# Estimation of GPS Differential Code Biases Based on Independent Reference Station and Recursive Filter

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## Abstract

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## 1. Introduction

## 2. Methods and Observations

#### 2.1. Carrier Phase Smoothing Pseudorange

#### 2.2. Local Ionospheric Modeling

#### 2.3. Model Initialization and Propagation

#### 2.4. Recursive Filter

#### 2.5. Reference Station Selection

- Estimating the satellite and receiver DCBs based on the recursive method described above with all observations.
- Estimating only the receiver DCBs with the initial satellite DCB estimates in the first step, by using the same recursive method. Note that the receiver DCBs in this step are determined by the median value of all the 15-min estimates, because in this step there is no ill-posed problem like in step 1.
- Comparing the receiver DCBs in the first and second step. If the difference between two receiver DCBs exceeds a threshold, we remove the corresponding station out of the set of reference stations and go back to step 1, until no station is removed in step 3. Then extract the final DCB estimates in this step. The threshold can be determined by the value of $\sigma $ in the last formula in (17).

## 3. Results and Analyses

#### 3.1. Experimental Data

#### 3.2. Evaluation of Local Ionospheric TEC Modeling

#### 3.3. Reference Station Selection

## 4. Discussion

#### 4.1. Comparision with CODE DCB Product

#### 4.2. Dependence on Solar Condition

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A comparison between the vertical total electron content (VTEC) derived from the recursive local model (orange line, denoted by recursive) and the global ionospheric model (GIM) from the Center for Orbit Determination in Europe (CODE) (blue line, denoted by CODE) over GODZ station on 15th January 2011.

**Figure 3.**Local horizontal gradient derived from recursive method over GODZ station on 15th January 2011.

**Figure 4.**Semi-monthly mean difference and root mean square (RMS) with respect to interpolated CODE GIM over 20 sample stations. The horizontal axis shows the name of stations.

**Figure 5.**Global distribution of selected ground receivers in geographic frame of reference. Green dots represent reference stations while red dots represent abandoned stations in step 3 on 17th January 2011.

**Figure 6.**Time series of GPS satellite DCBs estimated by the recursive method. (

**a**) shows the results with all station data and (

**b**) shows the results with reference station data. The horizontal axis represents the day of year in 2011.

**Figure 7.**The left axis represents the total numbers of DCB combinations ${N}_{0}$ (bar plot) in case 1 (red) and case 2 (green) and the right axis represents the weighted residuals (line plot) in Equation (17) in case 1 (blue) and case 2 (black). The horizontal axis represents the day of year in 2011.

**Figure 8.**Day scatter of satellite DCB estimates determined by recursive method after selecting reference stations and corresponding RMS values with respect to CODE DCB during the period of interests.

**Figure 9.**The difference of RMS and day scatter before and after selecting reference stations during the period of interests.

**Figure 11.**Mean difference and RMS with respect to interpolated CODE GIM over 20 sample stations during the disturbed period. The horizontal axis shows the name of stations. The missed stations are not labeled.

**Table 1.**Numbers of discarded stations located in low, mid and high latitudes during the second half of January 2011.

DOY | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Low(0°–30°) | 13 | 14 | 12 | 12 | 12 | 14 | 13 | 11 | 14 | 12 | 19 | 11 | 9 | 14 | 17 | 14 |

Mid(30°–60°) | 18 | 23 | 32 | 22 | 24 | 17 | 20 | 16 | 18 | 13 | 18 | 15 | 15 | 17 | 17 | 15 |

High(60°–90°) | 2 | 4 | 2 | 5 | 3 | 6 | 4 | 7 | 4 | 5 | 3 | 6 | 5 | 4 | 2 | 1 |

Total | 33 | 41 | 46 | 39 | 39 | 37 | 37 | 34 | 36 | 30 | 40 | 32 | 29 | 35 | 36 | 30 |

**Table 2.**Monthly mean difference with respect to CODE receiver DCBs (MAD) and day scatter (DS) of eight sample stations during the second half of January 2011. Note that the number denotes different cases. Case 1 represents the results using all stations and case 2 represents the results using reference stations. Unit: ns.

Station | GODZ | HERS | IQAL | PERT | QAQ1 | REYK | TIXI | YSSK |
---|---|---|---|---|---|---|---|---|

MAD 1 | 0.05 | 0.38 | 0.32 | 0.29 | 0.26 | 0.27 | 0.39 | 0.06 |

MAD 2 | 0.03 | 0.20 | 0.09 | 0.07 | 0.26 | 0.23 | 0.20 | 0.06 |

DS 1 | 0.31 | 0.67 | 0.42 | 0.82 | 0.21 | 0.29 | 0.45 | 0.30 |

DS 2 | 0.30 | 0.20 | 0.22 | 0.35 | 0.20 | 0.21 | 0.16 | 0.25 |

DSCODE | 0.12 | 0.09 | 0.12 | 0.22 | 0.12 | 0.07 | 0.16 | 0.20 |

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**MDPI and ACS Style**

Yuan, L.; Jin, S.; Hoque, M.
Estimation of GPS Differential Code Biases Based on Independent Reference Station and Recursive Filter. *Remote Sens.* **2020**, *12*, 951.
https://doi.org/10.3390/rs12060951

**AMA Style**

Yuan L, Jin S, Hoque M.
Estimation of GPS Differential Code Biases Based on Independent Reference Station and Recursive Filter. *Remote Sensing*. 2020; 12(6):951.
https://doi.org/10.3390/rs12060951

**Chicago/Turabian Style**

Yuan, Liangliang, Shuanggen Jin, and Mainul Hoque.
2020. "Estimation of GPS Differential Code Biases Based on Independent Reference Station and Recursive Filter" *Remote Sensing* 12, no. 6: 951.
https://doi.org/10.3390/rs12060951