# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting
## 4 Results (Geophysical Applications)
### Geodesics through a Salt Dome  
It is well known in seismics and in geophysical exploration that the presence of salt domes yield to blurriness in the imaging of subsalt regions. This is due to the high speed contrast between the salt body and its surroundings. Thus, a velocity model for a realistic geophysical model which considers a salt dome located at Perdido's Fold Belt is considered here, and is given by:  
\[v(x,z)=\left(1000-\frac{v_{1}(x,z)}{10^{13}}\right)+v_{2}(x,z); \tag{24}\]  
Figure 3: Refracted and reflected rays in a synthetic setup with two different homogeneous strata separated by an irregular interface modelled with a sawtooth function with the following initial speed conditions: (**a**) \(v_{x}=10\) m/s and \(v_{z}=10\tan(50^{\circ})\) m/s and, (**b**) \(v_{x}=10\) m/s and \(v_{z}=10\tan(51^{\circ})\) m/s.  
where  
\[v_{1}(x,z)=x(t)p_{01}+x(t)^{2}p_{02}+x(t)^{3}p_{03}+x(t)^{4}p_{04}+x(t)^{5}p_{05}+ z(t)p_{10}+\]  
\[x(t)z(t)p_{11}+x(t)^{2}z(t)p_{12}+x(t)^{3}z(t)p_{13}+x(t)^{4}z(t)p_{14}+z(t)^{2}p _{20}+\]  
\[x(t)z(t)^{2}p_{21}+x(t)^{2}z(t)^{2}p_{22}+x(t)^{3}z(t)^{2}p_{23}+z(t)^{3}p_{30}+ x(t)z(t)^{3}p_{31}+\]  
\[x(t)^{2}z(t)^{3}p_{32}+z(t)^{4}p_{40}+x(t)z(t)^{4}p_{41}+z(t)^{5}p_{50}+p_{00}\, \tag{25}\]  
and  
\[v_{2}(x,z)=4000\exp\left(-\left(\frac{\left(z(t)-x_{0}\right)^{4}}{2a_{x}^{2} }+\frac{\left(x(t)-y_{0}\right)^{4}}{2a_{y}^{2}}\right)\right)\, \tag{26}\]  
where \(p_{00}=2569\), \(p_{10}=-165.2\), \(p_{01}=-0.9672\), \(p_{20}=1113\), \(p_{11}=59.17\), \(p_{02}=-15.96\), \(p_{30}=1630\), \(p_{21}=-53.88\), \(p_{12}=34.73\), \(p_{03}=36.83\), \(p_{40}=-373.5\), \(p_{31}=-8.61\), \(p_{22}=-5.29\), \(p_{13}=-14.09\), \(p_{04}=5.541\), \(p_{50}=-470.2\), \(p_{41}=23.28\), \(p_{32}=-15.3\), \(p_{23}=-0.7027\), \(p_{14}=-0.9127\), \(p_{05}=-12.62\), \(x_{0}=172\), \(y_{0}=483\), \(a_{x}=500\), and \(a_{y}=50000\).  
The latter equations set a domain of \(0\leq x\leq 900\) km by \(0\leq z\leq 600\) km. The velocity model yields to the geodesic equations which is not here presented due to their extension. In order to assess the behaviour of geodesics in this seismic space, three experiments were performed as follows: a source of seismic rays at ground level (\(z=0\)) is located, in both the left and the right corners of the domain (\(x=0\) and \(x=900\)), as well as at the centre of the domain (\(x=450\)). In these locations of the source, the geodesic equations were solved several times by thoroughly sweeping the space of initial conditions, throwing geodesics each \(\pi/36\) rad (\(5^{\circ}\)). The results of refraction and reflection experiments are shown in Figures 4 and 5, respectively.  
Figure 4: _Cont._  
Figure 4: Refracted seismic rays (black lines) in a synthetic model for a salt dome, sweeping the domain with rays emitted each \(\pi/36\) (\(5^{\circ}\)), with source place at ground level at \((x,z)=(0,0)\) (**a**), \((x,z)=(450,0)\) (**b**) and \((x,z)=(900,0)\) (**c**). Original straight trajectories are omitted for the sake of clarity of the figure.  
Figure 5: Reflected seismic rays (black lines) in a synthetic model for a salt dome, sweeping the domain with rays emitted each \(\pi/36\) (\(5^{\circ}\)), with source place at ground level at \((x,z)=(0,0)\) **(a)**, \((x,z)=(450,0)\) **(b)** and \((x,z)=(900,0)\) **(c)**. Original straight trajectories are omitted for the sake of clarity of the figure.