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Published online by Cambridge University Press: 26 June 2025
We review applications of microlocal analysis (MA) to reflection seismology. In this inverse method one attempts to estimate the index of refraction of waves in the earth from seismic data measured at the Earth's surface. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem, with complications introduced by the heterogeneity and anisotropy of the Earth's crust. We describe several important developments using MA to generate these wave-solutions by manipulating the wavefields directly on their phase space. We also consider some recent applications of MA to global seismology.
1. Introduction
Microlocal analysis plays an increasingly important role in seismology, particularly in the imaging and inversion of seismic data. Here we consider imaging and inversion via the generalized Radon transform (GRT), concentrating on advances since the work of Beylkin [9], applying the work of Guillemin [51] and Taylor [98]. It is the aim of this exposition to connect microlocal analysis with seismology in the context of inverse scattering. The analysis of a related problem, the X-ray transform (see Greenleaf and Uhlmann [48; 49]) also contributes to the further understanding of the GRT in seismology. Microlocal analysis and the general theory of Fourier integral operators are described in the books by H6rmander [60; 61; 62], Duistermaat [43], and Treves [101; 102].
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