## Two-point ray-tracing and controlled initial-value
ray-tracing in 3-D heterogeneous block structures

**Petr Bulant**
### Summary

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.

### Whole paper

The manuscript of the paper is available in
PostScript (35022 kB !!)
and GZIPped PostScript (8741 kB !).

Journal of Seismic Exploration, **8** (1999), 57-75.

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