# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting
The Eikonal equation can be expressed by the Hamiltonian formalism [3; 15], allowing it to be decomposed into a series of first-order nonlinear ordinary differential equations, often simplifying its (numerical) solution with appropriate initial conditions, obtaining both the trajectories and the travel times [16]. It is clear that by applying the corresponding Legendre transforms, one can arrive at a Lagrangian formulation of the seismic ray theory [1]. Furthermore, it is possible to arrive at these formulations through variational principles, such as those of [PERSON] and [PERSON] [17]. From this formalism, there are very different approaches such as paraxial ray theory [15; 18], the study of the effects of structural interfaces [16; 19; 20], the study of the Green's function in inhomogeneous and weakly anisotropic elastodynamics [21], perturbation methods in ray theory [4; 12; 22; 23; 24; 25; 26; 27], handling around certain singularities [28; 29; 30; 31; 32], and even the study of \"chaotic\" rays using Lyapunov exponents [4; 33]. It should be noted that ray theory is usually not applicable to S-type shear waves, which require separate treatments [26; 34; 35]. Currently, the ray method can be used only for high-frequency signals and in models where heterogeneity does not exceed a certain degree. Thus, it is desirable to have available a more general extension of ray methods, both to simplify the theory and to lead to more precise results. In this sense, there is [PERSON]'s asymptotic ray theory introduced to seismology by [PERSON] [12], [PERSON] and [PERSON] [36], [PERSON] and [PERSON] [37], the theory based on the sum of Gaussian rays [38; 39], [PERSON]'s theory and Gaussian rays [40], the Gaussian packet sum method [4], the Kirchhoff surface integral method [12], and the one-way wave equation approach [41]. However, these extensions of ray theory only circumvent some of its original problems, hence, a new approach is required. Considering that conventional ray theory can be written using variational principles, one may wonder whether [PERSON]'s principle in seismic can be completely rewritten into mathematical language without any degree of approximation. In this sense, there is a transparent and physically direct formulation of geometrical optics that allows the optical [PERSON] principle to be translated into the formalism of differential geometry, an area of mathematics that provides intrinsically curved spaces (manifolds) with the power of differential and integral calculus [42]. [PERSON] et al. [43] show that the trajectories arising from [PERSON]'s principle behind geometrical optics are, mathematically speaking, equivalent to the geometrical problem of determining geodesic equations in an \"optical space\" that is not Euclidean (curved). In other words, the physical problem of determining the trajectories of a ray of light in the three-dimensional Euclidean space (\(\mathbb{R}^{3}\)) is changed by the paradigm of finding the geodesics (minimum length trajectories, generalisation of the concept of line in \(\mathbb{R}^{3}\)) in an intrinsically curved mathematical space (differential manifold), which abstracts the essence of the material media in which these rays are propagated. In this way, all the tools of differential geometry are available to treat light rays, and therefore, this approach is constituted as a very robust mathematical approach to geometric optics. Since electromagnetic and optical phenomena are essentially undulatory, the parallelism in the theory of rays is impeccable, finding that [PERSON]'s principle governs the phenomena of rays in both formalisms as they are identical in both cases. In this way, the demonstration by [PERSON] et al. [43] that the trajectories arising from the [PERSON]'s principle are equivalent to the problem of determining the geodesic equations in a non-Euclidean (curved) \"optical space\" can be extended to seismics. The purpose of this article is to extend the parallelism between geometrical optics and the differential geometry of [PERSON] et al.