Biases in the inversion

The main problem considered in the previous sections was how the limited view of the measurements affect our ability to estimate velocities in different directions. By assuming elliptic anisotropy it was necessary to estimate only two velocities: horizontal and vertical. Of course, this is too simple to describe the real complexities of the velocities in many cases but still, it is the first step beyond fitting the data with circles (isotropic tomography). We have seen that unless we constrain considerably the inversion (layered models) or we have measurements from a wide range of angles, it is difficult to estimate accurately and simultaneously S_x and S_z. Unfortunately, even if these conditions are satisfied, many other factors may affect the results. Among this factors we have:

• Picking errors. These errors may increase or decrease systematically the velocities, depending on which part of the first arriving wavelet has been picked. Picking before the correct value speeds up velocities whereas picking later slows them down. This may explain why in Figure 15 V_z is systematically 1 or 2 % faster than the sonic log.
• Well deviation. As we said before, the wells deviate in 3-D but we decided to work in 2-D. If the real 3-D variations in the medium are moderate, this is a good approximation but it may not be otherwise. When first testing our algorithm with real data the well deviation was not considered. We just substituted each well by a vertical one located at its average surface location. The results were (not shown) higher velocities (than those shown in Figure 12) where the wells were actually closer and lower velocities where the wells were actually farther apart. Considering the well deviation affected S_x more than S_z.
• Head waves vs. body waves. Although this may be considered a picking error, it affects primarily traveltimes at near offsets (small ray angles) in low velocity layers. These errors affect mainly the estimation of S_x because S_z does not use information from rays that travel at small angles. In principle, when head waves are inverted like body waves the estimated horizontal velocity turns out to be faster than the real one.
• Ray bending.

All the previous factors, when not considered appropriately, may produce artificially anisotropic results. For this reason and the ill-conditioning of the problem studied later, the estimation of small scale variations in velocity anisotropy is a difficult task.