PARABOLIC AND HYPERBOLIC PARAXIAL TWO‐POINT TRAVELTIMES IN 3D MEDIA¹

Abstract

SCHLEICHER, J., TYGEL, M. and HUBRAL, P. 1993. Parabolic and hyperbolic paraxial twopoint traveltimes in 3D media. Geophysical Prospecting 41,495-513.The 4 x 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.

New insights into the crustal structure of the North German Basin from reprocessing of seismic reflection data using the Common Reflection Surface stack

“…The CRS stack technique was recently established (Schleicher et al 1993;Tygel et al 1997;Müller 1999;Mann 2002). It is a macro-model independent multiparameter stacking tool for multi-coverage seismic reflection data that is related closely to the multifocusing method Landa et al 1999) and the polystack method .…”

Model-independent Travel-time Attributes for 2-D, Finite-offset Multicoverage Reflections

“…These are the two hypothetical eigenwaves introduced by HUBRAL (1983). The CRS travel-time moveout expression is another representation of the hyperbolic travel-time formula (see, e.g., SCHLEICHER et al, 1993), that is applied to source-receiver pairs in the vicinity of a fixed, central ZO ray.…”

Traveltime-based true-amplitude migration

“…His work is based on the paraxial ray approximation (Červený, 2001) and can be used to interpolate traveltimes for sources and receivers which are located in the bordering surfaces of the seismic system. Schleicher et al (1993b) link the Bortfeld theory to the ray propagator formalism and introduce a hyperbolic variant of the paraxial traveltime interpolation. Both methods are, however, restricted to source and receiver reference surfaces.…”

“…A variety of fast traveltime tools have been developed (for an overview, see, e.g., Leidenfrost et al 1999). Several methods for traveltime interpolation have also been proposed, e.g., parabolic and hyperbolic approximations (Ursin, 1982;Gjøystdal et al, 1984;Schleicher et al, 1993b) or Fourier (sinc) interpolation (Brokešová, 1996) to name only a few. Since the standard method to compute the weight functions is to perform dynamic ray tracing for each point under consideration, much effort has gone into finding simplified, more economic expressions for them, as shown by, e.g., Dellinger et al (2000) or Zhang et al (2000).…”