## 3 Experiments and Results
### Simulation Experiment  
In this study, the swept sinusoidal signal is adopted to validate the accuracy of the presented VC-AKF, as it can determine whether the proposed method is susceptible to changes in the frequency content of the signal [39]. Assume that the displacement is a sinusoidal swept signal with linear drift, of which time series are as follows:  
\[x(t)=\sin[(at+b)t]+ct \tag{17}\]  
where \(a=\pi/9,b=2\pi/5,c=0.1\). The time series of the corresponding velocity and acceleration can be obtained by the following formulas:  
\[\dot{x}(t)=(2 at+b)\cos[(at+b)t]+c \tag{18}\]\[\ddot{x}(t)=2a\cos[(at+b)t]-(2 at+b)^{2}\sin[(at+b)t] \tag{19}\]  
As previously mentioned, the data monitored by SM are generally acceleration information. To obtain co-seismic displacement, a double integration is required. Under the presence of linear drift, it may be challenging to accurately recover the actual ground displacement by performing double integration on acceleration data. The impact of linear drift is present in the acceleration information, but after double integration, this linear drift information will be lost. As shown in Equations (17)-(19), the original displacement contains linear drift information, but after obtaining the acceleration information through a second differential, the linear drift information has disappeared. Therefore, in theory, when displacement is obtained by a double integration from acceleration, the linear drift information will also be difficult to recover. To intuitively illustrate this phenomenon, this paper presents a simulated simple linear drift experiment to express that the integration of acceleration information will be difficult to restore the actual displacement. The experimental results are shown in Figure 1. The original displacement time series from the sinusoidal frequency sweep signal and acceleration-derived displacement time series from the acceleration integral are shown in the solid blue line and orange dashed line, respectively. It can be seen that the linear drift in displacements is not recognized by accelerometers after differentiation, which means the linear drift of the original signal cannot be recovered by accelerations after secondary integration. This leads to the loss of linear drift information and makes it difficult to accurately recover the actual ground displacement. To address this issue, this paper proposes combining SM data with high-frequency GNSS data. Thus, the high-rate GNSS displacement and SM acceleration loose integration will be conducted in this study to acquire displacement with both high-precision and high sampling rate. As this is the simulated experiment, to consider the multi-rate estimation case, it is assumed that the displacement sampling rate is 100 Hz and the acceleration sampling rate is 1000 Hz. In practice, such a high sampling rate is normally unnecessary for most seismic applications. However, it does provide a good data source for mathematical models that require large amounts of data for iterative calculations and capture very  
Figure 1: Original displacement time series (solid) and acceleration-derived displacement time series (dashed). high-frequency sudden response changes. A separate white noise is added to the original displacement and acceleration signals, i.e., \(SNR=5\). Initialize the constant noise statistical properties as \(q=1,R=0.1\). Since the exact time series of displacements are given, the results can be analyzed by errors from the original signal. Figure 2 indicates errors of the SKF, VC-AKF, and SKF with RTS smoothing (SKF + RTS). As shown in Table 1, the root mean square error (RMSE) is used as a statistical metric to measure the filter accuracy. The smaller the RMSE, the better the filter effect. Figure 2 and Table 1 highlight the following. (1) The influence of the system's prior noise covariance parameter on the multi-rate Kalman filter cannot be neglected.