## 2 Methods
### Fusion Algorithm  
For the DLSC algorithm, the weight ratio of the observation noise and the effective signal cannot be guaranteed to be consistent, although the problem of negative signal covariance is overcome. For the ALSC algorithm, the a priori weight ratio between the signal and the observation vector is consistent; however, only the positive signal covariance value will be intercepted to fit the empirical Gaussian function. This will affect the accuracy of the fitting parameters of the Gaussian function. In order to fully guarantee the reliability of the fitting and estimation results, and effectively solve the limitations of the above two problems existing in the DLSC and ALSC algorithms, we propose an adaptive least-squares collocation algorithm considering the distance scale factor (ADLSC). The new algorithm will overcome the problem of negative signal covariance, and ensure that the signal is consistent with the observed noise. The approximate frame diagram of the ADLSC is shown in Figure 1:  
The specific implementation steps are as follows:  
Figure 1: The frame diagram of the an adaptive least-squares algorithm considering the distance scale factor (ADLSC algorithm). (1) Solving the overall motion trend of the study area. A priori Euler rigid parameter \(\Omega\) (the random signal is not considered, or it is attributed to the observation noise) can be calculated according to the principle of least squares based on the GPS velocity. The model is as follows: \(V_{o}=\underset{q\times 1}{A}\underset{q\times 33\times 1}{\Omega}+\underset{q \times 1}{n}\). where \(V_{o}\) is the GPS horizontal velocity; \(n\) is observation noise. Then the residual components can be calculated based on the trend term \(A\Omega\). (2) Constructing parameters of the priori Gauss function. According to Equations (7) and (8), the parameters \(C(0)\) and \(C(d_{i})\) in the priori Gaussian function can be calculated based on the residual component solved in step (1). (3) Introducing the distance scale factor. According to Equation (14), by analyzing the RMS of the checking residual, the \(k\) value with the smallest RMS value will be selected by appropriately adjusting the \(R\) value. Then, the specific mathematical expression of the empirical Gaussian function can be determined. In this step, the introduced distance scale factor takes into account the scale of the study area, and will significantly improve data utilization. Because this strategy avoids negative cross-covariance variance, their guarantees a more accurate and reliable empirical Gaussian function. (4) Constructing the covariance matrix of observation signal based on the Gaussian function of the prior signal determined in step (3). The initial signal weight matrix \(P_{s}^{(0)}\) can be obtained by inverting the covariance. In order to analyze the influence of the noise variance on fitting and estimation results, it is usually assumed that the signal variance \(\sigma_{s}^{2^{(0)}}=1\) and the different noise variance \(\sigma_{e}^{2^{(0)}}\) will be given. Then the posterior Euler vector parameter \(\hat{\Omega}^{(0)}\) and the posterior signal \(\hat{V}_{s}{}^{(0)}\) can be further solved according to Equations (3) and (4). (5) Performing the adaptive iterative adjustments. The adaptive factor \(\alpha^{(0)}\) can be solved by using the VCEM according to Equations (16)-(18), which is used to adjust the signal weight matrix to obtain a new signal weight matrix \(P_{s}^{(1)}\). Then the adjustment process is carried out again, and a new posterior Euler vector parameter \(\hat{\Omega}^{(1)}\) and the posterior signal \(\hat{V}_{s}{}^{(1)}\) can be calculated. During \(K\) iterations, until the RMS in the checking residual is the smallest, the observed noise is consistent with the signal variance component, and the optimal Euler vector parameter \(\hat{\Omega}^{(K)}\) and the signal \(\hat{V}_{s}{}^{(K)}\) are obtained. (6) Solving the signal of the estimated points.