Geophysical Institute of the ASCR

Acoustic axes in strong and weak triclinic anisotropy
Calculation of acoustic axes in triclinic elastic anisotropy is considerably more complicated than for
anisotropy of higher symmetry. While one polynomial equation of the 6th degree is solved in
monoclinic anisotropy, we have to solve two coupled polynomial equations of the 6th order in two
variables in triclinic anisotropy. Furthermore, some solutions of the equations are spurious and must
be discarded. In this way we obtain 16 isolated acoustic axes, which can run in real or complex
directions. The real/complex acoustic axes describe the propagation of homogeneous/inhomogeneous
plane waves and are associated with a linear/elliptical polarization of waves in their vicinity. The most
frequent number of real acoustic axes is 8 for strong triclinic anisotropy and 4 to 6 for weak triclinic
anisotropy. Acoustic axes can exist even under an infinitesimally weak anisotropy, and occur when
slowness surfaces of the S1 and S2 waves touch or intersect. The maximum number of isolated
acoustic axes in weak triclinic anisotropy is 16 as in strong triclinic anisotropy. The directions of
acoustic axes are calculated by solving two coupled polynomial equations in two variables. The
degree of the equations is 6 under strong anisotropy and reduces to 5 under weak anisotropy. The
weak anisotropy approximation is particularly useful, when calculating the acoustic axes under
extremely weak anisotropy with anisotropy strength less than 0.1%, because the equations valid for
strong anisotropy might become numerically unstable and their modification, which stabilizes them,
is complicated. The weak anisotropy approximation can also find applications in inversions for
anisotropy from the directions of acoustic axes.
References
Vavryčuk, V., 2005a. Acoustic axes in triclinic anisotropy. J. Acoust. Soc. Am., 118, 647-653.
Vavryčuk, V., 2005b. Acoustic axes in weak triclinic anisotropy, Geophys. J. Int., 163, 629-638, doi: 10.1111/j.1365-
246X.2005.02762.x.
First-order ray tracing for smooth inhomogeneous weakly
anisotropic media
Ray tracing procedures represent basis of many seismic processing techniques as, e.g., prestack
Kirchhoff depth migration or AVO. Since majority of rocks are often anisotropic, and specifically
weakly anisotropic, development of ray tracing procedures for such kinds of media is desirable. Using
the perturbation theory, in which deviations of anisotropy from isotropy are considered to be the firstorder
quantities, we derived first-order ray-tracing equations for P waves, see Pšenčík and Farra
(2005). The traveltimes calculated along these rays are also of the first order. In order to increase the
accuracy of the traveltimes, a simple second-order correction calculated along rays can be used. All
equations are expressed in terms of weak anisotropy parameters, which represent a more natural
parameterization of weakly anisotropic media than standard elastic parameters. Only 15 weak
anisotropy parameters describing P-wave propagation are involved.
Results of numerical tests in medium whose horizontal axis of symmetry rotates with increasing
depth, called HTI-ROT are shown in Fig. 28. A VSP experiment is considered, in which the source
and the borehole are situated in a vertical plane. The borehole is vertical, the source is located on the
surface, in the distance of 1 km from the borehole. Results obtained with exact ray tracing and FORT
coincide so well that no differences are visible. The differences are made visible in Fig. 29, where the
relative travel time differences (from exact travel times) corresponding to the first- (black) and
second-order (red) traveltimes are plotted.
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