## 2 Methodology  
This study primarily adopts the moving load influence line for the rigid-frame structures. Let the influence line for the continuous rigid-frame bridges be denoted as \(\mathbf{I}(x)\). The displacement \(\mathbf{d}(x)\) under the action of a moving load is calculated as shown in Equation (1), where \(\mathbf{P}\) represents the moving load:  
\[d(x)=P\cdot I(x) \tag{1}\]  
According to kinematic principles, the response under impact loads, specifically the maximum displacement amplitude, is primarily dependent on the magnitude of the applied impulse, while the form of the impact load has less influence. Therefore, considering the impact of moving-load noise, the evaluation of the moving load is performed by comparing the area of the influence line with the area of the monitoring data, which is the basic concept of the influence line (IL), as shown in Equation (2) below. \[d_{i}=\int_{0}^{L}d(x)\mathrm{d}x=P\cdot\int_{0}^{L}I(x)dx=P\cdot a_{i} \tag{2}\]  
where \(\mathbf{d}_{i}\) is the area of the displacement data measured at the \(\mathbf{i}\)-th measurement point, and \(\mathbf{a}_{i}\) is the area of the \(\mathbf{i}\)-th point influence line. However, in the case of multi-span continuous beam bridges, as shown in Figure 2, and the influence line, as shown in Figure 4, the values of both \(\mathbf{d}_{i}\) and \(\mathbf{a}_{i}\) are very small, resulting in great errors in Equation (2). The application of IL will be extended by changing the area of the value as the absolute area, as expressed by the following equation:  
\[D_{i}=\int_{0}^{L}|d(x)|dx=\int_{0}^{L}P\cdot|I(x)|dx=P\cdot A_{i} \tag{3}\]  
where \(\mathbf{D}_{i}\) is the area of absolute displacement data measured at the \(\mathbf{i}\)-th measurement point and \(\mathbf{A}_{i}\) is the absolute area of the \(\mathbf{i}\)-th point influence line. In this case, Equation (4) can be named the method of the absolute area of the influence line (AIL). Considering the noise \(\varphi(x,t)\), which is the random function of bridge and time, it should be noted that the variable \(\mathbf{t}\) in the function of \(\varphi(x,t)\) is not independent, and it is linked to the location of \(\mathbf{x}\) and can be expressed as \(\mathbf{t}=\mathbf{x}/\mathbf{v}\) (\(\mathbf{v}\) is the velocity). In this case, Equation (4) can be named the method of the absolute area of the influence line (AIL). Considering the noise \(\varphi(x,t)\), which is the random function of the time and the location of the bridge, Equation (3) can be modified to Equation (4):  
\[D_{i}=P\cdot A_{i}+\lambda\cdot\varphi_{i} \tag{4}\]where \(\varphi_{i}=\int_{0}^{L}\varphi(x,t)\cdot I(x)\mathrm{d}x\) is the inner product of the noise and influence line, which relates to the distribution of the random noise, and \(\lambda\) is the parameter determined by the type of noise. If the random noise is a type of white noise, the parameter \(\varphi_{i}\) would be very small or even close to zero. Therefore, using Equation (4) can only obtain the value of identification. In fact, the noise is usually random and unknown; more monitoring data can be used to obtain the identification to mitigate the errors caused by the load noise and the measurement. By combining data from multiple monitoring points, the following equation can be derived:  
\[\left( \begin{array}{c}D_{1}\\ D_{2}\\ \ldots\end{array} \right)=P\cdot\left( \begin{array}{c}A_{1}\\ A_{2}\\ \ldots\end{array} \right)+\lambda\cdot\left( \begin{array}{c}\varphi_{1}\\ \varphi_{2}\\ \ldots\end{array} \right) \tag{5}\]  
When the number of monitoring points is more than two, Equation (5) becomes a system of multiple constraints, and the moving load \(P\) and the noise distribution parameter \(\lambda\) can be obtained using the least squares method. In this case, Equation (5) can be named as the method of AIL-Modified. From the discussion above, it can be concluded that the proposed method needs at least 2 measurement points.