## 3 Probability Modeling of Asphalt Pavement Responses
### Probability Calculation Procedure  
The combination of RSM and FORM was performed to calculate the probability distributions of the asphalt pavement responses. The probability calculation procedure was as illustrated, and the flowchart of the probability calculation is shown in Figure 2.  
Step 1: Pavement material parameters (i.e., modulus, Poisson's ratio, and density of each layer) and pavement structural parameters (i.e., depth of each layer) were inputted. In addition, the coordinates of the calculation point inside the pavement (\(x,y,\) and \(z\)) were determined. The target mechanical responses, which can be achieved as \(R_{t}\), were set. Step 2: The probability distribution of each random variable was inputted. The mean values of random variables were \(\boldsymbol{\mu}_{X}= \begin{bmatrix}\mu_{X_{1}}&\mu_{X_{2}}&\mu_{X_{3}}&\mu_{ X_{4}}\end{bmatrix} ^{\prime}\), and the standard deviations of random variables were \(\boldsymbol{\sigma}_{X}= \begin{bmatrix}\sigma_{X_{1}}&\sigma_{X_{2}}&\sigma_{ X_{3}}&\sigma_{X_{4}}\end{bmatrix} ^{\prime}\). Step 3: The initial iteration point \(\boldsymbol{X}= \begin{bmatrix}X_{1}&X_{2}&X_{3}&X_{4}\end{bmatrix} ^{\prime}\) was assumed. Generally, the mean values of each random variable were selected as the initial iteration point. Step 4: The expansion points \( \begin{bmatrix}X_{1}&X_{2}&X_{3}&X_{4}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}-f\sigma_{X_{1}}&X_{2}&X_{3}&X_{4}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}&X_{2}-f\sigma_{X_{2}}&X_{3}&X_{4}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}&X_{2}&X_{3}-f\sigma_{X_{3}}&X_{4}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}&X_{2}&X_{3}&X_{4}-f\sigma_{X_{4}}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}+f\sigma_{X_{1}}&X_{2}&X_{3}&X_{4}\end{bmatrix} ^{\prime}\), \( \begin{bmatrix}X_{1}&X_{2}&X_{3}&X_{4}+f\sigma_{X_{4}}\end{bmatrix} ^{\prime}\), and \( \begin{bmatrix}X_{1}&X_{2}&X_{3}&X_{4}+f\sigma_{X_{4}}\end{bmatrix} ^{\prime}\), were selected, where \(f=1\)-3. The values of the performance function \(Z_{k}\) (\(k=1,2,\cdots,9\)) at expansion points were calculated subsequently by Equation (7). The coefficient matrix A was established at expansion points. \[\boldsymbol{A}= \begin{bmatrix}1&\begin{bmatrix}X_{i}\end{bmatrix}_{1\times 4}& \begin{bmatrix}X_{i}^{2}\end{bmatrix}_{1\times 4}\\ 1&\begin{bmatrix}X_{i}\end{bmatrix}_{1\times 4}&\begin{bmatrix}X_{i}^{2} \end{bmatrix}_{1\times 4}\\ \vdots&\vdots&\vdots\\ \boldsymbol{0}&\begin{bmatrix}f\text{diag}\begin{bmatrix}\sigma_{X_{i}} \end{bmatrix}_{4\times 4}&\text{diag}\begin{bmatrix}f^{2}\sigma_{X_{i}}^{2}+2 fX_{i} \sigma_{X_{i}}\end{bmatrix}_{4\times 4}\end{bmatrix}\end{bmatrix}  \tag{9}\]  
Step 5: The fitting parameters of the response surface \(a\), \(b_{i}\), and \(c_{i}\) (\(i=1,2,\cdots,4\)) were solved by Equation (10). Meanwhile, the fitted results \(Z^{\prime}\) could be determined by Equation (8). \[ \begin{bmatrix}a&\begin{bmatrix}b_{i}\end{bmatrix}_{1\times 4}&\begin{bmatrix}c_{i} \end{bmatrix}_{1\times 4}\end{bmatrix} ^{\prime}=\boldsymbol{A}^{-1}\cdot \begin{bmatrix}Z_{k} \end{bmatrix} ^{\prime}{}_{1\times 9} \tag{10}\]  
Step 6: The reliability index \(\beta\) and design checking point \(\boldsymbol{X}^{*}= \begin{bmatrix}X_{1}^{*}&X_{2}^{*}&X_{3}^{*}&X_{4}^{*} \end{bmatrix} ^{\prime}\) were calculated through a series of iterations based on FORM. In this case, the mean values  
Figure 2: Flowchart of the probability calculation. of each random variable were selected as the initial design checking point. The iteration process can be expressed as Equations (11)-(13). Equivalent normalization of each random variable must be conducted before executing the FORM.