## 2 Methods  
### Traditional LSC Algorithm  
The LSC algorithm is an optimal estimation algorithm that simultaneously determines the tendency parameters and the signals with randomness [22]:  
\[\underset{q\times 1}{L}=\underset{q\times pp\times 1}{A}\underset{q\times m \times 1}{X}+\underset{q\times 1}{U}\underset{q\times m\times 1}{S}+\underset{q \times 1}{n} \tag{1}\]  
where \(L\) is the observation vector; \(X\) is the non-random tendency parameter; \(A\) is coefficient matrix, reflecting the contribution of \(X\) to \(L\); \(S\) is random signal, which includes the signal of observation points and unobserved points (estimated points) in the study area; \(U\) is the rectangular matrix, the left \(q\) columns of the \(U\) matrix is the unit matrix, corresponding to the observation points signal, the right \(m-q\) columns of the \(U\) matrix is null matrix, corresponding to the estimated points signal; \(n\) represents observation noise. The observed values \(L\) include the tendency part \(AX\) determined by fixed parameters, the random signal part \(S\), and the observation noise part \(n\). When the traditional LSC algorithm (TLSC) is applied to establish the GPS crustal horizontal velocity model, the Equation (1) can be rewritten as follows [13]:  
\[\underset{q\times 1}{V_{o}}=\underset{q\times 33\times 1}{A}\underset{q \times 33\times 1}{\Omega}+\underset{q\times m\times 1}{U}\underset{q\times 1}{V_{s}}+ \underset{q\times 1}{n} \tag{2}\]  
where \(V_{o}\) represents GPS horizontal velocity field; \(\Omega\) is a parameter of the [PERSON] rigid motion model, and \(A\) is the coefficient matrix that is calculated from the coordinates of the GPS stations; \(A\Omega\) represents the overall motion tendency of the study area; \(V_{s}\) is the effective random signal after deducting the overall motion tendency from \(V_{o}\), and \(U\) is the coefficient matrix. The adjustment solution of the tendency parameter \(\Omega\) and the signal \(V_{s}\) in Equation (2) can be calculated as follows [6]:  
\[\hat{\Omega}=\big{(}A^{T}\overline{C}^{-1}A\big{)}^{-1}A^{T}\overline{C}^{-1}V_ {o} \tag{3}\]  
\[\hat{V}_{s}=C_{uo}\overline{C}^{-1}\big{(}V_{o}-A\hat{\Omega}\big{)} \tag{4}\]  
\[\overline{C}=\big{(}C_{oo}+C_{nn}\big{)} \tag{5}\]  
where \(C_{oo}\) is the covariance of the \(V_{o}\), which is used to describe the spatial distribution correlation and dispersion among GPS stations; \(C_{nn}\) is the self-covariance of the observation error of \(V_{o}\), which is used to describe the dispersion of the GPS velocity itself and reflect the observation precision; \(C_{uo}\) is the covariance matrix between the estimated points signal and the observation points signal. The \(C_{nn}\) can be obtained from the GPS velocity field. \(C_{oo}\) and \(C_{uo}\) are unknown quantities, but can be determined from the functional relationship between signal covariance and the distance between different observation points. The covariance of signals between observation points will decrease with the increase of the distance between observation points. This feature is consistent with the Gaussian function, which is usually used as the signal empirical covariance function [13; 14; 15]:  
\[F(d)=a\ exp(-k^{2}d^{2}) \tag{6}\]  
where \(a\) represents the first undetermined parameter, and also represents the self-covariance of the observation signal; \(d\) is the distance between the observation points; \(k\) is the second undetermined parameter, denoting the smoothness of the continuous velocity field distribution. Normally, \(a\) and \(k\) can be obtained by least squares fitting [13].