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$Fayek & Kyser 2000 uraninite fractionations.pdf - Queen's University$

4.6 Comparison of the travel time tomography result and the full waveeld inversion from the third party synthetic elastic wave data. a) True velocity model used in elastic forward waveform modelling, b) traveltime tomographic image formed from the picked synthetic data, c) acoustic waveeld inversion of the elastic synthetic data, without trace normalization, d) acoustic waveeld inversion with tracenormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7 Starting model for waveeld inversions of the eld data (from anisotropic velocity tomography). . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.8 Preliminary full waveeld inversion image using non normalized crosshole part of the data only. . . . . . . . . . . . . . . . . . . . . . . . . 100 4.9 Preliminary full waveeld inversion image using non normalized VSP part of the data only. Short oset VSP data are excluded due to large amplitude variations. . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.10 Preliminary full waveeld inversion image using non normalized Field 2 data, including both crosshole and VSP sections of the data. Short oset data are excluded due to large amplitude variations. . . . . . . 101 4.11 Isotropic full waveeld inversion results with various values of smoothing parameter increasing from 0 (top left corner) to 100 (bottom right corner). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.12 Trade o curve showing RMS roughness vs RMS residuals for a suite of smoothing parameters. . . . . . . . . . . . . . . . . . . . . . . . . 104 4.13 Final isotropic full waveeld inversion result. . . . . . . . . . . . . . 104 4.14 Frequency domain eld data at 800Hz. Please see the text for a full description of this gure. The grey scale is a relative amplitude, from the maximum negative values through to the maximum positive values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11
4.15 Frequency domain modelled (predicted) data at 800Hz. See text for a full description of this gure. The grey scale is a relative amplitude, from the maximum negative values through to the maximum positive values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.16 Dierence between eld and modelled data at 800Hz. See text for a full description of this gure. The grey scale is a relative amplitude, from the maximum negative values through to the maximum positive values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.17 Inverted source signatures. These signatures were extracted as an integral part of the waveeld inversion scheme. The similarity of the VSP source signatures, apart from the known static shifts, gives credence to the robustness of the inversion scheme. . . . . . . . . . . 107 4.18 Isotropic inversion of synthetic data set from a homogeneous, anisotropic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.19 Data residuals for the waveform inversion runs on the acoustic syntetic elliptically anisotropic (2 percent) data by assuming: a) Isotropic data (underestimated level of anisotropy) b) 2 percent elliptical anisotropy (correct value) c) 4 percent eliptical anisotropy (overestimated value). 110 4.20 Anisotropic full waveeld inversion results with 0, 1, 2 and 3% elliptical anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.21 RMS residuals for each test anisotropy level. . . . . . . . . . . . . . 113 4.22 Final full waveeld inversion image using 2% elliptical anisotropy. . . 114 4.23 Frequency domain dierence eld (i.e., data residuals) at 800 Hz from the anisotropic inversion. See text for a full description of this gure. The grey scale is a relative amplitude, from the maximum negative values through to the maximum positive values. . . . . . . . . . . . 115 4.24 Final waveeld inversion images from both Fields 1 and 2, using 2% elliptical anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 12
Page 1 and 2: Frequency domain seismic forward mo
Page 3 and 4: Acknowledgements I would like to gr
Page 5 and 6: 2.5 Nested dissection ordering . .
Page 7 and 8: 6.1.4 Waveeld inversion . . . . . .
Page 9 and 10: 2.3 Two waydissected matrix S ~ = L
Page 11: 4.3 Two representative source gathe
Page 15 and 16: 5.3 Numerical dispersion of the new
Page 17 and 18: Chapter 1 Introduction Modelling th
Page 19 and 20: Pratt (1995) and Song (1995) showed
Page 21 and 22: parameters is a daunting task. Extr
Page 23 and 24: element methods. 1.2 The signicance
Page 25 and 26: this sparsity. As in time domain me
Page 27 and 28: where the complex \impedance" matri
Page 29 and 30: 1.4 Fourier transforms and frequenc
Page 31 and 32: in order to unambiguously reconstru
Page 33 and 34: If the forward Fourier transform is
Page 35 and 36: ased on data setfrom an underground
Page 37 and 38: therefore one of the main applicati
Page 39 and 40: 2.3 Solving linear equation systems
Page 41 and 42: are always calculated by the time t
Page 43 and 44: Case 2 2 S = ~ 6 4 4 0 0 0 .5 0 .79
Page 45 and 46: Figure 2.2: Two-way dissected nite
Page 47 and 48: (n/2*n/2)/2 L 55 (n/2*n/2)/2 n*n/2
Page 49 and 50: (to within anorder of magnitude) ne
Page 51 and 52: D 4 (n; 0) (2n) 2 =2 + 2(2n=2) 2 =
Page 53 and 54: The memory required to perform LU d
Page 55 and 56: are of theorder of hundereds of kil
Page 57 and 58: 10000 1000 Time (s) Sequential orde
Page 59 and 60: Chapter 3 Visco-acoustic frequency
Page 61 and 62: Figure 3.1: Finite dierence operato
Page 63 and 64:
However, analytical solutions do no
Page 65 and 66:
3.2.4 Lumped and consistent matrix
Page 67 and 68:
+4c=1 (3.9) which makes the minimiz
Page 69 and 70:
that the required memory is afuncti
Page 71 and 72:
1996). However quantitativeevaluati
Page 73 and 74:
(1991). The low velocity regions (1
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a) Time slice at 5s b) Time slice a
Page 77 and 78:
domain approach would have to gener
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1988). Reviews have been provided b
Page 81 and 82:
Figure 4.1: Grimsel Pass areal phot
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5 10 15 20 25 30 35 40 45 50 55 60
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160m BOUS 85.003 BOBK 85.004 BOBK 8
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will assume that rst n r n x n z (
Page 89 and 90:
where jxj represents represents the
Page 91 and 92:
or c J~ = S ~ ,1 G ~ (4.13) where F
Page 93 and 94:
Figure 4.6: Comparison of the trave
Page 95 and 96:
In order to initialize the waveeld
Page 97 and 98:
the trace-to-trace amplitude variat
Page 99 and 100:
manner these groups were identied.
Page 101 and 102:
km/s 4.80 4.85 4.90 4.95 5.00 5.05
Page 103 and 104:
4.6.2 Regularization tests From thi
Page 105 and 106:
RMS Roughness 0.8 0.7 0.6 0.5 5 10
Page 107 and 108:
Figure 4.15: Frequency domain model
Page 109 and 110:
y the best isotropic results. In al
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a) b) c) Figure 4.19: Data residual
Page 113 and 114:
Figure 4.20: Anisotropic full wavee
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km/s 4.80 4.85 4.90 4.95 5.00 5.05
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Figure 4.24: Final waveeld inversio
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can eect theimages (the wrong theor
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In essence, our new scheme is the e
Page 123 and 124:
where ! = 2f is the angular frequen
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dicult to formulate direct solvers
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! 2 u 0 + v 0 + ( +2) + ( + ) @ u
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\stars": ! @ 2 1 0 0 1 0 -2 0 @x 0
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free to vary from one node point to
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(see equations 5.24 and 5.25), exce
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1 1 0.8 0.8 a 0.6 0.4 b 0.6 0.4 0.2
Page 137 and 138:
Numerical dispersion curves for σ=
Page 139 and 140:
Ifollow closely the dispersion anal
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the opposite results. The fraction
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identied; it is also possible to id
Page 145 and 146:
Real data Acoustic modelling result
Page 147 and 148:
observed data. However, even with t
Page 149 and 150:
Chapter 6 Conclusions and further w
Page 151 and 152:
compensated for by the accuracy gai
Page 153 and 154:
(1992)). I have also shown how sens
Page 155 and 156:
Extensions to more complex cases Th
Page 157 and 158:
Although waveeld inversions have be
Page 159 and 160:
to nd more dicult targets. An incre
Page 161 and 162:
Bathe, K. J., and Wilson, E. L., 19
Page 163 and 164:
Cole, J. B., 1994, A nearly exact s
Page 165 and 166:
Lailly, P., 1984, Migration methods
Page 167 and 168:
Peng, C., Cheng, C. H., and Toksoz,
Page 169 and 170:
Smith, W. D., 1974, The application
Page 171 and 172:
Appendix A Dispersion analysis for
Page 173:
and 2 = ( 1 R 2 + 2 )( 1 + 2 R 2
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