Annual Report 2000 - WIT

Š
Š
Æ
³
Š
» •ÌÃ)Í
½
Í
³
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Î
Æ ¿ ‘ • Í » ³ ½ ¿ ³ ¢ Í ¢¢¢¢Î
Î
Ç
Š
Š
221
Vanelle, C., and Gajewski, D., 2000b, Traveltime based true amplitude migration: 70th
Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 961–964.
Vanelle, C., and Gajewski, D., 2000c, True-amplitude migration weights from traveltimes:
PAGEOPH (submitted).
Vidale, J., 1990, Finite-difference calculation of traveltimes in three dimensions: Geophysics,
55, 521–526.
PUBLICATIONS
Previous results concerning true-amplitude migration were published by (Vanelle and
Gajewski, 2000b) and (Vanelle and Gajewski, 2000a). A paper containing these results
has been submitted (Vanelle and Gajewski, 2000c).
APPENDIX A
The weight function in equation (4) contains the Hessian matrix of the difference between
diffraction (»¨¼ ) and reflection (»x ) traveltimes. To write !€ in terms of second
derivatives of traveltimes we will now derive expressions »¼ for »¨x and containing
first and second derivatives. Consider an arbitrary velocity model. Let sources be
positioned in a reference surface that we will denote the source surface. If the resulting
traveltime field for a source at the Š position
½ ¾
is single-valued, the traveltime
from a
½
point in the source surface and in a near
½ ¾ ½ª¿À“
vicinity of to a subsur-
Š
½ ½ ¿¾
Š
near can be expressed by a Taylor series Á ½Â• ½Äà ½ ¿ ¾
provided that
• ½ ¿ Ã ½ ¿¾
Š
Š
are small, the size of this small vicinity depends on the
¿
model
½
face Š point
Á Š and
under consideration and the required accuracy. For a multi-valued traveltime field the
Taylor expansion is valid if the different branches of the traveltime curve are treated
separately. Š As
» Š
½ ‘
½ ¾
lie in one surface the traveltimes are expanded into the surface
traveltime expansion looks as follows:
½Å¿¾ if Š
and
½
Š
in this variable. We also » expand into a surface Š in
½Å¿
, this will be the reflector surface,
or, more precisely, the reflector's tangent plane at Š
½ª¿
is on a curved reflector. The
Æ Á ¿ ¿ ÃÈÇ É Á ½ ¨ ½ ¦ Ê
Á ½
Š
«
Š Š Š
ÉÇ
¿ ¦ ¨ ½ ¿ ½ Ã Ë ½ ¨ ¦ §
Š Á Á Š Á Š
(A-1)
» ¾–Ã
» Š
½ ‘
Š
½ ¿ “–•
Æ ½
Š
« Á Š
Á Š
½ ¿
where the first order traveltime derivatives
%Ô ‘Õ •
‘ É “
(A-2)
ÐOÑÒ
Ð#ÓÑ
ÐOÑNÒ
Ð#ÓÑ