The main discontinuity of the field is located along the line if any of the following conditions is fulfilled (see Figure ):
Each operator defines a one-to-one mapping of a set of curves in the plane (x,y) into the set of curves in the plane . This mapping does not depend on the kernel and is determined only by the family of stacking lines .
There is a one-to-one correspondence between points of curves and (Figure ):
The curve is a special curve of a mapping :
The operator , with dual lines as stacking lines, produces the mapping inverse to :
that follows immediately from the symmetry of the two conditions in the first statement.
If the curve ends at the point N0, then the field has an edge discontinuity that is located along dual line (Figure ).
If the aperture X has the end a that does not depend on and belongs to the interval of determination of the curve , then the field has an edge discontinuity that is located along the dual line at .
If the value a depends on values and , then the field contains a discontinuity on the line satisfying the equation
(see Figure ).
be a traveltime curve of the reflected wave for this pattern and a given reflector. Does an IGO exist that (at given velocities) transforms the discontinuity on the line into a discontinuity coinciding with the position of the reflector?
Let us suggest that at the given velocity v(x,t) travel time inversion for the pattern (123) has a unique solution. This suggestion makes sense if it embraces situations such as CSP-pattern , CRP-pattern or zero-offset . We shall show that for homogeneous media a sufficient (but unnecessary!) condition is
The point we are looking for is the point on the isocron for which
where is a monotonically decreasing function. Then at condition (124) the sum
is also a monotonic function and it means that equation (125) has a unique solution. The condition (124) is essential as it follows from the example of CMP-pattern , when there are exactly two symmetrical solutions to equation(125). It is easy to notice that in the case of common-offset pattern
Uniqueness of travel time inversion means that there is a mapping of a set of travel time curves into the set of reflectors (at a given ). We pose the following general problem. Let be an arbitrary one-to-one mapping of an arbitrary set of lines in the plane (x,z) into a set of lines in the plane .Does the IGO exist for which ? If this operator exists, we shall say that operator realizes the mapping .Of course we must determine only the family of stacking lines : the kernal of the operator can be chosen arbitrarily.
It can be proved that there are only three alternatives:
Of course the second alternative is the most important for us.
Let us give an example of the second alternative. We assume that is the traveltime curve for an acceptable pattern and is a reflector on a depth section. It is easy to notice that the ray geometry depends only on the location of the reflector and the slope of the tangent (see Figure ). It immediately follows from this that all conditions for the existence of the second alternative are fulfilled, so there is an operator of integral geometry ( at ) that realizes the mapping of traveltime curves into reflector images.
It is evident that special lines of this mapping are traveltime curves for point reflectors.
We shall continue listing some general statements.
Let us define and such that
where is the position of a reflector on the depth section. We have shown above that is realized. The possibility of realizing the transformation of depth section into the time section t0(x) is obvious so the operator is realized. Stacking lines of the corresponding operator are travel time curves from the circular reflector with center at x and radius , because this reflector gives a point on the zero offset time (non-migrated) section, and consequently originates a special curve of mapping .Another example: cascaded migration. Let be determined by the system of stacking lines:
and by the system of stacking lines:
The system of stacking lines for the operator in the general case is determined by the formula
at where the function is derived from the equation
Application of all these equations to cascaded migration gives the system of stacking lines:
Now we shall give an example of the first alternative: the case of head waves (Figure ). It is easy to show that in the case of the CSP-pattern there is a unique solution of travel time inversion for head waves ( for a given v1 and v2 ). But touching of refractors does not guarantee touching of traveltime curves. So the mapping is not generally realized.
Nevertheless, it can be done for some special cases. For example, when is a set of planar refractors, this produces an operator which approximately realizes the mapping for refractors with small curvature.
In Chapter 7 we have testified that image ray technique does not guarantee exact time-to-depth migration if time migration was performed in a simpler model of medium. Does the correct time-to-depth migration algorithm in this case exist? The answer is positive. The mapping we are looking for is a product of three mappings: scaling operator that, with help of substitution (vc - continuation velocity for time migration), connects reflector images in time-sections with reflector images in depth section; mapping that connects with zero-offset travel-time curves in a simpler model of medium; and mapping that connects to reflectors in a true model. So . Mappings and are realizable, so their product is realizable too.
Finally, an example of the third alternative: CMP stacking. Traveltime is always an even function so any family of even curves guarantees touching at (Figure ). This means that CMP stacking always gives a section that is geometrically equivalent to t0(x).