## 2 Proposed Iterative Identification Method
Based on the Equation (5), the ground acceleration can be estimated as:  
\[\widetilde{\widetilde{\mathbf{X}}}_{g}(t)=-\mathbf{M}^{-1}(\mathbf{M}\mathbf{ \dot{x}}(t)+\widetilde{\alpha}\mathbf{M}\mathbf{\dot{x}}(t)+\widetilde{ \mathbf{Q}}\mathbf{\dot{x}}(t)+\widetilde{\mathbf{K}}\mathbf{x}(t)) \tag{9}\]  
where \(\widetilde{\alpha}\), \(\widetilde{\mathbf{Q}}\) and \(\widetilde{\mathbf{K}}\) are the estimated values of \(\alpha\), \(\mathbf{Q}\) and \(\mathbf{K}\), respectively. They can be determined with the aid of \(\widetilde{\theta}\) and structural topology. Notably, the estimated ground acceleration \(\widetilde{\widetilde{\mathbf{X}}}_{g}\) obtained from Equation (9) is an \(n\times s\) matrix in the form of  
\[\widetilde{\widetilde{\mathbf{X}}}_{g}(t)=\left[ \begin{array}{ccc}\widetilde {\widetilde{x}}_{g}^{1}(t_{1})&\widetilde{\widetilde{x}}_{g}^{1}(t_{2})& \ldots&\widetilde{\widetilde{x}}_{g}^{1}(t_{s})\\ \vdots&\vdots&&\vdots\\ \widetilde{\widetilde{x}}_{g}^{n}(t_{1})&\widetilde{\widetilde{x}}_{g}^{n}(t_ {2})&\ldots&\widetilde{\widetilde{x}}_{g}^{n}(t_{s})\end{array} \right] \tag{10}\]  
where \(\widetilde{\widetilde{x}}_{g}^{i}(t_{j})\) (\(i=1,\ldots,n;j=1,\ldots,s\)) denotes the ground acceleration distributed on the \(i\)-th DOF at the time step of \(t_{j}\). Step 3: Modify the estimated ground acceleration and form the corresponding inertia force vector \(\mathbf{\hat{P}}(t)\). For earthquake-excited building structures, the ground acceleration should be equally distributed on each floor of the building structure. It can be concluded that, if the unknown parameters are not stably converged to the actual ones, the estimated ground acceleration values shown in Equations (9,10) for different DOFs would not be close to each other. Based on this distribution characteristic, the following input modification process is presented in this study:  
\[\hat{\widetilde{x}}_{g}(t_{j})=\frac{1}{n}\sum_{i=1}^{n}\widetilde{\widetilde {x}}_{g}^{i}(t_{j}),(j=1,\ 2,\ \ldots,s) \tag{11}\]where \(\hat{\tilde{x}}_{g}\) stands for the modified ground acceleration. Obviously, the mean values of the estimated ground acceleration are used as the modified ones, and the corresponding inertia force vector can be determined as:  
\[\hat{\mathbf{P}}(t)=\left[ \begin{array}{cccc}\hat{\mathbf{P}}(t_{1})^{\mathrm{ T}}&\hat{\mathbf{P}}(t_{2})^{\mathrm{T}}&\cdots&\hat{\mathbf{P}}(t_{s})^{ \mathrm{T}}\end{array} \right]^{\mathrm{T}} \tag{12}\]  
where  
\[\hat{\mathbf{P}}(t_{j})=-\mathbf{M}1\hat{\tilde{x}}_{g}(t_{j})-\mathbf{M} \hat{\mathbf{x}}(t_{j}),\;(j=1,\;2,\;\ldots,s) \tag{13}\]  
Step 4: Form the response matrix \(\mathbf{H}(t)\), and obtain the updated parameters \(\hat{\theta}\) according to Equation (8). \[\hat{\theta}=\left[\mathbf{H}(t)^{T}\mathbf{H}(t)\right]^{-1}\mathbf{H}(t)^{ T}\hat{\mathbf{P}}(t) \tag{14}\]  
Step 5: Calculate the error between \(\hat{\theta}\) and \(\widetilde{\theta}\). Let \(\hat{\theta}\) be \(\widetilde{\theta}\), and repeat Steps 2-5 until the following convergence criterion is satisfied:  
\[\left|\frac{\hat{\theta}_{l}-\widetilde{\theta}_{l}}{\widetilde{\theta}_{l}} \right|\leq\varepsilon\;(l=1,\;2,\;\ldots,\;h) \tag{15}\]  
where \(\hat{\theta}_{l}\) and \(\widetilde{\theta}_{l}\) are the \(l\)-th element of \(\hat{\theta}\) and \(\widetilde{\theta}\), respectively; \(\varepsilon\) is a preset value. When the iteration procedure is complete, the ground acceleration can be directly found from Equation (11). Moreover, since both \(k_{i}\) and \(q_{i}\) are identified, the damping coefficient \(\beta\) can be determined as follows:  
\[\hat{\beta}=\frac{1}{n}\sum_{i=1}^{n}\frac{\hat{q}_{i}}{\hat{\tilde{k}}_{i}} \tag{16}\]  
where \(\hat{q}_{i}\) and \(\hat{k}_{i}\) are the identified values of \(q_{i}\) and \(k_{i}\), respectively.