Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave

Abstract

We report a newly discovered anomalous incident-angle of an elastically refracted P-wave, arising from a P-wave impinging on an interface between two VTI media with strong anisotropy. This anomalous incident-angle is found to be located in the post-critical incident-angle region corresponding to a refracted P-wave. Invoking Snell’s law for a refracted P-wave provides two distinctive solutions before and after the anomalous incident-angle. For an inhomogeneously refracted and elliptically polarized P-wave at the anomalous incident-angle, its rotational direction experiences an acute variation, from left-hand elliptical to right-hand elliptical polarization. The new findings provide us an enhanced understanding of acoustical-wave scattering and lead potentially to widespread and novel applications.

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. 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.

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.

Medium	α(n) (m/s)	β(n) (m/s)	ρ(n) (g/cm3)	Thomsen Parameters			Elastic constants (GPa)
				ε(n)	δ*(n)	γ(n)	C11
A-shale	2745	1508	2.340	0.103	−0.073	0.345	21.264	6.976	17.632	5.321	8.993
O-shale	4231	2539	2.370	0.200	0.000	0.145	9.397	15.824	42.426	15.278	19.709

The superscript n = {in, re} donates the incidence medium and refraction medium.

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. 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 .

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°.

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 .

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. Relationships of and versus θ⁽⁰⁾.

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

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

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).