## 2 Theoretical Foundation  
### Description of Ground Motion Fields  
The spatially variable earthquake motion can be considered as \(n\)-variate zero-mean nonstationary Gaussian vector process \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)]^{T}\) (the superscript \(T\) denotes the transpose), which can be characterized by its EPSD matrix given by [24]  
\[\mathbf{S}(\omega,t)=\begin{bmatrix}S_{11}(\omega,t)&S_{12}(\omega,t)&\cdots&S _{1n}(\omega,t)\\ S_{21}(\omega,t)&S_{22}(\omega,t)&\cdots&S_{2n}(\omega,t)\\ \vdots&\vdots&\ddots&\vdots\\ S_{n1}(\omega,t)&S_{n2}(\omega,t)&\cdots&S_{nn}(\omega,t)\end{bmatrix} \tag{1}\]  
where \(\omega\) is the circular frequency and \(t\) is the time; the diagonal element \(S_{jj}(\omega,t)\) (\(j=1,2,\cdots n\)) is the auto-EPSD of the \(j\)th element process and real function; the off-diagonal element \(S_{jk}(\omega,t)\) (\(j=1,2,\cdots n\); \(k=1,2,\cdots n\)) is the cross-EPSD between the \(j\)th and \(k\)th element processes and can be determined by  
\[S_{jk}(\omega,t)=\sqrt{S_{jj}(\omega,t)S_{kk}(\omega,t)}\gamma_{jk}(\omega) \tag{2}\]  
where \(\gamma_{jk}(\omega)\) is the coherence function between the \(j\)th and \(k\)th element processes.  
The EPSD matrix can be determined by the auto-EPSD of each element process and the coherence function between two-element processes. The auto-EPSD represents the time-varying amplitude and frequency content of earthquake motions, which can be calculated directly from recorded ground motion time histories or provided by an existing empirical or theoretical EPSD model [18; 25; 26]. The fluctuation in auto-EPSD at various sites may be used to account for the spatial variability of earthquake motions due to the 'attenuation effect'.  
The coherence function captures the spatial variability of earthquake motions caused by the 'incoherence effect', 'wave-passage effect' and'site effect'. Therefore, the coherence function can be represented as [1; 27]  
\[\gamma_{jk}(\omega)=\Big{|}\gamma_{jk}(\omega)\Big{|}e^{j\theta_{jk}(\omega)} \tag{3}\]  
\[\theta_{jk}(\omega)=\theta_{jk}^{w}(\omega)+\theta_{jk}^{s}(\omega) \tag{4}\]  
where \(|\cdots|\) denotes the module of a complex number; \(e\) and \(i\), respectively, denote the exponential function and imaginary unit; \(\theta_{jk}(\omega)\) is the complex phase of the coherence function; \(\Big{|}\gamma_{jk}(\omega)\Big{|}\) is the lagged coherence function describing the 'incoherent effect'; \(\theta_{jk}^{w}(\omega)\) is the phase describing the 'wave-passage effect' given by  
\[\theta_{jk}^{w}(\omega)=\frac{-\omega d_{jk}^{\bot}}{v_{app}} \tag{5}\]  
in which \(v_{app}\) is the wave velocity and \(d_{jk}^{\bot}\) is the distance between the \(j\)th and \(k\)th locations projected in the direction of the seismic wave propagation; \(\theta_{jk}^{s}(\omega)\) is the phase describing the'site effect' and given by  
\[\theta_{jk}^{s}(\omega)=\arctan\Biggl{\{}\frac{\text{Im}[H_{gj}(\omega)H_{gk}( -\omega)]}{\text{Re}[H_{gj}(\omega)H_{gk}(-\omega)]}\Biggr{\}} \tag{6}\]  
in which \(H_{gj}(\omega)\) and \(H_{gk}(\omega)\) are, respectively, the frequency response functions of the first filter corresponding to the \(j\)th and \(k\)th locations.  
Besides EPSD, \(\mathbf{x}(t)\) can also be characterized by the time-varying correlation function. The cross-correlation function between the \(j\)th and \(k\)th element processes can be represented by  
\[R_{jk}(t,t+\tau)=\int_{-\infty}^{\infty}\sqrt{S_{jj}(\omega,t)S_{kk}(\omega,t+ \tau)}\gamma_{jk}(\omega)e^{i\omega\tau}d\omega \tag{7}\]  
where \(\tau\) is the time lag.