# Estimation of the Antenna Phase Center Correction Model for the BeiDou-3 MEO Satellites

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Models

#### 2.1. PCO Parameters Model

#### 2.2. PV Parameters Model

#### 2.3. Solar Radiation Pressure Model

#### 2.4. A Priori Box-Wing Model for BeiDou-3 MEO

## 3. Strategies

## 4. PCOs and PVs Estimations

#### 4.1. Horizontal PCO Estimations

^{2}was added to the D

_{0}parameter (Figure 2c), the x-offset series appeared to be a stationary sequence that was almost irrelevant to the β-angle.

_{0}constraint on the precision of the orbit overlap when estimating the PCOs, three schemes of experiments were designed: (a) the precision of the overlap without estimating PCOs (Figure 3a); (b) estimating PCOs but not adding a constraint to the D

_{0}parameter (Figure 3b); (c) estimating PCOs and adding a constraint of 1 nm/s

^{2}to the D

_{0}parameter (Figure 3c). In Figure 2b, corresponding to the larger scatters of x-offset, |β| > 60°, the results were the gray-shaded area in Figure 3, and the RMS of the orbit overlap in along-track (A), cross-track (C), and radial (R) directions were computed and shown in Table 4. Compared with the results of estimating PCOs and non-constraint of D

_{0}parameter in scheme (b), the RMS of scheme (c) was reduced by 0.65 (6.8%), 6.36 (59.2%), and 0.4 (16.3%) cm, in A-, C-, and R-components, respectively. Therefore, it can be inferred that the correlation between the x-offset and the C-component of the orbit was higher than the A- and R-components for |β| > 60°. After adding the constraint to the D

_{0}parameter, the RMS of orbit overlap was close to the reference result (scheme a), and the differences between scheme (c) and scheme (a) were 0.79, −1.72, and 0.25 cm, for A-, C-, and R-components, respectively.

#### 4.2. Vertical PCO and PV Estimations

## 5. Validations

#### 5.1. Orbit Precision

#### 5.2. Clock Offset Precision

## 6. Conclusions

_{0}constraint to reduce the correlation. Therefore, a stationary sequence of the x-offset estimations that were weakly related to the β-angle was obtained, and the final estimation of the x-offset can be obtained for a relatively short period, rather than a half year.

^{2}was also applied to the D

_{0}parameter, and a stationary sequence with the average of −0.227 ± 0.044 m was obtained. Comparing the two averages and STDs, the average of x-offset estimations had a difference of about 0.1 m, and the stability was improved by 50%. For the y-offset parameter, whether or not the a priori SRP model was added had no significant effect on the final estimation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distributions of precise orbit determination (POD): magenta points are international GNSS Monitoring and Assessment System (iGMAS) stations, which can track all BeiDou-2/3 satellites; orange and yellow points are MGEX stations, of which orange points can provide BeiDou-3 data.

**Figure 2.**The x-offset estimations of the C21 using different solar radiation pressure (SRP) models. Figure (

**a**) is the pure ECOM1 model, Figure (

**b**) is the ECOM1+a priori SRP model, and Figure (

**c**) is the ECOM1+ a priori SRP model with a constraint of 1 nm/s

^{2}for the D

_{0}parameter. The blue line is the β-angle.

**Figure 3.**The RMS of the overlap of the C21 satellite. Figure (

**a**) is the result without estimating PCOs, Figure (

**b**) is the result estimating PCOs without adding a constraint for the D

_{0}parameter, and Figure (

**c**) is the result estimating PCOs with a constraint of 1 nm/s

^{2}for the D

_{0}parameter. The blue line is the β-angle. The black, red, and black points are along-, cross-, and radial-components, respectively. The gray-shaded area was the result for about |β| > 60°, where the RMS was significantly larger than others in Figure (

**b**).

**Figure 7.**The number of observations of the BeiDou-3 satellites. Orange and green bars are the number of observations provided by the MGEX and iGMAS networks, respectively; magenta and blue lines are the measured and expected ratios of MGEX to iGMAS, respectively.

**Table 1.**Coefficients of a priori box-wing model for the BeiDou-3 medium Earth orbit (MEO) satellites, the unit is nm/s

^{2}.

Types | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ | ${\mathit{a}}_{\mathbf{6}}$ | ${\mathit{a}}_{\mathbf{7}}$ |
---|---|---|---|---|---|---|---|

CAST | 5.99 | -0.32 | -1.18 | 11.10 | -0.53 | 0.21 | 110.62 |

SECM | 3.02 | 1.05 | 0.84 | 5.61 | 1.95 | 0.05 | 59.01 |

Items | Descriptions |
---|---|

Stations | About 17 iGMAS stations and 56 IGS/MGEX stations; |

Time period | From 214, 2018 to 140, 2019; |

Observation | Zero-difference phase and code observations; Elevation-dependent weight; elevation cutoff angle is 15°; |

Data arc | 72 h orbital arcs; |

Attitude model | Yaw-steering mode; |

Solar radiation pressure (SRP) | ECOM1+a priori box-wing model (Table 1); |

Inter-system biases (ISBs) | A constant parameter for each station per orbital arc, and zero-mean constraints were added for all ISBs; |

Ionosphere delay | Ionosphere-free linear combination GPS: L1/L2 BeiDou-3: B1I/B3I; |

Troposphere delay | ZTD parameters with an interval of 2 h; SAAS + GMF [28,29,30]; horizontal gradient parameters with an interval of 24 h; |

Station coordinates | Fixed to the IGSYYPWWWW.snx, where YY is the last two digits of the year, and WWWW is the corresponding GPS week; |

Receiver antenna | Fixed to igs14_WWWW.atx; |

Satellite antenna | GPS and BeiDou satellites are from igs14_2062.atx; |

Ambiguity | Fixed by adding double-difference constraint [31,32]; |

Eclipsing period | Removed. |

**Table 3.**Multi-GNSS Experiment (MGEX) phase center offsets (PCOs) of BeiDou-3 satellites refer to igs14_2062.atx, unit is mm.

Frequency | CAST | SECM | ||||
---|---|---|---|---|---|---|

x-offset | y-offset | z-offset | x-offset | y-offset | z-offset | |

C01 | −200 | 0 | 1460 | 40 | −10 | 1100 |

C02 | −200 | 0 | 1460 | 40 | −10 | 1100 |

C06 | −200 | 0 | 1180 | 40 | −10 | 1090 |

C07 | −200 | 0 | 1070 | 40 | −10 | 1090 |

Iono-free combination of B1I and B3I | −200 | 0 | 2004 | 40 | −10 | 1119 |

Scheme | A | C | R |
---|---|---|---|

(a) Without estimating PCOs | 8.08 | 6.10 | 1.81 |

(b) Estimating PCOs, without D_{0} constraint | 9.52 | 10.74 | 2.46 |

(c) Estimating PCOs, with D_{0} constraint of 1 nm/s^{2} | 8.87 | 4.38 | 2.06 |

**Table 5.**Average and standard deviation (STD) of the horizontal PCOs for each satellite, the unit is mm.

PRN | x-offset | y-offset | ||
---|---|---|---|---|

Average | STD | Average | STD | |

C19 | −228.28 | 33 | −6.09 | 59 |

C20 | −229.22 | 40 | −15.77 | 60 |

C21 | −211.47 | 44 | −8.56 | 61 |

C22 | −226.40 | 38 | −13.86 | 51 |

C23 | −230.12 | 39 | 1.74 | 52 |

C24 | −218.68 | 43 | −11.19 | 51 |

C25 | 37.69 | 48 | −12.54 | 62 |

C26 | 29.26 | 55 | −4.01 | 64 |

C27 | 18.08 | 49 | −7.54 | 33 |

C28 | 18.99 | 29 | −4.64 | 19 |

C29 | 19.46 | 36 | −11.63 | 38 |

C30 | 21.29 | 35 | −6.25 | 24 |

C32 | −165.81 | 63 | −8.00 | 52 |

C33 | −164.22 | 63 | −8.99 | 40 |

C34 | 44.66 | 30 | −6.63 | 21 |

C35 | 126.23 | 57 | −9.28 | 40 |

Type | x-offset | y-offset | ||
---|---|---|---|---|

Average | STD | Average | STD | |

CAST | −224.48 | 6 | −9.34 | 7 |

SECM | 24.00 | 10 | −7.81 | 4 |

CAST | C19 | C20 | C21 | C22 | C23 | C24 | C32 | C33 |
---|---|---|---|---|---|---|---|---|

z-offset | 2300.6 | 2339.1 | 2310.4 | 2333.1 | 2370.9 | 2426.5 | 2406.9 | 2459.5 |

STD | 126.9 | 128.9 | 109.8 | 121.5 | 127.2 | 136.0 | 117.8 | 119.0 |

SECM | C25 | C26 | C27 | C28 | C29 | C30 | C34 | C35 |

z-offset | 1429.1 | 1542.0 | 1627.3 | 1680.8 | 1739.7 | 1564.8 | 1567.0 | 1445.2 |

STD | 147.1 | 139.3 | 173.5 | 97.9 | 121.7 | 105.7 | 90.9 | 193.6 |

nadir angle/° | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

PVs | 3.20 | 1.57 | 1.68 | 0.73 | −0.75 | −1.16 | −1.66 |

STD | 2.22 | 1.13 | 1.01 | 0.75 | 0.39 | 0.29 | 0.41 |

nadir angle/° | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

PVs | −1.74 | −1.73 | −1.80 | −1.90 | −1.08 | 1.27 | 3.36 |

STD | 0.53 | 0.57 | 0.59 | 0.66 | 0.70 | 0.76 | 1.10 |

Type | z-offset | STD |
---|---|---|

CAST | 2274.22 | 35 |

SECM | 1520.94 | 51 |

C25 | 2341.41 | - |

C29 | 1695.62 | - |

PRN | S1 | S2 | S3 | ||||||
---|---|---|---|---|---|---|---|---|---|

A | C | R | A | C | R | A | C | R | |

C19 | 5.76 | 3.71 | 1.21 | 5.02 | 3.43 | 1.09 | 4.99 | 3.44 | 1.03 |

C20 | 6.06 | 3.64 | 1.36 | 5.35 | 3.48 | 1.18 | 5.04 | 3.51 | 1.14 |

C21 | 5.50 | 3.70 | 1.35 | 4.98 | 3.36 | 1.04 | 5.04 | 3.53 | 1.28 |

C22 | 5.60 | 3.73 | 1.20 | 5.09 | 3.50 | 1.14 | 4.86 | 3.40 | 1.15 |

C23 | 6.62 | 4.43 | 1.64 | 6.26 | 4.15 | 1.55 | 6.01 | 3.98 | 1.51 |

C24 | 6.83 | 4.51 | 1.64 | 6.49 | 4.54 | 1.59 | 6.34 | 4.44 | 1.47 |

C25 | 7.23 | 5.25 | 1.67 | 6.19 | 4.93 | 1.54 | 6.42 | 5.04 | 1.61 |

C26 | 6.74 | 4.52 | 1.47 | 6.56 | 4.50 | 1.44 | 7.08 | 4.60 | 1.53 |

C27 | 6.98 | 4.31 | 1.37 | 6.55 | 4.10 | 1.22 | 6.11 | 4.05 | 1.20 |

C28 | 6.63 | 3.89 | 1.18 | 5.26 | 3.58 | 0.88 | 5.12 | 3.55 | 0.85 |

C29 | 7.14 | 4.30 | 1.31 | 6.27 | 4.29 | 1.11 | 6.24 | 4.32 | 1.10 |

C30 | 6.47 | 4.39 | 1.21 | 5.63 | 4.27 | 1.11 | 5.87 | 4.23 | 1.11 |

C32 | 6.88 | 4.43 | 1.43 | 5.98 | 4.22 | 1.36 | 6.76 | 4.31 | 1.50 |

C33 | 6.79 | 4.59 | 1.47 | 6.53 | 4.35 | 1.39 | 6.71 | 4.50 | 1.53 |

C34 | 6.81 | 4.04 | 1.09 | 6.05 | 3.67 | 1.00 | 5.71 | 3.77 | 1.01 |

C35 | 10.16 | 6.34 | 2.76 | 10.20 | 6.55 | 2.68 | 9.52 | 6.16 | 2.61 |

Average | 6.76 | 4.36 | 1.46 | 6.15 | 4.18 | 1.33 | 6.11 | 4.18 | 1.35 |

PRN | S1 | S2 | S3 | S1-S2 | S1-S3 | |||||
---|---|---|---|---|---|---|---|---|---|---|

RMS | STD | RMS | STD | RMS | STD | RMS | STD | RMS | STD | |

C19 | 0.251 | 0.120 | 0.228 | 0.113 | 0.228 | 0.107 | 0.023 | 0.007 | 0.022 | 0.012 |

C20 | 0.255 | 0.114 | 0.232 | 0.115 | 0.216 | 0.105 | 0.024 | −0.001 | 0.040 | 0.009 |

C21 | 0.240 | 0.121 | 0.237 | 0.114 | 0.223 | 0.109 | 0.003 | 0.007 | 0.017 | 0.012 |

C22 | 0.249 | 0.126 | 0.241 | 0.125 | 0.236 | 0.113 | 0.008 | 0.001 | 0.013 | 0.013 |

C23 | 0.290 | 0.166 | 0.271 | 0.148 | 0.270 | 0.149 | 0.019 | 0.017 | 0.020 | 0.017 |

C24 | 0.275 | 0.149 | 0.264 | 0.134 | 0.264 | 0.140 | 0.011 | 0.015 | 0.011 | 0.009 |

C25 | 0.294 | 0.175 | 0.279 | 0.144 | 0.268 | 0.151 | 0.015 | 0.031 | 0.026 | 0.024 |

C26 | 0.301 | 0.174 | 0.281 | 0.136 | 0.266 | 0.141 | 0.020 | 0.038 | 0.035 | 0.033 |

C27 | 0.299 | 0.150 | 0.285 | 0.119 | 0.253 | 0.118 | 0.014 | 0.031 | 0.046 | 0.032 |

C28 | 0.267 | 0.120 | 0.240 | 0.100 | 0.234 | 0.094 | 0.026 | 0.020 | 0.033 | 0.027 |

C29 | 0.284 | 0.146 | 0.255 | 0.129 | 0.258 | 0.121 | 0.029 | 0.017 | 0.026 | 0.025 |

C30 | 0.285 | 0.134 | 0.265 | 0.116 | 0.261 | 0.113 | 0.020 | 0.019 | 0.024 | 0.022 |

C32 | 0.293 | 0.156 | 0.286 | 0.146 | 0.278 | 0.144 | 0.007 | 0.010 | 0.015 | 0.012 |

C33 | 0.290 | 0.151 | 0.258 | 0.143 | 0.258 | 0.133 | 0.033 | 0.008 | 0.032 | 0.018 |

C34 | 0.272 | 0.150 | 0.273 | 0.126 | 0.258 | 0.120 | −0.001 | 0.024 | 0.014 | 0.030 |

C35 | 0.340 | 0.248 | 0.329 | 0.220 | 0.329 | 0.228 | 0.011 | 0.028 | 0.011 | 0.020 |

Average | 0.280 | 0.150 | 0.264 | 0.133 | 0.256 | 0.130 | 0.016 | 0.017 | 0.024 | 0.020 |

PRN | S1 | S2 | S3 | S1-S2 | S1-S3 |
---|---|---|---|---|---|

C19 | 0.254 | 0.244 | 0.238 | 0.010 | 0.015 |

C20 | 0.242 | 0.237 | 0.238 | 0.004 | 0.004 |

C21 | 0.302 | 0.299 | 0.297 | 0.003 | 0.006 |

C22 | 0.303 | 0.298 | 0.292 | 0.005 | 0.010 |

C23 | 0.252 | 0.237 | 0.233 | 0.016 | 0.020 |

C24 | 0.243 | 0.245 | 0.232 | -0.002 | 0.011 |

C25 | 0.258 | 0.250 | 0.236 | 0.008 | 0.021 |

C26 | 0.233 | 0.214 | 0.210 | 0.019 | 0.022 |

C27 | 0.213 | 0.203 | 0.187 | 0.010 | 0.026 |

C28 | 0.153 | 0.154 | 0.141 | 0.000 | 0.013 |

C29 | 0.172 | 0.168 | 0.165 | 0.004 | 0.007 |

C30 | 0.171 | 0.165 | 0.162 | 0.006 | 0.009 |

C32 | 0.196 | 0.192 | 0.190 | 0.004 | 0.006 |

C33 | 0.238 | 0.231 | 0.233 | 0.006 | 0.005 |

C34 | 0.269 | 0.221 | 0.218 | 0.048 | 0.051 |

C35 | 0.394 | 0.383 | 0.368 | 0.011 | 0.026 |

Average | 0.243 | 0.234 | 0.227 | 0.009 | 0.016 |

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**MDPI and ACS Style**

Yan, X.; Huang, G.; Zhang, Q.; Wang, L.; Qin, Z.; Xie, S.
Estimation of the Antenna Phase Center Correction Model for the BeiDou-3 MEO Satellites. *Remote Sens.* **2019**, *11*, 2850.
https://doi.org/10.3390/rs11232850

**AMA Style**

Yan X, Huang G, Zhang Q, Wang L, Qin Z, Xie S.
Estimation of the Antenna Phase Center Correction Model for the BeiDou-3 MEO Satellites. *Remote Sensing*. 2019; 11(23):2850.
https://doi.org/10.3390/rs11232850

**Chicago/Turabian Style**

Yan, Xingyuan, Guanwen Huang, Qin Zhang, Le Wang, Zhiwei Qin, and Shichao Xie.
2019. "Estimation of the Antenna Phase Center Correction Model for the BeiDou-3 MEO Satellites" *Remote Sensing* 11, no. 23: 2850.
https://doi.org/10.3390/rs11232850