# Assessment of Earth Gravity Field Models in the Medium to High Frequency Spectrum Based on GRACE and GOCE Dynamic Orbit Analysis

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## Abstract

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## 1. Introduction

## 2. Dynamic Orbit Determination

- Simultaneous numerical integration of the variational equations and the equation of motion is performed based on a simplified force model. In the present analysis, we consider only the gravity field model, truncated to degree and order of 50. The effects of the empirical parameters are also included at this step.
- The design matrix of the estimator is formed based on the solution of the variations equations.
- Numerical integration of the equation of motion is applied based on the full force model as summarized in Table 1.
- The observation equations are formed based on the numerical solution of the previous step.
- The parameter estimation is then applied. One or two iterations steps may be required.

#### 2.1. Degree-Wise Cumulative Approach

#### 2.2. Orbit Resonances and Order-Wise Analysis

## 3. GRACE and GOCE Orbit Analysis

#### 3.1. Data Preprocessing

#### 3.2. Empirical Parameters

#### 3.3. Results

#### 3.4. Order Wise Analysis

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Principal objectives of satellite orbit determination [31]. The current orbit analysis applies the approach of dynamic orbit determination in a frequency-wise iterative scheme focusing on the discrepancies occurred by the gravity field models.

**Figure 2.**RMS variances of KBR data residuals to the GRACE estimated orbits based on time-variable gravity field models. The term d-range refers to the approach of eliminating the KBR range bias based on sequential range differences.

**Figure 3.**RMS variances of K-band ranging (KBR) data residuals to the Gravity Field and Steady-State Ocean Circulation (GRACE) estimated orbits based on static gravity field models. The term d-range refers to the approach of eliminating the KBR range bias based on sequential range differences.

**Figure 4.**RMS variances of Gravity Field and Steady-State Ocean Circulation (GOCE) orbit residuals and orbit differences in radial component between estimated orbit and reduced-dynamic orbit.

**Figure 5.**RMS variances of the GRACE inter-satellite ranging data residuals with respect to the order of the spherical harmonic coefficients.

**Figure 6.**RMS variances of GOCE orbit residuals with respect to the order of the spherical harmonic coefficients.

Satellite | GOCE | GRACE-A/GRACE-B |
---|---|---|

Orbit arc length | 1 day | 1 day |

Date | 28 May 2010 | 17 November 2009 |

Earth Rotation | IERS Conventions 2010 [35] | IERS Conventions 2010 |

EOP | 08 C04 [36] | 08 C04 |

Integrator | Gauss–Jackson 12th order | Gauss-Jackson 12th order |

Start Integrator | RKN7(6)-8 [37] | RKN7(6)-8 |

Integration step | 10 s | 10 s |

Observations | Kinematic positions [38] | Dynamic positions |

Gravity Field Model | various | various |

Planetary Ephemeris | DE423 [39] | DE423 [39] |

Solid Earth Tides | IERS Conventions 2010 | IERS Conventions 2010 |

Ocean Tides | FES2004 [40] | FES2004 |

Non-gravitational forces | - | Accelerometry data |

Relativistic effects | IERS Conventions 2010 | IERS Conventions 2010 |

Empirical parameters | Bias, 1-CPR (along & cross-track) | Bias, 1-CPR (along & cross-track) |

External Control | Reduced-Dynamic orbit (SST_PRD_2) [38] | K-band data (KBR1B) |

**Table 2.**Satellite data and maximum degree (static and time-variable part) of the used gravity field models.

Gravity Field Models | Degree | Time-Variable Degree | Satellite Data |
---|---|---|---|

GOCO05s [21] | 280 | 100 | GOCE, GRACE, CHAMP, LAGEOS, LAGEOS 2, Starlette, Stella, Ajisai, LARETS |

EIGEN-6S4 (v2) [59] | 300 | 80 | GOCE, GRACE, LAGEOS |

ITSG-Grace2014k [60] | 200 | 100 | GRACE |

GGM05G [61] | 240 | - | GOCE, GRACE |

GO_CONS_GCF_2_DIR_R5 [62] | 300 | - | GOCE, GRACE, LAGEOS |

GO_CONS_GCF_2_DIR_R4 [62] | 260 | - | GOCE, GRACE, LAGEOS |

**Table 3.**K-band data residuals based on GRACE dynamic orbit determination. Range residuals are provided for the cases of the KBR bias estimation and elimination through sequential range differences.

d/o | GOCO05s | EIGEN-6S4 | ITSG-Grace2014k | GGM05G | GOCE_DIR_R5 |
---|---|---|---|---|---|

Range residuals: RMS (mm) | |||||

120 | 6.20 | 6.38 | 6.15 | 6.17 | 6.33 |

150 | 5.80 | 6.00 | 5.80 | 5.92 | 5.95 |

180 | 5.77 | 5.96 | 5.82 | 5.90 | 5.91 |

Range residuals (sequential-differences): RMS (μm) | |||||

120 | 22.73 | 23.02 | 22.64 | 22.52 | 23.07 |

150 | 21.30 | 21.82 | 21.19 | 21.54 | 22.02 |

180 | 21.11 | 21.61 | 21.06 | 21.37 | 21.82 |

Range-rate residuals: RMS (μm/sec) | |||||

120 | 2.37 | 2.40 | 2.36 | 2.35 | 2.40 |

150 | 2.24 | 2.28 | 2.22 | 2.26 | 2.30 |

180 | 2.22 | 2.26 | 2.21 | 2.24 | 2.28 |

**Table 4.**GRACE-A orbit residuals based on dynamic orbit determination. The results are expressed by RMS in cm.

d/o | GOCO05s | EIGEN-6S4 | ITSG-Grace2014k | GGM05G | GOCE_DIR_R5 |
---|---|---|---|---|---|

Orbit residuals 3-D | |||||

120 | 17.11 | 16.73 | 17.07 | 16.58 | 16.10 |

150 | 17.08 | 16.71 | 17.04 | 16.55 | 16.09 |

180 | 17.11 | 16.74 | 17.06 | 16.58 | 16.12 |

**Table 5.**GOCE orbit residuals based on dynamic orbit determination and kinematic orbit positions as observations. The statistics results are expressed by RMS in cm.

d/o | GOCO05s | EIGEN-6S4 | GOCE_DIR_R4 |
---|---|---|---|

Orbit residuals 3-D | |||

120 | 56.78 | 57.22 | 58.3 |

150 | 51.77 | 52.31 | 53.4 |

180 | 56.24 | 56.46 | 57.6 |

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**MDPI and ACS Style**

Papanikolaou, T.D.; Tsoulis, D. Assessment of Earth Gravity Field Models in the Medium to High Frequency Spectrum Based on GRACE and GOCE Dynamic Orbit Analysis. *Geosciences* **2018**, *8*, 441.
https://doi.org/10.3390/geosciences8120441

**AMA Style**

Papanikolaou TD, Tsoulis D. Assessment of Earth Gravity Field Models in the Medium to High Frequency Spectrum Based on GRACE and GOCE Dynamic Orbit Analysis. *Geosciences*. 2018; 8(12):441.
https://doi.org/10.3390/geosciences8120441

**Chicago/Turabian Style**

Papanikolaou, Thomas D., and Dimitrios Tsoulis. 2018. "Assessment of Earth Gravity Field Models in the Medium to High Frequency Spectrum Based on GRACE and GOCE Dynamic Orbit Analysis" *Geosciences* 8, no. 12: 441.
https://doi.org/10.3390/geosciences8120441