A solution of an inverse problem in the 1 D wave equation application to the inversion of vertical seismic profiles

We deal with the inversion of a vertical seismic profile in 1D. A seismic source being located at the vicinity of the earth surface we measure the vibratory state at different depths in a well. We have to find the distribution of acoustic impedance versus depth from these measurements. The excitation resulting from the seismic source is unknown. So we have to identify both the distributed parameter (acoustic impedance) in the 1D wave equation and the Neumann boundary condition at one edge of the domain from an observation of the vibratory state in a part of the domain. The inverse problem is very close to the inversion of seismic surface data which was studied previously [2]. We shortly recall some mathematical results (uniqueness and stability of the solution) and the solution of the optimization problem which is here of large size (∼ 1500 unknowns). The numerical examples show the efficiency of the proposed solution and the interest of such an approach for the geophysicist: the redundancy available in the data allows a reliable inversion of strongly noise corrupted data provided that the proper mathematical constraints on the solution have been implemented to ensure stability.