Annual Report 2000 - WIT

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° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥ ç ç 92 original seismic section (provided the same macro-velocity model, the same measurement configuration, and the same ray codes are used; only primary events are considered here). The weight function for demigration can be calculated in an analogous way to the previously described migration weight function, and reads ¬ëê‹Å ¡ £ ì ¥ Ò Ò ¯ Å for the ray j· segments · and §j¨ where and denote once again the point-source geometrical spreading factors , respectively; is the angle between the halfangle direction – defined by the two ray segments – and the vertical · at (Jaramillo et al., 1998). Å Ò Ò ¯ –n—…˜ Ã€þ (8) IMPLEMENTATION As we have shown in the previous section, the knowledge about the Huygens surface and the isochrone (stacking surfaces) are fundamental for Kirchhoff-type migration and demigration. These surfaces can be built by the sum of traveltimes along the ray branches from the source to the depth point · and from · to the receiver . Thus, we need a priori information about the traveltimes (for the calculation of the Huygens surface or isochrone) and dynamic parameters (for the calculation of the weight function) to perform a true-amplitude migration or demigration. This is where the macro-velocity model comes into operation. The necessary information, i.e., the Green's function of the medium, is usually obtained by asymptotic dynamic ray tracing for a specified ray-code in the macro-velocity earth model, and storing the parameters in a database which we call Green's Function Table (GFT). This database is the link between the input and output space, see Figure 1. In other words, we need information about the wavefield from every shot and receiver position at the surface to every diffraction · point in the subsurface. Obviously, this is a large amount of information, and the GFT can exceed the input data by several orders of magnitude. In order to avoid computation and storage of redundant information, methods to optimize the creation of the GFT are explained by (Riede et al., <strong>2000</strong>) and (Hertweck, <strong>2000</strong>). Our imaging tool is written in C++ in an object-oriented way. According to the previously mentioned stacking integrals, we have to calculate the analytical signal and its derivative (3D) or half-derivative (2D). This is done in the frequency or wavenumber domain, respectively. Tapering of the input data is used to avoid boundary effects at the border of the aperture during the summation process. We multiply the input data (along the shot coordinates) by the function ÿ ¡ over a small range at the border, usually some traces. No tapering is needed within the shot gathers along the offset dimension. The actual number of traces which are used for tapering may be specified by the user. The complete taper function in 3D is §T¨ –n—˜
93 0 ksi1 demigration x ksi1 t Seismogram Traveltime 0 x migration z Green’s Function Table (GFT) z Depth-migrated image Figure 1: Scheme showing the migration and demigration procedure for 2D data. The input data is transformed to the output domain with the help of the Green's Function Table (GFT) which contains all necessary information to construct the stacking surfaces as well as the weight functions. plotted in Figure 2. The weight functions for true-amplitude migration or demigration are calculated during runtime. The necessary information is obtained from the GFT. To avoid operator aliasing, the operator itself is tapered or, in other words, the possible reflector dip which can be imaged is limited. The limit can be defined by the user according to his needs and the input and output grid spacings.
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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
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 [
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 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
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Page 152 and 153: 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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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
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196 Altogether, close to 260 shotpo
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198 Time (ms) 0 2 4 6 8 10 12 14 16
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200 of groundwater. ACKNOWLEDGEMENT
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202
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204 the wavefront curvature matrix
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û ò é from ã äñ to ã î§ñ
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é 208 Table 1: Median relative err
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210 Table 2: Median relative errors
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212 CONCLUSIONS We have presented a
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214 PUBLICATIONS Previous results c
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æVU ìÿå T ü ýWYXZX[ 9\]9^\ Þ
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218 weight functions from traveltim
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220 REFERENCES Bleistein, N., 1986,
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Š Ê ¦ Í ½ Í ’ » Ö ½ Ê
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!¥”“Z• Ç ¤ É¨§ — ‘
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226
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228 In this paper, we present a mor
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L • MM+,ON N N Ž ú R N N N ú
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232 Figure 3: The evolution of a WF
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234 Let us analyze next the computa
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236 CONCLUSIONS We have presented a
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238
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’ Ç r 240 traveltimes, a computa
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Ž Ž Ã Ž 242 Other two approxima
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“ Ã Æ ³ 244 0 0.5 [km] 0 0.5 1
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î r † Û(Û Ü Œ U Ú Ý Ü Û
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248 Ettrich, N., 1998, FD eikonal s
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ø Ð ¡ Ð÷ö Ð'ø ú Ð ¡ Ð'
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ø ü ø ø ý Þ do ø ý Ü
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ø ö ø ú ü ö ü ¡ ü
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‹ 256 transport equation. Neverth
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258 tivalued geometrical spreading
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260 .
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262
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… " ¯ ü ý +- 4=° ü ü +
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ö ô æ º Ò x ¹ v ý ñ
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268 NUMERICAL RESULTS We constructe
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¹ õ ' ' -š' ù ¹ ù ' ' -š
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£ -š' - 272 Ô- —- Figure 6: C
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274 REFERENCES Aldridge, D. F. (199
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276
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278 3. 4. True-amplitude imaging, m
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280 Research Group Hamburg (Gajewsk
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282 Stefan Lüth Robert Patzig Refr
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284 Landmark Graphics Corp. 7409 S.
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286 TotalFinaElf Exploration UK plc
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288 as research assistant at Geoeco
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290 Ingo Koglin is a diploma studen
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292 Matthias Riede received his M.S
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294 Svetlana Soukina received her d
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296 Yonghai Zhang received the Mast
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