# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting
## 3 Development  
### Riemannian Geophysical Modelling  
For the geophysical case, the differential manifolds will be generally of dimension \(d=3\) or \(d=2\), i.e., \(\mathbb{M}^{3}\) or \(\mathbb{M}^{2}\), respectively. Let a stratum of propagation velocity \(v=v(q^{i})\). **Definition 4**. : _The seismic refractive index \(n:\mathbb{M}^{d}\rightarrow\mathbb{R}\) is defined as a function of class \(C^{\infty}_{\mathbb{M}^{d}}\) such that_  
\[n=n(q^{i})=\frac{1}{v} \tag{10}\]  
**Lemma 1**. : _If \(n=n\big{(}q^{i}\big{)}:\mathbb{M}^{d}\rightarrow\mathbb{R}\) is a smooth function,_  
\[\mathbf{g}=n^{2}\delta\;, \tag{11}\]  
_is the Riemannian metric induced by \(n\) in \(\mathbb{M}^{d}\), where \(\delta\) is the Euclidean metric (with components \(\delta_{ij}\), the Kronecker delta) [43]._  
**Corollary 1**. : _The Riemannian seismic metric (11) naturally parametrizes the differential manifold \(\mathbb{M}\) in a Cartesian coordinate system, i.e., \(\big{\{}q^{i}\big{\}}\) are Cartesian coordinates._  
**Definition 5**. : _A seismic space \((\mathbb{M},\mathbf{g};n)\) is defined as a Riemannian manifold \((\mathbb{M},\mathbf{g})\), with \(\mathbf{g}\) the metric induced by the refraction index \(n\)._  
**Theorem 2**. : _(Fermat's Principle) A curve \(\kappa\) in a seismic space \((\mathbb{M},\mathbf{g};n)\) is a seismic ray if and only if \(\kappa\) is a geodesic in that space, i.e., if_  
\[\frac{D\kappa}{D\lambda}=0\;. \tag{12}\]The equivalence between Fermat's Principle and the statement of Theorem 2 is demonstrated by [PERSON] et al. [43]. For a seismic space \((\mathbb{M},\mathbf{g};n)\), the geodesic Equation (12) produce seismic rays which follow the laws of transmission/refraction. **Definition 6**. : _Let \((\mathbb{M},\mathbf{g};n)\) be a seismic space and let \(R^{k}_{lmn}\) be its Riemann curvature tensor. The reverse seismic space, or seismic space of inverse curvature, is defined as the seismic space \((\mathbb{M}^{\prime},\mathbf{g}^{\prime};n^{\prime})\) whose curvature tensor \(R^{\prime}\) is such that_  
\[R^{\prime}{}^{k}_{lmn}=-R^{k}{}_{lmn} \tag{13}\]  
\(\forall\;k,l,m,n=1,\cdots,d\). **Theorem 3**. : _Let \((\mathbb{M},\mathbf{g};n)\) be a seismic space of dimension \(d=2\). The seismic space induced by the inverse of the refractive index, \(n^{-1}\), i.e., \((\mathbb{M},\mathbf{g}^{-1};n^{-1})\), is the seismic space of inverse curvature to \((\mathbb{M},\mathbf{g};n)\) of dimension \(d=2\)._  
Proof. : As \((\mathbb{M},\mathbf{g};n)\) is a seismic space, in a Cartesian coordinate system \(\{q^{i}\}_{i=1}^{2}\), according to Equation (2), induces the following Christoffel symbols:  
\[\Gamma^{1}_{11}=\frac{\partial_{1}n}{n}\qquad\Gamma^{1}_{12}=\Gamma^{1}_{21}= \frac{\partial_{2}n}{n}\qquad\Gamma^{1}_{22}=-\frac{\partial_{1}n}{n}\]  
\[\Gamma^{2}_{11}=-\frac{\partial_{2}n}{n}\qquad\Gamma^{2}_{12}=\Gamma^{2}_{21}= \frac{\partial_{1}n}{n}\qquad\Gamma^{2}_{22}=\frac{\partial_{2}n}{n}\]  
Naturally, because \(d=2\), the curvature tensor (Equation (7)) induced by this Christoffel symbols has only one independent component,  
\[R^{1}_{212}=R^{2}_{121}=\frac{(\partial_{2}n)^{2}}{n^{2}}+\frac{(\partial_{1} n)^{2}}{n^{2}}-\frac{\partial_{2}^{2}n}{n}-\frac{\partial_{1}^{2}n}{n} \tag{14}\]  
inasmuch as \(R^{1}_{111}=R^{1}_{112}=R^{1}_{121}=R^{1}_{122}=R^{1}_{211}=R^{1}_{222}=R^{2} _{111}=R^{2}_{122}=R^{2}_{211}=R^{2}_{212}=R^{2}_{221}=R^{2}_{222}=0\) and \(R^{1}_{221}=R^{2}_{112}=-R^{1}_{212}\) by skew-symmetry. Now, due to the fact that the refraction index \(n\) is a smooth function (Definition 4), \(n^{-1}\) is so, so it induces the metric tensor (Lemma 1)  
\[\mathbf{g}^{\prime}=n^{-2}\delta=\frac{1}{n^{2}}\delta=\mathbf{g}^{-1}\]  
where the last equality follows because \(\mathbf{g}^{\prime}\) is the inverse of the metric tensor \(\mathbf{g}\) of seismic space \((\mathbb{M},\mathbf{g};n)\).