# Colocation in Time and Space of High-Precision Two-Way Optical and Microwave Observations for Calibration of a Microwave Ranging Link—The ACES Mission Case

^{*}

## Abstract

**:**

## 1. Motivation

#### Background

## 2. Simulation

#### 2.1. Simulation Software

#### 2.2. Clock and Time Distribution

#### 2.3. Orbit Propagation

#### 2.4. Atmosphere

#### 2.4.1. Ray-Tracing Technique

#### 2.4.2. Turbulence

#### 2.4.3. Dispersive Effects

#### 2.5. Data Set

## 3. Calibration Strategy

#### 3.1. Basic Workflow

#### 3.2. Optimum Approach

## 4. Assessment of Parameter Estimation

#### 4.1. Data Set

#### 4.2. A Priori Errors of the LSA Process

#### 4.3. “Best Possible” Test—Orbit

#### 4.4. “Best Possible” Test—Troposphere

## 5. Results and Discussion

#### 5.1. Optimum Approach

#### 5.2. Residual Analysis

#### 5.3. Time Synchronization Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACES | Atomic Clock Ensemble in Space |

AHM | Active Hydrogen Maser |

CNES | Centre national d’études spatiales |

ECMWF | European Centre for Medium-Range Weather Forecasts |

ELT | European Laser Timing |

ERA | ECMWF’s new atmospheric reanalysis |

ESA | European Space Agency |

GGOS | Global Geodetic Observing System |

IAG | International Association of Geodesy |

IERS | International Earth Rotation and Reference Systems Service |

ISS | International Space Station |

ITU-R | International Telecommunication Union—Radiocommunication Sector |

LNE-SYRTE | Laboratoire national de métrologie et d’essais—Système de Références Temps-Espace |

LSA | Least-Squares Adjustment |

MWL | Microwave Link System |

NOAA | National Oceanic and Atmospheric Administration |

NWM | Numerical Weather Model |

OD | Orbit Determination |

OmC | Observed minus Computed |

SLR | Satellite Laser Ranging |

SHM | Space Hydrogen Maser |

STEC | Slant Total Electron Content |

TDS | Timing Distribution System |

TEC | Total Electron Content |

TLE | Two-Line Elements |

VMFo | Vienna Mapping Function 3—Optical |

VMF3 | Vienna Mapping Function 3 |

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**Figure 3.**“Best Possible” test—Orbit. A posteriori residuals of 100 passes. Zero minutes at the x-axis corresponds to satellite culmination with respect to the ground station.

**Figure 4.**“Best Possible” test—Troposphere. Plots showing a posteriori errors of all passes. Zero minutes on the x-axis corresponds to satellite’s culmination with respect to the ground station. MWL a posteriori are shown in blue, ELT a posteriori in yellow.

**Figure 5.**The columns show the parameter differences and the OmC residuals of the four methods for all passes, where the tropospheric and range differences are presented in the first two rows and the MWL-Downlink OmC residuals are given in the last row. Zero minutes on the x-axis corresponds to the culmination of the satellite with respect to the ground station.

**Figure 6.**Boxplots of gradient contribution perpendicular to and along the orbit. The blue boxes contains the inner 50 percent of the contribution, where the red crosses show outliers.

**Figure 7.**In red: pass-wise differences between the estimated and simulated offsets of the clock differences (denoted as Est). In blue: pass-wise differences between the calculated and simulated offsets of the clock differences (denoted as Cal).

**Figure 8.**The figure shows the Allan deviation of the deviations of the estimated electronic delays in the downlink direction for methods M-2, M-3, and M-4. The units provided are picoseconds.

Parameter | Model | MWL | ELT |
---|---|---|---|

Troposphere | GFS [20] | X | X |

Orbit | TLE | X | X |

Ionosphere | NeQuickG [21] | X | - |

Clocks | Colored-Noise | X | X |

TDS Offset | Intertechnique Offset | - | X |

TDS Noise | Intertechnique Colored Noise | - | X |

Local Ties | Height Offset | - | X |

Sagnac | 1st & 2nd Order | X | X ^{1} |

Shapiro | - | X | X |

Measurement Noise | White Noise | X | X |

Electronic Delay | Static | X | - |

^{1}Sagnac effect is only relevant for ELT One-Way, not for Two-Way.

**Table 2.**Composition of noise for ACES and groundstation clocks as well as TDS and their respective magnitudes at 1 s.

Noise Type | ACES | Ground | TDS |
---|---|---|---|

White Phase | - | - | $1\times {10}^{-12}$ |

Flicker Phase | $1\times {10}^{-13}$ | $4\times {10}^{-13}$ | $6\times {10}^{-14}$ |

White Frequency ^{1} | $3\times {10}^{-14}$ | $3\times {10}^{-14}$ | - |

Flicker Frequency | $1\times {10}^{-15}$ | - | - |

Random Walk Frequency | - | $3\times {10}^{-17}$ | - |

^{1}$1\times {10}^{-13}$ for PHARAO. Other parameters are for SHM only.

Parameter | Description | LSA | Model |
---|---|---|---|

Troposphere ^{1} | VMF3 and VMFo | X | X |

Orbit | Common short arc | X | - |

Ionosphere | Corrected via STEC | - | X |

Clocks | Offset | X | - |

Sagnac | Based on initial orbit | - | X |

Shapiro | Based on initial orbit | - | X |

Electronic delays | Parameters of interest | X | - |

^{1}Estimated for MWL, corrected for ELT.

Method | MWL | ELT One-Way | ELT Two-Way | Wet-Gradient ^{1} |
---|---|---|---|---|

1 | X | - | - | - |

2 | X | X | - | - |

3 | X | X | X | - |

4 | X | X | X | X |

^{1}MWL only.

Parameter | 25th Percentile | Median | 75th Percentile | RMS |
---|---|---|---|---|

Radial | 1.05 | 4.83 | 8.32 | 7.97 |

Along-Track | 6.24 | 16.93 | 26.37 | 20.58 |

Cross-Track | −3.39 | 2.43 | 3.95 | 3.85 |

Range | −14.23 | −1.48 | 13.70 | 17.46 |

Trp-ELT | 0.06 | 0.19 | 0.37 | 0.42 |

Trp-MWL | 1.01 | 2.93 | 6.03 | 6.58 |

Ion-Down | −0.001 | −0.00006 | 0.001 | 0.002 |

Ion-Up | −0.002 | 0.00009 | 0.002 | 0.002 |

Range | 25th Percentile | Median | 75th Percentile | RMS |
---|---|---|---|---|

a priori | −14.235 | −1.475 | 13.703 | 17.461 |

a posteriori | −0.014 | −0.003 | 0.012 | 0.017 |

Troposphere | 25th Percentile | Median | 75th Percentile | RMS |
---|---|---|---|---|

a priori | 1.01 | 2.93 | 6.03 | 6.58 |

a posteriori | −0.18 | 0.02 | 0.23 | 0.49 |

**Table 8.**Characteristics of the electronic delay differences in the downlink and uplink directions as well as the clock differences of all passes. The means and standard deviations were calculated from the differences between the estimated and simulated parameters. The formal errors shown are the means of all formal errors from the least squares adjustment of each method. The units used are picoseconds [ps].

Mean | Mean Formal Errors | Standard Deviation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameter | M-1 | M-2 | M-3 | M-4 | M-1 | M-2 | M-3 | M-4 | M-1 | M-2 | M-3 | M-4 |

Downlink | - | 4.7 | 3.9 | 1.3 | - | 7.1 | 2.0 | 2.0 | - | 249 | 55 | 49 |

Uplink | - | 0.7 | 0.8 | −0.6 | - | 2.0 | 1.7 | 1.5 | - | 48 | 29 | 27 |

Clock | 950 | 2.1 | 1.6 | 1.1 | 0.08 | 4.2 | 1.7 | 1.5 | 0.01 | 143 | 14 | 12 |

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**MDPI and ACS Style**

Vollmair, P.; Schlicht, A.; Hugentobler, U.
Colocation in Time and Space of High-Precision Two-Way Optical and Microwave Observations for Calibration of a Microwave Ranging Link—The ACES Mission Case. *Remote Sens.* **2023**, *15*, 4897.
https://doi.org/10.3390/rs15204897

**AMA Style**

Vollmair P, Schlicht A, Hugentobler U.
Colocation in Time and Space of High-Precision Two-Way Optical and Microwave Observations for Calibration of a Microwave Ranging Link—The ACES Mission Case. *Remote Sensing*. 2023; 15(20):4897.
https://doi.org/10.3390/rs15204897

**Chicago/Turabian Style**

Vollmair, Peter, Anja Schlicht, and Urs Hugentobler.
2023. "Colocation in Time and Space of High-Precision Two-Way Optical and Microwave Observations for Calibration of a Microwave Ranging Link—The ACES Mission Case" *Remote Sensing* 15, no. 20: 4897.
https://doi.org/10.3390/rs15204897