Introduction

Concerning the interior of the Earth, it is a common understanding that the interior is composed of the regular sequences of isotropic thin layers of different properties. When the prevailing wavelength of a seismic wave is larger than the thickness of the individual layers, the sequences of thin layers behave anisotropically, whereas still transversely isotropic1,2,3,4. This macroscopically transversely isotropic medium with a vertical axis of symmetry is called a VTI medium5. In such a case, the mechanical property of a VTI medium can be described by the elastic stiffness tensor of a hexagonal crystal3,4,5,6,7. Based on these understandings, the influences of rock anisotropy on polarization, propagation and reflection/refraction of elastic waves have been studied and reported extensively7,8,9,10,11,12, e.g. the polarization direction of an elastic P-wave, which is different from propagation direction; the propagation velocity that is different from phase velocity; and the reflection/refraction coefficients, which vary with respect to the acoustic impendence and anisotropy of media.

The study on reflection of acoustic wave is very important to geophysics; for example, Nedimovic et al. analyzed the reflection signature of seismic and aseismic slip on the northern Cascadia subduction interface13 and Canales et al. discussed the seismic reflection images of a near-axis sill within the lower crust of the Juan de Fuca ridge14. Acoustic waves also bear many similarities to optical or electromagnetic waves in propagation, reflection, refraction and polarization. Grady et al. reported linear conversion and anomalous refraction for electromagnetic waves15. Genevet et al. studied the phenomena of anomalous reflection/refraction of light and its propagation with phase discontinuities16. Fa et al. predicted the existence of an anomalous incident-angle for an inhomogeneously refracted P-wave11.

In this paper, we show that there exists a physically significant anomalous incident-angle for the refracted P-wave. With this anomalous incident-angle, the incident-angle region can be classified into three sections: the pre-critical incident-angle region, the area between the critical incident-angle and the anomalous incident-angle and the post-anomalous incident-angle region. There are two distinctive phase velocity solutions before and after the anomalous incident-angle. For an inhomogeneously refracted elliptically-polarized P-wave, the anomalous incident-angle will cause an acute rotational-direction variation.

Results

Modeling

Considering a P-wave propagating in the x-z plane impinging on the interface (x-y plane) between two VTI media, the system can be described schematically as in Fig. 1.

Figure 1
figure 1

Polarization vector and wave-front normal for incident P-wave and induced waves at the interface.

The solid-lines with arrowhead indicate the phase velocity direction and the dashed-lines with arrowhead show the polarization direction; is a displacement and m = {0, 1, 2, 3, 4} denotes the incident P-wave, reflected P-wave, refracted P-wave, reflected SV-wave and refracted SV-wave, respectively.

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We performed calculations for two sedimentary rocks with some very well-known physical properties, as reported by geophysicists and given in Table 1 2,11,17. In this paper, we use anisotropic shale (A-shale) as the incidence medium and oil shale (O-shale) as the refraction medium. For this system, by calculation, we found that there is an anomalous incident-angle at .

Table 1 Anisotropic parameters and elastic constants for A-shale and O-shale.
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The elements of the elastic stiffness tensor, related to the anisotropic rock parameters, are given by Thomsen2,

For a harmonic acoustic-field, the wave displacements can be written as

In the equations above, θ(m) is either an incident-angle or a reflection/refraction angle and and are polarization coefficients; ϕ(m) is the phase shift for an induced wave relative to the incident P-wave and ϕ(0) is defined as 0°; R(m) is either the reflection or refraction coefficient for each induced wave and R(0) is defined as 1. For the refracted P-wave, the critical incident-angle is denoted by and the anomalous incident-angle is given as .

For the incident-angle range of , the reflection/refraction coefficients are real (not complex) and ϕ(m) is 0° or 180°. In the range of , the reflection/refraction coefficients are complex and ϕ(m) ∈ (−180°, 180°).

Verification of an anomalous incident-angle

The core existence of an anomalous incident-angle for an elastically refracted P-wave can be confirmed from Snell’s law. Based on the Christoffel equation, the solutions of the phase velocity for the incident wave (P-wave or SV-wave) and the four induced waves are given by3,8

where , , , , . Denoting the anomalous incident-angle as , the phase velocities of the refracted P-wave are for and for . They abide by Snell’s law such that for and for .

The reflection/refraction angles are calculated from the fourth-order polynomials of 11

where, , , , , , , , and . Eq. (13) can be used to calculate the refraction angles and determine the existence of the anomalous incident-angle, denoted by .

As shown in Fig. 2a, the value of is purely real for ; whereas for , it is purely imaginary, as shown in Fig. 2b. Figure 3a,b show that for θ(0) ∈ [0°, 90°], is real (not complex). In Fig. 3c,d, the value of is real for and purely imaginary for . For , both and are purely imaginary. Figure 4b,c show that is equal to for and is equal to for . The curve segment in Fig. 4b plus curve segment in Fig. 4c forms the curve in Fig. 4d, which is the same as that in Fig. 4a. Here, for , is purely real and is purely imaginary, so during the plotting of the relationship of versus θ(0), the computer takes the value of as zero automatically.

Figure 2
figure 2

Relationship between sinθ(2) and θ(0).

(a) For , is purely real. (b) For , sin θ(2) is purely imaginary. There is an obvious abnormality provided at .

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Figure 3
figure 3

Relationships of both and versus θ(0).

a and b show that is purely real for θ(0) ∈ [0°, 90°] and has a maximum at θ(0) = 62.04°; c and d show that is purely real for and is purely imaginary for . The modulus of has a maximum at θ(0) = 62.04°.

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Figure 4
figure 4

Relationships of versus θ(0).

stands for the phase velocity, , of the refracted P-wave for and the phase velocity, , of the refracted P-wave for .

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These results show clearly that Snell’s law is satisfied only if the phase velocity solution of the refracted P-wave is switched to from at θ(0) = 62.04°. And therefore, there is an anomalous incident-angle of . It resides in a region passing the critical incident angle , up to an incident angle 90°.

Verification of elliptically-polarized rotational direction change

Verification of elliptically-polarized rotational direction change can be achieved by invoking the so called energy balance principle. The polarization coefficients for the incident wave and the four induced waves are given by11

and

where the definitions of , and refer to those of Eq. (S2) in “Supplementary material for Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave”.

From Eqs. (14) and (15) we can obtain the expressions of polarization coefficients for the incident P-wave and the homogenous waves induced at the interface,

For the refracted P-wave is inhomogenous. Eqs. (14) and (15) provide two sets of solutions for the polarization coefficients (refer to Eqs. (S9) and (S10) in “Supplementary material for Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave”).

and

An alternative confirmation of the anomalous incident-angle may be achieved by looking at the z-component of Poynting vector, which can be obtained from the reflection/refraction coefficients11. Specifically, we look at the normalized z-component, , of the incident P-wave. We also look at the normalized real parts of z-components from the four induced waves:

Now, consider the polarization coefficients calculated from Eqs. (16) and (17) as those of the incident P-wave, reflected P-wave, reflected SV-wave and refracted SV-wave for θ(0) ∈ [0°, 90°] and the refracted P-wave for . Meanwhile, Eqs.(18) and (19) provide the polarization coefficients of the refracted P-wave for . Then the normalized real parts of z-components of Poynting vectors are plotted dashed-lines in Fig. 5a,b. It shows clearly that, for , the real part of is not identical to . Therefore, it is a violation of the energy balance principle. However, for , if we switch the calculation of polarization coefficients from Eqs. (18) and (19) to Eqs. (20) and (21), then the real part of is equal to , as shown by the solid-line in Fig. 5b, which abides the energy balance principle.

Figure 5
figure 5

Relationships of and versus θ(0).

(b) shows that the dashed-line segment stacks together with the solid-line segment for .

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For an inhomogenous refracted P-wave, the x-component of the polarization has a lag of 90° with respect to its z-component, which is defined as a left-rotational elliptical-polarized wave; otherwise, it is called a right-rotational elliptical-polarized wave. Figure 5 shows that a refracted P-wave is a linearly polarized wave for , a left-rotational elliptical-polarized wave for and a right-rotational elliptical-polarized wave for . There is an elliptically-polarized rotational direction change at the anomalous incident-angle .

Discussion

The current studies of the interface between two VTI media show that there is an anomalous incident-angle with respect to the refracted P-wave in the area . At such an incident-angle , the phase velocity of the refracted P-wave must be switched from to to satisfy Snell’s law. The inhomogeneously refracted P-wave experiences a sudden change from a left-rotational to a right-rotational elliptical-polarization.

It is worth noting that there is an anomalous incident-angle for the refracted P-wave, but no such an anomalous incident-angle θ(4) for the refracted SV-wave. As an example, let’s look at the interface between S-shale and C-sandstone. In this case, there are two critical incident-angles, i.e. and . The phase velocity of P-waves in S-shale is smaller than those of P-waves and SV-waves in C-sandstone (see Fig. 6). There is an anomalous incident-angle corresponding to the refracted P-wave at . However, even with the second critical incident-angle and the refracted SV-wave becoming an inhomogeneous wave for , we have not observed the existence of an anomalous incident-angle corresponding to the refracted SV-wave.

Figure 6
figure 6

The calculated phase velocity curves for sandstone shale (S-shale) and calcareous sandstone (C-sandstone).

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Additional Information

How to cite this article: Fa, L. et al. Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave. Sci. Rep. 5, 12700; doi: 10.1038/srep12700 (2015).