selecting reference anisotropic parameters

Anisotropy has been shown to exist in many sedimentary rocks Thomsen (1986). Most sedimentary rocks can be approximated by a transversely isotropic medium with a plane of symmetry. For vertically transversely isotropic (VTI) media, if we assume that the S-wave velocity is much smaller than the P-wave velocity, the dispersion relation can be computed as follows Tsvankin (2001):

$$\begin{eqnarray} k_z(v,\delta, \eta) & = & \frac{\omega}{v}\sqrt{\frac{\frac{\om... ...a(1+2\delta)}},\\ \eta & = & \frac{\varepsilon-\delta}{1+2\delta},\end{eqnarray}$$ (8) (9)

where v, $\delta$, $\varepsilon$ (and $\eta$) are the anisotropic parameters Thomsen (1986); Tsvankin (2001). In the VTI media, the vertical wavenumber k_z is a function not only of vertical velocity v, but also of $\delta$ and $\eta$ (or $\varepsilon$), which means that for one depth-level extrapolation, we must choose three parameters instead of one.

The 3D Lloyd's algorithm can be easily translated into the problem of selecting reference anisotropic parameters, because these three anisotropic parameters are always correlated and have similar distributions. These properties make the selection by Lloyd's algorithm quite effective. In our selection problem, we form the parameter vector $\mathbf{r}=(v, \delta, \eta)$ or $\mathbf{r}=(v, \delta, \varepsilon)$. Now the center of the 3D mass within a cluster is the reference-parameter vector $\mathbf{\hat{r}}$, and the boundaries of the 3D cluster define the regions with the same v, $\delta$ and $\eta$ (or $\varepsilon$).

We test our methodology on a real data set provided by ExxonMobil. Figure displays the corresponding anisotropy model also provided by ExxonMobil. Figure (a) is the vertical velocity model, Figure (b) is the $\delta$ model and Figure (c) is the $\eta$ model. As the 3D Lloyd's algorithm is hard to visualize, we show only a 2D procedure for selecting $\delta$ and $\eta$. Figures (a) and (b) are the input $\delta$ and $\eta$for a single depth level respectively. Figure (c) is their 2D histogram, where the horizontal axis is $\eta$, the vertical axis is $\delta$, and the amplitude is the value of the 2D histogram. Figure (a) shows the result by regular sampling, where 16 uniformly-selected points (4 $\delta$s and 4 $\eta$s) are overlaid with the 2D histogram (Figure (c)). It is quite clear that the points selected by regular sampling do not match well the distribution of $\delta$ and $\eta$.Figure (b) shows the result by using the modified Lloyd's alogrithm after 2 iterations, while Figure (c) is the result after 20 iterations. Note how well the result characterizes the 2D distribution of $\delta$ and $\eta$, and also note that after 20 iterations the number of $(\delta, \eta)$ is reduced from 16 to 10. Thus for this depth level and a given vertical velocity, we only need 10 depth extrapolations instead of 16, reducing the computational cost by almost 40 percent.

 

model
Figure 1 Vertical slice of the anisotropy model provided by ExxonMobil (the vertical axes are depths). (a) P-wave velocity; (b) anisotropic parameter $\delta$ and (c) anisotropic parameter $\eta$.

 

2D_histogram
Figure 2 The values of $\delta$, $\eta$ for a single depth slice and their 2D histogram. The values of (a) $\delta$ at depth=990 m and (b) $\eta$ at depth=990 m; (c) the 2D histogram computed from (a) and (b), the curves indicate the 2D distribution of $\delta$ and $\eta$.

 

overlay_lloyd
Figure 3 Comparison between the conventional method and the modified 2D Lloyd's algorithm. (a) The result of uniform sampling, overlaid with the 2D histogram from Figure (c); (b) the result of the modified 2D Lloyd's algorithm after 2 iterations and (c) after 20 interations. Note how accurately (b) and (c) characterize the distribution of $\delta$ and $\eta$