## 1 Introduction
Consequently, GNSSs have been widely applied in the field of seismic rupture process calculation [15; 16; 17; 18], fault slip model inversion [19; 20], rapid magnitude estimation [21; 22; 23], and real-time earthquake early warning (EEW) [24; 25]. In general, SM observations provide acceleration information and have the advantage of capturing unlimited near-field amplitude, allowing for high-frequency monitoring of short-duration intense shaking. This offers extremely important information for earthquake research; however, during strong shaking in large earthquakes, SM can be easily affected by surface motion. Compared with SM, one of the advantages of high-rate GNSSs is measuring ground displacements directly without drifts in an absolute reference frame [26]. Even in cases of (extra)large earthquakes, they can record more accurate displacements to estimate more precise magnitude. However, there are still several constraints and challenges for co-seismic displacement measurements with high-rate GNSSs. (1) During measurement, high-rate GNSSs are insensitive to weak surface deformation and do not pick up the primary seismic wave easily [9]. (2) Restricted by the bandwidth of receivers, high-rate GNSSs cannot achieve a sampling rate comparable to that of SM accelerometers (200 Hz or even 400 Hz) [27; 28; 29]. Therefore, SM accelerometers and high-rate GNSSs complement each other. By integrating and analyzing them together, it is possible to obtain high-precision positional information and high-sampling-rate broadband co-seismic displacement, thereby improving the accuracy and reliability of earthquake early warning systems. To maximize the advantages and minimize the weaknesses of SM accelerations and high-rate GNSSs, many methods have been proposed to combine GNSS displacements and SM accelerations at co-located sites. [PERSON] et al. [30] used a numerical inversion approach that considered the baseline shift in SM accelerations to estimate co-seismic displacements with the constraint of GNSS data. [PERSON] et al. [31] used linear trigonometric and time correlation to express the baseline shift in SM accelerations, and obtained the best fit with high-rate GNSS displacements. However, the above techniques were not employed for real-time applications because of their subjective process. [PERSON] et al. [32] derived the baselineshift in SM accelerations by comparing single-frequency GNSS displacements without pre-event trends or acceleration-integrated displacements, whereas the single-frequency GNSS was not suitable for long-term monitoring. At present, Kalman filters and particle filters (PFs) are popular research solutions for retrieving optimal estimates of co-seismic displacements in high-rate GNSS-SM accelerometer integration systems. In complex earthquake monitoring scenarios, such as when seismic waves propagate through different geological layers, the resulting displacement and acceleration can be affected by various nonlinear effects. PFs can be applied to estimate nonlinear dynamic systems. The use of PFs is a Bayesian method of estimating state variables in dynamic systems, particularly when dealing with nonlinearities and non-Gaussian noise. It approximates the posterior probability of the state using random samples (particles). But PFs require a large number of particles for accurate estimates, leading to increased computational and storage demands, especially in high-dimensional systems. The method is also sensitive to the distribution of initial particles, with improper initialization potentially causing biased results. Additionally, the performance of the PF method depends on the choice of parameters, such as the number of particles and resampling strategies [33; 34; 35; 36]. The Kalman filter not only supports real-time applications but also implicitly and automatically corrects baseline shifts in SM data. [PERSON] and [PERSON] [37] proposed a loose integration method by the standard Kalman filter (SKF) that considered GNSS displacements as observation quantities and SM accelerations as control quantities.