Annual Report 2000 - WIT

!¥”“Z• Ç ¤
ɨ§
—
‘
• Ç u
¡
ÿ Ú
²
£ ’
ý
Ç
¦
‘ %$ ’ ‘ ¥”“'& ¤ ¡ Ú ‘ ($ ’ ‘£¢ « »¨¼ Ú ‘ ($
Ú
Í
¢
224
tion introduced by (Bortfeld, 1989). (Hubral et al., 1992) give the normalized geometrical
spreading in terms of the matrices of second order derivatives of traveltimes. This
is
ú ´
‘
(C-1)
where œƒž œ1 the angles are the emergence
ܾ
Š angle at and the incidence Š angle at
and can be determined from the slownesses at ¸ these positions. is the KMAH-index of
the
Ý
Š
ray connecting .
œƒž9¨šA› œ1 ŸšA› £ |~} ¦ § î î
| ²#³
žãù
ݾ
For a 2.5-D situation we find that
|Ÿ}±¦ § • ¦ § ÚÚZû ¦ § ’’
(see Appendix B) and therefore
£
expression (C-1) can be reduced to the simple form
Ü
and Š
• Ç ü
¡
ú ´
(C-2)
œ·ž9ŸšA› œ1 ¨š› § ÚÚ î îo¦
|²#³
This result is no surprise since (Bleistein, 1986) found the relationship between the outof-plane
spreading commonly denoted by þ and the second order traveltime derivative
in out-of-plane direction. Thus we have
§ ’’
ž–ù
• Ç
¦ § ’’ þ
(C-3)
So far, however, it was common practice to determine the out-of-plane spreading from
the integral along the ray Š from
½ª¿
½
to Š
þ •$ÿ £ ܃¡
(C-4)
with Ü being the arclength and ¡ the velocity. We compute this quantity from traveltimes
and do not have to trace rays to determine þ .
APPENDIX D
Equations (4) and (5) are expressions for weight functions if the diffraction stack is
carried out over the aperture in
Ú
and
’
. Since in the 2.5-D case we have only data
from a single acquisition line (assumed to coincide with the
Ú
coordinate), we only
integrate
Ú
over . In this case we ¡ Ú ‘ ’¨‘£¢ “±• ¡ Ú ‘ ‰ ’ ‘£¢ “
have where the asterisk
denotes the stationary point. Inserting the according expression for the input traces (2)
into the stack integral (1) then leads to
¢ “
(D-1)
¦¥”“Z• Ã Ç ¤ ÿ© £
ɨ§
‘ ’¨‘ ¥”“ ü Ú
Í
Carrying out the integration over
’
following (Martins et al., 1997) we get
Ž
¢¢¢¢¢
¦ Ò ú Ò
ÿ © £ Ú Â’£" #
’ ‘ ¥”““ ¯ (D-2)