Application

We apply equation (2) on a five slanted planes model Vaillant and Biondi (2000). The algorithm is given in Appendix C. The dips are 0, 15, 30, 45 and 60, and their azimuth is 45. The velocity function is v(z)=1.5+.5z km.s^-1. Because of the velocity gradient, there is ray-bending. The ray beding induces a rotation of the soure-receiver axis which breaks the common-azimuth assumption. The steeper the plane, the stronger the bending of the rays and the wider the rotation of the azimuth of the reflection ($\beta$). Figures , and show the reflections for three different azimuths of the reflection. 2) for 3 different values of the reflection azimuth angle.

 

stack-cig-azim0-data1 Figure 5 The reflection azimuth angle ($\beta$) is 0.The angle gather associated to the horizontal reflector is well defined. For steeper reflectors, the quality of the gathers is deteriorated.

 

stack-cig-azim12-data1 Figure 6 The reflection azimuth angle ($\beta$) is $12^\circ$.It is a good estimation for the azimuth of the reflection dipping at $45^\circ$ at 1,150 meters. But it is not for shallower or deeper reflectors.

 

stack-cig-azim18-data1 Figure 7 The reflection azimuth angle ($\beta$) is 18. At this azimuth, the best defined gather is for a $60^\circ$ dipping reflector. This value of $\beta$ is not suitable for flatter reflectors.

Figure has been computed for a reflection azimuth set to zero ($\beta=0$). The best imaged reflectors are the least dipping ones. Indeed, when the reflector is close from the horizontal, the bending is not important and the rotation of the source ray with respect to the receiver ray is negligeable. The azimuth displayed in Figure is $12^\circ$. It means that when the rotation of the rays is $12^\circ$ at the reflection point then the reflector is well imaged. For such amount of rotation, the reflector must have a reasonable dip. That is what we observe in Figure : the plane with a $45^\circ$ dip displays a wide and sharp angle gather. In Figure , the azimuth selected is even bigger. It is for this amount of rotation that the angle gather for the $60^\circ$ dipping reflector is the best defined.