Next: Geometry of the main Up: Goldin: Method of discontinuities Previous: Zero-offset migration

# 12: OPERATOR OF INTEGRAL GEOMETRY (IGO)

We have connected migration with the idea of wave-field continuation. The great advantage of this approach is the possibility to introduce different continuous techniques'' to particular finite difference algorithms. Although this approach was successful in the case of both CSP and zero-offset patterns, it failed in other situations. However, it is not necessary to base migration on wave-field continuation even for the above particular patterns. We have integral transformations
 (118)
and

which we can perform in one step.

One may guess that equation (118) is a general form of the transformation of seismic records into the depth (migrated) section. It pushes us to investigate integral transformations of the form:
 (119)
This kind of transformation is typical not only in the problem of migration, but also in several other cases:

• slant analysis, when

• parabolic filtration, when

• seismic tomography, when

u0 (x,y) = s (x,y) - s0 (x,y)

(here s(x,z) is a slowness, is the equation of a ray with the starting point and end point , and ).
It is also possible to find other examples. Integral transform (119) belongs to the class of integral geometry operators.

In the K-operator case, the method of discontinuities was only one of the possible alternatives; another was the stationary phase technique. For the IGO operator, the method of discontinuities is the only approach that allows us to apply a relatively elementary mathematical concept.

Let us presume that

Again the main idea is that differentiation of the field , with respect to , allows us to transform a discontinuity of the field u0(x,t) into a -function. But in this situation it is not so simple because we have essentially used properties of discontinuities of arbitrary order and index. (Some of them were mentioned above.) Now I will add one more: Let t0 be the only solution of the equation , then

where , and .After that we get

Let's summarize the main concepts behind this technique. We shall consider the function as a family of curves in the plane (x,y) , that depends on the point of the plane (see Figure ).

The curve is called a stacking line. We shall suggest that a given family of stacking lines is regular. It means that for any point N =(x,y) and any value the system of equation
 (120)
with respect to M has one and only one solution (is a given open set). In other words, through the given point N in the given direction p runs only one stacking line.

If we consider only the first of the equations (120)
 (121)
(with fixed x and y) then we have a continuous line of points M satisfying the equation (121). This line is called a dual line (see Figure  ) and denoted as .

A discontinuity of wave-field u0(x,y) located along the line influences the appearance of several discontinuities of the field .We shall call the main discontinuity a discontinuity of the field with minimal order q'.

Next: Geometry of the main Up: Goldin: Method of discontinuities Previous: Zero-offset migration
Stanford Exploration Project
1/13/1998