A variational approach to an elastic inverse problem

Abstract

We present a variational approach to the seismic inverse problem of determining the coefficients C and ρ of the hyperbolic system of partial differential equations from traction and displacement data measured on the surface. A crucial point of our approach will be a transformation of the above system to an elliptic system of partial differential equations Thus, we transform the inverse problem for a hyperbolic system to an inverse problem for an elliptic system. We give a definition of the direct and inverse seismic problem, where we distinguish between the isotropic and anisotropic cases. Further, we develop the theoretical results that we need for a successful recovery procedure of the coefficients C and ρ in the isotropic case. Our approach consists of a minimization procedure based on a conjugate gradient descent algorithm. Finally, we present various numerical results that show the effectiveness of our approach.