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UBC Theses and Dissertations

Inversion of reflection seismograms Levy, Shlomo


A method for the estimation of impedance or pseudo-velocity sections from the information contained in CMP stacked sections, the corresponding stacking velocities and sonic and density logs (when available) is presented. The method relies on a linear programming approach for the reconstruction of full-band reflectivities, and utilizes linearized relations between the multiple free reflectivity functions and average or point-wise impedance or velocity values. The reconstruction procedure requires the solution of an underdetermined set of equations and hence a minimum structure condition is imposed on the desired solution. This condition guaranties the uniqueness of the obtained solution in the sense that it is the solution that features the least amount of impedance variations as a function of travel-time (or depth). Since the presented inversion yields minimum structure solutions, it is argued that features which appear on the obtained result are strictly demanded by the data and are not artifacts of the inversion scheme. A number of physical assumptions are required by the presented inversion. These are summarized below in point form: (1) The earth reflectivity function is non-white and can be reasonably represented by a sparse spike train. (2) The observed CMP stacked section is a reasonable representation of the multiple-free normal-ray section with reasonably correct relative amplitude relations. (3) The residual wavelet on the stacked section is to a good approximation a zero-phase wavelet with a relatively flat spectrum. (4) The estimated stacking velocities can be inverted to yield an acceptable representation of the averages of the true earth velocity model. Since in a realistic environment some of the above assumptions may be violated, all the corresponding relations in the presented inversion scheme include appropriate uncertainty terms. That is, all the information components considered in the inversion are satisfied only to within some prespecified error bounds. A number of possibilities for speeding up the inversion scheme are described. It is shown that utilizing the expected trace-to-trace coherency of seismic reflection data yields considerable reduction in computational efforts. Finally, a number of steps required for a successful completion of the inversion are described. In particular, the problems of preinversion data scaling and the correction of the residual wavelet's phase are discussed in some detail.

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