Discussion

I just showed that the inclusion of the velocity-depth trend increased significantly the match between the original and the inverted sonic log. The question now is: Why didn't we get an even better match in the sonic logs? An obvious explanation is that the sonic log in this case was sub-sampled to 10 feet which is still a very thin interval when it comes to the depth resolution that we can achieve with typical seismic data. In order to get a better match we would need higher frequencies, so there is always a limit in the quality of the match that we can reasonably expect to obtain in the inversion of the sonic log. To illustrate this point, Figure 5 shows the same comparison as in the bottom panles of Figure 4 except that the peak frequency of the Ricker wavelet was halved. Clearly, the lack of high frequencies have a very detrimental effect in the quality of the match of the sonic logs. See in particular the depth intervals between about 300-400 ft, 450-550 ft and 700-800 ft.

 

L2_inv_trend_low_freq
Figure 5 Comparison of reference and matched trace (left panel) and of original and inverted log using the L₂ norm when the velocity-depth trend was taken into account and the peak frequency of the Ricker wavelet was halved.

It should also be noted that the example here is not very realistic because I used the exact same approach to compute the seismic data to be matched and to compute the trial solutions. In a real situation the reference trace will be obtained from a real seismic survey whereas the trial solutions will still be computed with a procedure similar to the one used here (perhaps improved by allowing random noise or multiples or attenuation). Therefore, there will be reasons other than the velocity-depth trend or the frequency content of the data that will play a role. The estimation of the seismic wavelet, for example, is well-recognized as a very important issue when inverting for real data.