Introduction

Inverse scattering problems have to be solved in many scientific fields. Seismic inversion can be defined as the attempt to get a correct image of the subsurface from recorded seismograms (Claerbout, J.F., 1985). Iterative inversion tries to find the best fit between the registered data and synthetic seismograms generated by an earth model. It can be called a ``statistical'' approach because the best fit is defined in terms of statistics. Linear iterative inversion inverts by varying only the earth model to minimize the difference between observed and synthetic data in a Gaussian sense (Claerbout, J.F., 1989). Nonlinear iterative inversion also adapts the wave propagation matrix in each step of the iteration (Mora, P., 1987). In contrast to the iterative methods stand the non-iterative ``deterministic'' approach. Deterministic methods are also referred to as holography, migration, or tomography to distinguish them from the iterative, more statistical inversion methods (Mora, P., 1989). The theory of migration is based on the forward problem which can be solved exactly by the Kirchhoff integral. Migration constructs images by backpropagating only the homogeneous part of the registered wavefield. Migration algorithms have been programmed in different domains so that terms like Kirchhoff Migration (Schneider, W.A., 1978), the Phase-Shift Method (Gazdag, J., 1978), Stolt Migration (Stolt, R.H., 1978), and Radon Migration (Rueter, H., 1987) result. It can be shown that these migration methods are mathematically equivalent for a constant velocity medium (Tygel, M., and Hubral, P., 1990). Tomographic inversion methods gained in importance in seismology during the last decade. Traveltime and amplitude tomography describe the seismic wavefield by straight rays and, therefore, interpret the registered data as projections of the medium (Worthington, M.H., 1984). The projection and its inversion, the backprojection, can be described by the Radon transform pair (Deans, S.R., 1983). Diffraction tomography allows not only straight rays but all kinds of weak scatterer wavefields in a medium of constant background velocity (Woodward, M.J., 1989). Traditionally, reflection seismic inversion was based on rather heuristic ideas. A look over the shoulder of people working in non-destructive testing (Langenberg, 1986), on Radar techniques or medical tomography might be useful to achieve a deeper understanding of the theories and the vocabulary involved in the inversion business. A common misconception is that migration is an application of the Kirchhoff integral. Opposite starting points led to the development of tomography and migration. Many people are not aware of the relation between both. In this paper I will study the inversion problem in a general way. I introduce some terms used in the non-geophysical inversion theory and show how the different algorithms relate to each other. We will see that migration and diffraction tomography are nearly the same thing. Varying frequency and the angle of incidence for the signal and superposition of the different resulting images improve the quality of the reconstruction. Finally, we will come to the conclusion that today the nonlinear elastic inversion of seismic reflection and transmission data is the best algorithm that can be offered to image the earth's interior. First, the basics of the statistical and deterministic approach will be investigated.