A mathematical framework for inverse wave problems in heterogeneous media

@article{Blazek2013AMF,
  title={A mathematical framework for inverse wave problems in heterogeneous media},
  author={K. Blazek and C. Stolk and W. Symes},
  journal={Inverse Problems},
  year={2013},
  volume={29},
  pages={065001}
}
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 represent parametrically the spatially varying mechanical properties of materials. Rocks, manufactured materials, and other wave propagation environments often exhibit spatial heterogeneity in… Expand
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References

SHOWING 1-10 OF 73 REFERENCES
A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources
The determination of sources of acoustic wave motion in several dimensions from remote measurements is of considerable interest in many applications, and the underlying mathematical problem is quiteExpand
On the relation between coefficient and boundary values for solutions of Webster's horn equation
Webster’s horn equation is a normalized version of the one-dimension linear acoustic wave equation. It has been used extensively as a simple model for plane wave propagation in layered systems, andExpand
On the sensitivity of solutions of hyperbolic equations to the coefficients
The goal of this work is to determine appropriate domain and range of the map from the coefficients to the solutions of the wave equation for which its linearization or formal derivative is boundedExpand
On the relation between the velocity coefficient and boundary value for solutions of the one-dimensional wave equation
The one-dimensional acoustic wave equation is a simple model of wave propagation in layered media. As such, it is used in theoretical seismology to study the relation between sound velocity and theExpand
Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results
The nonlinear problem of inversion of seismic waveforms can be set up using least‐squares methods. The inverse problem is then reduced to the problem of minimizing a lp;nonquadratic) function in aExpand
On the modeling and inversion of seismic data
In this thesis we investigate some mathematical questions related to the inversion of seismic data. In Chapter 2 we review results in the literature and give some new results on wave equations withExpand
Acoustics of Porous Media
This book contains all that is currently known about the propagation of acoustic waves in porous media. Emphasis is on the physical aspects of wave propagation in porous media. Theory is developedExpand
Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary
Let Ω ⊂Rn be a smooth domain with boundary Γ. Let u be the solution of a second order hyperbolic scalar equation with homogeneous Neumann boundary conditions defined on Ω due to initial conditionsExpand
Methods of Mathematical Physics
Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues.Expand
Attractors of equations of non-Newtonian fluid dynamics
This survey describes a?version of the trajectory-attractor method, which is applied to study the limit asymptotic behaviour of solutions of equations of non-Newtonian fluid dynamics. TheExpand
...
1
2
3
4
5
...