Wave Inversion Technology - WIT

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241 Coherence Measure Properties In any application, coherency analysis is used to extract some information from the given data. Therefore, the main property of any coherence measure is its resolution capability determining the accuracy the desired information can be resolved. Resolution is a general attribute for coherence measures and is further influenced by a set of special properties as listed below. Properties of coherence measures: Resolving power Discrimination threshold Separation of events with equal zero-offset time Sensitivity to amplitude changes versus offset Sensitivity to sign/phase changes versus offset Noise level dependency (statistical significance) Enhancement of weak signals Dynamic range of display Stability Computational effort Some of these attributes are interrelated. The discrimination threshold, for example, can be a visual attribute and depends then on the dynamic range of values a coherence measure needs for display; however, the visual discrimination threshold may differ from thresholds numerical search algorithms need to discriminate events. The separation of events with equal zero-offset time is a challenging task for coherence measures. Theoretical, separation of events is better the wider the spatial range is chosen. In practice, however, a wider spatial range often collides with the small spread approximation of theoretical models. One method to improve this compromise is to introduce offset dependent weighting factors. The sensitivity of a coherence measure to amplitude, sign or phase changes versus offset depends on the theoretical properties of each measure. The noise level dependency rather is based on the practical properties of a coherence measure. For example, the normalized crosscorrelation measure is a statistical estimate of the normalized crosscorrelation function. The estimate is defined only for
242 a limited time range while the crosscorrelation function is defined for infinite records. The variance of the estimate varies with the length of the time gate, the bandwidth of signal and noise and with the signal-to-noise ratio (see Bendat and Piersol (1986)). It is evident that the statistical significance of the estimate shrinks with shorter time gates. In coherency analysis the operation windows are commonly short, thus the statistical significance is low. That is, coherence measures must proof their applicability in practice (see in addition Schneider and Backus, 1968). The enhancement of weak reflections is affected by the properties of the coherence measure and is normally supported by the normalization of measures. Stacked amplitude, for example, yields its highest sum for the reflection with the highest amplitudes, no destructive summation provided. Semblance, however, theoretically yields the same result for any scaled version of a signal. It may only yield lower values for weaker signals because of the reduced statistical significance, i.e., the lower S/N ratio for weaker reflections. Normalization reduces the dynamic range for display while with unnormalized measures the dynamic range is mainly occupied by the largest coherence value obtained for the reflection with the strongest amplitutes. To extract weak reflections by visual means it may be desirable to reduce the distance between the coherence values for different events. However, for numerical algorithms which search for local maxima it may be easier to find these maxima if the separation between the values is as great as possible. The possible resolution of coherence measures is limited by the discrete parameter space. If the discretization of the space is too coarse in combination with the resolving power of a measure an aliasing problem will occur (see Claerbout (1992)). That is, it is possible that a measure would yield a local maximum for a parameter value that lies right between two values from the discrete range of values. The computational effort of a coherence measure becomes a crucial point when moveout models with three or even more parameter dependencies are used. Any additional parameter increases the computation time with a factor equal to the number of its elements. Therefore, effective measures are needed. Also, one can reduce the computation time if only selected parameter combinations which yield local maxima are calculated. This requires numerical search algorithms which must be optimized to be effective. Although it is generally profitable to interpret the structure of the whole parameter space, visual examinations are less practicable the more parameters are involved. With modern technologies, however, such as virtual reality cubes, it is possible to visual investigate three or four-dimensional parameter sets in a reasonable effective way. In any case, since sophisticated moveout models becomes rather more complex and recent coherence measures also, the employment of massive parallel computer systems with highly efficient parallel computer algorithms is a requirement to manage forthcoming tasks.
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Wave Inversion Technology WIT Annua
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WIT Sponsors WIT Wave Inversion Tec
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with contributions from the WIT Gro
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Preface The third WIT (Wave-Inversi
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ii Vanelle C. and Gajewski D., Thre
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2 Müller and Shapiro extended the
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4 age data. Numerical results obtai
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Wave Inversion Technology, Report N
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9 t " ; t = ;() N R N ; R ! S R x
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11 1987), h B ( ~ M I )= 2 6 4 rT (
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13 200 (a) Time (ms) 250 300 350 40
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15 reflector, we can conceive its i
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19 0 X1 X2 X3 X4 Depth (m) 500 1000
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21 1 0 0.9 -0.05 0.8 -0.1 0.7 -0.15
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23 Interestingly enough, things do
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25 1 0 0.9 -0.05 0.8 -0.1 0.7 -0.15
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29 and x G = x m + h ; x o . The co
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31 SENSIBILITY ANALYSIS The most im
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33 CONCLUSIONS By using derivatives
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35 T 5 = T 6 = @ @ o = T 5 + T 6 (1
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37 0.8 Common−Offset Section (100
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40 INVERSION OF THE WAVEFIELD ATTRI
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42 X 0 α ~ α V 0 γ i (x ,z ) i i
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44 0 1 2 v = 1.5km/s 0 v = 4.5km/s
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46 second interface are slightly sc
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48 CONCLUSION We presented an inver
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50 .
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55 time in respect to the start tim
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57 Figure 3: The same as Figure 2,
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59 By analogy with (2) we will look
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61 CONCLUSIONS We have developed a
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65 will show later how the process
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67 and develops a series solution o
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69 For the 3-D case the results are
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71 0.3 0.2 relative fluctuations of
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73 It is now possible to describe s
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75 Pulse propagation in random medi
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77 Pulse propagation in random medi
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81 The standard staggered grid A st
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83 EXPERIMENTAL SETUP As described
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85 0 0.04 0.08 0.12 0.16 depth (m)
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87 to the 2D-model (SH-waves). The
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89 The first step to use this theor
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91 ACKNOWLEDGMENTS We wish to thank
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Modeling 93
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96 are so small that the resulting
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98 zero. In other words, he sought
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100 We simulated a common-shot expe
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102 Relative error (%) 60 40 20 0
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104
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106 for the isotropic elastic case.
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108 of body forces, and with a Gree
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110 THE STATIONARY-PHASE APPROXIMAT
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112 Note that for each preassigned
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114
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116 3 s s s s x r r x s ~x 3
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118 Claim two The second claim to b
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120 Figure 2: Local Cartesian, ray
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122 Here, q and q denote the in-p
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124 Hubral, P., Schleicher, J., and
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126 and Gajewski, 1996; Vinje et al
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128 Figure 2: Wavefronts computed w
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130 source rays wavefront Figure 6:
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132 ACKNOWLEDGEMENTS This work was
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134 and Lecomte, 1992; Qin et al.,
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136 which is another form of snell'
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138 NUMERICAL TESTS The figures bel
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140 SUMMARY The following give the
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142
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144 on, e.g., the wavefront or refl
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146 Hubral et al. (1992) do a simil
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148 Depth [km] 0.2 0.4 0.6 0.8 0.2
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150 0.2 0.4 0.6 0.8 Y-Offset [km] 0
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152 and assume a linear relationshi
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154 the transformation ^Q = ^R T s
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156 rock have been generated by Joh
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158 Those latter coefficients descr
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160 (9) by explicit difference oper
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162 a series of large-amplitude sho
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164 CONCLUSIONS We have presented t
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166
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168 is slower than computing the fu
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170 Figure 1 shows the original 3D
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172 Figure 3: Snapshot of the elast
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174
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176 COMPUTING THE EFFECTIVE MEDIUM
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178 Vp [ km/s ] 4 3.8 3.6 3.4 3.2 3
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180 Figure 3: Time history of fluid
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182
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184 as salt body). To use full wave
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186 teristic change in subsurface p
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188
Page 203 and 204: 190 CHEBYSHEV METHOD We use a Cheby
Page 205 and 206: 192 x P 1 f (x) r P 4 P 2 f (x) r+1
Page 207 and 208: 194 EXAMPLE In order to demonstrate
Page 209 and 210: 196 CONCLUSION We developed a metho
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Page 213 and 214: 200 The use of multi-parametric tra
Page 215 and 216: 202 depending on the original and c
Page 217 and 218: 204 (1999), the first part consists
Page 219 and 220: 206 0 0.7 1 0.6 Zero offset travelt
Page 221 and 222: 208 5 x 10−4 0 First reflector N
Page 223 and 224: 210 Müller, T., 1999, The common r
Page 225 and 226: 212 with respect to the physical pr
Page 227 and 228: 214 The problem under consideration
Page 229 and 230: 216 0 B @ TI mudshale =0:034, =0:
Page 231 and 232: 218 exact velocities for the horizo
Page 233 and 234: 220 ACCURACY OF SECTORIAL APPROXIMA
Page 235 and 236: 222 Sayers, C. M., 1994, P-wave pro
Page 237 and 238: 224 The forward model used for the
Page 239 and 240: 226 0.2 (a) 0.1 Amp(n) 0 −0.1 −
Page 241 and 242: 228 1 1 P(x,t) 0.5 P(x,t) 0.5 0 0 1
Page 243 and 244: 230 KALMAN - WIENER The integral eq
Page 245 and 246: 232 (2) Definition of the state vec
Page 247 and 248: 234 0.2 0.15 0.1 0.05 Amp(n) 0 −0
Page 249 and 250: 236 0 0 0 100 100 100 200 200 200 3
Page 251 and 252: 238 Mendel, J., Nashi, N., and Chan
Page 253: 240 One of the first methods which
Page 257 and 258: 244 Offset (km) 0.5 1.5 2.5 Figure
Page 259 and 260: 246 where u 0 j is the conjugate tr
Page 261 and 262: 248 CONCLUSION Coherency analysis o
Page 263 and 264: 250 Montgomery, D. C., and Runger,
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Page 267 and 268: 254 2. 3. 4. Macromodel determinati
Page 269 and 270: 256 Research students: Ingo Koglin
Page 271 and 272: 258 João Luis Martins Migration an
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Page 275 and 276: 262 Elf Exploration UK plc 30 Bucki
Page 277 and 278: 264 PGS Seres AS P.O. Box 354 Stran
Page 279 and 280: 266 Robert Essenreiter received his
Page 281 and 282: 268 L.W.B. Leite Professor of Geoph
Page 283 and 284: 270 Melanie Pohl is dealing with wa
Page 285 and 286: 272 The first two years of this tim
Page 287: 274 scientist in Karlsruhe and at C
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