Acoustic Waves at the Interface Between Different Types of Anisotropic-Strata

A good model needs to be as simple as possible while remaining close to physical reality, such as the strata model inside the earth. The strata layers can be macro-anisotropic but transversely isotropic, where some are vertically symmetric and the others not [12]. The macroscopic anisotropy and related properties have been modeled based on the so-called elastic tensor, which is usually reserved for the hexagonal crystals [1, 2]. Obviously, macroscopic anisotropy is significant for seismic waves with long wave-length regarding propagation, reflection, refraction, and polarization.

An incident P-wave from the VTI medium striking on the interface would induce several new waves at the interface, e.g., reflected P-wave, reflected SV-wave, refracted P-wave, and refracted SV-wave, as shown in  . At the interface between VTI and TTI media, a P-wave from VTI medium leads to reflection and refraction, which produces the reflected P-wave, reflected SV-wave, refracted P-wave, and refracted SVwave, corresponding to the index (1,2,3,4) The mechanical and physical properties of a general anisotropic medium can be described by an elastic stiffness tensor that may have up to 21 independent elements. The elastic stiffness tensors of the VTI and TTI media have only 5 independent elements [36,37]. Now, consider an incident P-wave from VTI to TTI medium with an incident angle (θ). For the TTI medium, Bond transformation can be used to convert its tilt-angle structure to the same tensor form of the VTI structure [24][25][26][27][28][29][30][31][32][33][34]; Kelvin-Christoffel equation can be solved for the reflected and refracted coefficients; Snell's law can be applied to determine the angles of reflection and refraction and to guarantee energy conservation.
Inside the incident VTI medium, Snell's law yields a fourth-order polynomial for the angles of reflection of the reflected P-wave and SV-wave [24,25], . ( Inside the refraction TTI medium, an eighth-order polynomial of the angles of refraction is obtained for the refracted P-wave and SV-wave [35], . . ( In equations (1)-(2), θ is an incident angle; the coefficients ( and ) are functions of the incident angle, the anisotropy, and the mechanical parameters of the media.

The Effects of TTI Tilting
Consider the interface between VTI and TTI media, where the TTI medium has a tilting-angle, φ. The typical mechanical and anisotropy parameters for the VTI and TTI media have been reported in the literature [13-14, 25, 35, 38-39]. Typically, the tilting level of the TTI medium shows a significant influence on the critical incident angle, the reflection and refraction coefficients, the magnitude of power density flux, and the state of polarization.
The critical incidence angle (θ ), corresponding to the refracted P-wave, has been affected significantly by the tilting level of the TTI medium and its anisotropy, as shown in Table 1. The tilting-angle and anisotropy of the TTI medium also influence the coefficients of the reflection and refraction. The calculated refraction coefficients and corresponding phase angles are shown in Figure 4. It should be noted that the reflection and refraction coefficients are not smooth continuous at the angle of critical incidence.  The polarization state of the refracted wave is also influenced by the tilting-angle and anisotropy of the refraction medium [29,31,40].
Prior to the critical incident angle, the polarization coefficients of the waves are all mathematically real numbers, including the incident P-wave, reflected P-wave, reflected SV-waves, refracted P-wave, and refracted SV-wave.
In the post-critical angle region, except for the refracted P-wave, the polarization coefficients are complex numbers for all waves. The polarization coefficients of the refracted P-wave have two possible solutions. interface. Additional work will lead to improved understanding of the interface between different types of anisotropic media, such as the interface between TVI and TTI anisotropic media. In practical applications for energy conversion, on the transient response of the acoustic-electric transducers, a similar issue may appear for acoustic scattering from spheres and cylinders [41][42][43].
The next steps may include the determinations of anomalous refraction if it ever exists, the effect of the competition between anisotropy and tilting-angle of the media, and the effect of anisotropy and the tiltingangle on polarization conversion. With respect to reflection and refraction, it is extremely desirable to achieve a better understanding of the relative importance of the anisotropy and TTI angle tilting, i.e., the competition between the anisotropy and angle-tilting of the media.