## 2 Principle of the Crack Detection and Lifetime Prediction Model
### Fusion Lifetime Prediction Model  
Reference [31] established a lifetime prediction model of coatings corresponding to crack length based on the assumption that interfacial oxidation is the major factor controlling the lifetime of coatings. In service, the residual stress in the ceramic layer increases with the number of thermal cycles. The accumulation of residual stress caused by oxidation promotes the length of cracks in the coating, and finally leads to the failure of the coating. In the real operation of a gas turbine, there are many sources of uncertainty in residual lifetime prediction, such as measurement error, randomness of load, change of working conditions, etc., so it is necessary to quantify and manage these uncertainties. Relevance Vector Machine (RVM) based on sparse Bayesian learning is a very suitable algorithm for residual lifetime prediction. It retains the idea of kernel mapping of the Support Vector Machine (SVM) and adopts some strategies to reduce the number of correlation vectors compared with SVM, which greatly reduces the prediction time while ensuring the prediction accuracy. Moreover, compared with traditional neural network and SVM, RVM can obtain probability output and better represent the uncertainty of residual lifetime prediction, so this paper uses RVM as one of the residual lifetime prediction methods. The steps of RVM are as follows:1. Input vector set (porosity, crack length, location, coating temperature) and target vector set (remaining useful lifetime RUL) are respectively formed for the data of each input image;
2. The radial basis function (RBF) was used as the kernel function, and the parameter of RBF was optimized by the five-fold cross validation method, with the objective of minimum mean square error;
3. RVM regression model was established by using the optimal parameter on each input image, and the optimal remaining lifetime value of the current image was obtained by prediction. 4. The average value of the remaining lifetime in a period of time was counted and the RUL value of the transition fragment was replaced by the mean value;  
The training of RVM can be completed by repeating 2-4 until it stops meeting certain convergence criteria, that is, when the change of mean square error is within the pre-set threshold. Due to the complex relationship between condition monitoring data and remaining lifetime, it is difficult to determine which prediction method will perform better in a specific situation. Therefore, a feasible way to fuse the lifetime prediction results under different conditions is to fuse multiple methods. Compared with a single prediction method, the fusion of multiple prediction methods can effectively improve the accuracy and robustness of prediction. In this paper, a RUL prediction fusion method based on information entropy is proposed, and the RUL prediction results using evidence theory regression, support vector machine and neural network are fused according to this method. The basic principle of the fusion method based on information entropy is that for each prediction method member, if the variability of the prediction error sequence is large, its corresponding weight should be set relatively small during synthesis. The specific calculation process is as follows [31]. \[e_{mt}=\left\{ \begin{aligned} & 1,\left|\frac{rul_{t}-rul_{mt}}{rul_{t}} \right|\geq 1;\\ &\left|\frac{rul_{t}-rul_{mt}}{rul_{t}}\right|,\left|\frac{rul_{t}- rul_{rt}}{rul_{t}}\right|<1\end{aligned} \right. \tag{4}\]  
In Equation (4), \(rul_{t}\) represents the ground truth RUL value of TBC at time \(t\), \(rul_{mt}\) represents the RUL value of TBC estimated by the physical model, and \(rul_{rt}\) is the RUL value of TBC estimated by RVM model.