A mathematical framework for inverse wave problems in heterogeneous media

Abstract

This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The coefficients of these time-dependent partial differential equations respresent parametrically the spatially varying mechanical properties of materials. Rocks, manufactured materials, and other wave propagation environments often exhibit spatial heterogeneity in mechanical properties at a wide variety of scales, and coefficient functions representing these properties must mimic this heterogeneity. We show how to choose domains (classes of nonsmooth coefficient functions) and data definitions (traces of weak solutions) so that optimization formulations of inverse wave problems satisfy some of the prerequisites for application of Newton's method and its relatives. These results follow from the properties of a class of abstract first-order evolution systems, of which various physical wave systems appear as concrete instances. Finite speed of propagation for linear waves with bounded, measurable mechanical parameter fields is one of the by-products of this theory. acoustics or elastodynamics, this property implies that appropriate traces of solutions are welldefined, provided that the right-hand-side or source term is minimally smooth in time (only!).Existence, uniqueness, and regularity of the solution as function of problem data -both right-hand side and (operator) coefficients -also follows from this fact.The abstract theory accomplishes even more than that, as it applies to a much wider range of dynamics than those commonly occurring in continuum mechanics, accommodating first-order systems with operator coefficients. This added generality creates no additional difficulty for the basic theory. It is in fact very useful: as will be explained in the Discussion section, it justifies certain infeasible-model methods for inverse problems in wave propagation.Another useful by-product of the theory is the finite speed of propagation property for hyperbolic systems with bounded, measurable coefficients, a result which so far as we can tell is new. This follows from the similar property for systems with smooth coefficients via the continuous dependence of solutions on the coefficients in the sense of convergence in measure.Precise statements of the main results for symmetric hyperbolic systems (1) appear in the next section, followed by a brief discussion of the related literature. The third section contains the definition of the class of abstract systems studied in this paper, and statements of the main results to be established concerning it. Proofs of these results follow in the next three sections.The seventh section shows how symmetric hyperbolic systems fit into the abstract framework, and contains proofs of the main theorems stated in the second section. The eighth section treats the hyperbolic case of linear viscoelasticity as an instance of the theory developed earlier. We end with a disc...

On the linearization of operators related to the full waveform inversion in seismology

“…Blazek, Stolk, and Symes studied abstract evolution problems like the following: Let U be a real separable Hilbert space. Find a U ‐valued function ξ = ξ ( t ), which solves (in a suitable sense) in which the right‐hand side ℓ = ℓ ( t ) is also U ‐valued.…”

“…System can be written in the abstract form when setting U = L 2 (Ω) 1 + d , , see ,Appendix A for the skew‐symmetry of P .…”

“…Bao extended this line of research and showed that the formal derivative is a Fréchet derivative for densities in a certain smoothness class. Very recently—during our work on the present article—Blazek, Stolk, and Symes came up with an elaborate consideration of first order hyperbolic systems, which cover the first order formulation of the aforementioned wave equation as a special case. Thus, some of our results can be obtained from under slightly stronger assumptions (we explain this in detail in the last section).…”

“…Very recently—during our work on the present article—Blazek, Stolk, and Symes came up with an elaborate consideration of first order hyperbolic systems, which cover the first order formulation of the aforementioned wave equation as a special case. Thus, some of our results can be obtained from under slightly stronger assumptions (we explain this in detail in the last section). Because their approach is very general and rather abstract, an independent, self‐contained, concrete study of the acoustic wave equation is justified, the more so as is still a widely used model in seismology.…”

Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity

“…Here, the mass density and the two Lamé parameters are sought. Fréchet differentiability of parameter-to-solution maps of abstract first order hyperbolic systems has been studied before by Blazek et al [1] using the technique of weak solutions. Indeed, our research was triggered by reading their article and with the present paper we complement and extend their work.…”