Wave characteristics in elliptical orthorhombic media

Abstract

An elliptical anisotropic medium is defined as a simplified representation of anisotropy in which the anelliptic parameters are set to zero in all symmetry planes. Despite of the fact that this model is rather seldom observed for real rocks, it is often used as a reference model. The P-wave equations for an elliptical anisotropic medium is well known. However, the S-wave equations have not been derived. Thus, we define all wave modes in elliptical orthorhombic models focusing mostly on the S-wave properties. We show that all wave modes in elliptical orthorhombic model are generally coupled and analyze the effect of additive coupling term. As the result, there is an S wave fundamental singularity point located in one of the symmetry planes depending on the relative magnitude of S wave stiffness coefficients.

The S waves geometrical spreading in elliptical orthorhombic media

“…η 1 = η 2 = η 3 = 0. Therefore, the conditions for all off-diagonal stiffness coefficients can be written similar to equation (3) (Stovas et al, 2021a) as follows:…”

“…Note that the coupling term f = 0 in all symmetry planes, and the coupling coefficient (Stovas et al, 2021a).…”

“…$${\eta _1} = {\eta _2} = {\eta _3} = 0$$. Therefore, the conditions for all off‐diagonal stiffness coefficients can be written similar to equation () (Stovas et al ., 2021a) as follows: $$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {c_{12}} = \sqrt {\left( {{c_{11}} - {c_{66}}} \right)\left( {{c_{22}} - {c_{66}}} \right)} - {c_{66}},\\ {c_{13}} = \sqrt {\left( {{c_{11}} - {c_{55}}} \right)\left( {{c_{33}} - {c_{55}}} \right)} - {c_{55}},\\ {c_{23}} = \sqrt {\left( {{c_{22}} - {c_{44}}} \right)\left( {{c_{33}} - {c_{44}}} \right)} - {c_{44}}. \end{array} \end{equation}$$…”

“…Note that the coupling term $$f = 0$$ in all symmetry planes, and the coupling coefficient $${f_0} = 0$$ if either $${F_P}$$ is isotropic $$({c_{11}} = {c_{22}} = {c_{33}})$$ or F S is isotropic $$({c_{44}} = {c_{55}} = {c_{66}})$$ (Stovas et al ., 2021a).…”

“…Most of the research works mentioned above deal with P waves only since the S‐wave propagation could be extremely complex in low‐symmetry anisotropic models. In this paper, we focus on a very simple example of intrinsically coupled S waves in an elliptical orthorhombic model, which was recently introduced by Stovas et al (2021a). In particular, we are interested in the behaviour of geometrical spreading in the vicinity of the fundamental singularity point.…”