Annual Report 2000 - WIT

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Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö » Ö Ö Ê ó Ò »Ë » Ö Ê ó »Ë » Z Ê Ö Ö Ò Ê ÷ “ “P£ Ê V Ö Ö Ò 140 DISCUSSION The main limitations of the extension of the SBRC to the case of heterogeneous media proposed here are apparently related to the validity range of equation (16). Roughly they can be formulated from the following consideration of the right hand part of equation (1) in a 1-D medium: Ö Ö Ö Ö (28) » Ü ‘ Ê Ò Our approach is expected to be valid if the following inequality is satisfied: Ö Ö óNZ Ì (29) ó] ó » This can be roughly reduced to the following: ó[V\ ó… ó…»](èó^Z number. Taking into account that ó,(!ó Ò ‘3é (» approximately we arrive at the following, rather simplified condition: Ì , where ( is the wave Ê Ò ó Ö ûÁéÈ (30) This inequality relates the gradient of the hydraulic diffusivity and the frequency of the pressure perturbation. It is rather typical for the geometric optic approximation. It shows, that if the frequency is high enough and the medium heterogeneity is smooth the above approximation can be applied. In the case of the step-function like pressure perturbation the frequency corresponding to the triggering front is accepted to be éÁ‘ . Using the equation of the triggering front in homogeneous poroelastic media (5) “¤£è¦ˆ the occurrence time of earlier events can be roughly approximated ˆ`_Ê Ò Bš § £!»ËŸ as . Note, Ê that denotes the distance from the injection source. Thus, inequality (30) can be reduced to the following one ó Ö È (31) This condition is a rather restrictive one. In addition, it shows that the smaller distance Ê the higher is the resolution of the method. In spite of the restrictive character of the inequalities above, we think that the geometric optic approximation is applicable to the propagation of microseismicity triggering fronts under rather common conditions. This is based on the causal nature of the triggering front definition. When considering the triggering front we are interested in a quickest possible configuration of the phase travel time surface for a given frequency. Thus, we are interested in kinematic aspects of the front propagation
a 141 only. The quickest possible configuration of the phase front is usually given by the Hamilton-Jacobi, i.e., eikonal equation. However, the conditions above necessarily take into account not only kinematic aspects of the front propagation but rather mainly dynamic aspects, i.e., amplitude of the pressure perturbation. In other words, the eikonal equation is usually valid in much broader domain of frequencies than those given by the inequalities above. Therefore, the method will give meaningful and useful results, at least semi qualitatively. To verify the assertion that we are able to describe the diffusion process and the kinematics of the evolution of the triggering front in a heterogeneous medium by the use of the eikonal solution, we performed some numerical tests. We solved the parabolic differential equation of diffusion (1) in two spatial dimensions with the help of a finite elements method (FE) implemented in the MATLAB R computing environment. We calculated the time-dependent pressure variation through a medium where the » diffusivity varies smoothly Ê in direction with a Gaussian profile. It changes from a background » ‘uÎÄÈ;ÍW¢ËÒ? value of to a minimal value »ú‘Î–È Ìè¢ËÒÆ of at the ¨ ¨ center. The half-width of this heterogeneity is approx. 200 m. The dimension of the computational mesh is 4000 m x 4000 m, and the source point is located at its center. As input signal we use a time-harmonic sinusoidal signal with a period of 400 h and 800 h respectively, multiplied with a boxcar function for the whole simulation time of 2400 h. The time increment in our ‡Pˆb_ Þ simulations was h while the elementary cell was of the order of 5 m x 5 m. We observe the pressure variation in a one dimensional section along the x-axis through the center of the model, where the source is located. In each point of observation we estimate the arrival time of the fourth resp. sixth zero-crossing of our quasi-periodic pressure signal. This was necessary to reduce the effects of the high-frequency components due to the finite character of the source signal. We compare this time with the corresponding theoretical eikonal solution ‘Sc¢d ä ˆ R%ë Œ°šŠëŸ È (32) Using the arrival time we also calculate the velocity Œ of the phase front at a given distance from the source point. Then we convert it into the diffusivity and compare the result with the exact diffusivity of the model. The diffusivity is calculated from the measured velocity of the phase front by »ÃšŠÊƒŸ ‘ ŒƒšŠÊƒŸFÒ £=e § • (33) e where is the dominant frequency of the source function (see eq. 4 and the comment below).
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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
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 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
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Page 200 and 201: 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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