Unified Ray and Amplitude Tracing ODE Systems for the Eikonal and Transport Equations via Contact Geometry
Studies in Applied Mathematics, vol.157, no.3, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 157 Issue: 3
- Publication Date: 2026
- Doi Number: 10.1111/sapm.70309
- Journal Name: Studies in Applied Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, ABI/INFORM, Aerospace Database, Applied Science & Technology Source, Compendex, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Business Source Ultimate (EBSCO), Engineering Source (EBSCO), Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Keywords: contact manifolds, eikonal equation, Hamiltonian systems, seismic waves, transport equation
- Middle East Technical University Affiliated: Yes
Abstract
High-frequency seismic wave descriptions are essential for fundamental geophysical applications, including earthquake hypocenter location, magnitude estimation, and traveltime–amplitude tomography. These descriptions rely on two coupled PDEs: the eikonal equation (governing traveltimes) and the transport equation (governing amplitudes). Solving both equations together provides sharper subsurface images and more reliable magnitude/location estimates. Although the transport equation has been transformed into an ODE in the classical literature, this reduction relies crucially on the isotropy assumption (i.e., the wavefront normal is parallel to the ray direction). An explicit, characteristic-based ODE formulation analogous to the one used for the eikonal equation—and valid without imposing isotropy—has been lacking. This study addresses that limitation by deriving a unified first-order ODE framework, using the principles of contact geometry, that simultaneously transforms both the eikonal and transport equations into coupled ODE systems. This formulation allows ray paths, traveltimes, and amplitudes to be solved together and explicitly as functions of ray time. In addition, we introduce a general methodology for computing the traveltime Laplacian—a key quantity in the amplitude equation—without assuming isotropy. We first apply the chain rule to the ray-coordinate mappings to derive an exact analytical expression for the Laplacian. We then provide a geometric interpretation that links this analytical expression to the local structure of the ray tubes in both the physical and momentum spaces, offering a robust way to compute amplitude evolution in anisotropic media. To validate the framework, we solve the coupled ray-tracing and amplitude-tracing ODEs analytically for benchmark models. For a homogeneous medium, our method analytically derives the classical amplitude decay proportional to (Formula presented.) in three-dimensional space and (Formula presented.) in two-dimensional space, explicitly linking our formulation to the physical principle of geometric spreading and energy conservation. For a linearly inhomogeneous and isotropic medium, we derive a new closed-form analytical solution for amplitude evolution along the ray. These analytical solutions demonstrate the accuracy and computational power of our unified framework.