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.
FROM A MATHEMATICS PERSPECTIVE, we have developed a unified frameworkfor nonlinear (broadband) waveform inversion and medium reconstructionthat go beyond linearization and separated treatments of inversescattering and tomography. Considering high-frequency waves, we showed that for certain mediumdiscontinuities (that is, conormal singularities of class$C^{1,alpha}$, $alpha > 0$) the reflected wave is more regular (inSobolev sense) than the incident wave. This allowed us to introduce ascattering series and to extract from the data constituents of aparticular order. We then developed an algorithm for the computation of general Fourierintegral operators (the canonical relations of which are graphs). Thealgorithm is based on the dyadic parabolic decomposition of phasespace and enables the discrete approximate evaluation of the action ofsuch operators on data in the presence of caustics. The formation ofcaustics is typically associated with low velocity zones. We adaptedthe algorithm to an RTM-based inverse scattering transform usingboundary data accounting for (realistic) partial illumination. We developed a direct (nonlinear) inversion method for thereconstruction of a metric (say, wavespeed variations) of a Riemannianmanifold using the shape operator associated with boundary diffractiontimes as the data. (This is a generalization of Dix' method to fullyheterogeneous wavespeed models.) We expect that the results can beapplied to microseismic data, for example, associated with aftershocksof earthquakes. For time-harmonic inverse problems we studied sequences of parameterfunctions generated by a nonlinear Landweber iteration and theconditions under which these converge, locally, to the solution. Weexpressed these conditions in terms of H""{o}lder/Lipschitz stabilityof the inverse maps, which ties naturally to the analysis of inverseproblems. We proceeded with establishing conditional Lipschitzstability estimates for the inverse boundary problem for the Helmholtzequation, allowing the presence of discontinuities. To enable the application of the above result on a regional scale, wecarried out research in fast algorithms. We considered thediscretization and approximate solution of the equation describingtime-harmonic qP polarized waves in TTI media and of the system ofequations describing time-harmonic (multi-component) elastic waves inorthorhombic media. We developed a massively parallel multifrontalsolver combined with Hierarchically SemiSeparable (HSS) matrixcompression. This allowed us to solve the mentioned equations on verylarge three-dimensional domains for a large number of differentsources. FROM A GEOSCIENCES PERSPECTIVE, we fundamentally improved our abilityto reconstruct geological media with non-smoothheterogeneity. Originally our geological target was the highlyheterogeneous crust beneath North America. The available data do not,however, provide sufficient spatial sampling of thewavefield. Instead, we applied our inverse techniques to data from thedensily spaced 'Hi-Climb' array in Tibet, to which end we incorporateda sparsity regularization. As a proof of concept we demonstrated thatthe interfaces that define the crust of the Tibetan plateau can beimaged with data from distant earthquakes. The preliminary resultsdemonstrate that the crust-mantle interface is (laterally) morecontinuous than had been inferred from earlier studies. The mentioned inverse techniques which we developed comprise (unknown)passive source RTM-based inverse scattering of mode converted waves(generalizing the notion of receiver functions), and a crosscorrelation power criterion for iterative RTM-based reflectiontomography using free-surface multiple reflections in teleseismic datafrom multiple sources (events).