# Estimation of Antarctic Ice Sheet Thickness Based on 3D Density Interface Inversion Considering Terrain and Undulating Observation Surface Simultaneously

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Forward Modeling Considering Terrain and UOS

#### 2.2. Inverse Ice–Rock Interface under Terrain and UOS

## 3. Synthetic Examples

#### 3.1. Simple Synthetic Example

#### 3.2. Complex Synthetic Example

^{3}. The gravity anomaly of this group model (Figure 6e) is calculated at the nodes of a $21\times 21$ grid with a 500 m grid step. And the relationship between the gravity anomaly and ice sheet thickness is also invisible.

## 4. Real Data Application

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Interpretation model of an ice sheet, considering terrain and UOSs. The UOSs are drawn with green lines, and the solid black dots represent the observed points. The ice sheet is divided into a series of juxtaposed vertical prisms in blue, and the parameter to be calculated is the elevation of the prisms’ bottom.

**Figure 3.**The models and gravity anomaly in synthetic example 1: (

**a**) The elevation of observation surface. (

**b**) The elevation of terrain. (

**c**) The elevation of ice–rock interface. (

**d**) The ice sheet thickness. (

**e**) The theoretical gravity anomaly due to these models.

**Figure 4.**The results of the simple models obtained by the conventional method: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the inverted thickness and the true value.

**Figure 5.**The results of the simple models obtained by the proposed method: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the inverted thickness and the true value.

**Figure 6.**The models and gravity anomaly in synthetic example 2: (

**a**) The elevation of observation surface. (

**b**) The elevation of terrain. (

**c**) The elevation of ice–rock interface. (

**d**) The ice sheet thickness. (

**e**) The theoretical gravity anomaly due to these models.

**Figure 7.**The results of the complex models obtained by the conventional method: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the inverted ice thickness and the true value.

**Figure 8.**The results of the complex models obtained by the proposed method: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the inverted thickness and the true value.

**Figure 9.**Location of the study area in Antarctica. The base map is the elevation of the Antarctic topography. The study area is bounded by a green rectangle.

**Figure 10.**The airborne geophysical data from the AGAP Project. Aerogravity data: (

**a**) the residual gravity anomaly caused by the ice–rock interface; (

**b**) the aircraft altitude. Radio-echo sounding data: (

**c**) ice surface elevation, (

**d**) ice bed elevation, and (

**e**) ice thickness.

**Figure 11.**The results obtained by the conventional method in the survey area: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the estimated ice thickness and the radar-derived ice thickness.

**Figure 12.**The results obtained by the proposed method in the survey area: (

**a**) The calculated gravity anomaly. (

**b**) The residuals of data fitting. (

**c**) The estimated ice thickness. (

**d**) The residuals between the inverted thickness and the radar-derived value.

Type | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Mean Value (m/mGal) |
---|---|---|---|

observation surface | 1233.0 | 2487.6 | 1487.9 |

terrain | 1016.1 | 1686.8 | 1142.9 |

ice–rock interface | −646.0 | 758.6 | 526.1 |

ice sheet thickness | 257.5 | 2230.5 | 616.8 |

gravity anomaly | −64.65 | −10.80 | −34.37 |

**Table 2.**Statistical misfit comparison between w/o considering the terrain and UOSs of synthetic example 1.

Result | RMSE (m/mGal) | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Minimum Error (m/mGal) | Maximum Error (m/mGal) |
---|---|---|---|---|---|

Without: gravity anomaly | 0.03 | −64.55 | −10.80 | −0.15 | 0.12 |

Without: ice thickness | 232.2 | 147.1 | 1357.5 | −1622.5 | 15.5 |

With: gravity anomaly | 0.01 | −64.59 | −10.80 | −0.03 | 0.06 |

With: ice thickness | 17.2 | 257.7 | 2109.0 | −151.3 | 44.60 |

Type | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Mean Value (m/mGal) |
---|---|---|---|

observation surface | 1586.0 | 4088.0 | 2245.7 |

terrain | 428.0 | 3338.0 | 1426.7 |

ice–rock interface | −929.7 | 1470.0 | 149.6 |

ice sheet thickness | 46.2 | 3373.0 | 1277.1 |

gravity anomaly | −108.73 | −9.87 | −60.30 |

**Table 4.**Statistical misfit comparison between w/o considering the terrain and UOSs of synthetic example 2.

Result | RMSE (m/mGal) | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Minimum Error (m/mGal) | Maximum Error (m/mGal) |
---|---|---|---|---|---|

Without: gravity anomaly | 0.44 | −108.56 | −9.88 | −2.01 | 1.93 |

Without: ice thickness | 491.4 | 118.1 | 2299.8 | −1858.2 | 627.9 |

With: gravity anomaly | 0.01 | −108.73 | −9.88 | −0.03 | 0.03 |

With: ice thickness | 38.9 | 45.8 | 3221.8 | −171.4 | 138.7 |

Type | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Mean Value (m/mGal) |
---|---|---|---|

residual gravity anomaly | −250.11 | −93.60 | −187.87 |

aircraft altitude | 2221.2 | 3095.0 | 2727.0 |

ice surface elevation | 2056.5 | 2741.1 | 2433.9 |

radar-derived ice bed elevation | −1484.7 | 1249.7 | −292.5 |

radar-derived ice thickness | 1178.1 | 3853.7 | 2726.4 |

**Table 6.**Statistical misfit comparison between w/o considering the terrain and UOSs of the survey area.

Result | RMSE (m/mGal) | Minimum Value (m/mGal) | Maximum Value (m/mGal) | Minimum Error (m/mGal) | Maximum Error (m/mGal) |
---|---|---|---|---|---|

Without: gravity anomaly | 0.18 | −250.03 | −93.66 | −0.59 | 1.05 |

Without: ice thickness | 79.2 | 1197.0 | 3832.6 | −548.7 | 123.5 |

With: gravity anomaly | 0.21 | −250.23 | −93.65 | −0.79 | 1.57 |

With: ice thickness | 19.8 | 1187.3 | 3876.1 | −172.6 | 86.4 |

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**MDPI and ACS Style**

Liu, Y.; Wang, J.; Li, F.; Meng, X.
Estimation of Antarctic Ice Sheet Thickness Based on 3D Density Interface Inversion Considering Terrain and Undulating Observation Surface Simultaneously. *Remote Sens.* **2024**, *16*, 1905.
https://0-doi-org.brum.beds.ac.uk/10.3390/rs16111905

**AMA Style**

Liu Y, Wang J, Li F, Meng X.
Estimation of Antarctic Ice Sheet Thickness Based on 3D Density Interface Inversion Considering Terrain and Undulating Observation Surface Simultaneously. *Remote Sensing*. 2024; 16(11):1905.
https://0-doi-org.brum.beds.ac.uk/10.3390/rs16111905

**Chicago/Turabian Style**

Liu, Yandong, Jun Wang, Fang Li, and Xiaohong Meng.
2024. "Estimation of Antarctic Ice Sheet Thickness Based on 3D Density Interface Inversion Considering Terrain and Undulating Observation Surface Simultaneously" *Remote Sensing* 16, no. 11: 1905.
https://0-doi-org.brum.beds.ac.uk/10.3390/rs16111905