A matching pursuit approach to the geophysical inverse problem of seismic travel time tomography under the ray theory approximation
Résumé
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.
Mots clés
Numerical Analysis (math.NA)
FOS: Mathematics
41A45
45Q05
65D15
65J20
65R32
68T05
86-10
86A15
86A22
inverse problems travel time tomography seismology matching pursuits numerical modelling MSC(2020) 41A45 45Q05 65D15 65J20 65R32 68T05 86-10 86A15 86A22
inverse problems
travel time tomography
seismology
matching pursuits
numerical modelling MSC(2020) 41A45
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