# The fixed angle scattering problem and wave equation inverse problems with two measurements

## TL;DR

This paper proves unique determination and stability for inverse wave problems using only two measurements, advancing understanding of fixed angle scattering and geophysical inverse problems.

## Contribution

It introduces new uniqueness and stability results for inverse wave problems with minimal measurements, employing Carleman estimates and novel analytical techniques.

## Key findings

- Unique determination of compactly supported potentials from two far field patterns.
- Lipschitz stability estimate for the inverse problem.
- Extension of Carleman estimate methods to new inverse problem settings.

## Abstract

We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the far field pattern generated by plane waves coming from exactly two opposite directions. This implies that a reflection symmetric potential is uniquely determined by its fixed angle scattering data. We also prove a Lipschitz stability estimate for an associated problem. Motivated by the point source inverse problem in geophysics, we show that a compactly supported potential is uniquely determined from boundary measurements of the waves generated by exactly two sources - a point source and an incoming spherical wave. These results are proved by using Carleman estimates and adapting the ideas introduced by Bukhgeim and Klibanov on the use of Carleman estimates for inverse problems.