β Scribed by Juan E. Santos
No coin nor oath required. For personal study only.
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 Fr e echet 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 Fr e echet 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 Fr e echet 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 presented.
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A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging p
## Abstract The oneβdimensional SchrΓΆdinger equation is considered when the potential is real valued, integrable, has a finite first moment, and contains no bound states. From either of the two reflection coefficients of such a potential the right and left reflection coefficients are extracted corr