# New Quality Control Algorithm Based on GNSS Sensing Data for a Bridge Health Monitoring System
## 2 Quality Control Algorithm and Quality Measures
### Quality Measures
#### 2.3.3 Reliability  
Deformation analysis involves two types of errors: measurement and deformation models [1]. To avoid misinterpretation of systematic errors or outliers in the measurements as deformation phenomena, screening of the measurements for outliers or systematic errors should be performed prior to the estimation of the deformation parameters. The concept of reliability originates from [PERSON] [4]. Generally, reliability assesses the capability of detecting outliers. In this respect, \"internal reliability and external reliability\" are distinguished. The internal reliability of a GNSS positioning solution is its ability to detect outliers by testing a hypothesis made with a specific confidence level (\(1-\alpha\)) and the power of test (\(1-\beta\)). In other words, higher reliability means that even small errors can be found, whereas lower reliability means that even larger errors cannot be detected. The other way of testing is through the external reliability of the GNSS positioning solution in informing of the impacts of undetected errors against a detected unknown vector or estimated coordinates. Hence, high external reliability means that the undetected errors from the quality control procedures have minimal impact on coordinates. The number of redundancy measures, the geometry of the tracking satellites, and error propagation all affect reliability. Therefore, if numerous redundant measurements or many tracking satellites exist, the reliability should improve.  
_Internal reliability_  
Internal reliability refers to the maximum number of undetectable errors and is simply represented by MDB. This variable is assessed by the lower values \(\
abla_{0}l_{i}\) of outlier \(\
abla l_{i}\), which can detect, through the data-snooping test, if any remaining errors heavily affect the final estimation of coordinates. MDB is the magnitude of the smallest bias detectable and is determined for a given confidence level. The critical value mentioned earlier can be simplified by diagonal elements of the weight matrix P (Q):  
\[|\
abla_{0}l_{i}|=\sigma_{0}\frac{\lambda_{0}}{\sqrt{P_{i}(Q_{v}P)_{ii}}}= \sigma_{0}\frac{\lambda_{0}}{\sqrt{P_{i}r_{i}}} \tag{54}\]  
According to this formula, the detectable outlier directly depends on the precision of the measurement or its variance (\(\sigma_{l_{i}}\)). Outliers occur when it is in the significance level of \(\alpha\) in type 1 errors and \(\beta\) in type 2 errors.  
\[\left|\frac{v_{i}}{\sigma_{v_{i}}}\right|\geqslant N_{\alpha+\beta} \tag{55}\]  
The power of test was suggested by [PERSON] [6] to be typically 20%. The result of the previous two rearranged equations is the following:  
\[|MDB|\geqslant\frac{\lambda_{0}\sqrt{Q_{v}}}{Q_{v}P}=\frac{\lambda_{0}\sqrt{Q _{v}}}{r} \tag{56}\]  
According to this equation, MDB is simultaneously dependent on the redundancy number and the covariance of the residuals, not on the measurement and residual.  
_External reliability_  
External reliability relates to the maximum effect of a possibly undiscovered observational outlier on the estimates of unknown parameters. It verifies whether an undetected error exists, and what impact the error would have on estimated coordinates [6]. Thus, external reliability can show the impact of MDB on estimated coordinates:  
\[\
abla_{0}\hat{x}=\left(A^{T}PA\right)^{-1}A^{T}e_{i}P\cdot\left(MDB\right)=Q _{\hat{x}}A^{T}Pe_{i}\cdot\left(MDB\right) \tag{57}\]  
where \(Q_{\hat{x}}\) is the posterior variance-covariance matrix of the estimated parameters.