Abstract

Inverse problems in wave propagation rely upon hyperbolic partial (integro-)differential systems to model physical wave motion. However, rocks, manufactured materials, and other natural and human-made wave propagation environments may exhibit spatial heterogeneity at a wide variety of scales. Therefore accuracy in modeling (hence in inversion) requires tha coefficient functions representing material parameter fields be permitted some degree of nonsmoothness. I will show how to formulate well-posed initial/boundary-value problems for hyperbolic systems with bounded and measureable coefficients, as instances of a class of abstract first-order evolution problems. This framework yields well-defined realizations of the mappings occurring in widely-used optimization formulations of inverse problems, and justifies the use of Newton’s method and its relatives for their solution. The finite speed of propagation for waves in material models with bounded and measurable heterogeneity also follows from this framework. Another useful by-product is a mathematical foundation for (unphysical) hyperbolic systems with operator coefficients, which are crucial components of a class of seismic inversion algorithms.

The content of this talk is the result of collaboration with Christiaan Stolk and Kirk Blazek.