# GPS Time Series Analysis from Aboa the Finnish Antarctic Research Station
## 2 Materials and Methods
### Data Processing
#### 2.3.2 PPP Combined Daily Solutions (RTKLIB)  
EKF is a recursive estimator that does not require all previous data to be saved and reprocessed when a new measurement is taken [61]. Thus, the estimated solution converges and its quality improves over time. PPP convergence time and positioning performance depend on several factors, such as the sampling interval of the precise products, the length of the observation session, the observation epoch rate, the observation quality, the receiver type, the multipath level, the geographical location, the number of tracked satellites and constellations, the positioning mode, and the ambiguity resolution [50; 62; 63]. For example, in static PPP, 20 min are needed for 95% of solutions to reach a horizontal accuracy of 20 cm [62]. Nevertheless, sub-centimetre accuracy levels are achievable for the 24-h data set [50; 62]. Figure 4 depicts the convergence process for the forward and backward solutions at Aboa on 1 January 2017. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Source** & **Error** & **Method of Correction** \\ \hline \multirow{4}{*}{satellite} & \multirow{2}{*}{orbit} & CODE a precise orbit products in standard product 3 (SP3) format with sampling rate of 15 min [52; 53] \\ \cline{2-3}  & & \\  & & \\ \cline{2-3}  & \multirow{2}{*}{clock offset} & CODE precise clock product in extended RINEX clock format (CLK) \\  & & with sampling rate of 5 s [52; 53] \\ \cline{2-3}  & \multirow{2}{*}{phase center offset and variations} & IGS ANTEX b maps [55] \\ \hline \multirow{4}{*}{atmosphere} & \multirow{2}{*}{Ionophere} & Ionosphere-free linear combination (1 st order term only). Higher order corrections were not applied. \\ \cline{2-3}  & & Saastamoinen model [56] for a-priori dry tropospheric delay using standard atmosphere. The wet tropospheric delay together with horizontal gradients were estimated. Global Mapping Function (GMF) [57] to estimate the delays along the line of sight. \\ \hline \multirow{2}{*}{receiver} & phase center offset and variations & IGS ANTEX maps [55] \\ \cline{2-3}  & differential code biases & not applied as the receiver reported P1 observable \\ \hline site & Solid Earth tide & IERS2010 conventions [58] \\ \hline \multirow{2}{*}{displacements} & Pole tide & CODE weekly ERP files [52; 53]; IERS2010 conventions [58] \\ \cline{2-3}  & Ocean Loading tide & FES2004 model c [59] \\ \hline \multirow{2}{*}{others} & Phase windup & wavelength dependent [60] \\ \cline{2-3}  & Phase ambiguity & estimated as float number \\ \hline \multicolumn{3}{l}{\({}^{\text{a}}\) [[ftp://ftp.aiub.unibe.ch](ftp://ftp.aiub.unibe.ch)]([ftp://ftp.aiub.unibe.ch](ftp://ftp.aiub.unibe.ch)), b [[ftp://igs.org/pub/station/general](ftp://igs.org/pub/station/general)]([ftp://igs.org/pub/station/general](ftp://igs.org/pub/station/general)), c [[http://holt.oso.chalmers.se/loading](http://holt.oso.chalmers.se/loading)]([http://holt.oso.chalmers.se/loading](http://holt.oso.chalmers.se/loading)).} \\ \end{tabular}
\end{table} 
Table 2: Methods of correction for various observation errors in our PPP data processing strategy. Typically, the filter is run in the forward direction and the solutions over the convergence period must be discarded. To cope with this, we used an unconventional way to run the filter. We run the filter twice, i.e. in both the forward and backward directions. This is possible only in post-processing mode. Next, the forward and backward solutions are joined into one combined solution using the inverse variance weighting approach. The following equation is used:  
\[X_{cmb}=\frac{X_{fwd}\cdot Q_{fwd}^{-1}+X_{bwd}\cdot Q_{bwd}^{-1}}{Q_{fwd}^{-1 }+Q_{bwd}^{-1}}\]  
where the \(X\) and \(Q\) denote the position vector and its associated variance co-variance matrix from the forward (\(fwd\)), backward (\(bwd\)) and combined (\(cmb\)) solution, respectively. As a result, the limitations coming from the convergence period are removed. The combined solution has a consistent standard deviation at each observation epoch.