Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
º<br />
¡ £<br />
ì ¥<br />
The stacking surface is the isochrone îZÖõ‚ê ¡ £<br />
ì ¥<br />
º<br />
§<br />
·<br />
·<br />
ã<br />
ã<br />
Ä<br />
§<br />
¡<br />
Æ<br />
î<br />
Ç<br />
Ç<br />
Ç<br />
Ç<br />
·<br />
§<br />
Ä<br />
º<br />
91<br />
True-amplitude demigration<br />
The true-amplitude demigration procedure is performed by a weighted Kirchhoff-type<br />
isochrone stack. The kinematic part of the transformation can easily be performed<br />
in a completely analogous way to true-amplitude migration as described in the last<br />
section. Rather than defining a grid · of points in the depth domain, we now<br />
¤f¥¦i§<br />
define<br />
a dense grid ã of points in the time domain, i.e., ¡ £ in the -volume in which<br />
we desire to construct the time section from the depth-migrated section. Each grid<br />
point together with the macro-velocity model defines an isochrone in the depth<br />
ã<br />
domain. The isochrone determines all subsurface points for which a reflection from a<br />
possible true reflector would be recorded at ã<br />
after traveling along a primary reflected<br />
ray from to ° . We now have to perform a stack along each isochrone on the<br />
depth-migrated section amplitudes. Then, we place the resulting stack value into the<br />
corresponding ã point . As in Kirchhoff migration, the isochrone-stack demigration<br />
will only result in significant values if the ã point is in the near vicinity of an actual<br />
reflection-traveltime surface. Elsewhere it will yield negligible results. The dynamic<br />
part can once again be performed by a multiplication of the amplitudes with a varying<br />
weight function.<br />
The summation can mathematically be expressed by<br />
¬ëêiÅ<br />
Æzí ì ¥î£§<br />
£ ¡ §<br />
ž (5)<br />
§j¨<br />
¡»ã<br />
¼ä½å¾£¾<br />
æ<br />
given by<br />
獏磎<br />
ó Ñ ôªÓ<br />
Ç ï‹ÉfðòñÍ Î<br />
(6)<br />
¡ £<br />
¤£§§l¨c¦<br />
¡ £<br />
¤'¥<br />
¡×Ø¡ £<br />
¤§%¥<br />
®nÄ<br />
§j¨c®<br />
§1Ù/®<br />
¥°<br />
for · ì<br />
£<br />
all ö points with in . As we can see here, the isochrone and<br />
¤f¥<br />
the Huygens<br />
surface are defined by the same ¡…£<br />
traveltime function . í ¡ ì ¥î§<br />
£<br />
The function ®nÄ<br />
represents the migrated reflector image to be demigrated and can be expressed as<br />
¡»·<br />
í ¡ £<br />
ì ¥î£§j¨<br />
í ©«¡ £<br />
ì §l¬<br />
¡ò÷<br />
îJVõn¯ø§§¥<br />
¦i§<br />
¡ where represents the ÷ ¡ ì §<br />
£<br />
analytic point-source wavelet, is the vertical<br />
stretch ¬ ¯ù¨<br />
·ú¡ ì ¥îû¨üõ‚¯<br />
£ ¡ ì §i§<br />
¬Uê‹Å<br />
£<br />
factor, and points define ·<br />
§¥Ü<br />
the actual reflector.<br />
is the true-amplitude weight function to ReÛ ¡ýã<br />
be specified. The value is the<br />
demigration result which is generally ã assigned to the point . However, if we want<br />
to chain true-amplitude migration and demigration, we have to use the complex value<br />
(Tygel et al., 1996).<br />
¡ýã<br />
(7)<br />
The true-amplitude weight function for demigration relates to the weight function<br />
for migration because the demigration should undo what the migration has done to the
Change language