## 3 Velocity Estimation Using Least Squares Analysis and Statistical Testing  
Annual and semi-annual components were included in the least squares estimation as the time dependent coefficients, and the unknown parameters a, b, c\({}_{1}\), c\({}_{2}\), d\({}_{1}\), d\({}_{2}\), and \(x_{\text{off}}\) were estimated using the regression equation:  
\[y(t)=a+b.t+c_{1}.\sin\left(\frac{2\pi t}{T_{1}}\right)+d_{1}.\cos\left(\frac{2 \pi t}{T_{1}}\right)+c_{2}.\sin\left(\frac{2\pi t}{T_{2}}\right)+d_{2}.\cos \left(\frac{2\pi t}{T_{2}}\right)+o.x_{off}+v(t) \tag{1}\]  
Figure 2: Continuous, monthly, and four-monthly time series of the IGS station GOLD from 2012 to 2016.  
where a is the intercept, b is the velocity, c\({}_{1}\) and d\({}_{1}\) are periodical constituents for the annual term whereas c\({}_{2}\) and d\({}_{2}\) are periodical constituents for the semi-annual term. In addition, \(T_{1}\) and \(T_{2}\) are periods of annual and semi-annual periodic components, which were taken to be 1.0 cycle and 0.5 cycles, respectively. \(x_{\mathit{off}}\) is an offset term and \(v(t)\) represents residuals. [PERSON] and [PERSON] (2018) performed a detailed study on the IGS continuous time series, where a wide spectrum of periodic components were investigated. However, here only the major annual and semi-annual periodicities of the seasonal variation were included in the Least Squares Estimation (LSE) analysis for simplicity. Similarly, we did not choose our stations from places in which the time series contain transients. This would have requested detailed modeling and station specific care. The reader should refer to [PERSON] and [PERSON] (2018, 2019) for transients in time series. The velocity b in Equation (1) was estimated for all; monthly, four-monthly, and continuous data. Figures 3 and 4 show the horizontal and vertical velocity fields of continuous and monthly sampled data. In addition, the other unknown parameters and their associated standard errors were also computed for n, e, and u components. Then we applied a Student's t-test as to whether the velocities from the monthly (or four-monthly) solutions differ significantly from those of the daily solutions. The formulation and the notation for the t-test from a multi-regression analysis is given in Table 1.  
The zero hypothesis was set to be H\({}_{0}\): b\({}_{\text{m}}\) = b\({}_{\text{d}}\) and the alternative hypothesis H\({}_{\text{A}}\): b\({}_{\text{m}}\) \(\
eq\) b\({}_{\text{d}}\). If, for a two-sided test, T exceeds \(\pm\)1.96 with \(\alpha\) = 0.05 then H\({}_{0}\) is rejected. In other words, the velocity estimated from monthly (or four-monthly) GPS campaigns is significantly different than that of the continuous GPS. In the Figures 3 and 4, antenna set-up errors were not yet imposed on the velocities. Obviously the estimated deformation rate could be biased due to this. Horizontal velocities were given with error ellipses having a 95% confidence level. The ellipse unit was 10 mm in the figure. Note that velocities from the continuous, monthly, and four-monthly time series were pretty much comparable (i.e., the direction and magnitude of black, blue, and red arrows coincided well). The detailed results of the statistical hypothesis testing, which are not obvious in Figures 3 and 4 are given in Figure 5. It is seen from the figure that a tremendous amount of success was gained over the study of [PERSON], [PERSON] and [PERSON] (2019) and [PERSON] and [PERSON] (2020). First of all, the effect of antenna set-up errors were clearly seen on all three components (c.f. the majority of red solid circles and plus signs over the blue ones).