Frequency domain seismic forward modelling: A tool for waveform ...

amplied instead of attenuated.
1.5 Overview of chapters in this thesis
This thesis begins with a discussion of the matrix solver used to generate
solutions to the frequency domain nite dierence matrix, equation (1.5). If the
matrix solver is inecient, no matter how good the nite dierence formulation
is, the costs involved will be prohibitive.
In Chapter 2 I will initially dene the
requirements expected from ecient matrix solvers. I will provide an analysis of
the structure of the matrix, and the manner in which this structure aects the
matrix solver in general, and the eects that various nite dierence operators will
have on this structure.
This will enable guidelines to be set for the appropriate
nite dierence operators in order that the computational costs can be kept low.
The technique of nested dissection, which optimises the initial sparsity pattern will
be described and quantitative estimates of the computation times and the storage
requirements will be given.
After analysing the general problem of the matrix solver, I will move on to
a specic nite dierence technique for visco-acoustic media in Chapter 3. I will
use the rotated nite dierence operators suggested by Jo et al. (1996). I will
present and analyze these operators, and then extend them to the heterogeneous
case. I will then discuss the combined use of the nested dissection method and the
implementation of the rotated nite dierence operators and prove that the scheme
is optimal, and that no improvement will be achieved by the use of higher order
spatial operators. Chapter 3 concludes with an example (based on a real wide-angle
experiment) that demonstrates the visco-acoustic modelling described and analyzed
in these initial chapters.
In Chapter 4 the application of the frequency domain visco-acoustic modelling
scheme as a tool for waveform inversion will be presented. The example presented is
33