# On the variants of SVM method for the estimation of soil elastic modulus from TSD model: Numerical parametric study

Abdelgader Abdelmuhsen, Jean-Michel Simonin, Franziska Schmidt, Denis

Lièvre, Alexis Cothenet, Amine Ihamouten

► To cite this version:

Abdelgader Abdelmuhsen, Jean-Michel Simonin, Franziska Schmidt, Denis Lièvre, Alexis Cothenet, et al.. On the variants of SVM method for the estimation of soil elastic modulus from TSD model: Numerical parametric study. Transportation Engineering, 2023, 13, pp.100187. 10.1016/j.treng.2023.100187. hal-04197395

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# On the Variants of SVM Method for the Estimation of Soil Elastic Modulus from TSD Model: Numerical Parametric Study

A. ABDELMUHSEN<sup>a,*</sup>, J-M. SIMONIN<sup>a</sup>, F. SCHMIDT<sup>b</sup>, D. LIÈVRE<sup>a</sup>, A. COTHENET<sup>a</sup> and
A. IHAMOUTEN<sup>a</sup>

## ARTICLE INFO

### Keywords:

- Soil Mechanics
- Subgrade Resilient Modulus
- Traffic Speed Deflectometer
- Deflections Velocity Slope
- Numerical Model
- Machine Learning
- Support Vector Machine

## ABSTRACT

The accurate estimation of the subgrade resilient modulus $M_R$ is a crucial element in designing durable and resilient pavements as it quantifies the soil's bearing capacity to withstand traffic loads without excessive deformation or failure. Conventional methods and devices that are used to estimate $M_R$, such as the Falling Weight Deflectometer (FWD) have several limitations due to operational and security constraints when measuring at the network level. To overcome these limitations, Traffic Speed Deflectometer (TSD) has been developed, which can continuously assess the pavement bearing capacity without requiring traffic control. However, processing the massive amounts of data generated by TSD requires specialized processing techniques such as Machine Learning (ML). Therefore, the primary objective of this paper is to develop an efficient and effective model that can accurately estimate $M_R$ by combining two techniques: (TSD + ML). The study uses numerical simulations of deflection velocity slope $S_V$ under TSD as the forward model. The research employs Support Vector Machine (SVM) algorithm for ML to classify and estimate $M_R$ modulus values from a synthetic and controlled dataset. SVM model exhibits robust performance in both classification and regression analysis, whereas, 95% of the discrepancies between estimated and tested values did not exceed $\pm$15 MPa. Advanced numerical validation, sensitivity analysis and parameter estimation methods were also conducted to optimize the model's performance. Overall, the approach proposed in this paper has the potential to make a significant impact in the field of subgrade evaluation and can facilitate the development of better road infrastructure.

## 1. Introduction

The Subgrade Resilient Modulus $M_R$ is an indicator measure of the stiffness of the soil or aggregate materials that forms the foundation of a pavement system. The evaluation of the soil's bearing capacity is a crucial factor in designing new pavements and overlays since it has a direct impact on the pavement's capacity to resist deformation and endure traffic loads. Infrastructure authorities rely on this assessment to guide their decisions regarding maintenance and rehabilitation planning for the Pavement Management System (PMS) at both network and project levels [1]. Due to the complexity of punctual and destructive laboratory testing procedures, highway agencies have been investigating Non-Destructive Techniques (NDT) to determine $M_R$. The Falling Weight Deflectometer (FWD) is widely used in the industry for this purpose. However, there are several operational limitations associated with FWD that should be taken into consideration. One of these limitations is that the FWD is stationary in nature, which makes it a relatively expensive and time-consuming device, particularly when network-level measurements are required. Moreover, FWD requires traffic management during the test, which means more difficulties in controlling traffic flow and concerns regarding workplace safety [1, 2].

To overcome these challenges, various devices have been developed in recent years for continuous bearing capacity

assessment. One such device is the Traffic Speed Deflectometer (TSD). TSD is becoming more attractive because it provides continuous measurements over long distances. It is also relatively fast (+80 km/h) and can survey large areas in a short amount of time, allowing for network-level pavement condition assessments. In addition, TSD does not require traffic control or lane closures during testing, which reduces the impact on traffic flow [3, 4].

Given that, TSD has the capacity to cover a significant section of roadways, the resulting production of massive amounts of data presents a considerable challenge for TSD operators. These challenges include the need for specialized methods and processing techniques to efficiently handle and extract meaningful information from the databases as conventional methods of data analysis are no longer sufficient [5].

To address this challenge, Machine Learning (ML) has gained immense popularity in tackling various data-related challenges. This is primarily due to its remarkable ability to learn patterns and gain insights from vast amounts of data. One of the key advantages of ML is that it automates tasks such as data cleaning and model selection. This automation enables the analysis of large data volumes, revealing hidden patterns and relationships that may not be immediately apparent. As a result, ML can produce better predictions and inform decision-making processes, particularly when rigorous feature engineering has been performed to ensure relevant databases are available with labeled and controlled discriminate features [6].

<footnote><sup>a</sup>Univ Gustave Eiffel, MAST-LAMES, Nantes campus, F-44344 Bouguenais, France, abdelmuhsen.abdelgader@univ-eiffel.fr</footnote>

<footnote><sup>b</sup>Univ Gustave Eiffel, MAST-EMGCU, F-77454 Marne-la-Vallée Cedex 2, France, franziska.schmidt@univ-eiffel.fr</footnote>

<footnote>*Corresponding author
ORCID(s): 0000-0002-7077-0591 (A. ABDELMUHSEN)</footnote>
Today, the limitations of the existing back-calculation methods, which rely on multilayer elastic theory and static loading characteristics derived from FWD data, underscore the need for alternative models. Consequently, there is an urgent disire to develop a new model that not only addresses these limitations but also provides improved capabilities and solutions, while bridging the existing research gap and making a substantial contribution to the state of the art [2, 3].

Therefore, this research study is motivated by tackling
two crucial challenging to address the existing research gap
and contribute to its remediation.

The first motivation stems from the disparities encoun-
tered when converting TSD measurements into FWD for
the purpose of estimating pavement moduli. The differences
in loading configuration, with FWD using a single plate
while TSD uses twin tires, as well as the distinct loading
mechanisms, with FWD utilizing an impulse load and TSD
employing a moving load, pose notable challenges in directly
applying conventional back-calculation techniques to TSD
data [3, 4].

Whereas, the second motivation stems from the increas-
ing adoption of TSD and the expansion of network-level de-
flection testing, which generate substantial volumes of data
that traditional methods struggle to handle effectively. Thus,
conventional approaches prove impractical in processing,
analysing and extracting meaningful insights from the TSD
big-data [5, 6].

As a consequence, this research paper introduces a innova-
tive model that aims to accurately estimate the Subgrade
Resilient Modulus *M*<sub>R</sub> by employing a combination of (TSD
+ ML) techniques. However, in contrast to conventional
methods that rely on deflection data, the applied approach
directly utilizes the deflection velocity slope *S*<sub>V</sub> to estimate
the soil elastic modulus *M*<sub>R</sub>, eliminating the need for its
numerical integration to obtain deflection values under the
TSD.

To accomplish this, the study uses as a first step a
model that simulates numerically the deflection velocity
slope S<sub>V</sub> under TSD as the foundation for this investigation.
In essence, the paper seeks to conduct a numerical study with
the use of a ML-based approach to estimate M<sub>R</sub> from TSD
data, via exploring the correlation between the input features
(deflection velocity slope values of seven TSD laser Doppler
sensors, Sn<sub>1</sub> to Sn<sub>7</sub>) and the predicted output variable M<sub>R</sub>.

The methodology of this paper has been structured into
various subsections in order to implement the proposed solu-
tion of accurately estimating the subgrade resilient modulus
*MR* under certain hypotheses. The primary objective of the
first section is to establish a numerical robust forward model,
that allows to construct a controlled physical representative
database based on the simulation of the TSD deflection
velocity slope *S*<sub>V</sub>. The forward model is based on *Burmister*
elastic linear isotropic approach with infinite in-plane lay-
ers as the analytical model [7]. Subsequently, in order to
enhance the model's precision and better replicate the real-
world noise observed during TSD measurement, a random
noise with a normal distribution that oscillates according to

the modeled TSD deflection is added. The additive white
Gaussian Noise (AWGN) produced through this process is
utilized to demonstrate the effects of real-world noise on
TSD data [8].

The second section of the paper is focused on utilizing
ML algorithm called Support Vector Machine (SVM) to
analyze and estimate the numerical M<sub>R</sub> modulus value from
the synthetic and controlled dataset [9].

Specifically, The library scikit-learn is used for ML in Python [10]. Support Vector Classification (SVC) is utilized to classify soil status based on their M<sub>R</sub> modulus values, whereas Support Vector Regression (SVR) is employed to achieve precise estimates of the M<sub>R</sub> modulus as a continuous value [11]. The classification of the M<sub>R</sub> is done according to the French catalog of structures and the French technical guide [12, 13].

In addition, the SVM process involved several steps,
beginning with the training of the models on a selected
portion of the database, known as the training data. Then the
performance of the model was evaluated on the rest of the
database, referred to as the testing data. To validate the ac-
curacy of the estimated M<sub>R</sub> modulus, statistical analysis was
conducted by comparing it with the deflection slope obtained
from Alizé2®, which is a commonly used back-calculation
software in the French industry [14]. This software employs
the Burmister model and the French infrastructure catalog.

The last section of the paper focuses on the numerical
validation and parameter estimation of the SVM process,
which involves tuning the classification and regression hy-
perparameters to optimize the SVM’s performance [15]. The
section also discusses the variability analysis to investigate
the impact of the variation in the elastic modulus of three
layers (E₁, E₂, and E₃) on the accuracy of M<sub>R</sub> estimation.

## 2. Forward Model

**2.1. Analytics-driven case study**

As it could be observed in (Fig 1) as well as in (Table 1), this paper is interested in the modulus column and particularly in M<sub>R</sub>. To accurately estimate M<sub>R</sub> modulus (Fig 1), it is important to create a reliable numerical forward model. This numerical forward model is developed by generating simulated data of S<sub>V</sub> under different M<sub>R</sub> values, which will be used to train the ML model and estimate M<sub>R</sub> based on TSD simulated measurements. The details of this process will be explained in the data processing subsection.

To create the numerical forward model, an analytical-
driven case study was conducted, utilizing predefined pa-
rameters (feature engineering) and dataset to develop the
controlled synthetic database. The purpose of choosing this
specific data frame was to conduct a comprehensive analysis
of classification and regression with the utmost accuracy to
categorize and estimate the M<sub>R</sub> value. (Table 1) illustrates
the characteristics of the data set in question.

The data presented in (Fig 1) shows that the Modulus
of Resilience *M*<sub>R</sub> value increased from 16 to 250 MPa. In
the first step, all other layers were kept for the time being
<caption>**Table 1**</caption>

The characteristics of the pavement structure That Utilized to
derive the forward model (The French catalog [12, 13])

<table><thead><tr><th>Layer (i)</th><th>Thickness (m)</th><th>Material</th><th>E<sub>i</sub> (MPa)</th><th>ν</th><th>Interface</th><th>°C</th></tr></thead><tbody><tr><td>1</td><td>0.06</td><td>HMA</td><td>7000</td><td>0.35</td><td>(1) bound</td><td>15</td></tr><tr><td>2</td><td>0.09</td><td>BC - g2</td><td>9300</td><td>0.35</td><td>(1) bonded</td><td>15</td></tr><tr><td>3</td><td>0.09</td><td>BC - g2</td><td>9300</td><td>0.35</td><td>(1) bonded</td><td>15</td></tr><tr><td>4</td><td>∞</td><td>Pf<sub>4</sub></td><td>16-250</td><td>0.35</td><td>(1) bonded</td><td>15</td></tr></tbody></table>

constant with a single modulus value to ensure consistency throughout the study and also to investigate later on, the sensitivity and the influence of each layer's modulus on the soil independently and distinctively. Furthermore, the temperature for all layers was kept constant at 15 (°C) to minimize the impact of temperature variations on the results.

In the subsequent sections, the research results will be elaborated upon in greater detail, with the aim of enhancing the understanding of the study's approach and findings. Special attention will be given to the outcomes of the ML analysis, including the classification and regression analysis that were conducted to assess their relationship with the $M_R$.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Structure of the pavement under study [13]. In scientific terminology from [12], (HMA) denotes a Hot Mix Asphalt. Similarly, (BC - g2) represents grade 2 aggregate bituminous concrete, while (pf4) refers to class 4 subgrade.</figcaption></figure>

## 2.2. Mechanical Behaviour Modelling

To simulate the pavement behavior under TSD system, a load superposition principle simulation was applied. The use of the superposition principle is justified as it aligns with the deflection behavior hypothesis assumed in this study, which is Burmister's elastic linear isotropic approach. This approach is a widely accepted and utilized method in pavement mechanics engineering for evaluating soil behavior under loading conditions [17].

In principle, TSD system consists of ten wheels (Fig 2a), therefore, the simulation involves summing the deflection value of all TSD wheels to calculate the deflection under the entire TSD system [3]. Once the deflection behavior of the entire TSD is simulated as shown (Fig 2b), $S_V$ is then calculated by taking the first-order partial derivative of the superpositioned deflection. This step allows for simulating the rate of change of the deflection behavior of the pavement under the entire TSD. Deflection under TSD and $S_V$ curves are shown (Fig 2b). To achieve the latter step, TSD load configuration and coordinates have to be defined as shown in (Fig 2a), which also shows the position of seven laser Doppler sensors $Sn_1$ to $Sn_7$ relative to the TSD wheel loads. These sensors measure $S_V$ beneath the TSD wheel loads. The distance of each sensor from the loading point is considered to be 0.1, 0.2, 0.3, 0.45, 0.6, 0.9, and 1.1 m, respectively. To this end, for the sake of simplifying the calibration process of TSD, a value of zero was artificially assigned to the reference sensor $Sn_8$ [16] which is located at the eighth position (3.8 m). However, this assumption causes a systematic bias in the TSD measurements. The impact of $Sn_8$ on TSD measurement uncertainty is illustrated in Figure 2b. The variance between the green curve ($Sn_8 = 0$) and the black curves ($Sn_8 \neq 0$) demonstrates the effect of $Sn_8$ on the uncertainty of TSD measurements. Furthermore, (Fig 2b) illustrates that the calculation of deflection is carried out at intervals of 1 cm for every 4 m offset, which is also supported by previous research [18]. The aforementioned facts play an instrumental role in enhancing our comprehension of the TSD numerical configuration process and the factors that are considered while simulating pavement deflection behavior.

# 3. Data Modelling and Processing

## 3.1. Numerical Dataset

This part introduces the numerical data collection and the characteristics of the input features used in this study. A dataset consisting of 235 instances of the simulated $S_V$, is generated as an equivalent value for varying $M_R$ modulus values that range from 16 to 250 MPa. The values of $Sn_1$ to $Sn_7$ represent the simulated TSD laser Doppler deflection velocity slope. $Sn_1$ is the sensor located closest to the loading point and $Sn_7$ is the farthest. The target variable $M_R$ represents the soil modulus of elasticity, which has to be estimated and categorized based on the values of $Sn_1$ to $Sn_7$.

The analysis of (Fig 3) reveals that there is increased variability in $S_V$ observed at the locations farther from
<figure><img src="image_3.png" /><figcaption>Figure 2: TSD Numerical Model: (a) TSD load configuration [16], (b) Numerical simulation of pavement behavior under TSD.</figcaption></figure>

the loading center. Hence, sensors positioned at the outer periphery of a deflection basin can provide more detailed information about the mechanical response and bearing capacity of the soil. This observation supports the findings of [19] which states that a correlation exists between deflection bowl parameters and mechanistic evaluations of pavement structures. These parameters are valuable in identifying sections with varying levels of structural capacity and behavior states.

## 3.2. Distribution of $M_R$ Classes

The French catalog of structures categorizes soil-bearing capacity into five different classes based on the modulus values [12, 13]. (Fig 4) displays the percentage distribution of the current elastic modulus $M_R$ value within each platform.

These classes are referred to as $pf_1$ to $pf_5$, with modulus values of 16-49 MPa, 50-79 MPa, 80-119 MPa, 120-199 MPa, and +200 MPa, respectively. This classification is used to evaluate the potential bearing capacity of a given soil type for construction purposes.

## 3.3. Additive Noise Effect

The purpose of adding Additive White Gaussian Noise (AWGN) signal to the syntactic data is to demonstrate the impact of real-world noise associated with TSD measurements. In other words, The measurements of TSD taken in real-world settings are susceptible to several sources of interference or unwanted noised signals, which can affect the precision of the collected data [18]. As could be observed from the classification and regression results section,
<figure><img src="image_4.png" /><figcaption>Figure 3: the characteristics of the input features $Sn_1$ to $Sn_7$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Percentage distribution of $M_R$ within each platform $pf_1$ to $pf_5$</figcaption></figure>

these sources of noise can compromise the accuracy of the obtained simulated measurements. Therefore, by adding AWGN to the numerical model, researchers can simulate this noise and study its effects on the model's performance.

To illustrate the effect of the noise on TSD measurements (Fig 5) shows the slope velocity with and without noise. In this context, three $S_V$ (100, 150, and 200 MPa) of different classes have been chosen from the data set. These curves come along with their corresponding $M_R$ value and the position of $Sn_1$ to $Sn_7$ values. The resulting $S_V$ of these simulations were analyzed to observe the impact of 15 (dB) noise addition. To this end, the addition of the noise to the numerical model improved its realism, by enabling to study of the impact of real-world noise on TSD measurements [18]. The noise addition effects on the performance of the proposed method will be studied and discussed in the next parts of the paper.

# 4. SVM Model: Protocol Setup, results and discussion

## 4.1. Train Test Split

The ultimate purpose of the SVM model is to establish a correlation between the input features $Sn_1$ to $Sn_7$ and the predicted output variable $M_R$. To initiate the SVM process, the dataset is randomly divided into two parts, with 70% of the data used for training the SVM model and the remaining 30% used for testing. This approach ensures that the results are reliable and can be compared fairly. The Python library sci-kit-learn is used to apply SVM model, and the topological structure of the SVM is determined by training different hyperparameters and monitoring their performances. The aim is to obtain an SVM model that can establish a strong correlation between the input features and the predicted output variable.

As shown in (Fig 6), the study involved training the SVM model using the Radial Basis Function (RBF) kernel, which was found to produce superior results compared to other types of kernels. The performance of the SVM model was evaluated using several parameters, including the Co-efficient of Correlation (R), Coefficient of Determination (R²) and the Root Mean Square Error (RMSE) between the predicted and actual $M_R$ values. The ultimate goal of the SVM model is to obtain a high (R²) value and a low (RMSE) value, indicating a strong correlation between the input features and the predicted output variable. Therefore, optimizing the performance of the SVM model via using rel-evant hyperparameters and metrics is critical in developing an effective and reliable SVM model [18]. To this end, it is worth to mention that, all calculations were performed on Google Colab, a cloud-based computing platform. The virtual machine that was utilized had an NVIDIA Tesla P100 GPU with 16 GB of memory, 12 GB of RAM, and 100 GB of storage, and it ran on Ubuntu 18.04 LTS. Overall, using cloud-based computing platforms can significantly reduce the time required for complex calculations compared to traditional computing methods, making it a more efficient and cost-effective solution.

## 4.2. Classification Analysis

In the present study, a classification model was developed and used to classify data into different classes based on their current elastic modulus values. (Fig 3) of this research displayed the percentage distribution of the current elastic modulus value $M_R$ in each class. To evaluate the performance of the model, a confusion matrix is used to allow relevant comparison between prediction and ground-truth results [20].

Accordingly, (Fig 7) shows a multi-class classification confusion matrix, where the majority of the entries in the matrix are along the diagonal, indicating correct predictions. This means that SVC model correctly identified most of the instances in each class. The off-diagonal entries represent the outliers made by the model. Upon analyzing the observations presented in (Fig 7), it becomes apparent that class 1 was predicted correctly for all instances, except for one instance
<figure><img src="image_6.png" /><figcaption>**Figure 5:** $S_V$ along with $S_{n_1}$ to $S_{n_7}$ representation: right figure (noise-Based) and left figure (noiseless)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Target variable $M_R$ Train-Test split</figcaption></figure>

that was misclassified as class 2. Similarly, for class 4, the model accurately predicted 26 instances but misclassified one instance as class 5. These misclassifications can be attributed to overlapping features and noise allocation to the raw data. Overlapping features can make it challenging for the model to distinguish between classes with similar characteristics, while noisy data may lead the model to classify instances based on inaccuracies, resulting in misclassifications.

To this end, The accuracy of the model, which is the proportion of correct predictions out of all predictions, is 0.95, meaning that the model is correct in almost 95% of its predictions. This is a good indication that the model is performing well on this particular test set.

In the context of evaluating a classification model, evaluation metrics [11] are used to assess the performance of the model in predicting the correct class labels for a given set of data. The metrics mentioned in this statement are commonly used in classification tasks and provide different measures of the model's performance which are Precision,

<figure><img src="image_8.png" /><figcaption>**Figure 7:** Confusion matrix for $M_R$ classification analysis</figcaption></figure>

Recall, F-score, Dice score, Accuracy and Support. Precision measures the proportion of true positive predictions to the total number of positive predictions. Recall measures the proportion of true positive predictions to the total number of actual positive instances. F-score is a weighted average of precision and recall. Dice score evaluates the overlap between predicted and true positive instances. Accuracy measures the proportion of correct predictions to the total number of instances. Support refers to the number of instances in each class and is useful for identifying imbalanced data-sets. These metrics evaluate the model's performance in predicting correct class labels and provide different measures of the model's strengths and weaknesses.
<figure><img src="image_9.png" /><figcaption>**Figure 8:** Evaluation metrics for a classification model analysis</figcaption></figure>

Based on the evaluation metrics (Fig 8), the SVC model appears to be performing well. The precision values for most classes are high, indicating that the model correctly identifies a large proportion of true positives. The recall values are also high, meaning that the model correctly identifies a large proportion of true positives out of all actual positives. The F-score, which is the harmonic mean of precision and recall, is also high for most classes, indicating a good balance between precision and recall and similarly the dice score.

### 4.3. Regression Analysis

This paragraph presents the outcomes of an analysis that evaluated the efficacy of using SVR to estimate the *M*<sub>R</sub> modulus based on deflection velocity slope values with and without noise. (Fig 9) and (Fig 10) illustrates the comparison between the results obtained from this analysis. The result analysis exhibits the impact of noise on the model's accuracy. The presence of noise increased the (RMSE) percentage and decreased the (R<sup>2</sup>) value, indicating that noise has a significant influence on the accuracy of the measurement.

However, even with the presence of the noise, it is evident from (Fig 10) that the SVR model performs strongly, as indicated by high values of (R) and (R<sup>2</sup>). Specifically, an (R<sup>2</sup>) value of 95% indicates a strong positive correlation between the estimated and actual *M*<sub>R</sub> values. Additionally, the (R<sup>2</sup>) value of 95% indicates that *S*<sub>V</sub> values can explain 95% of the variance in *M*<sub>R</sub>. Also, Root Mean Square Error (RMSE) value of 15% indicates the average distance between the predicted and actual values of *M*<sub>R</sub>. A lower (RMSE) value indicates better predictive accuracy. However, the fact that the (RMSE) value is not zero suggests that the model still has some degree of error, mainly due to the presence of noise in the data. In addition, the study's data-set has controlled characteristics, and as a result, the *M*<sub>R</sub> test prediction value implies a quasi-inversely proportional linear relationship between *S*<sub>V</sub> simulated measurement and *M*<sub>R</sub> modulus. This hypothesis indicates that as the *M*<sub>R</sub> increases, the *S*<sub>V</sub> decreases but it is expected to

be approximately linear. This relationship can be explained by the fact that a higher *M*<sub>R</sub> indicates a stiffer and stronger pavement soil structure that can better resist deformation under TSD loading. However, this relationship is likely to be influenced by other factors, such as the variation in the pavement structure, temperature, and loading conditions.

<figure><img src="image_10.png" /><figcaption>**Figure 9:** SVR (noiseless) Model Performance Analysis</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10:** SVR (noise-based) Model Performance Analysis</figcaption></figure>

Furthermore, to quantify the uncertainty in the SVR model a variance analysis was conducted and interpreted. The results of the analysis are presented in (Fig 11). The variance analysis involved comparing the predicted values to the actual test values, and it was observed that around 95% of the differences between the estimated and tested values were within the acceptable range of ±15 MPa. This range is considered acceptable based on the inherent variability of the data and the current state of the art in this domain [21].

Hyperparameters in a support vector regression (SVR) model are the parameters that cannot be learned from the data and need to be set before training the model. The hyperparameters in an SVR model include the kernel type, regularization parameter, and gamma [22].

The kernel type determines the type of non-linear transformation applied to the data in order to make it separable by a linear boundary. Popular kernel types include linear, polynomial, and radial basis function (RBF) kernels.
<figure><img src="image_12.png" /><figcaption>**Figure 11:** SVR (noise-based) Model Performance Analysis</figcaption></figure>

<table><caption>**Table 2**
SVR Grid search Hyperparameter Tuning</caption><thead><tr><td>Index</td><td>SVM Parameter</td><td>Value</td></tr></thead><tbody><tr><td>1</td><td>Kernel</td><td>RBF</td></tr><tr><td>2</td><td>ε precision</td><td>0.9</td></tr><tr><td>3</td><td>C tolerance parameter</td><td>2000</td></tr><tr><td>4</td><td>γ kernel scale factor</td><td>50</td></tr></tbody></table>

The regularization parameter (C) controls the trade-off between achieving a low training error and a low testing error by adjusting the penalty for misclassifications.

Gamma γ is a hyperparameter that is specific to RBF kernels and controls the shape of the decision boundary.

Choosing appropriate hyperparameters is crucial for achieving a well-performing model. The hyperparameters need to be tuned using techniques like cross-validation to avoid overfitting or underfitting the data.

Fine-tuning the hyperparameters of an SVM model [22] refers to the process of systematically adjusting these hyperparameters to achieve the best possible performance on a given dataset. This process involves training and evaluating the model with different combinations of hyperparameter values to find the optimal combination that produces the highest level of accuracy.

In this paper, Grid Search (GS) is a fine-tuning technique used to find the optimal combination of hyperparameters for a given model. In the context of (SVR), the hyperparameters to be tuned are shown in (Table 2).

The optimal SVR hyperparameter selected in this study is comparable to the hyperparameters reported in the literature for the same application, which were based on almost

similar results. This approximation may validate the choice of hyperparameter range used in this study as being acceptable, considering the inherent variability of the data and the current state of the art in this domain [23]. Based on the Grid search values presented in (Table 2), it can be inferred that the SVM model is using the RBF kernel, which is a popular choice for nonlinear tasks. The ε precision is set to 0.9, which means that any data points within the margin of tolerance (the epsilon tube). This helps the SVM model to be more flexible and better generalize to unseen data.

By setting the C tolerance parameter to 2000, the SVM model is designed to penalize errors significantly. A higher C value increases the risk of overfitting, as the model may excessively tailor itself to the training data and struggle to generalize to new data. Conversely, a lower C value may result in underfitting, where the SVM model is too simplistic and fails to capture the intricacies of the data. Therefore, the choice of C value requires a careful balance between achieving a low training error and a low testing error that is not over/under-fitting the model.

In the given table also, the γ value is 50, which means that the decision boundary will be relatively smooth, and the algorithm will give more weight to the training examples farther away from the decision boundary.

By fine-tuning the hyperparameters of an SVR model, it is possible to achieve an accuracy level of around 95%. This accuracy level is considered very good, which means that the model is able to make accurate predictions with a high degree of confidence. This implies that the fine-tuning process has led to the identification of the best combination of hyperparameters for the given dataset, resulting in a highly accurate and reliable model.

### 4.4. Effect of ($E_i$) Variability on the Accuracy of $M_R$ Estimation

To ensure the reliability of estimating the $M_R$ modulus, it is important to account for the variability in the $E_i$ modulus of the top layers. This can be achieved by accurately determining $E_i$ at each layer and incorporating this variability in the estimation process.

In this paper, the variability analysis was conducted to investigate the impact of the variation in the elastic modulus of three layers ($E_1$, $E_2$, and $E_3$) on the accuracy of $M_R$ estimation. The study involved defining the values of $E_i$ before and after varying each layer's elastic modulus by ±10% while keeping the soil modulus constant at 125 MPa.

The results of the study are presented in (Table 3) as well as (Fig 12) which shows the estimated values of $M_R$ and the variance percentage for each case.

The results demonstrate the following points:

* The elastic modulus $E_i$ of each layer has an impact on the estimated soil modulus. Any change in $E_i$ leads to a corresponding change in the estimated soil modulus. Specifically, an increase in $E_i$ results in an overestimation of the estimated soil modulus, while a decrease in $E_i$ leads to an underestimation of the estimated soil modulus. These findings suggest that
<table><caption>**Table 3**
Effects of variability in layers elastic modulus on the reliability of estimating the soil</caption><thead><tr><th>E<sub>i</sub> [MPa] (±10 %)</th><th>E<sub>i</sub> [MPa]</th><th>E<sub>i</sub> [MPa]</th><th>E<sub>i</sub> [MPa]</th><th>M<sub>R</sub> [MPa]</th><th>M<sub>R</sub>-Estimated MPa</th><th>V% = (σ/μ × 100)</th></tr></thead><tbody><tr><td>E<sub>i</sub> (+10 %)</td><td>7700</td><td>9300</td><td>9300</td><td>125</td><td>≈ 127</td><td>0.79%</td></tr><tr><td>E<sub>i</sub> (-10 %)</td><td>6300</td><td>9300</td><td>9300</td><td>125</td><td>≈ 122</td><td>1.21%</td></tr><tr><td>E<sub>i</sub> (+10 %)</td><td>7000</td><td>10230</td><td>9300</td><td>125</td><td>≈ 131</td><td>2.34%</td></tr><tr><td>E<sub>i</sub> (-10 %)</td><td>7000</td><td>8370</td><td>9300</td><td>125</td><td>≈ 118</td><td>2.88%</td></tr><tr><td>E<sub>i</sub> (+10 %)</td><td>7000</td><td>9300</td><td>10230</td><td>125</td><td>≈ 136</td><td>4.21%</td></tr><tr><td>E<sub>i</sub> (-10 %)</td><td>7000</td><td>9300</td><td>8370</td><td>125</td><td>≈ 111</td><td>5.39%</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Figure 12:** Eᵢ variability effects on M<sub>R</sub> estimation</figcaption></figure>

accurate estimation of the soil modulus requires careful consideration of Eᵢ of each layer.

* The results show that the Variance Percentage (V%), which is defined as the ratio of the Standard Deviation σ to the Mean μ. V% is important indicator in measuring the spread of data, and it increases as the variability in Eᵢ of any layer increases. This means that estimating M<sub>R</sub> becomes less reliable when there is more variability in the Eᵢ of any layer. Therefore, it is crucial to take into account the variability in

each layer's Eᵢ when estimating M<sub>R</sub> for accurate and reliable results.

* The results highlight also that the variability in E₃ of the third layer has a greater impact on the variability in the estimated M<sub>R</sub> compared to the variability in the first and second layers E₁ and E₂ respectively. This suggests that careful attention should be paid to E₃ of the third layer while estimating the soil modulus accurately.

## 5. Conclusion

This research contributes to fill the research gaps by exploring the implementation of Machine Learning (ML) to automate the analysis of Traffic Speed Deflectometer (TSD) data for the accurate estimation of the subgrade resilient modulus (M<sub>R</sub>) in road pavements.

* The forward model employed in this study involves employing TSD modeling and numerical simulation of deflection velocity slope S<sub>v</sub> to assess the subgrade resilient modulus (M<sub>R</sub>). This aspect of the research is pivotal to the estimation procedure, and to accomplish this, a computer program called Alizé2® software is used to generate a synthetic numerical database.

* The data processing part outlines the statistical characteristics of the input features, along with the procedures for data collection, pre-processing and analysis techniques.

* The impact of noise on the accuracy of (TSD) measurements was investigated using signal processing and optimization techniques. The analysis of the results revealed that the presence of noise had a substantial influence on the measurement accuracy. Therefore, there is a need for further improvements in TSD measurements to mitigate the detrimental effects of noise.

* In the machine learning section, the SVM model shows efficient performance in classification and regression analysis, with (R²) value of 95% and (RMSE) value of 14.8%, this results indicate that SVM model is reasonably accurate with some degree of error mainly due to the noise. On the other hand, the analysis of discrepancies between the estimated and tested values remained within the range of ±15 MPa.

* The paper's final section presents the findings of an investigation highlighting the importance to consider the variability in the elastic modulus Eᵢ of each layer when estimating M<sub>R</sub>. The results indicate that changes in Eᵢ introduce uncertainties in M<sub>R</sub> estimation, emphasizing the significance of accurate Eᵢ values. Moreover, the study reveals that the elastic modulus E₃ of the third layer has a greater impact on the variability of M<sub>R</sub> estimation compared to the first and second layers.
# 6. Future Works

In light of the applicability of these findings and as a perspective, integrating the proposed model into existing Pavement Management System (PMS) could lead to improve the efficiency of the pavement's Structural Health Monitoring (SHM).

Furthermore, the paper highlights the potential for future research to explore the use of other machine learning models to optimize the proposed approach's performance and potentially expand its application to other areas of pavement engineering.

However, it is essential to acknowledge the limitations of the study, such as the use of synthetic data, which may not completely reflect the real-world conditions. Therefore, as a complementary extension and future works, this paper will be followed by another study aimed at experimentally validating the numerical model.

In essence, the proposed approach outlined in this paper, along with the forthcoming one, possesses a remarkable capacity to substantially elevate the assessment of subgrade conditions. As a result, it paves the way for noteworthy advancements in road infrastructure, promising notable improvements in its overall quality and functionality.

# 7. Conflict of Interest

The authors have stated that they do not have any competing financial interests or personal relationships that may have influenced the research reported in this paper.

# 8. Acknowledgement

On behalf of all authors, the first author thanks Région Pays de la Loire (France) for their support and contribution to their academic pursuits, by co-financing their PhD thesis, which this paper is a part of it.

# References

[1] S.S. Park, A. Bobet, T.E. Nantung, Correlation between Resilient Modulus (M R) of Soil, Light Weight Deflectometer (LWD), and Falling Weight Deflectometer (FWD), (2018).

[2] D. Kim, Y. Ji, N.Z. Siddiki, Evaluation of in-situ stiffness of subgrade by resilient and FWD modulus, (2010).

[3] M. Nasimifar, R. Kamalizadeh, B. Heidary, The Available Approaches for Using Traffic Speed Deflectometer Data at Network Level Pavement Management System, Measurement. (2022) 111901.

[4] M. Zhang, R. Xiao, Y. Ma, X. Jiang, P.A. Polaczyk, B. Huang, Evaluating structural characteristics of asphalt pavements by using deflection slopes from traffic speed deflectometer, Construction and Building Materials. 365 (2023) 130052.

[5] M.S. Lasalle, G.W. Flintsch, S.W. Katicha, A.S. Brand, B.K. Diefenderfer, A Big-Data-Based Approach Towards Concrete and Composite Pavement Management and Rehabilitation, (2022).

[6] L. Zhou, S. Pan, J. Wang, A.V. Vasilakos, Machine learning on big data: Opportunities and challenges, Neurocomputing. 237 (2017) 350-361.

[7] D.M. Burmister, L. Palmer, E. Barber, T. Middlebrooks, The theory of stress and displacements in layered systems and applications to the design of airport runways, in: Highway Research Board Proceedings, 1944.

[8] X. Huang, Stochastic resonance in a piecewise bistable energy harvesting model driven by harmonic excitation and additive Gaussian white noise, Applied Mathematical Modelling. 90 (2021) 505-526.

[9] A. Roy, S. Chakraborty, Support vector machine in structural reliability analysis: A review, Reliability Engineering System Safety. (2023) 109126.

[10] O. Kramer, O. Kramer, Scikit-learn, Machine Learning for Evolution Strategies. (2016) 45-53.

[11] H. Huang, X. Wei, Y. Zhou, An overview on twin support vector regression, Neurocomputing. (2022).

[12] Service of Technical Studies for Roads and Motorways (SETRA), Central Laboratory of Bridges and Highways (LCPC), GUIDE TECHNIQUE, French Design Manual for Pavement Structures (English Version), SETRA, Paris, France, 1998.

[13] Association Française de Normalisation (AFNOR), NF 98-086, Dimensionnement des structures routières, application aux chaussées neuves [Design of road structures, application to new pavements], AFNOR, Paris, France, 2019.

[14] ALIZÉ 2 Software. Laboratoire Central des Ponts et Chaussées, Paris, France, 2021.

[15] R. Laref, E. Losson, A. Sava, M. Siadat, On the optimization of the support vector machine regression hyperparameters setting for gas sensors array applications, Chemometrics and Intelligent Laboratory Systems. 184 (2019) 22-27.

[16] F. Xiao, Q. Xiang, X. Hou, S.N. Amirkhanian, Utilization of traffic speed deflectometer for pavement structural evaluations, Measurement. 178 (2021) 109326.

[17] R. Ahlvin, Y. Chou, R. Hutchinson, The Principle of Superposition in Pavement Analysis, Highway Research Record. 466 (1973) 153-160.

[18] S. Katicha, G. Flintsch, B. Diefenderfer, Ten years of traffic speed deflectometer research in the United States: a review, Transportation Research Record. 2676 (2022) 152-165.

[19] E. Horak, Benchmarking the structural condition of flexible pavements with deflection bowl parameters, Journal of the South African Institution of Civil Engineering. 50 (2008) 2-9.

[20] X. Deng, Q. Liu, Y. Deng, S. Mahadevan, An improved method to construct basic probability assignment based on the confusion matrix for classification problem, Information Sciences. 340 (2016) 250-261.

[21] O. Elbagalati, M.A. Elseifi, K. Gaspard, Z. Zhang, Development of an artificial neural network model to predict subgrade resilient modulus from continuous deflection testing, Canadian Journal of Civil Engineering. 44 (2017) 700-706.

[22] I. Syarif, A. Prugel-Bennett, G. Wills, SVM parameter optimization using grid search and genetic algorithm to improve classification performance, TELKOMNIKA (Telecommunication Computing Electronics and Control). 14 (2016) 1502-1509.

[23] M. Svilar, I. Peško, M. Šešlija, Model for Estimating the Modulus of Elasticity of Asphalt Layers Using Machine Learning, Applied Sciences. 12 (2022) 10536.
Graphical Abstract

On the Variants of SVM Method for the Estimation of Soil Elastic
Modulus from TSD Model: Numerical Parametric Study

A. ABDELMUHSEN, J-M. SIMONIN, F. SCHMIDT, D. LIÈVRE, A. COTHENET, A. IHAMOUTEN

<figure><img src="image_14.png" /><figcaption>Figure 1: The research methodology of SVM model for backcalculation ($M_R$) moduli through numerical TSD deflection velocity simulation.</figcaption></figure>