Elastic wave-vector decomposition in heterogeneous anisotropic media |

where

are given in equation A-2 for the most general case of triclinic media and are functions of local stiffness and phase direction . Equations 21 and 22 can be derived from the generic solution of a cubic equation for , which corresponds to the Christoffel equation. They are valid for any locally homogeneous anisotropic medium with associated (Schoenberg and Helbig, 1997). With these analytical expressions, we can derive the sufficient condition for the occurence of a S-wave singularity. From and equations 21 and 22, it follows that in the singular direction ,

(23) |

or doing trigonometric expansions,

(24) |

which leads to the condition of

We propose to apply equation 25 to numerically detect the proximity of the point singularity for a given phase direction. This condition also allows us to create a filter with an adjustable effective area to smooth the polarization vectors around the singularity in order to attenuate the planar artifacts in wave-mode separation and wave vector decomposition. The variation of values of the left-hand side in equation 25 with different phase directions ( ) in the case of the example orthorhombic model is shown in Figure 3b. Similar plots for the triclinic model are shown in Figures 4 and 5.

ortoctant,orthos
a) Polarization vectors in the orthorhombic model of S1 (blue) and S2 (red) plotted on phase slowness surfaces to show how they rapidly rotate in the vicinity of point singularities. b)
for different phase directions (
) plotted on S1 phase slowness surface. Notice that the values turn zero at the locations corresponding to point singularities in circles and that there are two point singularities in the
plane, one in
planes, and one in the plane cut along
=
in this octant.
Figure 3. |
---|

t090p090s,t090p90180s,t090p180270s,t090p270360s
for different phase directions (
) plotted on S1 phase slowness surface the zenith angle
measured from vertical and
from the azimuthal angle a)
, b)
, c)
, and d)
measured with respect to
-axis. Notice that the values turn zero at the locations corresponding to singularities in circles.
Figure 4. |
---|

t90180p090s,t90180p90180s,t90180p180270s,t90180p270360s
for different phase directions (
) plotted on S1 phase slowness surface the zenith angle
measured from vertical and
from the azimuthal angle a)
, b)
, c)
, and d)
measured with respect to
-axis. Notice that the values turn zero at the locations corresponding to singularities in circles.
Figure 5. |
---|

Elastic wave-vector decomposition in heterogeneous anisotropic media |

2017-04-18