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\title{3-D GEOPHYSICAL DIFFRACTION TOMOGRAPHY}
\author{Vladimir N. Troyan}{
Institute of Physics,  University  of  St.~Petersburg,
%University emb.~7/9, 199034,
St.~Petersburg, Russia}

     The basis  of interpretation  problem is  the model  of the connection
between  measurements  data  being  usually digital      records and  the
desired  medium parameters  fields. Using  the      linear  model  of  signal
propagation  we construct the linear      functionals  (tomography  functionals)
having  the property of      influence  function  of  variation  of  parameter
fields on the      digital  sampling.  This  influence  function  is the result of
interaction of  the incoming   ($\varphi_{\mbox{in}}$) and   inverse  outgoing
($\varphi_{\mbox{out}}$) sounding
signal fields  such as ($\varphi_{\mbox{in}}$)
 and ($\varphi_{\mbox{out}}$)     are
determined  by the model of  the signal   propagation  in      reference
medium.  The tomography  functional  determines all      of the elements  of
the spatial region in the  sampling of the      experimental data. It should
be noted that in the traditional      ray tomography,  the tomography
functional is  singular and is      localized along the ray connection the
source and the receiver;      the weight along  the ray is  constant. Whereas
in  diffraction     tomography every  element of  the spatial  region has  its
own      weight, which is determined by the  interaction of the  fields
($\varphi_{\mbox{in}}$)
 and  ($\varphi_{\mbox{out}}$).
We  note that  the mathematical  methods      of
computational tomography  are based on  the methods for  the      solution
of   the   integral   geometry   problems, when data      (projections)  are
given  by  integrals of parameter functions on  manifolds  of  smaller
dimensions  (rays,  two-dimensional      surfaces); whereas  in diffraction
tomography, the  tomography      functional support belongs to R . The basic
content of the tomography      experiment is relevant to  the overlapping  of
the  tomography      functional  supports,  i.e.  information  on  the same
element      of the volume  is contained in  the whole set  of measurements.
The measurements are connected both with changes in the localization      of
the group of sources  and with the localization, orientation      of the
receiver and the sampling time for dynamic fields. The apriori
presentation can be determined technique, for instance,      by given a
certain type of spatial symmetry. The determined form of giving the a priori
information makes it possible to  reduce      the tomography  functional to  a
space  of smaller  dimensions.      If the medium is assumed a priori to be
strati-homogeneous, the      tomography  functional support  is
one-dimensional; there the      corresponding  tomography functional  is  the
Radon projection      of the   general tomography    functional to   the
transversal      direction.  We  consider  the  construction  and
interpretation    of tomography functionals for scalar wave equation and for
Lame      equation.

             To built an adequate model of the experimental material,      it  is
necessary  to  introduce  a quantitative measure of the      influence of the
variations of each parameter on the observation      field. For this purpose, we
introduce the notion of information      sensitivity.  We  consider  the
information sensitivity of the      observation field with respect to a fixed linear
functional  of      the field of parameters to be the limit of the derivative  with
respect to signal/noise  ratio with signal/noise  ratio tending      to zero
from the shannon information.  For the  represented numerical  experiments we
	     used the modified algorithm of diffraction tomography. This
algorithm      may be considered as generalized gradient method. The aim of
this      experiments was to investigate the resolving power of the
observation      system formed by regular net of 3-component geophones.

The research is supported by Grant No. 96-05 65904 from the Russian Foundation
for Basic Research.

\begin{thebibliography}{99}
\bibitem{troyan1} Troyan V. N.,   Ryzhikov G. A.  Three-dimensional   diffraction
tomography  functionals.  {\em Proc.  of  2-nd  Geophys. Congress,
Thessaloniki,} p.~340--348, 1994.
\bibitem{troyan2} Ryzhikov G. A.,   Troyan V. N.  Tomography and  inverse problems
of  remote  sensing.  {\em St.~Petersburg  University Press,
St.~Petersburg,}  220~p. (in Russian), 1994.
\end{thebibliography}
\end{document}
