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

Chapter 2
Solving frequency domain wave equations:
Numerical Considerations
2.1 Introduction
Seismic forward modelling can be formulated as a time domain inital value
problem or as a frequency domain boundary value problem (see Chapter 1 equation
(1.5)). Explicit initial value problems do not require a large amount of memory to
run, however the amount of computational time can be signicant if the number of
time steps or the number of sources is large. The numerical solution of boundary
value problems involve solving a large (usually sparse) system of linear equations
(i.e., the matrix S ~
in equation (1.5)). The cost of solving the system increases dramatically
as the number of equations increases. To perform full matrix inversion,
or Gaussian elimination on a large system of linear equations requires a signicant
amount of memory and CPU time. However, for sparse systems, savings can be
obtained by exploiting the sparsity, and further savings are realized when a large
number of right hand sides are involved (representing additional sources in the seismic
modelling case). The utility of dealing with multiple right hand sides is critical
in seismic inverse problems, in which only a limited number of frequencies for a
large number of sources may be required (Pratt and Worthington, 1990). This is
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