Non-flat interfaces

To include weak lateral heterogeneity from curved interfaces, we modify equation 4 and consider a Taylor expansion of $f_k$ with respect to the reference location $X=x_0$ as follows:

$$f_k(x_k) \approx f_k(X) + (x_k-X) \left.\frac{\partial f_k}{\partial x_k}\... ...(x_k-X)^2}{2} \left.\frac{\partial^2 f_k}{\partial x_k^2}\right\rvert_{x_k=X}~,$$ (13)

$$\approx F_k + (x_k-X)F'_k + \frac{(x_k-X)^2 }{2}F''_k~,$$

where $F_k$ denotes the vertical depth of the $k$-th interface evaluated at the reference $X$. $F'_k$ and $F''_k$ are its corresponding first- and second-order derivatives evaluated at $x_k = X$.