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

Figure 1.1: A Discrete representation of the forward modelling problem. The representation
is schematic; the assumption of two dimensions is not required at this
stage, nor is this ordering of the node points necessary. The waveeld (either a
scalar or a vector quantity) is sampled at each of the n x n z node points.
manner, although the diagonal block submatrices of S ~
are then no longer identical.
The same comment applies to the 2:5 , D method of Song and Williamson (1995),
in which a new diagonal block would be generated for each wavenumber considered.
By examining the solutions to equation (1.5) when the components of the
source vector, f i are replaced by a Kronecker delta, ij , it is clear that the columns
of S ~ ,1 must contain the discrete approximations to the Green's functions. Thus,
h
S ,1 = g
(1)
g (2)
~
::: g (nxnz) i
; (1.6)
where the column vectors g (j) approximate the discretized Green's function for an
impulse at the jth node. If the original physical problem is exactly reciprocal with
,1
respect to an interchange of source and receiver elements, then both S and S ~ ~
must be symmetric (not Hermitian) matrices. [In implementation S is often not
~
perfectly symmetric when certain (unphysical) absorbing boundary conditions are
implemented (Pratt and Worthington, 1990). This does not cause any problems].
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