*2.5. Estimation of Track Modulus Standard Deviation* (*USD*)

The estimation of the track modulus standard deviation from the *Yrel* data using statistical methods and curve-fitting approaches has not been successful for track section lengths shorter than 80 m [21]. Therefore, frequency characteristics of the deflection data are investigated in this study to increase the estimation accuracy of track modulus standard deviation. The coefficients associated with the *Yrel* frequency components are employed as one of the inputs to the ANNs, whose outputs are the track modulus standard deviation over different track section lengths. As demonstrated in Figure 7, *Yrel* and the track modulus data are divided into different subgroups based on various track section lengths (similar to the procedure used for estimating the track modulus average). Then, statistical analysis, fast Fourier transform, and a liftering technique are applied on the *Yrel* data in each subgroup to extract the average and standard deviation of the *Yrel* and average and the standard deviation of liftering the fast Fourier transform (FFT) coefficients. These parameters are used as the inputs of ANNs.

Figure 8a shows an example of the FFT coefficients of the *Yrel* data over a track section of 30 m for 81 models. As can be seen, the coefficients at higher orders are relatively small. This is undesirable for training the ANN due to possible bias. Therefore, the coefficients are processed using a liftering technique (Equation (6) to roughly normalize their variances) [41]:

$$X'(k) = \left(1 + \frac{L}{2}\sin\left(\frac{\pi(k+1)}{L}\right)\right) \cdot X(k), \ k = 0, \ \dots, N-1\tag{6}$$

where *L* is the sin lifter parameter, which is 50 in the current study, and *X*(*k*) is the FFT coefficients.

Once the liftering technique is applied (Figure 8b), the average and standard deviations of the lifted FFT are calculated using Equations (7) and (8) are used as two additional inputs for ANNs.

$$P\_1 = \frac{2}{N-1} \sum\_{k=0}^{(N-1)/2} \left| \mathbf{X} \prime(k) \right| \tag{7}$$

$$P\_2 = \sqrt{\frac{2}{N-1} \sum\_{k=0}^{N/2} \left( \left| \mathbf{X} \prime(k) \right| - P(1) \right)^2} \tag{8}$$

The architecture used for developing the network in this section has two hidden layers and 15 hidden nodes in each layer, similar to the network's architecture for estimating the track modulus average. The trained networks are used for estimating the track modulus standard deviation over different track section lengths and three accuracy measurements are reported in Table 4. In order to show that the current input–output pair is optimized, and two network architectures are trained (ANN-1 with four inputs, i.e., the average and standard deviation of *Yrel*, and the average and standard deviation of the lifted FFT; ANN-2 with two inputs, i.e., mean and standard deviation of *Yrel*). In each case, the two networks are trained and tested multiple times and the mean and standard deviation of the performance parameters are computed and reported in Table 4. For the case of 5 m section length, for instance, the networks' input, and output are first extracted based on the chosen section (5 m), then ANN-1 and ANN-2 networks are trained using the training data and tested against the data extracted from nine unseen FEMs.

**Figure 7.** Procedure for estimating the track modulus standard deviation (*USD*).

**Figure 8.** FFT of *Yrel*: (**a**) before liftering; and (**b**) after liftering.


**Table 4.** Estimation accuracy of the *USD* (no noise added, the standard deviation in the parenthesis).

\* Standard deviation of the estimation error.

From Table 4, the error values show that the standard deviation of track modulus (*USD*) can be estimated satisfactorily by both network configurations (ANN-1 and ANN-2). Even for the 10-m section length case, for instance, the coefficient of correlations between the actual *USD* and the one estimated by the two networks are very high, e.g., 0.83 and 0.82 respectively. However, the networks with four inputs (ANN-1) slightly outperform the one with two inputs (ANN-2) regardless of the section lengths. Specifically, the RMSE and MAPE are always smaller than those arising from the trained networks whose inputs are the statistical properties of *Yrel* only (ANN-2). Values estimated using the networks with four inputs have relatively high *R*<sup>2</sup> in all cases showing that the methodology is successful. In particular, the *R*<sup>2</sup> is as high as 0.94 for the case of the 25 m section length and the RMSE is 1.83 MPa, which is a relatively small error considering that the maximum standard deviation of the inputted track modulus in the FEMs is 31.05 MPa. Moreover, the first network (ANN-1) provides more reliable results as the standard deviation of RMSE remains stable (varying from 0.11 to 0.17 MPa) and lower than those of ANN-2. Therefore, combining FFT and statistical analysis to configure the input for the networks noticeably improves the estimation accuracy, and increases the stability of the ANNs, the mapping function between the *Yrel* characteristics and the track modulus standard deviation (*USD*). Most importantly, there is a big step forward in this paper compared to the previous study, where the *R*<sup>2</sup> coefficient is 0.748 even though the 40 m section length is used [21]. The performance of this estimation can be considered ineffective as the *R*<sup>2</sup> coefficient reduced significantly in shorter track segment cases (Table 4). Hence, considering the current results, it can be claimed that neural networks are more powerful for mapping the relationship between *Yrel* and track modulus, especially over the short track section lengths.

For more descriptive results, the strong correlation between the actual and estimated track modulus's standard deviation for the 25 m section length is demonstrated in Figure 9. As can be seen, the estimated standard deviations follow the same patterns as those of the actual values, which vary greatly from 3.2 to 31.05 MPa.

**Figure 9.** The actual track modulus standard deviation over the 25 m section length vs. the estimated values.

The effectiveness of the methodology is further validated by adding noise into the deflection data (*Yrel*). Similar to the procedure mentioned in the previous section, noise is added to the *Yrel* data from 90 models using Equation ((5). The dataset from 81 models is then used to train the networks using two approaches: networks with two inputs (average and standard deviation of *Yrel*) and networks with four inputs (average and standard deviation of *Yrel* and average and standard deviation of the lifted FFT). The developed networks are used to estimate the track modulus standard deviations over the different section lengths from the unseen *Yrel* data. The estimated values are compared with the standard deviation of track modulus inputted to FEMs and results are reported in Table 5. The results show that the proposed approaches work very well even when *Yrel* datasets are affected by noises. The *R*<sup>2</sup> is again higher than 0.90 when the 25 m or higher section lengths are utilized.


**Table 5.** Estimation accuracy of the *USD* (with noise added).
