## 4 Initial Parameters Selection  
As discussed above, the solution with augmented Tikhonov regularization method, six initial parameters need to be defined. Among these, the values of \(\lambda\) and \(\tau\) changes after each iteration, while those of \(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}\) do not. In order to understand the effect of the initial values of \(\lambda,\tau\), we run hundreds of simulations, where we noticed that the initial values of these parameters have little effect on the load identification algorithm. We also noticed that setting the parameters \(\alpha_{1},\alpha_{2}\) close to unity leads to stable results with accurate load identification. However, the choice of \(\beta_{1},\beta_{2}\) may lead to a significant change in the identification results. Therefore, in the following, we limit our discussion to the selection of these two parameters. The correlation coefficient (\(r\)) and relative error (\(RE\)) are used to estimate the load identification error in this paper. These are calculated as follows:  
\[r=\frac{\sum\limits_{i=1}^{n}\left[f_{idm}(i)-\overline{f}_{idm}\left|f_{true} \left(i\right)-\overline{f}_{true}\right|\right]}{\left\|\mathbf{f}_{idm}-\overline {f}_{idm}\right\|\cdot\left\|\mathbf{f}_{true}-\overline{f}_{true}\right\|}\times 1 00\%, \tag{29}\]  
Figure 1: Load identification flow chart. \[RE=\frac{\left\|f_{iden}(i)-f_{true}(i)\right\|}{\left\|f_{true}(i)\right\|}\times 1 00\%, \tag{30}\]  
where \(f_{iden}\) is the identified force, and \(f_{true}\) is the true force, while \(\overline{f}_{iden}\) and \(\overline{f}_{true}\) are the means of \(f_{iden}\) and \(f_{true}\), respectively. The values of \(\beta_{1},\beta_{2}\) can have a significant impact on the load identification results \(\left\|f_{k}\right\|_{2}^{2}\) and the residual norm \(\left\|H_{k}-y\right\|_{2}^{2}\). To analyze the impact of selecting the values of \(\beta_{1},\beta_{2}\) on the load identification, we study, in this section, a single degree of freedom system. The considered parameters for this system are \(m=1\) kg, \(k=1500\) N/m and \(c=15\) N-s/m. The load applied on the system is periodic defined by \(F=100*\sin(\pi*50*t)\). We also set the parameters \(\alpha_{1},\alpha_{2}\) to \(2\), which is relatively a small value while the parameters \(\lambda,\tau\) are set to \(100\) and \(1\times 10^{12}\), respectively. To understand the effect of the parameters \(\beta_{1}\) and \(\beta_{2}\) on the identified load, their values will be varied, and the identified load will be recorded. Figure 2 shows the changes in the correlation coefficient and the relative error for varying the values of \(\beta_{1},\beta_{2}\) when the signal noise ratio (SNR) of the response is \(30\) dB. It can be seen in the figure that the optimum range of values for \(\beta_{1}\) is from \(10^{-5}\) to \(10^{5}\), and, for \(\beta_{2}\), is a narrow band close to \(10^{4}\). In this range, the correlation coefficient is close to \(1\), while the relative error is close to \(0\). This suggests that the load identification accuracy is higher at this specific range of parameters \(\beta_{1},\beta_{2}\). Clearly, the optimum range for \(\beta_{1}\) is much wider compared to \(\beta_{2}\). Thus, choosing an optimum value for \(\beta_{2}\) is our next concern. Figure 2: (**a**) Correlation coefficient varies with the \(\beta_{1},\beta_{2}\). (**b**) Relative error varies with the \(\beta_{1},\beta_{2}\). In real world applications, because the load acting on the structures is unknown, it is not possible to define the correlation coefficient and the relative error. Therefore, to determine the optimum values for the parameter \(\beta_{2}\), we use the relation between the parameter \(\beta_{2}\) and the residual norm \(\|Hf-y\|_{2}^{2}\) of the identified load. Indeed, in the load identification process, we rebuild the dynamic responses according to the identified load for a given parameter \(\beta_{2}\) and then calculate the residual norm \(\|Hf-y\|_{2}^{2}\).