The ray series method and dynamic ray tracing system for three-dimensional inhomogeneous media

Abstract

The ray series method is used to study propagation of seismic waves in the three-dimensional media consisting of generally inhomogeneous layers separated by curved interfaces. The investigation is carried out with the help of the so-called ray centered corrdinate system which was proposed by Popov and Psencik (1978a). It is shown that in this system the principal components of the amplitude coefficients in the ray series for S waves do not rotate about the ray with respect to the basis vectors when the wave progresses, even though they rotate with respect to the unit vectors n^ and b^ along the direction of the normal and binormal to the ray, respectively. This considerably simplifies the final expressions for the amplitude coefficients for S waves, whose two principal components are decoupled in the ray-centered coordinate system. The ray-centered coordinate system is also applied to the eikonal equation in order to produce a dynamic ray tracing system consisting of three nonlinear ordinary differential equations of the first order determining the second derivatives of the time field and, in this manner, even the basic geometrical properties of the wave fronts (e.g., principal curvatures and geometrical spreading) along the ray. Several different modifications of the dynamic ray tracing system are presented. It is demonstrated that in the case of generally inhomogeneous two-dimensional media the dynamic ray tracing system reduces, under certain not too restrictive conditions, to the single first order differential equation of the Riccati type. Finally, the phase matching method is used to determine discontinuities of individual quantities in the dynamic ray tracing system when the wave is impinging on a curved interface separating two generally inhomogeneous media. Since all basic equations are presented in a computationally convenient matrix formulation, they can be readily employed for any numberical evaluation of dynamic properties of seismic waves propagating through structurally complicated media. As the paper describes all basic features of asymptotic ray theory (the name under which the ray series method is known on this continent), it can serve as a starting point for anyone wishing to develop computer programs for the computation of synthetic seismograms.