# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 1 Introduction  
As bridges across the world continue aging, structural health monitoring (SHM) systems are being established and operated in many developed countries [1, 2, 3, 4, 5]. In an SHM system, measurement data such as displacement, strain, acceleration, slope, temperature, and wind speed measured by each sensor are collected and managed. These measurement data can be used directly and indirectly to evaluate the conditions of a bridge. However, the use of measurement data remains ineffective because of the low development level of the utilization method for practical purposes. Therefore, a method for utilizing measurement data must be developed for the effective management of bridges. For the SHM of infrastructures, various structural responses, such as displacement, slope, and strain, are generated by an external load. Among them, displacement can be considered as one of the representative responses and can be used to evaluate the integrity of a bridge. It has been used for integrity evaluation by comparing it with thedeflection limit determined through various methods such as numerical analysis and load testing. Linear variable differential transformers (LVDTs) and laser Doppler vibrometers (LDVs) are frequently used to measure displacement, but they have limitations, such as the requirement of additional fixed reference points and sensitivity to the surrounding climate. To overcome these limitations, various studies have been conducted to indirectly estimate displacement using strain, slope, and acceleration data. Acceleration in the response of the structure has been widely used in the study of dynamic displacement estimation, owing to its low price and convenience. [PERSON] et al. estimated the displacement through the double integration of acceleration [PERSON] et al. (2018). To use the double integration method, the initial conditions of displacement and velocity should be considered. Displacement was estimated through iterative calculations, assuming that the initial displacement and average velocity were zero. On the other hand, [PERSON] et al. proposed a method for estimating displacement without initial conditions in a way that minimizes the least-squares error between the assumed acceleration and measured acceleration [PERSON] et al. (2018). The assumed acceleration is expressed as the central finite difference of displacement. The estimation method proposed in [PERSON] et al. (2018) was extended to increase the estimation accuracy and combine it with measurement data for other responses [PERSON] et al. (2018); [PERSON] et al. (2018); [PERSON] et al. (2018). However, because the spatial parameter is not included in the acceleration response, there is a limitation in that only the displacement where the sensor is installed can be estimated. Using the slope and strain data, which include spatial parameters, can overcome this limitation. [PERSON] et al. estimated displacement based on the shape superposition method using slope data [PERSON] et al. (2018). The shape function consisted of the power series and coefficient for the boundary condition, and the weight factors for each shape function were derived using the least-squares method. The estimation method of displacement using strain data was first proposed by [PERSON] et al. (2018) based on modal mapping. Modal mapping using the mode shape as a shape function is based on the superposition method. [PERSON] et al. applied a theoretical mode shape composed of sine functions [PERSON] et al. (2018) for general use of the estimation method proposed by [PERSON] et al. (2018), and its effectiveness was verified by empirical experiments on various structures. Subsequently, a study using mode shape, derived by frequency analysis of an finite element method (FEM) model, was conducted to improve the accuracy of the estimated displacement [PERSON] et al. (2018). It was mentioned that the mode shape derived from the FEM model was more useful than the theoretical model shape for estimating displacement. In addition, the modal mapping proposed by [PERSON] et al.