## 2 Bearing Load Testing Method Based on Redundant Strain and Jacking Data Fusion
### Data Fusion Processing
#### 2.3.2 Jacking Data Fusion Based on Bayesian Estimation
The Bayesian fusion rule obtains the posterior probability by linking prior probability, likelihood probability, and the probability of known sets, generally expressed as:  
\[p(x|z)=\frac{p(z|x)p(x)}{p(z)} \tag{26}\]  
where \(p(x)\) is the prior probability, \(p(z|x)\) is the likelihood function, \(p(z)\) is the known probability set, and \(p(x|z)\) is the posterior probability. Using the Bayesian fusion method, the fused jack load \(P_{0}\) and its PDF are obtained as:  
\[f_{P_{0}}(x)\propto f_{P_{0}^{\prime}}(x)f_{P_{0}^{\prime\prime}}(x) \tag{27}\]  
According to Bayesian theory, the fused mean \(\mu_{P_{0}}\) and variance \(\sigma_{P_{0}}^{2}\)[29] are:  
\[\left\{ \begin{array}{l}\mu_{P_{0}}=\frac{\sigma_{P_{0}}^{2}}{\sigma_{P_{0}^{ \prime}}^{2}+\sigma_{P_{0}^{\prime\prime}}^{2}}P_{0}^{\prime}+\frac{\sigma_{P_ {0}^{\prime}}^{2}}{\sigma_{P_{0}^{\prime}}^{2}+\sigma_{P_{0}^{\prime\prime}}^ {2}}P_{0}^{\prime\prime}\\ \sigma_{P_{0}}^{2}=\frac{1}{\sigma_{P_{0}^{\prime}}^{2}+\sigma_{P_{0}^{\prime \prime}}^{-2}}\end{array} \right. \tag{28}\]  
Let the theoretical jacking coefficient given by the alignment calculation book be \(K\), and the actual jacking coefficient be \(\overline{K}\). Assuming \(K\)\(\sim\)\(N(\overline{K},\sigma_{K}^{2})\), then according to Equation (19), the bearing load \(F_{J}\) by JM and its PDF are obtained as:  
\[f_{F_{J}}(x)=\left(2\pi\sigma_{F_{J}}^{2}\right)^{-1/2}\exp(-\frac{\left(x-F_ {J}\right)^{2}}{2\sigma_{F_{J}}^{2}}) \tag{29}\]  
These parameters can be calculated by:  
\[\left\{ \begin{array}{l}F_{J}=\mu_{P_{0}}K\\ \sigma_{F_{J}}^{2}=K^{2}\sigma_{P_{0}}^{2}+\mu_{P_{0}}^{2}\sigma_{K}^{2}+ \sigma_{P_{0}}^{2}\sigma_{K}^{2}\end{array} \right. \tag{30}\]  
Decision-Level Data Fusion Based on Bayesian Estimation  
Multi-sensor data fusion can be categorized by the level of fusion into data-level, feature-level, and decision-level fusion [30]. Data-level fusion directly integrates the raw data obtained from each sensor, which can maximize the preservation of the original data information but cannot be used for the fusion of heterogeneous sensors. Feature-level fusion is based on the integration of features extracted from the raw data. Decision-level fusion is an advanced form of fusion that integrates the target description results completed by each sensor individually, achieving a consistent description of the target. After separately processing redundant strain data and jacking data to obtain the bearing pre-decision load, the next step is to fuse the decision-level data based on the Bayesian estimation method to obtain the fused bearing load. The PDF of the \(F_{J}\) and the \(F_{s}\) are fused using the Bayesian fusion method to obtain the PDF of the fused bearing load. \[f_{F}(x)\propto f_{F_{S}}(x)f_{F_{J}}(x) \tag{31}\]  
Here, the fused bearing load \(F\) and its variance \(\sigma_{F}^{2}\) are:\[\left\{ \begin{array}{l}F=\frac{\sigma_{F_{I}}^{2}}{\sigma_{F_{S}}^{2}+\sigma_{F_{ I}}^{2}}F_{S}+\frac{\sigma_{F_{S}}^{2}}{\sigma_{F_{S}}^{2}+\sigma_{F_{I}}^{2}}F_{I}\\ \sigma_{F}^{2}=\frac{1}{\sigma_{F_{S}}^{2}+\sigma_{F_{I}}^{2}}\end{array} \right. \tag{32}\]