Inverse kinematical problems of reflection seismology - I. Classical problems in laterally inhomogeneous media

Abstract

Summary. The first non-trivial inverse problem for media with non-horizontal reflectors z=h(x, y) was set up for a model of the type: V=V(z), 0 ≤z≤h(x, y), and the possibility of reconstructing the functions h(x, y) and V(z) at z↦ (minh, maxh) was shown. In the alternative case, when h= constant and V=V(x) there is a unique solution. Only particular cases were considered for media with h= constant, v=V(x, z). In the second half of the 1970s, the conditional correctness of a number of inverse problems was proved and the important concept of a sufficient data system was proposed. Over the last 20 yr much attention has been paid to layered homogeneous media with curved interfaces, which are reflectors and refractors at the same time. The task of continuing the eikonals second derivatives played a very important role in this problem. Using the connection between the second derivatives of the CDP travel-time curve and the eikonal from a phantom source at the base of the normal ray (V. Chernyak, S. Gritsenko, T. krey) there were obtained formulae of the Dix type. Recently methods based on linearization using a small parameter were proposed for media with slightly curved interfaces. A number of iterative algorithms for optimization and inversion have been developed, which exploit advances in the solution of direct kinematical problems. The development of the theory of inverse problems and the statistical theory of interpretation has led to the creation of a general concept of multistep algorithms and their classification.