8 research outputs found
Dynamic inverse problem in a weakly laterally inhomogeneous medium
An inverse problem of wave propagation into a weakly laterally inhomogeneous
medium occupying a half-space is considered in the acoustic approximation. The
half-space consists of an upper layer and a semi-infinite bottom separated with
an interface. An assumption of a weak lateral inhomogeneity means that the
velocity of wave propagation and the shape of the interface depend weakly on
the horizontal coordinates, , in comparison with the strong
dependence on the vertical coordinate, , giving rise to a small parameter
\e <<1. Expanding the velocity in power series with respect to \e, we
obtain a recurrent system of 1D inverse problems. We provide algorithms to
solve these problems for the zero and first-order approximations. In the
zero-order approximation, the corresponding 1D inverse problem is reduced to a
system of non-linear Volterra-type integral equations. In the first-order
approximation, the corresponding 1D inverse problem is reduced to a system of
coupled linear Volterra integral equations. These equations are used for the
numerical reconstruction of the velocity in both layers and the interface up to
O(\e^2).Comment: 12 figure
Inverse problem for wave equation with sources and observations on disjoint sets
We consider an inverse problem for a hyperbolic partial differential equation
on a compact Riemannian manifold. Assuming that and are
two disjoint open subsets of the boundary of the manifold we define the
restricted Dirichlet-to-Neumann operator . This
operator corresponds the boundary measurements when we have smooth sources
supported on and the fields produced by these sources are observed
on . We show that when and are disjoint but
their closures intersect at least at one point, then the restricted
Dirichlet-to-Neumann operator determines the
Riemannian manifold and the metric on it up to an isometry. In the Euclidian
space, the result yields that an anisotropic wave speed inside a compact body
is determined, up to a natural coordinate transformations, by measurements on
the boundary of the body even when wave sources are kept away from receivers.
Moreover, we show that if we have three arbitrary non-empty open subsets
, and of the boundary, then the restricted
Dirichlet-to-Neumann operators for determine the Riemannian manifold to an isometry. Similar result is proven
also for the finite-time boundary measurements when the hyperbolic equation
satisfies an exact controllability condition
Inverse Problem in a Weakly Horizontally Inhomogeneous Layered Medium
In this paper we consider an inverse boundary-value problem for an acoustic wave equation in the half-space z> 0, Autt − div(B∇u) = 0, u|t<0 = 0, u|z=0 = δ(x)δ(t), (1) with the inverse data of the for