We know how to determine the image of a reflector's location on a migrated section. This location, of course, depends on the choice of the continuation velocity v_{c}. Being independent of a weight function , the location of caustics in the x,z-plane is determined by the system of equations
(102) |
(103) |
If at some velocity v_{c} = v_{ex} the caustic and the image cross each other at a special point , we call this the extremal velocity v_{ex}. Usually we have a continuous interval of extremal velocities: .Typically for each one can observe two or more special points. For some value v_{ex} = v^{*}_{ex} called the true extremal velocity, these special points unify in one point that appears as a bright spot.
Example: If the true model represents a homogeneous medium with velocity v and a planar reflector at depth h, the location of the reflector's image after CSP-migration is described by the system of equations (see Chapter 7):
To find the location of caustics, we substitute and .into system equations (102) and (103). So we have the system of equations
If for some and r we have and , then the reflector's image contains the special point .Calculations show that at v_{c} < v there is no special point on the image. At we have two symmetrical special points which merge together into the one bright point when (as shown in Figure ).
Instead of a complete investigation of the situation, we can choose some particular ray of the eikonal .It is not difficult to determine the meaning of for the given velocity v=v_{c} and the point belonging to the image and the ray ; we can easily check condition (103). Let us again consider CSP-migration with the true model containing a curved reflector in a homogeneous medium (Figure , solid line). Let us choose the ray which approaches the surface vertically. This ray is connected with a stationary point r_{s} of the function :
and we shall call this ray stationary. We shall denote h the true depth of the reflector and h_{m} the image depth for the x=r_{s}. If , then h_{m} satisfies(104) |
(105) |
The point belongs to a caustic if
(106) |
(107) |
Combining (106) and (107) we receive
(108) |
(109) |
As for we can use standard techniques based on the continuation of wave front curvatures along the ray. We shall omit all these calculations and give the final result:
(110) |
(111) |
(112) |
If the point is stationary for the reflector (), then
(113) |
If the reflector is planar and horizontal (k=0 and ), then
(114) |
v_{ex} = v | (115) |
Is this velocity v_{ex} the true extremal velocity? It is in all particular situations expressed by formulas (112) - (115).
In the zero offset case for arbitrary rays
and at If k=0 (planar reflector) we formally receive , but in this case the amplitude does not actually depend on the meaning of the velocity v_{c}.In the common mid-point case travel-time curves for a point reflector and for a flat reflector are the same. It means that in this situation v_{ex}=v.