Earth 2D Eikonal equations

The equations from section A.4 can be easily extended in order to account for earth's curvature, and correspond to[15]:

$$\frac{d\theta}{d\tilde{r}} = f_e\frac {1}{c}\frac{\partial{c}}{\partial{\tilde{z}}}- \frac {1}{c}\frac{\partial{c}}{\partial{\tilde{r}}}\tan\theta- \frac {1}{R_e} ,$$ (A.12)

$$\frac{d\tilde{z}}{d\tilde{r}} = f_e\tan\theta \hskip5mm ,$$ (A.13)

$$\frac{d\tau}{d\tilde{r}} = f_e\frac {\sec\theta}{c} .$$ (A.14)

where $\tilde{z}$ lies along the radius of the earth, $\tilde{z} = 0$ at the surface, $\tilde{z} = R_e$ at the earth's center, $R_e$ represents the earth's radius, $\tilde{r}$ is the distance traveled along a circular arc at the sea level, and

$$f_e = \frac {R_e - \tilde{z}}{R_e} .$$