Evaluation of Ground Displacements Caused by Installing Jet Grouted Columns Using Machine Learning Methods

During the jet grouting process, large volumes of high pressurized fluids injected into the soils will cause significant ground displacements, which may bring harmful impacts on surrounding environment. Therefore, it is essential to provide an accurate estimation of the ground displacement in the design stage. Based on multiple nonlinear regression (MNLR) and support vector regression (SVR), the prediction approaches are established, respectively. The column radius (Rc), Young’s modulus (E), and distance from column center to target point (LOA) are selected as the input parameters, while the displacement of target point A at the radial direction (δA) is taken as the output parameter. Comparisons results on the prediction performance of ground displacements indicate that the MNLR-based approach has a better prediction effect. The design charts of the MNLR-based approach for predicting the ground displacement are created, which will be helpful for the practicing engineers to get a quick estimation.

Because large volumes of high pressurized fluids are injected into the fine grained soils during the jet grouting process, it will cause significant ground displacements, which may bring harmful impacts on surrounding environment [36][37][38]. erefore, it is essential to provide an accurate estimation of the ground displacement in the design stage. However, most researchers mainly focused on the prediction of diameter and strength of jet grouted columns [39][40][41][42][43][44], and there are few published literatures related to ground displacement caused by installation of jet grouted columns. Shen et al. [31] proposed a method to calculate the lateral displacements caused by installing vertical jet grouted columns in clayey soils based on the analytical solution developed by Verruijt [45]. Wang et al. [8] proposed an approach to estimate the ground displacements caused by installation of horizontal jet grouted columns. However, the practical engineers may lack relative mathematical knowledge, which may bring difficulty in the use of these existed methods and will limit their widespread application. Intelligent approaches based on machine learning methods are becoming more and more popular in the field of engineering geology and geotechnical engineering [46][47][48][49][50], which can effectively overcome the difficulties of arbitrary assumptions adopted in the current methods and increase the confidence in predictions. Multiple nonlinear regression (MNLR) is a form of regression analysis in which the dependent variables are modeled by a nonlinear function of the independent variables [51][52][53]. MNLR can establish the models to describe the arbitrary relationships between the independent variables and the dependent variables, and this is different from traditional multiple linear regression (MLR), which is only used to obtain linear models. It has been proven that MNLR is an efficient prediction tool by many researchers in different fields (e.g., the field of economics, marketing, and engineering). As a new and efficient intelligent approach, the support vector machine (SVM) was produced on the basis of machine learning theory to solve the classification problems at the initial stage, and then, it was developed to deal with the regression problems after introducing the ε-insensitive loss function. e support vector regression (SVR) technique is based on the structural risk minimization (SRM) principle, which is not only to minimize the error on the training data but also to minimize a bound on the generalization error of a model [54].
In this paper, in order to control and mitigate environmental impacts due to installation disturbance of jet grouting, the machine learning methods including MNLR and SVR are taken as an attempt to propose an approach to predict the ground displacements caused by installing jet grouted columns. A series of field data on ground displacements caused by installing jet grouted columns were collected to conduct the analysis by MNLR and SVR. Finally, the design charts to estimate ground displacement are created to make it easy to use the proposed approach in this study for practical engineers.

Problem Description
Jet grouting is invented based on the combination of hydraulic mining technology and grouting technique. In the construction of jet grouting, high pressurized fluids (grout or water) are jetted from the nozzles that are fixed on the rod and have small diameters into the ground. en, the in situ soils below ground surface can be eroded by the high pressurized fluid jets, and an approximate cylindrical soilcement column will be formed by mixing the eroded soils with the injected grout. Figure 1 depicts the schematic view of ground displacements caused by installing a jet grouted column. As can be seen, the injection of large volumes of high pressurized fluids may cause an expansion effect in the internal stratum, which can induce movements in surroundings soil. Considering the jet grouting process, the factors influencing the ground displacements can be divided as follows: jetting parameters, soil properties, and distance to target point. Hence, the ground displacement caused by installing a jet grouted column can be expressed as a function of construction issues, soil properties, and distance to target point: δ A � f(jetting parameters, soil properties, distance to target point), (1) where δ A is the displacement of point A at the radial direction, as shown in Figure 1.
For the effect of jetting parameters on ground displacements, the withdrawal rate of the rod, nozzle diameter, number of nozzles, flow rate of injected fluid, rotation speed of the rod, and jetting pressure of fluid are mainly concerned. e values of ground displacements will be larger if the higher values of jetting parameters are given when other conditions are the same. For the effect of soil properties on ground displacements, the soil strength and soil stiffness are mainly included. e values of ground displacements will be smaller if the higher values of jetting parameters are introduced when other conditions are the same. In this study, column radius R c , Young's modulus E, and distance from column center to target point L OA are selected to represent the effects of jetting parameters, soil properties, and distance to target point on ground displacements, and their relationship can be expressed as follows: where R c is the radius of jet grouted column, E is Young's modulus of surrounding soils, and L OA is the distance from column center (point O) to target point A, as shown in Figure 1. In summary, to yield an accuracy prediction of ground displacements caused by installing a jet grouted column, a reasonable approach using MNLR and SVR should be capable of considering these three parameters: column radius R c , Young's modulus E, and distance from column center to target point L OA .

Data Preparation
Based on the above thought on influencing factors, considering the column radius R c , Young's modulus E, and distance from column center to target point L OA , 36 experimental data on ground displacements caused by installing a jet grouted column were collected from the published literature as regression data. Table 1 tabulates the collected experimental data on ground displacements caused by installing jet grouted columns. ese field data were obtained from two case histories in two different soils. Figure 2 shows the geotechnical profile and soil properties for two case histories conducted in the soft clay and silty clay, respectively. As can be seen, the physical properties and mechanical parameters can be easily determined.

Multiple Nonlinear Regression.
Multiple nonlinear regression (MNLR) is a type of regression analysis in which the dependent variables are modeled by a nonlinear function of the independent variables. MNLR can establish the models to describe the arbitrary relationships between the Note. R c : radius of jet grouted column; E: Young's modulus of soil; L OA : distance from column center to target point; δ A : measured displacement at the radial direction. Advances in Civil Engineering independent variables and the dependent variables, and this is different from traditional multiple linear regression (MLR), which is only adopted to obtain linear models. Based on the consideration of the influencing factors on ground displacements and equation (2), the following nonlinear function is proposed to conduct MNLR analysis:

Jet grouted column
where

Support Vector Regression.
e support vector machine (SVM) is a new and efficient artificial intelligence method, which was produced on the basis of statistical learning theory to solve the classification problems at the initial stage, and then, it was developed to deal with the regression problems after introducing the ε-insensitive loss function. SVR can be divided as the linear SVR and the nonlinear SVR. In many cases, it is not suitable using the linear SVR due to the real-world problem is very complex. e principle of SVR technique is introduced briefly in this section.
Taking a series of training data into account {(x 1 , y 1 ), . . ., (x n , y n )}, x ∈ R m , y ∈ R, where x is the input parameter, y is the output parameter, n is the number of collected data, R m is the m-dimensional vector space, and R is the one-dimensional vector space. Figure 3 shows the principle of a linear SVR technique, ε-insensitive loss function, and slack variables. As can be seen, the shaded area is called as the ε-insensitive tube. For these training data outside of the ε-insensitive tube, they will be given a nonzero slack variable. When the predicted data are inside the ε-insensitive tube, there will be no other differences, which mean that the value of ε-insensitive loss is zero. While the predicted data are outside the ε-insensitive tube, the value of the loss will be equal to be the magnitude of the difference between the estimated data and the tube radius ε [54]. e ε-insensitive loss function may be determined by the following equation: where L ε (y) is the loss function. e linear function for SVR can be given by the following equation in general: where b is the bias, w is the weight vector, and 〈w · x〉 is the inner product of w and x.
In SVR, there exists a main goal that is to find out a function f (x) that has the ability to minimize the complexity of model [54,55]. e aforementioned goal can be achieved by the minimization of the weight vector (w):

Advances in Civil Engineering
However, by the introduction of slack parameters ξ i , ξ * i (i � 1, . . ., n), the estimation error of training data outside the ε-insensitive tube could be incorporative. e function of convex optimization could be determined by the following equation: where C is the penalty parameter that is >0. In equation (8), the left term (1/2)‖w‖ 2 stands for the structure risk, while the right term C n i�1 (ξ i + ξ * i ) stands for the empirical risk. e penalty parameter C determines the trade-off between the term (1/2)‖w‖ 2 and the empirical risk. By introducing a Lagrange function, the function in equation (8) could be transformed as follows: where α * and α are Lagrange multipliers and L(α * , α) is the Lagrange function. When the Lagrange multipliers after optimization are obtained, the regression problem in equation (9) can be expressed as follows: where x r and x s are the support vectors, b 0 is the optimum value for the bias, and w 0 is the optimum value for the weight vector. During the training process of SVR technique, the values of some Lagrange multipliers could become zero, indicating that these training data could be irrelevant for the final regression analysis. Training modes with nonzero Lagrange multipliers are generally called as the support vectors [56]. e abovementioned SVR model is capable of solving the linear regression problems in general. By introducing a nonlinear kernel function, the following equations can be adopted to solve the nonlinear regression problems: Advances in Civil Engineering where K(x i · x) is the kernel function.
By the use of a map ϕ, the input variables could be mapped onto the feature space. e dot product of ϕ(x i ) · ϕ(x) is calculated by using a linear combination of the training data [57].
In this study, a structure of the SVR technique for prediction of ground displacements is established, as shown in Figure 4. SVR model has been implemented using Statistics and Machine Learning Toolbox in MATLAB 2019a environment with a SVR code. Radial basis kernel function is one of the most widely used kernel functions in SVR and is adopted to analyze the collected data for prediction of ground displacements in this study.

Accuracy of Predicted Values by MNLR and SVR
In order to evaluate the accuracy of predicted values of ground displacement by MNLR and SVR, five regression indices, namely, correlation coefficient (R 2 ), mean squared error (MSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and median absolute error (MEDAE), were adopted in this study. ese five regression indices can be determined as follows: where l i is the observed data, m i is the predicted data, n is the number of adopted data, l i is the average value of the observed data, and m i is the average value of the predicted data. Figure 5 shows the prediction performance of ground displacement with MNLR and SVR, in which training data and testing data are included. As can be seen, the prediction performance obtained by using MNLR (δ ref � 40 mm, a � 1.8, b � −1.4, and c � −1.7) is better, for training data: R 2 � 0.96, MSE � 13.32, MAPE � 50.5%, MEDAE � 3.2, and MAE � 3.3; for testing data: R 2 � 0.98, MSE � 14.05, MAPE � 12.5%, MEDAE � 3.1, and MAE � 3.0. While for the prediction performance obtained by SVR with radial basis function, these five regression indices are listed as follows: for training data: R 2 � 0.80, MSE � 38.61, MAPE � 35.01%, MEDAE � 3.0, and MAE � 0.84; for testing data: R 2 � 0.55, MSE � 59.8, MAPE � 21.5%, MEDAE � 5.82, and MAE � 5.13. From these comparison results, it is suggested that the MNLR has a better prediction performance and can be adopted to yield an accurate prediction of the ground displacement caused by installing a jet grouted column.

Design Charts to Estimate Ground Displacement Created by MNLR
Based on the analysis of prediction performance, it has been proven that the MNLR can be used to yield a good estimation of the ground displacement caused by installing a jet grouted column. In order to increase the application range of this MNLR method, the design charts for predicting the ground displacement caused by installing a jet grouted column for different cases are plotted, as shown in Figure 6, which will make it more convenient for the use of this MNLR method in engineering practice. As can be seen in this figure, the column radius R c varies from 0.25 m to 1.25 m, and Young's modulus E is within the range of 2500 kPa to 30000 kPa, while the distance from column center to target point L OA changes from 3 m to 18 m. ese design charts basically cover the most situations that can be encountered in engineering practice. Considering the construction issues and soil conditions in engineering site, the column radius can be estimated using the existed method [39,[41][42][43] firstly, and then, the practicing engineers can get a quick prediction of the ground displacement caused by installing a jet grouted column based on the design charts proposed in this paper. In engineering practice, it is always encountered that multiple rows of jet grouted columns are installed. For such cases, based on the assumption of the principle of superposition, the ground displacements induced by installing a row of columns can be determined by the sum of the displacement caused by each individual column, as shown in the following equation and Figure 7: where δ Asum is the ground displacements induced by installing a row of columns; i stands for the number of the jet grouted column being installed; and δ Ai is the displacement caused by installation of column i, which can be estimated by the design charts in this study.

Conclusions
Predicting the ground displacements caused by installing jet grouted columns is an important issue in the design stage of jet grouting, and large error in prediction may induce harmful results in engineering practice. Based upon the machine learning methods, prediction of the ground displacement caused during jet grouting is analyzed and discussed in this study, and the conclusions can be drawn as follows: (1) e factors influencing the ground displacements can be divided as jetting parameters, soil properties, and distance to target point. e column radius (R c ), Young's modulus (E), and distance from column center to target point (L OA ) are selected as the input parameters, while the displacement of target point A at the radial direction (δ A ) is taken as the output parameter. (2) e prediction approaches are established, respectively, based on MNLR and SVR. A nonlinear equation to express the relationship between the input parameters (R c , E, and L OA ) and the output parameter (δ A ) is proposed in this study. Comparisons results on prediction performance of ground displacements indicate that the MNLR-based approach has a better prediction effect. (3) Using the MNLR-based approach, the design charts for predicting the ground displacement induced by installation of jet grouted columns are created to increase the convenience for the use of the MNLR model in engineering practice. Based on the proposed design charts, the practicing engineers can get a quick estimation of the ground displacements caused by installing jet grouted columns.

Data Availability
e data used to support the findings of this research work are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.