## 1 Introduction
[PERSON] et al. [38] adopted the SKF to estimate broadband dynamic co-seismic displacements for shake-table experiments and the 2010 El Mayor-Cucapah earthquake. [PERSON] et al. [39] revised the SKF to obtain integrated waveforms and eliminate the aliasing problem existing in the 1 Hz GNSS signals of GSIS. [PERSON] and [PERSON] [40] upgraded the loose integration method with two parallel SKFs to decrease the influence of GNSS colored noise with uneven energy distribution across the frequency spectrum, in contrast to traditional Gaussian white noise. This includes commonly encountered types such as pink noise, blue noise, and red noise. Alternatively, there are also several tight integration methods by Kalman filters that add SM accelerations into the normal equation of GNSS positioning to improve resolving ambiguities [27; 41; 42; 43]. However, the optimality of the Kalman filter solution is determined largely by a priori knowledge of the noise statistics. The use of incorrect a priori knowledge may result in inaccurate estimation or even filter divergence. In GSIS, the observation noise variance from a GNSS is relatively easy to acquire and is comparatively homogeneous over a long period of time. Nevertheless, the system noise variance from SM is prone to nonlinear time variation because of its rotation and tilts, which means it is not suitable to fix to a certain a priori value. Therefore, the Rauch-Tung-Striebel (RTS) smoother and the adaptive Kalman filter (AKF) have been used to solve this problem [44; 45]. The RTS smoothing technique estimates the state minimum variance by all the output data in a fixed interval to reduce the effect of the time-varying system noise variance. Although the accuracy of results can be further improved, the RTS smoother is a causal system and requires more computational time and resources. The AKF methods have the benefits of determining and correcting incorrect state vectors and the statistical characteristics of system noise appropriately, which can be generally classified into four methods: Bayesian, maximum likelihood, correlation, and covariance matching. The Bayesian and maximum likelihood methods are based on probability statistics to calculate noise covariance. In the calculation, all data need to be used for multiple iterations, resulting in these methods being relatively cumbersome and inefficient for GSIS [46; 47]. The correlation method estimates the steady-state filter gain and the noise covariance from the autocorrelation function of the observation data or the innovation sequence. Because of the requirement of large data windows, the correlation method has high processing load. In addition, it is difficult to guarantee the positive definiteness of matrices, leading to biased results. In contrast, the covariance matching method is a recursive algorithm in real time that calculates a closer approximation to the actual state valuation by estimating the system noise variance vector with its own information [48; 49]. Compared with the other AKF methods, the covariance matching method is more compatible with the time-varying characteristics of the system noise in GSIS, and has unique advantages at data fusion for real-time applications and simple computation. In order to deal with the problem of the time-varying system noise variance from SM, a new AKF approach--variance compensation adaptive Kalman filter (VC-AKF)--is presented in this study to construct the loose integration of high-rate GNSS and SM records. VC-AKF is a covariance matching method that uses predicted residuals to estimate system noise covariance vectors in real time, and then substitutes valuations into the filtering process to correct original system noise covariance. The remaining chapters of this paper are organized as follows. First, the high-rate GNSS and SM loose integration system model is introduced, and the method based on VC-AKF is described particularly.