The well‐known instability of Kunetz’s (1963) inversion algorithm can be explained by the progressive manner in which the calculations are done (descending from the surface) and by the fact that completely different impedances can yield indistinguishable synthetic seismograms. Those difficulties can be overcome by using an iterative algorithm for the inversion of the one‐dimensional (1-D) wave equation, together with a stabilizing constraint on the sums of the jumps of the desired impedance. For computational efficiency, the synthetic seismogram is computed by the method of characteristics, and the gradient of the error criterion is computed by optimal control techniques (adjoint state equation). The numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise (tests include noise levels of 50 percent). The inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal as well as external multiple reflections. Reflection coefficients obtained by this method show an excellent agreement with well‐log data in a case where standard estimation techniques [deconvolution of common‐depth‐point (CDP) stacked and normal‐moveout (NMO) correction section] failed.
HEMON, Ch., 1978, Wave equations and models, Geophysical Prospecting 26, 790-821.The method of finite differences is applied to the computation of multi-dimensional synthetic seismograms. This paper gives a study of the mathematical and numerical formulations of the problem, the boundary conditions, the convergence conditions and how to simulate the source in both one solid or a liquid. It is shown that the numerical formulation chosen is valid both for direct and inverse problems (ie. for modeling and migration).This formulation makes it possible to use the normal incidence reflection coefficients for P and S waves, whether they travel horizontally or vertically. The examples shown have been chosen on purpose in order to be easily interpreted.They do not give a full idea of the possibilities of the algorithm which allows to consider non-planar interfaces, except close to the vertical axis. RÉSUMÉDans le cas de milieux héterogènes, on applique la methode des différences finies au calcul des sismogrammes synthétiques à plusieurs dimensions. On étudie successivement les formulations mathematique et numérique du problème, les conditions aux limites, la convergence de la solution et la simulation de la source, dans le cas solide comme dans le cas liquide. On montre que la formulation numérique adoptee convient aussi bien au probleme direct (modèles) qu'au problème inverse (migration) ; elle a également la particularité de faire apparaître explicitement les coefficients de réflexion des ondes P et S en incidence normale, qu'elle soit verticale ou horizontale.Les exemples présentés sont en général simples pour permettre une interprétation relativement facile, mais ils ne donnent qu'une petite idCe des possibilités du programme de calcul. 11 permet notamment de considerer des couches à interfaces de forme quelconque, sauf au voisinage de l'axe vertical.
Multispectral recordings used for remote sensing are not without analogy with seismic records from CDP field set‐ups. These seismic data may be regarded as “photographs’ of deep regions of the earth taken from various angles. The Karhunen‐Loève (K.L.) transformation is commonly used for multispectral data processing, where it helps emphasize some features of remote sensing information. The same method may be applied to seismic data processing. Signal‐to‐noise ratio is improved on synthetic or field examples when K.L. transformation is applied instead of conventional CDP stacking. Residual statics seem to be diminished by a significant factor.
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