# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions
## 4 Numerical Application to Bridge
### With Uncertainties  
This section examines the effect of the various errors that may occur during the measurement of the model updating results. Errors in the installation of the sensor were considered to be caused by the uncertainties related to the position and the direction of the sensor. Error in the measurement was considered as white noise. Each error was assumed to have a normal distribution \(N(0,\;\sigma)\), with a zero mean and standard deviation \(\sigma\). The standard deviation of the error in the installation of the sensor was set to 0 mm, 50 mm, and 100 mm for the position of the sensor and \(0^{\circ}\), \(2^{\circ}\), and \(4^{\circ}\) for the direction of the sensor considering the size of the error that may occur in a real situation. The standard deviation for the error with white noise was set to comparatively low levels of 0 m/m, \(1\times 10^{-6}\) m/m, and \(2\times 10^{-6}\) m/m under the assumption that the FBG sensor was installed. Model updating was performed 30 times for each considered case.  
The following error was computed for each analyzed case and averaged for all identical cases to obtain the mean error ratio for each case.  
\[e=\frac{1}{n_{seg}}\sum_{i}^{n_{seg}}\left|\frac{E_{updated,i}-E_{structure,i} }{E_{structure,i}}\right| \tag{56}\]  
where \(E_{structure,i}\) and \(E_{updated,i}\) are the elastic modulus and the updated elastic modulus of the \(i\)th segment of the bridge; and, \(n_{seg}\) is the number of segments (19 for the example). Figure 8 plots the mean error ratio for each case. The mean error ratio of the model updating was seen to increase with larger errors in the position of the sensor. On the other hand, the error relative to the direction of the sensor appears to have no significant effect on the mean error ratio of the model updating. Moreover,  
Figure 7: Model updating results: (**a**) model updating process; (**b**) comparison of elastic moduli of the structure and updated model; and (**c**) error ratio of the structure-updated model elastic moduli.  
a larger measurement noise increased the mean error ratio but seemed to be uncorrelated to the error in the installation of the sensor.  
In Figure 8a, the case where the standard deviation for the position error was 0 mm and the standard deviation for the direction was 0 degree, it corresponded to the case without uncertainties in sub-Section 4.1. In such a case, the mean error ratio was 2.4%, which was the one produced during the modal identification process as explained above. The mean error ratio rose to 3.8% according to the increase of the error in the installation of the sensor. Note that, if the error introduced by the modal identification process was discarded, the mean error ratio caused by the installation error dropped to a minimal level of 1.4%. The mean error ratio, in the case with a standard deviation of \(2\times 10^{-6}\) for the measurement error in Figure 8c, ran between 3.0% and 4.2%. Here also, the error ratio caused only by the uncertainties in the installation and measurement remained very minimal with a level of approximately 0.6% to 1.8%. Consequently, the method proposed in this study is also applicable in cases including sensor installation errors and measurement noises.