On the solution of an inverse scattering problem in seismic while-drilling technology Original Research Article

This paper concerns the solution of an inverse scattering problem arising in seismic exploration. The problem to be analyzed is the prediction of the acoustic wave speeds ahead of a drill-bit device located at the center of a well bore and regarded as a seismic source, from measurements of the displacements at recording positions located near the surface of the exploration well. The forward model is stated in terms of displacements, assuming cylindrical symmetry with the symmetry axis located at the center of the borehole. Absorbing boundary conditions are employed at the artificial boundaries. The inverse problem is formulated as the minimization of a quadratic cost functional, which is solved using a nonlinear iterative procedure known as quasilinearization that uses the Fréchet derivatives of the displacements with respect to the wave speed parameters to update the parameter values during the iteration. Approximations to the solution of the differential equations associated with the forward problem and the Fréchet derivatives, which are needed to discretize the optimization procedure, were obtained using an explicit finite element procedure. Results on the existence, uniqueness and regularity of the differential problem and the Fréchet derivatives are derived, as well as on their continuity with respect to the parameter. Convergence results for the iterative estimation procedure are also given. Numerical examples illustrating the application of the algorithm to estimate the wave speeds in model problems within the context of the seismic while-drilling technique are presente