Quasi-shear coupling in inhomogeneous weakly anisotropic media. Quasi-isotropic propagator matrices

Summary

In inhomogeneous weakly anisotropic media, the qS1 and qS2 waves do not propagate independently, but are mutually coupled. In this contribution, the Debye procedure is used to investigate the quasi-shear wave coupling. A ray Ω⁰ of an S wave in the background isotropic medium is considered. The displacement vector in the perturbed, weakly anisotropic medium along Ω⁰ is then expressed in the form U=Be⁽¹⁾+Ce⁽²⁾, where e⁽¹⁾ and e⁽²⁾ are two mutually perpendicular unit vectors, perpendicular to Ω⁰. The elastodynamic equation and the Debye procedure yield a system of two coupled linear ordinary differential equations of the first order along Ω⁰ for B and C. The 2x2 system matrix is frequency dependent, so that even B and C are frequency dependent. The system can be simplified, as the system matrix can be decomposed into two simpler 2x2 matrices: the average shear wave matrix and the shear wave splitting matrix. By a suitable substitution, the average shear wave matrix can be removed, so that the final system contains only the shear wave splitting matrix. The 2x2 propagator matrix of this system, called here the quasi-isotropic propagator matrix, is introduced. The propagator matrix equals the 2x2 identity matrix at an arbitrarily selected point S on Ω⁰, and represents the fundamental matrix of the system under consideration. As soon as the quasi-isotropic propagator matrix is known along Ω⁰, the solutions B(R) and C(R) at any point R on Ω⁰ can be obtained from B(S) and C(S) by a simple matrix multiplication. The properties of the quasi-isotropic propagator matrix are investigated. It is shown that the determinant of the quasi-isotropic propagator matrix equals unity along the whole ray Ω⁰ (Liouville's theorem), that the propagator matrix is symplectic and that it satisfies the chain rule. A great advantage of the quasi-isotropic propagator matrix is that it can be chained. The ray Ω⁰ from S to R may be divided into segments and the propagator matrix from S to R may be obtained as a product of propagator matrices along individual segments. The propagator matrices along individual segments may be computed in various ways (analytically, semi-analytically, numerically). Even segments in isotropic media, segments in strongly anisotropic media and segments across structural interfaces may be introduced. Finally, combining these expressions for qS waves with simpler (non-coupled) expressions for qS waves, very general expressions for approximate high-frequency Green functions in a 3-D laterally-varying structure containing curved interfaces are derived. The medium along individual segments of the ray Ω⁰ may be isotropic, weakly anisotropic or strongly anisotropic, and the wave under consideration may be multiply reflected and converted (containing qP and qS segments).

Introduction

Ray methods can be used to study the propagation of high-frequency seismic body waves both in isotropic and anisotropic inhomogeneous media. In anisotropic media, three independent waves and corresponding ray tracing and transport equations are obtained: qP, qS1 and qS2. The isotropic medium is a degenerate case of the anisotropic medium, with the rays and relevant travel times of qS1 and qS2 coinciding. Thus, only two waves, P and S, are obtained.

Let us now consider two inhomogeneous media, the first anisotropic and the second isotropic, close to the anisotropic medium in some sense. If the anisotropy in the first medium decreases, the first medium becomes closer to the second, isotropic medium. We shall speak of the case of vanishing anisotropy. If the anisotropy vanishes, the expressions for qP waves derived in anisotropic media yield smoothly the expressions for P waves derived in isotropic media. Thus, there is no problem with qP waves in weakly anisotropic media. The situation is, however, more involved for qS waves. If anisotropy vanishes, the expression for the superposition of qS1 and qS2 waves, derived in anisotropic medium, does not yield the expression for S waves, derived in isotropic medium. Thus, there is a conflict between the "anisotropic ray theory" and "isotropic ray theory" for S waves in weakly anisotropic media.

Consequently, the propagation of qS waves in inhomogeneous weakly anisotropic media requires a special theoretical treatment. In inhomogeneous weakly anisotropic media, the two components of qS waves are coupled. We speak of quasi-shear coupling (Coates and Chapman, 1990b). To derive equations for amplitudes of qS waves in inhomogeneous weakly anisotropic media, the perturbation methods are used. As a background medium, an isotropic medium close to weakly anisotropic medium under consideration, is considered. In the background medium, the rays of S waves can be computed by standard ray methods for inhomogeneous isotropic media. We select an arbitrary ray of the S wave and denote it Ω⁰. This ray Ω⁰ can be then used as a trajectory, along which the travel times and amplitudes of qS waves in inhomogeneous weakly anisotropic medium are computed. Some complications are connected with the fact that the ray method in the isotropic medium is a degenerate case of the ray method in the anisotropic medium. The consequence is that the travel-time perturbations are not linear in perturbations of elastic parameters, and that the system of two linear ordinary differential equations of the first order for the two components of qS waves is coupled. Moreover, the components of qS waves become frequency-dependent.

The coupled system of two linear ordinary differential equations of the first order for the two components of qS waves can be simplified considerably by certain substitutions, and solved in terms of 2x2 quasi-isotropic propagator matrices. Finally, very general expressions for amplitudes of qS waves in inhomogeneous weakly anisotropic media are derived. Those expressions can be even generalized and applied to any multiply reflected and converted elementary wave propagating in a 3-D laterally varying structure containing curved interfaces. Individual segments of the ray of this wave may be situated in isotropic, weakly anisotropic or even strongly anisotropic medium. Moreover, individual segments of the ray Ω⁰ may correspond to qP or qS waves. However, the method does not remove the singular behaviour of amplitudes in certain singular regions, such as caustics in background isotropic medium, and shear wave singularities in strongly anisotropic medium.

It should be noted that only the zero-order approximation of the ray method in inhomogeneous anisotropic and isotropic media are considered throughout this paper. Similarly, only the zero-order quasi-isotropic approximation in a weakly anisotropic inhomogeneous medium is studied. The higher-order approximation of the ray series method are not discussed here at all. It is likely that the higher-order term of the ray series may increase the accuracy of the ray-theoretical computations even in weakly anisotropic media, at least for some simple models. However, the main problem is that the higher-order terms of the ray series in complex environments are difficult to compute (with the exception of the additional components of the first-order term). Moreover, the initial conditions for the higher-order principal components are mostly unknown.

The contribution has a non-standard form. It consists of two sections, 3.9 and 5.4.6, which extend the manuscript on Seismic Ray Theory, see Research Report No. 5, Cerveny (1997). Section 3.9 is devoted to the perturbation methods for travel times, even in weakly anisotropic media. The reason why this section is included is that the theory of quasi-shear coupling, treated in Section 5.4.6, uses many terms and equations of Section 3.9. Section 3.9 presented here replaces the previous version of Section 3.9, included in Report No. 5, which was not written consistently and could be hardly used in the theory of quasi-shear coupling in weakly anisotropic medium. The references to equations and sections, not included in this contribution, refer to equations and sections of Report No. 5. The author apologizes that certain references may fail due to some recent changes in the manuscript.

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