Faithful Reeb Graph Reconstruction of a Tectonic Subduction Zone from Earthquake Hypocenters

An important problem in topological data analysis (TDA)$\text{unicode}x2014$of both theoretical and practical interest$\text{unicode}x2014$is to reconstruct the topology and geometry of an underlying (usually unknown) metric graph from possibly noisy data sampled around it. Reeb graphs have recently been successfully employed in abstract metric graph reconstruction under Gromov$\text{unicode}x2013$Hausdorff noise: the sample is assumed to be metrically close to the ground truth. However, such a strong global density guarantee is often unavailable, making the existing Reeb graph-based methods unusable. A very different yet more relevant paradigm focuses on the reconstruction of metric graphs$\text{unicode}x2014$embedded in the Euclidean space$\text{unicode}x2014$from Euclidean samples that are only Hausdorff-close. We relax the density assumption to give provable geometric reconstruction schemes, even when the sample is metrically close only locally, but still provide provable guarantees for the successful geometric reconstruction of Euclidean graphs under the Hausdorff noise model. We apply our graph reconstruction techniques to reconstruct earthquake plate tectonic boundaries from the global earthquake catalog. The SLAB2.0 model is a comprehensive spatial summary of all known subduction zone slabs on Earth. We reconstruct parts of the SLAB2.0 model from possibly noisy earthquake hypocenter data.