Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
v G ° à £§¦©¨ G ¬ à £§¦©¨ G ¾ à £§¦©¨ G ¿ à £§¦©¨ ' ' ' ' à à à à £ £ £ £ 0 ' 0 0 0 à à v v à à £ £ à ' ' £ ' à à · · & à · v à à £ & à 28 is formulated in such a way that the influence of the curved surface can be clearly recognized. THEORY According to (Schleicher et al., 1993), the hyperbolic traveltime approximation for a ray from ¢£ to ¢¤£ and from ¢¤£ to ¤:£ both on a curved surface, in the vicinity of a normal (zero-offset) ray from ¢Œ¤ to ¢ and from ¢ to ¢Œ¤ (Figure A-1) is given by (1) ›© ©¤ª«t¬®¯°Ÿ'(¬®¯# with (2) ±²¬®¯°QV¬®¯#Œ© ³ ©¥¦t§k |WŒ1©¨' c"µ$‡·¹¸ & ›´ (3) »5')Wº Wº²(» ¼ > This formula is valid for a 2D laterally inhomogeneus medium. is the velocity at the ZO location , is zero-offset (two-way) traveltime. and are the coordinates » & ¢ˆ¤ of ©½ the source Wº and receiver measured along the tangent to the curved surface ¢£ at ¤:£ . ¢Œ¤ The components °(|¬ |¾V|¿ of the so-called ÁÀ surface-to-surface propagator matrix for the two-way normal ray (see Appendix) are given by (4) £3¦©¨ ·…¸ 4 & (5) ,f. ¸ 4 · ,/. · à · 4 & (6) ,/. à £3¦6¨ ,/. ¸ ¸ · £3¦©¨ (7) £3¦©¨ ,f. where · ¸ is the angle of incidence of the normal ray (emerging on the curved measurement surface ¢Œ¤ at ) with the normal of the tangent to the curved surface at . Let us define à · as the surface curvature at the point ¢Œ¤ . £3¦6¨ and à £ are ¢Œ¤ the curvatures of the NIP-wave and the N-wave observed ¢Œ¤ at , respectively (Hubral, 1983; Jäger et al., 2001). · ¸ 4
· · v 0 B 0 £ & Ä Ã & v Ä Ã Ä 4 à £ · & · Ä Ã Ã £ à à · > à > ' > à à à £ 29 Inserting eqs. (2), (4), and (5) into eq. (1) we obtain © ,/. '(,/. ¥|¦P§ ` © ¸ c"µ$“· & (8) ·¹Å)v ¸ · ¸ · © ,/. '(,/. à à £3¦6¨ Here are wanted parameters that help solve a variety of stacking and ¸ inversion problems (Hubral, 1999). They can be obtained by modifying the presently existing 2D CRS stack formula (Jäger et al., 2001) that is obtained from eq. ÆB (8) by sub- . In the following, we discuss 3 particular reductions of formula (8) which are of practical relevance. stituting à · ¸ · ¸ · ·CÅ £§¦©¨ Particular cases Common-mid-point (CMP) gather For this case, v , the equation (8) reduces to © (9) This expression is commonly written as (Shah, 1973) Ç Pˆ‹ ,f. '(,f. 8¥¨ £§¦©¨ ¸ · ¸ · ·…Å (10) Ç tWŒŸ È where the normal moveout (NMO) velocity & £§8:9 is now given by 8¥¨ & £§8:9 & (11) For à · £§8:9 & this reduces to Shah's formula. For a 1-D model with a planar surface, £§¦©¨ ¸ ¸ · ©E à ,f. '(,f. =B TB straight normal ray and incidence ¸ angle g 8 · , where É g 8 · is the familiar root-mean-square velocity. É Common - shot gather For this case, , equation (8) reduces to »‡‹Wº²( this expression reduces to ÉW£§8:9 & 4 © & ¸ · ¸ · ¸ · i ·¹Å (12) ,/. Ç 1 ˆ ,/. ©Ê c"µ$“·…¸ ,f. £3¦6¨ where according to eq. (A-9) and (Hubral, 1983), is demonstrated that £3¦©¨ , with à à being the wavefront curvature of the reflected wave at ¢Œ¤ . ½ à ½
- Page 1 and 2: Wave Inversion Technology WIT Annua
- Page 3: Wave Inversion Technology WIT Wave
- Page 7: Copyright c 2000 by Karlsruhe Unive
- Page 11 and 12: i TABLE OF CONTENTS Reviews: WIT Re
- Page 13 and 14: Wave Inversion Technology, Report N
- Page 15 and 16: 3 theory. It relates properties of
- Page 17: Imaging 5
- Page 20 and 21: 8 two hypothetical eigenwave experi
- Page 22 and 23: 7 ¢¥£§¦©¨ ; < calculate sear
- Page 24 and 25: 7 7 searches ¡ HNJOL for ¢IHNJML
- Page 26 and 27: 14 Trace no. 1000 1500 2000 0.5 1.0
- Page 28 and 29: 16 CDP number 3100 3000 2900 2800 2
- Page 30 and 31: 18 REFERENCES Höcht, G., de Bazela
- Page 32 and 33: 20 INVERSION BY MEANS OF CRS ATTRIB
- Page 34 and 35: 22 For synthetic data, it is easy t
- Page 36 and 37: — J J J J J J G J - J J G
- Page 39: Wave Inversion Technology, Report N
- Page 43 and 44: 31 Dürbaum, H., 1954, Zur Bestimmu
- Page 45 and 46: Ã Ø Ì 0 0 Ì 4 0 Ã 0 G ' 4
- Page 47 and 48: Wave Inversion Technology, Report N
- Page 49 and 50: 0 0 ˜ ” Z ˜ ” Z 4 4
- Page 51 and 52: 39 strategy by combining global and
- Page 53 and 54: 41 elled data is presented in Figur
- Page 55 and 56: 43 0.2 Distance [m] 1000 1500 2000
- Page 57 and 58: 45 In Figure 10 we have the optimiz
- Page 59: 47 Gelchinsky, B., 1989, Homeomorph
- Page 62 and 63: § ¢ ø ø for a paraxial ray, ref
- Page 64 and 65: Œ § … Z ‹ Z § õ ø \ ø §
- Page 66 and 67: 54 As stated in Hubral and Krey, 19
- Page 68 and 69: § ó § 56 sequence. We made tes
- Page 70 and 71: 58 precision of the modeled input d
- Page 72 and 73: 60 Cohen, J., Hagin, F., and Bleist
- Page 75 and 76: Wave Inversion Technology, Report N
- Page 77 and 78: £ 65 The details of the theory inv
- Page 79 and 80: 67 of the figure. On both sides, a
- Page 81 and 82: 69 (a) (b) Depth (m) 600 800 Depth
- Page 83 and 84: 71 Langenberg, K., 1986, Applied in
- Page 85 and 86: È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð
- Page 87 and 88: Wave Inversion Technology, Report N
- Page 89 and 90: g ý g [ ^ â g [ g g g g g â h [
·<br />
·<br />
v<br />
0<br />
<br />
B<br />
0<br />
£<br />
<br />
&<br />
<br />
<br />
Ä<br />
<br />
Ã<br />
<br />
<br />
&<br />
v<br />
Ä<br />
Ã<br />
Ä<br />
4<br />
Ã<br />
<br />
£<br />
<br />
·<br />
<br />
&<br />
·<br />
Ä<br />
Ã<br />
<br />
Ã<br />
£<br />
Ã<br />
<br />
Ã<br />
·<br />
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Ã<br />
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><br />
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Ã<br />
Ã<br />
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£<br />
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29<br />
Inserting eqs. (2), (4), and (5) into eq. (1) we obtain<br />
©<br />
,/. <br />
'(,/.<br />
¥|¦P§ <br />
`<br />
©<br />
¸ c"µ$“·<br />
&<br />
(8)<br />
·¹Å)v<br />
¸ · <br />
¸ · <br />
©<br />
<br />
,/. <br />
'(,/.<br />
<br />
à Ã<br />
£3¦6¨<br />
Here are wanted parameters that help solve a variety of stacking and<br />
¸<br />
inversion problems (Hubral, 1999). They can be obtained by modifying the presently<br />
existing 2D CRS stack formula (Jäger et al., 2001) that is obtained from eq. ÆB<br />
(8) by sub-<br />
. In the following, we discuss 3 particular reductions of formula (8)<br />
which are of practical relevance.<br />
stituting à ·<br />
¸ · <br />
¸ · <br />
·CÅ<br />
£§¦©¨<br />
Particular cases<br />
Common-mid-point (CMP) gather<br />
For this case, v<br />
, the equation (8) reduces to<br />
©<br />
(9)<br />
This expression is commonly written as (Shah, 1973)<br />
Ç<br />
Pˆ‹ <br />
,f. <br />
'(,f.<br />
<br />
8¥¨<br />
£§¦©¨<br />
¸ · <br />
¸ · <br />
·…Å<br />
<br />
(10)<br />
Ç<br />
tWŒŸ È<br />
where the normal moveout (NMO) velocity & £§8:9 is now given by<br />
8¥¨<br />
& <br />
£§8:9<br />
&<br />
(11)<br />
For à ·<br />
<br />
£§8:9<br />
&<br />
this reduces to Shah's formula. For a 1-D model with a planar surface,<br />
£§¦©¨<br />
¸<br />
¸ · <br />
©E Ã<br />
,f. <br />
'(,f.<br />
=B<br />
TB<br />
straight normal ray and incidence ¸ angle<br />
g 8 · , where É g 8 · is the familiar root-mean-square velocity.<br />
É<br />
Common - shot gather<br />
For this case, , equation (8) reduces to<br />
»‡‹Wº²(<br />
this expression reduces to ÉW£§8:9<br />
& 4<br />
©<br />
& <br />
¸ · <br />
¸ · <br />
¸ · i<br />
<br />
·¹Å<br />
(12)<br />
,/. <br />
Ç<br />
1 ˆ<br />
,/.<br />
©Ê c"µ$“·…¸<br />
,f. <br />
£3¦6¨<br />
where according to eq. (A-9) and (Hubral, 1983), is demonstrated that £3¦©¨<br />
, with à à being the wavefront curvature of the reflected wave at ¢Œ¤ .<br />
½<br />
à ½
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