Frequency domain seismic forward modelling: A tool for waveform ...

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

4.5.4 Comparison of travel time and full waveeld inversion methods 96 4.6 Inversion of real eld data . . . . . . . . . . . . . . . . . . . . . . . . 97 4.6.1 Initial full waveeld inversion . . . . . . . . . . . . . . . . . . 98 4.6.2 Regularization tests . . . . . . . . . . . . . . . . . . . . . . . . 102 4.7 Isotropic results: Evaluation and verication . . . . . . . . . . . . . 102 4.7.1 Discussion of isotropic results . . . . . . . . . . . . . . . . . . 107 4.8 Anisotropic inversion of the eld data . . . . . . . . . . . . . . . . . . 110 4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter 5 Visco-elastic frequency domain seismic forward modelling119 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 Visco-elastic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2.1 Rotated nite dierences: Computational stars . . . . . . . . 122 5.2.2 Rotated nite dierences: Operators . . . . . . . . . . . . . . 125 5.2.3 Consistent and lumped mass terms . . . . . . . . . . . . . . . 128 5.2.4 Heterogeneous formulation . . . . . . . . . . . . . . . . . . . . 129 5.3 Numerical errors and optimization . . . . . . . . . . . . . . . . . . . . 132 5.3.1 Determination of optimal coecients . . . . . . . . . . . . . . 132 5.3.2 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . 134 5.3.3 Modelling in uids . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.4 Elastic modelling example . . . . . . . . . . . . . . . . . . . . . . . 141 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Chapter 6 Conclusions and further work 148 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.1.1 Matrix solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.1.2 Rotated nite dierence operators . . . . . . . . . . . . . . . . 150 6.1.3 Visco-elastic forward modelling . . . . . . . . . . . . . . . . . 150 5
6.1.4 Waveeld inversion . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.2.1 Developments in seismic modelling . . . . . . . . . . . . . . . 152 6.2.2 Developments in waveeld inversion . . . . . . . . . . . . . . . 155 Bibliography 159 Appendix A Dispersion analysis for visco-elastic modelling 170 6
Page 1 and 2: Frequency domain seismic forward mo
Page 3 and 4: Acknowledgements I would like to gr
Page 5: 2.5 Nested dissection ordering . .
Page 9 and 10: 2.3 Two waydissected matrix S ~ = L
Page 11 and 12: 4.3 Two representative source gathe
Page 13 and 14: 4.15 Frequency domain modelled (pre
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
Page 75 and 76:
a) Time slice at 5s b) Time slice a
Page 77 and 78:
domain approach would have to gener
Page 79 and 80:
1988). Reviews have been provided b
Page 81 and 82:
Figure 4.1: Grimsel Pass areal phot
Page 83 and 84:
5 10 15 20 25 30 35 40 45 50 55 60
Page 85 and 86:
160m BOUS 85.003 BOBK 85.004 BOBK 8
Page 87 and 88:
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
Page 111 and 112:
a) b) c) Figure 4.19: Data residual
Page 113 and 114:
Figure 4.20: Anisotropic full wavee
Page 115 and 116:
km/s 4.80 4.85 4.90 4.95 5.00 5.05
Page 117 and 118:
Figure 4.24: Final waveeld inversio
Page 119 and 120:
can eect theimages (the wrong theor
Page 121 and 122:
In essence, our new scheme is the e
Page 123 and 124:
where ! = 2f is the angular frequen
Page 125 and 126:
dicult to formulate direct solvers
Page 127 and 128:
! 2 u 0 + v 0 + ( +2) + ( + ) @ u
Page 129 and 130:
\stars": ! @ 2 1 0 0 1 0 -2 0 @x 0
Page 131 and 132:
free to vary from one node point to
Page 133 and 134:
(see equations 5.24 and 5.25), exce
Page 135 and 136:
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
Page 141 and 142:
the opposite results. The fraction
Page 143 and 144:
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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