Overview
A new integral ray transform, termed the head-wave transform, has been introduced and studied. This transform integrates a function along a specific type of piecewise linear path, which can be generalized to geodesic paths. The structure of these paths is relevant to the propagation of head-waves in media characterized by sharp variations in sound speed. A distinguishing feature of these ray paths is a middle segment that corresponds to gliding along a defined "gliding surface." The core contributions of this work include the derivation of inversion formulas and characterizations of the null space under different geometrical conditions pertaining to the gliding surface and distinct assumptions concerning the integrand function.
Research Context
The research is situated within the field of integral transforms, specifically focusing on ray transforms. Integral ray transforms involve integrating functions along paths, and in this particular case, the paths are modeled after physical phenomena. The motivation for defining the head-wave transform stems from the physical behavior of head-waves. These waves are observed in media where sound speed undergoes abrupt changes. In such scenarios, the ray paths of propagating waves often exhibit a piecewise linear structure, with a segment where the wave "glides" along an interface, referred to in this context as the gliding surface. Understanding and inverting such transforms is crucial for various applications where ray path information is used to reconstruct properties of the medium.
Approach
The research developed the head-wave transform as an integral ray transform. This involved defining the integral of a function along a path consisting of three distinct parts. The path geometry is piecewise linear, or more generally, geodesic. A central element of this path construction is the "gliding surface," along which the middle section of the ray path travels. The study proceeded by analyzing the mathematical properties of this newly defined transform. The primary analytical tasks were to establish methods for inverting the transform, meaning reconstructing the original function from its integrals along the specified paths, and to characterize its null space. The null space consists of functions that yield a zero transform, indicating that such functions cannot be uniquely recovered by the transform. These analytical derivations were performed under multiple sets of assumptions, which pertained to the specific geometry of the gliding surface and the characteristics of the integrand function. This multi-conditional analysis ensures the robustness of the derived formulas and characterizations across different scenarios relevant to head-wave propagation.
Findings
The principal findings center on two key aspects of the head-wave transform: inversion formulas and null space characterizations. The researchers successfully derived specific inversion formulas for this transform. These formulas outline how to reconstruct a function given its head-wave transform data. Complementary to the inversion formulas, the study also provided characterizations of the null space of the head-wave transform. These characterizations identify the class of functions that are not detectable by the transform. Both inversion formulas and null space characterizations were developed and presented under various assumptions. These assumptions specifically relate to the geometric properties of the gliding surface, which is the path segment traversed by the middle part of the ray, and to the nature of the function being integrated. The application of different sets of assumptions suggests a comprehensive treatment of the transform’s behavior under varying conditions relevant to its physical inspiration.