Anisotropy/velocity update

Anisotropy introduces error into seismically-estimated velocities. In a homogeneous, elliptically anisotropic medium with a single flat reflector, standard velocity analysis measures v_H, the horizontal wave velocity, rather than the vertical velocity v_V. An elliptically anisotropic medium is defined by the parameter

 

$$\mbox{\boldmath$\alpha_{aniso}$} = \frac{v_V}{v_H}$$ (10)

Equation (3) appears to be a complicated way to express the vertical misfit between $\bold H^{\prime}_i(x,y)$ and $\tilde{\bold H}_i(x,y)$, but it can be shown that $\alpha_{known}$ in Equation (3) is equivalent to $\alpha_{aniso}$ in Equation (10) for the simple medium discussed above. Given the flat reflector and constant $\alpha_{aniso}$ assumptions, Equation (8) computes $\alpha_{aniso}$ for all x and y, solely from the vertical seismic/well log misfit.

Though I have not yet been able to prove it, I believe that in a more complex medium, the $\alpha$(x,y) from Equation (8) is an ``RMS'' $\alpha$. From Equation (10) we obtain at the well locations the horizontal wave velocity v_H in the overlying medium. At other locations, the $\alpha$(x,y) from Equation (8) gives a reasonable estimate of v_H.

For multiple $\tilde{\bold H}_i(x,y)$, one could imagine vertically interpolating the $\alpha$(x,y) surfaces corresponding to each one, thus yielding a 3-D $\alpha$ cube which could then be subjected to a Dix-like inversion to obtain a 3-D ``interval'' $\alpha$ cube. The RMS $\alpha$ cube could be, after conversion back to time, used as a direct multiplicative correction on the stacking velocities.