The S waves geometrical spreading in elliptical orthorhombic media

Abstract

The presence of S waves singularity points in low‐symmetry anisotropic models significantly affects the topology of the slowness surfaces of S waves in the vicinity of these points and, consequently, results in complications in the geometrical spreading. Thus, we analyse the effect of a singularity point in a simple elliptical orthorhombic model with decoupled P wave.

On pathological orthorhombic models

“…The value of 𝑑 12 (in this case, the model is reduced to the degenerate ORT model, see Stovas et al, 2022b) is given by…”

“…When 𝑑 13 increases, the S1S2 and PS1 singularity lines diverge (Figure 9b), and PS1 singularity line is insensitive to changes in 𝑑 13 . The value of 𝑑 13 (in this case the model is reduced to the degenerate ORT model, see Stovas et al, 2022b) (32) The first factor in Equation ( 32) (PS1 singularity line) becomes a hyperbola (if 𝑐 44 (𝑀 23 − 𝑀 33 ) + 𝑐 66 (𝑀 11 − 𝑀 12 ) < 0) (Figure 11a) or an ellipse (if 𝑐 44 (𝑀 23 − 𝑀 33 ) + 𝑐 66 (𝑀 11 − 𝑀 12 ) > 0), and the second one (S1S2 singularity line) is an ellipse (if 𝑀 13 > 0) or a pure imaginary line (if 𝑀 13 < 0, our model).…”

“…The phase velocity along the S1S2 singularity line becomes constant (degenerate model according to Stovas et al (2022b)),…”

“…It means that the 'Pathological model I' section belongs to the class of degenerate orthorhombic (ORT) models. Stovas et al (2022b) have shown that for degenerate ORT model, the singularity line defined in phase domain results in two lines in the group velocity domain. By using the method of computation, the image of singularity line derived in Stovas et al (2022b),…”

“…The value of d 12 (in this case, the model is reduced to the degenerate ORT model, see Stovas et al., 2022b) is given by $$$$\begin{equation}{d_{12d}} = \sqrt {\frac{{\left( {{\Delta _{16}}{\Delta _{34}} - {\Delta _{45}}{\Delta _{65}}} \right)\left( {{\Delta _{26}}{\Delta _{35}} + {\Delta _{45}}{\Delta _{64}}} \right)}}{{{\Delta _{34}}{\Delta _{35}}}}} .\end{equation}$$$$…”