A Review of Inverse Problems in Seismic Tomography: Mathematical Approaches to Imaging Earth's Interior
Authors
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Kim Campbell
Horizon Labs
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Liam Parker
Crimson Education Trust
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Ian Ward
Horizon Labs
https://orcid.org/0009-0000-0000-0000
Abstract
Understanding the structure of Earth's deep interior relies on solving highly complex inverse problems using seismic wave data. This systematic review synthesizes mathematical and computational advancements in seismic tomography. We trace the evolution from linear, ray-based methods to the current frontier of Full-Waveform Inversion (FWI), a non-linear optimization challenge that leverages the complete information in seismic waveforms. The paper critically analyzes the mathematical underpinnings of each approach, including the choice of objective functions, regularization techniques (e.g., Tikhonov, total variation), and the optimization algorithms (e.g., L-BFGS, adjoint-state methods) required to navigate the high-dimensional, non-convex search space. We also explore the role of high-performance computing in making these methods feasible. By examining the successes and persistent challenges, such as cycle-skipping, this review provides a comprehensive roadmap for mathematicians and geophysicists working to create higher-fidelity images of Earth's mantle and core.
Copyright (c) 2026 Kim Campbell, Liam Parker, Ian Ward (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.