Spectral approach to geophysical inversion by Lorentz, Fourier, and Radon transforms

Geophysical inversion seeks to determine the structure of the interior of the earth from data obtained at the surface. In reflection seismology, the problem is to find inverse methods that give structure, composition, and source parameters by processing the received seismograms. The pioneering work of Jack Cohen and Norman Bleistein on general inverse methods has caused a revolution in the direction of research on long-standing unsolved geophysical problems. This paper does not deal with such general methods, but instead gives a survey of some production-type data processing methods in everyday use in geophysical exploration. The unifying theme is the spectral approach which provides methods for the approximate solution of some simplified inverse problems of practical importance. This paper is divided into two parts, one dealing with one-dimensional (1-D) inversion, the other with two-dimensional (2-D) inversion. The 1-D case treated is that of a horizontally layered earth (Goupillaud model) with seismic raypaths only in the vertical direction. This model exhibits a lattice structure which corresponds to the lattice methods of spectral estimation. It is shown that the lattice structure is mathematically equivalent to the sturcture of the Lorentz transformation of the special theory of relativity. The solution of this 1-D inverse problem is the discrete counterpart of the Gelfand-Levitan inversion method in physics. A practical computational scheme to carry out the inversion process is the method of dynamic deconvolution. It is based on a generalization of the Levinson recursion, and involves the interacting recursions of two polynomials P and Q. This paper treats only much simplified 2-D models. One 2-D method gives the forward and inverse solution for a horizontally layered earth (Goupillaud model) with slanting seismic raypaths. This method involves the Radon transform which is often called "slant stacking" by geophysicists. The other 2-D methods given in this paper are concerned with the process of wavefield reconstruction and imaging known as "migration" in the geophysical industry. A major breakthrough occurred in 1978 when Stolt introduced spectral migration which makes the use of the fast Fourier transform. Another method, the slant-stack migration of Hubral, is based on the Radon transform.