Two-point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media: Tests of accuracy

Umair bin Waheed, Ivan Psencik, Vlastislav Cerveny, Einar Iversen, Tariq Ali Alkhalifah

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16 Scopus citations


On several simple models of isotropic and anisotropic media, we have studied the accuracy of the two-point paraxial traveltime formula designed for the approximate calculation of the traveltime between points S' and R' located in the vicinity of points S and R on a reference ray. The reference ray may be situated in a 3D inhomogeneous isotropic or anisotropic medium with or without smooth curved interfaces. The twopoint paraxial traveltime formula has the form of the Taylor expansion of the two-point traveltime with respect to spatial Cartesian coordinates up to quadratic terms at points S and R on the reference ray. The constant term and the coefficients of the linear and quadratic terms are determined from quantities obtained from ray tracing and linear dynamic ray tracing along the reference ray. The use of linear dynamic ray tracing allows the evaluation of the quadratic terms in arbitrarily inhomogeneous media and, as shown by examples, it extends the region of accurate results around the reference ray between S and R (and even outside this interval) obtained with the linear terms only. Although the formula may be used for very general 3D models, we concentrated on simple 2D models of smoothly inhomogeneous isotropic and anisotropic (~8% and ~20% anisotropy) media only. On tests, in which we estimated twopoint traveltimes between a shifted source and a system of shifted receivers, we found that the formula may yield more accurate results than the numerical solution of an eikonal-based differential equation. The tests also indicated that the accuracy of the formula depends primarily on the length and the curvature of the reference ray and only weakly depends on anisotropy. The greater is the curvature of the reference ray, the narrower its vicinity, in which the formula yields accurate results.
Original languageEnglish (US)
Pages (from-to)WC65-WC80
Number of pages1
Issue number5
StatePublished - Sep 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We would like to express our gratitude to Joachim Mispel, three anonymous referees and the associate editor, Claudia Vanelle, for their stimulating comments. We are grateful to Seismic Wave Analysis Group (SWAG) of KAUST, Saudi Arabia, project "Seismic waves in complex 3-D structures" (SW3D), to Research Projects 210/11/0117 and 210/10/0736 of the Grant Agency of the Czech Republic and to Research Project MSM0021620860 of the Ministry of Education of the Czech Republic for support. E. Iversen acknowledges support from the Research Council of Norway and from the European Community's FP7 Consortium Project AIM "Advanced Industrial Microseismic Monitoring," Grant Agreement no. 230669.

ASJC Scopus subject areas

  • Geochemistry and Petrology
  • Geophysics


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