Numerical Modelling of Time-Harmonic Seismic Wave Fields in Simple Structures by the Gaussian Beam Method. Part I.

Gaussian beam summation method is used for numerical modelling of seismic wave fields in several simple types of models of media. Main attention is paid to the waves reflected from a plane interface, namely from the vicinity of a critical point. Comparison with exact solutions shows that the Gaussian beam summation method yields sufficiently accurate results even in the singular region of the critical point. By summation of Gaussian beams of waves reflected in the overcritical region even head waves are obtained. In the second part of this work, we shall investigate sensitivity of the results to various parameters, for example, to the initial width of a Gaussian beam, to the parameters of the summation of Gaussian beams, etc.     

Transformation relations for second-order derivatives of travel time in anisotropic media

In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.     

Ray amplitudes of seismic body waves in laterally inhomogeneous media

Summary . The most complicated part in the computation of ray amplitudes of seismic body waves in laterally inhomogeneous media with curved interfaces lies in the evaluation of the geometrical spreading. Geometrical spreading can be simply expressed in terms of the Jacobian J of the transformation from the Cartesian into ray coordinates. Several systems of ordinary differential equations to compute the function J are suggested. For general three-dimensional media, in which the velocity changes with all the three spatial coordinates, a system of three non-linear ordinary differential equations of the first order is derived. If the velocity does not depend on one coordinate, the system of equations reduces to only one non-linear differential equation. The initial conditions for these differential equations at point (or line) source and at points of intersection of the ray with curved interfaces are presented.     

Elementary seismograms of seismic body waves in dissipative media

Summary Approximate expressions for elementary seismograms of seismic body waves propagating in media with small causal absorption are derived. Special attention is devoted to modulated signals with a smooth envelope, for which especially simple formulae were obtained. The derived expressions give a good picture of all important effects of causal absorption, viz., the frequency dependent exponential decrease of amplitudes, the velocity dispersion related to absorption, and the decrease of the prevailing frequency.