The integration of Global Navigation Satellite Systems (GNSS) into Structural Health Monitoring (SHM) systems of super high-rise buildings is becoming increasingly prevalent, with a notable trend of combining GNSS with accelerometers to leverage the complementary nature of both technologies for estimating dynamic displacements. However, the multipath errors between the GNSS reference and monitoring stations give rise to the time-varying noises into the process of solving position, resulting in considerable fluctuations in the accuracy of displacement monitoring in complex environments, which impedes the development of GNSS-accelerometer integration techniques in the field of SHM. To address the issue, a variational Bayesian adaptive multi-rate algorithm (VBMRKF) is proposed to rapidly respond the time-varying noise for the GNSS system, thereby facilitating the fusion of GNSS and accelerometer data with greater accuracy. In this algorithm, a convergence criterion is established for approximation of posterior distribution in multi-rate process based on Evidence Lower Bound (ELBO) theory, which reduces the effects of noise accumulation on the accuracy of the approximation processes for the posterior distribution. Furthermore, a noise-sensitive factor is also heuristically proposed, which enables the algorithm to quickly adapt to fluctuations of noise characteristics during multi-rate processes. The numerical simulation results reveal that the algorithm has a higher estimation accuracy for dynamic displacements than the conventional algorithm, and it has a superior adaptability to systems with different frequencies. In instances where the initial settings of noise parameters are highly unreasonable, traditional algorithms even demonstrated a lack of convergence. Conversely, the normalized root mean square errors (NRMSE) of the displacement estimation results obtained by VBMRKF are consistently below 16 %, which also substantiates the robustness of this algorithm. In the presence of time-varying noise, the algorithm demonstrates the capacity to rapidly converge upon the distribution of the noise. Compared to the multi-rate Kalman filtering (MRKF) algorithm, the proposed algorithm achieves the highest reduction of 27.02 % in NRMSE, and compared to around 13.97 % NRMSE of the recent adaptive algorithm, the proposed algorithm also demonstrates significantly superior performance, reducing NRMSE by at least 5.13 %. In addition, regardless of whether the signal is time-varying or not, the algorithm is able to quickly approximate the distribution of noise and produces acceptable displacement estimation results with unreasonable initial settings of the noise parameters, demonstrating its robustness. Finally, the measured accelerometer and GNSS data are employed to successfully determine the two-dimensional dynamic displacement of the 632-metre-high Shanghai Tower at a distance of only 2.1 km from the eye of Typhoon Muifa (202212). A further analysis of the results demonstrates that the estimated displacements are more comprehensive in both time and frequency domains.
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