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
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:
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 ).
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
If we consider only the first of the equations (120)
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'.