Development of automatic prediction model for ground vibration using support vector machine

2.1. Ground vibration

For most sensitive equipment, manufacturers provide detailed criteria that define acceptable vibration level. These criteria are usually frequency-dependent, thus reflecting the vibration sensitivity of the internal components of an instrument. Based on the ground vibration characteristics induced by high-speed trains, a range of amplitudes (10 dB to 100 dB, ref. 1 micro-inch/sec) and frequencies (1 Hz to 100 Hz) are required to assess ground vibration. A frequency domain of 1/3 octave band for the center frequency range of 1 Hz to 100 Hz is adopted to describe the vibration level in dB and to evaluate the frequency effect. Ground vibration level is expressed in terms of the root-mean-square (RMS) velocity of the ground vibration level, which is defined in dB as follows:

where $VL$ is the vibration level, $v_m$ is the measured velocity, and $v_{ref}$ is the referred velocity ($v_{ref}=$2.54×10^-8 m/sec).

The overall vibration level is often used to evaluate the total vibration energy (e.g., [2, 7, 11]), and this parameter can be transferred from the RMS velocity of each 1/3 octave band vibration level, as shown in the calculation below:

where $V{L}_{oa}$ is the overall vibration level in dB, $f_k$ is the central frequency of each 1/3 octave band, and $VL\left(f_k\right)$ is the vibration level of each frequency of the 1/3 octave band.

2.2. Prediction model

For ground vibration prediction, the SVM algorithm is adopted to classify high-speed trains-induced vibration levels because this approach is probably the most widely used kernel learning algorithm [12]. The analysis of the SVM algorithm is mainly based on the vibration measurement database characteristics.

The basic concept of SVM is the application of nonlinear mapping from an input space to a high-dimensional feature space using a linear model to form a decision boundary. For a given training set {($x_1$, $y_1$), ($x_2$, $y_2$),…, ($x_n$, $y_n$)}, $x_i\in Rp$, $y_i\in \{-\text{1,}\text{1}\}$, where $x_i$ is a $p$-dimensional sample vector, and $y_i$ denotes the class labels of $x_i$, the supposed optimal separating hyperplane not only separates the two classes but also maximizes the margin between the classes if the samples are linearly separable. The hyperplane is written as:

where $<·>$ denotes the inner product, $w$ denotes the normal vector to the hyperplane, and $b$ is the distance from the origin to the hyperplane along the normal vector. The margin can be defined as:

The task of maximizing the margin can be summarized as:

Lagrange theory is used to solve the problem, which is defined as:

where $\alpha =\mathrm{}$(${\alpha }_1$,${\alpha }_2$,..., ${\alpha }_n$) is the vector of the Lagrange multipliers, and ${\alpha }_i\ge 0$, $i=\mathrm{}$0, 1, 2,..., $n$.

When the classes are not linearly separable, a new set of variables is introduced, namely:

where the variables ${\xi }_i\ge$0, $i=$ 1, 2,…, $n$, are the slack variables. ${\sum }_i{\xi }_i$ is the total amount by which constraints are violated. The slack variables are added to handle non-separable cases. Thus, ${\sum }_i{\xi }_i$ should be as small as possible for optimization.

This classifier is nonlinear in the original features but is linear in the expanded feature space. $x$ is replaced by $\mathrm{Ф}\left(x_i\right)$ for some nonlinear $\mathrm{Ф}$, such that the decision boundary is a nonlinear surface. SVMs do not produce a dot product $\mathrm{Ф}\left(x_i\right)\bullet \mathrm{Ф}\left(x_j\right)$ in a high-dimensional feature space. These products are replaced with a kernel function $K(x_i,x_j)$ instead. The kernel function $K(x_i,x_j)$ is a symmetric positive definite function that satisfies Mercer’s condition of $K\left(x_i,x_j\right)\ge$0. Function $K$ is analogous to a nonnegative definiteness for a matrix. Solve the optimization problem is easier using:

The most commonly used kernel function is the Gaussian radial basis function (RBF):

where $\gamma$ is the kernel parameter.

3.1. Background of high-speed trains

The trainset configuration and the related dimensions of Taiwan high-speed trains are shown in Fig. 1. Each Taiwan high-speed trainset consists of 12 train cars, where 10 cars are passenger cars and two locomotive cars. Each passenger car (PC) is 25 m long, whereas each locomotive car is 27 m long. Thus, the total length of the trainset is 304 m, as shown in Fig. 1.

Fig. 1Configuration of trainset for the Taiwan high-speed rail

3.2. Ground vibration measurement

The main measuring equipment consisted of accelerometers, integrator, and a data acquisition system. The following procedure is used for installing sensors to attach the ground:

a) Excavate a pit with proper dimensions which can install accelerometers.

b) Place the standard sand on the bottom of pit to even the excavated surface.

c) Compact the excavated surface and assure the surface in the horizontal level.

d) Place firmly accelerometers, which connect a steel plate as a firm base, on the ground.

e) Set the accelerometer direction: the $X$-direction is in the train passing; the $Y$-direction is perpendicular to the train passing; and the $Z$-direction is for the direction of gravity.

A set of measuring instruments and equipment for this study is shown in Fig. 2. The measured vibration accelerations included $X$-$Y$-$Z$ directions. Only the vertical component ($Z$ direction) is used in the subsequent discussion to simplify the process of vibration impact assessment.

Fig. 2Measuring equipment: (1) accelerometers, (2) integrator, (3) and (4) data acquisition system

The vibration measuring plan includes near-field and far-field measurements. The distance of the near-field vibration was set beside the alignment of high-speed trains. For far-field measurement, additional three measurement points in each site, which were in a straight line and perpendicular to the alignment of high-speed trains, are used to simultaneously measure the ground vibration when trains pass through the specific location. Fig. 3 shows the schematic diagram of embankment and a typical schematic layout of the measurement points. All these measurement points were employed to simultaneously measure the ground vibration when trains pass through the specific location. Before measuring, all equipment must be synchronized.

Fig. 3Schematic diagram of embankment and typical schematic layout of measurement site

a) Schematic diagram of embankment

b) Schematic measurement layout

The series of measurements performed in this study comprises different factors, such as geological conditions, embankment heights, and train speeds. Table 1 lists the basic information of these measurement sites. The measurement distances ($D_1$ to $D_4$) are 22 m to 203 m from the track center, and the selected embankments have various heights that range from 3.6 m to 6.8 m.

The geological conditions include sand/silt/clay soils, gravel, and rock. These conditions range from soft ground to hard ground. Ground shear wave velocity ($V_s$) is used as an indicator to describe soil stiffness. The $V_s$ value increases with increasing soil stiffness. The average $V_s$ taken from the ground surface to 10 m in depth ranges from 170 m/s to 650 m/s, which is representative of the surface wave analysis. Table 1 evidently shows that various geological conditions and ground shear wave velocities are covered.

3.3. Ground vibration characteristics

The measured results of ground vibration are shown in Table 2. Based on field measurements, the main characteristics that influence vibration levels include geological conditions, attenuation distance, and train speed. Fig. 4 presents the relationships of the overall vibration level with the train speed, ground shear wave velocity, and measurement distance. Evidently, the vibration level increases with the train speed. However, the vibration level decreases as the ground shear wave velocity and measurement distance increase.

Fig. 4Relationships of VLoa and various influence factors

a) Train speed

b) Shear wave velocity

c) Measurement distance

4.1. Prediction methodology

As mentioned in Section 2.2, the SVM kernel method is a supervised learning model that is popular for training known features. To construct such classification model, positive and negative data classes are provided in advance as training examples, after which a trained SVM model is constructed according to the selected features.

An SVM clustering tool (LIBSVM) is adopted in this study. The SVM_train and SVM_predict module in LIBSVM toolset developed by Lin’s lab [12, 13] are applied to train and classify train-induced vibration levels. An overview of SVM_train and SVM_predict module are presented in Fig. 5. In SVM_train module, all training data with defined features are trained to construct the trained model. In SVM_predict module, testing data were classified also by the previous model. The following sections discuss in detail the selection of classification features, the evaluation of different kernel transformation techniques, and the prediction results according to benchmark datasets.

Fig. 5LIBSVM’s training and predicting tool

4.2. Prediction process

Ground shear wave velocity, train speed, and measurement distance are the features for the SVM model used in this research. In this study, the RBF kernel type performs the best accuracy compared to other kernel types. Thus, the RBF is selected as the kernel type, and the function of multiclass classification is employed in this system. Based on the measured results of the possible minimum and maximum vibration levels, the SVM model output is classified into five groups of vibration level, as shown in Table 3. The vibration levels for Groups 1 to 5 are 30 dB to 40 dB, 40 dB to 50 dB, 50 dB to 60 dB, 60 dB to 70 dB, and more than 70 dB, respectively.

The training and prediction processes are presented in Fig. 6. In the training state, the system begins by reading the input data and then progresses to extracting and normalizing the same data to region (0, 1). The data transformed to a normal form are then entered by the system to LIBSVM, after which the training process begins. Train speed, ground shear wave velocity, and measurement distance is adopted as training and prediction features. The system uses a 10-fold cross validation. One fold is chosen as the validating fold, whereas the others are the training folds to build the hyperplane used in the validations. If testing results are better than the earlier outputs, the parameter is changed. This process is repeated several times until the system finds the result in the local maximum. The LIBSVM’s grid.py toolset is applied to implement the optimal parameters selection process. After completing the training process, the system uses a hyperplane to predict and classify the vibration level.

Fig. 6SVM prediction process