Machine Learning Techniques for Soil Characterization Using Cone Penetration Test Data

Abstract

Seismic response assessment requires reliable information about subsurface conditions, including soil shear wave velocity $(V_s)$. To properly assess seismic response, engineers need accurate information about $V_s$, an essential parameter for evaluating the propagation of seismic waves. However, measuring $V_s$ is generally challenging due to the complex and time-consuming nature of field and laboratory tests. This study aims to predict $V_s$ using machine learning (ML) algorithms from cone penetration test (CPT) data. The study utilized four ML algorithms, namely Random Forests (RFs), Support Vector Machine (SVM), Decision Trees (DT), and eXtreme Gradient Boosting (XGBoost), to predict $V_s$. These ML models were trained on 70% of the datasets, while their efficiency and generalization ability were assessed on the remaining 30%. The hyperparameters for each ML model were fine-tuned through Bayesian optimization with k-fold cross-validation techniques. The performance of each ML model was evaluated using eight different metrics, including root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), coefficient of determination ($R^2$), performance index ($PI$), scatter index ($SI$), $A10-I$, and $U_{95}$. The results demonstrated that the RF model consistently performed well across all metrics. It achieved high accuracy and the lowest level of errors, indicating superior accuracy and precision in predicting $V_s$. The SVM and XGBoost models also exhibited strong performance, with slightly higher error metrics compared with the RF model. However, the DT model performed poorly, with higher error rates and uncertainty in predicting $V_s$. Based on these results, we can conclude that the RF model is highly effective at accurately predicting $V_s$ using CPT data with minimal input features.

1. Introduction

Soil characterization plays a vital role in seismic response assessment and interpreting subsurface conditions for large-scale engineering projects. To properly assess seismic response, engineers need accurate and reliable information about subsurface conditions, including soil shear wave velocity ($V_s$), an essential parameter for evaluating the propagation of seismic waves [1,2,3,4,5,6]. Seismic-refraction and reflection methods using geophysical signal processing [7,8,9,10] measure $V_s$ at various depths and precision to produce a profile (preferably to bedrock) for later analysis. To measure $V_s$, an active source generates a wave and its travel time to one or more receivers is measured. The velocity results from knowing the time and distance traveled between the source and receiver. There are several $V_s$ measurement methods, including seismic cone penetration testing (SCPT) [7], Multi-Channel Analysis of Surface Waves (MASW) [8], Cross-hole testing [8], and down-hole testing methods. These techniques provide valuable information about subsurface conditions but become more complex with increasing soil layering.

Additionally, laboratory tests such as bender element [11], triaxial test [12], and resonant column tests [13] measure $V_s$ in different ways. These tests are conducted on soil samples collected from the site and offer a controlled environment for testing, providing detailed information on soil behavior under varying stress conditions. However, retrieving high-quality intact soil samples is a challenging task that requires specialist equipment [14]. Furthermore, it is important to note that the properties of collected samples may significantly change over time due to variations in stress conditions, temperature fluctuations, and moisture content.

A viable alternative is to correlate $V_s$ to cone penetration test (CPT) data, a relatively easier approach. Many empirical correlations have been developed over the past couple of decades to estimate $V_s$ from CPT data [15,16,17,18,19]. The CPT test involves pushing a cone-shaped instrument into the ground at a constant rate while measuring the resistance of the soil. Two measurements are typically taken during this test: cone tip resistance ($q_c$) and sleeve friction ($f_s$) [20,21]. The CPT test provides continuous and reliable soil data, making it an efficient and cost-effective method in geotechnical engineering practice. This wealth of CPT data has attracted the attention of many geotechnical researchers to further improve the prediction accuracy of $V_s$ employing machine learning (ML) algorithms [22,23,24,25,26]. ML algorithms have shown great promise in accurately predicting $V_s$ from CPT data. The ML algorithms can learn complex relationships between input variables (e.g., $q_c$ and $f_s$ records) and output variables (e.g., soil $V_s$) from large datasets without the need for explicit mathematical models.

Many ML algorithms, such as gradient boosting, random forest, support vector machine (SVM) artificial neural network (ANN), and decision trees (DT), have been used in various geotechnical applications, including soil classification [27,28,29,30,31,32,33], $V_s$ prediction [23,24,25,26,34], liquefaction analysis [35,36,37,38,39,40], stability analysis [41,42,43,44,45], and settlement prediction [46,47,48]. The application of ML algorithms in geotechnical engineering has shown promising results in terms of efficiency and accuracy. For example, Tsiaousi et al. [25] successfully employed an ANN model to characterize soil stratigraphy and predict $V_s$. This study demonstrates how ML approaches can be used to improve soil characterization and prediction of important geotechnical parameters. Assaf et al. [24] and Riyadi et al. [49] have also used ML algorithms, including RF and XGBoost, to predict $V_s$. Their findings confirm that ML models can achieve high accuracy and performance in predicting $V_s$. Previous research has also shown that SVM performs well in predicting $V_s$ [50,51]. These studies collectively demonstrate the potential of ML algorithms in improving the accuracy of $V_s$ prediction in geotechnical engineering applications.

The aim of this study is to improve the prediction of $V_s$ using various ML algorithms with minimal input features. Four ML algorithms, namely RF, SVM, DT, and eXtreme gradient boosting (XGBoost), are employed to predict $V_s$ from CPT data. The study also aims to minimize the need for expensive and time-consuming fields or laboratory measurements. The development of ML models can lead to higher accuracy and performance in predicting $V_s$. The improvement in the accuracy of $V_s$ prediction has significant implications for site response assessment and seismic risk reduction. By utilizing ML to predict $V_S$, this study has the potential to enhance existing knowledge and inspire future research in the field of ML applications for soil characterization.

The rest of this document is organized as follows: Section 2 discusses dataset preprocessing and visualization, Methodology and performance metrics are described in Section 3, Section 4 describes the ML models, and Section 5 presents the results. Finally, Section 6 outlines the main results of the study and concludes by suggesting future research.

2. Datasets Preprocessing and Visualization

The dataset used in this study was obtained from a previously published dataset [52]. This study utilized 61 CPT soundings, each containing over 1000 $q_c$ and $f_s$ recordings. These data sets were collected from various regions of Austria, including the Vienna Basin, Gastein Valley, and Zell Basin. The data is publicly accessible and can be downloaded from the following link: https://www.tugraz.at/en/institutes/ibg/research/computational-geotechnics-group/database/ (accessed on 12 May 2023). The CPT datasets were preprocessed before applying ML training and testing techniques. The preprocessing step involved removing outliers from the data. Specifically, outliers were identified and removed from both the $q_c$ and $f_s$ values in the raw CPT data. Any data point that exceeded twice the interquartile range $\left(IQR\right)$, where $IQR$ is the difference between the third quartile ($Q3$) and the first quartile ($Q1$), was considered an outlier. Next, the target variable, which in this case was the shear wave velocity, $V_s$, was estimated using Equation (1) [16]. Subsequently, the datasets were divided into a training set and a testing set, with a ratio of 0.7:0.3 for training and testing purposes.

$$V_S=\sqrt{\left(\frac{q_c-{\sigma }_{v0}}{p_a}\times {10}^{0.55I_c+1.68}\right)}$$

(1)

where $q_c$ represents cone tip resistance, ${\sigma }_{v0}$ represents total overburden pressure, $p_a$ represents atmospheric pressure, and $I_c$ represents soil behavioral type index estimated as follows:

$$I_c={\left(3.47-log{(\left(q_c-{\sigma }_{v0}\right)/\sigma {\prime }_{v0})}^2+{(log{F}_r+1.22)}^2\right)}^{0.5}$$

(2)

$$F_r=f_s/\left(q_c-{\sigma }_{v0}\right)\times 100$$

(3)

where $f_s$ represents sleeve friction, $\sigma {\prime }_{v0}$ is the effective overburden stress, and $F_r$ represents normalized friction ratio.

The statistical summaries of both the training and testing datasets considered in this study are presented in

Table 1. Statistical summary of training and testing datasets.

Features	Unit	Class	Training Dataset					Testing Dataset
			Mean	SD	Min	Max	Count	Mean	SD	Min	Max	Count
$D$	m	Input	12.42	8.88	0.01	40	79,579	12.38	8.78	0.01	40	34,104
$q_c$	MPa	Input	4.89	3.60	0.01	17	79,579	4.87	3.59	0.01	17	34,104
$f_s$	kPa	Input	42.91	35.35	0.07	142	79,579	42.88	35.39	0.10	142	34,104
$R_f$	%	Input	1.56	6.11	0.00	1121	79,579	1.57	6.28	0.00	1083	34,104
$V_s$	m/s	Target	166.76	55.89	10.06	322	79,579	166.55	55.57	9.93	322	34,104

. To gain further insights into the relationship between the input features and the target variable ($V_s$), scatter plots are presented in Figure 1. Each scatter plot indicates the correlation between an individual input feature and the target variable. In addition, Figure 2 shows the frequency distribution of the input features and target variable, providing a visual representation of their distribution patterns. Furthermore, box plots of both the input features and the target variable are presented in Figure 3, offering an overview of their distribution.

The interdependencies among input features in ML models can lead to overfitting and decreased efficiency. To assess the correlation between each input feature, a Pearson’s correlation analysis was conducted. Figure 4 displays the correlation coefficients among the input features in the dataset. The correlation coefficients range from −0.08 to 0.51, indicating a combination of weak to moderate correlations among the features. The absence of highly correlated features in the correlation analysis suggests a lower risk of overfitting, as no redundant features were observed.

3. Methodology

This section outlines the training and testing procedures for the ML models used to predict $V_s$. Four ML models, namely RF, SVM, DT and XGBoost, were trained for this purpose. Each ML model was trained on the training datasets, with $q_c$, $f_s$, friction ratio ($R_f$), and soil depth ($D$) serving as input features and $V_s$ as target variable (output). To optimize the performance of these models, the hyperparameters of each model were fine-tuned using a model-based Bayesian optimization technique. To ensure that the models can generalize well to new data, the commonly used k-fold cross-validation techniques were employed. This involves dividing the data into k subsets, training the model on k-1 subsets, and evaluating their performance on the remaining subset. The root mean squared error (RMSE) was used as the evaluation metric to assess the models’ accuracy. In addition to hyperparameter tuning, permutation feature importance and/or recursive feature elimination techniques were applied using the optimized models. This technique was used for the identification and removal of irrelevant features, if present, in the input features. Once the irrelevant features were eliminated, the hyperparameter tuning process was repeated with the updated features to further enhance the models’ performance. Finally, the performances of the optimized models were assessed using the testing dataset. The entire process of training and testing the models is presented in Figure 5, providing a visual representation of the workflow.

The performance of optimized ML models was evaluated using multiple statistical metrics such as root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), coefficient of determination $\left(R^2\right)$, scatter index ($SI$), and performance index ($PI$) (see

Table 2. Performance indices used to evaluate the efficiency of the models.

Metrics	Best Performance	Equations	Equation No.
Root mean squared error	Lower value	$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{\left(X_i-{\stackrel{⌢}{X}}_i\right)}^2}{n}}$	(4)
Mean absolute error	Lower value	$MAE=\frac{1}{n}\times \left|{\displaystyle \sum _{i=1}^n}{X}_i-{\stackrel{⌢}{X}}_i\right|$	(5)
Mean absolute percentage error	Lower value	$MAPE=\frac{1}{n}\times \left|{\displaystyle \sum _{i=1}^n}\frac{\left(X_i-{\stackrel{⌢}{X}}_i\right)}{X_i}\right|\times 100\%$	(6)
Coefficient of determination	unity	$R^2=1-{\displaystyle \sum _{i=1}^n}\frac{{\left(X_i-{\stackrel{⌢}{X}}_i\right)}^2}{{\left(X_i-\overline{X}\right)}^2}$	(7)
$A10-I$	unity	$A10-I=\frac{n10}{n}$	(8)
Scatter index	Lower value	$SI=\frac{RMSE}{\overline{X}}$	(9)
Performance index	Lower value	$PI=\frac{RMSE}{\overline{X}\times \sqrt{R^2+1}}$	(10)
Uncertainty at 95% confidence level	Lower value	$U_{95}=\sqrt{S{D}^2+RMS{E}^2}$	(11)
$SI$ [64,65]	SI < 0.05: excellent precision (EP), 0.05 < SI < 0.1: good precision (GP), 0.1 < SI < 0.15: fair precision (FP), SI > 0.15: poor precision (PP)

n is total number of datasets, $X_i$ is the actual value of the ith observation, ${\stackrel{⌢}{X}}_i$ is the predicted value of the ith observation, and $\overline{X}$ is mean of target variable. $n10$ is the number of samples with actual/predicted value between 0.90 and 1.10, $U_{95}$ is uncertainty with 95% confidence intervals, and $SD$ is standard deviation of residuals (the difference between target $V_s$ and predicted $V_s$).

). Many researchers utilized these indices to evaluate the predictive performance of different ML models [53,54,55,56,57,58,59,60]. The RMSE measures the average magnitude of the errors between the predicted and actual values, indicating the model’s predictive accuracy. A lower RMSE indicates better model performance. MAE estimates the average absolute difference between the predicted and actual values. Like RMSE, a lower MAE indicates better model performance. MAPE represents the average percentage difference between the predicted and actual values. A lower MAPE signifies better model accuracy. $R^2$ measures the proportion of the variance in the target variable ($V_S$) that can be explained by the model, with values closer to 1 indicating a better fit. Furthermore, a newly proposed engineering index ($A10-I$) was used to evaluate the predictive performance of the models [55,59,61,62,63]. In an ideal model, the value of $A10-I$ is expected to be one. The $A10-I$ has significance in engineering as it represents the proportion of samples that fall within $\pm 10\%$ deviation from the predicted values compared with the target value. Additionally, the efficiency of the models was evaluated using uncertainty analysis at 95% confidence level ($U_{95}$) [64,65].

4. Machine Learning Models

4.1. Random Forests

RF is an ML algorithm that has been widely used for classification and regression tasks [66,67]. It is an ensemble method that combines multiple decision trees to improve predictive accuracy and reduce overfitting. RF has several advantages over other machine learning algorithms, including its ability to handle high-dimensional data, nonlinear relationships between variables, and missing values [68]. In addition, it provides measures of variable importance that can be used for feature selection and interpretation [69].

RF classification and regression can be implemented in R using various packages such as randomForest [68] and ranger [70]. The randomForest package is one of the most widely used packages for RF classification and regression in R. It provides a simple interface for building and evaluating RF models and supports both classification and regression tasks. The ranger package is another popular package for RF classification and regression in R. It is designed to be faster and more memory-efficient than the randomForest package, and supports parallel processing [70].

The ranger package provides several hyperparameters that can be tuned to improve the performance and robustness of RF models. To tune these hyperparameters, one common approach is to use cross-validation. This involves splitting the data into training and test sets, fitting the model on the training set with different combinations of hyperparameters, and evaluating the performance on the validation set. Bayesian optimization is one of the most efficient methods for hyperparameter tuning. It uses a probabilistic model to predict the performance of different hyperparameter configurations based on previous evaluations [71].

4.2. Support Vector Machine

SVMs have gained immense popularity in the field of machine learning due to their ability to solve both classification and regression problems effectively. SVMs work by constructing hyperplanes that can optimally separate data points belonging to different classes or predict target variables with maximum margin. One of the most significant advantages of using SVMs is their ability in handling high-dimensional datasets and nonlinear relationships between variables [72].

In R, e1071 package [73] is commonly utilized to implement SVM models for regression and classification tasks. The package provides options for tuning hyperparameters such as the kernel function, regularization parameter, and cost parameter. One important consideration when using SVMs is their robustness to outliers and noise in the data. Outliers influence the position of the hyperplane and lead to poor generalization performance. To address this issue, Bayesian optimization can be utilized to increase the model’s performance and robustness. Bayesian optimization has been shown to be effective at tuning hyperparameters in various machine-learning algorithms, including SVMs [71].

4.3. Decision Trees

The DT algorithm is commonly used for both classification and regression tasks. The DT algorithms recursively partition data into subsets based on the values of input features and then assign labels to each subset based on the majority class or average value of the target variable. The resulting tree structure can be used to make predictions on new data by traversing the tree from the root node to a leaf node that corresponds to a specific class or value. According to Quinlan [74], decision trees are particularly useful for problems with discrete-valued output variables and can handle both categorical and continuous input features. They are also easy to interpret and visualize, making them a popular choice for exploratory data analysis and decision-making tasks. Figure 6 presents a sample decision tree structure to provide insights into the relationships and decision-making process within the data, aiding in understanding and interpreting the model’s predictions. The node numbers are depicted within the boxes, while the input features are represented by the variables (see Section 1 and Section 3). The green leaves in the figure represent the target value, $V_s$.

The implementation of the DT algorithm for regression tasks is usually performed using the rpart package [75]. The package provides ranges of DT hyperparameters, including complexity parameters, maximum number of trees, minimum number of splits, etc., that can be tuned through grid search or Bayesian optimization. DT algorithms have been used for a variety of geotechnical applicatons, including classification [76,77] and soil parameters predictions.

4.4. eXtreme Gradient Boosting

Recently, the XGBoost algorithm has gained popularity due to its high accuracy and efficiency. XGBoost is an ensemble method that combines multiple weak learners such as decision trees into a single strong learner [78,79]. The algorithm iteratively adds decision trees to the model, with each tree attempting to correct the errors of the previous trees.

XGBoost package [78] is usually utilized to implement the XGBoost regression model in R. The XGBoost package also offers support for hyperparameter tuning, which can significantly improve the model’s performance. It provides a range of options for tuning its hyperparameters, including learning rate (eta), maximum depth of each tree (max_depth), number of trees, and regularization parameters (alpha and gamma). Bayesian optimization can be used for hyperparameter tuning in XGBoost.

5. Results and Discussion

In Section 2, we indicated that the datasets were randomly split into training and testing datasets. The training datasets were used to train the ML models, while the testing datasets were used to evaluate the efficiency of each model in predicting $V_s$. In this section, we will discuss the results obtained from training and testing ML models. All the ML models were trained and tested using a personal computer with 8GB RAM and Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz 1.50 GHz processor (Intel Co., Santa Clara, CA, USA). The performance of each ML model was evaluated using the multiple performance metrics listed in Table 2.

5.1. Hyperparameter Optimization Results

The hyperparameters of each model were fine-tuned using Bayesian optimization with a k-fold cross-validation strategy. Specifically, we used 10-fold cross-validation with the RMSE as the evaluation metric for fine-tuning the hyperparameters. The goal was to minimize the RMSE, as lower values indicate better performance. The maximum number of iterations for the fine-tuning process was set to 100 for each model. Figure 7 illustrates the convergence behaviors of the ML models during the fine-tuning process. It shows how the performance metric (RMSE) changed over the iterations. We observed that all the ML models reached stable results within 100 iterations.

Table 3. Optimized hyperparameters.

ML Models	Tuned Hyperparameters
	Names	Ranges	Optimized Values
RF	Number of variables, mtry	1–4	3
	Minimum node of tree	1–30	2
	Maximum depth of tree	2–100	64
	Number of trees in the forest	1–30	12
SVM	Penalty parameter, Cost	0.1–100	58.75
	Kernel coefficient, gamma	0.01–10	9.44
	Margin of tolerance, Epsilon	0.01–1	0.026
	Kernel type	radial	radial
DT	Complexity parameter, cp	0.001–1	0.001
	Maximum depth of trees	1–30	20
	Minimum number of splits	2–20	5
	Minimum number of observations at terminal node, minbucket	2–20	6
	Maximum number of splits at node, maxcompete	1–20	9
XGBoost	Learning rate, eta	0.01–1	0.26
	Loss reduction term, gamma	0.01–10	3.79
	L2 regularization term, lambda	0.01–1	0.38
	L1 regularization term, alpha	0.01–1	0.83
	Number of boosting rounds, nrounds	1–100	84
	Maximum depth of trees	2–10	9
	Fraction of samples for each tree, subsample	0.1–1	0.79

presents the hyperparameters of the ML models along with their optimized values. The optimized values represent the set of hyperparameters that yielded the best average performance according to the RMSE metric.

5.2. Performance of ML Models

Figure 8 illustrates actual $V_s$ and predicted $V_s$ using the optimized ML models, along with $\pm 10\%$ error lines (red lines). The green lines show a match between actual and predicted $V_s$ values. The results demonstrate that all ML models, except for the DT model, achieved excellent predictive accuracy on both training and testing datasets with high $R^2$ and $A10-I$ score values of 1. This shows that the RF, SVM, and XGBoost models can explain all the variance in the $V_s$ using the given features. Furthermore, the scatter plots for these models show that many data points are closer to the error bounds, indicating that the models performed well. In contrast, the DT model achieved lower $R^2$ and $A10-I$ values ranging from 0.94 to 0.95 and from 0.77 to 0.78 on the testing and training data, respectively.

The residual plots of ML models are shown in Figure 9, illustrating a random distribution of points around the horizontal orange line at y = 0 (line of zero error). This indicates that the model’s predictions are unbiased and have captured the underlying patterns in the data. Additionally, the frequency distributions of residuals (green bars) are shown in the figure. The distribution is approximately symmetric, indicating that the errors are normally distributed, a desirable property. To gain more insight into the performance of the ML models, a further comparison is carried out in the following subsection.

5.3. Comparisons of ML Models

Table 4. Summary of evaluation results for each ML model using training and testing datasets.

Models	Train dataset								Rank
	A10 − I	RMSE	$R^2$	PI	SI	MAE	MAPE	U95
RF	1 (1)	0.46 (1)	1 (1)	0.002 (1)	0.003 (1)	0.24 (1)	0.17 (1)	1.24 (1)	1
SVM	0.998 (2)	1.11 (2)	1 (1)	0.005 (2)	0.007 (2)	0.37 (2)	0.28 (2)	3.07 (2)	2
DT	0.78 (3)	13.1 (4)	0.95 (2)	0.06 (4)	0.08 (4)	10.27 (4)	7.23 (4)	36.20 (4)	4
XGBoost	1 (1)	1.68 (3)	1 (1)	0.007 (3)	0.01 (3)	1.29 (3)	0.87 (3)	4.65 (3)	3
Models	Test dataset								Rank
	A10 − I	RMSE	$R^2$	PI	SI	MAE	MAPE	U95
RF	1 (1)	0.96 (1)	1 (1)	0.004 (1)	0.006 (1)	0.50 (2)	0.36 (2)	2.66 (2)	1
SVM	0.998 (2)	1.36 (2)	1 (1)	0.006 (2)	0.008 (2)	0.38 (1)	0.31 (1)	2.3 (1)	2
DT	0.77 (3)	13.2 (4)	0.94 (2)	0.06 (4)	0.08 (4)	10.34 (4)	7.31 (4)	36.48 (4)	4
XGBoost	1 (1)	1.86 (3)	1 (1)	0.008 (3)	0.01 (3)	1.40 (3)	0.94 (3)	5.16 (3)	3

provides a summary of the evaluation results for the ML models. Based on our results, the DT model exhibited lower accuracy, as evidenced by a lower $R^2$ on both training and testing data. The DT model recorded RMSE of 13.06 and 13.16, MAE of 10.27 m/s and 10.34 m/s, MAPE of 7.27% and 7.31%, and $R^2$ of 0.95 and 0.94 on the training and testing datasets, respectively. Additionally, Spider charts were utilized to visualize and assess each model’s efficiency relative to others (Figure 10). The spider chart shows that the DT model significantly diverged towards higher RMSE, MAPE, and MAE on both training and testing datasets in comparison to other ML models. The RF, SVM, and XGBoost models outperform the DT model in terms of RMSE, MAE, MAPE, and $R^2$. All RF, SVM, and XGBoost models have lower error values and higher $R^2$ scores, indicating higher accuracy and better performance in predicting $V_S$ from the input features.

Figure 11 illustrates the performance of ML models in predicting $V_{s }$ as evaluated using four performance metrics: $U_{95 }, A10-I, PI, \mathrm{and} SI$. The results indicate that the DT model achieved lower performance in comparison to other models, as evidenced by its higher $U_{95 }, PI, \mathrm{and} SI$ scores and lower $A10-I$ values. On the other hand, the RF model demonstrated exceptional performance, outperforming the other models in terms of these performance indicators. In terms of $SI$, the RF, SVM, and XGBoost models achieved excellent precision (EP), with $SI<0.05$ on both the training and testing datasets. In contrast, the DT model achieved good precision with $0.05<SI<0.1$ on the training and testing datasets. Overall, the RF model ranked first, outperforming the other three ML models, while the DT model ranked fourth. The SVM and XGBoost models ranked second and third, respectively.

To further assess the performance of the ML models, a comparison was made between model-predicted $V_S$ values and estimated $V_S$ values based on existing empirical correlation. A correlation model was selected to estimate $V_S$ from CPT soundings. Equation (12) [80] was utilized for the estimation of $V_S$ from CPT soundings.

$$V_S={10}^{0.31I_c+0.77}\times \sqrt{(q_c-{\sigma }_{v0})/p_a}$$

(12)

where $V_S$ is soil shear wave velocity, $I_c$ is soil behavior type index, $q_c$ is cone tip resistance, ${\sigma }_{v0}$ is total overburden pressure, and $p_a$ is atmospheric pressure.

This correlation model served as a benchmark for evaluating the accuracy and reliability of the ML models’ predictions. Figure 12 illustrates the models’ predictions (red) alongside the profiles of estimated $V_S$ values (black) based on the empirical correlations. The results of this comparison indicate a high level of agreement between the predicted $V_S$ values and the estimated $V_S$ values. This demonstrates that the ML models can produce accurate predictions in line with the established correlations.

6. Conclusions

This study utilized various ML algorithms, including RF, SVM, DT, and XGBoost, to predict $V_S$ from CPT data. To train and test these ML models, we used a previously published open-source CPT dataset. The hyperparameters of each ML model were fine-tuned through Bayesian optimization with cross-validation techniques. Eight performance metrics, namely $RMSE, MAE, MAPE, R^2, A10-I, SI, PI \mathrm{and} U_{95}$, provided quantitative evaluation of the models. Based on our results, the following conclusion can be drawn:

• The RF model outperformed the other ML models, achieving the lowest error metrics on both the training and testing datasets. Specifically, it achieved an $RMSE$ of 0.46 and 0.96, an $MAE$ of 0.24 m/s and 0.5 m/s, and an $MAPE$ of 0.17% and 0.36%, respectively. The model also demonstrated low scatter, with $SI$ values of 0.003 and 0.006, and $PI$ values of 0.002 and 0.004 on the training and testing datasets, respectively. Additionally, the RF model achieved $R^2$ and $A10-I$ values of 1 on both datasets, indicating a perfect fit. Furthermore, the RF model recorded the lowest uncertainty, with a $U_{95}$ value of 1.24 on the training dataset.
• The SVM and XGBoost models also exhibited strong performance, with slightly higher error metrics compared with the RF model. These two models ranked second and third, respectively, following the RF model, which achieved the highest performance. However, the DT model performed poorly, with higher error rates and uncertainty in predicting $V_s$.
• The RF model demonstrated its overall superior performance and high accuracy in predicting soil $V_s$, even when trained with minimal input features. Hence, owing to its excellent performance across multiple metrics, the RF model can be integrated into a software package for rapid and accurate prediction of soil $V_s$.
• In summary, while this study relied solely on CPT data for training ML models, it is important to recognize the limitations of the CPT, particularly its primary suitability for fine-grained soils. To further enhance the application of ML models in soil characterization, future research should consider incorporating experimental results and data for coarse-grained soil types.