Abstract
The starting point of this work is the inversion of vertical seismic profiling (VSP) data. The usual processing of VSP data by inverse techniques is restricted to 1D propagation model. In this case, the parameters to identify are the acoustic impedance as function of travel time and the seismic source so that we have as unknowns two functions of one variable and as data a function of two variables, the time and the depth positions of geophones. The problem is thus largely overdetermined and an elementary mathematical analysis can be made. The source is modelled as a boundary condition at the top of the geophones zone. So this boundary condition replaces the true source function and the medium parameters above the geophones zone. The question asked by V. Richard from IFP was the “management” of this unknown source when 3D propagation effects are taken into account in horizontally layered medium where the propagation equations are parametrized by the k parameter of the Hankel transform. Now we think that the answer is that it is impossible to work round the fact that there are at least two unknown functions, the source and the medium parameters above the geophones zone. During this study, we have searched for some non local boundary conditions and this was the opportunity to obtain some results on exact transparent conditions for 3D propagation in 1D media (preliminary communication was made by Petit and Cuer (1994)) and on the discretization of such conditions in the acoustic case (preliminary communication was made by Cuer and Petit (1995)). This is the mathematical substance of this work in which the Poisson summation formula is used to prove the stability of a discrete non local boundary condition.
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Cuer, M., Petit, J.L. (1997). On the inverse seismic problem for horizontally layered media: Subsidiary study. In: Chavent, G., Sabatier, P.C. (eds) Inverse Problems of Wave Propagation and Diffraction. Lecture Notes in Physics, vol 486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105777
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DOI: https://doi.org/10.1007/BFb0105777
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