Knowing Earth?s internal structure on a range of length scales is necessary to understand natural hazards (earthquakes, volcanoes), exploit subsurface energy resources, and understand the long-term geological evolution of our planet. Seismic waves emitted by earthquakes or man-made sources represent the most direct and precise probes of Earth?s interior. Traditionally, seismologists recognize two types of non-destructive method for determining medium properties from measurements made at its boundary. Tomography (which in concept is similar to medical computer-aided tomography) aims to constrain smooth medium variations from transmitted seismic waves, whereas inverse scattering aims to constrain non-smooth heterogeneity (edges, interfaces) from reflected, refracted, or diffracted waves. These methods have revolutionized our understanding of Earth?s structure but have not yet reached their full potential. An important issue is that for practical and technical reasons they used to be treated separately. Indeed, (local) linearization and the use of asymptotic theory prevent internally consistent interpretation of seismic data with different (but complementary) sampling properties. Building on expertise in seismology, inverse theory, and microlocal and harmonic analysis, the proposed research aims to develop a unified theoretical framework for (nonlinear, full wave) inversion and medium reconstruction, where tomography and inverse scattering are no longer treated separately. The new methods can lead to more accurate seismic exploration for oil and gas but the main geoscience motivation is to study the crust and mantle beneath North America with data provided by USArray, the seismology component of EarthScope, a nationwide, multi-year geosciences project funded by NSF.
From a mathematical sciences perspective the challenge is to develop a unified analysis of and computationally efficient algorithms for full wave inversion of the elastic wave equation and Cauchy or partial boundary data (here, broad-band waveforms measured at Earth?s surface). The proposed research extends the PIs previous research on inverse scattering and multi-scale tomography; it aims to transition from inverse scattering with the (asymptotic) generalized Radon transform to a full waveform analogue, to develop a nonlinear illumination correction and partial reconstruction approach and a (complementary) analyses for the (transient) time-domain formulation and (multi-)frequency (?fixed energy?) formulation, and to study wave constituents associated with (multiple) scattering off complex structures (edges, for example). In view of application to USArray data we aim to generalize receiver function analysis, characterize sharp transitions (such as the crust-mantle interface, the lithosphere-asthenosphere boundary, and interfaces associated with subduction zones), and develop nonlinear reflection and transmission tomography to constrain physical properties of the mantle beneath North America.
