3-D ray tracing, especially the computation of ray-theory travel times at nodes of dense grids of points has become quite important in recent years. The possibility to compute the travel times is of principal importance in imaging of depth structures, seismic tomography or hypocenter determination techniques. Several methods based on decomposition of the model volume into ray cells, and on further interpolation within individual ray cells, were introduced.
In the wavefront tracing method (Vinje et al. 1996a, 1996b) the entire wavefront is represented by a triangular mesh with a ray at each node. The wavefront is time-step by time-step propagated through the model. The space between two consecutive wavefronts is decomposed into ray cells, each of them being bounded by a triangle at the older wavefront, by the corresponding triangle at the newer wavefront, and by segments of the three rays connecting the triangles. The receivers located within the ray cells are assigned values of interpolated travel times and other quantities. If there are some difficulties with the interpolation, new rays may be computed, and the ray cell may be split into smaller cells.
In the recursive seismic ray modelling method by Moser and Pajchel (1997), the computation is organized in "depth-first search" rather than "breadth-first search", as it is in the wavefront tracing method. The computation of a selected ray cell is followed by the computation of the cell in the next time step. At each interface, ray tubes corresponding to all the elementary waves under consideration are recursively generated. This procedure is continued until the whole ray tube, composed of individual ray cells, is computed, including reflections, transmissions and diffractions. From the point of view of interpolation, the main difference as compared to the wavefront tracing method is that the neighbouring ray cells are not available when interpolating in a ray cell.
In the controlled initial-value ray tracing method (Bulant 1997a,b) just ray tubes corresponding to a prescribed elementary wave are considered, without splitting at structural interfaces. The model volume is decomposed into ray tubes in the first step, and the interpolation within the ray cells is carried out in the second step. This is because the method was proposed as a post-processing method to a two-point ray tracing algorithm based on the triangularization of the domain of ray take-off parameters (for details, see Bulant 1997a,b). Thus the system of ray tubes and ray cells is given before the interpolation process takes place. Each ray cell corresponds to the space in the ray tube, limited by two planes which approximate wavefronts or structural interfaces. The values of the interpolated quantities at the vertices of a ray cell are available for interpolation.
Whereas the above-mentioned methods differ in the manner of the decomposition of the model volume into ray cells, they all contain some procedure of interpolation of the computed quantities to the receivers (usually nodes of a receiver grid) located within a single ray cell. We propose bilinear and bicubic interpolation schemes. The bilinear interpolation is easy and robust. It incorporates both the decision, whether a receiver lies in the ray cell, and the interpolation of quantities computed along the rays to the receiver. The scheme uses only the values of quantities at the vertices of the ray cell. Thus the scheme should be applicable in any modelling method based on interpolation within ray cells. If the partial derivatives are known in addition to the functional values at the vertices of the ray cell, the bicubic interpolation scheme may be used to increase the accuracy of the interpolation. The bicubic interpolation is designed for travel times and the bilinear interpolation for other quantities.
The image of the paper in GIF 150dpi (1080 kB !) is designed for an instant screen preview.
The paper is available in PostScript (10480 kB !!, colour figures) and GZIPped PostScript (2260 kB !, colour figures).
Bulant, P. & Klimes, L.: Interpolation of ray theory traveltimes within ray cells. Geophysical Journal International, 139 (1999), 273-282.