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A first approach to learning a best basis for gravitational field modelling

Volker Michel, Naomi Schneider

TL;DR

The paper tackles the ill-posed problem of downward continuation in Earth gravity-field modelling by extending Regularized Functional/Inverse Problem Matching Pursuit with a learnable dictionary. It introduces Learning RFMP (LRFMP), which optimizes over an infinite pool of trial functions composed of spherical harmonics and Abel–Poisson kernels to obtain a compact, effective dictionary that yields sparse, accurate gravity models. Numerical tests on EGM2008 and GRACE data show that the learnt dictionary outperforms manually chosen dictionaries in both data- and model-error metrics, while reducing storage demands. The approach promises improved physical interpretability and efficiency for gravity-field inversion and lays groundwork for extending dictionary learning to other multiscale, ill-posed inverse problems in geodesy and beyond.

Abstract

Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a non-linear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.

A first approach to learning a best basis for gravitational field modelling

TL;DR

The paper tackles the ill-posed problem of downward continuation in Earth gravity-field modelling by extending Regularized Functional/Inverse Problem Matching Pursuit with a learnable dictionary. It introduces Learning RFMP (LRFMP), which optimizes over an infinite pool of trial functions composed of spherical harmonics and Abel–Poisson kernels to obtain a compact, effective dictionary that yields sparse, accurate gravity models. Numerical tests on EGM2008 and GRACE data show that the learnt dictionary outperforms manually chosen dictionaries in both data- and model-error metrics, while reducing storage demands. The approach promises improved physical interpretability and efficiency for gravity-field inversion and lays groundwork for extending dictionary learning to other multiscale, ill-posed inverse problems in geodesy and beyond.

Abstract

Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a non-linear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.

Paper Structure

This paper contains 20 sections, 3 theorems, 56 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some mathematical tools for learning a dictionary
  3. Trial functions under consideration for dictionaries
  4. Basic principles of linear ill-posed inverse problems
  5. The learning approach
  6. About learning dictionaries for inverse problems
  7. A first learning algorithm
  8. Determination of candidates
  9. Additional features for practice
  10. Numerical results
  11. Setting of the experiments
  12. Results
  13. Conclusion and Outlook
  14. Acknowledgement
  15. Gradient of the objective function with respect to Abel--Poisson kernels
  16. ...and 5 more sections

Key Result

Theorem 7

The minimization of the Tikhonov functional in the $n$-th step of the RFMP with respect to a trial function $d$ and a real coefficient $\alpha$ as seen in eqTikh is equivalent to the maximization of $\mathrm{RFMP}(\cdot;n)$ with respect to a trial function $d$.

Figures (3)

  • Figure 1: Examples of trial functions. Left: spherical harmonic. Right: Abel--Poisson kernel for a fixed centre $\xi$ (location of the maximum).
  • Figure 2: Schematic representation of the basic learning algorithm.
  • Figure 3: Results of the RFMP with a manually chosen dictionary and the learnt dictionary. Upper row: Results for EGM2008 data. Lower row: Results for GRACE data. Left: Solution. Middle: Absolute approximation error of RFMP with arbitrary dictionary. Right: Absolute approximation error of RFMP with learnt dictionary. 3000 iterations allowed in all experiments. All values in $\mathrm{m}^2/\mathrm{s}^2$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 7
  • proof
  • Definition 9
  • Theorem 10
  • ...and 2 more