The inverse scattering problem for cavities with impedance boundary condition

Abstract

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ∂ D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ∂ D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.

The interior inverse scattering problem for cavities with an artificial obstacle

“…The interior inverse scattering problems are fairly new research topics for reconstructing the shape of a cavity from the measurements taken on a curve or surface inside the cavity [10,11,[17][18][19][20][21][22][23]. These problems occur in many industrial applications of non-destructive testing where both the sources (incident waves) and measurements (scattered waves) are inside the cavity [11,19].…”

“…The uniqueness of the inverse problem has been established in [17,19,23] for the Dirichlet boundary condition, in [17,20] for the impedance boundary condition and in [10] for the mixed boundary condition. Here we note that these uniqueness results are proved under Assumption 1.1.…”

The MFS for the identification of a sound-soft interior acoustic scatterer

“…This inverse mathematical formulation models the determination of the extent of a reservoir from the data obtained by lowering a transmitter-receiver combination through a bore hole into the reservoir [23]. This interior problem is apparently more difficult than the usually addressed exterior scattering problem because all the scattered waves are now trapped inside the domain and are repeatedly reflected off its boundary [23,22], see, also, [24]. In addition, this problem is still nonlinear and ill-posed and therefore difficult to solve.…”

“…In addition, this problem is still nonlinear and ill-posed and therefore difficult to solve. In particular, in [23] the determination of the boundary of a sound-soft scatterer was achieved using the linear sampling method while the same method was used for the recovery of an impedance boundary in [24]. In [12], the structural integrity of a cavity was tested using the Kirsch-Kress approach applied to the analytical extension of the scattered field to the interior of the tested domain via the solution of the corresponding Cauchy problem.…”

“…The more general and difficult interior inverse scattering problem given by equations (2.2), (2.7) and the boundary condition (2.3) with the impedance boundary operator (2.4d), where λ may or may not be known, [24], is deferred to a future study.…”