Solving the inverse shortest path problem for earthquakes’ motion

Abstract

According to Fermat’s principle, seismic waves follow paths of least travel time. Thus, shortest path algorithms such as Dijkstra’s can be used to determine these paths. Conversely, inferring the parameters of a mathematical program from an observed optimal path defines the inverse shortest path problem, an area within inverse optimisation. With its wide range of applicability, inverse optimisation has attracted considerable interest. One of the earliest topics in this field was the inverse shortest path problem, with Burton and Toint (1992) laying its foundations. Since then, this problem has been explored across several domains, with various mathematical formulations and algorithms proposed. This dissertation examines the inverse shortest path problem in depth, reviews its theoretical foundations, and applies it to a seismological case study. Three algorithms are employed to solve the problem: the column generation algorithm, a quadratic programming algorithm, and a deep inverse optimisation algorithm using a modern deep learning framework. The aim is to estimate the weight vector using these algorithms, thereby reconstructing the mathematical program that defines the shortest paths taken by seismic waves. To the best of the author’s knowledge, this is the first study to apply the inverse shortest path problem to local seismic data from the Maltese Islands and Sicily.