## 2 Materials and Methods  
### High-Rate GNSS and SM Loose Integration System Functions  
During an earthquake, the ground is treated as a continuous dynamics system. High-frequency GNSS provides absolute three-dimensional displacement information and is not affected by tilt or baseline drift. When tilt occurs in SM (strong-motion accelerometers), the data typically exhibit anomalies in the low-frequency range, as the false acceleration caused by the tilt results in displacement drift that persists even after the shaking stops. GNSS data do not have this kind of low-frequency drift. Therefore, this paper integrates GNSS data to effectively mitigate the impact of tilt effects in SM data on measurement results. The algorithm presented in this paper takes into account and avoids the issue of baseline deviation caused by the tilt and rotation of the SM sensor, which can make it difficult for the integrated displacement data to accurately reflect the true surface displacement. Moreover, once accurate surface displacements in the three directions have been obtained, displacements in other directions can be derived through appropriate angular projections. Due to the mutual independence of each, the state of motions can be defined by the following first-order linear differential equation:  
\[x(t+1)=Ax(t)+Ba(t)+C\alpha(t) \tag{1}\]  
\[x(t)=\begin{bmatrix}d(t)\\ v(t)\end{bmatrix},A=\begin{bmatrix}0&1\\ 0&0\end{bmatrix},B=\begin{bmatrix}0\\ 1\end{bmatrix},C=\begin{bmatrix}0\\ 1\end{bmatrix} \tag{2}\]  
where the state vector \(x(t)\) consists of the displacement \(d(t)\) and the velocity \(v(t)\). \(a(t)\) is the acceleration. \(\alpha(t)\) is the system noise with distribution \(\alpha(t)\sim(0,Q)\). The system noise stochastic model is expressed as:  
\[Q=\begin{bmatrix}0&0\\ 0&q\end{bmatrix} \tag{3}\]  
In this study, if the above first-order linear differential equation is discretized, its state equation coefficient matrix will be changed. Therefore, it is assumed that the discrete data sampling interval is \(\tau\), the discrete GNSS displacement is \(z\), and the SM acceleration is \(a\). Then, the discrete state equation and observation equation at epoch \(k\) can be established as:  
\[x_{k}=A^{S}x_{k-1}+B^{S}a_{k-1}+\alpha_{k-1} \tag{4}\]  
and:  
\[z_{k}=H_{k}x_{k}+\beta_{k} \tag{5}\]  
where \(k=1,2,3,\ldots,N^{+}\). The SM acceleration \(a_{k-1}\) is regarded as a constant in the equal time sampling interval. \(H_{k}\) is a designed matrix describing the relationship between observations and state vectors. The system noise \(\alpha_{k-1}\). and the GNSS displacement noise \(\beta_{k}\) with a distribution \(\beta_{k}\sim(0,R)\) are observation noises. In Equation (4), the noise derives only from the SM acceleration time series, and thereby the system noise variance matrix is given as follows:  
\[Q=\begin{bmatrix}q\tau_{q}^{3}/3&q\tau_{a}^{2}/2\\ q\tau_{a}^{2}/2&q\tau_{a}\end{bmatrix} \tag{6}\]  
and the coefficient matrix can be described as:  
\[A^{S}=\begin{bmatrix}1&\tau_{a}\\ 0&1\end{bmatrix},\ B^{S}=\begin{bmatrix}\tau_{a}^{2}/2\\ \tau_{a}\end{bmatrix} \tag{7}\]  
where \(q\) is a priori system noise variance. \(\tau_{a}\) is the sampling interval of SM.  
In Equation (5), the displacement observations are mainly GNSS displacement time series, and the observation noise variance matrix can be determined as:  
\[R=\frac{r}{\tau_{d}} \tag{8}\]  
where \(r\) and \(\tau_{d}\) are the variance and sampling interval of GNSS displacements, respectively.  
According to Equations (4)-(8), the loose integration system of high-rate GNSS and SM can be established by the SKF.