Seismology- and Geodesy-Based Inverse Problems Crossing Scales, with Scattering, Anisotropy and Nonlinear Elasticity

William Marsh Rice University, Houston TX

Investigators

Abstract

The investigator and his collaborators study inverse problems in seismology and geodesy crossing many scales. These inverse problems are associated with systems of partial differential equations describing elastic waves and elastostatics. Their key advances in these problems pertain to reaching real earth material properties and their complex heterogeneities through the consideration of scattering, anisotropy and nonlinear elasticity. The application goal is rigorous reconstruction, hierarchically mapping Earth's interior, locally and possibly globally, honoring these material properties, in conjunction with dislocations and fault shapes encoded in co-seismic deformation. The investigator exploits important developments in novel sensor design and data acquisition. Differentiating the features representing the observed hierarchical data complexity, expected to be associated with different scattering regimes, the investigator distinguishes different types of nonlinear inverse problems. The research promises to lead to innovative technologies for interpreting the rich information contained in seismic and geodesic (Global Positioning System) data crossing scales on the one hand, and new directions in the analysis of inverse problems underpinning modern data science on the other hand. The studies benefit natural resources management including hydrofracking and geothermal energy, hazard analysis and, moreover, provide possible approaches for planetary exploration with very few sensors. They also provide keys for important further insight on how processes at the surface are coupled to processes in Earth's deep interior. The research program offers a unique interdisciplinary educational experience for the students involved giving them a much broader appreciation of the importance of novel techniques and real-life implications. The investigator and his collaborators develop a composite analysis of seismology and geodesy based inverse problems. Following the hierarchical data complexity, they (1) analyze spectral rigidity of spherically symmetric planets, and then include angular variations through perturbation and semiclassical analysis, as well as inverse problems for (Rayleigh and Stoneley) waveguide coupling at and near phase boundaries; (2) study geometric inverse problems for the local mixed geodesic ray transforms on rank-4 tensors in Riemannian geometry, (broken) geodesic ray transforms in Finsler geometry and then boundary rigidity, and with visible orientable geodesic spheres as well as the geometric inverse problem with microseismicity data; (3) analyze hyperbolic inverse boundary value problems with piecewise smooth stiffness tensors (with associated interfaces or domain partitions), partial data and improved stability estimates using unbalanced, complex optimal transport for optimization-based reconstruction; (4) study uniqueness and conditional stability for the recovery of a heterogeneous dislocation and fault shape from geodesy boundary data; and (5) develop and analyze inverse problems in nonlinear elasticity in sedimentary rocks in the presence of discontinuities using paired Lagrangian distributions and Strichartz estimates. The implicit connections between these inverse problems come into play as they probe one planet, while contributing to a deeper understanding of information content in the exponentially growing data volumes that are available through modern data enters. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.