Annual Report 2000 - WIT

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— J J J J J J G J – J J G 24 Examples of such a weight function are a boxcar, a triangle or the cosine function. The second step is to fit Ž a polynomial of th order to the points within the window using weighted least ‘“’ squares Z with weights . denotes all points of one event where ” is the maximum number of points of that picked event. • >b>b> • represents the th weight function corresponding to the Z th point. This initial fit, – , is the locally weighted regression. Now, J ' the residual ( ) is calculated to get a — J new weight function, , that has large weights for small residuals and small weights for large residuals. The fitted values are calculated again with a new set of weights, , which are multiplied with the original data. The last step ‘“’ is repeated several times and the result is the robust locally weighted regression. The number of iterations is given by the parameter . The length of the smoothing window is †‚” obtained by , ”˜/6š#›W˜ where is rounded to the nearest integer neighbour and is a œ factor between zero and one. If is close to zero, the window length for smoothing is short. Thus, the curve of fitted points is more roughly. If gets closer to one, more œ points are considered for the fit. That leads to a smoother curve. This filter was Ÿž designed to V¡ gain the best fit for data for which is a J random variable with mean zero and constant scale. >b>b> , where ž is a smooth function and ¡ J In contrast to the robust locally weighted regression, a spline approximation, a spline interpolation, or a local polynomial regression shows a basic problem. They cannot merge the local parts such that the resulting curve is smooth in first and second order. That means the endpoints and/or their first derivative from one local part to the next must be continuous. To overcome this problem, e.g., a global polynomial regression could be computed for the whole reflector with a certain order. Choosing then a small order yields a polynomial that cannot follow high frequency fluctuations and a large order might lead to a polynomial with more fluctuations than demanded. The robust locally weighted regression is also a kind of a polynomial fitting procedure.There, the problem of smoothness is solved by improving the weight functions.Hence, the robust locally weighted regression is in favour, see Figure 3 (b). #” ” REAL DATA EXAMPLE Figure 2 (b) shows a small simulated ZO subset of a real data-set. We had picked ten events between CMP number 150 and 480 of which we used seven to generate the two macro-velocity models in Figure 1 (a) and (b). This CMP window was chosen because there we were able to pick continuous events for times between 0.3 and 3.2 seconds. As mentioned before, some events could not be followed continuously to the left or right because scattering prevails or another event crosses the picked one. We could not pick continuously at smaller times because parts of the events were muted. The arrows in Figure 2 (a) and (b) indicate some picked events used for the inversion. At later times in Figure 2 (b), the noise is too strong to find more continuous events.
25 Both macro-velocity models or smoothed versions of them can be used for migration. The velocity for the half-space beneath the last interface has to be chosen manually. CONCLUSIONS To our knowledge, it is the first time that a data-driven model is presented which is not a result of an iterative improvement of an initial model. The model is calculated directly from the picked data and the model independent CRS stack attributes. We also presented a sophisticated smoothing algorithm, the robust locally weighted regression. This algorithm is suited best to prepare the CRS stack attributes in order to ensure a stable inversion. REFERENCES Cleveland, W. S., 1979, Robust locally weighted regression and smoothing scatterplots: Journal of the American Statistical Association, 74, 829–836. Hubral, P., 1983, Computing true amplitude reflections in a laterally inhomogeneous earth: Geophysics, 48, no. 8, 1051–1062. Jäger, R., Mann, J., Höcht, G., and Hubral, P., 2001, Common-reflection-surface Stack: Image and attributes: Geophysics, 66, no. 1. Majer, P., <strong>2000</strong>, Inversion of seismic parameters: Determination of the 2-D iso-velocity layer model: Master's thesis, Universität Karlsruhe, Germany.
Page 1 and 2: Wave Inversion Technology WIT Annua
Page 3: Wave Inversion Technology WIT Wave
Page 7: Copyright c 2000 by Karlsruhe Unive
Page 11 and 12: i TABLE OF CONTENTS Reviews: WIT Re
Page 13 and 14: Wave Inversion Technology, Report N
Page 15 and 16: 3 theory. It relates properties of
Page 17: Imaging 5
Page 20 and 21: 8 two hypothetical eigenwave experi
Page 22 and 23: 7 ¢¥£§¦©¨ ; < calculate sear
Page 24 and 25: 7 7 searches ¡ HNJOL for ¢IHNJML
Page 26 and 27: 14 Trace no. 1000 1500 2000 0.5 1.0
Page 28 and 29: 16 CDP number 3100 3000 2900 2800 2
Page 30 and 31: 18 REFERENCES Höcht, G., de Bazela
Page 32 and 33: 20 INVERSION BY MEANS OF CRS ATTRIB
Page 34 and 35: 22 For synthetic data, it is easy t
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 and 84: 71 Langenberg, K., 1986, Applied in
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Wave Inversion Technology, Report N
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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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¨ º Ý È · § À · Á · Á ¼
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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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Å éÙó ó ».-.‹œì/-jì¨‹
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136 TRIGGERING FRONTS IN HETEROGENE
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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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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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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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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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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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218 weight functions from traveltim
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220 REFERENCES Bleistein, N., 1986,
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226
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228 In this paper, we present a mor
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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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‹ 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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