## 3 Methodology  
As mentioned in Section 2.2, particle filtering is a type of particle-based Kalman filter and can be applied to a predictive model with high nonlinearity. In addition, particle filtering is known as an advanced technique for supplementing the prediction accuracy of the Bayesian method. In particular, considering the characteristics of particle filtering based on Bayesian probability theory, model variables with high probability are resampled with weights, and model variables with low probability are relatively resampled through an update process that reflects the actual measured data. In other words, the variables with higher probability among the model variables are resampled through the update process, and accordingly, the prediction accuracy is improved for the target data (or model) with higher reliability of the measured data. Due to these characteristics, the particle filtering technique is also called \"Sequential Monte Carlo\" [33]. In this study, the regression analysis parameters, \"\(a\)\" and \"\(b\)\", as states, were used in the \"augmented particle filter\" [30; 31; 32]. Furthermore, in our problem, all equations were prepared based on a previous study [33]. Based on the concrete pavement condition index (CPCI), which is one of the various condition indices of concrete pavement, the following equations were established as the first step to utilize particle filtering:  
\[CPCI_{k}=CPCI_{k-1}+a_{k}t+b_{k}, \tag{1}\]  
\[a_{k+1}=a_{k}+w_{ak}, \tag{2}\]  
\[b_{k+1}=b_{k}+w_{lk}, \tag{3}\]  
\[y_{k}=CPCI_{k}^{measured}-CPCI_{k}\sim CPCI\Big{(}0,\sigma_{CPCI}^{2}\Big{)}. \tag{4}\]  
In Equations (1) to (4), \(a_{k}\) and \(b_{k}\) are regression coefficients. In this study, \"\(a\)\" and \"\(b\)\" are uniform distributions, set in the range considering (a) the slope of the regression analysis of the actual detection data and (b) the uncertainty of the initial CPCI. \(w_{ak}\) and \(w_{lk}\) are white noise vectors of a Gaussian distribution with a mean of 0. In addition, \(CPCI_{k}\) represents the future CPCI and is set as an objective function of \(k\) corresponding to a time variable. \(t\) denotes the age of concrete pavement, and \(CPCI_{k}^{measured}\) denotes the measured CPCI. Therefore, \(y_{k}\) is a concrete pavement condition exponential function, which can be expressed as \(CPCI\big{(}0,\sigma_{CPCI}^{2}\big{)}\). Consequently, the relative likelihood can be expressed as Equation (5). The relative likelihood is given as follows [33]:  
\[q=P\Big{(}CPCI_{k}^{measured}-CPCI_{k}\Big{)}\sim\frac{1}{\sqrt{2\pi}\sigma_{ CPCI}}exp\left\{-\frac{1}{2}\left(\frac{CPCI_{k}^{measured}-CPCI_{k}}{ \sigma_{CPCI}}\right)^{2}\right\}, \tag{5}\]  
So far, no case has been found regarding the quantitative error rate that occurs during PMS pavement surface scanning image analysis. Therefore, 5-10% of the error rate that is commonly recognized in the PMS field of Korea was used as a measure of data validity for the application of particle filter; thus, \(\sigma_{CPCI}\) was assumed to be 0.1 (10%). In addition, the relative likelihood can be normalized as expressed in Equation (6). \[q_{i}=\frac{q_{i}}{\sum_{j=1}^{N}q_{j}}. \tag{6}\]  
In Equation (6), \(N\) denotes the number of particles, indicating that the sum of the relative likelihoods equals 1. New particles can be calculated based on the relative likelihoods \(q_{i}\). This procedure can be performed using several different methods. In this study, we applied [34]. To prevent the situation where the initial CPCI is biased toward a specific distribution, a random number (\(r\)) was set to a uniform probability distribution between 0 and 1. \[\sum_{i=1}^{j-1}q_{i}<r\leq\sum_{i=1}^{j}q_{i}. \tag{7}\]  
Through the process of Equation (7), \(a_{j}\) and \(b_{j}\) can be reselected. \(q\) denotes the measurement data. At this time, the probability value increases as the CPCI approaches the mean, and the probability decreases toward the tail of the normal distribution.