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Integral (126) describes the situation in the linear tomography when
rays can be parametricized as a function of and

| |
(129) |

The general problem of tomography is the reconstruction of the field *u*_{0}(*x*,*y*)
using the given field . We shall reduce this problem to a
simpler one: to reconstruct the discontinuities of the field *u*_{0}(*x*,*y*). The
possibility of this reduction follows from the fact that discontinuities are
a model for the most vivid particularities of the object.
So our task is to find such an operator , that

Let us introduce a dual IG-operator
which has the family of dual lines as a system
of stacking lines and an arbitrary kernel *w*^{'}=*w*^{'}(*M*,*N*).
It follows from the results of Chapter 12 that for any kernel *w*^{'}, the
operator gives geometrical reconstruction of the original
discontinuity of the field *u*_{0}(*x*,*y*) since this operator realizes the mapping
.It is clear that the operator transforms the dynamics of a discontinuity of the field in the
same way as the operator did. Then, according to formula (126)
| |
(130) |

where
We have taken into account that

Now, combining formulae (128) and (130), we have:
where

The operator acts like
and the operator acts like
.
This means that in order to reconstruct the
order and the index of discontinuities we shall apply the correction filter
.
It can be shown that independently
on the curve

| |
(132) |

consequently, the correction filter is
. It follows from equations (131)
and (132) that
But
since .
The kernal *w*^{'} must be chosen from the condition

I have proven that
*dd*^{'}=*FG*

where
Consequently,
| |
(133) |

So we have shown that
| |
(134) |

where *w*^{'} is expressed by Formula (133).
For example, let

| |
(135) |

then
If equation (135) is an equation of rays in ray tomography, then according
to equation (129)
and
Figures - give an illustration of reconstruction of the
step-function

given at the square (equation (126)).
Figure a shows the field . Figure b is a result of application
of the simplest operator at .
Figure c
is a result of application of the operator with *w*^{'}
chosen according to equation(135), And Figure d gives the field
It is interesting that Kirchhoff's operators of the forward and reverse
wave-field continuation originate a symmetrical pair of asymptotically inverse operators
and with

In the n-dimensional case we have the following analog of formula (134)

| |
(136) |

where the kernel *w*^{'} is expressed by the same formula (133) but *F* and
*G* are determinants of the correspondent matrices.
The formula (111) and (113) were published first by Gr. Beylkin (1984)
who applied a very different approach connected with the theory
of pseudodifferential operators.

** Next:** 15: COMPARISON OF PRE-
** Up:** 12: OPERATOR OF INTEGRAL
** Previous:** Dynamics of the main
Stanford Exploration Project

1/13/1998