** Next:** ITERATIVE INVERSION
** Up:** Kneib: Migration, tomography, seismic
** Previous:** Kneib: Migration, tomography, seismic

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.

** Next:** ITERATIVE INVERSION
** Up:** Kneib: Migration, tomography, seismic
** Previous:** Kneib: Migration, tomography, seismic
Stanford Exploration Project

1/13/1998