Annual Report 2000 - WIT

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È ¸ â ¹ Ë ¾œ¿ ¸ » ¸ Ç õ Ë ê È » â ô õ ü ü ü ü ü ü ü ¹ â â â Ë ¼ ö ¿ ¹ ¾ » ¾ ¾ õ ü ü ü ü ü ü ü ü ¼ ¼ ô Ç Ë ô 72 for the 2.5D case in constant-velocity media. In the »Ò¿ ¸ »¤¼bß ¾ ¹ Ë above ¿ formula, , is the amplitude of the event with pulse ¹ Ë ¾ ¸ ¸ Ë ¾ ¹ , is the traveltime curve ¼bà ¸ and Ó¾ shape found in the seismic trace Ó'¿ at . Also, denotes the partial derivative in the Ì Ç Ë is the isochron function defined ÍŠÎ implicitly Ï ¼Ó¾ direction of the normal ¹ by ¹ Ë ¼ ¹ »¤¼%Ñã¾S¿äÓœ¿ ¸ constant (A-2) ¼ »Ê¼ á to Ç and Ñ ¸º¹ where ¹ Ë ¼ ¹ ¸ ¸ ¹ Ë is the traveltime å ¾ from the source-point to »Ê¼%½³¾ »¤¼b½Š¾ the depth-point ¸º¹ Ë . To garantee correct amplitude recovery, the ¾ kernel and back to the æ receiver-point is selected as Ë ¾S¿èç¡éëêeì¡íNî³ïbðñ ¹ (A-3) ¸ ¹ »Ê¼ ¸º¹ point-source geometrical-spreading factors along the ray åœö segments öøæ ans , respectively. ñ Moreover, represents the angle the normal Ç to makes with the vertical makes with the isochron normal ö at . ç¡é Finally, is the modulus of the Beylkin determinant, Ó -axis, and ò denotes the incidence angle that the incoming ray åœö íNî³ï ðò óžô‡óžõ where ê is the medium velocity at ö ¿ ¸²¹ Ë ¼ Ç ¾¾ , and ó÷ô and ó÷õ are the the »¤¼%Ñ ¸º¹ »¤¼ ¹ Ë ¼ ¸ ö ¹ Ë ¼ ¸ ö (A-4) ýãþ ¿äù’úNûSü Í Û ýãþ ç¡é ¸ ¹ In the 2.5D case, and assuming, without loss of generality, that ß and à are the simmetry axes, we may write ¹ Í³ÿ ýãþ where ý þ ¿ ¸ Í þ ¼%Í ¡ ¼%Í£¢B¾ is the spatial gradient operator. and æ ¾÷¿ ¸ ¾ Ë ¸ Ë Ë ¾ Ë , å ¾÷¿ ¸ » ¸ Ë ¾B¼¥¤Š¾ , Ì Ç ¼¥¤Š¾ ¸ Ë . We also define the midpoint and ¾ half-offset coordinates, ¸ ¹ » ¿ ¸ »¤¼¥¤Š¾ , Ì ¸ ¹ ¹ Ë ¾÷¿ ¸ § » À » ¨ Þ (A-5) ¦”¿ In the above integral (A-1), the normal direction is not properly defined, since the -coordinates have different dimensions. In order to overcome this problem, ¨ ¼ and ç - and ¹ Ë Ó we must change the scale of the -axis. For constant ê , we define the new depth Ó coordinate © ¿ Ó as , so that it has the same dimension (length) as the -axes. In the new coordinates, the traveltime surface is given by ¹ ¿ Ë ¾ . The integral (A-1) © is therefore, ê Ç ¸ ¹ à ¹ Ë ¼ Ç ¾¾ž¼ Å È (A-6) Û ¿ ¸ Í Û ¼%Í³ÿ¨¼%Í£B¾ ¹ where . Note that now the non-unitary surface á ¿ normal ý ¼¤ ¼ºÀ µ ¾ is well defined. Moreover, from now on, the prime will indicate derivative Ç ê »¤¼%½Š¾‰¿ µ ·5¸ ÂÙÃªÄÆÅ with respect to the horizontal-coordinate Ë . Ë ¼ ¹ ê © ¸ »¤¼ Ë ¾Ì ¸ Ë ¾ ¹ á ý ¹ Ñ ¸ »¤¼ Û ¾ Ö+× Í£¢NÏ ¸ ½[À Ñ ¸ »¤¼
È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð ð ô ¼ ¿ »ÀB¦É¾ ¸ » ¸ ô Ë ð Ó Ë ¼ ¹ ¾¿ + © ð ò íî¨ï ì ê » § õ ð ¼ . ð ó ç ð ô ô § Ñ9 óžô » ¸ ô À » À/¦É¾ ¸ ð * ó ð õ õ § » õ 2 ð ð ð À õ À À ó à ð ð + ð ô õ § Þ ð ó ð õ ô »ÀA¦É¾ ¸ ¨ ð Ñ>* »À/¦É¾ ¸ ¾ Þ ð ì < ¼ Þ ¼ 73 To make use of the simmetry of the 2.5D situation, we transform integral (A-6) into the frequency domain, where it reads ¸ ¢ ¾ ¢ Ä Å Â¨Ã à Å È Here, denotes the Fourier transform of . Ï The isochron ½ ¿]Ñ ¸ »¤¼ Ë ¼ ¹ ê © Ë ¼ ¹ ¾) Þ Ç (A-7) ·5¸ »¤¼¢ ¾œ¿ ¸ »¤¼ ¾"# Ö+× ú%$'&(‡À ¢ Ñ ¸ »¤¼ Ë ¾2Ì ¸ Ë ¾ ¹ á! ý ¹ Ñ ¸ »¤¼ Û Ë ¼ ¹ ¾ corresponding to a constant time, can be easily evalu- © ated as the half-ellipsoid of revolution with center at ¸ ¦ ¼¥¤ ¼¤Š¾ and semi-axes (A-8) * ¿ ê In symbols, we have the isochron ç ð Þ ¨ ¿ © ¨ ¼ and + ¿-, * (A-9) Ñ ¸ »¤¼ µ À The Beylkin determinant (A-4), in this case, reduces to (see Martins et al., 1997) ¨ Ñ » » (A-10) 43 µ § µ 10 ç¥é with óžô óžõ65 and (A-11) ¿87 ¸ » § Ñ ¿87 ¸ » § Ñ À »5¾ À »5¾ ó÷õ ó÷ô Note that we do not need to íNî³ïð ò compute , since it will be canceled in the computation of the kernel (A-3). The last quantity we need íNî³ïbðñ is , which is given by ð Þ ¿ µ § ¸ (A-12) ¸ Ë ¾¾ Therefore, the expression for the kernel (A-3) is ê Ç ð Þ íNî³ï ð ñ ¸ »¤¼ Ë ¼à’¾S¿ ¨ µ » » (A-13) § ¸ ¸ Ë ¾¾ ;: ê Ç óžõ We also must compute the gradiente ý Û Ñ . After some algebraic manipulation, we find à Ñ *@ À (A-14) Í³ÿºÑ¿À ¼ and Í? Ñ¿ where, ç¡ð (A-15) ¾ ¸ * ¾ §
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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
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: 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: 71 Langenberg, K., 1986, Applied in
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
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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
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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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