## 3 Measurement Errors of PLLs
### MEMS Accelerometer-Aided PLLs  
When a strong earthquake occurs, the GNSS receiver and the MEMS accelerometer at the same location bear the same dynamic stress. Hence, the dynamic information measured by the MEMS accelerometer can aid GNSS tracking loops. The signal transfer relationship of the MEMS accelerometer-aided PLLs in a Laplace domain is shown in Figure 6. Compared with Figure 3, the MEMS accelerometer information assists PLLs based on the concept of feed-forward [27]. The velocity is obtained by the integration of the accelerometer measurement, and maps to the line of sight (LOS) between the satellite and receiver to obtain the Doppler estimation information \(f_{aid}(s)\)[28]. With the aid of the accelerometer feed-forward information, the tracking loop only needs to undertake the error of the aiding information \(\Delta f_{Acc}(s)\), instead of dynamic stress. For the GNSS/MEMS accelerometer deeply coupled system, the error sources of the aiding information include the initial velocity error \(\delta V_{N}(0)\) after correction of GNSS/accelerometer integration update, the accelerometer bias, the accelerometer scale factor, etc. Our previous research showed that \(\delta V_{N}(0)\) was the main factor of the aiding information impacting the PLL phase error [29]. Based on the transfer relationship between the aiding information error and the loop tracking error, the loop tracking error caused by the aiding information in time domain can be deduced as follows:  
\[\sigma_{faid}=\frac{180}{\pi}\sqrt{\frac{2\pi\cdot\delta V_{N}(0)}{\lambda\cdot e \cdot\omega_{n}}}(^{\circ}) \tag{6}\]  
where \(\lambda\) is the carrier wavelength, and \(\omega_{n}\) is the natural radian frequency. \(\sigma_{faid}\) is a random constant value in each update interval and follows the Gaussian distribution. Since \(\sigma_{faid}\) is uncorrelated with that due to thermal noise and oscillator noise, the error caused by aiding information is the geometric  
Figure 5: Phase errors of second-order PLL as a function of signal strength. Figure 6: Error transfer model of microelectromechanical systems (MEMS) accelerometer-aided PLLs. sum relationship to that caused by thermal noise and oscillator. Therefore, compared with Equation (1), the mean squared error of total errors of MEMS accelerometer-aided PLL can be written as follows:  
\[\sigma_{PLLaid}=\sqrt{\sigma_{tPLL}^{2}+\sigma_{rv}^{2}+\sigma_{rA}^{2}+\sigma_ {gaid}^{2}} \tag{7}\]  
Taking a typical MEMS sensor STIM 300 as an example [30], we obtain the statistic of \(\delta V_{N}(0)\) based on field tests, which is about 0.01 m/s. Based on above error model, the phase error of MEMS accelerometer-aided PLLs can be analyzed. When the movement acceleration is 2 g, the CN0 is 40 dB-Hz, and the coherent integration time of the PLL is 1 ms, Figure 7 shows phase errors of MEMS accelerometer-aided PLL as a function of PLL bandwidth. Compared with Figure 4, the phase error caused by the aiding information is much less than that caused by the dynamic stress, which also increases with the bandwidth narrowing. The impact of the thermal noise and crystal oscillator keep the same. The accelerometer-aided PLL with a bandwidth of more than 2 Hz can keep the total phase error less than the threshold, which is much narrower than the stand-alone loop bandwidth. And with the accelerometer aiding, the total phase error significantly reduces. When the strong earthquake occurs, the GNSS signal may become weak. To obtain an accurate and stable carrier phase under a strong earthquake, it is necessary to study the relationship between the loop bandwidth and the tracking error under a weak signal. Figure 8 shows phase errors as a function of PLL bandwidth when the CN0 is down to 30 dB-Hz. Compared with Figure 7, the phase error caused by thermal noise clearly increases. When the loop bandwidth is 5 Hz, the total phase error is the smallest. Figure 8: Phase errors of MEMS accelerometer-aided PLL with a CN0 of 30 dB-Hz.