Theory shows that even in a simple 1-D *v*(*z*) medium, CAM is not
perfectly accurate: the stationary-phase approximation used in the
derivation of CAM Biondi and Palacharla (1996) imposes the relation among ray
parameters indicated below that constrains rays to keep the same
azimuth at each depth step (Figure 1):

(1) |

ray-comaz
Ray geometry imposed by
common-azimuth constraints: both receiver and source rays keep the
same azimuth at each depth step.
Figure 1 |

We choose to test the behavior of rays and the accuracy of CAM approximations in a synthetic medium where the velocity varies linearly. In such a medium, ray trajectories can be computed analytically, as well as all ray parameters. Figure 2 illustrates the geometry of the problem. With those notations, ray curvature can be expressed as Aki and Richards (1980):

(2) |

(3) |

Figure 2

The ratio is also the horizontal component of the slowness vector along the ray, which therefore is also a constant:

(4) |

After some calculation (see Appendix), the equation of the circle of
radius *R* passing through point source with initial incident angle
is, in the plane :

(5) |

For any given triplet of points (S,P,R), respectively source, image point and receiver locations, there exist only two circles satisfying equation (5) that form the complete ray path. Figures 3 to 5 illustrate such ray paths. We can verify that the projections of the source ray and the receiver ray on the cross-line plane do not coincide in general and therefore break the assumption of azimuth conservation in CAM downward-continuation imposed by relation (1).

ray_exmpl1
Example of 3-D analytical ray
tracing, with the three projections of both rays on vertical and horizontal
planes. Source and receiver location are indicated with solid
stars. The reflection point and its three projections are represented
by a circle. Offset is 3000m (in-line). Velocity is .Figure 3 |

ray_exmpl2
Same geometry as before. The only
difference is in the velocity: . The source ray has
overturned because of the stronger velocity gradient.
Figure 4 |

ray_exmpl3
Velocity law is . The reflection point is at an equal distance from source
and receiver: the problem is symmetrical and azimuth is conserved at
each depth step.
Figure 5 |

However, with too strong a velocity gradient, rays quickly start to overturn (Figure 4). The corresponding reflection cannot be imaged with one-way wave propagation methods, such as CAM and the other wave-equation migration methods we discuss in this paper.

4/28/2000