## 2 Theory Background and Methodology
To design the MA filter, the filter's window is selected through trial and error, and the mother-wavelet name and wavelet order are selected based on previous studies and the signal noise ratio for the measurements. In the current study, a 70-point MA filter was used to remove strains' noises. The [PERSON]'s theory of the MA filter is applied to decrease the shift between the measured and smoothed responses [31]. For the wavelet denoising, the discrete wavelet transform (DWT) was used to transform measured strain signals into decomposition levels, as follows [38]:  
\[W_{\psi}f(j,k)=2^{-j/2}\int\limits_{t=0}^{N-1}f(t)\overline{\psi}(2^{-j}t-k) \tag{5}\]  
where k and j are integer numbers; \(W_{\psi}f(j,k)\) is the discrete wavelet coefficient of scale \(a=2^{j},\ b=2^{j}k\), while \(a\) is scale or frequency factor and \(b\) is a time factor. \(\overline{\psi}(t)\) represents the complex conjugate function of mother wavelet (\(\psi(t)\)), where, the Daubechies (db) wavelet function is utilized. The DWT generated two sets of functions viewed as high-pass and low-pass filters. Thereby, the measured data was separated into one containing its trend (the approximate (A)) and one comprising high frequencies contents (the detail (D)) using Equation (5). The detail coefficient was reconstructed to estimate the dynamic response of strain and acceleration measurements. The MA filter was used to extract semi-static strain and displacement measurements; and the wavelet method was applied to extract accurate dynamic measurements. In both filters, the double filtration process was applied to extract the dynamic strain measurements. In this respect, the filtration cycle can be increased based on the performance of structures. The filtration and prediction methods were used to extract time-series displacement and acceleration of steel bridges using strain measurements. Figure 2 illustrates the developed model flow chart to extract the behavior of steel bridges in time and frequency domains using strain measurements. The displacement and acceleration responses were used for model design, while the strain measurements were used to extract the displacement and acceleration in the testing stage. The following steps were considered to design the model. First, the MA filter was used to estimate the semi-static strain and displacement responses from the measured strain and displacement, respectively. From both semi-static responses, the regression model was designed for the semi-static displacement responses of the bridge. Second, the dynamic strain responses were detected by subtracting the semi-static response from the measured strain signals. Then, the dominant frequency of dynamic strains and acceleration responses of bridge elements were calculated and assessed. If both frequencies are the same, then the wavelet filter can be applied to remove the remaining noises, else, the MA filter should be applied again to remove the remaining semi-static response. At the same time, the wavelet filter can be used to extract the filtered acceleration response. Finally, from the relation between the extracted dynamic strain and acceleration, the dynamic transformation function can be estimated. To assess the performance of the model design, statistical indices were used. The coefficient of determination (R\({}^{2}\)), root mean square error (RMSE) and mean absolute error (MAE) values were calculated and assessed. The R\({}^{2}\) was used to measure the relation between observed and predicted values, where low values of RMSE and MAE reflect high accuracy of the model.