Wave Inversion Technology - WIT

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Wave Inversion Technology, Report No. 3, pages 125-133 3-D traveltime computation using a hybrid method R. Coman and D. Gajewski 1 keywords: traveltime, wavefront construction, FD eikonal ABSTRACT A hybrid method for computing multi-arrival traveltimes in 3-D weakly smoothed media is presented. The method is based on the computation of first-arrival traveltimes with a finite-difference eikonal solver (FDES) and the computation of later arrivals with the wavefront construction method (WFCM) . The detection and bounding of regions where later arrivals occur is done automatically. WFCM is only used in complex models. If no triplications are present, only FDES is used. The complexity of the model is automatically investigated, without user intervention. The applicability of the method to a model with a triplication is demonstrated. The hybrid method is a better alternative to WFCM, since it is faster and has a comparable accuracy. INTRODUCTION Three-dimensional (3-D) traveltime computation is commonly done with finitedifference eikonal solvers (FDESs) or with ray-tracing methods (e.g., Vidale, 1990; Cervený, 1985). Only ray-tracing methods permit computation of multi-valued arrivals which occur in complex models, but for a prestack Kirchhoff migration this computation is very time consuming. Thus, a faster computation of 3-D multi-valued traveltimes is required. FDESs provide a fast and robust method of first-arrival traveltime computations (Vidale, 1990; van Trier and Symes, 1991; Sethian and Popovici, 1999). However, in complex velocity structures, first arrivals do not necessarily correspond to the most energetic wave, and other arrivals can also be important for accurate modeling and imaging (Geoltrain and Brac, 1993; Ettrich and Gajewski, 1996). Multiple arrivals are traditionally computed with ray-tracing methods. The most suitable implementation of ray tracing for computing a large number of two-point problems is the wavefront construction method (WFCM) (Vinje et al., 1993; Ettrich 1 email: coman@dkrz.de 125
126 and Gajewski, 1996; Vinje et al., 1996; Lambaré et al., 1996). Unfortunately, the computational efficiency of WFCM is low (Leidenfrost et al., 1999). Our aim is a more efficient computation of multi-valued 3-D traveltimes in weakly smoothed media. We combine FDES and WFCM to a hybrid method, taking advantages of the computational speed of FDES and using the ability of WFCM to compute multi-valued traveltimes. This idea was also used by Ettrich and Gajewski (1997) by presenting a related 2-D hybrid method. The corresponding 3-D hybrid method is not only an extension from 2-D, because many problems ( e.g., the searching and bounding for regions where later transmitted arrivals occur) require new algorithms in 3-D. The method that we present consists of four steps: 1. Computing of first-arrival traveltimes with a FDES, 2. Searching for triplications, 3. Bounding the regions where later arrivals occur, 4. Computing of later arrivals with WFCM. After the computation of first-arrival traveltimes, the hybrid method will automatically realize if and where later arrivals occur. If no triplication is detected, computation with a fast FDES is sufficient. The implementation of the first and the last step is simple; the problems in developing a hybrid method are: the identification of zones where triplications occur, and the setting of initial conditions for WFCM. We developed a new algorithm for the automatic detection and bounding of regions where later arrivals occur, and adapt a 3-D FDES (Vidale, 1990) and a 3-D WFCM ( based on the implementation by Ettrich and Gajewski,1996) to the needs of the hybrid method. THE FDES-WFCM HYBRID METHOD We describe the four steps of the method by means of an example. The 3-D velocity model used in the example is a cube with a spherical inclusion in the middle (Figure 1). The 10 10 10 km model used has a background velocity of 3:0 km=s. In the middle of the model is a spherical inclusion with a diameter of 3:5 km, where the velocity smoothly decreases to 1:5 km=s in the center of the inclusion. The source is located at coordinates x =5km, y =5km, z =0:5 km. We choose this velocity distribution because it leads to a triplicated wavefront. The computed multi-valued wavefronts at three different traveltime steps are shown in Figure 2 . The bottom wavefront displays a triplication.
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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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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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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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44 0 1 2 v = 1.5km/s 0 v = 4.5km/s
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48 CONCLUSION We presented an inver
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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
Page 88 and 89: 75 Pulse propagation in random medi
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Page 94 and 95: 81 The standard staggered grid A st
Page 96 and 97: 83 EXPERIMENTAL SETUP As described
Page 98 and 99: 85 0 0.04 0.08 0.12 0.16 depth (m)
Page 100 and 101: 87 to the 2D-model (SH-waves). The
Page 102 and 103: 89 The first step to use this theor
Page 104 and 105: 91 ACKNOWLEDGMENTS We wish to thank
Page 106: Modeling 93
Page 109 and 110: 96 are so small that the resulting
Page 111 and 112: 98 zero. In other words, he sought
Page 113 and 114: 100 We simulated a common-shot expe
Page 115 and 116: 102 Relative error (%) 60 40 20 0
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Page 121 and 122: 108 of body forces, and with a Gree
Page 123 and 124: 110 THE STATIONARY-PHASE APPROXIMAT
Page 125 and 126: 112 Note that for each preassigned
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Page 129 and 130: 116 3 s s s s x r r x s ~x 3
Page 131 and 132: 118 Claim two The second claim to b
Page 133 and 134: 120 Figure 2: Local Cartesian, ray
Page 135 and 136: 122 Here, q and q denote the in-p
Page 137: 124 Hubral, P., Schleicher, J., and
Page 141 and 142: 128 Figure 2: Wavefronts computed w
Page 143 and 144: 130 source rays wavefront Figure 6:
Page 145 and 146: 132 ACKNOWLEDGEMENTS This work was
Page 147 and 148: 134 and Lecomte, 1992; Qin et al.,
Page 149 and 150: 136 which is another form of snell'
Page 151 and 152: 138 NUMERICAL TESTS The figures bel
Page 153 and 154: 140 SUMMARY The following give the
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Page 157 and 158: 144 on, e.g., the wavefront or refl
Page 159 and 160: 146 Hubral et al. (1992) do a simil
Page 161 and 162: 148 Depth [km] 0.2 0.4 0.6 0.8 0.2
Page 163 and 164: 150 0.2 0.4 0.6 0.8 Y-Offset [km] 0
Page 165 and 166: 152 and assume a linear relationshi
Page 167 and 168: 154 the transformation ^Q = ^R T s
Page 169 and 170: 156 rock have been generated by Joh
Page 171 and 172: 158 Those latter coefficients descr
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Page 175 and 176: 162 a series of large-amplitude sho
Page 177 and 178: 164 CONCLUSIONS We have presented t
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Page 181 and 182: 168 is slower than computing the fu
Page 183 and 184: 170 Figure 1 shows the original 3D
Page 185 and 186: 172 Figure 3: Snapshot of the elast
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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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184 as salt body). To use full wave
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186 teristic change in subsurface p
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190 CHEBYSHEV METHOD We use a Cheby
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200 The use of multi-parametric tra
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202 depending on the original and c
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204 (1999), the first part consists
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206 0 0.7 1 0.6 Zero offset travelt
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210 Müller, T., 1999, The common r
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214 The problem under consideration
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222 Sayers, C. M., 1994, P-wave pro
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242 a limited time range while the
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248 CONCLUSION Coherency analysis o
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250 Montgomery, D. C., and Runger,
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256 Research students: Ingo Koglin
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258 João Luis Martins Migration an
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262 Elf Exploration UK plc 30 Bucki
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264 PGS Seres AS P.O. Box 354 Stran
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266 Robert Essenreiter received his
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268 L.W.B. Leite Professor of Geoph
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