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

(RAM) memory, andtheunderlying numerical approximation must betuned tothe
matrix solver for minimum overall computational costs. These problems must all be
considered together, as the each choice at each stage aects the choice, at the next
stage.
Iterative solvers are usually considered to be the best way of solving positive
denite linear systems (for a denition of positive denite equations see for example
George and Liu (1981)). The main advantage of iterative matrix solvers is that full
advantage can be taken of the initial matrix sparsity. As a result, the amount of
memory required is small (of the order of n x n z ). The problem with frequencydomain
forward modelling of the seismic problem is that the matrices arising from
the nite dierence equations are not always positive denite.
For example, the
absorbing boundary conditions often used are not physical - they are used only
because we are attempting to model innite media by using a nite model. As a
result, due to ill conditioning, iterative methods either do not converge, or converge
too slowly to be considered appropriate to solve the system. The other problem with
iterative solvers is that they are not suitable for systems with multiple right hand
sides. The computational costs for iterative solvers increases in linear proportion to
the number of right hand sides. In the problems I am going consider, the number
of right-hand sides can be signicant. Direct methods, which are able to solve the
problem eciently for multiple right hand sides, are therefore more eective than
the iterative ones in this case.
Direct methods for solving linear equation systems require signicantphysical
memory.
Matrices produced by nite-dierence (or nite-elements) methods are
always sparse, but the sparsity pattern is not preserved by most direct matrix solvers.
The sparsity pattern of the initial matrix depends on the nite dierence operator
used and on the grid ordering used. In this chapter I will initially concentrate on
the eect of grid ordering, and then move on to consider the eect of the size (i.e.,
the order) of the nite dierence operator on the memory requirements.
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