# A Second-Order Time-Difference Position Constrained Reduced-Dynamic Technique for the Precise Orbit Determination of LEOs Using GPS

^{1}

^{2}

^{*}

## Abstract

**:**

^{−5}ms

^{−2}) acting on the LEOs are ignored. Therefore, only using the Earth’s gravity model is good enough for the proposed RD_STP method. All unmodeled dynamic models (e.g., luni-solar gravitation, tide forces) are treated as the error sources of the STP pseudo-observation. In addition, there are no pseudo-stochastic orbit parameters to be estimated in the RD_STP method. Finally, we use the RD_STP method to process 15 days of GPS data from the GOCE mission. The results show that the accuracy of the RD_STP solution is more accurate and smoother than the kinematic solution in nearly polar and equatorial regions, and consistent with the RD solution. The 3D RMS of the differences between the RD_STP and RD solutions is 1.93 cm for 1 s sampling. This indicates that the proposed method has a performance comparable to the RD method, and could be an alternative for the POD of LEOs.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Preparation

#### 2.2. Kinematic Observation Equation

_{k}) at time epoch t can be described as follows [33]:

_{1}; and ${\epsilon}_{{P}_{i},L}^{{G}_{k}}$ and ${\epsilon}_{i,L}^{{G}_{k}}$ are the code and carrier phase observation errors (e.g., multipath), respectively. Generally, ${\overrightarrow{r}}^{{G}_{k}}$ and $d{t}^{{G}_{k}}$ can be obtained from the precise ephemeris and clock products released by CODE or the International GNSS Service (IGS) [34]. The relativistic effect and phase wind-up effect (due to the relative rotation between a transmitting and receiving antennas), which can be calculated using known correction models, are not shown in Equation (1). Moreover, for simplicity, subscript L, superscript G

_{k}, ${\rho}_{L,0}^{{G}_{k}}$, and $d{t}^{{G}_{k}}$ are omitted in some of the following equations, without causing confusion.

#### 2.3. New Dynamic Pseudo-Observation Equation

#### 2.4. Function Model of the RD_STP Method

#### 2.5. Characteristics of the RD_STP Method

^{−5}ms

^{−2}) can be taken into account for integrating the STPs into the proposed RD_STP method as well, if a stronger dynamic constraint on the POD solutions is desired. Finally, there are no dynamic parameters or pseudo-stochastic parameters to be estimated in the proposed method.

#### 2.6. Steps of the RD_STP Method for POD

## 3. Results

#### 3.1. Accuracy Analysis of the Integrated STP Pseudo-Observations

#### 3.1.1. Influence of the Sampling Interval

^{−5}ms

^{−2}, the error of the computed STP is less than 0.01 m, even for a sampling interval of 30 s. For a sampling interval of 10 s, the accuracy of the a priori accelerations should be better than 10

^{−4}ms

^{−2}. Fortunately, the errors or magnitudes of the accelerations of many dynamic models of LEOs are below 10

^{−4}ms

^{−2}, which will be discussed in the following section.

#### 3.1.2. Magnitude and Error of the a Priori Dynamic Models

^{−6}ms

^{−2}. Therefore, even if we ignore these known conservative force models, the influence on the computed STP is less than 1 mm for a sampling interval of less than 30 s.

^{−5}ms

^{−2}, it cannot be ignored in computing the STP for an accuracy requirement of 0.01 m. However, the maximum degree of the EGM, which could satisfy the accuracy requirement of computing the STP, should be analyzed. We use the EGM2008 (up to degree and order 180) [41] as the reference model to analyze the omission errors of the accelerations for different maximum degrees (60, 90, 120, and 150), which are shown in Figure 4. According to Figure 4, the omission errors of the EGM2008 model up to degree and order 90 are less than 10

^{−5}ms

^{−2}for GOCE, which indicates that the EGM up to degree and order 90 is sufficient for computing the STPs with an accuracy of 0.01 m when Δt

**=**30 s.

^{−5}ms

^{−2}. Therefore, for integrating the STPs with an accuracy of 0.01 m, the non-conservative forces acting on the LEOs can be ignored as well, which usually cannot be neglected in dynamic or RD orbit determinations [42,43].

#### 3.1.3. Accuracy Analysis of Integrating the STPs of GOCE and GRACE

**,**the differences between the results that correspond to EIGEN5S and EGM2008 are negligible, even with a sampling interval of 300 s and a maximum degree and order up to 150. The RMS of the STP errors increases as the sampling interval increases. Further, although the RMS of the STP errors decreases as the degree increases, the reduction after 90 degrees is very limited, consistent with the results shown in Figure 4. Therefore, only EGM2008 up to degree and order 90 is set as the only a priori dynamic model in the real data processing.

^{−5}ms

^{−2}. Therefore, we set ${\sigma}_{{\ddot{\overrightarrow{r}}}_{M}}={10}^{-5}{\mathrm{ms}}^{-2}$. Based on Equation (3), the corresponding error of $\Delta \Delta d{\overrightarrow{r}}_{L}$ with a sampling interval of 30 s is 9 mm, which is much larger than the “true” errors shown in Figure 5. Thus, the assumption of a stochastic model for $\Delta \Delta d{\overrightarrow{r}}_{L}$ is moderate, which will reduce the influence of the a priori dynamic model on the RD_STP solution.

#### 3.2. POD of GOCE Satellite Based on the RD_STP Method

#### POD Results and Analysis

^{−5}ms

^{−2}. According to Figure 9, the differences are less than 0.5 mm for the R, T, and N directions, illustrating that the RD_STP POD solutions are not so sensitive to the a priori gravity field model, and the assumption of the stochastic model for the STP pseudo-observation is reasonable. This outcome is consistent with the conclusion given in Section 3.1.3. Moreover, the differences in the radial direction are larger than the ones in the other directions, which is reasonable because the acceleration differences between EGM2008 and EIGEN5S in the radial direction are biggest.

## 4. Conclusions

^{−5}ms

^{−2}over a short time span (e.g., $2\Delta t=60\mathrm{s}$), the accuracy of the integrated STPs is better than 0.01 m, which is good enough for the POD of LEOs. Therefore, compared with the traditional dynamic/RD method, the minor perturbation forces with a magnitude less than 10

^{−5}ms

^{−2}(such as luni-solar gravitation, tidal forces, and atmospheric drag) acting on the satellite can be ignored and treated as the error sources of the STP pseudo-observation. Further, only the Earth’s gravity field model (such as EGM2008, EIGEN5S) up to degree and order 90 is sufficient for the POD because its omission errors are less than 10

^{−5}ms

^{−2}. This outcome means that the dependence of the RD_STP method on Earth’s gravity field model is reduced compared with the traditional dynamic/RD method.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Bertiger, W.I.; Bar-Sever, Y.E.; Christensen, E.J.; Davis, E.S.; Guinn, J.R.; Haines, B.J.; Ibanez-Meier, R.W.; Jee, J.R.; Lichten, S.M.; Melbourne, W.G.; et al. GPS precise tracking of TOPEX/POSEIDON: Results and implications. J. Geophys. Res. Space Phys.
**1994**, 99, 24449–24464. [Google Scholar] [CrossRef] - Luthcke, S.B.; Zelensky, N.P.; Rowlands, D.D.; Lemoine, F.G.; Williams, T.A. The 1-Centimeter Orbit: Jason-1 Precision Orbit Determination Using GPS, SLR, DORIS, and Altimeter Data Special Issue: Jason-1 Calibration/Validation. Mar. Geod.
**2003**, 26, 399–421. [Google Scholar] [CrossRef] - Cerri, L.; Berthias, J.P.; Bertiger, W.I.; Haines, B.J.; Lemoine, F.G.; Mercier, F.; Ries, J.C.; Willis, P.; Zelensky, N.P.; Ziebart, M. Precision Orbit Determination Standards for the Jason Series of Altimeter Missions. Mar. Geod.
**2010**, 33, 379–418. [Google Scholar] [CrossRef] - Flohrer, C.; Otten, M.; Springer, T.; Dow, J. Generating precise and homogeneous orbits for Jason-1 and Jason-2. Adv. Space Res.
**2011**, 48, 152–172. [Google Scholar] [CrossRef] - Bisnath, S. Precise Orbit Determination of Low Earth Orbiters with a Single GPS Receiver-Based. Geometric Strategy. Ph.D. Dissertation, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, NB, Canada, 2004; 143p. [Google Scholar]
- Jäggi, A.; Hugentobler, U.; Beutler, G. Pseudo-Stochastic Orbit Modeling Techniques for Low-Earth Orbiters. J. Geod.
**2006**, 80, 47–60. [Google Scholar] [CrossRef] [Green Version] - Jäggi, A.; Beutler, G.; Bock, H.; Hugentobler, U. Kinematic and highly reduced-dynamic LEO orbit determination for gravity field estimation. In International Symposium on Earth and Environmental Sciences for Future Generations; Springer Science and Business Media LLC: Berlin/Heidelberg, Germany, 2008; pp. 354–361. [Google Scholar]
- Jäggi, A.; Hugentobler, U.; Bock, H.; Beutler, G. Precise orbit determination for GRACE using undifferenced or doubly differenced GPS data. Adv. Space Res.
**2007**, 39, 1612–1619. [Google Scholar] [CrossRef] - Jäggi, A.; Dach, R.; Montenbruck, O.; Hugentobler, U.; Bock, H.; Beutler, G. Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination. J. Geod.
**2009**, 83, 1145–1162. [Google Scholar] [CrossRef] [Green Version] - Montenbruck, O.; Andres, Y.; Bock, H.; Van Helleputte, T.; Ijssel, J.V.D.; Loiselet, M.; Marquardt, C.; Silvestrin, P.; Visser, P.; Yoon, Y. Tracking and orbit determination performance of the GRAS instrument on MetOp-A. GPS Solut.
**2008**, 12, 289–299. [Google Scholar] [CrossRef] [Green Version] - Bock, H.; Jäggi, A.; Meyer, U.; Visser, P.; Ijssel, J.V.D.; Van Helleputte, T.; Heinze, M.; Hugentobler, U. GPS-derived orbits for the GOCE satellite. J. Geod.
**2011**, 85, 807–818. [Google Scholar] [CrossRef] [Green Version] - Bock, H.; Jaggi, A.; Beutler, G.; Meyer, U. GOCE: Precise orbit determination for the entire mission. J. Geod.
**2014**, 88, 1047–1060. [Google Scholar] [CrossRef] - Ijssel, J.V.D.; da Encarnacao, J.T.; Doornbos, E.; Visser, P. Precise science orbits for the Swarm satellite constellation. Adv. Space Res.
**2015**, 56, 1042–1055. [Google Scholar] [CrossRef] - Jäggi, A.; Dahle, C.; Arnold, D.; Bock, H.; Meyer, U.; Beutler, G.; Ijssel, J.V.D. Swarm kinematic orbits and gravity fields from 18 months of GPS data. Adv. Space Res.
**2016**, 57, 218–233. [Google Scholar] [CrossRef] - Montenbruck, O.; Hackel, S.; Ijssel, J.V.D.; Arnold, D. Reduced dynamic and kinematic precise orbit determination for the Swarm mission from 4 years of GPS tracking. GPS Solut.
**2018**, 22, 79. [Google Scholar] [CrossRef] - Montenbruck, O.; Hackel, S.; Jäggi, A. Precise orbit determination of the Sentinel-3A altimetry satellite using ambiguity-fixed GPS carrier phase observations. J. Geod.
**2018**, 92, 711–726. [Google Scholar] [CrossRef] [Green Version] - Švehla, D.; Rothacher, M. Kinematic positioning of LEO and GPS satellites and IGS stations on the ground. Adv. Space Res.
**2005**, 36, 376–381. [Google Scholar] [CrossRef] - Geng, J.; Teferle, F.N.; Meng, X.; Dodson, A.H. Kinematic precise point positioning at remote marine platforms. GPS Solut.
**2010**, 14, 343–350. [Google Scholar] [CrossRef] - Li, J.; Zhang, S.; Zou, X.; Jiang, W. Precise orbit determination for GRACE with zero-difference kinematic method. Chin. Sci. Bull.
**2010**, 55, 600–606. [Google Scholar] [CrossRef] - Weinbach, U.; Schön, S. Improved GRACE kinematic orbit determination using GPS receiver clock modeling. GPS Solut.
**2013**, 17, 511–520. [Google Scholar] [CrossRef] - Baur, O.; Bock, H.; Höck, E.; Jäggi, A.; Krauss, S.; Mayer-Gürr, T.; Reubelt, T.; Siemes, C.; Zehentner, N. Comparison of GOCE-GPS gravity fields derived by different approaches. J. Geod.
**2014**, 88, 959–973. [Google Scholar] [CrossRef] - Zehentner, N.; Mayer-Gürr, T. Precise orbit determination based on raw GPS measurements. J. Geod.
**2016**, 90, 275–286. [Google Scholar] [CrossRef] [Green Version] - Montenbruck, O.; van Helleputte, T.; Kroes, R.; Gill, E. Reduced dynamic orbit determination using GPS code and carrier measurements. Aerosp. Sci. Technol.
**2005**, 9, 261–271. [Google Scholar] [CrossRef] - Švehla, D.; Rothacher, M. Kinematic and reduced-dynamic precise orbit determination of low earth orbiters. Adv. Geosci.
**2003**, 1, 47–56. [Google Scholar] [CrossRef] [Green Version] - Colombo, O.L. The dynamics of global positioning system orbits and the determination of precise ephemerides. J. Geophys. Res. Space Phys.
**1989**, 94, 9167. [Google Scholar] [CrossRef] - Beutler, G.; Brockmann, E.; Gurtner, W.; Hugentobler, U.; Mervart, L.; Rothacher, M. Extended Orbit Modeling Techniques at the CODE Processing Center of the International GPS Service for Geodynamics (IGS): Theory and Initial Re-sults. Manuscr. Geod.
**1994**, 19, 367–386. [Google Scholar] - Yunck, T.; Wu, S.-C.; Wu, J.-T.; Thornton, C. Precise tracking of remote sensing satellites with the Global Positioning System. IEEE Trans. Geosci. Remote Sens.
**1990**, 28, 108–116. [Google Scholar] [CrossRef] - Wu, S.C.; Yunck, T.P.; Thornton, C.L. Reduced-dynamic technique for precise orbit determination of low earth satel-lites. J. Guid. Control Dyn.
**1991**, 14, 24–30. [Google Scholar] [CrossRef] - Bruinsma, S.; Loyer, S.; Lemoine, J.M.; Perosanz, F.; Tamagnan, D. The impact of accelerometry on CHAMP orbit determination. J. Geod.
**2003**, 77, 86–93. [Google Scholar] [CrossRef] - Visser, P.; Ijssel, J.V.D. Aiming at a 1-cm Orbit for Low Earth Orbiters: Reduced-Dynamic and Kinematic Precise Orbit Determination. Space Sci. Rev.
**2003**, 108, 27–36. [Google Scholar] [CrossRef] - ESA. GOCE l1b Products User Handbook; GOCE-GSEG-EOPGTN-06-0137; Tech. Rep.; 2008. Available online: https://earth.esa.int/c/document_library/get_file?folderId=14168&name=DLFE-772.pdf (accessed on 13 April 2018).
- ESA. GOCE Level 2 Product Data Handbook; go-ma-hpf-gs-0110; Tech. Rep.; 2014. Available online: https://earth.esa.int/documents/10174/1650485/GOCE_Product_Data_Handbook_Level-2 (accessed on 13 April 2018).
- Hauschild, A. Basic observation equations. In Springer Handbook of Global Navigation Satellite Systems, Chapter 19; Teunissen, P., Montenbruck, O., Eds.; Springer: Berlin/Heidelberg, Germany, 2017; pp. 561–582. [Google Scholar]
- Johnston, G.; Riddell, A.; Hausler, G. The International GNSS Service. In Springer Handbook of Global Navigation Satellite Systems; Teunissen, P.J.G., Montenbruck, O., Eds.; Springer International Publishing: Cham, Switzerland, 2017; Volume 1, pp. 967–982. [Google Scholar] [CrossRef]
- Ditmar, P.; Sluijs, A.A.V.E.V.D. A technique for modeling the Earths gravity field on the basis of satellite accelerations. J. Geod.
**2004**, 78, 12–33. [Google Scholar] [CrossRef] - Beutler, G. Methods of Celestial Mechanics; Springer: Berlin/Heidelberg, Germny, 2004. [Google Scholar]
- Bock, H.; Jäggi, A.; Švehla, D.; Beutler, G.; Hugentobler, U.; Visser, P. Precise orbit determination for the GOCE satellite using GPS. Adv. Space Res.
**2007**, 39, 1638–1647. [Google Scholar] [CrossRef] - Case, K.; Kruizinga, G.; Wu, S.C. GRACE Level 1B Data Product User Handbook JPL Publication D-22027. 2010. Available online: https://earth.esa.int/c/document_library/get_file?folderId=123371&name=DLFE-1408.pdf (accessed on 17 April 2018).
- Montenbruck, O.; Garcia-Fernandez, M.; Williams, J. Performance comparison of semicodeless GPS receivers for LEO satellites. GPS Solut.
**2006**, 10, 249–261. [Google Scholar] [CrossRef] - Petit, G.; Luzum, B. IERS Conventions (2010) (No. IERS-TN-36); Bureau International Des Poids Et Mesures Sevres: Sèvres, France, 2010. [Google Scholar]
- Pavlis, N.K.; Holmes, S.A.; Kenyon, S.C.; Factor, J.K. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J. Geophys. Res. Space Phys.
**2012**, 117. [Google Scholar] [CrossRef] [Green Version] - Ijssel, J.V.D.; Visser, P. Determination of non-gravitational accelerations from GPS satellite-to-satellite tracking of CHAMP. Adv. Space Res.
**2005**, 36, 418–423. [Google Scholar] [CrossRef] - Beutler, G.; Jäggi, A.; Mervart, L.; Meyer, U. The celestial mechanics approach: Application to data of the GRACE mission. J. Geod.
**2010**, 84, 661–681. [Google Scholar] [CrossRef] [Green Version] - Förste, C.; Flechtner, F.; Schmidt, R.; Stubenvoll, R.; Rothacher, M.; Kusche, J.; Neumayer, H.; Biancale, R.; Lemoine, J.M.; Barthelmes, F.; et al. EIGEN-GL05C—A New Global Combined High-Resolution GRACE-Based Gravity Field Model of the GFZ-GRGS Cooperation; General Assembly European Geosciences Union: Vienna, Austria, April 2008; Geophys Res Abstr 10: EGU2008-A-03426. [Google Scholar]

**Figure 2.**Processing scheme of the reduced-dynamic second-order time-difference position (RD_STP) method for the POD of LEOs from GPS observations.

**Figure 3.**(

**a**–

**f**) One-day accelerations (in X/Y/Z direction) from the known conservative force models for the gravity field and steady-state ocean circulation explorer (GOCE) satellite: moon (

**a**), sun (

**b**), ocean tide (

**c**), ocean pole tide (

**d**), solid tide (

**e**), and solid pole tide (

**f**).

**Figure 4.**(

**a**–

**d**) Omission errors of GOCE’s accelerations (in X/Y/Z direction) of gravity field model EGM2008 up to different maximum degrees and orders: 150 (

**a**), 120 (

**b**), 90 (

**c**), and 60 (

**d**), compared with the model up to degree and order 180.

**Figure 5.**Errors of STPs (in X/Y/Z direction) integrated with EGM2008 up to degree and order 90 with a 30 s sampling interval for LEOs: GOCE and GRACE A/B, on 25 November 2009. The RD POD solutions are used as the true orbit.

**Figure 6.**Three-dimensional root mean square (3D RMS) of the GOCE’s integrated STP errors corresponding to two different models (EIGEN5S and EGM2008) with different maximum degrees and orders (30, 40, 50, 60, 70, 80, 90, 120, and 150) and different sampling intervals (30, 60, 90, 120, 180, 240, and 300 s). The RD POD solutions are used as the true orbit.

**Figure 7.**(

**a**–

**c**) Differences between the RD_STP, KIN, and RD (the reference orbit) POD solutions: R (

**a**), T (

**b**), and N (

**c**), at day of year (DOY) 330, 2009.

**Figure 8.**Daily mean and RMS of the differences between the RD_STP and the RD (the reference orbit) POD solutions for DOY 318 to 332 of 2009.

**Figure 9.**Differences of the RD_STP solutions based on the EGM2008 and EIGEN5S models up to 90 d/o, at DOY 330 of 2009.

**Figure 10.**Radial orbit differences (compared with the RD solutions) of the KIN solutions and the RD_STP solutions based on the different stochastic models of the STP pseudo-observations at DOY 319 of 2009 (top). Zoomed view in 3 h (bottom).

**Table 1.**Characteristics of the precise orbit determination (POD) methods of low Earth orbiters (LEOs).

Kinematic | Dynamic | RD | RD_STP | |
---|---|---|---|---|

GPS observations | Yes | Yes | Yes | Yes |

Integration and integral arc length | No | Yes | Yes, usually 30 h [11] | Yes, 60 s or less |

Minor perturbation force models | No | Yes | Yes | Usually no |

Earth’s gravity field model | No | Yes | Yes | Yes, lower-degree |

Pseudo-stochastic/dynamical parameters | No | Yes | Yes | No |

**Table 2.**Summary of dynamical and measurement models that are employed for the orbit determination of GOCE.

Items | Bernese GPS Software KIN and RD POD [11] | RD_STP POD |
---|---|---|

GPS measurement model | Undifferenced ionosphere-free phase | Undifferenced ionosphere-free phase |

igs05.atx | igs05.atx | |

GOCE PCOs + PCVs | GOCE PCOs + PCVs | |

CODE final GPS ephemerides and 5 s clocks | CODE final GPS ephemerides and 5 s clocks | |

Elevation cut-off 0° | Elevation cut-off 0° | |

10 s/1 s (RD/KIN) sampling | 1 s sampling | |

Gravitational forces | EIGEN-5S (120 × 120) [44] Solid Earth, pole and ocean tides luni-solar-planetary gravity N/A for KIN PSO | EGM2008(90 × 90) [41] |

Non-gravitational forces | Empirical constant N/A for KIN PSO | N/A |

Estimation | Batch least squares | Batch least squares |

**Table 3.**Mean RMS of the differences between the solutions (KIN-10 s, KIN-30 s, RD_STP-10 s, and RD_STP-30 s) and the RD solutions for DOY 318–332, 2009. RD_STP solutions are with the stochastic models ${\sigma}_{\Delta \Delta d{r}_{L}}={10}^{-5}\Delta {t}^{2}(\mathrm{m})$.

Mean-RMS (cm) | R | T | N | 3D |
---|---|---|---|---|

KIN-10 s | 2.63 | 2.28 | 2.42 | 4.25 |

KIN-30 s | 2.74 | 2.42 | 2.55 | 4.47 |

STPRD-10 s | 1.52 | 1.33 | 1.50 | 2.53 |

STPRD-30 s | 1.69 | 1.46 | 1.63 | 2.77 |

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**MDPI and ACS Style**

Wei, H.; Li, J.; Xu, X.; Zhang, S.; Kuang, K.
A Second-Order Time-Difference Position Constrained Reduced-Dynamic Technique for the Precise Orbit Determination of LEOs Using GPS. *Remote Sens.* **2021**, *13*, 3033.
https://doi.org/10.3390/rs13153033

**AMA Style**

Wei H, Li J, Xu X, Zhang S, Kuang K.
A Second-Order Time-Difference Position Constrained Reduced-Dynamic Technique for the Precise Orbit Determination of LEOs Using GPS. *Remote Sensing*. 2021; 13(15):3033.
https://doi.org/10.3390/rs13153033

**Chicago/Turabian Style**

Wei, Hui, Jiancheng Li, Xinyu Xu, Shoujian Zhang, and Kaifa Kuang.
2021. "A Second-Order Time-Difference Position Constrained Reduced-Dynamic Technique for the Precise Orbit Determination of LEOs Using GPS" *Remote Sensing* 13, no. 15: 3033.
https://doi.org/10.3390/rs13153033