Abstract

The research described in this thesis covers various aspects of
the forward calculation of seismic ray fields and ray field maps.
The central theme is the solution of problems encountered in
smooth but complex media, i.e., media that give rise to wave front
folding and associated multi-pathing of rays. The ultimate aim of
the presented
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material is to enhance the efficiency of seismic
inverse methods, by enhancing the efficiency of the forward
calculations. Particular emphasis is placed on the applicability
of the ray tracing results to seismic inverse methods.
After an overview of seismic ray theory in Chapter 2, a novel
approach to the calculation and representation of ray field maps
is introduced in Chapter 3. The approach is particularly useful in
cases where ray field maps are needed for a dense distribution of
sources at an acquisition surface, as in reflection seismics and
borehole tomography. For such source distributions it is suggested
to construct a single ray field map in an extended space of
spatial coordinates and angles, rather than a number of maps in
the spatial domain for a range of acquisition coordinates.
The ray field map in the position/angle domain is single-valued,
regardless of the complexity of the medium and the ray field
information is organised by angles at depth rather than by points
of emergence at the surface, which makes the maps particularly
suitable for use in modern seismic imaging methods. An important
result is that, in contrast to what is commonly assumed, obtaining
this information does not require the tracing of rays up towards
the acquisition surface. Instead, existing algorithms that trace
downwards can be adapted to work in the position/angle domain,
leading to a considerable gain in efficiency.
Interpolation is an important tool in both the construction and
the application of ray field maps. A new technique for accurate
interpolation using derivative information is presented in Chapter
4. It is a hybrid of extrapolation to arbitrary order and linear
interpolation, and combines the advantages of both methods.
Through a modification of the coefficients of the Taylor
expansion, extrapolations from a number of locations can be
combined to obtain a polynomial order of accuracy that is one
higher than that of a single conventional Taylor expansion.
In Chapter 5 a ray field construction and mapping algorithm is
developed that extends and refines existing wave front
construction methods. For ray field mapping in the spatial domain
two refinements are proposed that enhance the accuracy and the
completeness of the maps. The applicability in the position/angle
domain is investigated as well, with the unfortunate conclusion is
that ray field construction in its current form is not suitable
for that domain, due to the type of deformation in the geometrical
structure of the ray field.
A successful algorithm for the calculation of ray field maps in
the position/angle domain is developed in Chapter 6. It is based
on the observation that in the position/angle domain the ray field
maps are single-valued and that the geometrical spreading is very
limited. This implies that the two most important reasons for
developing wave front construction methods in the spatial domain
are absent in the position/angle domain. Instead, it is possible
to use the more primitive - but more efficient - paraxial ray
methods.
The one-to-one mapping between position/angle coordinates and ray
field coordinates can be exploited in practical applications.
Calculations that are typically performed in terms of ray field
coordinates can now be performed in terms of position/angle
coordinates and the other way around. Appendices A and B show that
this may be advantageous in tomography and the and the forward
calculation of ray fields maps directly on a grid in the
position/angle domain.
Finally, Appendix C presents an algorithm for the calculation of
ray fields in smooth 2-D media, using a pseudo-spectral expansion
of the wave front. This line of research was abandoned in favour
of the ray field map methods described above. Nevertheless, it is
presented in the thesis because its development provided useful
insights for the ray field map approach and some of its features
may be useful in other applications.
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