Annual Report 2000 - WIT

More documents

Recommendations

Info

’ Ç r 240 traveltimes, a computation by a first order perturbation method embedded into the FD scheme is possible. Perturbation method embedded into an FD scheme for weak anisotropy was considered by (Ettrich, 1998) for the 3-D case and by (Ettrich and Gajewski, 1998) for the 2-D case. For application to anisotropic media an elliptically anisotropic reference model gives higher accuracy but using an isotropic reference model gives higher speed of calculation. To minimize errors which are inherent in the perturbation approach one should choose the reference medium as close as possible to the given anisotropic medium. In the paper by (Ettrich et al., <strong>2000</strong>) the problem for the approximation of an arbitrary anisotropic medium by an ellipsoidal medium were derived. This approximation works well for P-anisotropies of up to 10 % when the polarization vector is substituted by the phase normal. We want to apply the FD perturbation method to strong anisotropy. Following (Burridge et al., 1993) there are three possibilities to simplify general anisotropy to ellipsoidal symmetry and we use these possibilities for the construction of a reference ellipsoidal medium. Taken the physics of wave propagation into account we minimize the average of differences between the Christoffel matrices of the anisotropic medium and the ellipsoidal medium. This paper presents first results in this approach. FD APPROXIMATION OF THE EIKONAL EQUATION If one type of k wave (quasi l Ú -wave, quasi l ’ - or -wave) in an elliptically anisotropic medium is defined by three phase velocities ( , , ’ ¡ Ž ) along each axis in the crystal coordinate system and by three angles ¡#Ú , ¡ , ) describing the orientation of the crystal (m n coordinate system with respect to the global Cartesian coordinate system, the slowness surface (eikonal equation) for this wave reads: ‘Qq “Z• o po 3è q “ o where is the slowness vector, is a 3 symmetric matrix, denotes scalar product. In the crystal coordinate system this matrix is a diagonal matrix and reads •à¡ q ² ³Ö ³ts ³Ö s q Ž m ¡ n where is the Kronecker-delta. In the global Cartesian coordinate sys- is a function of , , and , , . , tem q and r ‘ (1) ¡ where is a rotation matrix which defines the orientation of the crystal coordinate system with respect to the global Cartesian coordinate system. q are connected by q • ¡vu The eikonal equation allows for the computation of one slowness vector component if the others are known and for the computation of the phase velocity for a given phase direction. In our algorithm the approximation of the eikonal equation from (Ettrich, 1998) is used. If there are traveltimes in gridpoints 0, 3, 4, 5, 6, 7 and 8 (see Fig. 1)
x x Ž Ž • • • • • • Ç Ç Ç Ž Ç “ ’ ’ ’ ’ ’ ’ ’ Ž ’ Ž Ž ’ Ž ’ Ž ’ ’ ’ ’ Ž ’ “ ’ ’ 241 Figure 1: Scheme of expansion. The timed points are given by the black circle. The computation starts at point 0 with the minimum traveltime. Using traveltimes in points 0, 1, 2, 3 and 4 the traveltime in 8 can be calculated (scheme 3). Using traveltimes in points 0, 2, 3, 4 and 8 the traveltime in 6 can be calculated (scheme 2). To find the traveltime in points 9 the traveltimes in points 0, 3, 4, 5, 6 and 7 are used (scheme 1). z 7 9 y 8 x 6 5 4 1 0 3 2 the ansatz used by (Ettrich, 1998) for the searched traveltime ¢\w at the corner point of a cubic cell is: ¢\w Ã ¢ ¾O“Z•yxâ¾ « x®Ú ¤ ¢\# Ã ¢.z “ « ¢.{ Ã ¢\| “ ¯^« x ’ ¤ ¢\# Ã ¢} “ « ¢\[ Ã ¢\| “ ¯ « ¤ ¢\[ Ã ¢ « ¢.{ Ã ¢} “ ¯^« x }¥¤ ¢ Ã ¢} “ « ¢\[ Ã ¢.{ “ ¯ « (2) x #¥¤ ¢\[ Ã ¢\# “ « ¢\| Ã ¢} “ ¯#« x [¤ ¢.{ Ã ¢\# “ « ¢\| Ã ¢ ¯ ‘ where: xâ¾ ’ É ¡ ~€ ’ É ¡ }ã« ’ Ãø¡ [ ’ Ãø¡ Ú x®Úö• “ ‘ ^ Ã ’ É ¡ }ã« ’ Ãø¡ # ’ Ãø¡ ’ xâ¾ ’ É ¡ ~€ x ’ “ ‘ ^ Ã ’ É ¡ #–« ’ Ãø¡ [ ’ Ãø¡ Ú xâ¾ ’ É ¡ ~€ ’ “ ‘ ^ Ã x } ’ Ã¬¡ } ’ Ã ¡ Ú xâ¾ ’ É ¡ ~€ ’ “ßÃøŒ ^ ‘ (3) ’ xâ¾ ’ É ¡ ~€ ’ Ã¬¡ [ ’ Ã ¡ ’ “ßÃøŒ x # ^ ‘ xâ¾ ’ É ¡ ~€ ’ Ã¬¡ # ’ Ã ¡ Ú “ßÃøŒ x [ ^ ‘ ’ z ’ { ¡ xâ¾ and is traveltime in gridpoints with number and point 0 is the gridpoints ³ with minimum ¢ traveltime, , ‘ ¨ ‘ c are the velocities in -, - and J Ô -direction ÔÄ• ‚ ¡ ³ (directions 1, 2, 3) and in direction of the diagonals J Ã ‚ , J Ã ‚ and Ã ‚ (directions 3, 4, 5). Formula (2) with coefficients (3) is analogous formula 1 from (Vidale, 1990), called scheme 1, applied to the majority of grid points.
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 and 40:
Page 41 and 42:
· · v 0 B 0 £ & Ä Ã & v
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:
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:
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:
Page 89 and 90:
g ý g [ ^ â g [ g g g g g â h [
Page 91 and 92:
g § » ¹ [ƒŽ g 79 We now subst
Page 93 and 94:
ê 81 ing was realized by an implem
Page 95 and 96:
ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
Page 97:
85 on a second-order approximation.
Page 100 and 101:
88 kinematic (related to traveltime
Page 102 and 103:
¨ º Ý È · § À · Á · Á ¼
Page 104 and 105:
° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
Page 106 and 107:
94 1 taper value ksi1 0 ksi2 Figure
Page 108 and 109:
96 0 1 2 3 4 5 Depth [km] CMP [km]
Page 110 and 111:
98 CONCLUSION As a generalization o
Page 112 and 113:
100 weight during the stacking proc
Page 114 and 115:
102 are not illuminated for every o
Page 116 and 117:
104 obtained from Zoeppritz' equati
Page 118 and 119:
106 PreSDM of Porous layer 0.358 re
Page 120 and 121:
108 REFERENCES Gassmann, F., 1951,
Page 122 and 123:
110 et al., 1992), Hanitzsch (1997)
Page 124 and 125:
112 consecutive wavefronts). Howeve
Page 126 and 127:
114 Numerical example We test the W
Page 128 and 129:
116 Versteeg, R., and Grau, G., 199
Page 130 and 131:
118 indicator. To extract elastic p
Page 132 and 133:
€ b b 120 S where is the position
Page 134 and 135:
É É É É 122 The À constant (da
Page 136 and 137:
124 0 Distance [ m ] 1000 2000 3000
Page 138 and 139:
126 Kosloff, D., Sherwood, J., Kore
Page 140 and 141:
128
Page 142 and 143:
130 penetration distances and a pot
Page 144 and 145:
and » ˆ Ö Ö is the scalar hydra
Page 146 and 147:
Å éÙó ó ».-.‹œì/-jì¨‹
Page 148 and 149:
136 TRIGGERING FRONTS IN HETEROGENE
Page 150 and 151:
Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
Page 152 and 153:
Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
Page 154 and 155:
142 300 250 200 a) eikonal equation
Page 156 and 157:
144 Shapiro, S. A., and Müller, T.
Page 158 and 159:
» i » m ×]Ú ‘Œƒšj'Ÿlk hM
Page 160 and 161:
148 RESULTS Soultz-sous-Forêts Inv
Page 162 and 163:
…„ƒ 150 for the smaller ellips
Page 164 and 165:
152 Figure 5: The cloud of events f
Page 166 and 167:
…„ƒ Î Î ÎÄÈ‹•¨ Î-È
Page 168 and 169:
156 straightforward. The new approa
Page 170 and 171:
158
Page 172 and 173:
160 1982). There is also the so-cal
Page 174 and 175:
{ ³ u ´ ´ ´ ´ -y ›-^œ ´ ´
Page 176 and 177:
Ï u u ¬ 164 To summarize, we deri
Page 178 and 179:
Ý ì á ä é å¤æDçzè ä é
Page 180 and 181:
ø M ã = ã Ù ø ä Ú ã Ù ø
Page 182 and 183:
ã Q c ã ä 170 a=10m; std.dev.=8%
Page 184 and 185:
172 Frankel, A., and Clayton, R. W.
Page 186 and 187:
174 It is well-known that inhomogen
Page 188 and 189:
ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
Page 190 and 191:
Kneib, 1995 285-6000 superposition
Page 192 and 193:
180 parameter: Hurst coefficient pa
Page 194 and 195:
182 REFERENCES Bourbié, T., Coussy
Page 196 and 197:
184 1999) in multiple fractured med
Page 198 and 199:
‹œž Ÿ¢¡¤£¦¥ƒ§©¨ ‘
Page 200 and 201:
j jÆÅÇ[È”u j jÎÅÇ[È”uw
Page 202 and 203: 190 Davis, P. M., and Knopoff, L.,
Page 204 and 205: 192 properties from the measured se
Page 206 and 207: 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
Page 214 and 215: 202
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
Page 236 and 237: !¥”“Z• Ç ¤ É¨§ — ‘
Page 238 and 239: 226
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 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 258 and 259: î r † Û(Û Ü Œ U Ú Ý Ü Û
Page 260 and 261: 248 Ettrich, N., 1998, FD eikonal s
Page 262 and 263: ø Ð ¡ Ð÷ö Ð'ø ú Ð ¡ Ð'
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
Page 288 and 289: 276
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
Page 308:
296 Yonghai Zhang received the Mast
show all

Annual Report 2000 - WIT

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?