Inversion with the Velocity-depth trend

The velocity-depth trend of Figure 3 was used to constrain the possible values of velocity for each sample of the sonic log. Figure 4 shows a comparison similar to that of Figure 2 but considering the velocity-depth trend. The match for the seismic traces did not change much, since it was already very good without the trend, although a comparison of the top left panels in Figures 2 and 4 does show some improvement in the trace match (for example between times 0.04 and 0.065 s). The match of the sonic log, however, changed significantly, and now, without the distraction of the lack of the trend, we can see that the inverted sonic log matches the original log very well, except for depths between about 700 and 800 feet where the match is a little off in depth. Similar situation can be seen in the bottom panels which show the same comparison when the L₂ norm was employed. The conclusion is similar as for the L₁ norm, that is, the match of the sonic logs increases greatly, although it is still far from perfect, illustrating the limitations of the method.

 

L1L2_inv_with_trend
Figure 4 Results for inversion with the velocity-depth trend. Comparison of the synthetic trace and the inverted trace (top left panel) and the original log and the inverted log (top right panel) using the L₁ norm. The bottom panels show similar comparisons using the L₂ norm.