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## Application to tomography

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 u0(x,y) using the given field . We shall reduce this problem to a simpler one: to reconstruct the discontinuities of the field u0(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 u0(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

 q''=q+1, (131)

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