We derive formulae for the approximate computation of two-point paraxial traveltimes (traveltimes between two points) for points arbitrarily chosen in a paraxial vicinity of a reference ray computed in a smoothly varying inhomogeneous anisotropic medium containing structural interfaces. The formulae have a form of the Taylor expansion in Cartesian coordinates of the two-point paraxial traveltime or its square to the quadratic terms. The coefficients of the expansion depend on quantities obtained by ray tracing in Cartesian coordinates and by dynamic ray tracing in ray-centred coordinates. Alternatively, the dynamic ray tracing can be performed in Cartesian coordinates. The advantages of the approach based on dynamic ray tracing in ray centred coordinates are its efficiency and elimination of possible complications that may arise from the redundant fundamental solutions of dynamic ray tracing in Cartesian coordinates (the ray-tangent and non-eikonal solutions). As a by-product, we also obtain simple formulae for the slowness vectors at the two points in the paraxial vicinity of the reference ray. They belong to a paraxial ray passing through these points. Potential applications of the proposed formulae consist in the fast and flexible two-point traveltime calculations from sources to receivers specified in Cartesian coordinates and situated close to a reference ray, along which dynamic ray tracing has been performed. The formulae for the paraxial slowness vectors can be used in two-point ray tracing.
Body waves, seismic anisotropy, theoretical seismology, wave propagation.
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