## 3 Results and Analysis  
To verify the rationality of the new algorithm in real cases, we selected the Sichuan-Yunnan block located in southern China [15] and used long-term GPS velocities to test the ADLSC algorithm. The GPS measurements were taken from the Crustal Movement Observation Network of China (CMONOC I) [26] and the Tectonic and Environmental Observation Network of Mainland China (CMONOC II) [27]. The horizontal GPS velocities used in this study were with respect to stable Eurasia based on the ITRF2008 reference frame and included a total of 81 stations from 1999 to 2013 (Figure 2) [28]. These stations contain 22 continuously operating GPS stations and 59 campaign-mode GPS stations. The latter have reinforced concrete monuments with forced-centering apparatus for GPS antennas. In each survey, dual-frequency GPS receivers and choke ring antennas were used, and each station was occupied for at least three days, to ensure the quality of the data. As the stations ran at least five observation campaigns over a 10-year span, highly accurate crustal horizontal velocity fields can be obtained. Among the 81 GPS monitoring data values, 12 GPS points are randomly selected as checking points and the remaining 69 GPS points are fitting points. The checking points are not included in the calculation of unknown parameters. Then, the velocity at each fitting and checking point can be estimated using different collocation algorithms, and the residuals between the estimated and observed velocity can be also calculated. The RMS is used to assess the quality of the results, as follows:  
\[RMS=[\frac{1}{m}\sum_{i=1}^{m}(d_{i})^{2}]^{1/2} \tag{22}\]  
where \(m\) represents the numbers of fitting and checking points; \(d_{i}\) is the residual between the estimated and observed velocity at each fitting and checking point. We use four collocation algorithms to calculate the RMS of the fitting points and checking points, namely, the TLSC, DLSC, ALSC, and ADLSC algorithms (Table 1). Figure 2: The GPS velocities for 1999-2013 (mm/a) and the location of the Sichuan-Yunnan block. Red rectangle outlines the study area in mainland China (**a**). Gray solid lines represent the major faults [29] (**b**). Red triangles represent major cities (capital cities) (**b**). Black arrows are the observed GPS velocities, and the red color at the GPS station represents the checking points (**b**). Compared to the other three algorithms, the ADLSC has the smallest RMS both in the fitting points and checking points (Table 1). It indicates that the ADLSC has the optimal fitting effect and estimation effect. Figure 3 shows the residual between the estimated and observed velocity of each checking point. The overall amplitude of the residuals derived from ADLSC is relatively smaller and closer to zero (Figure 3). For the DLSC and ALSC, the RMS are about 18.7% and 12.1% better than the TLSC in the east direction, and are about 6.9% and 33.3% better than the TLSC in the north direction, respectively. However, for the ADLSC, the RMS is about 19.9% and 44.6% better than the TLSC in the east and north directions, respectively (Table 1). In addition, we can see that there are no significant differences in program running time among the four algorithms, which are within two seconds (Table 1). The subtle differences may be the adaptive computing is an iterative process, which will take a relatively long time. Figure 4 shows the residuals between the estimated and observed velocity of the fitting points. The overall amplitude of the residuals derived from ADLSC is also relatively smaller, closer to zero (Figure 4), and has the smallest RMS value (only has 1.96 mm/a to the east and 0.54 mm/a to the north). In addition, we further conduct correlation analysis on the fitting and observed velocity in the east and north directions to comprehensively test the fitting effect of each algorithm (Figure 5).