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) |

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) |

- slant analysis, when
- parabolic filtration, when
- seismic tomography, when
*u*(_{0}*x*,*y*) =*s*(*x*,*y*) -*s*(_{0}*x*,*y*)*s*(*x*,*z*) is a slowness, is the equation of a ray with the starting point and end point , and ).

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
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) |

If we consider only the first of the equations (120)

(121) |

A discontinuity of wave-field *u _{0}*(

1/13/1998