# Antarctic Time-Variable Regional Gravity Field Model Derived from Satellite Line-of-Sight Gravity Differences and Spherical Cap Harmonic Analysis

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## Abstract

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_{max}= 20, corresponding to the SH degree and order up to 60. The comparison of the radial gravity on the Earth’s surface map across Antarctica with the corresponding radial gravity components of the L2 JPL is carried out using local geo-potential coefficients. The results of this comparison provide evidence that these basis functions for K

_{max}= 20 are valid across the entirety of Antarctica. Subsequently, the analysis proceeds using LGD data obtained from the Level 1B product of GFO by transforming these LGD data into the SCHA coordinate system and applying them to constrain the SCHA harmonic coefficients up to K

_{max}= 20. In this case, several independent LGD profiles along the trajectories of the satellites are devised to verify the accuracy of the local model. These LGD profiles are not employed in the inverse problem of determining harmonic coefficients. The results indicate that using regional harmonic basis functions, specifically spherical cap harmonic analysis (SCHA) functions, leads to a close estimation of LGD compared to the L2 JPL. The regional harmonic basis function exhibits a root mean square error (RMSE) of 3.71 × 10

^{−4}mGal. This represents a substantial improvement over the RMSE of the L2 JPL, which is 6.36 × 10

^{−4}mGal. Thus, it can be concluded that the use of local geo-potential coefficients obtained from SCHA is a reliable method for extracting nearly the full gravitational signal within a spherical cap region, after validation of this method. The SCHA model provides significant realistic information as it addresses the mass gain and loss across various regions in Antarctica.

## 1. Introduction

- GRACE (Gravity Recovery and Climate Experiment) was a joint mission between NASA and the German Aerospace Center. It consisted of two satellites in a polar orbit that provided highly accurate measurements of the Earth’s gravity field. Research such as [2,3,4,5,6] used GRACE data to estimate regional gravitational field models.
- Champ (Challenging Minisatellite Payload) was a German satellite that operated between 2000 and 2010. It also provided highly accurate gravity measurements and has been utilized by research such as [7] to estimate regional gravitational field models.
- GOCE (Gravity field and steady-state Ocean Circulation Explorer) was a European Space Agency mission that operated from 2009 to 2013. It was designed specifically for gravity measurements and provided highly accurate data. Research such as [8,9,10,11] used GOCE data to estimate regional gravitational field models.

^{−4}mGal [14]. Ref. [12] presented a novel method of using GRACE data to discover geophysical mass changes in terms of LGD that are not directly detectable by monthly Level 2 or mascon solutions. [13] calculated instantaneous LGD using low latency (1–3 days) to demonstrate the possibility of identifying water storage change as fast as feasible with only a few days of la. [14] demonstrated the sensitivity of LGD LRI observations to detect high-frequency oceanic mass variability in the Argentine Basin and the Gulf of Carpentaria as well as sub-monthly variations in surface (river) water in the Amazon Basin [14].

_{max}, specifically 20, 30, 40, and 50. The evaluation was performed on a grid coordinate system with the Earth’s surface radial coordinate over Antarctica, with a capsize = ${30}^{\circ}$. This analysis provided insights into the effectiveness of the SCHA model in capturing the complexity of the time-variable gravity field with higher accuracy.

## 2. Spherical Cap Harmonics Analysis

_{max}, which defines the spatial resolution (half of the minimum wavelength) of the geo-potential field model as [39]:

## 3. Line-of-Sight Gravity Difference (LGD)

**g**, in the spherical cap coordinate system is presented by the following equation [14,37,41]:

## 4. Numerical Analysis

#### 4.1. The Study Area

#### 4.2. Antarctic Time-Variable Regional Gravity Field Model

- In the first scenario, we first use the LGD data obtained from L2 JPL, and then we apply the SCHA basis function with capsize = ${30}^{\circ}$ and K
_{max}= 20. It corresponds to degree 60 in spherical harmonic and captures the majority of the time-variable gravity signal acquired during the GFO mission. This study discovers that by utilizing regional geo-potential coefficients, the radial component of the Earth’s gravitational field on the surface can be precisely determined, supporting the reliability of the regional harmonic basis functions in the study region. - In the second scenario, after validation in the first scenario, the SCHA model is used to generate a local geo-potential model using LGD data from GFO Level 1B product to demonstrate the effectiveness of the regional harmonic basis function (SCHA) in providing more information about the gravity field of Antarctica, particularly in time-variable gravity fields. Moreover, we analyze the time-variable local geo-potential model using various degrees of harmonics. We calculate geo-potential coefficients of SCHA for different degrees of harmonics over Antarctica using LGD data to evaluate the performance of each degree in estimating the time-variable gravity field.

#### 4.2.1. First Scenario: Antarctic Time-Variable Regional Gravity Field Model Using LGD Data Obtained from the L2 JPL

_{max}= 20.

#### 4.2.2. Second Scenario: Antarctic Time-Variable Regional Gravity Field Model Using LGD Data from GFO Level 1B Data

_{max}= 20 (441 number of unknown parameter; according to Equations (10) and (27) the spatial resolution of the SCHA technique in this area would be ${\mathsf{\Lambda}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=2\times 30/20={3}^{\circ}=333\mathrm{k}\mathrm{m}$ spatial resolution) and a cap size of 30.

^{−9}, while the RMSE value of GFO’s L2 JPL is 6.3598 × 10

^{−9}. Specifically, the RMSE of the L2 JPL is approximately twice that of the SCHA model, demonstrating the superior performance of the SCHA model at the same wavelength to estimate LGD data over Antarctica.

_{max}= 20. Additionally, LGD data obtained from GFO’s L2 JPL in January 2019 were also calculated to serve as a comparison. The variation of the gravity field was then depicted using these data.

_{max}equal to 20, 30, 40, and 50 over Antarctica using LGD data obtained from Level 1B GRACE-FO product data. Next, we use these coefficients to calculate LGD data on the control profiles for each degree of harmonics. The LGD data obtained from the SCHA model is compared with the LGD data obtained from the L2 JPL model. To do so, we choose the corresponding degree of harmonic, which means that for SCHA with K

_{max}= 20, we use the L2 JPL with n = 60, which has the same wavelength (~300 km), and for K

_{max}= 30 (222 km), 40 (160 km), 50 (132 km) we use n = 96 (~222 km) as the degree harmonic that has near wavelength to the SCHA models. Therefore, we analyze our local model compared to the L2 JPL model with the corresponding degree of harmonic in terms of wavelength.

_{max}= 50). As can be observed, there is stronger correlation in the variations of GOCO06S and SCHA (K

_{max}= 50) in the control profile than in the L2 JPL model.

_{max}= 20, 30, 40, and 50. To achieve this, the radial component of the local gravity model on the grid coordinates with the Earth’s surface radial coordinate over Antarctica with capsize = ${30}^{\circ}$ is calculated using SCHA coefficients with the different K

_{max}. The result of this calculation is presented in Figure 11.

_{max}in the SCHA model, more detail in the radial components of the local gravity model emerges. However, by increasing the number of K

_{max}, yellow strips appear. The reason for an increase in the yellow strips primarily relates to the spatial resolution of the SCHA model and the overfitting problem. In our solution, the spatial resolution of two SCHA models with K

_{max}= 40 and 50 and capsize = ${30}^{\circ}$ are 166 km and 133 km, respectively, which are far away from the standard resolution of the GFO model. Therefore, we conclude that yellow strips are defined as fake signals. Moreover, the SCHA model provided realistic information about mass gain in the north of Antarctica, particularly across Dronning Maud Land and Enderby Land. These mass redistributions are insignificant in the L2 JPL result. Additionally, it revealed mass loss across Totten Glacier and Ninnis Glacier in East Antarctica and the Amundsen Sea Sector in West Antarctica [47,48]. These findings demonstrate the potential of the SCHA model to enhance our understanding of time-variable gravity fields over Antarctica.

## 5. Conclusions

_{max}can be obtained as a trade-off between providing precise estimation of LGD data and preventing the overfitting problem.

^{−4}mGal in measurement. (2) The region-specific constraints can be implemented. (3) Temporal resolution of the gravity recovery can be flexible and adjusted depending on the satellite coverage. (4) The SCHA model provides realistic information as it addresses mass gain and loss across various regions in Antarctica. Overall, this research highlights the potential of SCHA in accurately modelling the Earth’s gravity field and enhancing our comprehension of time-variable gravity fields over Antarctica.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spherical cap coordinate system [38].

**Figure 2.**A schematic description of inter-satellites’ LOS gravity difference [40].

**Figure 3.**Relation between the base vector in the SCHA coordinate system and the base vector in the spherical coordinate [41]. ${P}_{0}({\mathsf{\Phi}}_{{P}_{0}},{\mathsf{\Lambda}}_{{P}_{0}})$ is the position of the new pole in a geocentric coordinate system, and $Q$ is a position of an arbitrary point which is defined in the spherical cap coordinate system parameters $Q(\theta ,\lambda )$.

**Figure 6.**The scenarios of the project [13].

**Figure 7.**(

**a**) LGD value obtained from SCHA model; (

**b**) the difference between SCHA model and the data (LGD data obtained from L2 JPL); (

**c**) radial component from SCHA model; and (

**d**) the difference between SCHA model and the data (radial component data obtained from time-variable geo-potential model).

**Figure 9.**(

**Left**) Radial gravity component of L2 JPL model on the r = R; and (

**Right**) radial gravity component from SCHA model on the r = R.

**Figure 10.**A comparison of the SCHA model (red), the L2 model (green), and the LGD data (blue) for various degrees of the SCHA and L2 JPL and GOCO06S models (n = 300) (pink).

**Figure 11.**Comparison of the radial gravity component of the time-variable local gravity model in January 2019 for different K

_{max}; left to right: K

_{max}= 20, 30, 40, 50.

**Table 1.**RMSE and the mean value (mGal) of differences of LGD and the radial component between SCHA model and GFO Level 2.

RMSE | Mean | ${\mathit{K}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ | |
---|---|---|---|

LGD | 1.5735 × 10^{−6} | 8.0426 × 10^{−7} | 20 |

Radial component | 4.0089 × 10^{−4} | 2.0707 × 10^{−4} |

Data Set | Reference Field | True Field | Filter | Altitude | Length |
---|---|---|---|---|---|

1 | ITG GRACE 2010 (d/o 60) | CSR-R05b August 2002 (d/o 60) | DDK6 | 500 | 10 days |

2 | ITG GRACE 2010 (d/o 60) | CSR-R05 August 2002 (d/o 60) | DDK4 | 400 | 10 days |

3 | ITG GRACE 2010 (d/o 60) | CSR-R05 August 2002 (d/o 60) | DDK2 | 300 | 10 days |

4 | Static gravity and better-known geophysical signals such as ocean tide as for March 2004 | NASA GSFC mascon March 2004 (d/o 90) | - | 487 | 1 month |

5 | Static gravity and better-known geophysical signals such as ocean tide for March 2016 | NASA GSFC mascon March 2016 (d/o 90) | - | 377 | 1 month |

**Table 3.**RMSE and mean values (mGal) of differences of LGD obtained from geo-potential models (SCHA model GFO Level 2) and that obtained from GFO Level 1B on the control profile.

Method | RMSE | Mean | ${\mathit{K}}_{\mathbf{m}\mathbf{a}\mathbf{x}}(\mathit{S}\mathit{C}\mathit{H}\mathit{A})/\mathit{n}(\mathit{S}\mathit{H})$ |
---|---|---|---|

SCHA | 3.7133 × 10^{−4} | 2.2307 × 10^{−4} | 20 |

L2 | 6.3598 × 10^{−4} | −2.9193 × 10^{−5} | 60 |

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**MDPI and ACS Style**

Feizi, M.; Raoofian Naeeni, M.; Flury, J.
Antarctic Time-Variable Regional Gravity Field Model Derived from Satellite Line-of-Sight Gravity Differences and Spherical Cap Harmonic Analysis. *Remote Sens.* **2023**, *15*, 2815.
https://doi.org/10.3390/rs15112815

**AMA Style**

Feizi M, Raoofian Naeeni M, Flury J.
Antarctic Time-Variable Regional Gravity Field Model Derived from Satellite Line-of-Sight Gravity Differences and Spherical Cap Harmonic Analysis. *Remote Sensing*. 2023; 15(11):2815.
https://doi.org/10.3390/rs15112815

**Chicago/Turabian Style**

Feizi, Mohsen, Mehdi Raoofian Naeeni, and Jakob Flury.
2023. "Antarctic Time-Variable Regional Gravity Field Model Derived from Satellite Line-of-Sight Gravity Differences and Spherical Cap Harmonic Analysis" *Remote Sensing* 15, no. 11: 2815.
https://doi.org/10.3390/rs15112815