Inverse problem for Goupillaud-layered earth model and dynamic deconvolution

This paper presents, in a tutorial form, some analytical inversion techniques for the Goupillaud-layered earth model. Finding the reflection coefficients from the reflection seismogram is the inverse problem for the model. For this reason we present a thorough description of the inverse problem for the Goupillaud model, two solutions to solve the inverse scattering problem using linear discrete equations and the solution obtained using the classic dynamic deconvolution method. The inversion is achieved using Robinson's polynomials P-k(z), Q(k)(z), A(k)(z) and their reverse polynomials, as well as some properties of the model (the Lorentz transformation and the Einstein subtraction formula). The method of dynamic deconvolution, which makes the inversion of the model very simple computationally, is based on the physical structure of the reflection seismogram. We present the classic dynamic deconvolution algorithm for the non-free-surface Goupillaud model to show that the dynamic deconvolution method can provide efficient discrete procedures for the inversion. For this reason, though the inverse dynamic deconvolution procedures are old algorithms, they could be useful today for solving inverse scattering problems arising in exploration geophysics and various fields.