Grid computations of rays and travel times of seismic body waves play a very important role in seismological applications, particularly in seismic exploration. Several methods of such computations have been proposed. The different methods may yield different travel times in certain regions of the model. The differences are caused mainly by different concepts of travel times. Definitions of four different types of travel times of seismic body waves in non-dispersive inhomogeneous layered and block structures are introduced. They are as follows: 1) first arrival travel times, 2) constrained first arrival travel times, 3) zero-order elementary travel times, 4) elementary travel times. Appropriate algorithms of computation of individual travel times are described. There are four basic classes of such computations: a) "finite-difference" methods, b) network ray tracing (also called the shortest path ray tracing), c) wavefront tracing, d) methods based on the numerical ray tracing, supplemented by suitable interpolation/extrapolation procedures. The advantages and limitations of these methods are discussed. Particular attention is devoted to the fourth method, where the interpolation/extrapolation is performed by weighting paraxial ray approximations. In the method, the standard initial value ray tracing is performed and supplemented by the dynamic ray tracing. The computed rays must cover the target zone with a sufficient density, even though irregularly. At any point of the target zone, the travel time is then calculated using the weighting paraxial approximations from the closest ray points. Several alternatives of the method of weighting paraxial approximations are proposed and numerical examples are presented. The accuracy of the method is compared with other approaches. It is shown that the method is computationally very efficient and yields results of a high accuracy.
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