Annual Report 2000 - WIT

ò
Ý
ã þ ¥¿åLK
D
û
Ý
ò
ýÝ
þ =OF å Þ ü ã
, ü
ã
û ¥ å Þ K
,
§
ä¨= Þ §
ô
ý
þ
Ý
Ý
ý
Þ
=OF þ
can be expressed by , ã þ ==
and ã ,
þ
þ == Þ ã
û
ñ
ý
ã þ =E
Ý
ñ¦=
ñ ò
þ
û
207
as they are shown in Figure 1. We ò é
insert ò and into the parabolic expansion (1)
respectively. Building the sum and the difference of the resulting expressions yields
the following result:
Þâò é
ò ñ
(6)
ú²ñ¦= å
9 ã ==„å ò é ô ò
For the hyperbolic form we find a similar solution. ò é
Inserting ò ,
to
ú€ñ¦=wå
ò Ý
Þâò Ý é
ã == å ò Ý
Ý 9
ò Ý é Þ
=
ò Ý
ú Ý
and ò ñ
into (4) leads
(7)
ó î=
ó î Ý
= Þ
ò ñ ó î=
B The - and ã ú€ñ
-components ã of and can be found in the same way by varying 4
respectively . Varying both îE
and leads to ã =E
; ã EF
and ã F=
follow accordingly.
The determination of the îF - and î= -components of îE ø•ñ
is straightforward: instead
ã
B 5
and ã
ò ñ ó î Ý
of varying the receiver position we use different source positions. For the -, BHB -, 5 B -
and 5G5 5
-components of ã
B
both source and receiver positions have to be varied. But this
does not yet give us the -components. Unless we compute also traveltimes for sources
at different depths – which we do not intend to – another approach is needed. At this
4
point we make use of the eikonal equation to express the -component of the slowness
vector 4 ø•ñ
as ã
ø•ñFqåJI Þ ø Ý ñ¦= Þ ø Ý (8)
ñ¦E Ý
where is the velocity at the source, provided that the source lies in the top surface
,
of the model. Otherwise we have to insert a sign in (8). Since second order traveltime
derivatives are also first order derivatives of slownesses we can ã rewrite ã and to
ø¥ §
äHM § ¤N
ú/ §
䨥OM § %
ø9¥ §
î¨#M § ¤
(9)
åLK
If we now substitute ø•ñF in equation (9) by (8) we can compute the second order derivatives
of ò with respect to äF and îF from the already known 5 -B -matrix elements and
derivatives of the velocity. Since we assume the velocity field to be smooth, the velocity
derivatives can be determined with a second order FD operator on the coarse grid.
To give an example the matrix ã
þ =E
element as
¦
ø•ñ¦=
ø•ñF
ø•ñ¦E
ø•ñF
(10)
ø•ñF
This expression will not yield a result for ã þ =OF
if ø•ñF equals zero. This case has, however,
no practical relevance for the applications that the method was developped for. The
ã and is straightforward.
derivation of the remaining 4 -components of ã
, ë