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

as sequential grid ordering. Sequential ordering just involves starting for example,
in the top left corner of the grid and numbering the grid points in the rst row
sequentially up to n x (where n x is number of grid points in x direction). We then
move to the next row, repeat the procedure, and continue in this manner until we
run out of grid points. Imagine our problem is dened on a grid of n x by n z nodes:
If each node is coupled only to it's immediate neighbours (as in nite dierence
equations arising from second order nite dierence operators), the initial matrix
S ~
will only have non-zero elements on the main diagonal, on the two neighbouring
sub-diagonals, and on two sub-diagonal bands at a distance of n z diagonals away
from the main diagonal.
In general any nite dierence operator will produce a
symmetric sparsity pattern in the initial matrix
S ~
. This does not imply that the
matrix itself will be symmetric. This depends on the boundary conditions and on
the type of nite dierence operators used.
Now let us examine the way in which LU decomposition can be performed.
The algorithm is relatively simple (more details can be found, for example, in Peres
et al. (1992)): Let i;j be elements of the starting matrix S ~
, i;j be elements of
matrix L ~
and i;j be elements of matrix U ~
. The algorithm proceeds as follows.
Set i;i =1
For each j =1;2;:::;N carry out the following two procedures:
First, for i =1;:::;j
i;j = i;j ,
Second, for i = j +1;:::;N
Xi,1
k=1
i;k k;j : (2.5)
0
1
i;j = 1
j,1
X
@ i;j , i;k k;j
A : (2.6)
j;j
k=1
Once the element a i;j is used the value is not required any more, so the same memory
location can be used to store the corresponding i;j or i;j . Values i;j and i;j
39