Spherical Earth mode

The spherical Earth mode can be invoked by setting is_sphere = T. This mode assumes that input/output velocities and depths are defined with the spherical coordinate. Before forward computations, velocities, depths, and densities are adjusted for the cartesian coordinate via the Earth-flattening approximation as follows:

$$z_{cart} = R_E\ln\left(\frac{R_E}{r}\right) ,$$ $$\alpha_{cart} = \alpha_{sphe} \left(\frac{r}{R_E}\right),$$ $$\beta_{cart} = \beta_{sphe} \left(\frac{r}{R_E}\right),$$ and $$\rho_{cart} = \rho_{sphe} \left(\frac{r}{R_E}\right),$$

where \(z\), \(\alpha\), \(\beta\), and \(\rho\) represent depth, P-wave velocity, S-wave velocity, and density, respectively. \(R_E\) is the Earth's radius and \(r\) is a distance to the origin.