## 2 Proposed Iterative Identification Method  
The governing equation of motion of an \(n\) degree-of-freedom (DOF) linear structure under earthquake can be expressed as:  
\[\mathbf{M\ddot{x}}(t)+\mathbf{C\dot{x}}(t)+\mathbf{Kx}(t)=-\mathbf{M}1\ddot{x }_{g}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) represent the \(n\times n\) mass, damping, and stiffness matrices of the structure, respectively; \(\ddot{\mathbf{x}}(t)\), \(\dot{\mathbf{x}}(t)\) and \(\mathbf{x}(t)\) denote the structural acceleration, velocity, and displacement responses, respectively; \(\mathbf{1}\) denotes an \(n\times 1\) matrix with all the elements being equal to 1; and \(\ddot{x}_{g}(t)\) is the time history of ground acceleration. In this study, the Rayleigh damping assumption is adopted:  
\[\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K} \tag{2}\]  
where \(\alpha\) and \(\beta\) are mass-proportional and stiffness-proportional damping coefficients, respectively. Substituting Equation (2) into Equation (1) results in  
\[\mathbf{M\ddot{x}}(t)+(\alpha\mathbf{M}+\beta\mathbf{K})\dot{\mathbf{x}}(t)+ \mathbf{Kx}(t)=-\mathbf{M}1\ddot{x}_{g}(t) \tag{3}\]  
Here, the mass matrix \(\mathbf{M}\) is assumed to be known, and then Equation (3) can be rearranged as:  
\[\alpha\mathbf{M\dot{x}}(t)+\beta\mathbf{K\dot{x}}(t)+\mathbf{Kx}(t)=-\mathbf{ M}1\ddot{x}_{g}(t)-\mathbf{M\ddot{x}}(t) \tag{4}\]  
Notably, since both \(\beta\) and \(\mathbf{K}\) are the parameters to be identified, their coupled products would introduce a source of non-linearity into the identification procedure when the LSE technique is used. To handle this situation, Equation (4) can be rewritten as:  
\[\alpha\mathbf{M\dot{x}}(t)+\mathbf{Q\dot{x}}(t)+\mathbf{Kx}(t)=-\mathbf{M}1 \ddot{x}_{g}(t)-\mathbf{M\ddot{x}}(t) \tag{5}\]  
where \(\mathbf{Q}\) can be viewed as a transformation matrix (\(\mathbf{Q}=\beta\mathbf{K}\)) with the same dimension of \(\mathbf{K}\). By rearranging Equation (5), the identification equation can be expressed as follows:  
\[\mathbf{H}(t)\theta=\mathbf{P}(t) \tag{6}\]in which \(\mathbf{H}(t)\) is the response matrix with the dimension of \(ns\times h\) (\(ns=n\times s\)); \(h\) and \(s\) are defined as the number of unknown coefficients and sample points to be used, respectively; \(\mathbf{P}(t)\) is \(ns\times 1\) inertia force vector; \(\theta\) is the \(h\times 1\) (\(h=2\times n+1\)) unknown coefficients vector defined as:  
\[\theta=[k_{1},\ldots,k_{n},\alpha,q_{1},\ldots,q_{n}]^{T} \tag{7}\]  
where \(k_{i}\) (\(i=1,\ldots,n\)) denotes the stiffness of the \(i\)-th floor; \(q_{i}=\beta\times k_{i}\) (\(i=1,\ldots,n\)) is the element-wise coefficient in \(\mathbf{Q}\) matrix. The unknown parameters can then be estimated by LSE as follows:  
\[\theta=\Big{[}\mathbf{H}(t)^{\mathrm{T}}\mathbf{H}(t)\Big{]}^{-1}\mathbf{H}( t)^{\mathrm{T}}\mathbf{P}(t) \tag{8}\]  
It follows from Equation (8) that the input should be known for the identification. However, in some practical situations, for example an earthquake-excited building, the measurements of loadings are sometimes unavailable. Therefore, the parameters cannot be directly identified according to Equation (8) in this case. Additional information or supplementary equations are required for the identification. Based on the spatial distribution characteristics of the ground motion in earthquake-excited structures, an input modification process is proposed in this study. A time-domain iterative identification procedure is then developed for the simultaneous identification of structural parameters and unknown ground motion. The details are given below:  
Step 1: Assign the initial values of the unknown parameters named as \(\widetilde{\theta}\). Step 2: Estimate ground acceleration by using \(\widetilde{\theta}\) and structural response measurements.