Analytical Expressions Of Phase And Group Velocities For QP Waves Propagating In Strong Anisotropic Media Of Arbitrary Symmetry

Nowadays, it has been universally recognized that the existence of anisotropy in Earth materials is rather common——from the crust to the core. In the crust, periodic thin layers (PTL) and preferred orientation cracks and fractures are the predominant factors responsible for the arising of seismic anisotropy. In the upper mantle, the lattice-preferred orientation (LPO) can give rise to strong anisotropy. The core is anisotropic either, which may be caused by its rotation behavior. It is revealed by recent surveys about the anisotropy of the core that there may be an innermost core inside the core (in the literature, the innermost core is also called the most inner core, inner inner core, or inner-most inner core). Seismic anisotropy has become a very valuable information source and an effective tool for studying the Earth. The research on seismic anisotropy has been the front and the focus in both earthquake seismology and exploration seismology. The relevant research covers the theory of anisotropic elastic waves, wavefield modeling, forward and inverse modeling of traveltimes, AVO analysis, velocity analysis, inversion of elastic constants, shear-wave splitting, inversion of the physical properties of the materials and the structures of the crust, the upper mantle and the core, and so forth. Some people think that the research about seismic anisotropy is developing into a special anisotropic seismology.The characteristics of propagation of elastic waves in anisotropic media are quite different from those in isotropic media. One such example is that in isotropic media, phase and group velocities always coincide, while in anisotropic media they generally diverge from each other in both magnitudes and directions, and their divergence appears obviously in even weakly anisotropic media. The phase and group velocities in anisotropic media coincide only in some special directions which are very rare. For studying the behavior of elastic waves in anisotropic media, phase velocity is one of the most fundamental and important physical quantities. Mathematically, phase velocity is the eigenvalue of the Christoffel matrix and can be derived from the Christoffel characteristic equation. Many other physical quantities are derived from the phase velocity and many physical laws (e.g. Snell's law) are represented by it. However, the phase velocity is difficult to extract from observations. Instead, what is obtained in practice is usually the ray velocity (i.e. the velocity of energy propagation). In perfectly elastic anisotropic media, group velocity is equivalent to the velocity of wavefront and ray velocity and thus it becomes a very important quantity either. It is one of the fundaments for implementing many geophysical applications, such as anisotropic ray tracing, earthquake location, tomography, inversion of elastic constants, calculation of reflection and transmission coefficients, and so on. The expression of group velocity can be derived from the relationships between the phase and group velocities. However, the exact expression of phase velocity in anisotropic media is a very complicated formula in terms of elastic constants and phase angles and thus is difficult to be utilized in the analytical study and practical applications. The derivation of group velocity requires calculating the derivatives of phase velocity. Consequently, no meaningful exact expression exists for group velocity. To overcome the difficulties, in the past decades, many authors have endeavored to develop simpler explicit approximations for the phase and group velocities. Some authors expand the exact expressions into Taylor series, Fourier series (trigonometric functions) and spherical harmonics functions to derive the approximations by truncating the series. There are also some others making use of the perturbation method introduced from the quantum mechanics. However, the previous work is mainly developed for simple cases (e.g. the elliptical anisotropy, transverse isotropy and the case of symmetry plane) and under the assumption of weak anisotropy. Naturally, their accuracy and applicable ranges are limited. In contrast to the previous work, the high-order approximations proposed in this thesis, especially those of V (φ,θ) and V_g(φ_g),θ_g), are applicable for arbitrary symmetry and quite accurate for strong anisotropy. Such explicit formulae are effective and useful tools for developing theoretical basis of geophysical applications, offering physical insights into the dependence of wave attributes on the elastic constants, providing theoretical supports for analyzing wave phenomenon, and so forth. The following is a brief presentation of the main contents and conclusions of this thesis.In Chapter one, the development history of the study on seismic anisotropy is reviewed, with a particular interest on the analytical study of the phase and group velocities during the past decades. Then, a short description about the contents of the following chapters is given.In Chapter two, the theory of plane waves propagating in generally anisotropic media is introduced systemically. Starting with basic physical laws of thermodynamics, wave attributes describing the propagation behaviors of plane waves in anisotropic media, including phase velocity, phase slowness, group velocity, group slowness, polarization etc., are introduced step by step, with discussion on their physical significances and correlations.In Chapter three, the squared phase velocity of qP waves is divided into a linear part and a part of higher degree. Then it is substituted into the Christoffel characteristic equation in the wave-vector coordinates to produce a cubic equation of the higher-order term (the wave-vector-coordinate system is set up with its x 3 axis along the wave vector). From this equation, a fraction-style approximation for the higher-order term and a high-order approximation for the squared phase velocity can be easily obtained. Utilizing the phase-velocity approximation, the high-order approximation for the group velocity can be derived. Then series expansions of multi-variable functions are applied to the derived approximations in terms of phase angles to generate the approximations in terms of group angles. All the derived formulae in this chapter are in the form of a linear part plus a higher-order fraction. The high-order approximations are not so more complicated than the linear approximations but much more accurate than them. Some properties of qP waves can be deduced from the theoretical analysis in chapters two and three:The three body waves propagating along a longitudinal direction are all pure-mode, namely, the qP wave is purely longitudinal and the two qS waves are purely transverse. In such a direction, the phase and group velocities of qP waves coincide and don't vary locally.The phase and group velocities of qP waves have the same extreme points, of which the directions are longitudinal.The linear approximations of V (φ,θ) and V_g(φ,θ) (phase velocity and group velocity in phase angles) are generally lower than the exact values. They coincide if and only if the wave propagates along a longitudinal direction.The linear approximations of V(φ_g,θg) and V_g(φ_g,θ_g) (phase velocity and group velocity in group angles) are generally beyond the exact values. They coincide if and only if the wave propagates along a longitudinal direction.The high-order approximation of V (φ,θ) for TI media is usually higher than the exact value except in a longitudinal direction.In general, the polarization direction of the qP wave diverges from the wave vector, so does the group direction. The divergences can be used as an evaluation for the strength of anisotropy.The divergence between the phase and group directions is greater than the divergence between the phase and polarization directions except in longitudinal directions.The divergences between the phase, group and polarization directions can be used to evaluate the relative errors of the approximations.In Chapter four, to illustrate the accuracy of the derived formulae, several models of different symmetry and strength of anisotropy are used for numerical calculation. The data are listed in tables and plotted. The numerical results accord well with the theoretical analysis in the previous chapters. Moreover, as shown below, some additional conclusions can be made from the tables and figures.Unlike that of V (φ,θ), other high-order approximations for TI symmetry are lower than the exact values in some directions and higher than them in some other directions.Unlike those for TI symmetry, high-order approximations of V (φ,θ) for other symmetry classes are not necessary to be always higher or lower than the exact values.The high-order approximations perform much better than the corresponding linear ones.The accuracy of all the approximations becomes lower with the growth of the strength of anisotropy.The strength of anisotropy defined as (V_max - V_min)/V_ave×100%, which is frequently used in the literature, cannot represent the maximum relative errors of the approximations very well. The reason may be that the definition represents actually the overall strength of anisotropy instead of the local ones.In a descendant order of the accuracy of the approximations of the same order, they read V (φ,θ), )V_g(φ_g,θ_g), V_g(φ,θ) and V(φ_g,θ_g).The high-order approximations of V (φ,θ) and V_g(φ_g,θ_g) are applicable for very strong anisotropy, while the performance of the approximations of V_g(φ,θ) and V_φg,θg is much worse than those of V (φ,θ) and V_g(φ_g,θ_g).Lastly, the main conclusions of this text are summed up in Chapter five.