## 3 Methodology
### Structure Deterioration Identification
#### 3.2.4 Establishment of Bridge Model Update Process
The expression for the correction result is as follows:  
\[\text{arg}_{\theta}\text{min}e(\theta,M)=\text{arg}_{\theta}\text{min}{\sum_{i }\alpha_{i}}\left|\frac{y_{i}(\theta,M)-\hat{y}_{i}}{\hat{y}_{i}}\right| \tag{9}\]  
where \(\theta\) represents the updated parameters, M represents the finite element model to be updated, \(\text{e}(\theta,\text{M})\) represents the total error value, \(\hat{y}_{i}\) represents the test data for the i-th indicator, \(y_{i}(\theta,\text{M})\) represents the design parameter data corresponding to the test data, and a represents the weight value of the i-th indicator. To quantify the uncertainty of the test data, the Bayesian model updating theory is introduced, with its basic formula being the following:  
\[P(\text{p}\,|\text{D})=\frac{P(\text{D}\,|\text{p}\,)P(\text{p})}{P(\text{D})} \tag{10}\]  
where \(\text{P}(\text{P}|\text{D})\) represents the posterior distribution, \(\text{P}(\text{D}|\text{p})\) represents the likelihood function, \(\text{P}(\text{p})\) represents the prior distribution, and \(\text{P}(\text{D})\) represents the probability of the observed data. Before this, we need to determine the static parameters of the bridge, such as material strength indices, deformation indices, stress distributions, etc., and define a range of numerical values for each parameter. Subsequently, all the static parameters and their value ranges are constructed into an initial matrix:  
\[\mathrm{P}= \begin{bmatrix}P_{1}&\mathrm{min}&p_{1}&\mathrm{max}\\ P_{2}&\mathrm{min}&p_{2}&\mathrm{max}\\ \vdots&\vdots&\vdots&\vdots\\ P_{n}&\mathrm{min}&p_{n}&\mathrm{max}\end{bmatrix}  \tag{11}\]  
where \(\mathrm{P}_{i,\mathrm{min}}\) and \(\mathrm{P}_{i,\mathrm{max}}\) represent the minimum and maximum values of the i-th parameter, respectively. Then, based on the observed crack positions and deformation positions, the error function E(p) is constructed as follows:  
\[E(\mathrm{p})=\sum_{i=1}^{n}{(R_{i}^{\mathrm{obs}}-R_{i}^{\mathrm{sim}}( \mathrm{p}))}^{2} \tag{12}\]  
where \(\mathrm{R}_{i}^{\mathrm{obs}}\) represents the observed response data, and \(\mathrm{R}_{i}^{\mathrm{sim}}(\mathrm{p})\) represents the simulated response of the model under the parameter p. In this framework, the adaptive nested sampling method (ANS) is selected to initialize sample points, generate initial sample points in the parameter space, and calculate their likelihood values. Subsequently, nested sampling is performed, sampling in areas with higher likelihood values to generate new sample points, and then updating the probability distribution of the model parameters based on the new sample points. After convergence judgment, if the likelihood value distribution of the sample points no longer significantly changes, it is considered that the algorithm has converged. Finally, based on Bayesian updating and ANS results, the posterior distribution of parameters is analyzed, the range of each parameter is narrowed, and the posterior distribution of parameters is reflected in the parameter matrix:  
\[\mathrm{P}= \begin{bmatrix}P_{1}&\mathrm{updatedmin}&P_{1}&\mathrm{updatedmax}\\ P_{2}&\mathrm{updatedmin}&P_{2}&\mathrm{updatedmax}\\ \vdots&\vdots&\vdots&\vdots\\ P_{n}&\mathrm{updatedmin}&P_{3}&\mathrm{updatedmax}\end{bmatrix}  \tag{13}\]  
In conclusion, through the above steps, corresponding registration methods can be used to register the overall point cloud data structures of bridges from two or more different years based on feature point clouds. This results in the displacement along the bridge's longitudinal axis (x-axis), transverse axis (y-axis), and vertical axis (z-axis), as well as the rotation around the three coordinate axes. Furthermore, a transformation matrix is obtained, and the nearest neighbor search is employed to find the nearest neighbor points of each point in the point cloud array. These two nearest neighbor points are then used as corresponding points.