Assuming that the residual errors of satellite orbit and clock could be neglected, the ionosphere-free (L₃) combinations can be written as follows: (1) L j = ρ j + δ + b j + T j + ɛ j

The superscript of “j” indicates tracked satellite. L^j is the ionosphere-free phase observation of j; ρ^j is geometric range between receiver and satellite; δ is the receiver clock error; b^j is the ambiguity of L₃ phase observation; T^j is the tropospheric delay; ε^j is the phase noise of L₃ observation.

For one receiver tracking two satellites (j,i) simultaneously, the single-differenced measurements between satellite can be used to eliminate the receiver clocks. By taking differences of L₃ observation of satellite i and j, we obtain: (2) L j , i = L i − L j = ρ j , i + b j , i + T j , i + ɛ j , i

Equation (2) is the defined as the satellite-differenced (SD) equation. Assuming there are no cycle slips between two adjacent epochs, the ambiguity term b^j,i in Equation (2) can be further eliminated by differencing the SD observations at the adjacent epoch n and n−1 (n = 2…n₁, n₁ is total number of defined epochs): (3) Δ L j , i ( n ) = L j , i ( n ) − L j , i ( n − 1 ) = Δ ρ j , i ( n ) + Δ T j , i ( n ) + Δ ɛ j , i ( n )where “Δ” indicates the ED operator. Equation (3) can be re-written as: (4) Δ ρ j , i ( n ) = Δ L j , i ( n ) − Δ T j , i ( n ) − Δ ɛ j , i ( n )

Equation (4) is defined as the SDED based ambiguity-free equation.

Although the main parts of the tropospheric delay are canceled by forming differences between adjacent epochs, there are still residual components in Equation (4). Due to the correlation between parameters, it is difficult to obtain a quick and reliable estimation of both coordinates and tropospheric parameters. A better solution is to model tropospheric delay rather than to estimate it. Following the RTK method, the modeling of atmospheric delay has been studied by a large number of researchers [22–24]. The general idea is the interpolation based on a reference network. The atmospheric delays at reference stations could be accurately estimated while they could be modeled at user stations. In practice, the wet part of tropospheric delay (Zwd) of user stations is interpolated, while the dry part (Zhd) is corrected using a standard Saastamoinen model. The interpolation model [23] is: (5) Zwd = ∑ m = 1 k ( Z w d m ⋅ P m ) / ∑ m = 1 k P m (6) P m = 1 l mwhere k is the number of reference stations; Zwd_m is Zwd of reference station m; P_m is weight; l_m is the distance between the reference and user stations.

Mathematically, the SDED method is sensitive to the epoch-wise un-modeled errors (UMEs). These errors include satellite related errors (orbits and clocks) and appear to be similar at a regional scale. At reference stations, we could calculate the remaining un-modeled errors. With estimated tropospheric delay, fixed coordinates of the reference station, clocks and orbits, the SDED UMEs of reference stations can be retrieved from the SDED observations. Similar to Equation (5), the SDED UMEs at user station can be computed based on the inverse distance weighted model: (7) UME j , i ( n ) = ∑ m = 1 k ( UME j , i ( n ) m ⋅ P m ) / ∑ m = 1 k P m

Substituting the interpolated tropospheric delay corrections and computed UMEs into Equation (4), and linearizing Equation (4), Equation (4) can be re-written in a matrix form: (8) ( a j , i ( n − 1 ) − a j , i ( n ) ) ⋅ X = Δ l 0 j , i ( n )where a^j,i(n − 1) and a^j,i(n) are SD design matrices. The unknowns are the coordinates of the user station. From Equation (8), we know that two epochs of observations are sufficient to get coordinate estimations when there are more than three observation equations (i.e., four satellites are observed simultaneously).

The coordinate estimation using Equation (8) could be used as the starting point for kinematic positioning thereafter, i.e., after observing in static mode for a short period, a user station could run kinematically afterwards. The strategy for user kinematic PPP positioning is as following: SDED starting points + epoch-wise coordinates differences. In this strategy, the SDED coordinates from Equation (8) are used as known coordinates and epoch-wise coordinates differences (relative coordinates) are derived following the ED described in Bock et al. [4]. The final absolution coordinates are epoch-wisely accumulated using the initial and relative coordinates. The precision of relative coordinates is normally at mm level, therefore the precision of kinematic coordinates is similar to that of the starting points.

Taking the Zwds estimated at the five reference stations, the Zwds of the station SHBS are interpolated with the method described in Section 2.2. Comparison between estimated and interpolated Zwds is illustrated in Figure 3. In Figure 3, the Zwds of all stations (including SHBS) are first estimated by setting up a Pice-Wise-Constant (PWC) parameter at an interval of 1 h. Afterwards, Zwds of SHBS are interpolated based on the estimated Zwds of the other stations and are compared to the previous estimation. The RMS is 2.26 mm.

The SDED UMEs of the all the stations can be retrieved using Equation (4). Using the retrieved SDED UMEs at the five reference stations, SDED UMEs at SHBS are interpolated according to Equation (7). Figure 4 illustrates for satellite pair PRN20 and PRN28 the retrieved SDED UMEs in 2 h with an interval of 10 min. Comparing the interpolated and estimated UMEs, we found very good agreement with RMS of 3 mm, which validates the idea of UMEs interpolation of user station.

Figure 5 illustrates the SDED UMEs for all satellites with PRN20 being the reference satellite. It is observed that SDED UMEs are sometimes more than 2 cm, e.g., PRN02, PRN04, PRN10, PRN13, PRN17 and PRN32.

SDED-based differential PPP was performed for SHBS for the whole week. As two epochs of data are sufficient to derive station coordinates, we split the 12-h observations into 72 10-min and 48 15-min sessions for each day. Each session thus contains only two epochs of data with the interval at 10-min or 15-min. Differential PPP was carried out to verify the robustness and efficiency of the proposed SDED approach. The estimators of Least Square (LS) and Robust Estimation (RE) were tested by different strategies. The difference between strategy 1 and 6 is in the tropospheric delay handling: it is corrected using models in strategy 1, while it is being estimated in strategy 6. Table 1 presents for each tested strategy the success rate of differential PPP under the sampling of 10 min and 15 min. The success rate is defined as the case that the difference between the estimated and the known coordinate components are less than 10 cm. For comparison, the traditional PPP (NORM-PPP) method was tested using the observations (sampling at 30 s) within the same time window and the results were listed in Table 1 as well.

The success rate shown in Table 1 of different strategies indicates that the residual tropospheric delays affect the positioning results, although they can be partly eliminated by the epoch and satellite difference. The study strategy 1 is actually the normal SDED PPP [21], its success rate is only 36% due to the correlation between tropospheric parameters and coordinates. Applying the tropospheric delay corrections, the success rate of LS improves from 36% to 52%. Another improvement of 6% (in case of LS) and 32% (in case of RE) of success rate is achieved when we further apply the UME corrections. This improvement validates the contribution of UME estimation. Comparing the results of Robust Estimation to that of Least Square estimation, we see that success rate has notable improvement as bad SDED observations are down weighted in the robust estimation. Comparing the success rate of 10 min and 15 min, there is a general 10% improvement in success rate when the interval between the two epochs is increased to 15 min. By correcting tropospheric delay and UMEs, all sessions implementing differential PPP successfully meet the threshold and reach the desired accuracy.

Position RMSs are calculated based on results from sessions with 10-min and 15-min intervals. Table 2 shows for each tested strategy the RMS (in cm) of coordinates with respect to the known coordinates in the North, East and Up directions at the interval of 10 min and 15 min, respectively.

Similar to the success rate discussed in previous section, the position results show also the improvement due to the corrections of tropospheric delays and UMEs. From the results, we see RE performs better than LS in all cases with the best result improved from 2.47, 3.95, 5.78 cm to 2.21, 3.93, 4.90 cm in the North, East and Up directions, respectively. Similarly, the cases where UME corrections are applied behave better than the others. All the cases of differential PPP have a better precision than NORM-PPP with the best strategy (strategy 6) has around 50% improvement in all three coordinate components.

To demonstrate the RTK applications of our new SDED approach, SHBS is used as a kinematic station. We use the first RTK strategy described in Section 2.3 to derive kinematic PPP coordinates. The first two epochs of data sampling at 15 min is used to derive coordinates at the starting point, data sampling is then set to 30 s in RTK PPP thereafter. Figure 6 shows the RTK coordinates differences for DOY 011 with respect to the known coordinates. We notice that there are offsets of few centimeters for each component, which are introduced by the SDED initial coordinates. Removing the offsets, the coordinate standard deviation (STD) is of 0.80, 1.34, 0.97 cm in the North, East and Up directions.