Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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È<br />
¸<br />
â<br />
¹ Ë ¾œ¿ ¸ » ¸<br />
Ç<br />
õ<br />
Ë<br />
ê<br />
È<br />
»<br />
â<br />
ô<br />
õ<br />
ü ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
¹<br />
â<br />
â<br />
â<br />
Ë ¼ ö<br />
¿<br />
¹<br />
¾<br />
»<br />
¾<br />
¾<br />
õ<br />
ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
ü<br />
¼<br />
¼<br />
ô<br />
Ç<br />
Ë<br />
ô<br />
72<br />
for the 2.5D case in constant-velocity media. In the<br />
»Ò¿ ¸ »¤¼bß ¾<br />
¹ Ë<br />
above<br />
¿<br />
formula, ,<br />
is the amplitude of the event with pulse<br />
¹<br />
Ë ¾ ¸ ¸ Ë ¾<br />
¹<br />
, is the traveltime curve<br />
¼bà<br />
¸<br />
and<br />
Ó¾<br />
shape found in the seismic trace Ó'¿<br />
at . Also, denotes the partial derivative in the<br />
Ì<br />
Ç<br />
Ë<br />
is the isochron function defined ÍŠÎ implicitly<br />
Ï ¼Ó¾<br />
direction of the normal ¹<br />
by<br />
¹ Ë ¼ ¹ »¤¼%Ñã¾S¿äÓœ¿ ¸<br />
constant (A-2)<br />
¼<br />
»Ê¼<br />
á<br />
to Ç and Ñ ¸º¹<br />
where<br />
¹ Ë ¼ ¹ ¸<br />
¸ ¹ Ë<br />
is the traveltime å ¾<br />
from the source-point to<br />
»Ê¼%½³¾<br />
»¤¼b½Š¾<br />
the depth-point ¸º¹<br />
Ë<br />
. To garantee correct amplitude recovery, the<br />
¾<br />
kernel<br />
and back to the æ<br />
receiver-point<br />
is selected as<br />
Ë ¾S¿èç¡éëêeì¡íNî³ïbðñ<br />
¹<br />
(A-3)<br />
¸ ¹<br />
»Ê¼ ¸º¹<br />
point-source geometrical-spreading factors along the ray åœö segments öøæ ans , respectively.<br />
ñ Moreover, represents the angle the normal Ç to makes with the vertical<br />
makes with the<br />
isochron normal ö at . ç¡é Finally, is the modulus of the Beylkin determinant,<br />
Ó<br />
-axis, and ò denotes the incidence angle that the incoming ray åœö<br />
íNî³ï ðò<br />
óžô‡óžõ<br />
where ê is the medium velocity at ö ¿ ¸²¹<br />
Ë ¼ Ç ¾¾<br />
, and<br />
ó÷ô<br />
and<br />
ó÷õ<br />
are the the<br />
»¤¼%Ñ ¸º¹<br />
»¤¼<br />
¹ Ë ¼ ¸<br />
ö ¹ Ë ¼ ¸<br />
ö<br />
(A-4)<br />
ýãþ<br />
¿äù’úNûSü<br />
Í Û ýãþ<br />
ç¡é<br />
¸ ¹<br />
In the 2.5D case, and assuming, without loss of generality, that ß and à are the simmetry<br />
axes, we may write ¹<br />
ͳÿ ýãþ<br />
where ý þ ¿<br />
¸ Í þ ¼%Í ¡ ¼%Í£¢B¾<br />
is the spatial gradient operator.<br />
and æ<br />
¾÷¿ ¸ ¾ Ë ¸ Ë Ë ¾ Ë<br />
, å ¾÷¿ ¸ » ¸ Ë ¾B¼¥¤Š¾<br />
,<br />
Ì Ç<br />
¼¥¤Š¾ ¸ Ë<br />
. We also define the midpoint and<br />
¾<br />
half-offset coordinates,<br />
¸ ¹<br />
» ¿ ¸ »¤¼¥¤Š¾<br />
, Ì ¸ ¹<br />
¹ Ë ¾÷¿ ¸<br />
§ »<br />
À »<br />
¨ Þ (A-5)<br />
¦”¿<br />
In the above integral (A-1), the normal direction is not properly defined, since the<br />
-coordinates have different dimensions. In order to overcome this problem,<br />
¨ ¼<br />
and ç<br />
- and ¹ Ë Ó<br />
we must change the scale of the -axis. For constant ê , we define the new depth<br />
Ó<br />
coordinate © ¿ Ó<br />
as , so that it has the same dimension (length) as the -axes. In<br />
the new coordinates, the traveltime surface is given by ¹<br />
¿ Ë ¾<br />
. The integral (A-1)<br />
©<br />
is therefore,<br />
ê Ç<br />
¸ ¹<br />
à<br />
¹ Ë ¼ Ç ¾¾ž¼<br />
Å È<br />
(A-6)<br />
Û ¿ ¸ Í Û ¼%ͳÿ¨¼%Í£B¾ ¹<br />
where . Note that now the non-unitary surface<br />
á ¿<br />
normal<br />
ý ¼¤ ¼ºÀ µ ¾<br />
is well defined. Moreover, from now on, the prime will indicate derivative<br />
Ç ê<br />
»¤¼%½Š¾‰¿ µ ·5¸<br />
ÂÙêÄÆÅ<br />
with respect to the horizontal-coordinate Ë .<br />
Ë ¼<br />
¹<br />
ê ©<br />
¸ »¤¼<br />
Ë ¾Ì ¸ Ë ¾ ¹ á ý ¹<br />
Ñ ¸ »¤¼ Û<br />
¾ Ö+× Í£¢NÏ ¸ ½[À<br />
Ñ ¸ »¤¼
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