The mapping between the depth and the vertical traveltime domain are defined by the following transformation of coordinates:
This transformation implies the following relationships between the partial derivatives of the traveltime that appear in the eikonal equation (1):
Substituting these partial derivatives in the eikonal equation (1) we derive the focusing eikonal equation
This property is easily demonstrated by performing the change of variable from z to defined in equation (2) in the integral that defines in equation (5). After this change of variable the expression for as a function of becomes
In the companion paper 1997 we analyze the errors caused by assuming equal to when the condition of equation (7) in not exactly fulfilled.
The previous result demonstrates that, as long as the condition of equation (7) is satisfied, reflection data can be focused without knowledge of the mapping velocity, and thus that the focusing step and the mapping step can be performed sequentially.
Notice that in an horizontally stratified medium the focusing eikonal becomes the eikonal for an elliptical anisotropic medium with normalized vertical ``velocity'' equal to 2. If the velocity is laterally varying, neglecting is equivalent to neglecting the thin-lens term in finite-difference time migration Hatton et al. (1981). Raynaud and Thore 1993 used this approximation to trace rays in the domain.
The presence of the differential mapping factor in the focusing eikonal makes the separation of the mapping and the focusing processes imperfect. Therefore the eikonal in equation (1) should be properly called quasi-focusing. An interesting development would be to substitute for the transformation of variables defined by equations (2) and (3) a different transformation of variables for which would be uniformly zero even in a laterally varying medium. We speculate that this transformation of variable is the one induced by the image rays Hubral (1977).
Finally we should notice that the expressions for evaluating given in equation (5), or even in equation (8), are not convenient when working in the domain, because they require the evaluation of spatial derivatives in the .It can be easily demonstrated that can be evaluated using the following expression,