# Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica

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## Abstract

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## 1. Introduction

## 2. Material and Methods

#### 2.1. Methodology

#### 2.1.1. The Improved Point-Mass (IPM) Solution

#### 2.1.2. The Surface Integral Considering the Exact Kernel

#### 2.2. GRACE Products

#### 2.2.1. GRACE Level-2 Data

#### 2.2.2. GRACE Level-3 Data

#### 2.3. Experimental Design

## 3. Results

#### 3.1. Assessment of OPM and IPM by Means of a Closed-Loop Simulation

#### 3.2. Inversion of GRACE Level-2 Data Using the IPM Modeling

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Surface Integral via Analytical Integration over the Radial Direction

## Appendix B. The Expression for the Remainder

## References

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**Figure 1.**Flow chart showing the experiment used for assessing the original (zeroth-order) and improved point-mass (third-order) solutions as per Equation (10) as well as a study case over Antarctica using GRACE Level-2 data as given by ITSG-Grace2018.

**Figure 2.**The map shows the division of the Antarctica Ice Sheet (AIS) into the ice front (defines the coastlines) and grounding line (defines the landmasses) with the limits established by Rignot et al. [53]. The AIS base map also shows the partitioning of the sphere into equal-area spherical panels used in GSFC-M as defined by Luthcke et al. [24], in which the respective geometrical centers (red circles) depict the Taylor points ${Q}_{0}$. The insert illustrates the geometry of an individual equal-area spherical panel in view of the equations presented in Section 2.1.1 and Section 2.1.2.

**Figure 3.**(

**a**) Spatial patterns of the root-mean-square (RMS) of the series of each one of the 41,168 equivalent water height panels from GSFC-M based on the period of January 2003 to December 2015. (

**b**) Spatial patterns of the RMS of the forwarded gravitational potential at 500 km altitude according to the radial integral in Equation (20) using the GSFC-M monthly fields of equivalent water height.

**Figure 4.**Root-mean-square errors (RMSEs) between the input mass changes from GSFC-M and those inverted from the OPM and IPM solutions over Antarctica. The panels (

**a**,

**b**) shows the RMSE regarding the recovering of mass changes only over the continents for OPM and IPM solutions, respectively. Similarly, to panels (

**a**,

**b**), panels (

**c**,

**d**) show the summary considering the results of the inversion accounting for a global set of mass changes.

**Figure 5.**Linear trends derived from the mass changes referring to the best fit of the series considering constant, linear trend, annual, semi-annual, and 161-days components regarding the period from January 2003 to December 2015. Panels (

**a**–

**d**) show results based on improved point-mass (IPM) and mascon solutions from CSR (CSR-M), GSFC (GSFC-M), and JPL (JPL-M), respectively.

**Figure 6.**(

**a**) The areal-weighted average time series of mass changes for the AIS for the different solutions expressed as equivalent water height (e.w.h.) in mm. (

**b**) The box plots summaries of e.w.h. series in panel (

**a**) at which the whiskers indicate the maximum and minimum range of e.w.h. (

**c**) Time series of the error noise defined as the residuals of the de-trended and de-seasoned series minus low-pass filtered residuals (i.e., the high-pass filtered residuals). (

**d**) The same as panel (

**b**), however, for the error noises. The boxes show the interquartile (first and third quartile) ranges of variation of the values, the segments inside the respective boxes sow the median of the values, and the crosses indicate the outliers.

**Table 1.**Summary of quantitative statistics for the assessment of OPM (zeroth-order approximation) and IPM (third-order approximation) solutions based on a closed-loop experiment.

Domain of the Inversion | Solution | Domain of the Evaluation | |||||
---|---|---|---|---|---|---|---|

Land | Global | Antarctica | |||||

RMSE (mm) | RSR (Unitless) | RMSE (mm) | RSR (Unitless) | RMSE (mm) | RSR (Unitless) | ||

Land | OPM | 23.82 | 0.29 | - | - | 58.94 | 0.52 |

IPM | 2.90 | 0.03 | - | - | 9.46 | 0.08 | |

Global | OPM | 2.41 | 0.03 | 1.68 | 0.04 | 3.83 | 0.03 |

IPM | 0.16 | 0.00 | 0.13 | 0.00 | 0.21 | 0.00 |

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## Share and Cite

**MDPI and ACS Style**

Ferreira, V.G.; Yong, B.; Seitz, K.; Heck, B.; Grombein, T. Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica. *Remote Sens.* **2020**, *12*, 3197.
https://doi.org/10.3390/rs12193197

**AMA Style**

Ferreira VG, Yong B, Seitz K, Heck B, Grombein T. Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica. *Remote Sensing*. 2020; 12(19):3197.
https://doi.org/10.3390/rs12193197

**Chicago/Turabian Style**

Ferreira, Vagner G., Bin Yong, Kurt Seitz, Bernhard Heck, and Thomas Grombein. 2020. "Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica" *Remote Sensing* 12, no. 19: 3197.
https://doi.org/10.3390/rs12193197