Annual Report 2000 - WIT

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î r † Û(Û Ü Œ U Ú Ý Ü Û Ô † ‘ £ÒÒ Û Ü ~ î r ¿ Ú Ã UÌËÎÍ ¿ ‘ £ÒÒ © † ¿ × ¿ r ¿ Ë U r ¿ UÌËÎÙ Ô Ô © Ð I & Ð Ð z Ý à Ý…ß c § á † z › Ô Ô ¿ Ü Û Û z Û í Ð ß z & ¿ Ð Ð Ò ‘ £ÒÒ£6 ¥ { Ô ¿ ¸ Ò ‘ £¨ÒÒ£S î ¿ ¿ ~ © I & Ð Ð Ð Ô & Ð Ív Ð Ô Ù Ð Ü Û Ð & & Ü ß z Ýãß(ä § Í Ò † å Û Ý…ß(ä § Ív † zâà Ô Ô ¿ ‘ Ý › Ô Ô Û Û } Ô î ¿ Ô Ý Û Ü ¥ Ô Ô Û Ü z ß Ü à Ý ¿ Û Û z ë è r Ô Û Ý Ü Ô × Ý Ô Ô å z î Ý ¿ Ü ß Ý } º § î%ï 246 As a result we get the following nonlinear system for four unknown parameters : U £ ÚÚ , £6 , £ Ú and r £ Ú † Ê (5) ´ÏÑÐ £¨Ò* …Ö £¨ÒÒ £S (ÓÕÔ £6 † Ê Ò ÔØ× £6 Ô Ð £Ò* (Ö £ÒÒ Ô £¨ÒÒ † Ê Ò £S Ô £Ò* Ö ÐÚ £¨ÒÒ Ô ÐÚ ÐÚ Ü Þ › ¥ Û(Û ¿ÅÝ £6 £¨Ò* 6’ É £ Ò* ’ where: £Ò* Ý…Þ ‘ £¨ÒÒ £S 6’ Ý…Þ Ù † Ý…ß(ä Ý…ß c § ¿ÅÝ…Þ Ê Ò †æå Û…ç ç Ò Ýéè † Û…ç ¥ Û…ç å ç ¥ Ý…ß(ä § Ê † The remaining of £S¥¥ parameters £ }} , £Sêê and can be linearly expressed £¨ÒÒ by £S , £Ò* , and ë : r Ý…ß(ä § ¿ z ër £6¥¥ † ‘ £¨ÒÒ z £¨Ò* › Ò £S S’ ¿ z ¿ z ë (6) £ }}ì† z à Ý ‘ £ÒÒ £S 6’ z £¨Ò* z à Ý r Ü ß In (5) and (6) we used the abbreviations: £6êê † ‘ £ÒÒ › ¥ £S 6’ z £Ò* } ¿ Û…î ¥ î%ï › Ò † z § › † Ò î%ï º § › ¥ † î ¥ ¿Èä z ¸ î ¥ Ô î ¨§ ç † î Ò î Ò î ê › ¥:§ ç Ò † › ¥S’6§ ç ¥ † ë † r Ý ë § å Ò † – î ¿ÅÝ – ¿ÅÝ – Ò Ẽ˜ Ò ÒÒÒÒ6§ ¿ÅÝ – Ẽ˜ ¨§ î † – ¿ÅÝ – ¿ – ¿ÅÝ – ¿ – ¿ÅÝ – ¿ – ¿ÅÝ – î ¥ † – ¥ Ẽ˜ ¥ ¥¥¥¥¨§ }ð† – Ẽ˜ Ò Ẽ˜ Ò Ò*¥¥£Ò ¥¥ ¥ Ẽ˜ ¥ ¥£ÒÒ*¥ ¥ Ẽ˜ ¥ ¥ ¥¨§ ¿ÅÝ – ¿ – ¿ÅÝ – ¿ ~ – ¿ ~ – † – ¿ – Ò Ẽ˜ Ò Ò* £Ò Ẽ˜ £ÒÒ* ÒÒ* Ò* £Ò* ¨§ î€ï ÒÒ* Ò* £Ò* 6’6§ î ê † ~ ‘ – ¿ÈÝ £S £Ò* 6’QÉ £ Ò* ’ £Sêê§
Ð Ð Ð † ‘ £ÒÒ ‘ £ÒÒ † ‘ Ý £ÒÒ † Ô ¿ ¿ ¿ Ô Ô ‘ £ÒÒ ‘ £ÒÒ ¿ ‘ £ÒÒ ¿ ¿ ¿ Ò ‘ £ÒÒ.£6 Ò ‘ £ÒÒ.£6 Ô Ô Ô 247 and: ¿ÈÝ ¿ÅÝ £6 £ Ò* ’ó¨§ ÐÚ £S £¨Ò* Q’ É Òòñ £S £Ò* S’ É £¨ÒÒ ¿ÈÝ ¿ÅÝ £6 £ Ò* ’ó¨§ ÐÚ £S £¨Ò* Q’ É Òòñ £ÒÒ £Ò* S’ É £6 ¿ÅÝ ¿tÝ To approximate an anisotropic medium by an elliptical one, the non-linear system (5) are ë ÐÚ should be solved by a numerical method. When four parameters £¨ÒÒ , £6 , £Ò* and ô Ò€ñ Ò ‘ £ÒÒ£6 £6 £¨Ò* 6’ É £Ò* £6 £Ò* S’ É £ Ò* ’ó§ £¨Ò* computed the remaining parameters £S¥¥ , £Sõõ and £6êê are determined by (6). Solving the non-linear system inside each cell of the FD-grid is too expensive. The proposed way, therefore, is suitable for a piece-wise homogeneous anisotropic medium when it is possible to use the elliptical reference medium for each layer. The ellipsoidal medium can be used for calculation of traveltimes by an FD eikonal solver with embedded perturbation scheme. CONCLUSION The presented algorithm provides a method for the efficient computation of the first arrival traveltimes for 3-D elliptically anisotropic media. The maximum relative error does not exceed 0.4 %. The calculation technique for approximating an arbitrary anisotropic medium by an ellipsoidal medium was considered. As a result we get the nonlinear system for defining parameters of a reference medium. Future work must be devoted to solving this system for embedding a first order perturbation method to the FD scheme in order to consider an arbitrary anisotropic medium. This technique is suitable for piece-wise homogeneous anisotropic medium. REFERENCES Burridge, R., Chadwick, P., and Norric, A. N., 1993, Fundamental elastodynamic solutions for anisotropic media with ellipsoidal slowness surfaces: Proceedings of the Royal Socity London A, 440, 655–1681. Ettrich, N., and Gajewski, D., 1998, Traveltime computation by perturbation with FDeikonal solvers in isotropic and weakly anisotropic media: Geophysics, 63, 1066– 1078. Ettrich, N., Gajewski, D., and Kashtan, B., <strong>2000</strong>, Reference ellipsoids for anisotropic media: submitted to Geophys. Pros.
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Wave Inversion Technology WIT Annua
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Wave Inversion Technology WIT Wave
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Copyright c 2000 by Karlsruhe Unive
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i TABLE OF CONTENTS Reviews: WIT Re
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Wave Inversion Technology, Report N
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3 theory. It relates properties of
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Imaging 5
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8 two hypothetical eigenwave experi
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7 ¢¥£§¦©¨ ; < calculate sear
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7 7 searches ¡ HNJOL for ¢IHNJML
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14 Trace no. 1000 1500 2000 0.5 1.0
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16 CDP number 3100 3000 2900 2800 2
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18 REFERENCES Höcht, G., de Bazela
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20 INVERSION BY MEANS OF CRS ATTRIB
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22 For synthetic data, it is easy t
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— J J J J J J G J - J J G
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· · v 0 B 0 £ & Ä Ã & v
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31 Dürbaum, H., 1954, Zur Bestimmu
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Ã Ø Ì 0 0 Ì 4 0 Ã 0 G ' 4
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0 0 ˜ ” Z ˜ ” Z 4 4
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39 strategy by combining global and
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41 elled data is presented in Figur
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43 0.2 Distance [m] 1000 1500 2000
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45 In Figure 10 we have the optimiz
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47 Gelchinsky, B., 1989, Homeomorph
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§ ¢ ø ø for a paraxial ray, ref
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Œ § … Z ‹ Z § õ ø \ ø §
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54 As stated in Hubral and Krey, 19
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§ ó § 56 sequence. We made tes
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58 precision of the modeled input d
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60 Cohen, J., Hagin, F., and Bleist
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£ 65 The details of the theory inv
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67 of the figure. On both sides, a
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69 (a) (b) Depth (m) 600 800 Depth
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71 Langenberg, K., 1986, Applied in
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È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð
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g ý g [ ^ â g [ g g g g g â h [
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g § » ¹ [ƒŽ g 79 We now subst
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ê 81 ing was realized by an implem
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ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
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85 on a second-order approximation.
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88 kinematic (related to traveltime
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¨ º Ý È · § À · Á · Á ¼
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° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
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94 1 taper value ksi1 0 ksi2 Figure
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96 0 1 2 3 4 5 Depth [km] CMP [km]
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98 CONCLUSION As a generalization o
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100 weight during the stacking proc
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102 are not illuminated for every o
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104 obtained from Zoeppritz' equati
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106 PreSDM of Porous layer 0.358 re
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108 REFERENCES Gassmann, F., 1951,
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110 et al., 1992), Hanitzsch (1997)
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112 consecutive wavefronts). Howeve
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114 Numerical example We test the W
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116 Versteeg, R., and Grau, G., 199
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118 indicator. To extract elastic p
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€ b b 120 S where is the position
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É É É É 122 The À constant (da
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124 0 Distance [ m ] 1000 2000 3000
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126 Kosloff, D., Sherwood, J., Kore
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128
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130 penetration distances and a pot
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and » ˆ Ö Ö is the scalar hydra
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Å éÙó ó ».-.‹œì/-jì¨‹
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136 TRIGGERING FRONTS IN HETEROGENE
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Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
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Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
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142 300 250 200 a) eikonal equation
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144 Shapiro, S. A., and Müller, T.
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» i » m ×]Ú ‘Œƒšj'Ÿlk hM
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148 RESULTS Soultz-sous-Forêts Inv
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…„ƒ 150 for the smaller ellips
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152 Figure 5: The cloud of events f
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…„ƒ Î Î ÎÄÈ‹•¨ Î-È
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156 straightforward. The new approa
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158
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160 1982). There is also the so-cal
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{ ³ u ´ ´ ´ ´ -y ›-^œ ´ ´
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Ï u u ¬ 164 To summarize, we deri
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Ý ì á ä é å¤æDçzè ä é
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ø M ã = ã Ù ø ä Ú ã Ù ø
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ã Q c ã ä 170 a=10m; std.dev.=8%
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172 Frankel, A., and Clayton, R. W.
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174 It is well-known that inhomogen
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ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
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Kneib, 1995 285-6000 superposition
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180 parameter: Hurst coefficient pa
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182 REFERENCES Bourbié, T., Coussy
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184 1999) in multiple fractured med
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‹œž Ÿ¢¡¤£¦¥ƒ§©¨ ‘
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j jÆÅÇ[È”u j jÎÅÇ[È”uw
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190 Davis, P. M., and Knopoff, L.,
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192 properties from the measured se
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194 discussion of these problems ca
Page 208 and 209: 196 Altogether, close to 260 shotpo
Page 210 and 211: 198 Time (ms) 0 2 4 6 8 10 12 14 16
Page 212 and 213: 200 of groundwater. ACKNOWLEDGEMENT
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Page 216 and 217: 204 the wavefront curvature matrix
Page 218 and 219: û ò é from ã äñ to ã î§ñ
Page 220 and 221: é 208 Table 1: Median relative err
Page 222 and 223: 210 Table 2: Median relative errors
Page 224 and 225: 212 CONCLUSIONS We have presented a
Page 226 and 227: 214 PUBLICATIONS Previous results c
Page 228 and 229: æVU ìÿå T ü ýWYXZX[ 9\]9^\ Þ
Page 230 and 231: 218 weight functions from traveltim
Page 232 and 233: 220 REFERENCES Bleistein, N., 1986,
Page 234 and 235: Š Ê ¦ Í ½ Í ’ » Ö ½ Ê
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Page 240 and 241: 228 In this paper, we present a mor
Page 242 and 243: L • MM+,ON N N Ž ú R N N N ú
Page 244 and 245: 232 Figure 3: The evolution of a WF
Page 246 and 247: 234 Let us analyze next the computa
Page 248 and 249: 236 CONCLUSIONS We have presented a
Page 250 and 251: 238
Page 252 and 253: ’ Ç r 240 traveltimes, a computa
Page 254 and 255: Ž Ž Ã Ž 242 Other two approxima
Page 256 and 257: “ Ã Æ ³ 244 0 0.5 [km] 0 0.5 1
Page 260 and 261: 248 Ettrich, N., 1998, FD eikonal s
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Page 264 and 265: ø ü ø ø ý Þ do ø ý Ü
Page 266 and 267: ø ö ø ú ü ö ü ¡ ü
Page 268 and 269: ‹ 256 transport equation. Neverth
Page 270 and 271: 258 tivalued geometrical spreading
Page 272 and 273: 260 .
Page 274 and 275: 262
Page 276 and 277: … " ¯ ü ý +- 4=° ü ü +
Page 278 and 279: ö ô æ º Ò x ¹ v ý ñ
Page 280 and 281: 268 NUMERICAL RESULTS We constructe
Page 282 and 283: ¹ õ ' ' -š' ù ¹ ù ' ' -š
Page 284 and 285: £ -š' - 272 Ô- —- Figure 6: C
Page 286 and 287: 274 REFERENCES Aldridge, D. F. (199
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Page 290 and 291: 278 3. 4. True-amplitude imaging, m
Page 292 and 293: 280 Research Group Hamburg (Gajewsk
Page 294 and 295: 282 Stefan Lüth Robert Patzig Refr
Page 296 and 297: 284 Landmark Graphics Corp. 7409 S.
Page 298 and 299: 286 TotalFinaElf Exploration UK plc
Page 300 and 301: 288 as research assistant at Geoeco
Page 302 and 303: 290 Ingo Koglin is a diploma studen
Page 304 and 305: 292 Matthias Riede received his M.S
Page 306 and 307: 294 Svetlana Soukina received her d
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296 Yonghai Zhang received the Mast
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