Amplitudes

The amplitude calculation requires that the slowness models are smooth. I apply the algorithm to compute the amplitudes for a medium having a velocity function that increases linearly with depth. The velocity gradient is 1.5 1/sec. Figure shows the amplitudes computed for a line source having an isotropic radiation pattern. The computed amplitudes for a point source having an isotropic radiation pattern are shown in Figure . Comparing these two figures, we see that for a point source, the amplitudes decay faster than for a line source. Both figures show that large errors appear in a narrow region just below the source position. These errors are due to the approximation of using constant velocity cells in polar coordinates. To solve this problem, I have proposed a new approach called the local paraxial ray method (Zhang, 1991). I expect that the local paraxial ray method can handle the amplitude calculation for slowness models containing reasonably large contrasts.

 

ampgrals
Figure 5 Amplitudes for a model with a velocity function that is linearly increasing as a function of depth. The source is a line source with an isotropic radiation pattern, located at the origin. The right and bottom panels show cross sections of the amplitudes as functions of depth and horizontal distance, respectively. The dashed lines indicate the positions of the two cross sections.

 

ampgraps
Figure 6 Amplitudes for a model with a velocity function that is linearly increasing as a function of depth. The source is a point source with an isotropic radiation pattern, located at the origin. The right and bottom panels show cross sections of the amplitudes as functions of depth and horizontal distance, respectively. The dashed lines indicate the positions of the two cross sections.