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High-dimensional experiments for the downward continuation using the LRFMP algorithm

Naomi Schneider, Volker Michel, Nico Sneeuw

TL;DR

This work addresses the downward continuation problem, aiming to recover the Earth's surface gravitational potential from satellite measurements, a classic exponentially ill-posed inverse problem. It extends Learning Inverse Problem Matching Pursuit (LIPMP) methods by employing a Learning Regularized Functional Matching Pursuit (LRFMP) that combines global spherical harmonics with local Abel–Poisson kernels and wavelets, enabling high-dimensional inversion with over 500k data points. Key innovations include a closed-form Sobolev inner product for efficient learning, avoidance of Slepian-function learning due to computational costs, and demonstrated feasibility on GRACE-FO December 2022 data and EGM2008, achieving a relative data error near 5% and a relative RMSE around 17.5%. The results show sparse, accurate reconstructions in regions with strong local structure and establish a scalable framework for gravity-field inversion on large satellite datasets, with code and data access provided for reproducibility and further research.

Abstract

Time-dependent gravity data from satellite missions like GRACE-FO reveal mass redistribution in the system Earth at various time scales: long-term climate change signals, inter-annual phenomena like El Nino, seasonal mass transports and transients, e. g. due to earthquakes. For this contemporary issue, a classical inverse problem has to be considered: the gravitational potential has to be modelled on the Earth's surface from measurements in space. This is also known as the downward continuation problem. Thus, it is important to further develop current mathematical methods for such inverse problems. For this, the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed within the last decade. Their unique feature is the combination of local as well as global trial functions in the approximative solution of an inverse problem such as the downward continuation of the gravitational potential. In this way, they harmonize the ideas of a traditional spherical harmonic ansatz and the radial basis function approach. Previous publications on these methods showed proofs of concept. Here, we consider the methods for high-dimensional experiments settings with more than 500 000 grid points which yields a resolution of 20 km at best on a realistic satellite geometry. We also explain the changes in the methods that had to be done to work with such a large amount of data. The corresponding code (updated for big data use) is available at https://doi.org/10.5281/zenodo.8223771 under the licence CC BY-NC-SA 3.0 Germany.

High-dimensional experiments for the downward continuation using the LRFMP algorithm

TL;DR

This work addresses the downward continuation problem, aiming to recover the Earth's surface gravitational potential from satellite measurements, a classic exponentially ill-posed inverse problem. It extends Learning Inverse Problem Matching Pursuit (LIPMP) methods by employing a Learning Regularized Functional Matching Pursuit (LRFMP) that combines global spherical harmonics with local Abel–Poisson kernels and wavelets, enabling high-dimensional inversion with over 500k data points. Key innovations include a closed-form Sobolev inner product for efficient learning, avoidance of Slepian-function learning due to computational costs, and demonstrated feasibility on GRACE-FO December 2022 data and EGM2008, achieving a relative data error near 5% and a relative RMSE around 17.5%. The results show sparse, accurate reconstructions in regions with strong local structure and establish a scalable framework for gravity-field inversion on large satellite datasets, with code and data access provided for reproducibility and further research.

Abstract

Time-dependent gravity data from satellite missions like GRACE-FO reveal mass redistribution in the system Earth at various time scales: long-term climate change signals, inter-annual phenomena like El Nino, seasonal mass transports and transients, e. g. due to earthquakes. For this contemporary issue, a classical inverse problem has to be considered: the gravitational potential has to be modelled on the Earth's surface from measurements in space. This is also known as the downward continuation problem. Thus, it is important to further develop current mathematical methods for such inverse problems. For this, the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed within the last decade. Their unique feature is the combination of local as well as global trial functions in the approximative solution of an inverse problem such as the downward continuation of the gravitational potential. In this way, they harmonize the ideas of a traditional spherical harmonic ansatz and the radial basis function approach. Previous publications on these methods showed proofs of concept. Here, we consider the methods for high-dimensional experiments settings with more than 500 000 grid points which yields a resolution of 20 km at best on a realistic satellite geometry. We also explain the changes in the methods that had to be done to work with such a large amount of data. The corresponding code (updated for big data use) is available at https://doi.org/10.5281/zenodo.8223771 under the licence CC BY-NC-SA 3.0 Germany.
Paper Structure (21 sections, 1 theorem, 48 equations, 4 figures, 2 tables)

This paper contains 21 sections, 1 theorem, 48 equations, 4 figures, 2 tables.

Table of Contents

  1. Keywords
  2. MSC(2020)
  3. Acknowledgments
  4. Introduction
  5. Notation
  6. Mathematical basics
  7. Trial functions
  8. The LRFMP
  9. Main approach
  10. Aspects for upscaling
  11. Numerics
  12. General setting
  13. Results and evaluation
  14. Conclusion and Outlook
  15. Funding
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $k\in \mathbb{N}_0, q\in [0,1[$ and $\tau \in [-1,1]$. Then it holds

Figures (4)

  • Figure 1: Examples of dictionary elements. Upper row: Fully normalized spherical harmonic $Y_{2,1}$. Lower row: Abel--Poisson kernel $K(x,\cdot)$ (left) and Abel--Poisson wavelet $W(x,\cdot)$ (right) both for $x=(0.9,0,0)^\mathrm{T}$ on a global scale and zoomed-in on their extremum.
  • Figure 2: Distribution of kinematic orbits.
  • Figure 3: RDEs and RRMSEs along the iterations.
  • Figure 4: Left column: the respective truncated gravitational potential. Middle column: the obtained approximation. Right column: the absolute approximation error. The upper row contains these visualizations regarding the EGM2008 data. The lower row contains the respective ones regarding the GRACE FO data. All values in $\mathrm{m}^2/\mathrm{s}^2$. The scales are adapted for better comparability.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof