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PARABOLIC AND HYPERBOLIC PARAXIAL TWO‐POINT TRAVELTIMES IN 3D MEDIA 1
Author(s) -
SCHLEICHER J.,
TYGEL M.,
HUBRAL P.
Publication year - 1993
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.1993.tb00581.x
Subject(s) - paraxial approximation , propagator , scalar (mathematics) , isotropy , mathematical analysis , inversion (geology) , geometry , physics , mathematics , optics , geology , mathematical physics , seismology , beam (structure) , tectonics
A bstract The 4 × 4 T ‐propagator matrix of a 3D central ray determines, among other important seismic quantities, second‐order (parabolic or hyperbolic) two‐point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two‐component vector. Here, we provide parabolic and hyperbolic paraxial two‐point traveltime approximations using the T ‐propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three‐component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial‐ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.