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

More documents

Recommendations

Info

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

Contents Abstract 1 Acknowledgements 2 List of Figures 7 Chapter 1 Introduction 16 1.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 The signicance of the frequency-space domain . . . . . . . . . . . . . 22 1.3 Forward modelling in the frequency-space domain . . . . . . . . . . 24 1.4 Fourier transforms and frequency domain modelling . . . . . . . . . 28 1.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.2 Sampling and the Sampling Theorem . . . . . . . . . . . . . . 29 1.4.3 Anti time-aliasing . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.4 Reduced time and the Fourier transform shifting property . . 31 1.5 Overview of chapters in this thesis . . . . . . . . . . . . . . . . . . . . 33 Chapter 2 Solving frequency domain wave equations: Numerical Considerations 35 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Solving linear equation systems: bottlenecks . . . . . . . . . . . . . . 36 2.3 Solving linear equation systems with multiple right hand sides . . . . 38 2.4 Matrix \ll in" and ordering schemes . . . . . . . . . . . . . . . . . . 41 3
2.5 Nested dissection ordering . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Operators and memory requirements . . . . . . . . . . . . . . . . . . 49 2.7 Comparison of band and nested dissection ordering . . . . . . . . . . 52 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 3 Visco-acoustic frequency domain acoustic forward modelling using rotated nite dierence operators 58 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Forward modelling using rotated nite dierence operators . . . . . . 60 3.2.1 Second order frequency-domain seismic modelling . . . . . . . 60 3.2.2 The rotated operator concept . . . . . . . . . . . . . . . . . . 61 3.2.3 Finite dierence scheme in homogeneous media . . . . . . . . 63 3.2.4 Lumped and consistent matrix terms . . . . . . . . . . . . . . 64 3.2.5 Determination of optimal coecients . . . . . . . . . . . . . . 64 3.2.6 Discussion of savings with rotated operators . . . . . . . . . . 67 3.2.7 Extension to the heterogenous case . . . . . . . . . . . . . . . 68 3.3 Improvements acheived by rotated nite dierence operators . . . . . 70 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 4 Frequency domain waveeld inversion example 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Site description: Grimsel Rock Labaratory . . . . . . . . . . . . . . . 79 4.3 Waveeld inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 Waveeld inversion theory . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Ecient calculation of the gradient direction . . . . . . . . . 89 4.5 Processing of third party synthetic data . . . . . . . . . . . . . . . . . 91 4.5.1 Travel time tomography . . . . . . . . . . . . . . . . . . . . . 94 4.5.2 Full waveeld inversion . . . . . . . . . . . . . . . . . . . . . 94 4.5.3 Full waveeld inversion of trace-normalised data . . . . . . . . 95 4
Page 1 and 2: Frequency domain seismic forward mo
Page 3: Acknowledgements I would like to gr
Page 7 and 8: 6.1.4 Waveeld inversion . . . . . .
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?