It is well known, that even in quite simple seismic models it is almost never sufficient to perform an initial-value ray tracing on a dense grid followed by interpolation. Alternatively, choosing an arbitrary curve and obtaining a two-point ray by bending is problematic in view of multivaluedness. If the model produces multivalued arrivals, we need some kind of a more advanced two-point ray tracer to find all two-point rays for all the receivers. Different methods are currently being used (e.g. Hanyga 1996, Lucio et al. 1996, Vinje et al. 1996a, 1996b). Our solution to the two-point ray tracing problem is a sophisticated shooting program, based on an accurate search for the subdomains of ray take-off parameters, where the two-point rays may occur. The subdomains may be found by a special triangularization of the 2-D domain of ray take-off parameters. The two-point rays may be then identified inside triangles whose all three vertices are formed by rays starting in same subdomain and arriving into a receiver area.
Once the domain of ray take-off parameters is triangularized, we can use the triangles mentioned above to construct ray tubes through the model volume, and to interpolate travel times and other quantities inside the tubes, like in the wavefront tracing method (Vinje et al. 1993). The described controlled initial-value ray tracing method is a useful extension of two-point ray tracing programs based on triangularization, and it may be very easily applied, once the domain of ray take-off parameters is appropriately triangularized. It allows accurate interpolation of wavefield attributes, e.g. ray-theory travel times, amplitudes and other quantities, at dense grids located in the seismic model. This might be of interest for, e.g., 3-D travel time tomography or ray-theory amplitude migration.
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