Transformation of spatial and perturbation derivatives of travel time at a curved interface between two arbitrary media

Ludek Klimes


We consider the partial derivatives of travel time with respect to both spatial coordinates and perturbation parameters. These derivatives are very important in studying wave propagation and have already found various applications in smooth media without interfaces. In order to extend the applications to media composed of layers and blocks, we derive the explicit equations for transforming these travel-time derivatives of arbitrary orders at a general smooth curved interface between two arbitrary media. The equations are applicable to both real-valued and complex-valued travel time. The equations are expressed in terms of a general Hamiltonian function and are applicable to the transformation of travel-time derivatives in both isotropic and anisotropic media. The interface is specified by an implicit equation. No local coordinates are needed for the transformation.


Ray theory, Hamilton-Jacobi equation, eikonal equation, travel time (action), spatial derivatives of travel time, perturbation derivatives of travel time, reflection or refraction at curved interfaces, anisotropy, bianisotropy, heterogeneous media, paraxial approximation, Gaussian beams, wave propagation.

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Stud. geophys. geod., 60 (2016), 451-470.