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Travel Time Tomography

Inverse Problems are problems where causes for a desired or observed effect are to be determined. They arise in all fields of science and technology. An important example is to determine the density distribution inside a body from measuring the attenuation of X-rays sent through this body, the problem of "X-ray tomography."

Two types of inverse problems have intimate connections with a
Riemannian manifold with boundary in differential geometry. For
the first type of problem one considers solutions of the
Laplace equation on the manifold. One imposes a function *f*
everywhere on the boundary. Then there is a unique solution *u*
of the Laplace equation with these values on the boundary and
this determines the normal derivative of the solution at the
boundary. The question is how to determine the metric at each
interior point in the manifold if we know the normal
derivative on the boundary for all boundary values *f*. This
problem was solved by Matti Lassas and Gunther Uhlmann in 2001.
An example application is Electrical Impedance Tomography (EIT)
which has been proposed recently as a useful diagnostic tool.
In this case the question is to determine the internal
conductivity of a body by making voltage and current
measurements at the boundary. EIT might be an useful inverse
technique for early breast cancer detection since the
conductivity of breast tumor is significantly higher than of
normal tissue.

The second, seemingly unrelated inverse problem is to determine
the metric at each interior point in the manifold if we know
the geodesic distance *I(x,y)* between any pair of points *x*,*y* on
the boundary. This is called the boundary rigidity problem.
For example, our knowledge of the Earth's interior is
indirectly derived from surface measurements. Scientists use
the information provided by earthquakes to penetrate deeply
into its interior. Mathematically, this is an inverse problem:
Can one determine the sound speed of the Earth from travel
times of seismic waves? In this case the sound speed depends on
direction as well as position and is modeled as a Riemannian
metric. Finding the sound speed at each interior point is just
the boundary rigidity problem.

A connection between these two quite different inverse problems
was proposed, as a conjecture, at the lecture by Uhlmann at the
IMA summer school on on "Geometrical Methods in Inverse
Problems and PDE Control" in the summer of 2001. Leonid Pestov
and Uhlmann started to work on this problem at the IMA workshop
and in 2005 solved it for two-dimensional manifolds. They
showed that the geodesic distance data on the boundary from the
second problem enables one to to determine the map relating
the function *f* and its normal derivative on the boundary from
the first problem, hence to determine the metric. The result
has now been extended to manifolds in higher dimensions.