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A matching pursuit approach to the geophysical inverse problem of seismic travel time tomography under the ray theory approximation

Naomi Schneider, Volker Michel, Karin Sigloch, Eoghan J. Totten

Abstract

Seismic travel time tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linear(ized) inverse problem, it maps differences between observed and expected wave travel times to interior regions where waves propagate faster or slower than the expected average. The Earth's interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions. Within the past decade the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov-Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven its applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for travel time tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial travel time delays obtained from a contrived model for velocity differences. The corresponding code is available at https://doi.org/10.5281/zenodo.8227888 under the licence CC-BY-NC-SA 3.0 DE.

A matching pursuit approach to the geophysical inverse problem of seismic travel time tomography under the ray theory approximation

Abstract

Seismic travel time tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linear(ized) inverse problem, it maps differences between observed and expected wave travel times to interior regions where waves propagate faster or slower than the expected average. The Earth's interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions. Within the past decade the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov-Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven its applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for travel time tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial travel time delays obtained from a contrived model for velocity differences. The corresponding code is available at https://doi.org/10.5281/zenodo.8227888 under the licence CC-BY-NC-SA 3.0 DE.
Paper Structure (30 sections, 91 equations, 9 figures)

This paper contains 30 sections, 91 equations, 9 figures.

Table of Contents

  1. Keywords
  2. MSC(2020)
  3. Acknowledgments
  4. Introduction
  5. Notation
  6. Seismic travel time tomography
  7. Trial functions
  8. Definition and examples
  9. Tesseroid-based finite element hat functions
  10. Polynomials
  11. A dictionary
  12. Regularization terms
  13. The (Learning) Inverse Problem Matching Pursuits
  14. The (Learning) Regularized Functional Matching Pursuit
  15. The learning add-on
  16. ...and 15 more sections

Figures (9)

  • Figure 1: Example of an FEHF. We show $N_{(5096.8,0,0),(955.65,\pi/4,0.25)}$ for the depth slices (left, middle and right) at the following radial distances to the centre of the Earth in km: 3193.1, 3482.0 and 3770.9 (first row), 4059.8, 4348.7 and 4637.6 (second row), 4926.5, 5215.4 and 5504.3 (third row), 5793.2, 6082.1 and 6371.0 (last row). The colour scales are adjusted for a better comparability.
  • Figure 2: Example of a polynomial. We show $G_{2,2,1}^{\mathrm{I}}$ for the depth slices (left, middle and right) at the following radial distances to the centre of the Earth in km: 3193.1, 3482.0 and 3770.9 (first row), 4059.8, 4348.7 and 4637.6 (second row), 4926.5, 5215.4 and 5504.3 (third row), 5793.2, 6082.1 and 6371.0 (last row). The colour scales are adjusted for a better comparability.
  • Figure 3: Resolution test input model for P-velocity consisting of two plumes below the Volcanic Eifel and Yellowstone volcano. We show the depth slices (left, middle and right) at the following radial distances to the centre of the Earth in km: 3193.1, 3482.0 and 3770.9 (first row), 4059.8, 4348.7 and 4637.6 (second row), 4926.5, 5215.4 and 5504.3 (third row), 5793.2, 6082.1 and 6371.0 (last row).
  • Figure 4: Distribution of ISC-EHB meta-data rays in the given depth ranges. Depth in km as the distance to the Earth's centre.
  • Figure 5: Absolute approximation error for the approximation of the plumes model given in \ref{['fig:resolutiontests']}. We show the depth slices (left, middle and right) at the following radial distances to the centre of the Earth in km: 3193.1, 3482.0 and 3770.9 (first row), 4059.8, 4348.7 and 4637.6 (second row), 4926.5, 5215.4 and 5504.3 (third row), 5793.2, 6082.1 and 6371.0 (last row). The colour scales are adjusted for a better comparability.
  • ...and 4 more figures