https://orcid.org/0000-0002-2294-9626
Abstract
The acoustic orthorhombic model is widely used in seismic modeling and processing of P-wave data. However, the anisotropic acoustic models have so called S-wave artifacts (1 artifact in transversely isotropic acoustic medium and two artifacts in orthorhombic acoustic medium). I show that S-wave artifacts can have one singularity point that results in complications in polarization field and the group velocity surface. The conditions of the existence of this point are defined in terms of anellipticity parameters. This singularity point and its group velocity image are the objects of my analysis.
1. Introduction
The singularity points play very important role for wave propagation in low-symmetry media (Grechka and Obolentseva 1993, Vavrycuk 2001, Stovas et al. 2021a, 2023). Even in elliptical orthorhombic (ORT) media, the singularity point results in complication of the group velocity surface of S-waves (Stovas et al. 2021a). They result in internal refraction cones on the group velocity surface, complications in the polarization fields (Alshits and Shuvalov 1984), reflection/transmission coefficients (Jin and Stovas 2020a,b), ray tracing (Bakker 1998, Cerveny 2001, Vavrycuk 2001), and geometrical spreading (Payton 1992, Klimes 2002, 2010, Stovas et al. 2021a, 2022). The singularity points can also be an important characteristic of effective ORT models defined from porous media with meso-scale fractures (Pang et al. 2022). The acoustic anisotropic models (with transversely isotropic and ORT symmetry) are very popular for seismic modeling, data processing, and inversion (Alkhalifah 1998, 2000, 2003, Stovas 2015, 2018). Despite that acoustic models are developed for P-wave propagation, the corresponding equations also support S-wave artifacts in transversely isotropic (Grechka et al. 2004, Jin and Stovas 2018, 2019) and ORT (Xu and Stovas 2019) media.
In this paper, I define the position of the singularity point in an acoustic ORT medium, the conditions for its existence, the topology of the slowness surface in vicinity of this point, and the image of this point in the group velocity domain.
2. Acoustic orthorhombic media
The acoustic ORT medium is defined by Alkhalifah (2000, 2003). The detailed analysis of the wave propagation in acoustic ORT model can be found in Stovas (2015, 2018), Abedi et al. (2019), and Abedi and Stovas (2020).
The classic definition of the stiffness coefficient matrix for acoustic ORT medium from elastic one is given by following transformation
where the aneliptic parameters ηj are defined in individual |$k - l$| symmetry planes (Alkhalifah 2003),
by using existing stiffness coefficients |${c}_{44}$|, |${c}_{55}$|, and |${c}_{66}$|. Then, on the last step, the on-axes S-waves’ phase velocities are set to zero, i.e. |${c}_{44} = {c}_{55} = {c}_{66} = 0$|. This paper uses the symmetric anelliptic parameters |${\eta }_j$| proposed in (Stovas et al., 2021b). In this case, the non-diagonal stiffness coefficients are given by
For numerical examples, we use the “acoustic version’” of the standard ORT model (Schoenberg and Helbig 1997) with non-zero stiffness coefficients: |${c}_{11} = 9\,GPa,\,\,\,{c}_{22} = 9.84\,GPa,\,\,\,{c}_{33} = 5.94\,GPa,\,\,\,{c}_{12} = 3.6\,GPa,\,\,\,{c}_{13} = 2.25\,GPa,\,\,\,{c}_{23} = 2.4\,GPa.$| The symmetric anelliptic parameters |${\eta }_j$| computed for the standard ORT model are: |${\eta }_1 = 0.135,\,\,\,{\eta }_2 = 0.208,\,\,\,{\eta }_3 = 0.138$|.
From Sylvester's conditions, Equation (2), and the normal polarization condition, we have
In terms of anelliptic parameters, the conditions in Equations (4a) and (4b) imply
and the condition in Equation (4c) gives
The characteristic equation for acoustic ORT can be computed from the eigen-value problem for the corresponding Christoffel matrix and given by
where |${p}_j,\,\,j = 1,2,3$| are the slowness vector projections.
The solution of Equation (6) for vertical slowness projection is given by the rational equation (Alkhalifah 2003, Stovas 2015),
where
The slowness surface computed from Equation (7) for the acoustic standard ORT model is shown in Fig. 1. One can see one inner surface (P-wave) and two other surfaces (S-wave artifacts). For reasons of brevity, I annotate the S-wave artifacts as S1- and S2-waves.
The slowness surface for acoustic ORT medium given by Equation (7). The P-wave slowness surface is the inner one.
Solutions of equations |${f}_1 = 0$| and |${f}_2 = 0$| are, respectively, given by
The solutions given by Equations (9) and (10) are illustrated in Fig. 2 with different anelliptic parameters. We select two models: model 1 with |${\eta }_2 = 0.208$| and |${\eta }_3 = 0.138$| (black dot), and model 2 with |${\eta }_2 = 0.06$| and |${\eta }_3 = 0.05$| (gray dot). Model 1 has singularity point in between the symmetry planes, and model 2 has no singularity point. The propagation zones for the P-, S1-, and S2-waves are notated. The S1- and S2-waves can have one joint point (singularity point) that is shown by the dot in Fig. 2a.
Solutions of equations |${f}_1 = 0$| and |${f}_2 = 0$| given by Equation (9) (red line) and Equation (10) (blue line) for model 1 with |${\eta }_1 = 0.135$|, |${\eta }_2 = 0.208$| and |${\eta }_3 = 0.138$| (a) and model 2 with |${\eta }_1 = 0.135$|, |${\eta }_2 = 0.06$|, and |${\eta }_3 = 0.05$| (b). The propagation zones for P-, S1-, and S2-waves are indicated by corresponding letters. The position of singularity point in model 1 is shown by a black dot.
The position of singularity point can be found from the system of equations for three non-diagonal minors of the Christoffel matrix (Schoenberg and Helbig 1997) adjusted for acoustic medium,
with the solution
The solution in Equation (12) provides the position of singularity point (in between the symmetry planes) on the slowness surface given by Equation (7). Note that |${p}_3( {{p}_1 = {p}_{1s},{p}_2} ) = {p}_3( {{p}_1,{p}_2 = {p}_{2s}} ) = {p}_{3s}$|.
From the solution in Equation (12), we can obtain the conditions for existing of singularity point,
which result in the following conditions for anelliptic parameters |${\eta }_j$|,
where |$j,k,l = 1,2,3$|. In Fig. 3, we plot the physically realizable model condition Equation (4b) (inside the region shown by dashed line) and the singularity point conditions, and Equations (14a) and (14b) (inside the region shown by solid line) with fixed |${\eta }_1 = 0.135$|. These models are also distinguished in Fig. 2a and b.
The zones in |$( {{\eta }_2,{\eta }_3} )$| with |${\eta }_1 = 0.135$| that correspond to physically realizable model (inside the region shown by dashed line) and the zone resulting in singularity point (inside the region shown by solid line). The black and gray dots correspond to models 1 and 2, respectively.
The change in position of singularity point due to change in |${\eta }_3$| with fixed |${\eta }_1$| and |${\eta }_2$| is illustrated in Fig. 4. Note that the bounds in inequalities, Equation (14a) and (14b) correspond to situations when one coordinate of singularity point [given by Equation (12)] tends to be infinite.
Position of singularity point depending on different anelliptic parameters: |${\eta }_3$| (solid line), |${\eta }_2$| (dashed line) and |${\eta }_1$| (dotted line). When changing given anelliptic parameter, two other parameters are fixed according to model 1. The black dot shows the position of singularity point for model 1.
3. The topology of the slowness surface in the vicinity of singularity point
To analyze the topology of the slowness surface in the vicinity of singularity point, I start with the series of Christoffel polynomials in the vicinity of the slowness vector corresponding to singularity point |${{\bf p}}_S = {({p}_{1s},{p}_{2s},{p}_{3s})}^T$|,
where the gradient, the second-order derivative matrix, and the third-order tensor are, respectively, given by
The in-singularity point is |$\nabla F( {{{{\bf p}}}_S} ) = 0$|. The elements of the second-order derivative matrix are given by
The determinant of matrix |$\det {{\bf M}} = - 128{c}_{12}{c}_{13}{c}_{23} \ne 0$|, therefore, this singularity point belongs to the class of conical degeneracy (Shuvalov 1998). Within the range of existing singularity point defined by Equation (14a) and (14b), the matrix |${{\bf M}}$| has one negative and two positive eigenvalues, which means that the corresponding quadratic form |${( {{{\bf p}} - {{{\bf p}}}_S} )}^T{{\bf M}}( {{{{\bf p}}}_S} )( {{{\bf p}} - {{{\bf p}}}_S} )$| shapes the hyperboloid (Fig. 5). The elements of the third-order tensor are given by
The phase domain cone illustrating the behavior of S1- and S2-wave surfaces in vicinity of singularity point.
For further analysis, I introduce the new coordinate system associated with the singularity point,
where |${p}_r$| is the local radial horizontal slowness projection measured from singularity point, and |$\phi $| is the local phase azimuth angle taken at this point.
Then the vertical slowness projection in the vicinity of the singularity point behaves as
where the coefficients |$a( \phi )$|, |$b( \phi )$|, and |$c( \phi )$|
where
The coefficient |$a( \phi )$| is plotted versus the phase azimuth |$\phi $| in Fig. 6a. We see that |$a( \phi ) = 0$| for |$\phi = 0,{\pi / 2},\pi ,{{3\pi } / 2}$|, which indicates the irregularity of this point. For the critical azimuth angle given by equation
the coefficient |$a \to \infty $|.
Coefficients |$a( \phi )$| (a) and |${K}_{ - 1}( \phi )$| (b) shown by polar plots.
The Gaussian curvature is controlling waves amplitudes via the relative geometrical spreading factor. The equation for Gaussian curvature is given by
where |$adj( {{\nabla }^2F} ) = ( {{r}_{ij}} )$| is the adjoint matrix with the elements as follows
where |${m}_{ij}$| are given in Equation (17).
To compute the Gaussian curvature in vicinity of singularity point, we need to define the behavior of the first-order derivatives |${f}_j,\,\,\,j = 1,2,3$| and the elements of adjoint matrix |${r}_{ij},\,\,\,i,j = 1,2,3$| in vicinity of singularity point (Appendix). Substituting Appendix Equations (A.1)–(A.3) into Equation (23) results in the following behavior of the Gaussian curvature,
where the coefficients |${K}_{ - 1}$| and |${K}_0$| are given by
with
The dominant term |${K}_{ - 1}( \phi )$| given in Equation (26a) is shown in a polar plot versus local azimuth angle |$\phi $| in Fig. 6b. The azimuthal behavior of |${K}_{ - 1}( \phi )$| shows three specific azimuthal directions of abnormal behavior of Gaussian curvature.
The Gaussian curvature is plotted in |${p}_1$|- |${p}_2$| coordinates in Fig. 7. One can clearly see the P, S1, and S2 propagation zones, the gaps (non-propagating zones), and the position of singularity point. The Gaussian curvature |$K( {{p}_r,\phi = 0} ) = K( {{p}_r,\phi = {\pi / 2}} ) = 0$|. For the critical value of azimuth angle, |$\phi \to {\phi }_c$| and |${p}_r \to 0$|, |$K \to 0$|. If |$\phi \ne {\phi }_c$|, |$\phi \ne 0,\,\,{\pi / {2,\,\,\pi ,}}{{\,3\pi } / 2}$|, then |$K \to \pm \infty $|.
The Gaussian curvature of the slowness surface plotted versus horizontal slowness projections. The white zones indicate the gaps in the wave propagation.
4. The group velocity image of singularity point
If |$\nabla F( {{{{\bf p}}}_S} ) \ne 0$|, the group velocity is given by (Fedorov 1968)
If |$\nabla F( {{{{\bf p}}}_S} ) = 0$| (singularity point), the group velocity image is defined by the second-order derivative matrix |${{\bf M}}$| and can be obtained by intersecting of cone given by |${{\bf V}}{{{\bf M}}}^{ - 1}{{{\bf V}}}^T = 0$| and the plane |${{{\bf p}}}_S{{\bf V}} = 1$| (see Fig. 8). Since |${{{\bf M}}}^{ - 1} = {{adj( {{\bf M}} )} / {\det {{\bf M}}}}$|, this approach can be used if |$\det {{\bf M}} \ne 0$| (conical degeneracy).
The group velocity cone given by equation |${{\bf V}}{{{\bf M}}}^{ - 1}{{{\bf V}}}^T = 0$| (magenta), the plane |${{{\bf p}}}_S{{\bf V}} = 1$| (green), and their intersection (group velocity ellipse) shown by a bold black line.
The corresponding quadratic form [using the elements of the second-order derivatives matrix given in Equation (17)] is given by
where |$\Delta {p}_1 = {p}_1 - {p}_{1s},\,\,\,\Delta {p}_2 = {p}_2 - {p}_{2s},\,\,\,\Delta {p}_3 = {p}_3 - {p}_{3s}$|. Equation (28) defines cone in the phase domain illustrating the behavior of the S1- and S2-wave slowness surfaces in vicinity of singularity point (Fig. 5). The origin of coordinate system corresponds to position of singularity point. The right- and left-hand side sheets correspond to S2- and S1-waves.
The cone in the group velocity domain can be defined by using the scaled group velocity projections,
where |${W}_1 = {m}_{23}{V}_1,\,\,\,{W}_2 = {m}_{13}{V}_2,\,\,\,{W}_3 = {m}_{12}{V}_3$|.
The shape of the group velocity image of singularity point is given by interception of cone and the plane, respectively, given by
This intersection gives the ellipse (Fig. 9), which is tangent to all three symmetry planes 2–3, 1–3, and 1–2 in the points, respectively, given by,
The image of singularity point in the group velocity domain (ellipse shown by solid line). The dashed line illustrates projection of this ellipse on horizontal symmetry plane. The black dots indicate the points of the ellipse and the symmetry planes. The gray dot indicates the center of the ellipse.
The coordinates of these points are summarized in Table 1. In Fig. 10, we see the group velocity ellipse computed for three different values of |${\eta }_3$|. One can see that with increase in |${\eta }_3$|, the group velocity ellipse is rotated in 3D space and the plane where the ellipse is located becomes less steep. The limits for position of singularity point and its group velocity image computed for extreme values of anelliptic parameter |${\eta }_3$| from Equation (14a) and (14b) are shown in Table 2. We see that in these cases, the one of the slowness projection goes to infinity and the corresponding group velocity collapses in one of the symmetry planes.
The image of singularity point and its projection on horizontal symmetry plane computed for fixed values of anelliptic parameters |${\eta }_1 = 0.1$| and |${\eta }_2 = 0.15$|, and three different values for anelliptic parameter |${\eta }_3$|: (a) |${\eta }_3 = 0.12$|, (b) |${\eta }_3 = 0.15$|, and (c) |${\eta }_3 = 0.18$|.
The specific points on the group velocity ellipse for given values of the local azimuth angle. The first three rows correspond to the points where the ellipse is tangent to the symmetry planes 2–3, 1–3, and 1–2, respectively. Rows 4–6 correspond to the point with the maximum values of |${V}_1$|, |${V}_2$|, and |${V}_3$|, respectively. Row 7 shows the position of the group velocity ellipse.
| Local azimuth | |${V}_1$| | |${V}_2$| | |${V}_3$| | |
|---|---|---|---|---|
| 1 | |$0,\pi $| | |$0$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| |
| 2 | |$\frac{\pi }{2},\frac{{3\pi }}{2}$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$0$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| |
| 3 | |$- \arctan \frac{{{m}_{13}}}{{{m}_{23}}}$| | |$\frac{{{m}_{13}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$0$| |
| 4 | |$- \arctan \frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{{m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{d}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| |
| 5 | |$- \arctan \frac{{{m}_{13}{p}_{3s}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{d}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| |
| 6 | |$\arctan \frac{{{p}_{2s}}}{{{p}_{1s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{2s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{12}{m}_{13}p_{1s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{d}$| |
| 7 | Center of the ellipse | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{{2d}}$| |
| Local azimuth | |${V}_1$| | |${V}_2$| | |${V}_3$| | |
|---|---|---|---|---|
| 1 | |$0,\pi $| | |$0$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| |
| 2 | |$\frac{\pi }{2},\frac{{3\pi }}{2}$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$0$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| |
| 3 | |$- \arctan \frac{{{m}_{13}}}{{{m}_{23}}}$| | |$\frac{{{m}_{13}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$0$| |
| 4 | |$- \arctan \frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{{m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{d}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| |
| 5 | |$- \arctan \frac{{{m}_{13}{p}_{3s}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{d}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| |
| 6 | |$\arctan \frac{{{p}_{2s}}}{{{p}_{1s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{2s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{12}{m}_{13}p_{1s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{d}$| |
| 7 | Center of the ellipse | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{{2d}}$| |
The specific points on the group velocity ellipse for given values of the local azimuth angle. The first three rows correspond to the points where the ellipse is tangent to the symmetry planes 2–3, 1–3, and 1–2, respectively. Rows 4–6 correspond to the point with the maximum values of |${V}_1$|, |${V}_2$|, and |${V}_3$|, respectively. Row 7 shows the position of the group velocity ellipse.
| Local azimuth | |${V}_1$| | |${V}_2$| | |${V}_3$| | |
|---|---|---|---|---|
| 1 | |$0,\pi $| | |$0$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| |
| 2 | |$\frac{\pi }{2},\frac{{3\pi }}{2}$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$0$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| |
| 3 | |$- \arctan \frac{{{m}_{13}}}{{{m}_{23}}}$| | |$\frac{{{m}_{13}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$0$| |
| 4 | |$- \arctan \frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{{m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{d}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| |
| 5 | |$- \arctan \frac{{{m}_{13}{p}_{3s}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{d}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| |
| 6 | |$\arctan \frac{{{p}_{2s}}}{{{p}_{1s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{2s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{12}{m}_{13}p_{1s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{d}$| |
| 7 | Center of the ellipse | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{{2d}}$| |
| Local azimuth | |${V}_1$| | |${V}_2$| | |${V}_3$| | |
|---|---|---|---|---|
| 1 | |$0,\pi $| | |$0$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}$| |
| 2 | |$\frac{\pi }{2},\frac{{3\pi }}{2}$| | |$\frac{{{m}_{12}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$0$| | |$\frac{{{m}_{23}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| |
| 3 | |$- \arctan \frac{{{m}_{13}}}{{{m}_{23}}}$| | |$\frac{{{m}_{13}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$\frac{{{m}_{23}}}{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}$| | |$0$| |
| 4 | |$- \arctan \frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{{m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{d}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )d}}$| |
| 5 | |$- \arctan \frac{{{m}_{13}{p}_{3s}}}{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{3s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{d}$| | |$\frac{{{m}_{12}{m}_{23}p_{2s}^2}}{{( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )d}}$| |
| 6 | |$\arctan \frac{{{p}_{2s}}}{{{p}_{1s}}}$| | |$\frac{{{m}_{13}{m}_{23}p_{2s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{12}{m}_{13}p_{1s}^2}}{{( {{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}} )d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{d}$| |
| 7 | Center of the ellipse | |$\frac{{{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}}}{{2d}}$| | |$\frac{{{m}_{13}{p}_{1s} + {m}_{23}{p}_{2s}}}{{2d}}$| |
The extreme cases for position of singularity point and corresponding group velocity image. The first two rows correspond to minimum value of anelliptic parameter |${\eta }_3$| in the case of |${\eta }_2 > {\eta }_1$| and |${\eta }_1 > {\eta }_2$|, respectively. The third row corresponds to maximum value of |${\eta }_3$|. Note that the ratios for |${m}_{jk}$| in equations for the group velocity image should be computed as the limits.
| |$p_{1s}^2$| | |$p_{2s}^2$| | |$p_{3s}^2$| | Group velocity image | |
|---|---|---|---|---|
| |${\eta }_3 \to \frac{{{\eta }_2 - {\eta }_1}}{{1 - 2{\eta }_1}}$| | |$\frac{{1 - 2{\eta }_1}}{{2{c}_{11}( {{\eta }_2 - {\eta }_1} )}}$| | |$\infty $| | |$\frac{1}{{2{\eta }_1{c}_{33}}}$| | |${V}_2 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{23}}}{{{m}_{12}}}{V}_1$| |
| |${\eta }_3 \to \frac{{{\eta }_1 - {\eta }_2}}{{1 - 2{\eta }_2}}$| | |$\infty $| | |$\frac{{1 - 2{\eta }_2}}{{2{c}_{22}( {{\eta }_1 - {\eta }_2} )}}$| | |$\frac{1}{{2{\eta }_2{c}_{33}}}$| | |${V}_1 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{13}}}{{{m}_{12}}}{V}_2$| |
| |${\eta }_3 \to {\eta }_1 + {\eta }_2 - 2{\eta }_1{\eta }_2$| | |$\frac{1}{{2{\eta }_2{c}_{11}}}$| | |$\frac{1}{{2{\eta }_1{c}_{22}}}$| | |$\infty $| | |${V}_3 = 0,\,\,\,{V}_2 = \pm \frac{{{m}_{23}}}{{{m}_{13}}}{V}_1$| |
| |$p_{1s}^2$| | |$p_{2s}^2$| | |$p_{3s}^2$| | Group velocity image | |
|---|---|---|---|---|
| |${\eta }_3 \to \frac{{{\eta }_2 - {\eta }_1}}{{1 - 2{\eta }_1}}$| | |$\frac{{1 - 2{\eta }_1}}{{2{c}_{11}( {{\eta }_2 - {\eta }_1} )}}$| | |$\infty $| | |$\frac{1}{{2{\eta }_1{c}_{33}}}$| | |${V}_2 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{23}}}{{{m}_{12}}}{V}_1$| |
| |${\eta }_3 \to \frac{{{\eta }_1 - {\eta }_2}}{{1 - 2{\eta }_2}}$| | |$\infty $| | |$\frac{{1 - 2{\eta }_2}}{{2{c}_{22}( {{\eta }_1 - {\eta }_2} )}}$| | |$\frac{1}{{2{\eta }_2{c}_{33}}}$| | |${V}_1 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{13}}}{{{m}_{12}}}{V}_2$| |
| |${\eta }_3 \to {\eta }_1 + {\eta }_2 - 2{\eta }_1{\eta }_2$| | |$\frac{1}{{2{\eta }_2{c}_{11}}}$| | |$\frac{1}{{2{\eta }_1{c}_{22}}}$| | |$\infty $| | |${V}_3 = 0,\,\,\,{V}_2 = \pm \frac{{{m}_{23}}}{{{m}_{13}}}{V}_1$| |
The extreme cases for position of singularity point and corresponding group velocity image. The first two rows correspond to minimum value of anelliptic parameter |${\eta }_3$| in the case of |${\eta }_2 > {\eta }_1$| and |${\eta }_1 > {\eta }_2$|, respectively. The third row corresponds to maximum value of |${\eta }_3$|. Note that the ratios for |${m}_{jk}$| in equations for the group velocity image should be computed as the limits.
| |$p_{1s}^2$| | |$p_{2s}^2$| | |$p_{3s}^2$| | Group velocity image | |
|---|---|---|---|---|
| |${\eta }_3 \to \frac{{{\eta }_2 - {\eta }_1}}{{1 - 2{\eta }_1}}$| | |$\frac{{1 - 2{\eta }_1}}{{2{c}_{11}( {{\eta }_2 - {\eta }_1} )}}$| | |$\infty $| | |$\frac{1}{{2{\eta }_1{c}_{33}}}$| | |${V}_2 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{23}}}{{{m}_{12}}}{V}_1$| |
| |${\eta }_3 \to \frac{{{\eta }_1 - {\eta }_2}}{{1 - 2{\eta }_2}}$| | |$\infty $| | |$\frac{{1 - 2{\eta }_2}}{{2{c}_{22}( {{\eta }_1 - {\eta }_2} )}}$| | |$\frac{1}{{2{\eta }_2{c}_{33}}}$| | |${V}_1 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{13}}}{{{m}_{12}}}{V}_2$| |
| |${\eta }_3 \to {\eta }_1 + {\eta }_2 - 2{\eta }_1{\eta }_2$| | |$\frac{1}{{2{\eta }_2{c}_{11}}}$| | |$\frac{1}{{2{\eta }_1{c}_{22}}}$| | |$\infty $| | |${V}_3 = 0,\,\,\,{V}_2 = \pm \frac{{{m}_{23}}}{{{m}_{13}}}{V}_1$| |
| |$p_{1s}^2$| | |$p_{2s}^2$| | |$p_{3s}^2$| | Group velocity image | |
|---|---|---|---|---|
| |${\eta }_3 \to \frac{{{\eta }_2 - {\eta }_1}}{{1 - 2{\eta }_1}}$| | |$\frac{{1 - 2{\eta }_1}}{{2{c}_{11}( {{\eta }_2 - {\eta }_1} )}}$| | |$\infty $| | |$\frac{1}{{2{\eta }_1{c}_{33}}}$| | |${V}_2 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{23}}}{{{m}_{12}}}{V}_1$| |
| |${\eta }_3 \to \frac{{{\eta }_1 - {\eta }_2}}{{1 - 2{\eta }_2}}$| | |$\infty $| | |$\frac{{1 - 2{\eta }_2}}{{2{c}_{22}( {{\eta }_1 - {\eta }_2} )}}$| | |$\frac{1}{{2{\eta }_2{c}_{33}}}$| | |${V}_1 = 0,\,\,\,{V}_3 = \pm \frac{{{m}_{13}}}{{{m}_{12}}}{V}_2$| |
| |${\eta }_3 \to {\eta }_1 + {\eta }_2 - 2{\eta }_1{\eta }_2$| | |$\frac{1}{{2{\eta }_2{c}_{11}}}$| | |$\frac{1}{{2{\eta }_1{c}_{22}}}$| | |$\infty $| | |${V}_3 = 0,\,\,\,{V}_2 = \pm \frac{{{m}_{23}}}{{{m}_{13}}}{V}_1$| |
The most practical plane to analyze the ellipse parameters is the horizontal symmetry plane. The projection of the ellipse defined by Equation (30) on the horizontal symmetry plane is
where the position of the center of ellipse and coefficient |${a}_0$| are defined as
where
and the coefficients |${\alpha }_{jk},\,\,{\beta }_{jk},\,\,\,{\gamma }_{jk}$| are
Note that |${\beta }_{02} = {\alpha }_{02}$| and |${\gamma }_{20} = {\beta }_{20}$|.
Let us analyze the properties of the ellipse versus local azimuth angle defined in Equation (19). The group velocity projections can be defined from Equation (27),
where F is given in Equation (6). By using Equation (A.1), the group velocity projections can be computed as follows:
where
The group velocity projections are plotted versus local azimuth |$\phi $| in Fig. 11. If |$\phi = 0$| or |$\phi = \pi $|, the group velocity ellipse reaches the symmetry plane 2–3, If |$\phi = {\pi / 2}$| or |$\phi = {{3\pi } / 2}$|, the group velocity ellipse reaches the symmetry plane 1–3, for the critical azimuth defined by Equation (22), the group velocity ellipse reaches the symmetry plane 1–2. The corresponding points are given in Equation (31).
The group velocity projections computed from Equation (24) versus local azimuth |$\phi $| computed for the acoustic standard ORT model. The projections |${V}_1$|, |${V}_2$|, and |${V}_3$| are shown by red, blue, and black colors, respectively.
At azimuth |$\phi = \arctan ( { - {{( {{m}_{12}{p}_{2s} + {m}_{13}{p}_{3s}} )} / {{m}_{23}{p}_{3s}}}} )$|, we have the point with the maximum group velocity projection |${V}_1$|,
At azimuth |$\phi = \arctan ( { - {{{m}_{13}{p}_{3s}} / {( {{m}_{12}{p}_{1s} + {m}_{23}{p}_{3s}} )}}} )$|, we have the point with the maximum group velocity projection |${V}_2$|,
and at azimuth |$\phi = \arctan ( {{{{p}_{2s}} / {{p}_{1s}}}} )$| (which coincide with the global azimuth of singularity point), we have the point with the maximum group velocity projection |${V}_3$|,
The results from Equation (39a)–(39c) are summarized in Table 1. The S1 and S2 group velocity surfaces are shown in Fig. 12.
The S1 (a) and S2 (b) group velocity surfaces. Computations are performed for the acoustic standard ORT model.
5. Conclusions
The conditions for existence of singularity point for S1- and S2-wave artifacts in acoustic ORT medium are defined. It is shown that this singularity point is located in between the symmetry planes and has conical degeneracy. The position of this point is controlled by anellipticity parameters defined in all symmetry planes. The topology of the slowness surfaces in vicinity of singularity point is analyzed showing the anomaly in the Gaussian curvature and, consequently, the relative geometrical spreading for S1- and S2-wave artifacts. The image of singularity point in the group velocity domain is given by the ellipse located in a specific plane. The parameters of this ellipse are defined. The projections of the group velocity ellipse on symmetry planes are computed. The projection on the horizontal symmetry plane is of importance since it defines (after normalization) the projection of the ellipse on the xy plane (acquisition plane).
Acknowledgements
Alexey Stovas acknowledges the GAMES project for financial support.
Conflict of interest statement
There is no any conflict of interests.
Funding
Funded by Research Council of Norway, grant 294404.
Appendix. The first- and second-order derivatives of Christoffel polynomial computed in vicinity of singularity point
In vicinity of singularity point, the first-order derivatives behave as
The second-order derivatives behave as
and
where |${g}_{ijk},\,\,\,i,j,k = 1,2,3$| are the elements of the third-order tensor given in Equation (18).
