAT,1 ,  WAT,2 ,  WAT,3 , and  WAT,4 are the Aweighted sound power levels for a large vehicle, a medium vehicle, a light truck, and a car, respectively.  eq(WA) is the measured equivalent sound power level obtained from previous research [13,14].In this optimization problem, the A-weighted sound power levels can be defined as given in Table 3.Thus, the coefficients of  1 ,  2 ,  3 , and  4 must be determined via the HS algorithm.The slope is fixed in the ASJ models for both surface types (DGA and PA); therefore, the slope given in Table 3 is fixed at 30.
(2) The HM matrix is initially filled with randomly generated solution vectors up to the HMS, together with the corresponding function values of the random vectors ():

𝑓 (𝑋 HMS
(3) A column vector of the newly generated harmony, , is improvised utilizing the following three mechanisms: (a) random selection, (b) memory consideration, and (c) pitch adjustment.In the random selection, the value of each decision variable,    , in the column vector is randomly chosen within the range of values with a probability of (1-HMCR).HMCR (which is between 0 and 1) is the rate at which a single value is chosen from the historical values stored in the HM.The value of each decision variable selected by memory consideration is examined in terms of pitch adjustment.This operation uses the PAR parameter (which is the rate of the necessary pitch adjustment according to the neighboring pitches) with a probability of HMCR × with a probability of HMCR × PAR × 0.5    −  × , with a probability of HMCR × PAR × 0.5    , with a probability of HMCR × (1 − PAR) .
If the newly generated column vector is better than the worst harmony in the HM, based on evaluation of the objective function, the newly generated column vector is included in the HM, and the existing worst harmony is excluded from the HM.(4) If the stopping criterion (or maximum number of improvisations) is satisfied, the computation is terminated.Otherwise, Steps 3 and 4 are repeated.

Application to Parameter Estimation for Noise Prediction
Models.In order to estimate the parameters of noise prediction models based on the objective function of (8), noise measurements for three different road surfaces were obtained from previous research, which was conducted on a test track [13,14].The vehicle velocities and hourly traffic flows are listed in Table 4.
To apply the HS algorithm to parameter estimation for the noise prediction models, the four coefficients were determined for each road surface type (stone mastic asphalt (SMA) surface, 30 mm transversely tined Portland cement concrete surface (30 mm trans.), and 18-mm longitudinally tined Portland cement concrete surface (18 mm long.)), as shown in Figure 2 and Table 5, based on the training data from Table 4.In this way, the coefficients of the noise prediction models, which are dependent on the road surface type, can be updated via the HS algorithm.As a result, noise prediction models can be provided for various surface types by using the HS algorithm to update the ASJ model equations.
Another set of testing data (given in Table 6) was used to evaluate whether or not the updated noise prediction models  (with the four coefficients estimated by the HS algorithm) provided results consistent with measured noise levels.The predictions and measurements are compared in Table 7; good agreement was noted for all three surface types.Using the original ASJ model, the prediction value is resulted in a same noise level regardless of surface types and in bad agreement when compared with the measured noise levels.
Finally, the A-weighted sound power levels for the individual octave bands were estimated for the SMA, 30 mm trans., and 18 mm long.surface types, utilizing the parameters estimated via the HS algorithm and (1).For example, Figure 3 shows the A-weighted sound power levels for the octave bands for the three different surface types in the case of the large vehicle speeding at 80 km/h.

Conclusions
In this study, it was shown that the optimization problem related to updating the noise prediction models for several surface types could be solved using an HS algorithm process.
The process involves (a) obtaining measurements for different road surfaces, (b) estimating the coefficients of the noise prediction models using this measurement set as training data, and (c) evaluating the estimated coefficients using another measurement set as testing data.When this procedure was utilized, an evaluation of the parameters of the traffic noise prediction model yielded good agreement between predicted and measured sound power levels.

Figure 2 :
Figure 2: Minimizing the error function of (8) through the HS algorithm.

Table 1 :
Vehicle types used in this study.

Table 2 :
A-weighted sound power levels ( WA ) in dB for a densegraded asphalt (DGA) surface.
*  is a velocity.
() is the objective function,  is the set of decision variables   ,  is the number of decision variables, and   is the possible range of values for the th decision variable; that is,   ≤   ≤   , where   and   are the respective lower and upper bounds for the th decision variable.To estimate the parameters of a noise prediction model, the following minimization function can be used:Minimize       eq(WA) −   eq(WA)     subject to   ∈   = 1, 2, . . ., ,

Table 3 :
A-weighted sound power levels ( WA ) in dB for the different surface types.
*  is a velocity.

Table 4 :
Vehicle velocities and hourly traffic flows.

Table 5 :
Determination of the model coefficients via the HS algorithm.

Table 6 :
Vehicle velocities and hourly traffic flows used as evaluation data.

Table 7 :
Comparison of predicted and measured traffic noise levels.