Second-order traveltimes and first-order spreading of reflected/transmitted P waves in inhomogeneous, weakly anisotropic media

Ivan Psencik and Veronique Farra


First-order ray tracing (FORT) and dynamic ray tracing (FODRT) for P waves (Psencik and Farra, 2005, 2007) provide approximate, but simple and quite accurate way of computing traveltimes and geometrical spreading of seismic waves propagating in inhomogeneous, weakly anisotropic media and even in media of moderate anisotropy. FORT is a technique based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be small quantities, used further in the perturbation procedure. The basic idea of FORT is to replace the exact eigenvalue of the Christoffel matrix, which controls P-wave ray tracing and dynamic ray tracing, by its first-order counterpart.

We extend the applicability of FORT and FODRT to layered media. Basic step in this extension is introduction of a rule (Snell's law) for the determination of first-order slowness vectors of reflected or transmitted P waves into FORT. Transformation of FODRT quantities across an interfaces is controlled by formally the same equations as in the exact case (Farra and Le Begat, 1995), with exact quantities replaced by their first-order counterparts. Therefore, in the following, we concentrate on description of traveltime computations including transformation of slowness vectors at an interface, and for spreading computations we refer to Psencik and Farra (2007) and Farra and Le Begat (1995).

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In: Seismic Waves in Complex 3-D Structures, Report 22, pp. 139-142, Dep. Geophys., Charles Univ., Prague, 2012.