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Abstract

The importance of ray parameters as first integrals of the ray equation does not appear to have been sufficiently recognized in Geophysical theory or applications. In fact, the existence of ray parameters in fairly general propagation situations can be exploited to obtain explicit exact solutions for the rays by integration of first- (rather than second-) order differential equations. Making use of Noether’s theorem, which associates conserved quantities with symmetries with the Lagrangian afforded by Fermat’s variational principle, we derive ray parameters for a class of inhomogeneous media, denoted as pseudo-linear, whose characteristics are adjustable to represent a wide variety of structural features in a realistic and computationally convenient way. Exact solutions for the rays are obtained in a surprisingly wide variety of speed profiles, this permitting a qualitative and quantitative investigation of certain phenomena, such as the existence of a definite zone of accessibility from a given source, the exact location of the reflexion points in certain cases, and other general features. In particular, a sub-class of pseudo-linear media provide exact ray solution for complex geological profiles that can be interpreted as dipping layers, synclines or anticlines, according to the numerical value of controllable parameters in the mathematical formulas.
... In geometrical optics, the conserved quantity (9) is a generalized raypath parameter (e.g. [3]) while in classical mechanics, this conserved quantity is the generalized momentum (e.g. [9]). ...
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