# Application of Riemannian Seismic Ray Path Tracing in Salt Dome Prospecting
## 6 Conclusions
Subsurface Heterogeneity: Future work can focus on improving the ability of differential geometric methods to model highly heterogeneous subsurface environments, such as those with significant variations in rock properties. * Dynamic modelling: Developing dynamic models that account for changes in geological structures over time, such as those caused by tectonic activity or fluid extraction, can provide more accurate predictions for petroleum exploration. 4. _Incorporating Physical Processes:_  
1. Wave-Propagation modelling: Further research can enhance the modelling of wave propagation in complex media by incorporating additional physical processes, such as anisotropy and viscoelasticity, into differential geometric frameworks. * Thermal and Mechanical Properties: Integrating thermal and mechanical property variations into differential geometric models can improve theunderstanding of how these factors influence seismic wave behaviour and reservoir dynamics. * **Field Applications and Validation. *
* (a) _Case Studies and Field Trials:_
* Real-World Applications: Conducting case studies and field trials to validate differential geometric methods in various geological settings, such as offshore oil fields and complex onshore basins, will be essential for demonstrating their practical value. * Benchmarking Studies: Comparing the performance of differential geometric methods with traditional techniques in real-world scenarios can help identify best practices and areas for improvement. (b) _Industry Collaboration:_
* Collaborative Projects: Encouraging collaboration between academia, industry, and government agencies can facilitate the development and implementation of differential geometric methods in petroleum exploration. * Standardisation: Establishing industry standards for the application of differential geometry in geophysical modelling can promote wider adoption and ensure consistency in practice. Conceptualization, G.Y. and [PERSON]-G.; methodology, [PERSON]-G.; software, [PERSON] and [PERSON]-G.; validation, [PERSON]-G. and G.Y. ; formal analysis, [PERSON]-G. and G.Y. ; investigation, [PERSON]-G. and G.Y. ; resources, [PERSON]-G.; data curation, [PERSON]-G. and M.G.O.-d.-C.; writing--original draft preparation, [PERSON]-G., G.Y. and A.T.-A. ; writing--review and editing, [PERSON]-G., G.Y. and A.T.-A. ; visualization, [PERSON]-G. and M.G.O.-d.-C.; supervision, [PERSON]-G. and A.T.-A. ; project administration, [PERSON]; funding acquisition, [PERSON]. All authors have read and agreed to the published version of the manuscript. This research was partially funded by Secretaria de Investigacion y Posgrado, Instituto Politecnico Nacional grant numbers 20242752, 20240894 and 20241163, as well as EDI and PIFI grants. The APC was funded by Secretaria de Investigacion y Posgrado, Instituto Politecnico Nacional. Not applicable. There are no data available associated with this research. G.Y. acknowledges Consejo Nacional de Ciencias, Humanidades y Tecnologia (CONAHCyT) for scholarship grant, as well as to Instituto Mexicano del Petroleo for the support performing this research. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} Blackboard-bold for & Differential manifolds; for example \(\mathbb{M}^{d}\) for \(d\)-dimensional manifold \\ Bold for tensors & Tensors, for example \(\mathbf{g}\) for metric tensor \\ \(\vec{U}\) & Vector tangent to a point of \(d\)-dimensional manifold \\ \(X\),\(Y\),\(Z\) & Elements of differential manifolds \\ \(\{q^{i}\}_{i=1}^{d}\) & Global functions in \(d\)-dimensional manifold \\ \(T_{p}(\mathbb{M})\) & Vector space tangent to a manifold at the point \(p\) \\ \(g_{ij}\) & \(i,j\)-th component of the tensor \(\mathbf{g}\) \\ \(\Gamma^{l}_{mn}\) & \(l,m,n\)-nth Christoffel symbol \\ \end{tabular}