# Effect of Surface Mass Loading on Geodetic GPS Observations
## 3 Results and Discussion
### Noise and Velocity Analysis  
Errors in the mass loading model are currently difficult to quantify. Nonetheless, they will certainly enter into our estimates of surface deformation [24]. In this case, aside from investigating the RMS reduction of GPS position time series, discussing the effect of mass loading corrections on the velocity and noise estimates is equally important. In this study, we adopted the Hector software to determine the velocity and its uncertainty with pre-assumed power-law noise and white noise [25]. The software can sufficiently describe the stochastic properties of most daily GPS position time series. The noise parameters, including the spectral indices and noise amplitudes, were also estimated before and after the AOH corrections by using the maximum likelihood estimation to analyze the change in noise properties. The accuracy of velocity is calculated based on the following formula [26]:  
\[m_{v}\approx\pm\sqrt{\frac{A_{PL}^{2}\cdot\Gamma(3-K)\cdot\Gamma(4-K)\cdot(N-1)^ {K-3}}{\Delta T^{2-\frac{K}{2}}\cdot\left[\Gamma\left(2-\frac{K}{2}\right) \right]^{2}}} \tag{6}\]  
where \(m_{v}\) is the accuracy of velocity, \(A_{PL}\) is the amplitude of noise, \(K\) is the spectral index, \(\Gamma\) is the gamma function, \(N\) is the number of data in the time series and \(\Delta T\) is the sampling interval.  
The middle histogram in Figure 6 demonstrates that the power-law noise amplitude is mostly decreased when AOH corrections are applied. The small amplitude of noise after AOH corrections indicates the small scatter of residuals, and consequently a considerably stable solution for the determination of site velocity. Meanwhile, the bottom histogram in Figure 6 indicates that the spectral index shifts from a flicker noise (\(k=-1\)) to white noise (\(k=0\)) in most sites. Hence, the loading signals may be superimposed on the stochastic process realization. The removal of AOH deformation for GPS stations causes a reduction in median velocity error from 1.17 to 0.87 mm/year in median power-law noise amplitude from 14.99 to 11.84 mm/year-k/4 and in the median spectral index from \(-0.84\) to \(-0.81\) mm/year-k/4. The results indicate that AOH corrections altered the noise properties of GPS height time series. Based on Equation (6), changes in the noise properties will certainly influence the estimates of velocity uncertainties. Accordingly, AOH corrections will potentially affect the velocity-field determination, and thus, its geodynamical interpretation. From the top histogram in Figure 6, we can see that for 24 out of the 30 sites the velocity uncertainty estimates decrease after  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Site** & **ARMS\%** & **Site** & **\(\Delta\text{RMS}\%\)** & **Site** & **\(\Delta\text{RMS}\%\)** \\ \hline ANKR & 30.2\% & KUNM & 20.1\% & TASH & 30.2\% \\ ARTU & 45.3\% & LHAZ & 27.3\% & TEHN & 38.0\% \\ BIL & 28.8\% & NOVM & 38.7\% & TISE & 18.0\% \\ BIFS & 23.7\% & NRIL & 34.2\% & URUM & 29.7\% \\ BOR1 & 35.6\% & ONSA & 17.1\% & VILL & 3.0\% \\ CHAN & 28.4\% & PADO & 25.7\% & WUHN & 9.0\% \\ CHUM & 33.9\% & POLV & 42.6\% & YAKT & 14.6\% \\ GRAZ & 27.6\% & POTS & 24.1\% & YEBE & 18.1\% \\ IRKT & 26.2\% & SHAO & 4.2\% & ZECK & 38.3\% \\ KIRU & 18.9\% & SOFI & 14.0\% & ZWE2 & 39.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: The RMS reduction ratio in percent when correcting GPS heights with AOH displacements.  
AOH corrections. However, for 6 out of the 30 sites (KUNM, PADO, SOFI, TLSE, VILL and YEBE) the velocity uncertainty increases due to the increased amplitude of noise and/or the decreased spectral index after AOH corrections. This indicates that errors or noises are introduced when implementing the mass loading correction, suggesting that the correction of GPS heights with the mass loading model is potentially feasible but not sufficient.