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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 become a very hot topic in GNSS research and applications. However, it usually takes about several tens of minutes in order to obtain positions with better than 10 cm accuracy. This prevents PPP from being widely used in real-time kinematic positioning services, therefore, a large effort has been made to tackle the convergence problem. One of the recent approaches is the ionospheric delay constrained precise point positioning (IC-PPP) that uses the spatial and temporal characteristics of ionospheric delays and also delays from an

Precise Point Positioning (PPP) was firstly proposed by Zumberge

Bar-Sever

Since then, real-time PPP services have been considered a hot topic in GNSS research and development. On the one hand, large efforts have been made to improve the accuracy of the IGS precise orbit and clock products, from 30 cm to 40 cm in the early stages to an optimization of 2.5 cm for orbits and from 1 ns to 2 ns to better than 0.1 ns for clocks (Ye [

One of the major concerns in real-time PPP is that usually it takes about 30 min in order to obtain positions with accuracy better than 10 cm. The PPP positioning accuracy and convergence are mainly influenced by the observing geometry between the station and GPS satellites (Li and Shen [

As is well known, the first-order ionosphere delay can be eliminated by forming an ionosphere-free observation. Although ionospheric delays in phase and range are expressed by the same ionospheric delay parameter, it is eliminated as different ones for phase and range. Furthermore, the spatial and temporal characteristics of the ionospheric delays and an available a priori correction model, which are implemented as constraints to enhance PPP using single-frequency observations (Beran

After a brief introduction of the observation equations, the mathematical model of the ionosphere delay constrained PPP (IC-PPP) is presented, with details on the ionospheric constraints and DCB parameterization. Then, the data processing scenarios are illustrated for assessing the impact of the quality of ionospheric corrections and the effect of receiver DCB and its estimation. Results from a large GPS data set will be presented and discussed.

In order to discuss the details of the IC-PPP model, we first introduce the basic GNSS observation equations. Then, an approach to generate satellite-specified ionospheric corrections based on dense regional reference networks is discussed. Of course the temporal and spatial constraints imposed on ionospehric parameters and the DCB parameterization are also presented to complete the IC-PPP algorithm.

The observation equations of the pseudorange and carrier-phase at frequency band _{i}_{r}_{s}_{trop}_{ion}_{r}_{dcb}_{i}_{P}_{L}

In traditional PPP, ionosphere-free phase (LC) and range (PC) based on dual-frequency pesudoranges and carrier phases are used to eliminate the first order ionospheric delay. The residual high order ionospheric delay is usually less than 1% (Hernandez-Pajares

From

In order to avoid aforesaid disadvantages of the LC-PPP using

Dual-frequency GNSS observations at ground networks are the basic information for reconstructing ionospheric delay models for both ionosphere study and precise positioning. The ionosphere delay models could be generated at global or regional scales, corresponding to the coverage of the reference networks.

The global model is usually expressed in the form of spherical harmonic functions or grids, for example, the global ionospheric map (GIM) by CODE (Schaer [

For a LOS path of an observed satellite, the position of the ionosphere pierce point (IPP),

Due to the inaccuracy of the assumption and the mapping function, and the limited station density, the Root Mean Square (RMS) of a global model is usually of about 0.0∼0.9 m (Hernández-Pajares

For the regional model, PPP is undertaken for all the reference stations with known coordinates and even receiver DCBs, so that slant ionospheric delays for each LOS can be calculated and serve as ionospheric model. As illustrated in

First of all, the calculated slant ionospheric delay from an _{ion}_{1} and

For the regional model, the STD of the interpolated slant delay can be estimated according to the binterpolation method and the variance of slant ionospheric delay from PPP technique, usually,

In addition to the

Besides the ionospheric delay, the receiver DCB must also be handled differently in LC-PPP and IC-PPP. As usual, satellites DCB must be corrected using the values associated with the clock product. In LC-PPP, the receiver DCB biases all LC ranges by a constant which is absorbed by the receiver clock parameter, therefore, we do not have to consider it. By the way, the DCB of PC measurement is also defined as zero (Dach

In order to evaluate the impact of the quality of ionospheric delay corrections and the receiver DCBs, three PPP modes are employed in the experimental test: PPP using ionosphere-free observations (LC-PPP), PPP using raw observations with ionospheric delay constraints,

With the above-mentioned PPP modes, data from the IGS global network and data from the Crustal Movement Observation Network of China (CMONOC) are processed. For the IGS network, GIM data provided by IGS is used to calculate ionospheric delay correction as constraints in IC-PPP and IC-PPP + DCB, while for the CMONOC network, a reference network is defined for constructing regional ionospheric correction as explained in Section 2.2 and then applied for the client stations as ionospheric constraint for IC-PPP and IC-PPP + DCB. The details of the IGS and CMONOC networks and data sets will be presented Sections 4 and 5, respectively, together with results.

For each of the network, the estimated station positions and convergence time are compared with the known values and against each other, respectively, for assessing their performance. For the IC-PPP and IC-PPP + DCB modes, the estimated ionospheric delays are all interpreted for validating their advantages.

In PPP solutions, the weight of pseudoranges and carrier phases at different elevations are calculated using the following formula (Gendt

For the IGS network, about 300 IGS stations are selected and data from the days 024 to 040, 2012 at the sampling rate of 30 s are processed to evaluate the performance of the three PPP approaches

The daily estimated station coordinates of the three processing scenarios are compared with the related IGS weekly solutions. The Root Mean Square (RMS) of the coordinate differences in the NEU is shown in

The differences between IGS and IC-PPP + DCB derived DCBs are shown in

To test PPP convergence performance, the daily data are divided into 12 sessions each of two hours. Seven days' data at 300 IGS stations are processed in two-hour sessions, so that a total of 24,521 re-convergence sessions should be involved after removing those sessions missing data. However, there are some sessions that failed in PPP convergence within one hour, and then are removed too. As the last column in

The larger convergence percentage of IC-PPP + DCB compared with that of the IC-PPP demonstrates that the receiver DCB has a strong impact on IC-PPP convergence. However, it is unexpected that the 93.10% of the LC-PPP solution is slight better than that of IC-PPP + DCB. The possible explanation might be the quality of the GIM is not good enough during this period for mitigating the range noise for better positioning accuracy.

To further study the convergence, the convergence time for stations located in different latitudes are shown in

To show the effect of the receiver DCB, the convergence time against the receiver DCB is plotted in

There is an obvious trend in the IC-PPP where DCB is ignored. It indicates that the convergence time becomes longer as the receiver DCB increases. The trend disappears in the IC-PPP + DCB solution where the receiver DCB is estimated. The mean convergence time improvement from IC-PPP to IC-PPP + DCB is about 7.3 min, nearly 30% as a percentage. As

In order to investigate the impact of the ionospheric correction model on the PPP performance, the Crustal Movement Observation Network of China (CMONOC) is exploited since it can provide more continuous GNSS tracking sites for PPP using regional ionospheric model. There are about 160 stations on DOY 218∼224, 2012 that are selected and divided into two groups: a reference network comprising about 85 stations with a inter-station distance of about 320 km and the others as PPP test stations as shown in

The reference network is used to generate the satellite-specified slant ionosphere delays with the IC-PPP + DCB solution. In the processing, the satellite DCB is calibrated using the IGS products and the station coordinate is fixed to the IGS-like weekly solution. Furthermore, forward and backward filtering are carried out, so that the derived ionospheric delays could achieve an accuracy of better than 2.0 TECU. Then the slant delays at the test stations can be calculated by the linear interpolation of the estimated slant delays of the nearby reference stations. These satellite-specified corrections are referred as to China Regional Model (CRM). Correspondingly, GIM is also used to provide the

To assess the quality of ionospehric corrections, the data of the 75 test stations is also processed in the same way as for the 85 sites, so that the slant ionospheric delays can be directly estimated from the observations. The estimated delays can be served as reference values to assess the quality of the interpolated corrections from CRM and GIM. As

The initial positioning accuracy of IC-PPP is mainly affected by the level of pseudorange noises, residuals of

The coordinate estimates of the first three epochs of all the test sessions at all sites are compared with the ground true and the residuals are plotted in

From

The analysis above shows that the convergence of the IC-PPP + DCB should be faster than that of LC-PPP if the quality of the

The impact of the quality of ionospheric model corrections and receiver DCBs on the convergence of the IC-PPP is investigated through the analysis of a large amount of data. In IC-PPP solution, receiver DCB has significant influence on its convergence. The bigger the DCB, the slower the PPP converges. Estimating receiver DCB in IC-PPP solution is a proper way to overcome the problem. The results, which are derived from 300 IGS sites using GIM as

The accuracy of the

Therefore, we strongly suggest that receiver DCB should be estimated in current IC-PPP and regional satellite-specific ionospheric correction models should be utilized in order to speed up its convergence for wider applications.

The authors would like to thank the reviewers for their beneficial comments and suggestions. This work was supported partly by Key Program of National Natural Science Foundation of China (41231064), the National Basic Research Project of China (Grant No. 2009CB72400205), the National High Technology Research and Develop Program of China (2012AA12A206). We also thank the CMONOC authorities for providing the data for this study.

The authors declare no conflict of interest.

Interpolation of the slant ionospheric delay of a client station using the estimated slant delays of three closest reference stations.

The distribution of the 300 IGS sites used.

Histogram showing the differences of slant ionospheric delays and receivers' DCB between IGS published results and IC-PPP + DCB derived results at the selected 300 IGS sites.

Percentage of PPP results converged to 10cm in horizontal components in different time spans.

Convergence time for different latitude zones for the three processing scenarios IC-PPP (

Relationship between receivers DCB magnitude and PPP convergence time in IC-PPP +DCB (

Distribution of the 160 selected CMONOC stations. Red stars represent the reference stations for generating regional ionospheric corrections and green dots indicate PPP test stations.

RMS of the residuals of the interpolated slant delays from GIM (black) and CRM (red) with respect to the reference slant delays.

Initial horizontal positioning results for the five PPP schemes. The position of the first three epoches are counted and plotted for all the convergence trials.

Initial height results for the five PPP schemes. The position of the first three epochs are counted and plotted for all the convergence trials.

Success rates of convergence into 10 cm in horizontal components of the five PPP schemes.

Mean convergence time into 10 cm in both North and East components of five PPP solutions using CMONOC data (minutes).

Parameter schemes for IC-PPP estimation no italic units.

Position | Static/Kin | Static/Kin | Static/Kin | 10 m each component | |

Receiver clock | White noise | White noise | White noise | 300 m | |

Troposphere delay | ZTD | ZTD | ZTD | 20 cm + 1 cm/
| |

Receiver DCB | Absorbed | Ignored | Random walk | 15 cm + 1 cm/
| |

Ionosphere delay | Eliminated | Slant Delay + Constraint | Slant Delay + Constraint | 30 cm + 1 cm/
| |

Ambiguities | LC | L1, L2 | L1, L2 | ||

Number of observation parameter | N × 2/N + 5 | N × 4/3 × N + 5 | N × 4/3 × N + 6 |

300 IGS Sites' distribution at different latitudes.

1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

|B| < 10 | |B| > 10 |
|B| > 20 |
|B| > 30 |
|B| > 40 |
|B| > 50 |
|B| > 60 | |

13 | 31 | 32 | 71 | 64 | 45 | 44 |

The overall RMS (in meters) of the coordinate differences of 300 IGS sites in NEU directions of the three processing scenarios.

IC-PPP + DCB | 0.0038 | 0.0062 | 0.0139 |

IC-PPP | 0.0039 | 0.0091 | 0.0135 |