First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

Phase shift at caustics along rays in anisotropic media

In isotropic ray tracing, the ray approximation to the wavefield undergoes a phase shift when the ray crosses a caustic. The cumulative number of such phase shifts along a ray is usually called the KMAH index. The sign of these phase shifts is prescribed by the sign of the angular frequency in combination with the sign convention used for the Fourier transformation. In isotropic media the KMAH index always increases by one or by two, depending on the type of caustic crossed. For (quasi-)shear waves in anisotropic media the KMAH index may decrease. This is the case if the associated slowness sheet is locally concave in one or two of its principal directions of curvature.     

Weak-contrast reflection/transmission coefficients in weakly anisotropic elastic media: P-wave incidence

We present approximate displacement and energy PP and PS reflection/transmission coefficients for weak-contrast interfaces in general weakly anisotropic elastic media. The coefficients were obtained by applying first-order perturbation theory and then expressed in a compact and relatively simple form. The formulae can be used for arbitrary orientations of the incidence plane and interface, without the need to transform the elasticity parameters to a local Cartesian coordinate system. The accuracy of the approximate formulae is illustrated for the PS reflection coefficient for two synthetic models. For these models, we also study the possibility of using the approximate PP reflection coefficient in the inverse problem.