Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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§<br />
¦<br />
¦<br />
¦<br />
¢¢¢<br />
§<br />
Ë<br />
Ê<br />
Ë<br />
¨š›óò ÐÐ<br />
’<br />
ò<br />
¢¢¢<br />
Ë<br />
Ë<br />
Ù<br />
¢¢¢<br />
¦<br />
Í » •<br />
ê Í<br />
›Nõ ÐOô ò ’ Ù<br />
ò<br />
Ù<br />
§<br />
æ<br />
¢¢¢<br />
<br />
Ù<br />
Ë<br />
<br />
223<br />
instead of an expansion into reference surfaces. The procedure of determining the corresponding<br />
3-D slowness ¦ Æ<br />
vectors ¦ Æ¿<br />
and and matrices<br />
¦ Ê ‘ ¦¦ Ë<br />
and ¦ ¦ §<br />
from multi-fold<br />
traveltime data sampled on a coarse cartesian grid is described in detail in (Vanelle<br />
¦<br />
¦ ¦ Ê ‘ ¦¦ Ë ¦ §<br />
2è<br />
Ê ‘é¦ § ¦<br />
Š<br />
and Gajewski, <strong>2000</strong>a) as well as in (Gajewski, 1998). Once and are known,<br />
the desired 2 matrices and can be computed by projecting them from the<br />
cartesian coordinates into the reference surfaces. Since we assume a velocity model to<br />
compute traveltimes, we can also make use of this to extract the reflector position and<br />
geometry from it.<br />
In a situation with a 2.5-D symmetry as considered in the numerical examples the<br />
components of the slowness vectors<br />
and ¦<br />
simplify.<br />
Let the out-of-plane direction have index 2 – coincinding with<br />
Æ ‘<br />
Š<br />
Æ ¿<br />
and second order derivative matrices ¦ Ê ‘–¦ Ë<br />
ê the -axis of the cartesian system used for the determination of<br />
¦ Ê ‘ ¦¦<br />
and ¦ ¦ § ¦<br />
¾ë•$Ü ’ ¾ì•íÝ ’ ¾ì•+Þ ’ ¾<br />
the ê -position of the sources and receiver line. Then we have<br />
ê<br />
¦ § ’’î ï Ñ • ¦¦ § ïïHî ï Ñ ‘ ¦ Ë ’’Aî ï Ñ • ¦¦ Ë ïïHî ï Ñ<br />
£ ¦ Ê ’’Aî ï Ñ • ¦¦ Ê ïï#î ï Ñ<br />
– with<br />
(B-1)<br />
ØÙ<br />
From the symmetry we can easily see that ê the -components of the slownesses vanish<br />
ê ¾<br />
at :<br />
• Í »<br />
(B-2)<br />
» Í<br />
ê ž ¢ Í ï Ñ<br />
From this follows that the matrices ¦ Ê ‘ ¦<br />
and ¦<br />
¢¢¢¢<br />
Furthermore, for ê^ê the - or -components we get<br />
ÉAÉ<br />
Í ê<br />
¢ ¢¢¢¢<br />
ï Ñ<br />
consist only of diagonal elements.<br />
¢¢¢¢¢<br />
ï Ñ<br />
§ ’’ ¦<br />
ï Ñ<br />
• ¦ Ë ’’<br />
ï Ñ<br />
•ÌÃ ¦ Ê ’’<br />
ï Ñ<br />
(B-3)<br />
and the<br />
§ §<br />
ï ¦<br />
sign of is<br />
’’Aî ï ’’<br />
Ñ “Z• «<br />
Ç positive; i.e., .<br />
Ñ ›ð<br />
If the source-receiver line is equal to the -direction of the cartesian system from the<br />
input traveltimes, the 11-components are computed as follows:<br />
½<br />
§ ÚÚñ• ¦¦<br />
à ¦¦ §<br />
Ë ÚÚö• ¦¦<br />
Ã É ¦¦ Ë<br />
« ¦¦<br />
ŸšA› ô÷ô ›õ ÐOô ›Nõ òø¨šA›óò<br />
ÐÐ<br />
ÚÚö• ¦¦ ÐÐ ‘ Ê<br />
(B-4)<br />
ò where is the inclination angle of the reflector's tangent plane against the sourcereceiver<br />
line ( -coordinate).<br />
½<br />
APPENDIX C<br />
Equation (5) contains the geometrical spreading u which we will now express in terms<br />
of traveltime derivatives. Equation (A-1) is equivalent to the paraxial ray approxima-
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