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This section presents a short discussion of the special treatment
required by the boundaries of the computation domain. These boundaries
are of two kinds: exterior boundaries, represented by the edges of the
computational domain, and interior boundaries, represented by the
triplication lines.
Because of the centered finitedifference scheme, HWT cannot be used
at the boundaries of the computational domain. This means that the boundaries
need to be treated differently from the rest of the domain. Also, the
centered finitedifference scheme cannot be used when the wavefronts
create triplications. Triplications represent points of discontinuity
of the derivative along the wavefront, and, therefore, the centered
finitedifference representation of the derivative is inappropriate.
Figure 4 describes a point of triplication represented
in both the physical (Cartesian) domain (left) and the ray
coordinate domain (right).
One possible solution for the boundaries is to make a local
approximation of the wavefront. Instead of
considering the actual points on the wavefront, we can create an
approximate wavefront that is locally orthogonal to the ray arriving
at the cusp point, as depicted in Figure 5.
We can then pick an appropriate
number of points (two in 2D or four in 3D) on this approximate
wavefront, and use the HWT scheme without any change.
A new search for the cusp points is then needed on the new
wavefront before we can proceed any further.
cusp
Figure 4 The centered finitedifference
representation of the derivative along the wavefront cannot be used
at the cusps. These points represent discontinuities in the
derivative, and need to be treated separately.

 
hrt
Figure 5 The centered finitedifference
representation of the derivative along the wavefront cannot be
used at the cusps. Instead, we can use a local approximation of the
wavefront as a plane locally orthogonal to the ray arriving at the
cusp.

 
Next: Examples
Up: Sava: 3D HWT
Previous: Review of HWT theory
Stanford Exploration Project
4/20/1999