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
coordinate 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 numerical evaluation of dynamic
properties of seismic waves propagating through structurally
complicated media. As the paper decribes 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.

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