# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 4 Validation of Results
### Sensor Placement Method  
#### 4.2.1 Methods of Sensor Placement  
To ensure the accuracy of the deformed shape estimation, the response data should be measured at an appropriate location. Research on the OSP method must been conducted in various fields before the need for SHM can be recognized. Representatively, [23] applied six different OSP methods to estimate the deformed shape of a suspension bridge and compared the estimation results. The OSP methods used for comparison were effective independence (EI), EI-driving point residue (EI-DPR), kinetic energy method (KEM), variance  
Figure 14: Required number of sensors by MSF and SSF (NMAPE < 5%): (**a**) displacement, (**b**) slope, (**c**) strain. method (VM), eigenvalue vector product (EVP), and nonoptimal driving point (NODP). As a result of comparisons, the estimation results obtained based on EI-DPR showed the best performance. The EI-DPR method has been used in various studies on sensor placement methods [24; 25]. Therefore, in this study, a sensor placement method for deformed shape estimation using SSF and multi-response data was analyzed based on the EI method. The EI sensor placement method [26] was developed to maximize both the spatial independence and signal strength of the shape function by maximizing the determinant of the associated Fisher information matrix. To maximize the determinant of the Fisher information matrix, the effective independence distribution (EID) \(E_{D}\) is derived as a diagonal of Equation (19). The \(i\)th component of \(E_{D}\), described in Equation (20), indicates the fractional contribution to the shape function at the \(i\)th sensor location. A sensor location noted as the lowest index of \(E_{D}\) is eliminated from the candidate locations, and this procedure is repeated until the remaining number of candidate locations is the same as the determined number of sensors. \[[E]\ =\ [\Phi]\Big{(}[\Phi]^{T}[\Phi]\Big{)}^{-1}[\Phi]^{T} \tag{19}\]  
\[E_{D_{i}}\ =\ diagonal([E])_{i}. \tag{20}\]  
[PERSON] and [PERSON] proposed a driving point residue (DPR) coefficient to overcome the limitations of the EI method, which allows the selection of sensor locations associated with low energy content [27]. The DPR coefficient, as described in Equation (21), is a weight factor that considers the modal contribution at each sensing location. The \(j\)th modal frequency \(\omega_{j}\) is applied to consider a higher weight factor for the higher-order mode. Then, \(E_{D}\) from Equation (20) can be transformed into Equation (22) by multiplying it by the DPR coefficient. EI-DPR is a method that concentrates sensor positions in high-energy-content regions. \[DPR_{i}\ =\ \sum_{j=1}^{N_{sf}}\frac{\Phi_{ij}^{2}}{\omega_{j}}\ (MSF) \tag{21}\]  
\[E_{D_{i}}\ =\ diagonal([E])_{i}\times DPR_{i}. \tag{22}\]  
Although the EI and EI-DPR methods have been verified by various researchers, there is a limitation, in that they cannot be applied to an SSF with \(N_{sf}>N_{md}\) because of the \(\Big{(}[\Phi]^{T}[\Phi]\Big{)}^{-1}\) term in Equation (19). In addition, when using various response data, the contribution of overlapping and interference affects the relationship between response data. Therefore, in this paper, the DPR coefficient for the SSF and distance coefficient \(d\) are proposed as Equations (23) and (24) to estimate the deformed shape using the SSF. To derive the DPR coefficient for SSF, the maximum value of the absolute \(j\)th shape function \(\big{|}\Phi_{j}\big{|}_{max}\) is used instead of \(\omega_{j}\). The distance coefficient \(d_{i}\) is the minimum value among the distances from the \(i\)th candidate location to each sensor location previously selected. The sensor locations previously selected contain locations that have a constant certain response value, such as the boundary for displacement. \(E_{D}\), proposed in this study, was calculated using Equation (24) by multiplying the distance coefficient using Equation (22).