## 4 Discussion
### Advantage of the New Algorithm for Treating Functional Model Error  
From the perspective of functional model error, the problem of unreliable covariance function can be attributed to systematic model errors that can be solved using the residual re-fitting method. From the perspective random model error, the unreliableness of the covariance function can be partially adjusted by the variance factor, which can re-adjust the priori weight ratio between the signal and the observation vector based on the VECM. In this study, the ADLSC algorithm is used to study the unreliable covariance function in the fitting and estimation of GPS crustal horizontal velocity. In practice, it is hard to say whether the unreliable covariance will result in systematic errors or random errors, but both of them should exist. That is to say, the internal correlation of the variance-covariance of signal is inaccurate, or the variance-covariance of observation is inconsistent with the signal. In order to solve the above-mentioned problem, the ADLSC algorithm first introduces the distance scale factor to weaken the influence of systematic errors by \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multicolumn{2}{c}{**Fitting Points RMS (mm/a)**} & \multicolumn{2}{c}{**Checking Points RMS (mm/a)**} & \multicolumn{2}{c}{**Running Time**} \\ \cline{2-5}  & **(E)** & **(N)** & **(E)** & **(N)** & **(s)** \\ \hline TLSC & 2.38 & 1.59 & 2.84 & 1.88 & 1.29 \\ DLSC & 1.89 & 1.42 & 2.32 (18.3\%) & 1.56 (17.0\%) & 0.81 \\ ALSC & 2.01 & 1.29 & 2.71 (4.6\%) & 1.35 (28.2\%) & 1.42 \\ ADLSC & 1.87 & 1.14 & 2.30 (19.0\%) & 1.11 (41.0\%) & 1.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMS of the fitting points and checking points using four algorithms (Experiment 3). Figure 7: Optimal distance scale factor of the ADLSC in the (**a**) east direction and (**b**) north direction. improving the function model, which plays an important role in adjusting the inter-correlation of signals. Secondly, the ADLSC algorithm combines the ALSC algorithm to weaken the influence of random error and adjust the overall coordination relationship between the variance-covariance of the signal and the observation. We showcase the advantage of the ADLSC in dealing with functional model errors using the signal variance results estimated by the first Helmert VCEM calculation because this depends entirely on the signal covariance function model established earlier. The results are shown in Table 5, Figure 8, and Figure 9.  
It can be seen from Table 5 and Figure 8 that the estimated signal variance results by the first Helmert VCEM calculation of the ADLSC are all less than that of the ALSC under different noise levels. When the noise levels are \(\sigma_{0}^{2}=0.1\), \(\sigma_{0}^{2}=1\), and \(\sigma_{0}^{2}=10\), the signal variance of the ADLSC are about 87.3%, 83.8%, and 71.2% better than the ALSC in the east direction (Figure 8a), and are about 43.4%, 12.2%, 17.2% better than the ALSC in the north direction, respectively, (Figure 8b). The above results clearly show that the strategy of introducing distance scale factor can significantly reduce the signal variance estimate value and obtain more reliable signal fitting and estimation results. Figure 9 shows that the improvement of the east and north fitting accuracy is significantly reduced with the increase of iteration times. Because the contribution of the first iteration is the largest in the whole adaptive iteration calculation, the first iteration calculation is directly related to the parameters of the covariance function model. Figure 9 also shows that the distance scale factor can effectively weaken the systematic error of the model (Figure 9a,b), and plays an important role in the subsequent adaptive fitting estimation. It further indicates the rationality of the ADLSC algorithm that considered the unreliable covariance function as a combination of systematic error of the model and the random