Laterally varying velocity

Following the approach of Lynn and Claerbout (1982) and Grechka and Tsvankin (1999), we assume that the group slowness in each sublayer only varies laterally and can also be approximated as a Taylor expansion with respect to the reference location $X=x_0$ as follows:

$$w_k(m_k) \approx w_k(X) + (m_k-X) \left.\frac{\partial w_k}{\partial m_k}\... ...(m_k-X)^2}{2} \left.\frac{\partial^2 w_k}{\partial m_k^2}\right\rvert_{m_k=X}~,$$ (14)

$$\approx W_k + (m_k-X)W'_k + \frac{(m_k-X)^2 }{2}W''_k~,$$

where $W_k$ denotes the group slowness in the $k$-th layer evaluated at the reference $X$. In the case of anisotropic media, equation 14 implies the spatial variation of the vertical group slowness with respect to the horizontal coordinate $m_k$, which follows from the choice of vertical reference ray in the reference 1-D anisotropic medium.