To appreciate the effect that the salt body has on the wavefronts, I will show 3-D plots of the wavefront ISO-surfaces, again using VIS5D, as they penetrate through the salt body. The complex, yet continuous wavefronts include body waves, as well as head-waves.

Figure 5

Figure 5 shows the eight ISO-surface plots of the wavefront at different times along with either a vertical or depth slice of the velocity field. The wavefront expands with time and its shape alters from the typical spherical shape as it penetrates through the salt body. The high salt velocity causes portions of the wavefront to bulge and expand, especially those parts that have penetrated through the salt. The top flanks of this bulge are dominated by the head-wave arrivals, which smoothly connect the portions of the wavefront that penetrated through the salt with those that traveled only through sediments. Clearly, these head-wave flanks have generally low curvature, especially along the vertical plane, indicative of their low amplitude. The low curvature of head-waves in the Cartesian coordinates will translate to high curvature in the polar coordinates. As mentioned above and by Alkhalifah and Fomel (1997), propagating high curvature waves is a weakness of the fast marching approach, especially when propagating diagonally with respect to the coordinate system. As a result, head-waves traveltimes, as we will see later, will include large errors in spherical coordinate implementation of the fast marching method.

The bottom flanks of the wavefronts in Figure 5 also have low curvature on the vertical plane. This portion of the wavefront corresponds to waves that traveled through the high velocity salt, and as a result experienced a lot of spreading. These flanks also replace another, but more important, solution of the Eikonal equation, waves traveling directly from the source, without going through the salt. These waves, however, arrived a little later, and as a result were discluded from the finite-difference solution of eikonal equation solution. Such arrivals and their implications will be discussed further in the head-waves and first arrivals section.

7/5/1998