Annual Report 2000 - WIT

ø
ü
ø
ø
ý
Þ
do
ø ý
Ü
Ü
Ü
Ü
Þ
ü
Ü
ø
Ü
ù
!
Ü
ü
ù
ø
!
Ü
ü
ü
Þ
Ü
ü
Ü
252
, a plane wave would be simply
. Introducing the finite difference notation for a cartesian grid with indices
! and
and
ü
and the constant grid intervals !
describes the shape of the wave front at
ü ý
ö
ö
(8)
è è è
!
!
è è è
and replacing the derivative of
difference operator yields ö
with respect to
ü
by a simple first-order forward
ø ý
ø
ü ý
ø ý
#"¡ü
#"ø
ü ý
ü
ö$&%(' $*),+- ý
ö$&%(' $*)
è (9)
This form allows in principle the successive computation of traveltimes from level
for all grid points in the model. As pseudo-code this would look as
follows:
to ü ú
1. initialise plane wave
do
ø ý
end do
ö$&%('.- ý
#"ø
2. extrapolate forward
do
ü ý
#"¹ü
level
by (for instance) a central difference operator
/
compute flux function
#"ø
compute (derivative of T with respect to
/
) from known traveltimes at
ø
/
compute new traveltime at level
ý10 þ
$&%(' $*)
ü ú
end do
end do
ö$&%(' $*),+- ý
ö$&%(' $*)
Unfortunately, this simple approach fails. For instance kinks in the wavefront will
introduce unstable oscillations if a central difference operator is used to compute the
value of . This problem itself is well known in the literature on the numerical solution
of hyperbolic conservation laws. Fortunately, a variety of numerical methods have
been developed which avoid this problem. The main idea is to use one-sided difference
operators for the computation of and then to decide which is the correct one in terms