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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Precise Point Positioning (PPP) has been demonstrated as a simple and effective approach for user positioning. The key issue in PPP is how to shorten convergence time and improve positioning efficiency. Recent researches mainly focus on the ambiguity resolution by correcting residual phase errors at a single station. The success of this approach (referred to hereafter as NORM-PPP) is subject to how rapidly one can fix wide-lane and narrow-lane ambiguities to achieve the first ambiguity-fixed solution. The convergence time of NORM-PPP is receiver type dependent, and normally takes 15–20 min. Different from the general algorithm and theory by which the float ambiguities are estimated and the integer ambiguities are fixed, we concentrate on a differential PPP approach: the satellite- and epoch differenced (SDED) approach. In general, the SDED approach eliminates receiver clocks and ambiguity parameters and thus avoids the complicated residual phase modeling procedure. As a further development of the SDED approach, we use a regional augmentation network to derive tropospheric delay and remaining un-modeled errors at user sites. By adding these corrections and applying the Robust estimation, the weak mathematic properties due to the ED operation is much improved. Implementing this new approach, we need only two epochs of data to achieve PPP positioning converging to centimeter-positioning accuracy. Using seven days of GPS data at six CORS stations in Shanghai, we demonstrate the success rate, defined as the case when three directions converging to desired positioning accuracy of 10 cm, reaches 100% when the interval between the two epochs is longer than 15 min. Comparing the results of 15 min' interval to that of 10 min', it is observed that the position RMS improves from 2.47, 3.95, 5.78 cm to 2.21, 3.93, 4.90 cm in the North, East and Up directions, respectively. Combining the SDED coordinates at the starting point and the ED relative coordinates thereafter, we demonstrate the performance of RTK PPP with standard deviation of 0.80, 1.34, 0.97 cm in the North, East and Up directions.

It has been more than ten years since precise point positioning (PPP) theory was proposed [

The above mentioned strategies focus on the PPP ambiguity resolution. Following the strategy of satellite- and epoch difference (SDED) [

Network augmented PPP (Net-aPPP) [

Assuming that the residual errors of satellite orbit and clock could be neglected, the ionosphere-free (_{3}

The superscript of “^{j}^{j}^{j}_{3}^{j}^{j}_{3}

For one receiver tracking two satellites (_{3}

^{j,i}_{1}_{1}

Although the main parts of the tropospheric delay are canceled by forming differences between adjacent epochs, there are still residual components in _{m}_{m}_{m}

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

Substituting the interpolated tropospheric delay corrections and computed UMEs into ^{j,i}(n − 1)^{j,i}(n)

The coordinate estimation using

According to

The preprocessing of phase data is based on the Melbourne-Wuebbena and Geometry-free combination to detect cycle slips and outliers. For real-time applications, the orbit and clock are from the predicted ultra-rapid orbits and real-time estimated clocks. The correction models including the phase wind-up, Earth tides, relativistic effects, antenna phase center offset and variation,

The robust estimation [_{i}_{i}_{i}_{/}σ_{0}_{i}_{0}_{0}_{1}_{0}_{1}

In order to test the proposed method, one week (DOY 011 to DOY 017, 2009) data from six stations (SHBS, CMMZ, SHQP, SHJS, LGXC and SSJG) of Shanghai Continuous Operation Reference Station (CORS) network are used. Each station has 12-h observations for each day. The station SHBS is taken as user station and the other five stations are used as reference stations to derive the correction information. Data sampling is 30 s and elevation cut-off angle is set to 9 degrees. The reference and user stations are shown in

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

The SDED UMEs of the all the stations can be retrieved using

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.

The success rate shown in

Position RMSs are calculated based on results from sessions with 10-min and 15-min intervals.

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.

This study introduces a differential PPP approach based on a regional reference augmentation network, where ambiguity and receiver clock are removed by the SDED model. The corrections of tropospheric delay and UME at user station are interpolated from the estimated Zwds and retrieved UMEs of reference stations. In addition, the Robust Estimation is implemented to overcome the defect of the weakened geometry caused by the SDED model.

The approach presented in the paper has been validated using one-week data from a regional network in Shanghai. From the experiment, we see the interpolated Zwds and UMEs have an accuracy of 2.26 mm and 3 mm, respectively. Using the interpolated corrections, differential PPP is performed with estimators of Least Square and Robust Estimation. The position results and success rate indicate that the RE performs better than the LS. The PPP position accuracy of RE is at the cm level for an interval longer than 10 min. The success rate of RE with an interval longer than 15 min reaches 100% and position accuracy reaches 2.21, 3.93, 4.90 cm in the North, East and Up directions. Comparing the results from our new approach to that of NORM-PPP, we see remarkable improvements in both convergence and precision.

The differential PPP approach improves PPP positioning with shorter convergence-time and more reliable results. In real-time applications, the user station observes firstly in static mode for around 15 min and runs kinematically afterwards. Results show the kinematic PPP coordinates precision (STD) is of 0.80, 1.34, 0.97 cm in the North, East and Up directions, with offsets of a few centimeters in each component.

This research is supported by the National Natural Science Foundation of China (NSFC) (No. 40974018) and 100 Talents Programme of The Chinese Academy of Sciences. Anonymous reviewers are acknowledged for their valuable comments and suggestions.

Flow chart of the SDED based differential PPP software.

Stations selected from Shanghai CORS and their distribution. Solid triangle shows the reference stations, Circle is user station.

Differences between estimated and interpolated Zwds of SHBS.

SDED un-modeled errors of SHBS using satellite pair of PRN20 and PRN28, where Estimated UME is calculated following

Satellite- and epoch differenced un-modeled errors.

SDED & ED combined kinematic PPP positioning.

Strategies, Success rate (in %) within 10-min and 15-min time window.

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LS | 1 | No | 36 | 27 |

2 | Zwd | 52 | 63 | |

3 | UME and Zwd | 58 | 77 | |

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RE | 4 | Zwd | 70 | 81 |

5 | UME and Zwd | 85 | 100 | |

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NORM-PPP | 6 | 46 | 60 |

RMS (in cm) of different strategies static PPP coordinates with respect to the known coordinates in the North, East and Up directions.

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LS | Zwd | 7.68 | 10.19 | 12.47 | 4.13 | 5.96 | 6.20 |

UME and Zwd | 7.49 | 9.11 | 11.35 | 2.47 | 3.95 | 5.78 | |

| |||||||

RE | Zwd | 6.11 | 8.43 | 10.02 | 2.44 | 3.82 | 5.62 |

UME and Zwd | 5.35 | 6.50 | 8.12 | 2.21 | 3.93 | 4.90 | |

| |||||||

NORM-PPP | 7.93 | 10.23 | 12.56 | 4.51 | 7.86 | 8.91 |