# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting  
[PERSON]  
1 Instituto Mexicano del Petroleo, Mexico City 07730, Mexico  
2 Centro de Desarrollo Aeroespacial, Instituto Politecnico Nacional, Mexico City 06610, Mexico  
3 Smart Solutions TS, Mexico City 06600, Mexico  
4 Departamento de Sistemas y Computacion, Tecnologico Nacional de Mexico/IT. Merida,  
Merida, Yucatain 97118, Mexico  
5 Correspondence: [PERSON]SS]  
[PERSON]  
2 Centro de Desarrollo Aeroespacial, Instituto Politecnico Nacional, Mexico City 06610, Mexico
are crucial for a more detailed understanding of seismic data. Also, it does not examine the amount of scattered energy from ray propagation throughout surfaces [5]. These omissions can limit the theory's applicability in practical seismic exploration scenarios where comprehensive and precise modelling is essential [3; 4]. While ray theory remains a cornerstone in seismic research and exploration, its approximations and inherent limitations necessitate careful consideration and often, supplementary methods to achieve accurate seismic imaging and interpretation in diverse geological settings. Starting from an intuitive description of seismic propagation along special trajectories (rays), based on the laws of geometric optics [6], it has become a highly sophisticated method. The first seismological applications date back to the end of the 19 th century, through the application of kinematic laws and straight rays to estimate travel times [3]. [PERSON] [7] was one of the pioneers in developing dynamical ray theory, also focusing on amplitudes and waveforms. The \"modern\" series theory (of rays) is obtained by calculating solutions to the equations of elastodynamics [8], and were obtained for the first time by [PERSON] and [PERSON] [9] and afterwards by [PERSON] and [PERSON] [10] for non-homogeneous isotropic media, and later by [PERSON] [11] for anisotropic media. [PERSON] [11] proposes that high-frequency waves propagate with an oscillatory time dependence, which induces a solution of the wave equation by separation of variables. For the temporal part, a function for the travel time is made manifest, and the spatial part is expanded into a series of inverse powers of the angular frequency, whose coefficients are vector amplitude coefficients and are usually complex. For perfectly elastic media, travel times and vector amplitude coefficients are usually independent of frequency [3]. By introducing series expansion to the equations of elastodynamics, approximations of different orders to the ray method are obtained. However, the complexity of the equations increases considerably with increasing order of approximation, since higher-order amplitude coefficients are inaccurate and unstable, due to their high sensitivity to the fine details of the medium [3]. In this context, sensitivity refers to the extent to which higher-order amplitude coefficients depend on small variations in the properties of the medium. This means that even very small changes or imperfections in the medium can cause large variations in the calculated amplitude coefficients. As a result, higher-order approximations become less accurate and more prone to errors, making them unstable and difficult to use reliably in practical applications. In this way, only the zero-order coefficient, and at most the first-order coefficient, have been used in seismological applications. This fact has limited the use of the ray approximation in its conventional form. There are other seismic ray series such as Chapman [12], which expand in terms of particle velocity and traction, but they have not been popularised for ray tracing. The zero-order ray method allows us to arrive at the well-known Eikonal equation, a nonlinear first-order partial differential equation for the travel time, the polarisation vector (which indicates the direction of the displacement vector) [3], and the transport equation that allows calculating amplitudes [13; 14].