Introduction

Traveltime maps are useful for Kirchhoff-type migration and tomographic velocity inversion. There are various methods of obtaining the traveltimes that differ in terms of speed, accuracy, and ability to find multiple arrivals Audebert et al. (1994). Some of the methods, such as those based on the solution of the eikonal equation, provide the results directly on the regular grid. Other methods, like paraxial ray tracing, provide the traveltimes at irregular points and, therefore, require the data to be interpolated between adjacent rays.

In this report, we propose a method of computing the traveltimes at the points of a regular grid by interpolating between adjacent rays. This procedure is based on the physical continuity of the traveltime field and uses rays generated by paraxial ray tracing Beydoun and Keho (1987); Cervený (1987) or Huygens wavefront tracing Sava and Fomel (1997). The paraxial ray tracing method, though not very robust when applied to models with big velocity contrasts, is very accurate in estimating times, and therefore it is expected to produce reliable traveltime maps.

The difficulty of interpolating traveltimes in media with complex velocities is that the rays do not follow smooth and uniform paths. In areas of high velocity variation, they bend and cross each other making the interpolation extremely difficult, if possible at all.

To avoid such a situation, we use additional parameters to separate the rays so that they no longer intersect. For example, in the 2-D case, we can associate with each point of a ray the value of the take-off angle. We convert the 2-D problem, in which the points of the rays are described by their x and z coordinates, into a 3-D one, in which the points of the rays are described by their x, z and p parameters. The rays then appear to be ``stacked'' in the p dimension in equally spaced planes, defining a continuous surface whose bending can represent the multiple, successive arrivals at a given location (Figure 1).

 

concept
Figure 1 The expansion in the take-off angle direction. Each ray has associated the value of the take-off angle, and therefore can be represented in a 3-D coordinate system (x-z-p) in a separate plane. The rays are split and don't cross, which facilitates the interpolation. The right side represents a plot of three rays in the 2-D plane. The left side represents the same rays after the expansion in the p dimension.

A similar approach is possible in the 3-D case, where we describe the points on the rays by their x, y, z, p, and q coordinates, where p and q are the two take-off angles (azimuth and inclination).