Parameter Estimation in a Hyperbolic Partial Differential Equation with a Focused Source as Initial Value

We study the problem of recovering the continuously varying wave speed in the one-dimensional wave equation with a focused source as initial data. In this paper this inverse problem is transformed into a parameter estimation problem, which can be solved efficiently. The wave speed can be recalculated by solving an ordinary differential equation of second order where the parameter of the transformed inverse problem enters as a coefficient. We present a regularized finite difference scheme inversion for the stable recovery of the solution of the transformed parameter estimation problem, which combined with the solution of the ordinary differential equation, gives an estimation for the sound speed.