Annual Report 2000 - WIT

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â Í ð Í ^ g ð Í [f} Í’» ^ g ^ g ¸ ¸ ¸ g ð Í Í ð ^ g u g g g Í Í â h Í ¿ Í h Í’»€ Ú þ Ö Ú þ% h Í [f} Í’» ¸ ¿ ˆ ^ g ‰ ð Í ¸ h ð h ‰ g Í Í Í ¿ ð â ` ð h Í Í ð g § ` Í [f} Í’» h Ë Í 2 h Í [f} Í’» ` Í [f} Í’» ` Í [ Í¡» < Ü § 78 The derivations of » [ are given by ^ ¸ dif Ø Ú Ûir s[Ü dif knmpoEq Þ (6) ö][ ¼ih'¾ with respect to the components of ¹ ` § ö][ ¼ih'¾S¿ ÏS hj ¨ ÃÉÄ_ Ä 0 Since [f} Í’» [ , we observe that » [ is not a function of ¹ dif âfe âf~ ¼ (7) where we have used the notation for the diffraction-traveltime function along rays connecting an arbitrary source-receiver pair defined by to an arbitrary point on the reflector ¹ . Fermat's principle states then that \ [} Í¡» â~ Í’» [f} Í’» [f} âf~ ¿ ¤ Þ (8) Differentiating equation (6) a second time yields âf~ âf~ ` § ÏN hj‚ ö][ ¼ih'¾›¿ ¨ ÃªÄ_ Ä : Í¡» [} Í’» [ Í’» [f} Í’» [ âf~ âf~ âf~ Ø Ú Ûr s[Ü ¼ dif (9) mpoEq k § ¸ ` § h'¾ Í¡» [ Í’» [f} Í’» [ Í’» [ where we have again used equation (7). Í’» [ƒ} In high-frequency approximation, equations (6) and (9) are dominated by âf~ the highest-order non-vanishing terms in . ö|[ At , the first derivative of vanishes due to Fermat's principle, equation (8). Thus, we find in this approximation, h ö|[ ¼ih'¾S¿c¤ (10) and â~ h'¾ ¸ Ë ¼ ` ØƒÚ ¾iknm„o…q ¹ Ûr s dif ö|[ Þ (11) öA[ ¼ih'¾S¿ ÏS hj Í’» [f} Í’» [ Í’» [f} Í¡» [ The asymptotic evaluation of equation (11) is completely parallel to that of integral (2) and yields âf~ ¨ Ã Ä_ Ä hj‡y ¸ ¹ Ë ¼ öA[ ¾ (12) Ï† Here, we recognize the Hessian ˆ matrix thus write in matrix form, as defined by Hubral et al. (1992). We Í¡» [} Í’» [ Í’» [f} Í’» [ [ ¿ hjŠy ¸ ¹ Ë ¼ öA[ ¾ Þ (13) Ï†
g § » ¹ [ƒŽ g 79 We now substitute equations (4), (10) and (12) in the Taylor series (3) to obtain Back in the time domain, this reads » ¹ ˆ [ ‰ ^X¸ ö][ ¼‹h'¾¿ Œ µ hj‡y ¸ ¹ Ë ¼ ö|[ ¾ Þ (14) ÏS µ ¨ h or, at Óœ¿c¤ , » ¹ b ÏN Ó" Ž y ¸ ¹ Ë ¼ öA[ ¾ ¼ (15) [ ^ ¸ ¼Ó¾œ¿ Œ7ÏNOÓ" § µ ¨ öO[ » ¹ ˆ [ ‰ ^X¸ ¾S¿ ö][ ^ ¸ ¼¥¤Š¾S¿ Œ7ÏSC¤E § µ ¨ ö][ » ¹ ˆ [ ‰ » ¹ b Ï†C¤E Ž y ¸ ¹ Ë ¼ ö|[ ¾ Þ (16) [ This result can again be interpreted as a first-order Taylor expansion of ^¸ » ¹ ˆ [ ‰ » ¹ Šy ¸ ¹ Ë ¼ öA[ ¾ Þ (17) [ ¾œ¿]ÏN µ ö][ ¨ The physical interpretation of this result is straightforward. Since is zero outside the interval ¤B‘ ÓN‘ â'’ , the influence of the migrated wavefield at ÏN ends at that Ó" öA[ particular point ö][ , where µ » ¹ ¹ ˆ ¨ [ » [ ¿ Þ (18) ‰ This is exactly the definition of the time-domain Fresnel zone (Hubral et al., 1992). Thus, the area affecting the reflected field in the öA[ vicinity of is the area of the paraxial Fresnel öA[ zone at . Let us appreciate the meaning of this result. We know that forward wave propagation distributes the information scattered from each “diffraction ö point” in the seismic data over one projected Fresnel zone, which is therefore the minimum aperture for seismic Kirchhoff prestack depth migration (Schleicher et al., 1997). The present result tells us that migration smears the information pertaining to each depth point is smeared in the migrated section over a paraxial Fresnel zone. To undo the effects of wave propagation, that is, to recover the migrated image with a perfect resolution, however, it should smear this information over the true time-domain Fresnel zone. Thus, we can expect that the migrated data will show a lateral resolution that is roughly equivalent to the quality of the paraxial approximation of the Fresnel zone. In the next section, we confirm this conclusion with a simple numerical experiment. â'’ SYNTHETIC EXAMPLE To demonstrate the lateral resolution of seismic Kirchhoff depth migration, we have devised the following simple numerical experiment. Consider a horizontal interface
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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
Page 39 and 40: Wave Inversion Technology, Report N
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 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 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 89: g ý g [ ^ â g [ g g g g g â h [
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
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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
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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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Annual Report 2000 - WIT

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