Annual Report 2000 - WIT

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248 Ettrich, N., 1998, FD eikonal solver for 3-d anisotropic media: 68th Ann. Int. Mtg. SEG, Expanded Abstracts, 1957–1960. Fedorov, F. I., 1968, Theory of elastic waves in crystals: Plenum Press, New York. Vidale, J. E., 1990, Finite-difference calculation of traveltimes in three dimensions: Geophysics, 55, 521–526.
Wave Inversion Technology, <strong>Report</strong> No. 4, pages 249-259 First arrival traveltimes and amplitudes by FD solution of eikonal and transport equation S. Buske 1 keywords: modeling, FD, first arrival, eikonal equation, transport equation ABSTRACT This paper describes the development and the implementation of a combined firstarrival traveltime and amplitude computation procedure. The schemes described here are based on a finite difference solution of the eikonal equation and the transport equation. After formulation of both equations as hyperbolic conservation laws the eikonal equation is solved numerically by a third-order ENO-Godunov scheme for the traveltimes and the transport equation is solved by a first-order upwind scheme for the amplitudes. The schemes are implemented in cartesian coordinates for a plane wave and in polar coordinates for a point source, respectively. The resulting traveltimes are highly accurate and amplitudes are smooth even in the case of complex models, for instance the Marmousi model. INTRODUCTION First-arrival traveltimes are still a popular and widely used tool for prestack Kirchhoff migration and tomographic applications. A variety of methods have been developed to solve the eikonal equation in order to efficiently obtain these traveltimes. (Vidale, 1988, 1990) has been one of the first to approximate the eikonal equation by finite differences and to solve it along expanding rectangles around the source. This method is very efficient and can be coded easily but it may fail to compute the correct solution for certain models. (Podvin and Lecomte, 1991) used a similar FD approximation, which incorporates head waves and diffracted waves. Due to its specific implementation this method is unconditionally stable but less efficient than Vidale's scheme. Another way of obtaining first arrivals are the so called 'shortest path' methods which are mainly based on geometric considerations (Moser, 1991; Klimejs and Kvasnijcka, 1994). The most interesting alternative in terms of the paper presented here has been published by (van Trier and Symes, 1991) for 2-D and (Schneider, 1995) for 3-D models. In this case the eikonal equation is formulated as a hypberbolic conservation 1 email: buske@geophysik.fu-berlin.de 249
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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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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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31 Dürbaum, H., 1954, Zur Bestimmu
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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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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
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° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
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94 1 taper value ksi1 0 ksi2 Figure
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96 0 1 2 3 4 5 Depth [km] CMP [km]
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98 CONCLUSION As a generalization o
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100 weight during the stacking proc
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102 are not illuminated for every o
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104 obtained from Zoeppritz' equati
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106 PreSDM of Porous layer 0.358 re
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108 REFERENCES Gassmann, F., 1951,
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110 et al., 1992), Hanitzsch (1997)
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112 consecutive wavefronts). Howeve
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114 Numerical example We test the W
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116 Versteeg, R., and Grau, G., 199
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118 indicator. To extract elastic p
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€ 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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136 TRIGGERING FRONTS IN HETEROGENE
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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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148 RESULTS Soultz-sous-Forêts Inv
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152 Figure 5: The cloud of events f
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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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Ï u u ¬ 164 To summarize, we deri
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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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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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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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Page 212 and 213: 200 of groundwater. ACKNOWLEDGEMENT
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Page 216 and 217: 204 the wavefront curvature matrix
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Page 220 and 221: é 208 Table 1: Median relative err
Page 222 and 223: 210 Table 2: Median relative errors
Page 224 and 225: 212 CONCLUSIONS We have presented a
Page 226 and 227: 214 PUBLICATIONS Previous results c
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Page 230 and 231: 218 weight functions from traveltim
Page 232 and 233: 220 REFERENCES Bleistein, N., 1986,
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Page 240 and 241: 228 In this paper, we present a mor
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Page 248 and 249: 236 CONCLUSIONS We have presented a
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Page 280 and 281: 268 NUMERICAL RESULTS We constructe
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Page 284 and 285: £ -š' - 272 Ô- —- Figure 6: C
Page 286 and 287: 274 REFERENCES Aldridge, D. F. (199
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Page 290 and 291: 278 3. 4. True-amplitude imaging, m
Page 292 and 293: 280 Research Group Hamburg (Gajewsk
Page 294 and 295: 282 Stefan Lüth Robert Patzig Refr
Page 296 and 297: 284 Landmark Graphics Corp. 7409 S.
Page 298 and 299: 286 TotalFinaElf Exploration UK plc
Page 300 and 301: 288 as research assistant at Geoeco
Page 302 and 303: 290 Ingo Koglin is a diploma studen
Page 304 and 305: 292 Matthias Riede received his M.S
Page 306 and 307: 294 Svetlana Soukina received her d
Page 308: 296 Yonghai Zhang received the Mast
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