A variational approach to an elastic inverse problem

We present a variational approach to the seismic inverse problem of determining the coefficients C and p of the hyperbolic system of partial differential equations (j,k,l)Sigma partial derivative/partial derivative x(j) (C(i,j,k,l)(x)partial derivative/partial derivative xi u(k)(x,t)) = p(x)partial derivative(2)/partial derivative t(2) u(i), 1 <= i <= n, 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 -Sigma del(.) (Ci,k del uk(x,s))+ ps(2) u(i)(x,s) = 0 1 <= i <= n. 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 p 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.